Texture evolution in equal-channel angular extrusion

Progress in Materials Science 54 (2009) 427–510

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Texture evolution in equal-channel angular extrusion Irene J. Beyerlein a,*, László S. Tóth b a b

Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Laboratoire de Physique et Mécanique des Matériaux, Université de Metz, Ile du Saulcy, 57045 Metz, France

a r t i c l e

i n f o

Article history: Received 12 June 2008 Received in revised form 13 January 2009 Accepted 23 January 2009

a b s t r a c t The focus of this article is texture development in metals of fcc, bcc, and hcp crystal structure processed by a severe plastic deformation (SPD) technique called equal-channel angular extrusion (ECAE) or equal-channel angular pressing (ECAP). The ECAE process involves very large plastic strains and is well known for its ability to refine the grain size of a polycrystalline metal to submicron or even nanosize lengthscales depending on the material. During this process, the texture also changes substantially. While the strength, microstructure and formability of ECAE-deformed metals have received much attention, texture evolution and its connection with these properties have not. In this article, we cover a multitude of factors that can influence texture evolution, such as applied strain path, die geometry, processing conditions, deformation inhomogeneities, accumulated strain, crystal structure, material plastic behavior, initial texture, dynamic recrystallization, substructure, and deformation twinning. We evaluate current constitutive models for texture evolution based on the physics they include and their agreement with measurements. Last, we discuss the influence of texture on post-processed mechanical response, plastic anisotropy, and grain refinement, properties which have made ECAE, as well as other SPD processes, attractive. It is our intent to make SPD researchers aware of the importance of texture development in SPD and provide the background, guidance, and methodologies necessary for incorporating texture analyses in their studies. Ó 2009 Elsevier Ltd. All rights reserved.

* Corresponding author. Tel.: +1 505 665 2231; fax: +1 505 665 5926. E-mail addresses: [email protected] (I.J. Beyerlein), [email protected] (L.S. Tóth). 0079-6425/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.pmatsci.2009.01.001

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Contents 1. 2.

3.

4.

5.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Severe plastic deformation (SPD) processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1. Equal-channel angular extrusion (ECAE). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2. Definitions of deformation and velocity gradient in ECAE. . . . . . . . . . . . . . . . . . . . . . . 2.1.3. Simple shear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4. Distortions and rotations of material elements in ECAE . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5. Multiple passes: rotations and strain path changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6. Relationship to other deformation processes: drawing, rolling, plane strain compression, torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.7. Other SPD techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Ideal texture orientations in simple shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. Fcc crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2. Bcc crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3. Hcp crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Ideal texture orientations in ECAE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ECAE texture measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Cubic textures in ECAE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Influence of die angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2. First-pass textures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3. Influence of route and pass number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4. Fcc example: copper. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5. Bcc example: IF-steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.6. Deviations from ideal texture positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.7. Texture strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.8. Saturation of texture with pass number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.9. Effect of stacking fault energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.10. Effect of deformation twinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.11. Effect of substructure evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.12. Effect of temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Hcp textures in ECAE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Slip and twinning modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. First-pass textures and effect of die angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3. Effect of route and pass number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4. Effect of alloying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5. Effect of deformation twinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.6. Effect of temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Effect of initial texture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. ECAE of single crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Cubic polycrystals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3. Hexagonal polycrystals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4. Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Dynamic recrystallization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1. Cubic metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2. Hexagonal metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling texture evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Polycrystal models for SPD processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Full constraints (FC) Taylor polycrystal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Visco-plastic self-consistent (VPSC) polycrystal model . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Assessment of model performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Polycrystal model performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VPSC predictions using the simple shear (SS) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Alternative deformation models for ECAE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. 2D and 3D finite element models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1. Corner gaps and plastic deformation zones (PDZ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2. Inhomogeneous deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3. Three-dimensional effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

429 431 431 431 431 432 433 433 435 435 436 436 436 436 438 439 441 441 442 442 443 445 446 448 449 449 449 450 452 453 453 453 456 456 457 457 458 458 459 463 465 465 466 467 468 468 469 469 469 470 470 471 471 472 473 474

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6.

7. 8.

5.2. Nonphysical factors affecting assessment of model performance . . . . . . . . . . . . . . . . . . . . . . . 5.3. Texture calculations with finite element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Heterogeneity in texture evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Route A: heterogeneity and end-effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6. Analytical flow models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7. Texture calculations with analytical flow models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extended texture calculations including grain-scale and subgrain-scale effects . . . . . . . . . . . . . . . . . . 6.1. Grain–grain interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Grain shape effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Substructure models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Deformation twinning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5. Application: Modeling texture evolution in route C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1. Grain–grain interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2. Inhomogeneous deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3. Grain hardening and substructure evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.4. Imperfect reversal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6. Sensitivity of texture to microstructural evolution (grain refinement) . . . . . . . . . . . . . . . . . . . 6.7. Influence of texture on post deformation of SPD materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding remarks and recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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475 476 477 478 479 483 484 486 486 489 490 491 492 492 492 494 499 500 502 503 503

1. Introduction This article examines the state-of-the-art in texture knowledge and prediction in the field of severe plastic deformation (SPD). The commonly used processes in SPD are equal-channel angular extrusion (ECAE), also frequently called equal-channel angular pressing (ECAP), high pressure torsion (HPT), accumulated roll bonding (ARB) and conshearing. The emphasis here will be on ECAE because, to date, SPD studies of texture evolution have mostly treated this process. Material texture is defined as a microstructural property that describes the orientation distribution of the grains constituting a polycrystalline aggregate. Severe plastic deformation processes, which involve large strain levels, and in some cases, frequent strain path changes, are expected to lead to substantial changes in texture. The post-processed deformation texture can influence many aspects of material behavior, such as strength, work hardening, plastic anisotropy, formability, grain refinement, and fracture. Most of these structural properties of interest cannot be fully understood without knowledge of the texture of the SPD-processed metal. Of all these properties, texture is most often associated with plastic anisotropy. Plastic anisotropy describes the dependence of the plastic deformation response on the sense and direction of loading. In a polycrystal some grains have a ‘hard’ or ‘soft’ orientation with respect to the specified loading direction. Consequently, a loading state that samples more of the relatively hard grains than soft ones leads to higher deformation stresses and vice versa. Even in the simplest pure metals, such as initially weakly textured Cu or Al, it has been shown that the texture resulting from an SPD process can contribute to substantial plastic anisotropy, e.g., [1–5], and tensile-compression strength asymmetry [4]. The attractiveness of SPD techniques is the synthesis of bulk ultra-fine grained (one micron or less grain size), or in some cases, nano-grained (tens of nanometers) metals, via an SPD-induced grain refinement process in an initially coarse-grained metal. Although not a serious consideration to date, it can be argued that texture plays an important role in grain refinement. Grain refinement by SPD is closely related to the development of deformation substructures, and texture and substructural evolution are strongly coupled. As a polycrystal metal deforms, the texture evolves because the crystal lattices of the individual grains rotate to preferred positions characteristic of the applied deformation. The rate at which texture evolves (or the grains rotate) depends on the slip distribution within the active slip systems. For a given grain, the choice of active slip can vary depending on its substructure or the presence of twins. In turn, substructural evolution (including twinning) and the eventual

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transformation of substructure to subgrains, depend on the number and choice of active systems, which are partly determined by crystallographic orientation. A discussion on SPD textures is complicated because many factors influence texture development. These factors can be broadly categorized as those related to (i) the deformation conditions, (ii) material parameters (microstructural mechanisms), or (iii) the starting (initial) texture. In the ECAE process, the deformation conditions (i) include the die angle U and the corner angle W (see Fig. 1 [6]), the number of passes N, the route between passes, pressing rate, back pressure, lubrication and temperature. The space of material parameters (ii) is equally as large, including crystal structure, microstructure, deformation mechanisms, chemical composition, phases, etc. Finally, the initial texture (iii) plays an equally important role in the final state of the deformed structure. Sometimes small differences in pre-processing that change the initial texture and microstructure can produce large differences in the final texture under the same deformation process. On the basis of these three categories, an attempt is made in this article to examine the role of each in texture evolution. This analysis is conducted from both an experimental and theoretical perspective. Models for texture evolution will be presented and analyzed. Undoubtedly, while it is important to at least have a measurement of the texture, it is even more important to be able to predict it for several reasons. Reliable texture prediction is required for understanding and prediction of anisotropic mechanical behavior. It can help in designing for better control and energy efficiency during forming processes. (‘‘Texture control” is widely regarded as one of the most important applications of texture modeling.) Texture prediction can also be used as a tool for determining dominant microscale mechanisms or certain characteristics of the applied deformation. For instance, deformation twins which reorient parts of the grain and result in large misorientation across their boundaries can be detected by a texture measurement [7,8]. As another example, it has also been used to help isolate the mechanisms in the creep of SPD processed aluminum [9]. The contents of this article can be summarized as follows. The first section provides the background necessary for understanding the mechanics of ECAE and hence for understanding its texture development. The second part reviews methods of texture characterization and measurement and follows with application to experimental textures reported in the literature. Next, we discuss the effects of deformation mechanisms, initial texture, microstructure, and various other material and processing factors on texture evolution. We follow with a comprehensive review of the models involved in texture prediction, ranging in lengthscale from the macro- and meso- to the microscale, and a critical

IN

RE-INSERTION

OUT

Fig. 1. ECAE die geometry and coordinate system employed in this paper [6].

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assessment of their performance. We conclude with a rather broad set of discussion topics, from texture heterogeneity to texture effects on grain refinement and mechanical properties.

2. Background 2.1. Severe plastic deformation (SPD) processes 2.1.1. Equal-channel angular extrusion (ECAE) Equal-channel angular extrusion (ECAE) [10,11] is by far the most well studied and widely used SPD technique to date. This metal forming process was invented by Segal in the early 1970s [12]. Starting in the 1990s, it quickly became well known as an effective technique for grain refinement of a coarse-grained polycrystalline metal [13–15]. The advantages of ECAE are that near net shaped, relatively large (mm to cm) samples of fine-grained, and in some cases even nano-grained, metals with very low porosity can be formed. ECAE is a discontinuous process, involving insertion and reinsertion of the sample in a die that has two channels with equal cross section intersecting at an angle U (see Fig. 1), varying from 60° to 150°. During the extrusion, the sample undergoes no change in cross section, and therefore it can be reinserted into the die to impose more plastic strain. It is not uncommon for a sample to experience three or more extrusions. Each pass invokes a strain path change to the sample because reinsertion requires rotating the sample to align it back with the entry channel. ECAE samples are typically square or circular in cross section. Therefore, one can also rotate the sample clockwise or counterclockwise about the billet-axis prior to each reinsertion. Four standard sequences of strain path changes have been developed since the conception of ECAE, referred to as routes A, Ba, Bc, and C. (Ba and Bc have also been referred to as B and D, respectively, e.g., [10].) Processing routes are not limited to these, particularly for circular cross-section billets, and indeed alternative routes have been devised. Previous texture studies for large strain deformation involved torsion testing, drawing or rolling; none of these deformation processes, however, are similar to ECAE (see later Section 2.1.6 for more detail). There are two major differences. First, each ECAE pass imposes a large amount of shear. Second, each consecutive pass unavoidably imposes a strain path change from the previous one. After several passes, the sample has been sheared in different directions, accumulates extreme strains (>300%), and experiences substantial microstructural evolution such that the original coarse-grained microstructure has been erased. The deformation characteristics, as a result of possible variations in die geometry (such as rounding the outer corner by angle W, Fig. 1) and routes, will be detailed in Sections 3.1.3, 3.2.3 and 5. ECAE offers a unique deformation technique to study the mechanisms driving texture evolution and to push and improve polycrystal models for texture evolution. Most texture measurement and modeling tools have never been applied to such severe strain regimes and frequent strain path strain sequences. Because of the complexity of the multi-pass deformation process, ECAE textures need to be characterized differently than those of conventional processes. Compared to the other SPD techniques, most of this article will be devoted to reviewing the current understanding of texture development in ECAE and the relationship between ECAE textures and microstructure (including grain refinement) and mechanical properties. 2.1.2. Definitions of deformation and velocity gradient in ECAE In order to understand texture evolution in a material, one must first understand the deformation being imposed to it. For this, the reader needs to be familiar with a few basic definitions in mechanics. For modeling texture evolution, the applied deformation is best described in the form of a deformation gradient F and velocity gradient L. The deformation gradient F relates the final deformed configuration to the starting one. Let x = f(X) be a one-to-one mapping of the original point X to a point x in the deformed body, where X and x are vectors. F is defined as the gradient of f and is

F ¼ rf

or F ij ¼

@xi @X j

ð1Þ

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and

xi ¼ F ij X j :

ð2Þ

The velocity gradient L is given by

Lij ¼ @ u_ i =@xj ;

ð3Þ

which is related to F by

_ 1 ; L ¼ FF

ð4Þ

where i, j = 1 ,..., 3 and u_ is the displacement rate vector. The symmetric part of L is the strain rate tensor e_ and the skew part is the rigid rotation rate tensor x_ . If the deformation is inhomogeneous, then L will depend on position X in the undeformed billet, or if the deformation is continuously varying, then it will depend on time t. Both situations are encountered in SPD processes. 2.1.3. Simple shear model Segal and co-authors [10,11,16] proposed the simple shear (SS) model as the most fundamental description for the deformation mode in an ECAE extrusion. The SS model assumes that the material experiences simple shearing on the intersection plane of the two channels (concentrating the shear in a narrow planar region). The conditions of the model are the following: no rounding of the inner and outer corner regions, perfectly plastic material, and no friction. Even though these conditions are rarely met in practice, the SS model can be a good first approximation of the deformation mode and useful when modeling texture development. It is also helpful in understanding the strain path changes experienced by the sample during multiple-pass ECAE and during deformation sequences involving ECAE plus subsequent mechanical testing or forming operations. It follows from the SS conditions that F and L are functions of U only. The lower U, the higher the accumulated shear strain per pass. When expressed in a coordinate system (x0 –y0 –z) lying parallel to the intersection plane (see Fig. 1), F and L are:

2

1 2 cotðU=2Þ 0

6 F ¼ 40 0

1 0

2

3

7 05 1 x0 ;y0 ;z

0 c_ 0

3

6 and L ¼ 4 0

0

7 05

0

0

0

ðalong intersection planeÞ; x0 ;y0 ;z

ð5Þ where the only non-zero component of the velocity gradient is Lx0 y0 ¼ c_ , and c_ > 0 is the applied R shear rate in the shear plane. After a single pass, the accumulated shear c ¼ c_ dt is [10]

pffiffiffi

c ¼ 2 cotðU=2Þ and ev M ¼ c= 3:

ð6Þ

For the purposes of multiple-pass ECAE modeling, in which the direction of shearing with respect to the sample changes from pass to pass, it is best to express the deformation in a fixed laboratory coordinate system, such as the x–y–z system shown in Fig. 1. In this coordinate system, F and L transform to

2

2 þ cos U

 sin U

0

3

6 7 F ¼ 4 ðcos U þ 1Þ2 = sin U  cos U 0 5 ; 0 0 1 x;y;z

2

sin U

6 L ¼ c_ 4 cos U þ 1 0

cos U  1 0  sin U 0

3

7 05 : 0 x;y;z

ð7Þ

Note that when expressing simple shear deformation in this fixed system, all the in-plane components become non-zero. The fixed coordinate system in Fig. 1 is highly convenient and will be used throughout this article. Accordingly, F and L in Eq. (7) are strongly recommended when applying the SS assumption in polycrystal modeling. With the deformation expressed in the form of F and L, one can begin to model texture evolution in a given material. We will return to this topic of texture development under SS later in Section 4.4. Next, we describe how F can be used to determine certain measures of the deformation, such as the stretches and rotations, strains, and strain rates experienced by a material point during ECAE.

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2.1.4. Distortions and rotations of material elements in ECAE It is useful to visualize the deformation during ECAE using an initially cubic or spherical element. For example, the initial cube can be used to calculate the distortions expected in a rectangular grid scribed on the sample surface, while the initial sphere is commonly used to represent the starting grain shape and to calculate its changes during deformation. What is often not realized is that a cube and sphere will deform differently in the same ECAE die. Under the conditions of the SS model and using F in Eq. (7), the inclination angle hr of a deformed cube with respect to the axis of the exit channel (also called the extrusion direction ED in this work) is:

   U hr ¼ cot1 2 cot : 2

ð8Þ

Under the same boundary conditions, an initial sphere will deform into an ellipsoid with its long axis inclined to ED with the following angle:

     U U U  cot ¼ : hs ¼ tan1 cosec 2 2 4

ð9Þ

Note that hs and hr are different. For example, for the most common die angles, U = 90° and 120°, hr is 26.6° and 40.9°, respectively, while hs is significantly smaller, 22.5° and 30°, respectively. Fig. 2 illustrates the differences for the U = 90° die, for which the shear value is c = 2.0. This figure also demonstrates the large rotations induced in ECAE. The segments of the square (identified with small circles) that are initially vertical become horizontal, while those that are initially horizontal (crosses) become inclined laterally with the angle hr. This rotation comes from the skew part of F. Continuum theory calculations in Eqs. (8) and (9) are expected to only approximate the distortions of actual grains in a polycrystal. In reality, grains are neither cubes nor spheres. They do not deform as homogeneous continuum elements but along specific planes and directions governed by their crystal lattice. Moreover, due to material variation, such as initial texture and grain distribution topology, the shapes of the deformed grains will assume different inclinations. Statistical averages of inclination angles over several grains, in fcc, bcc, and hcp metals alike, however, are found to be close to the analytical values of hr and hs expressed in Eqs. (8) and (9) [2,7,17–24]. For instance, in Han et al. [2], the elongated direction of Fe grains was on average 27.6° from the ED after the first pass. Similarly Beyerlein et al. [22] observed that Cu grains were inclined on average 27° and Ag grains were at an angle of 22° [7] after the first pass. In Shin et al. [24], the Ti grains elongated to 30° from the ED. 2.1.5. Multiple passes: rotations and strain path changes The conventional application of ECAE involves multiple passes through the die to accumulate large amounts of strain. In addition to the deformation imposed by each pass, a multi-pass ECAE process includes one or two rigid body rotations of the sample between consecutive passes. The first one, which is inevitable, is defined by U. In order to begin each subsequent pass, the sample has to be rotated around the TD (or z-axis) so that the ‘head’ of the sample is reinserted into the die. Because of 1.5 1 0.5 0

shear shear

-0.5 -1 -1.5 -3

θr

θs -2

shear plane -1

0

1

2

3

Fig. 2. Deformation of an original circle (or sphere) inscribed in a square (or cube) under the ideal case of simple shear along the intersection plane of a U = 90° die.

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this rotation, ECAE is discontinuous. The second rotation is applied about the longitudinal axis of the sample. This rotation is optional and is not related to U. It also defines the ECAE route. Motivated for die channels with square cross-sections, three types of routes have been devised and are the ones commonly used today, called; A, B and C. They involve sample axis rotations of 0°, ±90° and 180°, respectively. Route B decomposes further into two routes, Ba and Bc. In route Ba, the sense of the 90° rotation is reversed in every pass. In route Bc, the 90° rotation keeps the same sense in every pass. Multi-pass ECAE is not limited to these four routes and the methodologies presented in this article can be applied to any set of rotations between passes. One, however, will find very little deviation in the literature, and for this reason, our applications will also consider only these four. All routes impose a strain path change and changing the rotations between ECAE extrusions changes the strain-path-change sequence. A common misconception is that route A imposes monotonic deformation. Route A still represents a strain path change; in route A, the velocity gradient is rotated by 90° while the strain rate tensor remains the same [25]. (Deformation and texture are governed by the velocity gradient, not by the strain rate alone.) In route B, there is an additional ±90° rotation of the applied velocity gradient. In route C, the velocity gradient is reversed after each pass. The rotations induced in multi-pass ECAE deformation can be modeled in two ways. One way is to rotate the die for each pass and extrude the billet through a network of connecting dies. In this method, the simulation is performed in a reference system fixed to the sample and L changes from pass to pass. The other way is to rotate the sample while keeping the die fixed [26,27]. In both schemes, it is necessary to first select the orientation of the ECAE die and the order in which these rotations are applied.1 If done properly, they should lead to the same texture. For both, we will use the fixed laboratory coordinate system in Fig. 1. The first method is described for the case of simple shear in which L is homogeneous Eq. (5). In this case, the alterations in L with each pass can be expressed analytically. Suppose the exit die is aligned with the horizontal x and the entry die varies according to U as in Fig. 1. The first rotation is about the x-axis and depends on the ECAE route Z through the rotation angle /(Z, N). The second rotation is clockwise by h = (p  U) about the z-axis for reinsertion of the ‘head’ of the sample into the entry die. The rotation tensors for the first and second rotation are

2

3 1 0 0 6 7 RI ðZ; NÞ ¼ 4 0 cos /ðZ; NÞ  sin /ðZ; NÞ 5; 0 sin /ðZ; NÞ cos /ðZ; NÞ

2

3 cos h  sin h 0 6 7 RII ¼ 4 sin h cos h 0 5: 0 0 1

ð10Þ

For example, for route A, /(A, N) = 0; for route Bc, /(Bc, N) = p/2; and for route C, /(C, N) = p for all N. For route Ba, /(Ba, N) = p/2 for odd N and p/2 for even N. These assignments assume that counterclockwise rotations are positive. The deformation gradient for two passes F2 in any route is F2(Z) = F1RII(Z,N)RI(Z,N)F1. This procedure can be repeated to obtain FN after N passes and for any route, as was done in2 [28]. With FN(Z), one can find, for instance, hr and hs after any route Z and pass N. For two passes of route A, F2(A) = F1RIF1, whereas that for monotonic loading to the same strain would be simply F2 = (F1)2. Under both the two-pass route A and the monotonic deformation, the element will stretch and elongate, but its final shape and inclination relative to the shear plane will not be the same, see Fig. 14 in [28]. In the second method, the location of the shear plane does not change during the ECAE process and L retains its orientation, just as in the laboratory. To simulate a rigid rotation of the billet between passes, both the crystallographic and morphological (grain shape) texture are rotated relative to a fixed global coordinate system. This method has been used to successfully simulate all four routes in [27,29,30–32] for all types of L, from homogeneous SS to inhomogeneous finite element (FE) deformations. When the deformation is inhomogeneous, one needs to further account for the change of material positions in the billet from pass to pass when modeling texture evolution [33]. In routes C, Bc, and Ba, material points in the top and bottom layers differ with pass number, while they are maintained in route A. This consistency has made route A easier to model than the rest. In this article, 1

The selection does not matter. It is only necessary that it is known and fixed. In [28] the order of the rotations were switched and derived assuming the entry channel is vertical and the exit channel varied according to U. The result will be the same. 2

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the second method will be the preferred one for texture calculations because it models more clearly the operational sequence during the experiment. For instructive purposes, F and L have been derived thus far assuming ideal SS ECAE deformation. Other analytical and numerical methods for calculating F and L in non-ideal ECAE deformation conditions will be reviewed in Sections 5.1 and 5.7. 2.1.6. Relationship to other deformation processes: drawing, rolling, plane strain compression, torsion It has been of interest in some studies to compare textures from ECAE to those from other deformation processes, like rolling, compression, drawing, or torsion. A comparison between ECAE and torsion, for instance, was carried out by Tóth [26] for textures in copper. In this case, the two deformation processes are not fundamentally that different. Tóth’s analysis [26] elucidated some similarities, as well as some differences, especially those concerning the tilts of the ideal components from their expected positions. Unlike torsion, the remaining processes mentioned above have no relationship with ECAE. In an attempt to draw a relationship between two deformation states or two deformation textures obtained by different processes, one has to be very careful to compare not only the textures (after applying a suitable rigid rotation to coincide them) but also to make sure that the imposed deformation mode involves the same plastic rotations. In other words, texture development depends on the velocity gradient (rotations and stretches), not just on the strain rate tensor (stretches only). Consider, for example, a comparison between rolling and ECAE. By applying a 45° rotation, the textures apparently become similar. This initial step only considers the similarities of the strain tensors in their rotated state. Considering further their velocity gradients reveals that the two deformations are completely different and hence would lead to different textures. In doing so, one would find that it is impossible to transform the velocity gradient in ECAE (whether assuming simple shearing on one plane or a set of rotating planes) into the necessary velocity gradient in rolling. The differences in the velocity gradients can lead to the complete removal of certain texture components in one deformation mode, while   maintaining their strength in another mode. For example, the A1 and A2 components can have very different strengths in simple shear, while these same components are equally strong in rolling. Other features can also be strikingly different, like the nature of the rotation field around the ideal positions. In rolling, all ideal components are located within a convergent rotation field [34], while in torsion the rotation field is convergent on one side and divergent on the other side of the ideal position [35]. This feature naturally leads to tilts of the ideal components of the texture in shear, while there is no such effect in rolling. In summary, comparisons based on strain rate tensors alone are superficial and physically unfounded. 2.1.7. Other SPD techniques Although this article concentrates on ECAE textures, for completeness, we end this background section with a brief review of the main deformation characteristics of three other SPD processes: conshearing, high pressure torsion, and accumulative roll bonding. Reference is made to selected works containing texture analyses. Currently, compared to ECAE, the number of texture studies for these processes is significantly less and more are highly recommended. The main principles presented in this article for ECAE can be applied to these SPD processes. 2.1.7.1. Conshearing. The ‘conshearing’ process is a continuous version of ECAE developed for sheet metals. In this technique, the sheet is fed into the ECAE die by a central roll with the help of a series of satellite rolls. The main purpose of this technology is to refine the microstructure and improve the rvalue of the sheet via the shear texture that is introduced by the ECAE die [36–38]. 2.1.7.2. High pressure torsion (HPT). HPT is a special SPD technique used to achieve even higher strains than those possible with ECAE. It has been applied since 1993 [39]; however, it was first proposed by Bridgeman in 1952 [40]. Shear strains as high as 100 can be readily obtained, but in theory, there is no limit. The sample is a thin disk, usually not more than 1 mm in thickness. It is placed between two flat surfaces and then compressed by a pressure of at least 0.5 GPa, and up to about 12 GPa. Shear strain is introduced into the disk by rotating one of the two pressing surfaces with the help of friction.

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With this SPD technique, much smaller grain sizes can be obtained than in ECAE, e.g., 20–50 nm depending on the material. Similar to conventional torsion of a rod, the shear strain is proportional with the distance from the center of the disk. Not surprisingly, shear textures develop during HPT; however, they are quite weak. More details can be found in [39–44]. A new SPD technique, which can be considered a variant of HPT but suitable for thin-walled tubes, has been presented recently in [45]. 2.1.7.3. Accumulative roll bonding (ARB). ARB is a recently developed rolling process that produces submicron grain sizes. In ARB, SPD is introduced into the sheet without decreasing the thickness of the sheet. This is achieved by cutting the rolled sheet into two, stacking them, and rolling again. In this way, the thickness is kept if the rolling reduction in the previous rolling cycle was 50%. Between two consecutive rolling passes, the sheet surfaces that will be in contact during rolling require special preparation. The cohesion of the intermediate surfaces is made by diffusion which requires higher temperatures during the process. The textures are typically rolling textures in the middle part of the sheet but are strong shear textures near the outer surfaces in contact with the rolls. The intensity of the shear components reduce rapidly with larger numbers of ARB passes. ARB textures were studied in [46–55]. 2.2. Ideal texture orientations in simple shear As most severe plastic processes involve simple shear, it is important to be familiar first with the textures that develop in simple shear. Simple shear textures depend on crystal structure and their ideal components have been determined for fcc [6,56–58], bcc [6,59], and hcp metals [60]. We refer the reader to these articles as well as to [61,62] for fundamental aspects of texture characterization. 2.2.1. Fcc crystal structure The behaviors of the ideal orientations of fcc crystals have been examined in detail in [35,56,57]. They can be described by two partial fibers that are defined by {1 1 1}||SP (A fiber) and (h1 1 0i||SD  A (B fibre), where SP is the shear plane and SD is the shear direction. The A fiber contains the A, A; 1    and A2 ideal components, while the B fiber contains A, A, B, B, and C. Fig. 3a shows the fcc fibers  positions. and ideal components in a {1 1 1} pole figure. The two fibers are connected at the A and A The relative intensities of the ideal components depend on the symmetry of the test. Simple shear involves a two-fold symmetry around the axis perpendicular to both the SP normal (SPN)  and the B and the SD vector. Under this symmetry, the A component has the same intensity as A    component the same intensity as B. However, because the A1 and A2 components are not symmetric with respect to a two-fold symmetry operation, they can develop different intensities. C is self symmetric. In Sections 3.1.2 and 3.1.3, these symmetries are discussed in further detail with respect to ECAE textures. 2.2.2. Bcc crystal structure The ideal orientations for bcc materials in simple shear (torsion) were identified in [58]. Like in fcc metals, they are defined by two fibers: {1 1 0}||SP and (h1 1 1i||SD). The former contains the F, J, J, E, and  components. (In negative simple shear, which is  ideal components and the latter the D1, D2, E, and E E relevant for ECAE, the D1 component is very weak [6,59,63].) Note that these fibers and ideal orientations can be related to those for an fcc metal by an exchange of the slip planes {h k l} and slip directions hu v wi. Fig. 3b shows the location of these fibers and the ideal components in a {1 1 0} pole figure. 2.2.3. Hcp crystal structure Only recently have the ideal orientations for hcp materials in simple shear been mapped systematically in a theoretical study [60]. Five fibers were found in Euler space defined for a shear applied on the y plane and in the x direction. They were defined with the help of Euler angles:

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437

Fig. 3. Ideal orientations of torsion (simple shear) textures in for (a) an fcc metal in a {1 1 1} pole figure [35] and (b) a bcc metal in a {1 1 0} pole figure [6].

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B  fiberð0 ; 90 ; 0—60 Þ;

basal planekshear plane;

P  fiberð0 ; 0—90 ; 30 Þ;

haikshear direction;







Y  fiberð0 ; 30 ; 0—60 Þ;

ð11Þ

C 1  fiberð60 ; 90 ; 0—60 Þ; C 2  fiberð120 ; 90 ; 0—60 Þ These fibers have very different ‘‘orientation persistences” (see more in [60] about orientation persistence) depending on the relative critical strengths of the slip families for a given hcp material. For example, for Mg, the B fiber is the most stable followed by the P fiber. 2.3. Ideal texture orientations in ECAE Ideal texture orientations in ECAE are defined directly from those in simple shear, assuming that simple shearing takes place along the intersection plane of the two channels in a negative sense [10]. Thus, in all crystal structures, when an ideal orientation g = (u1, u, u2) is expressed in the die reference system x–y–z, see Fig. 1, the transformation simply involves an additional U/2 rotation to u1:

g U ¼ ðu1 þ U=2; /; u2 Þ:

ð12Þ

Tables 1 and 2 list the ideal ECAE components for fcc and bcc structures for the U = 90° die angle [6,63]. The subscript in this table signifies that their definitions depend on the die angle U. According Table 1 Ideal ECAE orientations and fibers for fcc materials and a U = 90° die [6]. Notation

A1U A2U AU U A BU U B CU a b

Euler anglesa (°)

Miller indicesb

Fibers it belongs to

u1

u

u2

ND

ED

TD

80.26/260.26 170.26/350.26 9.74/189.74 99.74/279.74 45 225 45/165/285 105/225/345 135/315 45/225

45 90 45 90 35.26 35.26 54.74 54.74 45 90

0 45 0 45 45 45 45 45 0 45

 ½8 1 1

 4 ½1 4

[0 1 1]

{1 1 1}U

 4 ½1 4

 ½8 1 1

[0 1 1]

{1 1 1}U

[9 1 4] 1 1  5 ½1

 ½1 1 1 5 1  4  ½9  9  ½7 2 61 5 1  1  4 ½1  ½2 2 3

 1 2 ½1  1 2 ½1  1 1 ½1  1 1 ½1  1 0 ½1

f1 1 1gU h1 1 0iU f1 1 1gU h1 1 0iU h1 1 0iU h1 1 0iU h1 1 0iU

½1 5 4 1 1 6  1 9 2 ½7 [3 3 4]

Given in the u2 = 0° and 45° sections only. Approximate values.

Table 2 Ideal ECAE orientations and fibers for bcc materials and a U = 90° die [63]. Notation

D1U D2U EU U E JU JU FU a b

Euler anglesa (°)

Miller indices

b

Fibers it belongs to

/1

/

/2

ND

ED

TD

99.74/279.74 9.74/189.74 170.26/350.26 80.26/260.26 135 315 15/135/255 75/195/315 45/225 135/315

45 90 45 90 35.26 35.26 54.74 54.74 45 90

0 45 0 45 45 45 45 45 0 45

[1 1 8]

 ½4 4 1

 0 ½1 1

h1 1 1iU

 ½4 4 1

[1 1 8]

 1 0 ½1

h1 1 1iU

[9 1 4] 1  4  ½9 [1 5 4 1 1] 5 1  1  4 ½1

1 1  5 ½1  ½1 1 1 5 6  1 9 2 ½7  9  ½7 2 6 1 2  3 ½2

 1 2 ½1  1 2 ½1  1 1 ½1  1 1 ½1  1 0 ½1

h1 1 1iU ; f110gU h1 1 1iU ; f110gU f1 1 0gU f1 1 0gU f1 1 0gU

Given in the u2 = 0° and 45° sections only. Approximate values.

[3 3 4]

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439

to Eq. (12), the ideal orientations of textures generated by tooling with two different die angles, U1 and U2, will differ by a rotation of (U1  U2)/2 about the TD axis. To aid texture characterization, it is useful to know the location of the ideal ECAE orientations in pole figures and ODF sections. For fcc and bcc structures, they are presented in Fig. 4 in three projections, on the TD (z-plane), ND (y-plane), and ED (x-plane) for both U = 90° and 120° [6]. As for their appearance in an ODF presentation, an example is shown in Fig. 5 for the fcc case in two sections of Euler space in (a) simple shear, (b) a U = 90° die and (c) a U = 120° die. These two sections contain all the ideal components and the entire B fiber (in the u2 = 45° section), but not the whole A fiber. The ECAE texture components in hcp metals, derived from applying the transformation formula (12) to the simple shear fibers defined in Eq. (11), are given in [64] and shown in pole figure format in Fig. 6 for a U = 90° die. ECAE textures after multiple passes are not expected to be directly related to classical simple shear textures and are much more difficult to characterize. The sample in multi-pass ECAE experiences shear deformation on different planes and/or different directions with each new pass, and not monotonically in the same plane and in the same sense of direction as assumed in classical simple shear texture characterization. As discussed in Section 2.1.5, in multi-pass ECAE the sample can undergo one or two rotations about different axes between consecutive passes. Both crystallographic orientations and the morphological axes of each grain are altered before each new pass [27]. Consequently, when the sample is reinserted into the die for the next pass, grains which have been preferably oriented by the deformation of the previous extrusion will be displaced into an orientation that is usually not an ideal position (in theory). The rotation associated with the reinsertion of the sample into the die involves a shift of the ODF in the laboratory system. For all routes with the exception of route A, this shift is complex. For route A, which involves a single rotation around TD, only the u1 angle changes and this ODF shift can be easily expressed as a function of U;

g U ðRoute AÞ ¼ ðu1  U; /; u2 Þ

ð13Þ

where orientation g = (u1, u, u2) in the above is the ideal ECAE component after the previous pass and gU (Route A) is the orientation after the TD rotation and before the next pass in route A. Generally in all routes, all texture components shift to unstable positions before each new pass due to the rigid body rotations applied between passes. There is one exception. In the case of a U = 120° die and fcc metals,  components replace each other before each new pass in route A. the B and B To characterize the textures in multiple passes, it is best to identify and use components that are independent of route or amount of strain. For this reason, the ideal components identified from the simple shear test given in Tables 1 and 2 are suitable. This technique appears to work reasonably well in characterizing the evolution of texture from pass to pass and various routes in cubic metals, e.g., [6,65]. It is also possible to characterize texture evolution by identifying new ideal fibers that repeatedly appear after subsequent passes. This technique has been attempted as well, but again only in cubics [31,32,63,65,66]. For Cu, Li et al. [23,65,67] found that regardless of route and pass number the resulting textures could be described well by the f1–f3 fibers in Table 3. Only the intensities of the components along these fibers varied with the number of passes and route. Similarly, for bcc structures, three fibers (b1–b3) were identified [63], and these are also given in Table 3. It was found that these too could universally describe textures from different die angles, pass number, and routes A, Bc, Ba, and C in IF steel [31,32]. They are given in Table 3 only for U = 90°. Like the ideal ECAE components, the f1–f3 fibers or b1–b3 fibers for other U are similar in nature, only differing by a rotation h = U/2 about the TD axis.

3. ECAE texture measurements Texture evolution is governed by three main factors: applied deformation (which depend on U, N, route), deformation mechanisms (e.g., slip, twinning systems), and initial texture. Generally, texture analyses aiming to ascertain one of these factors require knowledge of the other two. For instance, if one seeks to use the texture analysis to determine the active deformation mechanisms, the nature

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Fig. 4. (1 1 1) pole figures on the TD (z) ND (y) and ED (x) planes showing the ideal orientations of ECAE textures defined by the simple shear model in (a) fcc and 90° die; (b) fcc and 120° die; (c) bcc and 90° die; and (d) bcc and 120° die, Li et al. [6]. The subscript h is U in the present article.

of the applied deformation and initial texture must be known. Textures measured in ECAE-processed cubic and hcp materials are examined in this section with reference to these three factors, although not exactly in a separated fashion.

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φ2 = 0° φ1

0°

Φ

A1*

(a) Simple shear:

φ2 = 45°

direction of rigid body rotation 360° 0°

C

A2*

A1 *

C

A

A2*

φ1

90°

A2*

A1 *

A2*

C

A1*

A

C

C

B

B

C

A2*

A

B A2 *

shift by Φ/2 = 60°

A1*

B

B fibre

C

A2*

B

B A1*

C

shift by Φ/2 = 45°

B

(c) ECAE 120°:

A

B fibre

A1*

B A2*

C

shift by Φ/2 = 45°

(b) ECAE 90°:

A

B fibre

B

360°

B fibre

B

B

B

A1 *

C

A2*

B A1*

shift by Φ/2 = 60°

A1*

A2 *

A

C

B fibre

A

B fibre

B

B

B

B

A1 *

C

A2 *

B

B A1*

A2 *

C

B A1*

Fig. 5. Relationships between the ideal positions in fcc simple shear (a) and ECAE deformation for (b) U = 90° and (c) 120° in orientation space using two different sections, u2 = 0° and u2 = 45°. Note that any subscript U or h used previously to signify ideal ECAE orientations for a given U from ideal simple shear orientations have been removed in this figure. Recall that for U = 90° the ideal orientations are shifted U/2 = 45° and for U = 120°, they are shifted U/2 = 60°.

C1

2: ND C2

2’: NSD

2’: NSD

1’: SD B

1’: SD P

P1 Y

P

C1

2: ND P1

B

Y

C2

P1 P

1: ED

1: ED

P

P1 P1

C1

Y

Y B

C2

a: (0002)

P1 P1

b: (10 10)

Fig. 6. Ideal components and fibers in textures of magnesium under ECAE loading in (00.2) (a) and (10.0) (b) pole figures [64].

3.1. Cubic textures in ECAE 3.1.1. Influence of die angle The die angles U tested thus far range from as low as 60° [68] to as high as 157.5° [69]. By far the most common die angle is 90°, unless the material is hard to deform, in which case a larger U is needed.

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Table 3 Definition of fibers in fcc (f1, f2, f3) [65] and bcc (b1, b2, b3) ECAE textures [63]. Notation of fiber

Orientations in fiber

f1 f2 f3 b1 b2 b3

 U  A f1 1 1g partial fiber A1U  AU =A U 2U  U =AU h1 1 0i partial fiber;A  U =AU  A f1 1 1g partial fiber  U =BU  A CU  B U U 1U  U h1 1 0i partial fiber;AU =A  U  A f1 1 1g partial fiber  U  AU =A C U  BU =B U U 2U D2U–EU–D1Uh1 1 1iU partial fiber F U  J U  EU f1 1 0gU partial fiber;EU  D2U h1 1 1iU partial fiber U h1 1 1i partial fiber;  E U  JU  F U f1 1 0g partial fiber D1U  E U U

The die angle U influences texture evolution primarily in two ways. The first effect is a change in the ideal positions of the components (derived from the simple shear components) as discussed in Section 2.3 and as predicted from Eq. (12). In theory, the expectation is that two textures one developed by U1 = 120° and the other by U2 = 90°, will differ by a rotation of (U1U2)/2 = (12090°)/ 2 = 15° about the TD. This expectation agrees well with several observations, e.g. see [6,31,70] for fcc, bcc, and hcp examples as well as Sections 3.2.2–3.2.3. The second effect is the degree of texture evolution; by decreasing U the deformation per pass increases, and so the ideal components appear stronger. This effect is more apparent after a large number of passes [31]. However, this effect not only involves a change in the overall intensities with U but also in the relative intensities of the components. For example, by decreasing U from 120° to 90°, the D1U component becomes significantly stronger in IF steel [31]. Another example is in Cu, where the CU component becomes nearly absent when the die angle is increased from 90° to 120° [30]. The die angle U can also have other, not exactly minor, effects, on texture evolution. First, the rotation to reinsert the head into the entry channel made between passes has to be done in all routes and depends on U. A different U changes this rotation which naturally leads to different ‘entry textures’ at the start of each pass. Second, changing U may affect the size of the plastic deformation zone (PDZ). For instance, it was found in [31] that because the material filled the 120° die better than in the 90° die (both W = 0), the PDZ was smaller, causing the deformation to closely resemble simple shear. 3.1.2. First-pass textures We begin with textures developed after one pass, which do not have a specified route name, as it is the same for each route. In most cases, ECAE textures after one pass can be well described by the intensities of the ideal ECAE components presented in Tables 1 and 2 and Figs. 4–6. Due to the large plastic strain imposed by one ECAE extrusion, crystal orientations will normally approach the ideal ECAE positions. In analyzing textures, an important question is the so-called sample symmetry with respect to the imposed macroscopic deformation. If the initial texture was random or had monoclinic symmetry, the monoclinic symmetry of simple shearing is achieved in the first-pass texture. The monoclinic symmetry refers to invariance about a 180° rotation around the TD axis of the die. Only in ideal simple shear in the intersection plane of the channels one can expect to maintain monoclinic symmetry. Any deviation in deformation, such as rounded and asymmetric flow in the plastic zone, may lead to deviations from monoclinic symmetry. For simple shearing textures of interest here, monoclinic symmetry in first-pass textures has been observed in samples processed in 90° dies [22,25,32,33,65,66,71–74], and in 120° dies [31,75] for Cu, Al, Ni, and steel. When, on the other hand, the initial texture is relatively strong and does not have monoclinic symmetry, monoclinic symmetry will not be observed in the first-pass texture [72,76]. A non-random initial texture can happen easily for metals containing large 0.1 to 1 mm-sized grains or ones that experienced heavy pre-processing. Initial texture effects are non-negligible and will be discussed further in Section 3.3. 3.1.3. Influence of route and pass number Next to the die angle U, ECAE route and pass number significantly impact texture evolution. Their influences have been studied most comprehensively in cubic materials.

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The question of symmetry in multi-pass ECAE textures is particularly important and the answer depends on the selected route. In ideal conditions when near simple shear takes place in ECAE, any rotation of the sample around the TD axis, which is the axis of symmetry of the simple shear process, does not change the texture symmetry. The TD axis can, therefore, be called the ‘texture-symmetryaxis’ of the process, however, only for routes A and C. Sample rotation around any other axis leads immediately to the loss of monoclinic symmetry. This is why in routes A and C, monoclinic symmetry of the texture is maintained for many passes as long as the sample deformation remains homogeneous in the sample where the measurement was taken. Monoclinic symmetry in route A textures has been reported in several studies: Gholinia et al. 12 passes in a 90° die and 15 passes in a 120° die in Al–Mg [77]; up to three passes in Tóth et al. [74] and four passes in Li et al. [67] in Cu; and four passes in Li et al. [31,32] in IF steel. In routes Bc and Ba, after the first pass, the initially monoclinic symmetry is lost in subsequent passes. As explained above, the reason for this is that the texture is rotated between passes around an axis which is not the texture-symmetry-axis of the process. The textures developed in these routes are highly dependent on pass number, e.g., [65]. These features make textures developed by routes Bc and Ba relatively difficult to characterize. Route C is considered the reversal route. In route C, the plane of shearing remains the same, but every pass reverses the shear direction of the previous one. At least in the first few passes, grain morphologies, as seen in optical microscopy, change accordingly. In the first pass, grains severely distort and elongate, while in the second, they more or less restore their original shape, as seen for example in [10,23,78]. If their crystallographic orientations were to behave in the same way, texture evolution would be cyclic in route C: even-pass textures would resemble the initial texture and odd-pass textures would resemble the first-pass texture. However, there is much experimental evidence to the contrary. In a wide range of materials processed by route C, Cu [23,65,79,80], Al and Al alloys [66,81,82], Fe [83], IF-steel [25,31,32], Ti [80,84], and Zr [85]), textures after even-numbered passes (usually after two or four passes) exhibit a monoclinic symmetry (shear-like texture) which is similar to that of the first pass. Typically these are, however, weaker than those after the odd-numbered passes, and in some cases can contain a mix of shear components and components of the initial texture. The lower the strain per pass, the less pronounced the effect, as seen for example in pure Al processed at 500 °C using a U = 120° die, in which the second-pass texture was similar to the initial texture [86]. Some studies have referred to a shear-like texture after a reversal as a ‘retained’ texture, as if some grain orientations are retained, unchanged during reverse deformation. Texture components in evenpasses are obtained by deformation, see for example discussion in [87,88], and Section 6.5, where modeling textures in large strain shear reversals will be discussed. To be more concrete, we discuss these trends in Cu and IF steel as representative examples of fcc and bcc metals, respectively. 3.1.4. Fcc example: copper Using both (1 1 1) pole figures and ODFs, Figs. 7 and 8 show the neutron diffraction measurements of texture development in fcc copper processed by routes A, Bc and C using a rounded corner U = 90° die [65,67]. The initial texture before ECAE was weak. The monoclinic symmetry can be well identified in the pole figures (Fig. 7) for routes A and C while it is absent in route Bc. In the ODF sections shown in Fig. 8, for monoclinic symmetry, the range of the angle u1 could have been restricted to 0° < u1 < 180°, however, it was extended for the full range of u1 to show the intensity relations between the non-cen and B=B,  are particularly suitable tro-symmetric components. The two texture components, the A=A for determining if a texture exhibits monoclinic symmetry. Namely, in monoclinic symmetry the  as well as the B and B  components, are equal, respectively, while in the abintensities of the A and A, sence of that symmetry, they are different. When monoclinic symmetry is not present, the range of u1 has to be 0–360°, like for routes Ba and Bc. It is recommended that the ODF is always calculated and presented in the full range of u1 in ECAE because the symmetry should not be assumed a priori. Nevertheless, it can be verified a posteriori. It is evident in both the pole figure and ODF representations that all the ideal ECAE components appear with the exception of a weak A2U. Moreover, with an ODF presentation of textures (Fig. 8)

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Bc:

C:

Fig. 7. (1 1 1) pole figures of textures in Cu in routes A [67], Bc, and C [65] in a 90° die with an outer corner angle of w = 37°. Axis x (ED) is horizontal and y (ND) is vertical. Isolevels: 1/1.4/2/2.8/4 mr.

involving many sections, the main orientation fibers f1–f3 (Table 3 [65]) for an fcc metal can be identified in the first pass.  U components strengthen in In route A, the main component in subsequent passes is A1U. The BU =B the second pass and decreases slightly in the third pass. It is important to mention that when the initial texture is strong, texture evolution may be quite different, particularly in the first few passes. Take for example, the work of Suwas et al. [29], in which the initial texture was a strong h1 0 0i fiber texture. Like the initially weak Cu in Figs. 7 and 8, A1U became the main component in the second pass and  U strengthened progressively. However, due to this particular initial texture, the Cu component BU =B was very strong in the first pass and completely absent by the third pass.  U is the strongest, A1U is relatively weak, and the AU and CU compoIn route Bc, after two passes, A nents are absent. After four passes, A1U is the strongest and CU reappears [65]. Measurements in [65] revealed that the texture after eight and 16 passes are similar to those of four, with only minor differences in the second-order features. Due to the complexity in the deformation imposed by this route, textures in Bc are difficult to characterize. This challenge was overcome in [65], where it was found

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N=1

445

Ν=4, Bc

φ2

φ2

Fig. 8. u2 = constant ODF sections of textures measured in copper after one and four passes in route Bc [65]. Contours isolevels: 1.4/2/2.8/4/5.6/8 mr. The f1–f3 fibers (Table 3) connect ideal positions in the full ODF. The subscript h is U in the present work.

that the texture evolution in route Bc, up to 16 passes, could be well characterized and understood based on the main orientation fibers f1–f3 (Table 3). There it was shown that only the location and variation of orientation intensities along these fibers f1–f3 change with pass number (and route). In route C, texture evolution is not cyclic as one may first expect due to the cyclic nature of the imposed macroscopic deformation. Instead, textures in subsequent passes are all very similar to the first U orientations, together pass texture, depicting main texture components near the C U ; A1U ; AU ; BU =B with a much weaker A2U component [65]. The strengths of these components change with pass number. Compared to those for the odd-numbered passes, the textures for the even-numbered passes  U but a stronger A component. show slightly weaker A1U , CU and BU =B 2U 3.1.5. Bcc example: IF-steel Figs. 9 and 10 show the (110) pole figures and ODFs as measured by X-ray diffraction of IF steel processed after one to four passes by routes A, Bc, Ba, and C using a sharp U = 90° die [32]. Like Cu, this material also had an initially weak texture. Consequently, in the first-pass texture, all the ideal ECAE bcc components shown in Fig. 4c and the main orientation fibers b1–b3 for IF-steel can be clearly identified (see Table 3 for the definition of the b1–b3 fibers). The pole figures in Fig. 9 show that texture evolves with pass number in a manner that depends on route. Generally, routes A and C produce textures with a monoclinic sample symmetry and routes Bc and Ba do not. The ODFs in Fig. 10 reveal more details [32]. Due to the symmetry found in routes A and C, one only needs to show sections for 0° < u1 < 180° in Euler space. Compared to the one-pass texture, the fourpass route A texture depicts higher orientation densities in the b1 fiber (or h1 1 1iU ). For route C, cyclic texture evolution is apparent in the pole figures of Fig. 9 from the similarities between textures after two and four passes and, to a lesser extent, between one and three passes. Generally both the {1 1 0}U and h1 1 1iU fibers are complete in each pass, with the primary texture components located between the FU and JU/J U orientations along the {1 1 0}U fibers. In routes Bc and Ba, the orientation densities along the three fibers b1–b3 vary with pass number [32]. In route Bc, texture evolution is gradual. U Þ become less apparent than the By passes three and four, the {1 1 0}U fibers ðF U  J U =J U  EU =E U -FU deh1 1 1iU fiber. As shown in the texture after four passes, the b3 fiber passing through D1U  E picts the most complete orientation distribution with its main texture components located between U . In route Ba, the {1 1 0}U partial fiber becomes less apparent than the h1 1 1i fiber with D1U and EU =E U pass number. As shown in Fig. 10, the most complete fiber is the b1 (or h1 1 1iU fiber passing through U  D1U , with the maximum orientation density near E U . Interestingly for this route, texture D2U  E evolution is cyclic. The three-pass texture is approximately the same as the two or four-pass texture, but rotated by 180° about the TD, a relationship that Li et al. [32] attributed to the alternating ±90° rotations of route BA billet longitudinal axis during each two successive passes.

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A Levels:

1.0

1.3

Ba 1.6

2.0

2.5

Bc 4.0 3.2

5.0

6.4C 8.0

N=1

N=2

N=3

N=4

ND ED Fig. 9. (1 1 0) pole figures of experimental textures in IF-steel [32] after 1–4 passes (N) of ECAE via routes A, Ba, Bc and C. Isolevels: 1.0/1.3/1.6/2.0/2.5/3.2/4.0/5.0/6.4/8.0 mr.

3.1.6. Deviations from ideal texture positions It is possible in some instances for the positions of the ECAE shear components to deviate from the expected theoretically ideal positions [6,22,73,74]. We refer to these rotational deviations as ‘tilts’. They can amount to as large as 20° [22,74]. There is no singular reason for them and the various causes of these deviations are addressed below. Deviations from ideal positions are not particular to ECAE and have been observed in torsion testing. In torsion testing, at relatively small strain levels (<3), the preferred orientations shift towards the anti-shear direction [57,58]. In ECAE processing, tilts from the ideal ECAE components can, therefore, be expected. We mention that tilting about the torsion axis has also been attributed to lengthening of the billet during torsion testing or to the axial stresses in fixed-end torsion [56,57]. While lengthening/ shortening is not likely to happen in ECAE because of the very constrained nature of the deformation process, the axial stresses might be relevant. In some cases, tilts in ECAE textures can be relatively small (i.e., a few degrees) and dependent on the component. Because of these reasons, their precise measurement is only possible in an ODF representation of the texture. Such quantitative measures using ODFs have been reported, for instance, in Cu [72,74,89]. The main tendency in the ECAE of Cu is that in the first pass the tilts are small, while they become very high (up to 20°) in the subsequent passes (route A) [72,74]. The above comparison with torsion testing only applies when the condition of ideal simple shearing is achieved; however, as discussed in Section 5.1, this condition does not occur often. In reality, the material is usually not rigid plastic or isotropic, and the die geometry does not have sharp inner and

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Fig. 10. u2 = constant ODF sections of textures in IF-steel in a 90° sharp corner die [32] for the first pass and the fourth pass in all four routes. Contours: 1.4/2/2.8/4/5.6/8 mr. The b1–b3 fibers (Table 3) connect ideal positions in the full ODF.

outer corners with frictionless surfaces as assumed by the simple shear model. As a result, the deformation deviates from this ideal deformation mode (see Sections 5.1 and 5.3) and occurs instead over a broad plastic deformation zone. This deviation can result in the largest tilts. FE and analytical flow models that capture realistically the non-ideal deformation characteristics are able to reproduce these tilts in the textures (see Sections 5.3 and 5.7, as well as [22,23,28,72–74]).

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Table 4 Summary of recent ECAE hcp studies Material

Passes/routes

Processing Temp, U/w

References

Conclusions*

Zr702 Zr702 Zr702 Zr702 Zr702 Be Be Be Ti64 a-Ti CP-Ti CP-Ti CP-Ti Pure Ti CP-Ti CP-Ti CP-Ti CP-Ti Mg-AZ61 Mg–3.3%Li Mg–4Li Mg-AZ31 Mg-ZK60 Mg-ZK60 Mg- WE43 Mg-AZ31 Mg-AZ31 Mg-pure

4/A, Ba, C 1st pass 2–4/Bc,C 1st pass 4/Bc

RT, 135°/45° 350 °C, 90°/20° 350 °C, 90°/20° 350 °C, 90°/20° 350 °C, 90°/20° 425 °C, 120° 425 °C, 90° (10 1/s) 425 °C, 90° preheat 900 °C, 90° 400 °C, 90° 250–550 °C, 90°/20° 350 °C, 90°/20° 350 °C, 90°/20° 200 °C, 90°/37° 350 °C, 90°/20° 350 °C, 90°/20° 350 °C, 90°/20° 350 °C, 90°/20° 275 °C, 90°/30° 250 °C, 90° 260 °C, 90° 200° and 300 °C, 90° 260–325 °C, 90° 200 °C, 90°/110°/135° 260–325 °C, 90° 200 °C, 90°/37° 250 °C, 90°/37° 250 °C, 90°

[115] [85] [85] [116] [117] [67] [116] [88] [118] [84] [119] [120] [120] [121] [24] [24] [24] [24] [122] [123] [118,124] [118,124] [118,124] [271] [118,124] [135] [135] [28,60]

Prismatic, TTW, CTW Mostly basal Basal, prismatic, pyramidal  1}CTW Basal, prismatic, {10 1 Basal Basal Basal, prismatic Basal, prismatic Prismatic Basal, twinning (not specified)  1} CTW Prismatic > basal, pyramidal hc + ai, {10 1  CTW {1011} Prismatic Basal, prismatic f10 1 1Þ CTW Prism, pyramidal Basal, twinning (not specified) Prismatic Basal, prismatic Basal, prismatic, pyramidal hc + ai Basal, prismatic Basal, prismatic, pyramidalhc + ai Basal, prismatic, pyramidal hc + ai Basal, prismatic, pyramidal hc + ai Basal, prismatic, pyramidal hc + ai  2} TTW Basal, prismatic, {10 1 Basal, pyramidal hc + ai Basal

2A, 2C 1–2/A,C 2/C 1st pass 1st pass 2/Bc 1–4/Bc 1st pass 2/C 2/A 2/Bc 8/A, Bc 4/A, Bc 1–4/A 1–4/A 1–4/A 1st pass 1–4/A 1st pass 1st pass 1–4/A, Bc, C

*

Deformation mechanisms will be dependent on the method used to determine them. Some works used modeling, others used microscopy, and a few, both. The table summarizes results reported in the corresponding reference and are not reflective of our independent interpretation of their results. The results listed are brief, and the reader is encouraged to consult the reference for more details on the dependence of deformation mechanism on route and pass number. TTW stands for tensile twinning and CTW stands for compression twinning. We do not claim that this list is complete and some relevant works may be missing.

Thus far, we have mentioned only physical reasons for these tilts. Other sources can be the polycrystal model itself. The texture simulated by the visco-plastic self-consistent (VPSC) model (See Section 4.4) using the simple shear deformation history were slightly tilted about the TD from the texture simulated by the FC model, improving agreement with measurement [32,63,65]. Also changes in latent hardening coefficient in the single crystal hardening law are known to change the tilt as well [23,90]. 3.1.7. Texture strength Another monitored feature of texture analyses is texture strength. In some applications it is desirable to minimize texture strengths and reduce texture development (or achieve nearly random textures) in ECAE. Texture strengths are often measured by a texture index T or by the maximum intensity level found in the texture, normalized by that of random. The texture index T characterizes the overall intensity of a texture and is defined as follows:

T¼

Z

½f ðgÞ2 dg:

ð14Þ

where f(g) is the orientation density function and g is the orientation defined by the Euler angles: g = (u1, u, u2). Texture strength is usually reported as a function of N and route. When examining texture strengths it is good to keep in mind that texture strength can also be sensitive to any local heterogeneities sampled in the measurements, the measurement technique, the area or volume scanned, the number of grains in measurement, and the initial texture. In general, strengths of ECAE textures do

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not vary significantly with route or pass number, unless there is an unusually strong initial texture. In this case texture strength drops significantly in the first pass and then stabilizes with N (see Section 3.3.2). Because maximum values of intensity are sensitive to many factors, it is difficult (and perhaps fruitless) to find meaningful trends. Therefore, when literature results are compared, one will find substantial differences in texture strengths with N. Using neutron diffraction Li et al. [65] observe a slight increase in texture strength in Cu up to 16 passes. There are some cyclic variations as well; the textures after eight and 16 passes depict a similar strength but they are much stronger than that after four passes. Also, using neutron diffraction, Baik et al. [82], see the opposite trend; a weakening in route Bc with pass number N. Similarly, but using X-ray diffraction, Ferrasse et al. [78] also observe a weakening in Cu with N. However, in an Al alloy, they find that texture strengths fluctuate in routes Ba and Bc, more so than in routes A and C, and then stabilize after four passes. Using EBSD, Cao et al. [91], observe a strengthening in Al with N. Using X-rays on IF steel, Li et al. [31,32] do not observe any strong weakening or strengthening trend with N. 3.1.8. Saturation of texture with pass number It is possible that texture evolution can appear to ‘saturate’ after many passes (say beyond four to eight) to a steady-state texture that is specific only to the processing route. ‘Saturation’ in this case would not imply that grains have rotated to stable orientations and remain there. This cannot happen as the texture is always rotated between passes from a nearly stabilized orientation into an unstable position. The textures after many passes become very similar due to the repetition of the deformation path. 3.1.9. Effect of stacking fault energy It has been found that fcc materials with different stacking fault energies (SFEs) lead to different torsion textures. SFE is a material property, which varies in pure metals and can be lowered by alloying (e.g., brass), see for instance [92]. Al (167.5 mJ/m2), Ni (128 mJ/m2), and Cu (78 mJ/m2) are high to medium SFE materials. Low SFE fcc materials would include gold (45 mJ/m2) and silver (19–22 mJ/ m2). High to medium SFE materials tend to have their slip directions align with the shear direction (h1 1 0i), leading to a strong B fiber. As the SFE decreases, the C component gets replaced by a {112}h1 1 0i component [61]. Hughes et al. [93] find that the B component strengthens with large deformations for a wide range of SFE and the C component strengthens for the high to medium SFE  and A or A increase materials, while it weakens in a low SFE metal. Also for the low SFE material A=A 1 2  with strain (where the latter depends on the direction of the shear: A2 for negative shear and A1 for positive shear). These torsion test results can, at best, be applied to one-pass ECAE textures. Most of these texture differences in torsion testing were observed after monotonic loading up to equivalent strains greater than 2, much higher than those induced by a single pass of ECAE. Obtaining such high strains in ECAE requires at least two passes and involves at least one strain path change. SFE can only significantly influence texture evolution when it induces a transition in deformation mechanisms. The propensity of cross slip, climb, stacking fault formations, subgrain formation, shear banding and twinning can all be affected by SFE. Twinning and shear banding, in particular, result in large misorientations across their boundaries. As a consequence, texture can be altered significantly in two ways: (1) if at least 5–10% of the grains have been reoriented because of these mechanisms and (2) if their boundaries act as barriers to dislocation slip leading to anisotropic hardening of the crystal, which changes slip activity, and in turn, texture evolution. 3.1.10. Effect of deformation twinning SPD texture evolution in cubic metals which deform primarily by slip is reasonably well understood to date; however, the same is not true in metals which deform by both slip and twinning. Twinning has been observed at a small scale in SPD-processed fcc metals, usually in nano-sized grains [94–96] or close to regions of concentrated strain, such as shear bands [96,97], or low stacking fault metals [95,98,99]. Moreover, systematic SPD texture analyses under low temperatures and high deformation rates, when twinning may be promoted, are lacking.

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Fig. 11. Experimental ODFs and {1 1 1} pole figures corresponding to ECAE passes 1, 2 and 3 for Ag [95]. Isovalues in ODF: 0.8, 1, 1.4, 2, 2.8, 4, 5.6, 8 mr. Isovalues in pole figures: 0.8, 1, 1.3, 1.6, 2, 2.5, 3.2, 4, 5, 6.4 mr.

The only work we are aware of is recent work [95] on room temperature ECAE of Ag. As one of the lowest SFE fcc metals, Ag is known to both slip and twin. The {1 1 1}h1 1 2i twin reorients the lattice 60° with respect to the original matrix. To elucidate the role of twinning, the authors compared the textures of Ag and Cu, generated under the same ECAE die, temperature, and route A. In the first pass in  U components are the strongest, followed by the A2U and CU components, and Ag (Fig. 11), the BU =B finally the A1U component, the weakest. In contrast, in Cu, which does not twin, the CU, A1U, and  U components are equally strong, while the A2U component is the weakest. In the subsequent BU =B  U component becomes stronger, the A2U component maintains approxpasses in Ag (Fig. 11), the BU/B imately the same strength, and the A1U and CU components become weaker. In Cu, on the other hand, the CU and A2U components weaken. 3.1.11. Effect of substructure evolution Due to the severe plastic deformation, subgrain microstructures develop. There are two generic lengthscales of substructure. The higher scale consists of shear and deformation bands [78,81, 100–106], which are on the order of 10–100 lm apart. The lower scale consists of finer dislocation structures arranged into cellular patterns or as long extended subboundaries, spaced approximately 1–0.1 lm apart, e.g., [107]. Both lengthscales of substructure develop in metals processed by ECAE. Substructure of all types can affect texture evolution, either directly, by reorienting a volume fraction of the grains, or indirectly, by creating directional barriers for slip. The boundaries promote slip in parallel glide planes or hinder slip in intersecting ones. While these substructure elements do not deform independently from their surrounding neighborhood, they do use different slip systems. Consequently, with severe plastic straining, the misorientation across sub-boundaries can increase.

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Lower-scale substructure evolution is often used to explain why measured texture intensities are lower than those predicted by standard polycrystal models or to explain the irreversibility in texture evolution under forward and reverse shearing (see Section 6.5). The notion that substructure evolution can impact texture evolution certainly makes sense. The substructure elements within a grain are misoriented, and thus a grain develops a texture of its own. Their misorientation can change with straining even after substructural sizes have saturated. The effects of misorientation on slip activity are certainly not reversible even under a reversal reload, such as two-pass route C. Results from a recent EBSD study on ECAE Cu [23] that show that the misorientation across subgrains increases from one to two passes of route C (when examining the center of the sample) are not surprising. Similarly in twopass route C IF-steel, cell sizes are found to decrease or remain the same after one pass regardless if U = 90 ° or 120° [31]. A similar result was found in hcp Mg ZK60 [108]. Higher-scale deformation bands [109,110] and shear bands [101,111] (contained within grains3) can greatly influence texture as their misorientations can become high. Their boundaries do not lie along crystallographic slip planes. They provide high-angle boundaries that can contribute to grain refinement and recrystallization (Section 3.4), two processes that can change texture. Misorientations across deformation bands of 45–50° for U = 90° after 4 passes and 30° for U = 135° after eight passes of route Bc have been measured in Al [112]. These misorientations are higher after even-numbered passes of route C than after the first pass [81]. Shear band misorientation as high as 40° to 60° have been measured in one-pass single crystal Nb [102]. Shear band misorientation in an Al–Cu alloy were reported to vary widely from 10° to 90° [106]. The coupling between texture and subgrain microstructures becomes stronger at large strains: grain orientation can influence the morphology of the substructure formed (e.g. [113,114] and substructures can, in turn, affect directional hardening and slip activity and hence grain re-orientation. The relationship between grain orientation and substructure has been studied extensively, however, usually after small strains and most easily in single crystals. For polycrystalline metals, we only briefly mention a few works, starting with that of Huang on axially strained polycrystalline Cu. Using TEM, Huang [113] associated one of three types of substructure morphologies to grain orientation. Type 1 grains developed a single set of crystallographic boundaries; type 2, a cellular structure; and type 3, two sets of boundaries. The type of substructure can be correlated to the slip plane activity, which is determined by crystallographic orientation and material hardening. Type 1 has only one slip plane that dominates (planar slip). Type 2 activates three or more planes (multi-slip). Type 3 has two slip planes that accommodate grain deformation. Similar studies in the SPD literature are lacking. As reviewed in Section 3.3.1, some studies have processed single crystals via ECAE, where a strong link between substructure evolution and the initial orientation of the crystal was found. In polycrystalline Cu, Xue et al. [107] studied substructural evolution during one ECAE pass using TEM. Their schematic of the observed progression of substructure development in two representative grain orientations, denoted as grains A and B, is provided in Fig. 12. They found that the best grain orientations for grain refinement are those like grain A, that promote a transition from planar slip on one plane to another during the course of extrusion. The first slip activity builds dislocation boundaries parallel to the primary plane and the second activity builds upon it another intersecting set of boundaries, fragmenting the grain. The type of textures and subgrain morphologies that are best for grain refinement under the strain path changes of multi-pass ECAE has yet to be determined or explored. The strain path change causes the ‘entry texture’ of each pass to be different from that produced at the end of the previous pass. Also, the ‘entry’ microstructure changes from pass to pass. Together the changes in the entry texture and microstructure ensure that with each new pass, the grains and subgrains will likely activate slip systems which lie on different slip planes or act in the opposite sense as those in the previous pass. New substructure can locally channel through the old substructure, build up on it, or further stabilize it, depending on the grain orientation.

3 The bands discussed in this section are subgrain structures and are not the macroscopic shear bands which cut across several grain boundaries [11].

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Two grains with different orientations

A

B

Unit Cell

O

(a) Main Slip Plane

Main Shear Plane

A

O’ B (b) Second Activated Gliding Microbands

(c)

(d)

A: Two sets

Second Activated Slip Systems

B: One set

Fig. 12. A schematic of the substructural evolution during the first pass of ECAE [107] in (a) two grains with different crystal orientations, (b) both of which form cellular substructures early in the extrusion, but before they approach the intersection plane (c) grain A has formed lamina subboundaries not aligned with the central (main) plane of simple shearing, whereas grain B has formed lamina subboundaries closely aligned with it. (d) After passing through the deformation zone, both grains form subboundaries on slip planes aligned closely with the intersection plane (i.e., active glide planes). For grain A, this means that two sets of intersecting subboundaries have formed by the end of the extrusion, but for grain B, only one.

Deformation twinning (Section 3.1.10) is another texture-related (i.e., orientation dependent) deformation mechanism which can aid grain refinement. Twinning can help grain refinement by introducing highly misoriented twin boundaries and by reorienting parts of the crystal favorably for slip, thereby promoting slip-induced deformation microstructures. With this in mind, one can take advantage of the orientation dependence of twinning. For instance, Beyerlein et al. [95] found that grains near the A1U-orientation twin readily (see previous Section 3.1.10). Initial textures and strain path changes (i.e. billet twists) that place grains near this orientation will promote twinning and hence, possibly grain refinement. 3.1.12. Effect of temperature For cubic metals, temperature effects on texture evolution are not well studied; indeed most ECAE processing of cubics takes place at room temperature because of their good ductility. Nonetheless, we mention one study by Chakkingal and Thomson [86], who tested pure Al at 500° C. They observed a more diffuse texture with increasing pass number and suspected dynamic recrystallization to occur in both routes A and C. The effects of recrystallization will be discussed further in Section 3.4. An alternative explanation concerns temperature effects on dislocation slip. Texture

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development slows down at higher temperatures as less dislocation slip is needed because of diffusion processes. The higher thermal energy also enables dislocations to participate in plastic relaxation, which promotes the development of stress-free, low-energy, high-angle boundary (HAB) dislocation structures, which eventually form subgrains. Grains and subgrains alike rotate towards the preferred ideal ECAE shear orientations during extrusion. As a result, the texture components are the same but more diffuse due to subgrain evolution. Nonetheless, it should be emphasized that to achieve better grain refinement, lower processing temperatures are recommended. Higher temperatures can promote dynamic recrystallization processes which operate by grain boundary motion and lead to larger grain sizes. 3.2. Hcp textures in ECAE 3.2.1. Slip and twinning modes A list of recent ECAE hcp studies is summarized in Table 4. One can see that the processing parameters – U, W, the testing temperature, routes, number of passes and alloy composition – vary considerably in these studies. The table also provides the main slip and twinning mechanisms identified in each work. Generally, the active mechanisms in Ti, Zr, Mg and Be were found to be basal slip and prismatic slip in different proportions together with less or negligible activity of pyramidal hc + ai4 slip and deformation twinning, e.g., [5,8,70,88,116,118,124]. The considerable variation in the reported mechanisms is not alarming. The active mechanisms are expected to vary with c/a ratio, electronic structure, processing temperature,5 strain rate, alloying content (Ti64 vs. CPTi vs pure Ti), and crystal orientation (e.g., texture in a polycrystal) [126]. The latter leads to a large dependency of deformation behavior on initial texture and processing route. In the first pass the initial texture effect would be particularly noticeable, for instance, when one initial texture promotes twinning, while another promotes primarily slip. After the first pass, the ‘entry’ texture for the second pass (and subsequent passes) will depend on ECAE route. As in the first pass, a different entry texture in subsequent passes can invoke different slip and/or twinning modes in hcp crystals. In addition, the resistances to slip and twin propagation for each mode harden differently with strain so that their relative activities in the first pass may not be repeated in subsequent passes. To demonstrate, consider a ser 1g compressive twins (for 250–550 °C) in ies of studies by Shin and co-workers who observed f1 0 1 commercially pure Ti after the first pass, and none after the second [127] and fourth passes of route Bc [119]. Twinning in the first pass was due to the initial texture, wherein the basal poles were strongly aligned along the billet axis. Part of the variation in deformation mechanisms observed in Table 4 for the same material can be attributed to the methods used to determine them. Many of these were either indirectly inferred through texture characterization or modeling, or directly identified experimentally. Some of these studies used high resolution microscopy which examines local behavior, while others used polycrystal model predictions of texture which assess average behavior. A more reliable assessment is possible by using both model interpretations and experimental evidence, as was done, for instance, in [5,8,88]. In the following, we discuss the effects of ECAE processing conditions, (die angle, route, pass number), alloying, twinning, temperature, initial texture (Section 3.3), temperature, and recyrystallization (Section 3.4) on texture evolution in hcp metals. 3.2.2. First-pass textures and effect of die angle As representative examples, we consider texture evolution in pure Ti (transition metal c/a = 1.586) [121] and pure Be (simple metal c/a = 1.581) [70,88]. In the starting Ti material, all the basal poles were uniformly distributed normal to the billet axis with a maximum strength of 4 mr. The initial texture of the Be was nearly random. Figs. 13 and 14 display the (0001) pole figures for Be and Ti, respec-

4

Ti and Zr tend to use first-order pyramidal hc + ai and Mg and Be to use second-order pyramidal hc + ai. There are two main factors determining the temperature of the billet during ECAE processing. One is the processing temperature controlled by the operator and the other one is that due to adiabatic heating of the material under deformation, which can be a significant contribution as the pressing rate increases [125]. 5

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Fig. 13. The (0 0 0 1) pole figures for pure Be processed at 425 °C using (a) a 120° die for one to four passes of route Bc [70] and (b) a 90° die for pass one, after to two passes of routes A and C [88], and four passes of route Bc [70]. Contour levels: 0.8/1.0/1.5/ 2.0/3.0/4.0 mr.

tively. The U, W, routes, pass numbers, temperature, and measurement technique (XRD or neutron diffraction) are indicated in the figure or figure caption. First-pass textures are the first pole figures in each row of Figs. 13 and 14(a). All show a monoclinic symmetry, where the symmetry plane is misaligned from the normal of the intersection (main shear) plane, denoted as SPN. From the ND-axis the theoretical SPN lies at 60° for U = 120° and 45° for U = 90°. In contrast, from the ND axis, the maxima of the ECAE textures lie at roughly 35–45° for the Be U = 120° die and 22° for the Be U = 90° die as shown in Fig. 13. For the first-pass Ti texture in Fig. 14(a), the maxima appear to be split, lying at 11° and 33° from the ND axis. Still the midpoint between these two maxima 22° is similar to the pure Be processed using the same die angle U = 90°. The degree of misalignment of the maxima is found to depend on U, material, and possibly initial texture. For the most part, U can explain the main differences. In the case where the same material is used, such as for the initially random, pure Be processed by U1 = 120° and U2 = 90° in Fig. 13, the maxima differed by 13–23° after one pass and 15° after four passes of route Bc, [70,88], which agrees reasonably well with the expectation that the difference should be (U1U2)/2 = (12090°)/2 = 15°. A possible reason that the fourth-pass textures agree better with theory than the first-pass textures is that for the former, both the U1 = 120° and U2 = 90° dies had the same rounded corner (W > 0), whereas for the first-pass textures, the dies had different W: the U1 = 120° die was rounded and U2 = 90° die sharp. Reasons for the second-order differences, like small differences in the intensity distribution along the symmetry plane, between the two metals can be related to initial texture and the relative activities of basal, prismatic, and pyramidal slip. The existence of a symmetry plane and similar deviations from the SPN have been identified in the first-pass textures for several other hcp materials: Mg alloys [118,122,124], Ti [24,80,119,127], Be

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(a)

ND

ND

ND

ND

ED

1 200ºC

2A 200ºC

2C 200ºC

4Bc 200ºC

ND

ND

(b)

ND

ND

ED

4Bc 250ºC

4Bc 300ºC

4Bc 350ºC

4Bc 400ºC

Fig. 14. The (0 0 0 1) pole figures for pure Ti using a 90° die for (a) pass one, after to two passes of routes A and C, and four passes of route Bc at 200 °C and (b) four passes of route Bc at four different temperatures higher than 200 °C. Contour levels: 1/2/2.8/4.0 mr. Data is taken from [121].

[70,88,116] and Zr [5,85]. In CP-Ti [24] the first-pass texture was a strong basal texture, 9.5 mr, with  1g deformation poles 35° from the y-axis, in spite of a strong initial texture and occurrence of f1 0 1 twinning. In [5], pure Zr samples with two different starting textures were processed at room temperature, and irrespective of the initial texture, the basal poles in the first-pass texture rotated 25° away from the y-axis (see ahead to Figs. 18 and 19). Depending on initial texture, the orientation of the basal poles in AZ31 (an Mg alloy) after one pass ranged from 20° to 40° from the y-axis [118]. The consistency of this result in spite of material variation in c/a ratio, slip mode activity, alloying content and initial texture is interesting. Ti and Zr deform predominantly by prismatic slip and Mg and Be by basal slip. Both slip modes, basal slip and prismatic slip, share the same slip direction, that is, the closedpacked hai direction. Therefore, it is tempting to attribute similar first-pass textures to the dominance of hai slip, regardless if it occurs on the prismatic or basal plane. Recent modeling work, however, suggests that the common feature in first-pass texture of hcp metals may be the result of pronounced activity of basal slip and secondly of prismatic and pyramidal slip [5,88]. Poor texture predictions for pure Zr were obtained if hai slip were forced to occur primarily on prismatic and secondly on basal. For some hcp metals, such as Zr and Ti, the notion of basal slip is in conflict with several experimental results and observations. Consider for instance, Zr. Previous single crystal and polycrystal studies involving low to moderate strain levels [8,126,128–130] show that basal slip is unexpected for pure Zr below 800 K. However, measured first-pass ECAE-Zr textures were similar to those of other hcp materials known to deform primarily by basal slip, such as Be and Mg [70,88,116,118]. Recent texture analyses of Zr and its alloys after ECAE have confirmed that these textures can only be produced by significant basal slip activity [5,85,117,131]. The unexpected prevalence of basal slip in Zr could be a consequence of the severe plastic deformation and deserves further study.

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3.2.3. Effect of route and pass number As in cubics, texture development in hcp materials vary greatly with processing route and pass number for a given route. To begin this discussion, we recall the Be and Ti examples in Figs. 13 and 14. Fig. 13 displays the (0001) pole figures for pure Be processed at 425 °C using (a) a 120° die for passes one to four of route Bc and (b) a 90° die for pass one, after two passes of routes A and C, and four passes of route Bc. Fig. 14(a) shows the same as Fig. 13(b) but for pure Ti at 200 °C. Textures in row Fig. 14(b) are the (0001) pole figures for pure Ti after four passes of route Bc at four different temperatures higher than 200 °C. Most textures in Fig. 13 and Fig. 14 have approximately the same intensity, 2.8 mr, with perhaps the exception of the first three textures in Fig. 13(b) for Be which appear slightly stronger than the rest: the first-pass, second-pass route A, and second-pass route C samples processed using a 90° die. These three Be samples were processed using the same sharp corner die and their textures measured by XRD. The remaining samples shown: the Be samples processed using a 120° die (Fig. 13(a)), the fourth-pass route Bc sample using a 90° die Fig. 13(b), and all Ti samples in Fig. 14, were processed using the same rounded corner die and their textures were measured using neutron diffraction. Both the difference in W, as well as in measurement technique, can explain the slightly higher texture intensity. In route Bc, for both Be and Ti, Fig. 13(a) U = 120°, Fig. 13(b) U = 90° and Fig. 14(a) U = 90°, the textures have evolved significantly after the second to fourth passes, being noticeably different in feature from that in the first pass. The monoclinic symmetry of the first pass is lost by the second pass. As mentioned earlier, the textures of the fourth-pass Be samples in Fig. 13(a) and (b) differ by a 15° rotation which can be explained by differences in their U. However, for the same U, the fourth-pass Bc textures of Be and Ti are different. Possible reasons are differences in initial texture and deformation mechanisms. As mentioned, in ECAE, the starting texture for Be was random, whereas for Ti it was not. Be activates primarily basal slip [70,88], whereas Ti would active both basal and prismatic slip. In routes A and C, texture evolution in Be and Ti (Figs. 13(b) and 14(a) U = 90°) is not significantly different between the first to the second pass and between the two metals. All textures exhibit a monoclinic symmetry and maximum intensity around 2.8–4.0 mr. For Be the main differences are found in the location of the maxima: 22°, 16° and 45° from the ND-axis in the first pass, second-pass route A and second-pass route C, respectively. Modeling in [88] was also able to reproduce these measurements well. The model suggested that basal slip was the main slip mode in the first and second passes of these two routes, being more active in the second than in the first. Basal slip became more active because the first-pass texture and subsequent rotations made the entry texture for the second pass preferable for basal slip. Prismatic slip and pyramidal hc + ai slip were secondary, and twinning was not observed in microscopy nor was it activated in simulation, even though it was made available to the model material. Like Be, for Ti, the location of the maxima also change between the first and second passes. There is, interestingly, another difference: the maxima are no longer split after two passes. Although no modeling analysis of these particular Ti textures has been performed, studies on Ti and Zr suggest that prismatic slip is also very likely [5,24,115,127]. However, similar texture evolution between the pure Ti and pure Be after two passes of route A and C (Fig. 14(a)) suggests that basal slip was also an active slip mode. For hcp metals, the deformation mechanisms can vary with route. Material differences influenced texture evolution more so in route Bc than in routes A and C. The strain path sequences and associated texture changes for route Bc could have easily promoted different types of slip modes and/or slip activities in Be vs. Ti, whereas those associated with routes A and C, promoted similar ones (i.e. predominantly basal slip). More experimental evidence and modeling work are needed to validate this hypothesis. 3.2.4. Effect of alloying Alloying can change the critical strengths that activate slip and twinning, leading to differences in their relative activity and hence texture development. The effect of alloying was studied by Agnew et al. [118] who examined five different Mg alloys with different initial textures. To separate the two effects, polycrystal simulations were performed. The results suggest that adding Li promotes

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the activity of non-basal slip systems in the Mg–4Li alloy which led to a different texture compared to AZ80 and AZ31. At the same time, the latter materials developed the same textures although their initial textures were different. Based on these results, they concluded that alloying makes a greater impact on the evolution of the texture than the initial texture. It is, however, not evident that the effect of alloying is larger than the effect of the initial texture. To verify this, experiments should be conducted with the same alloy but with different initial textures. 3.2.5. Effect of deformation twinning Compared to cubic metals, twinning in hcp metals is much more prevalent and larger in scale (lm). In hcp metals, there are many twin types, and twin-reorientation depends on twin type and the c/a  2gh1 0 1  1i,  which for Mg ratio of the hcp metal [126]. The most commonly observed twin is f1 0 1 (c/a = 1.63), for instance, reorients the lattice by 86.6°. Texture evolution in hcp metals cannot be understood without knowledge of the operative twinning modes and the twin fraction. There are several factors which govern the propensity for twinning: c/a ratio, alloying, temperature, strain rate, grain size, strain level, and crystallographic orientation. Observations of twinning in hcp metals in SPD have been reported in commercially pure Ti and Ti– 6Al–4V [24,127,132] processed at high temperature (350 °C and above), pure Zr [5] processed at room temperature, and Mg alloys [108,133] processed at intermediate (warm ECAE) temperatures (200– 300 °C). There was no evidence of twinning, however, in pure Be processed at 425 °C and U = 90° [88,116]. In Ti and Ti–6Al–4V ECAE-processed at high temperatures the twin type observed is  1g-twinning [24,127,132,133], primarily in the first pass. f1 0 1  1g-twinning is known to appear f1 0 1 in Ti single crystals at high temperatures above 400 °C [134]. In ECAE of Ti–6Al–4V, it was demonstrated through VPSC polycrystal modeling that the texture evolution after one pass was inaccurate  1g-twinning [131]. Likewise in ECAE of Zr [5], which deforms by prismatic, without considering f1 0 1  2g-twinning, f1 0 1  2g-twinning was necessary in order to basal, and pyramidal hc + ai slip and f1 0 1 accurately predict one-pass textures with different initial textures (see Section 3.3.3). Lapovok et al. [108] studied the effect of grain refinement during routes A and C and in subsequent room temperature deformation of Mg ZK60. It was found that under lower processing temperatures (200 °C vs. 300 °C) twinning was maintained up to eight passes but decreased with pass number thereafter. Due to grain refinement, twinning was suppressed during post-deformation of the ECAE-pre-strained material, unlike in the un-processed one. Due to the dependence of twinning on crystal orientation, the activation of deformation twinning is expected to depend on initial texture, ECAE route, and pass number. Shin et al. [24] studied texture development in CP-Ti processed by routes A, Bc and C at 90°. The initial texture contained basal poles  1g compressive strongly aligned along the billet axis. In the first pass, this initial texture induced f1 0 1 twinning.6 These were less than 1 lm in width. In subsequent passes of routes C and Bc, they found that twinning did not occur, while in subsequent passes of route A, they observed secondary twinning in the first-pass twins. Regarding dislocation slip, in all cases, prismatic slip was prevalent followed equally by basal and pyramidal hc + ai slip. In the second pass of route Bc, for instance, deformation was mostly accommodated by prismatic slip [127]. A texture analysis by Choi et al. [115] was performed on CPZr, processed using U = 135°, W = 45° at room temperature for routes A, Bc and Ba. Using OIM/EBSD they found predominantly tensile twins at low pass numbers and a mix of tensile twinning plus compressive twinning at higher pass numbers (although the route was not specified).

3.2.6. Effect of temperature When the temperature alters the deformation mechanisms (such as from twinning to slip), the relative activities of these mechanisms, or the fraction of recrystallized grains, noticeable transitions in the texture evolution with temperature will occur. In this regard, the most interesting temperature effects are found in hexagonal metals where the activation and relative activities of deformation mechanisms are sensitive to temperature. 6 Compressive twinning was prevalent in the first stages of the extrusion when the normal strain component along the billet axis is compressive. It later changes to tensile as the billet travels around the die corner.

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Most hcp temperature-effect studies have been performed on Mg alloys. Temperature is expected to affect Mg alloys. At room temperature, the primary slip mode in Mg alloys is basal slip. Increases in temperature (like the addition of alloying elements, such as Li and Al [123]), can lower the differences between the critical strengths of basal slip and non-basal slip modes, thus promoting relatively more activity of the latter. This effect can be further enhanced in ECAE if the strain path change between passes places the texture in an orientation that does not favor basal slip. In AZ31 Mg, Yoshida et al. [135] followed the texture evolution during one pass at two different temperatures, 250° and 300° C. At both temperatures, the dominant deformation mechanism changed during the course of one pass. Although basal slip is by far the easiest slip mode, the initial texture induced non-basal slip activity in the initial stages of deformation. At the lower temperature, tensile  2Þ twinning and prismatic were active, but at the higher temperature pyramidal hc + ai was acð1 0 1 tive. In the remainder of the extrusion at both temperatures, the deformation mechanism transitioned to basal slip. As a result of the different mechanisms occurring in the early stages of the extrusion, the first-pass textures were different. The low-temperature texture had its basal poles aligned with the zaxis (TD) and 30° from the y-axis (ND), whereas the high temperature texture had its basal poles aligned with the y- and z-axes. With a similar extruded initial texture but at 200° C (with backpressure) Agnew et al. [124] also found that basal poles were approximately 30° from the ND-axis in AZ31, whereas at a higher temperature, 300° C (but with a different initial texture), the basal poles were aligned closer the ND-axis. A theoretical study conducted in [118] supports the assessment by Yoshida et al. [135] that prismatic dominated the initial stages of ECAE deformation processed at 250° C and pyramidal at 300° C. It was found that increased prismatic activity tends to stabilize the basal poles aligned with TD while increased pyramidal activity brings poles closer to the ND and causes a split in the poles. For the same range of temperatures, little effect of temperature is found in Ti. Similar textures were observed after four passes of route Bc in pure Ti between 200° C and 400° C [121] (see textures in Fig. 14(b)). Yu et al. [119] also found that in commercially pure Ti the texture features were unchanged when the temperature increased; only the texture intensities lowered with temperature. Apparently, for the temperature range studied in these examples, there was no transition in the activated deformation mechanisms in Ti.

3.3. Effect of initial texture Ususally the starting sample will have a texture that is different from random as a result of the manufacturing process. Regardless of its strength, the initial texture is, however, not retained after the first pass of ECAE. Even in route C – where it may be expected that the texture is recovered after every even-numbered pass – the initial texture is generally not recovered. Nevertheless, the initial texture has an effect on the evolution of the texture during ECAE. It can promote the strengthening of certain texture components and the reduction of others. This section investigates the effect of initial texture on texture evolution in ECAE. The most prononced effects can be observed in the ECAE of single crystals, which will be investigated first. We then follow with sections dedicated to cubic and hexagonal close packed polycrystalline metals. 3.3.1. ECAE of single crystals Experiments on single crystals (SX) processed by ECAE have been reported only recently. The first study is from Wang et al. [103] who extruded high purity Cu single crystals in a 90° die up to five passes in route C. The misorientation distributions were measured and analyzed [103,104]. Because the study focused on dynamic recrystallization under large imposed strains, the initial orientations were, unfortunately, not specified. Later more studies were performed in which the initial orientation was documented and shown to make a noticeable impact on texture and microstructural evolution. Fukuda et al. [136,137] extruded SX Al in a 90° die with a 30° rounded outer corner. In [136] the initial orientation was such that a (1 1 1) plane of the crystal coincided with the ideal simple shear plane of the die and a [110] slip

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direction was parallel to the shear direction. The purpose of this test was to put the crystal already into an ideal stable position with respect to ECAE shearing (the ‘A’ ideal orientation). During the extrusion, it rotated about 60° in the direction of shear, which would happen if the orientation were unstable, not stable, as was originally intended. Close inspection of their figures reveals that the reason for the large rotation was that the initial orientation was actually off from the ideal one they intended and located on the divergent side of the rotational field of shear, making the crystal orientation unstable. Consequently, the crystal ended up in the vicinity of the B ideal orientation, which is about 60° away from A. Some A components were, however, found within the shear bands and stringers that formed during the extrusion. In a second work, Fukuda et al. [137] extruded, in the same die, another SX Al, which was initially 20° rotated CW around the TD axis with respect to the previous one. During the extrusion, this orientation rotated approximately 40° CCW around TD. (Note that their X–Y–Z reference system in their Figs. 1 and 6 was not right-handed, so for this discussion we use our axis notation system which is right-handed: ED, ND, TD.) Fukuda et al. [138] repeated the same experiments on SX Cu for the same two initial orientations. In the first sample, which was oriented for ideal single slip, they found a similar large rotation leading to B type shear components. As for the second one, the rotation was 20°, in contrast to 40° for SX Al. In this case, the crystal stopped at an ideal A type shear component during the extrusion. Miyamoto et al. [100,139] extruded cylindrical-shaped SX Cu samples with seven different initial orientations in a 90° die. Shear bands and twinning activities were reported. Depending on the initial orientation, the substructure was either heterogeneous with shear bands and mechanical twins (group I), deformation bands (group II), or homogeneous as a result of uniform crystal slip (group III). Although valuable in providing useful experimental information, these works lack accurate interpretations primarily because they do not perform a crystal plasticity analysis or modeling. Modeling would be very helpful in understanding and interpretating the main mechanisms of the material behavior. In some cases, unfortunately, even the analyses are faulty. Fukuda et al. [136] interpreted their results based on ‘a similitude’ of the resulting texture component to the Brass component in rolling. This explanation, however, does not hold simply because the deformation imposed by ECAE is simple shearing and not rolling (see Section 2.1.6). For the initial orientation used, simple shear would operate only two slip systems, while in rolling the number would be four. Fukuda et al. [137,138] used a different approach, interpreting these results with the help of so-called ‘shear factors’ which is based only on the geometrical position of the slip system with respect to the applied shear. Several other SX studies [100,139,140] have also employed shear factors to analyze the resulting textures. Shear factors, however, cannot give precise information on slip activity, the latter being dependent on the stress state as well. For example, for the B orientation, the correct theoretical ratio of the slips in the two active slip systems is 1:3 [57], while the shear factors lead to 1:2. In addition to employing shear factors, the work of Han et al. [140] on texture evolution of SX Al during ECAE also contained numerous errors in both presentation and interpretation of the results, as later shown in [141,142]. The main experimental error concerned the initial orientations of the crystals that were far from their assumed orientations, while the theoretical discussion was faulty in terms of the assumed deformation mode in ECAE. They actually assumed that the main deformation mode is simple shear perpendicular to the intersection plane of the two channels. Such a contradictory proposition was also made in [143] based only on texture measurements. In that work, the main error is that the Miller indices of the ideal orientations lying in the direction of shear are mixed up with the Burgers vectors of dislocations. In order to understand the crystal orientation changes, full crystal plasticity simulations are needed. An example of this approach can be found in a series of works by Goran et al. [144–146]. Goran et al. examined the texture and the microstructure in pure Ni single crystals oriented initially in [010]||TD, [001]||ND [144,145] and in [011]||TD, [311]||ND orientations [146]. Strong heterogeneities were found in terms of local textures and shear band features [147].

3.3.2. Cubic polycrystals Due to the manufacturing process that shaped the billet, initial textures in cubics can vary from nearly random (1.2–1.4 mr) (multiply random) to moderate and axisymmetric (2–3 mr) and to strong (>5–6 mr) with no symmetry. Their effect on texture evolution in polycrystals is analyzed

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below. Here it is assumed that the cubic polycrystals deformed by slip, not twinning or grain boundary sliding. It is also assumed that dynamic recrystallization does not take place. Dynamic recrystallization is treated in Section 3.4. Initial texture effects in cubics are commonly ignored and any texture differences have been attributed instead to material differences, such as stacking fault energies (SFE). Evidence, however, exists showing that initial texture can have an effect, even after large deformations. Early work on torsion testing of fcc materials show that initial texture effects can prevail up to shears of c = 2 [148]. The authors explained the differences between the torsion textures of four different fcc materials by those in their initial texture, rather than by large differences in their SFE (Al, Cu, Ag, and brass). As another example, Vogel et al. [71] studied the first-pass textures of pure Cu, Ni, and Al. With nearly random initial textures, Cu and Ni exhibited strong Ah and a weak Bh fiber after one pass, whereas with a swaged, asymmetric initial texture, Al showed only Ah and Bh fibers. They suggested that these differences might be attributed to both initial texture and other material differences, such as the normalized SFE and initial grain size. (The normalized SFE – the ratio of the SFE to the product of the shear modulus and Burgers vector – are nearly equal for Ni and Cu, but much lower than that of Al.) To address this, a second study was conducted using a polycrystal model, which only accounted for initial texture effects and not material differences [73]. Results in [73] suggested that the observed differences in texture evolution between Al and Cu up to four passes (route Bc) could be fully explained by initial texture effects. This result implies that differences in grain and subgrain microstructures of the two materials [149], although important for mechanical properties, are not reflected in the texture evolution (at least up to four passes). Regarding initial texture effects, a useful rule of thumb resulted from the work of Stout et al. [148]: when the initial texture is more intense than 1/10th of that developed by the deformation, the initial texture effects will persist up to large strains. Textures generated from simple shearing, like torsion testing and ECAE, are generally not considered strong (fmax = 2.0–6.0). By this rule, even a weak initial texture (1.5–3.0 mr) can possibly impact texture evolution in ECAE, approximately up to the first and second passes. Relatively stronger textures (>5.0–6.0 mr), therefore, can be expected to affect texture evolution and require much larger strains to erase. In this case, the less strain per pass, the more passes required to remove the effects. In addition to total accumulated strain, ECAE processing requires consideration of processing route. It is possible that ECAE routes will vary in their abilities to remove initial texture effects due to the strain path changes associated with each. Ferrasse et al. [66] studied the initial texture effect in processing routes A and Bc (also known as D). To generate different starting textures, they pre-processed an Al–0.5Cu alloy in different ways and then annealed both, producing a strong texture 21.7 mr and a weak one 2.6 mr with a slight grain size difference. For both routes, the stronger one led to higher maximum texture intensities up to six passes. After both routes and up to eight passes, texture intensities ranged from 2 to 5 mr for the weak one and 2–17 mr for the strong one. For route Bc, their inverse pole figures suggested that by four passes, the two initial textures generated similar components. Initial texture strength affected the strength more so than the components of the deformation textures after large N. In contrast, Suwas et al. [72] found that despite a strong h1 0 0i starting fiber texture, after the third pass the textures in route A of Cu became similar to those that develop starting from a near random texture. Texture modeling can advance understanding of these results. To demonstrate systematically the influence of route and die geometry on initial texture effects, we calculate, using the VPSC polycrystal model (Section 4.4), texture evolution in ECAE with one of three starting textures: random, a weak swaged texture (2.41 mr), and a strong rolled texture (5.24 mr). Two die angles, U = 90° and 120°, and three routes, A, C, and Bc, are considered. In these calculations, ideal simple shearing is used, implicitly assuming sharp corner angles and no work-hardening. A co-rotation scheme [27,150] is applied to slow down texture evolution and provide intensities similar to those typically found in measurements [65,71,76]. Figs. 15 and 16 show the U = 90° results and Fig. 17 the U = 120° results. Figs. 15–17 show that in all cases the one-pass textures are significantly affected. Even the U = 90° extrusion is not sufficient to wipe out the effects (Fig. 15) even though the strain is almost twice as high as in the U = 120° die. Instead, three to four passes of routes Bc and A with U = 90°, are needed to remove any visible effects from the initial texture (Fig. 16). What remain are second-order differences in the

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relative intensities of some components and overall intensities of the textures. This trend agrees with the results of Ferrasse et al. [66] and Suwas et al. [72]. Also in agreement with these measurements, the calculated ECAE textures with the two initial starting textures (Fig. 16) tend to be stronger than those with no initial (random) texture. These predictions imply a strong initial texture effect for route C. As shown, in route C, the two- and four-pass textures are very similar to the initial in intensity and components. Likewise, a theoretical study [63] on a hypothetical bcc metal found that the initial texture effect to be the strongest for route C, less so for routes A and Bc, and notably, the least for route Ba. The calculations in Fig. 17 suggest that when texture evolution is slowed down due to less accumulated strain in the U = 120° die compared to the 90° die, the initial texture effects can last for large N. For the larger die angle U = 120° (Fig. 17), the initial rolled texture considered still has an effect after four passes (route Bc). Similar results can be expected for high processing temperatures or low strain levels per pass due to large W. Li et al. [31,32] processed IF-steel by route C using both U = 90° and U = 120°. Only for the latter die angle was the second-pass texture similar to the starting texture. A few points may be raised in light of the experimental and theoretical results discussed above. First, initial texture effects in ECAE processing of cubics have often been neglected or falsely assumed to be

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erased after a single extrusion. It is possible, however, that they are not observed often in measurement because it is common to process samples with U = 90°, for greater than four passes, and with route Bc, conditions which tend to wipe out any effect of initial texture, as implied by calculations in Figs. 15–17. Second, it is difficult to isolate them in practice. Samples typically have different initial textures because of their pre-processing histories. Therefore, in addition to texture differences, samples begin with other microstructural differences, such as grain sizes, grain shapes, and dislocation densities. There are a few occasions when initial texture effects are especially pronounced. Clearly one of these is when the initial texture is strong (>6 mr) and the number of ECAE passes is less than four. Initial texture effects can also prevail when the strains per pass are low and/or the deformation is close to ideal SS [31]. Of the four routes, route C is likely to be the most susceptible to initial texture effects. However, for route C, the effects are typically not observed to be as great as that expected or predicted, because the original texture components after even-numbered passes (see Section 6.5 for further discussion) are rarely recovered. Last, initial textures that are asymmetric about the billet long axis can present difficulties in texture analyses. In this case, unless one is extremely careful to mark the initial orientations prior to processing, each sample will have a different initial texture, potentially preventing any meaningful comparison between samples of different routes, pass number, and other processing conditions. In this way, the sample begins with the same microstructural state and pre-processing history and the only difference is the orientation of the crystals with respect to the loading. When the initial texture effects are unknown it is recommended that comparisons be made between ECAE textures across studies after approximately three to four passes. 3.3.3. Hexagonal polycrystals As mentioned in Section 3.2.2, hexagonal crystals can deform by a variety of slip and twinning modes [126,151]: basal, prismatic, pyramidal hai, and first-order and second-order pyramidal hc + ai  2g-, f1 0 1  2g-, as well as f1 1 2  1g-twinning, to name a few. The operative modes are slip, and f1 1 2 determined by many factors, with crystal orientation and c/a ratio, being, undoubtedly, the most

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important ones. Because of the strong coupling between crystal orientation and deformation mode, the initial texture effect is expected to be more pronounced in hcp metals than in cubics. In ECAE of hcp metals, it is often the case that the initial texture is relatively strong but after one or more passes, the texture intensity reduces. DeLo et al. [152] observed a reduction from 12.0 mr in the initial texture of Ti64 to 4.0 mr after the first pass. Similarly Yu et al. [153] observed a reduction from 9.0 mr in the initial texture of CP-Ti to 4.0 mr after the first pass, and Kim et al. [122] a decrease from an initial axisymmetric texture of 7.0 mr in Mg AZ61 to 5.6 mr after the first pass and then to 2.8 mr after the second. Some exceptions in which the initial texture was not strong in the hcp material can be cited, such as the pure Be in [70,88] (Fig. 13) with a nearly random initial texture. The basal-slip dominated texture produced in the first pass had a texture intensity of 2.6 for U = 120° [70] and 4.0 for U = 90° [88]. Agnew et al. [118] investigated the effect of initial texture on the ECAE texture development of Mg alloy (AZ31) samples. The samples had one of two initial textures, either extruded or plate. After ECAE, from the former texture the basal poles aligned 20° left from the y-axis (parallel to the entry channel), whereas from the latter, they were more diffuse and closer to the y-axis up to four passes of route A. A theoretical study using VPSC showed that the ECAE extrusion will promote nearly all basal slip in a polycrystal with an initially random texture and an increasing amount of non-basal activity in the same polycrystal with an initial (non-random) texture. Another study [153] attempted to study initial texture effects by using two different hcp materials: commercially pure (CP) Ti and CP-Zr. These two metals had different starting textures. After processing using the same die and conditions (350° C), both materials had their basal poles concentrated approximately 35° from the y-axis. However, in CP-Zr, they were more concentrated at an intensity of 8 mr, whereas in CP-Ti, they were much more diffuse at 6 mr. Because there are larger differences among hcp metals than fcc metals, one cannot attribute such differences in texture evolution solely to initial texture effects. In a recent study [5], ECAE processed pure Zr exhibited significant initial texture effects on texture evolution and plastic anisotropy of one-pass material. The starting metal had its basal poles strongly aligned in one direction. A different starting texture was produced by changing the orientation of the sample with respect to the die channel when it was first inserted into the die. In one case, the basal poles were aligned along the ED (Fig. 18a) and in the other, they were aligned along the TD (Fig. 19a). After one pass, the basal poles were tilted 25° away from the y-axis ND in both cases (Figs. 18b and 19b). However, in the second case, a portion of the basal poles remained spread about the TD (Fig. 19b). Model interpretations suggest that the active modes changed due to initial texture: basal slip is the most active and prismatic slip the second most active when the basal poles were initially aligned with the ED (case one) and vice versa when they were initially aligned with the TD (case two). As in the ECAE Mg alloy textures studied by Agnew et al. [118], the non-basal slip activity was responsible for the texture components along the TD found in the second case (Fig. 19). A similar investigation is currently underway for AZ31B Mg [133].

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Fig. 19. Basal pole figures of the measured textures in the case in which the basal poles were aligned along the TD before the extrusion, as shown in (a), (b) after single pass ECAE, and after post-ECAE compression to 30% strain (c) along ED, (d) along ND, and (e) along TD [5]. Contours levels are 0.5/1/2/3.5/5.5/8 mr.

3.3.4. Recommendations In summary, it is important that the initial texture is reported in experimental studies and is used as input into theoretical models. Some of the discrepancies across texture observations of similar materials could be due to differences in initial texture. For hcp materials or metals that twin, where the type of deformation mechanism is dependent on crystal orientation, an ECAE texture analysis cannot be performed without knowledge of the initial texture. The best way to study the initial texture effect involves comparing samples of the same material with different starting textures due to an initial rotation of the sample, such as in [5,133] and Figs. 18 and 19. Another way is by comparing measurements to polycrystal model calculations that are able to isolate the influence of the initial texture [73,76]. 3.4. Dynamic recrystallization Dynamic recrystallization (DRX) during ECAE can have a significant impact on texture. DRX may take place at temperatures lower than the temperature of static recrystallization (SRX) because the stored energy that accumulates during large plastic strain lowers the recrystallization temperature. It has been shown, for example, that DRX can occur below room temperature in high purity Al samples pre-deformed by torsion at liquid nitrogen temperature [154]. DRX can take place by two mechanisms: 1) nucleation and growth of new grains and 2) continuous conversion of low-angle dislocation cell boundaries into high-angle boundaries (HAGB) forming subgrains. The first mechanism is frequently called discontinuous DRX (DDRX) while the second continuous DRX (CDRX), see in [155]. DDRX is characterized by coarsening and subdivision of subgrains during continuous shearing in the ECAE process. During DDRX, the HAGBs, as well as grain boundaries supporting large differences in dislocation densities across them, can have some mobility (albeit limited). DDRX is favored by high purity, low SFE, high temperatures, and low strain rates. During CDRX, dislocations produced by strain hardening accumulate in low-angle grain boundaries (LAGB) that are converted into HAGBs at larger pass numbers. CDRX is expected to take place in high SFE materials, like Al. Previous understanding for Al was that it cannot recrystallize dynamically; only dynamic recovery (DRV) can take place. However, CDRX is now widely accepted in Al and even DDRX is possible in high purity Al ([156], see below). One study has found that CDRX does not saturate even after 16 passes in Al in routes A, Ba, Bc, and C [157]. The main interest in severe straining by ECAE (or any SPD technique) is that it can produce ultrafine grain structures with average grain sizes from submicron to a few microns [11,13,15]. The refined microstructure that forms during ECAE can be attributed to DRX because DRX is a strain-induced process in which the initial grains are replaced by new ones. Whether it is discontinuous or continuous depends on material and processing parameters. CDRX or DDRX can govern the smallest grain size possible in a material. While both processes can lead to a steady and small average grain size, grains sizes tend to be much larger after DDRX than after CDRX to the extent that DDRX can

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possibly even destroy the refined microstructure achieved in SPD. Also the shape change of the grains in DRX does not follow the shape imposed by the continuous shearing but stabilizes at a given aspect ratio. In this section we focus on the effects of DRX on texture development rather than SRX. Our main interest is to examine the interaction between crystallographic slip and DRX during severe plastic deformation. Static recrystallization, on the other hand, occurs after the deformation has been applied, the sample has been unloaded, and usually, when the temperature has been increased from the deformation temperature over a period of time. SRX studies are performed relatively frequently because ECAE followed by SRX can be used as a tool for grain refinement. After severe plastic deformation, the strongly strained and frequently non-equilibrium microstructures are usually stabilized by subsequent annealing. Generally, when the purity level is high, it is easier for discontinuous SRX to occur. For instance, it has been observed to easily happen after ECAE testing (Al:[158], Cu: [103]). We refer the reader to the following for SRX studies on cubics [103,159–173] and on hexagonals [117,174–176]. Below we discuss DRX first in ECAE-processed cubic metals, then hexagonal metals. Texture can be radically influenced by DRX in cubic materials, while in hexagonal the deformation texture is mostly retained. 3.4.1. Cubic metals The first study that addressed texture in ECAE resulting from DRX used an Al–Mg alloy, A5056 [177]. Very sharp textures – above 35 mr – were obtained after eight passes in route A at 200 °C that resulted from a continuous increase of the texture strength with pass number. Not considering simple shearing as the ECAE deformation, they interpreted the textures based on the main texture components of rolling (Copper and Brass) [178]. The onset of DRX was only observed above 300 °C in ECAE of A5056, however, the grain size was much finer compared to a conventional extrusion process at the same equivalent strain. Sun et al. [177] also found that ECAE is much more effective in grain refining with DRX than other deformation processes; they compared microstructure development in pure Al using ECAE route Bc to cyclic extrusion–compression. The texture, however, was examined locally in a small area (only 6.6  10 lm and hence termed micro-texture) using inverse pole figures and found to be nearly random. Aluminum of commercial purity was extruded at 500 °C in [86] in routes A and C in a 120° die up to five passes. Relatively weak textures were obtained in which the contribution of DRV was clearly identified with the help of TEM. The texture strength decreased as a function of passes approaching a random texture. The difficulty in the interpretation of the microstructure examined in multiple passes at high temperature is that SRX may happen during reheating of the samples between passes [86]. Similar decrease of the texture index with ECAE passes was observed in [164] in an Al0.5Cu alloy and was attributed to the occurrence of CDRX. The texture components that are formed by DRX are usually the same as the deformation texture components. The main texture component that can originate from DRX and that is different from the deformation texture components is the cube component. It appears, however, in rotated position in the direction of the simple shear process. This effect was first observed experimentally by Tóth et al. [179] in torsion of Cu and was also successfully simulated by polycrystal modelling including DRX [180,181]. Skrotzki et al. [156] obtained a strong oblique cube component in high purity Al (Fig. 20) which was completely recrystallized at room temperature ECAE during the first pass. The rotation angle of the cube slightly increased as a function of pass number but decreased locally as a function of the distance from the top of the extruded billet, see Fig. 21. The formation of the rotated cube component was interpreted by the occurrence of a DDRX mechanism in pure Al in spite of its high SFE. Accordingly, the nuclei are assumed to be produced in the non-rotated cube position. During the shear process they progressively grow and undergo a shape change that rotates their shape axes in the direction of macroscopic shear. The DRX cube orientation, however, appears in rotated position because the cube rotates with the rigid body spin and continues to rotate with the same speed as the non-rotated cube even in rotated positions. The process leads to a maximum intensity of the cube at a specific angle of rotation and a stabilized aspect ratio together with characteristic grain shape axis directions. The rotated cube component was also observed in CP Al [182], and polycrystalline Cu [89] and Ni [183]. The Ni textures could also be interpreted with the help of DDRX [183], while in the

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Fig. 20. Textures in (2 0 0) pole figures of high purity Al deformed by ECAE at room temperature in a 90° sharp corner die [156] measured on the TD plane. (a) Initial and deformed global textures measured by neutron diffraction (contour lines: 1–6 and 8 mrd). (b)–(d): Local textures measured by EBSD in the (b) top, (c) middle and (d) bottom part of the sample (contour lines: 2, 4, 6, 8, 10, 12 and 14 mr). ED direction is X and ND direction is Y.

textures of pure Cu, the CDRX mechanism was proposed [89]. In the latter work, polycrystal simulations without a DRX model could reproduce the deformation texture components; however, the rotated cube component was not obtained – another indirect proof that this component does not originate from slip but from DRX. Ni has a SFE between Cu and Al (see Section 3.1.9), thus its DRV rate is slower than that of Al, and consequently, the rate of CDRX is also lower leading to smaller grain sizes. Compared to Cu, Ni produces a more homogeneous microstructure and therefore is a good model material for studying DRX in ECAE processes [184].

3.4.2. Hexagonal metals There are just a few studies for texture variations due to DRX in hexagonals during ECAE [64,123,133,185–187]. In general, in this crystal structure, the DRX components are all the same as the deformation texture components; only the intensity distribution changes when DRX takes place [185]. (Recall that in cubics it is only the rotated cube that is different from the deformation texture components.)

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Fig. 21. Deviation (in terms of a rotation about the TD) of the cube component from its symmetry position as a function of the position within the billet and pass number in 5 N Al deformed by ECAE at room temperature [156].

Liu et al. [123] reported the occurrence of DRX in a Mg–3.3%Li alloy at 350 °C ECAE. The extent of DRX was found to be dependent on ECAE route; 100% for four-pass route Bc and 70% for four-pass route A. Although textures were measured, the effects caused by DRX were not examined in detail. RX was observed in ECAE of pure Mg [186], however, it was pointed out that it might have happened after deformation – that is, by static RX at room temperature – because of the low homologous temperature of Mg. Beausir et al. [64] examined the effect of RX in pure Mg at 250 °C. Polycrystal plasticity modeling (considering slip only) was successful in explaining the evolution of the texture in all routes meaning that the occurrence of DRX did not produce any component that differs from the deformation texture components. This is also related to the fact that such fibers were observed which have their rotation axis around the c-axis of the hexagon; such fibers keep their axis if DRX happens by growth of nuclei that are nucleated at a ±30° c-axis rotation with respect to the parent grain. In another work, in torsion of pure Mg (which is also simple shear) the effect of RX on the textures was clearly correlated to the Taylor factor of the deformation process [187]. 4. Modeling texture evolution 4.1. Polycrystal models for SPD processes Polycrystal (PX) models have been developed to predict texture development and to calculate the stress–strain response of polycrystalline materials, considering texture evolution, evolving grain shapes, and single crystal hardening. There are basically two types of PX models used to predict texture evolution: Taylor-type (full constaints, FC, or relaxed constraints, RC) and self-consistent (SC) models [188–190]. In these, a grain is defined by its orientation, and its volume fraction with respect to the volume of the whole aggregate. As texture is a statistical property, large numbers of grains have to be considered in the representation of the polycrystal. It is best if the number is at least 1000. The PX results apply best when there is at least 20–100 grains in the sample cross section. Early applications of polycrystal modeling to ECAE have shown that modeling texture evolution under large strains and strain path changes has not been straightforward. First of all, it is well known that the simulated textures evolve much more quickly than measured textures, yielding higher intensities for the same deformation. The discrepancy in intensities can grow with straining and hence become noticeable in ECAE textures. Second, at large strains, certain features of the texture may not be predicted. Reasons given are typically grain–grain interactions and subgrain structure evolution which are not modeled in the homogenization procedures used in the PX models listed above. The subgrain scale inhomogeneities in crystal orientation and hardening that result prevail at larger strains and their effects become pronounced under strain path changes.

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The two commonly used models in ECAE texture calculations are FC Taylor and visco-plastic selfconsistent (VPSC) and these are briefly described below. The reader is referred to [61,188–190] for a complete description of their formulation. 4.1.1. Full constraints (FC) Taylor polycrystal model The most widely used PX model is the FC model. In the FC model it is assumed that the strain (or strain rate) that a grain experiences is exactly the same as the imposed macroscopic strain (or strain rate). It is an upper bound model for the stress state. It is readily applicable for crystal structures containing at least five independent slip systems, such as cubic materials. This assumption also applies best at relatively small strains before plastic inhomogeneities develop due to subgrain-induced anisotropy. The FC model can become rapidly inaccurate at large strains, even for cubic materials. The RC model tries to account for specific grain shapes by relaxing some of the strain-rate components, depending on the strain mode. The RC model can be successful at large strains but not applicable at small strains. This model still assumes that some of the strain-rate components in the grains are identical to their corresponding macroscopic values. 4.1.2. Visco-plastic self-consistent (VPSC) polycrystal model The VPSC model is well suited to model large strain behavior of polycrystals. It accounts for the plastic anisotropy of each grain, as well as that of the polycrystal. It neglects elasticity which is justified at large strains. Each grain is modeled as an ellipsoidal inclusion which is embedded into the so called Homogeneous Equivalent Medium (HEM) with the average properties of the whole polycrystal. VPSC is given its name because the visco-plastic compliance tensor of the polycrystal is determined in each strain increment in a self-consistent manner. With the help of this tensor and by solution of the Eshelby inclusion problem, the stress and strain state of each individual crystal is determined. For self consistency, the macroscopic stress and strain rate of the HEM are set equal to the average of the stresses and strain rates of all the individual grains, each of which, in turn, is governed by its interaction with the HEM. Unlike the FC Taylor model, the stress and strain rate of each grain can deviate from the corresponding macroscopic quantities, as well as deviate from each other. One consequence is that each grain changes its form according to its local velocity gradient and deformation history. Another important consequence of the self-consistent formulation is that a grain no longer needs to have five independent slip systems to deform. This feature is especially important for non-cubic crystal structures, like hexagonals, where the number of available slip systems could be less than five on the basal (3) slip or prismatic (3) slip planes. In VPSC modeling it is, therefore, possible that a given grain deforms solely by basal slip while the other grains are deforming also on non-basal slip systems or by deformation twinning.

4.2. Assessment of model performance Assessing the predictive capability of texture models involves comparing calculated textures with measured ones. In most ECAE textures studies, comparisons between simulated and measured textures are usually qualitative. They are frequently performed by a subjective, visual comparison. This approach is suitable as long as the textures are presented in the same way. In case of pole figures or ODFs, the use of the same isolevels, reference system and figure size makes the comparison more objective. The choice of sections in Euler space is also important. (Most often in the literature an insufficient number of sections are presented.) Naturally, when the textures are presented differently, the comparison becomes difficult and the conclusions less meaningful. The best solution would be for the whole texture community to use a standardized presentation of textures. For a quantitative comparison, the normalized texture index T^ d of the difference ODF (DODF) can be used:

R ½fsim ðgÞ  fexp ðgÞ2 dg Td T^ d ¼ : ¼ R T ½fexp ðgÞ2 dg

ð15Þ

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Here Td is the texture index of the DODF, i.e., fsim(g)  fexp(g). Td itself is a measure of the error in the predicted texture. Smaller Td means a better approximation of the experimental texture. The use of T^ d permits comparing textures with different reference textures, such as those generated from different materials or deformation conditions. Td and T^ d were employed in [29,31–33,65,73]. Simulations with values less than 1.0 can be considered to be good textures in comparison with experimental measurement. While values as low as 0.2 can be obtained in simulations of the first pass, this value can increase with the pass number, depending on the model used, whether self-consistent or Taylor [29,31,32,65]. Deviations can be on the order of 10 in T^ d . 4.3. Polycrystal model performance Results obtained with the FC model have not compared well with measurements made in cubic materials processed by ECAE [31,32,65,66,74,82,191–193]. This was first studied by Agnew et al. [193] where the simulated textures obtained with VPSC and FC models were compared to experiments in Cu up to four passes of route A. The FC model led to much sharper textures than the VPSC model (or the measurements) and some features in the third and fourth passes found in the measurement were obtained by the VPSC model, but not FC model. Ferrasse et al. [66] presented FC textures simulated for an aluminum alloy processed via routes A, BC and C up to four passes, depicting significant differences in features and intensity as compared to X-ray measurement. They see good agreement for the first pass only, albeit with much stronger intensities. Another study, carried out by Baik et al. [82] on pure Cu processed via route A, BC and C, also revealed large discrepancies between simulated and experimental neutron diffraction textures after four passes. Even after one pass, the texture was missing some important components observed in the measurement. In Li et al. [65] the model performance was evaluated with the help of the T^ d quantity (see Eq. (15) in Section 4.2) by comparing the FC and VPSC predictions to experiment in route Bc copper. It was found that the VPSC model performed much better up to 16 passes, irrespective of the macro-deformation model used (simple shearing or finite element). The VPSC model was able to reproduce both the relative intensities and the positions of the main texture components of the textures. Although the quality of the agreement decreased progressively with pass number during the first four passes, it did not get worse with further passes. Similar results were obtained for bcc IF steel processed by U = 90°. VPSC performed much better than FC Taylor for all pass numbers and all routes [32] and FC became much worse with each pass but VPSC did not. The FC model does not perform poorly in every case. For Cu processed by route A is it found that the FC model can predict the location of the texture components relatively well, but only when a deformation field more accurate than simple shearing was used [31,32,67,192]. For IF steel processed by U = 120°, the performance of FC-Taylor and VPSC was similar in routes A, C, and even Bc. For U = 120°, the deformation was closer to ideal simple shearing7 and lower in intensity than U = 90°. 4.4. VPSC predictions using the simple shear (SS) model Using a standard VPSC model in conjunction with the SS model (Section 2.1.3) for an initially random textured and non-hardening fcc metal, it was demonstrated in [27] that its texture develops as expected with respect to the strain path changes associated with each processing route. In route A, in which the shear planes cross from pass to pass but their shear plane normals remain in the plane of the ECAE die (TD-plane), the texture has the monoclinic symmetry of simple shear and progressively strengthens with increasing pass number. Route C imposes a shear reversal every consecutive pass, thus the initial texture is nearly recovered in even passes, while a one-pass shear texture appears in odd passes. In routes Bc and Ba the shear plane and shear plane normals of the previous passes change and rotate out of the TD-plane from pass to pass. As a result, these routes lead to mixed-type

7

The material likely filled the die in the 120° case but formed a corner gap in the 90° case.

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textures that vary substantially from pass to pass up to four to eight passes. After a large number of passes texture evolution in these routes are found to repeat themselves every two (route Ba, see Section 3.1.5 and [32]) or every four (route Bc) passes [65]. These simulated features of the simple shear model agree reasonably well with the experimental observations for routes A, Bc, and Ba, however, the agreement is poor for route C. As will be shown in the next sections, much better agreement can be obtained for route C by using more realistic deformation fields for the extrusion process.

5. Alternative deformation models for ECAE Up to now, the applied ECAE deformation has been modeled as simple shearing acting along the intersection plane in the negative sense. The simple shear model is preferred for its simplicity and ability to give reasonable texture results. When processing conditions deviate from the ideal (zero friction, perfectly plastic material and sharp inner and outer corners), however, the deformation field generated during an ECAE extrusion becomes much more complex and can no longer be represented as simple shearing concentrated along the intersection plane. For calculating texture and microstructural evolution correctly, it is imperative that such deviations and other details of the macroscopic flow history are known and are represented accurately in the polycrystal calculation. Otherwise, meaningful comparisons with measurement or predictions of sample mechanical properties cannot be made. Polycrystal models for texture require the applied deformation to be described in terms of F and L (see their meaning in Section 2.1.2). To calculate F and L under these general processing conditions, finite element (FE) simulations and analytical flow models can be used in place of the simple shear model. For texture calculations, both are good alternatives, with the finite element technique being more computationally intensive but more accurate than the analytical models. In this section, these two approaches and their results on deformation and texture will be discussed. 5.1. 2D and 3D finite element models In this section, we examine the deformation fields calculated from 2D and 3D FE simulations on ECAE. Although less common, FE analyses have been performed on other SPD techniques, such as constrained groove pressing [194] and equal-channel multi-angular pressing [195]. These are not discussed here, but the issues addressed in this section will apply to these and other SPD techniques, where, as always, deformation characteristics must be known for accurate texture calculation. Inhomogeneity develops along the billet axis (x-direction), from the top to bottom of the sample (ydirection), and across (z-direction). When the strain field is inhomogeneous, the accumulated strain varies from point to point, and it cannot be described well by a single value. An inhomogeneous strain field can be readily handled in numerical simulations, such as finite element. Most FE studies look at one or a combination of several factors on the effective strain, strain rate, and/or effective stress fields. Through FE simulation of the ECAE process, several factors related to processing, tooling, and material have been shown to cause deformation heterogeneity to various degrees. This information is valuable for design as well as for defining processing limits when producing large volumes of homogeneously refined material. Undoubtedly, spatial variations in strain and stress lead to undesirable variations in microstructure and texture. Local regions of no deformation lead to no texture and microstructural changes, whereas local regions of high deformation can lead to premature fracture. When comparing different processing situations, common measures are the average bulk strain, average strains in the x, y, and z (ED, ND, TD) directions, maximum strain, minimum strain, and volume (or area) of homogeneous material. A useful measure is the following heterogeneity factor Ci [196] (excluding the two ends):

C i ¼ ðepmax  epmin Þ=epave : pmax ; pmin

pave

ð16Þ

where e e and e denote, respectively, the maximum, minimum and average of the equivalent plastic strains across the billet thickness. This expression has been extended to the x, y, and z directions [197]. Other measures of hetereogeneity have also been proposed [198].

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5.1.1. Corner gaps and plastic deformation zones (PDZ) Most of the deformation and sources of heterogeneity develop as the material flows through the corner region. Two widely studied deformation characteristics are corner gap formation and the plastic deformation zone (PDZ). A corner gap is a region in the outer corner of the die where the workpiece has separated from the die surface [199]. Its size is measured by the angle We about the inner corner (see Fig. 22 [200]) that subtends the billet’s points of contact with the entry and exit channel. For a die with an outer corner angle W P 0 (with W = 0 being a sharp corner), when W < We there is a free-surface gap and when W > We there is not. The larger the outer rounded corner angle W, the smaller the corner gap and the more likely the material fills the die [196,200]. Other factors shown by FE to increase the corner gap are large strain hardening rates [196,198,199,201], low friction or frictionless conditions [196,199,202,203] (see Fig. 22), small U angles, high strain rate sensitivity [204], no backpressure [202,205], and large initial yield strains [201]. Interestingly a narrower exit channel than entry channel [206] was found to slightly reduce the corner gap. Also, a patented die with a sliding bottom [207] helps to reduce the corner gap [208]. The effect of the sliding bottom was modeled in the FE simulation [208] by increasing the friction coefficient. When a free-surface gap forms (incomplete filling, We > 0) or the outer corner of the die is rounded (W > 0), FE simulations [196,203,200,209] observe that plastic deformation takes place in a broad PDZ spread about the intersection plane. One of the best ways to visualize the PDZ in FE simulations is by fields of equivalent strain rate or of individual strain rate components. Examples of PDZs from FE are shown in Fig. 23, assuming no friction and either (a) an elastic-perfectly plastic or (b) a strain-hardening material. Fig. 23c shows the PDZ when friction is taken into account (l = 0.1) for a strain hardening material. These calculations demonstrate the usual effect of work hardening and friction seen in many FE simulations: the perfectly plastic material fills the die (Fig. 23a) but the hardening one (Fig. 23b) does not. Friction helps to fill the die better (Fig. 23c). The PDZ has two parts, each with a distinctly different mode of deformation. As shown in Fig. 23, a large portion of the PDZ can be represented by a fan emanating from the inner corner, and the angle b subtended by the PDZ is one measure of its size [11,22]. The deformation within the fan-shaped portion is simple shearing in sequentially rotating shear planes [11,28] (see Section 5.6). It is observed that b increases with W or We, whichever is larger [196,209,210]. A larger b leads to less accumulated strain after the extrusion [196,198,209]. In the lower region of the PDZ, the fan shape idealization breaks down. The material flows from the entry to the exit channel by rigid body rotation. Within

Fig. 22. Comparison of the measured corner gaps formed during ECAE in a 90° die under conditions of low friction and high friction (from [200]). The definition of we discussed in the text is also indicated.

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β

(a)

(b)

(c)

Fig. 23. Maps of the strain rate component Dxx in finite element calculations for a (a) perfectly plastic material without friction, (b) strain hardening material without friction, and (c) strain hardening material with a friction coefficient of 0.1 (from [22]). An estimate for an effective value of the ideal fan size b is indicated.

the lower region, most of the deformation occurs at the two points where the material leaves the entry channel and enters the exit channel. The normal components of the strain rate change sign across the upper shearing region and lower rotating region of the PDZ [22,28]. The PDZ formed by a corner gap (We > 0) and by tool geometry (W > 0) are not exactly the same. When a corner gap forms We > W > 0 the path of flow in the outer region is not symmetric, but parabolic-like, [199,202,211]. The strain rate distribution within the PDZ becomes asymmetrical and skewed towards the exit channel [22,23,31,199]. Any processing and material factors, such as friction, material hardening, backpressure, that affect the size of a corner gap We > 0 will also affect b. As shown for example in Fig. 23b, b is slightly larger with a corner gap than b in Fig. 23a without a gap. When the flow path is instead dictated by the geometry of the tooling W > We > 0, the strain field within the PDZ is symmetric about the inner corner [22,196]. W dictates b and not strain hardening or friction [196,211]. FE simulations have shown that these differences significantly affect the local characteristics of the deformation near the bottom of the workpiece compared to the rest of the workpiece [22,202]. Consequently, texture measurements made in the central region are less sensitive to how the PDZ was formed. 5.1.2. Inhomogeneous deformation The number of factors that can impact deformation heterogeneity is large, too large that experimental exploration of the entire parameter space would be prohibitive. FE simulation has shown that factors affecting heterogeneity include U, W, material behavior (hardening, initial yield, softening, rate sensitivity, thermal–mechanical properties), processing route, pre-form shape, friction, backpressure, and cross-sectional shape. In this parameter space, trade-offs exist, such that a universal optimal processing condition for homogeneity cannot be recommended. Nonetheless, as discussed below, some general guidelines can be provided. With the large number of FE studies conducted in the past decade, heterogeneity in deformation in all three directions after the first pass has been thoroughly characterized. Along the x-direction (billet axis) most of the inhomogeneous deformation occurs at the head and tail ends. The central region is considered the most uniform and useful portion of the workpiece. The size of the tail region is not as well studied because it requires running the FE simulation to complete extrusion and such plots are most often not shown. The head region is found to scale with billet thickness w. In [196], it was shown that the head region also increases with W and strain hardening. The sizes of both end regions grow with multiple passes, as will be discussed later. These two end regions are often disregarded in strain analyses but because they must be avoided when sampling the material for microstructural (e.g., texture) measurements, their size must be known. Shear banding can also lead to non-uniformity in the x-direction. Flow-softening behavior can result in a sequence of macroscopic shear banding along the billet axis. These bands are relatively thin regions of high strain and low stress, separated by regions of low strain and high stress, both aligned roughly along the intersection plane. They have been observed in a few FE studies [201,211,212], but

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studied in detail in [212] where a fine mesh was used. In these cases, the outer corner was sharp. Lapovok et al. [213] modeled shear banding in AZ31 using an analytic perturbation technique based on a constitutive equation that included a strain gradient term. The effect of a rounded corner on shear banding for a flow-softening material has not been studied. It would also be of interest to examine deformation textures separately within the banded regions and in-between. For this, spatial resolutions below 1 lm3 would be required. Heterogeneity in the through-thickness direction (y-direction) from top to bottom is the most widely studied and is found to be linked to flow properties. There are three possible flow characteristics: (i) natural abrupt flow around a sharp corner die associated with ideal conditions, (ii) curved flow, and (iii) forced abrupt flow around a sharp corner die. Case (i) is considered ideal. Case (i) leads to the most homogeneity and is possible to achieve under frictionless conditions, zero-hardening or mild-strain hardening materials, and large-angle dies 90° < U [31,196,203,209]. Curved flow in case (ii) is induced by a geometric rounded corner (W > We) or free-surface corner gap (W < We) and results in inhomogeneous deformation. For this discussion, it is useful to define Wmax = max(W, We). The heterogeneity in y grows with Wmax = max(W, We). The strain in the central portion of the billet decreases with an increase in Wmax, relative to that when Wmax = 0. The top and bottom layers deform differently than the center. Shearing is typically enhanced in the top layer. The bottom layer, on the other hand, experiences significantly less deformation and large rotations. This lower-undeformed region can encompass an appreciable fraction of the billet thickness (5–20%), increasing with Wmax [22,33,196,203,200,210,214,215]. In this region, the elements are practically undistorted. The development of this ‘no-shear zone’ has been validated by experimental observations in Plasticine billets and metals billets with grids, which showed little to no deformation in the lower region, and likewise, by EBSD measurements on processed samples, which showed little to no grain deformation and dislocation accumulation [22,33,216] also in this lower region. The curvature allows for motion largely by rigid-body rotation and little deformation (mainly at the contact points), which is more energetically favorable than severe plastic shearing. This leads to a faster flow rate in the outer region than in the rest of the material deforming by shear [214]. Forced abrupt flow in case (iii) may lead to complete filling but not necessarily to homogeneity in deformation. It is possible to force complete filling of the die corner region by increasing friction, applying backpressure, using a strain-softening material, or possibly even re-designing the pre-form shape [196,201,205,205–207,215,217,218]. The material has difficultly flowing around the sharp corners, and strain localization along the bottom layer is promoted [202]. Consequently, deformation non-uniformity along y is enhanced. Most of these results are obtained from simulations of a 90° die. Recently it was shown by FE that the impact of friction is greater when U < 90° [206]. With friction, an acute angle die leads to much more heterogeneity than U > 90° [219]. Unlike friction and strain-softening materials, backpressure has the advantage of hindering cracking and damage formation and enabling the processing of low ductility metals [218]. 5.1.3. Three-dimensional effects There are considerably more studies which use 2D than 3D FE; however, the number of 3D ECAE models being developed is growing [197,203,209,211,220–225]. Some obvious benefits of a 3D calculation over a 2D calculation are evaluation of heterogeneity in the z-direction, simulation of samples of varying cross-sectional shape (e.g. circle vs. square), and simulation of multi-pass ECAE. Although some routes, such as routes A and C can be modeled in 2D plane strain conditions using a network of inter-connecting dies (e.g., [226]), not all routes can be modeled this way, such as routes Bc and Ba. For passes greater than say four, this 2D multi-pass network can become rather large and clumsy and it is not clear if steady-state is achieved at any point in the process. 3D FE evaluation of strain heterogeneity has been performed in a few studies [203,209,211,222,224]. In [203,211] the effects of 3D were quantified by directly comparing 2D and 3D simulations in the same study. Generally 3D models predict lower accumulated strains than 2D and than those predicted by geometrical considerations (compared to Eq. (20)) [203,209,222,224]. Regarding heterogeneity in the x and y directions, these 3D simulations predict similar trends as those described above for 2D simulations. Temperature gradients were slightly higher in the 3D simulations,

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although the character of the temperature fields was the same [211]. The differences between the two models increased as processing parameters deviated from ideal [203]. For rounded corner dies, the undeformed bottom layer was larger in 3D than 2D [203]. Regarding inhomogeneous deformation in the z-directions, several interesting findings have been reported from 3D FE simulations. The heterogeneity in z can be higher than that in y [225] or lower [222]. As the outer corner radius increased, the heterogeneity in the z-direction decreased [203]. For a circular cross section, the deformation may not be symmetrical about the center plane [197]. Last, the heterogeneity in z changes with pass number [223]. With multiple passes, deformation heterogeneity is not removed (see for example [220,223,205]. In [223], 3D finite element was used to study the progression of heterogeneity for the standard ECAE four routes up to four passes. The billet lengths were relatively short (approximately 45w, where w is the width) and heterogeneity grew with pass number for all routes with the exception of route Bc [224]. It was found that route C leads to the most homogeneous billet with a central region approximately one billet width w in length [205]. A uniform central region did not develop with the other routes after four passes. Even when the tooling, temperature, pressing rates, and friction conditions are kept constant, changes in material behavior and strain path changes associated with ECAE can help to retain deformation hetereogeneity from pass to pass. Material hardening (or softening) leads to heterogeneity and can change with accumulated strain and with microstructure and texture evolution. All material points undergo a strain path change from pass to pass, regardless of the route and these strain path changes can induce changes in the hardening response. Even if hardening were to saturate later and lead to homogeneous deformation in subsequent passes, this would not erase the heterogeneity generated in the first few passes. The reason has to do with the exchange of material points with respect to the ECAE die from pass to pass. For example, the top layer (bottom) of the first pass becomes the bottom (top) layer in the second pass of route C or the side layer in the second pass and bottom layer in the third pass of route Bc [65]. In route A, such an exchange does not occur from pass to pass. 5.2. Nonphysical factors affecting assessment of model performance In many sections to follow, we will examine the ability of texture models to predict texture evolution in ECAE. Before doing so, we believe it critical to reveal some problems related to model performance assessment. Several factors, that have little relation to the capability of the model, have affected comparisons between texture prediction and measurement made in many studies. One must carefully screen for them before meaningful conclusions can be made from reported results. As described below, such unphysical factors exist in experiments as well as in simulations. Let us first consider the measurement technique and execution. First of all, there are several techniques that have been used in ECAE studies to measure texture: X-ray diffraction, neutron diffraction, Orientation Imaging Microscopy/Electron Back-Scattering Diffraction (OIM/EBSD), and synchrotron. With X-rays and OIM/EBSD, while only small sample volumes close to the sample surface can be examined, spatially resolved measurements are possible [22,23,33]. In all techniques, the number of grains studied can strongly influence the measured intensities. Secondly, different techniques can lead to different results for the same material. Comparing, for example, results obtained from neutron diffraction, which is a bulk measurement, with those from OIM, which is a surface measurement, usually finds lower intensities from the former. The difference may be up to a factor of two. Next, surface measurements carried out by X-ray diffraction or OIM are sensitive to topological heterogeneities, and thus, to the selection of the scan location or the size of the scan area. In choosing a scan location or size, one must be aware of texture gradients that may exist across the billet. This is especially relevant when the texture is measured on the side-face (the ND-ED plane), where usually there is a relatively strong texture gradient. In this case, an average texture is measured and a comparison with results from simulations is less precise. This effect can be eliminated by simply measuring the texture on a TD-ED section of the billet where there is a much larger central portion, with little to no texture gradients. Synchrotron measurements can also capture small dimensions but with larger grain numbers for a good statistical representation. In such measurements, the penetration depth below the surface and the irradiated volume are larger than in conventional X-ray or OIM. Last, as always, one must not

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take measurements near the billet ends, where the deformation and hence texture are significantly different than those of the center, as will be discussed in Section 5.4. In particular, billets that are too short are susceptible to growing end effects (see Section 5.5). As a result, texture results may appear to be falsely dependent on billet length, which can range from as short as 4 cm [25] to as long as 15 cm [118] for a 10 mm diameter billet. Numerical studies are also subject to non-physical effects, such as the generated number of grains in the PX model, or possible dependencies on discretization, e.g., the coarseness of mesh in FE calculations. It has been shown in Li et al. [65] that by using a larger number of grain orientations to represent the initial texture, e.g. 8000 vs. 1000 grains, more details of the initial texture components are retained in simulation. Unfortunately, it is often the case that less than 1000 grains are used in polycrystal simulations. There are some instances where FE calculations of ECAE are not reliable. Strain calculations will be less accurate close to sharp corners, in coarse meshes (e.g., approximately 10 elements or less across the sample thickness in a 2D simulation), prior to reaching steady-state, and when the friction coefficient is too high. Regarding the latter, as discussed in [196], the FE simulation should not use a friction coefficient that exceeds a maximum allowable value lm, defined as the maximum value of l above which either the FE simulation cannot be completed or steady-state cannot be achieved. This idea has been applied in [197], but for the most part is un-tested in FE studies. lm increases as W increases and as the length of the billet or exit channel decreases. The relationship with billet length explains why in order to perform an FE simulation with large l, researchers have shortened the billet to lengths which may be impractical (less than 5 times the width). Otherwise, results from most FE simulations using a high l are reported in cases when steady-state was not achieved. In the other extreme, in frictionless conditions, l = 0 [214], it was found that the sample length must exceed four times its width in order to reach steady-state. Unless stated otherwise, in this article, we have not included FE results affected by these problems. While the outer corner angle W > 0 has been considered an important engineering variable, the inner corner angle is not and is typically considered to be nearly sharp. However, it cannot be modeled as such; such a discontinuity can cause artificial localizations. It has been shown that choice of inner corner angle can alter the deformation near the top and bottom surfaces [196]. 5.3. Texture calculations with finite element There are two ways in which finite element can be used in polycrystal texture calculations. The computationally efficient method is to decouple the FE simulation from the polycrystal model that calculates texture. In a separate FE simulation, a continuum material model is used instead. The time variation of the deformation and velocity gradient from the FE is then imported into the polycrystal model for texture prediction. By this method, the texture contribution to plastic flow during the process is neglected. Specifically the FE predictions lack contributions from any physics that may be featured in the polycrystal model and evolve with deformation, such as grain–grain interactions, substructure development, or grain refinement. The polycrystal model, however, does benefit from a more accurate description of the deformation. The second and potentially more accurate way is to directly couple FE and the polycrystal model, where each FE element is a polycrystal. This method is clearly more computationally intensive but may not necessarily lead to more accurate results than the first method if too few grains per element and/or too few elements are used or if the polycrystal model is too simplistic, like the FC Taylor model. Most FE-polycrystal modeling efforts use the first method; that is, a separate FE simulation is performed as another method of calculating F and L for texture calculations. Compared to simple shear [31,65,73,227,228], agreement between calculated and experimental textures improves greatly when the more realistic FE-derived deformation is used. For example with an FE-calculated deformation history, VPSC predicted the texture evolution in Cu much better in route C up to four passes, route Bc up to 16 passes [65], and route A up to five passes (U = 120°) [227], compared to when an simple shear history is used. This improvement occurs despite the fact that grain refinement is not modeled, the FE history is decoupled from the polycrystal calculation (i.e., the texture contribution to anisotropy is neglected), and the anisotropic response of Cu due to subgrain microstructure is not considered.

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The differences between an FE-derived history and simple shear are more marked when modeling texture evolution during route C [23,31]. Significant improvement occurred in the route C texture prediction because the FE-calculated deformation accounted for changes in material response from pass to pass. Because the deformation characteristics of the first pass were not the same as that of the second pass, the second-pass deformation did not represent a perfect reversal of the first pass deformation. Incidentally, such slight differences in deformation have less of an impact on the first-pass texture. Recall the two FE-calculated PDZs shown in Fig. 23, in which the one from (a) an elastic-perfectly plastic material is symmetric about the intersection plane and the one from (b) a strain hardening one is skewed about the intersection plane. Importing either FE strain path history found in Fig. 23a,b into VPSC leads to one-pass textures that vary through the thickness of the sample and are similar in features and strengths to EBSD measurements [22]. Also compared to route C, these differences in deformation also have little impact on subsequent passes of routes A, Ba, and Bc. There have been a few studies which directly couple FE simulation and texture modeling with promising success. A multi-scale model combining dislocation density-based hardening [229], Taylor polycrystal modeling, and 2D FE simulation was developed in [82,191] and used to predict texture evolution in Cu up to four passes of routes A and C and two modified routes. The model results could capture the heterogeneity in texture evolution and agreed well with measurement in all cases. In [221], a 3D crystal-plasticity FE (CPFE) method using multiple grains per element predicted textures in good agreement with measurement and better than Taylor or standard VPSC and similar in predictive capability to an improved VPSC scheme [27]. Because the CPFE textures using one grain per element were not as good, any improvements by the multi-grain CPFE were attributed to grain–grain interactions. In summary, success in using the de-coupled method emphasizes the importance of proper representation of the main flow characteristics and how they change from pass to pass. Coupled methods can lead to more accuracy if multiple grains are contained in each element. So far, these efforts consider cubic crystal structures. Any errors introduced in using the de-coupled method would be enhanced for metals with low symmetry crystal structures, such as Mg and Ti, which are plastically anisotropic and in which the coupling between texture and deformation mechanisms is strong. Therefore, research in this direction is strongly recommended.

5.4. Heterogeneity in texture evolution Inhomogeneous starting textures or inhomogenoeus deformation can lead to inhomogeneous textures. Inhomogeneous starting textures will influence texture evolution mainly the first few passes (see Section 3.3). Inhomogenoeus deformation, on the other hand, is likely to influence texture evolution all passes. Inhomogeneous deformation predicted by 2D and 3D FE simulation can be imported into polycrystal simulations for predictions of heterogeneity in texture evolution. As discussed in the previous Section 5.2, inhomogeneous deformation results from a rounded corner W > 0 die or the formation of free surface corner gaps. In this case, the most important inhomogeneity develops in the lower region of the billet. The development of this lower region, as well as its deformation characteristics, has been investigated in several 2D or 3D FE simulations, e.g., [22,33,196,230] and modeled analytically by [28,231]. (For more, see Sections 5.2 and 5.6 and references therein.) Its deformation mode is distinctly different from the intense plastic shearing of the upper portion: plastic deformation is small and material flow occurs by large rigid body rotations. Consequently, the texture evolution in this region will be different than the rest of the billet. Precise measurement of the texture gradient is possible with synchrotron [232]. Such results were presented in [89,232,233] for copper, in [232,234] for Ni single crystal and in [232,235] for silver. Texture heterogeneity was measured in Ti and Cu processed by route C using X-ray diffraction [80,84] and in Cu using OIM [23]. Both studies found that after one pass, the bottom texture, unlike the texture in the center of the billet, was very spotty because the number of grains in the scanned area was relatively few. Apparently, the microstructure in the bottom region of the sample was less refined due to little plastic deformation and more rigid body rotations. Because the strains are low, texture development in this region will be more sensitive than the central region to material and processing

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conditions, such as initial texture, surface friction, and material changes, than texture development in the center. Although the size of this undeformed zone increases proportionally with either W or corner gap size, modeling suggests that their effects are not the same. A more heterogeneous texture was predicted in Cu after one pass [82] when the outer corner was rounded than when it was sharp, despite the fact that in the sharp corner die, a corner gap had formed. In the model, texture evolution in the bottom layer will strongly depend on the amount of rigid body rotation, which can be large. Any over- or underestimation of the rigid body rotation in the deformation model (e.g. FE simulation) can lead to large discrepancies between prediction and measurement. This sensitivity was demonstrated in [22]. In using an analytical model for the deformation in this region [28], the amount of rotation can be easily determined by fits with either EBSD/OIM texture measurements or FE/polycrystal simulations. Because this inhomogeneous deformation persists in every pass, in multi-pass ECAE, textures in the top, middle and bottom layers become distinctly different with N and the texture gradients depend on route. For example in route A, the locations of material points in the billet are unchanged from pass to pass, so after every pass the middle to upper portions will have shear-like textures and the bottom portion will have a texture close to the initial one. In route C, however, the bottom and top material points are interchanged, and therefore the effect of any deformation heterogeneity is expected to diminish, albeit slower than the material in the center. Similarly, in routes Bc and Ba, the outer layers change their orientation in the die from pass to pass, possibly creating an outer perimeter of microstructure and texture different from the center. Using FE simulation combined with polycrystal modeling, Li et al. [23] predicted in route A that with an increase of N, the texture gradient increased whereas in route C the top and bottom textures become similar, but remain different from those in the middle. For the even-numbered passes of route C, these predictions are in qualitative agreement with the textures measured by Tidu et al. [79] after 14 and 16 passes (U = 120°) and by Li et al. [23] after two passes (U = 90°) at different positions across the width of an OFHC copper billet. In [84] the two-pass route C Cu displayed large variations with the top and bottom having more dispersed textures than in the center. The material was more homogeneous by the fourth pass. On the other hand, in the same work [84], they found that Ti had more uniformity than Cu, which was attributed to higher processing temperatures and a finer grain size. These factors help to reduce the work hardening in Ti causing the deformation to be uniform during the extrusion. A second-order (minor) inhomogeneity is the variation in texture within the fan-shaped portion of the PDZ. One consequence of PDZ deformation is the generation of a ‘‘transition” zone, lying between the fan region and lower rigid-body rotation region, which has texture characteristics of both [22]. Another is the slight variation in the ‘tilts’ in the texture components from the top of the billet to the end of the transition zone. The tilts in the shear components about the TD (away from the y axis) across the thickness have been measured in one-pass samples. In the Ti of [84] they were larger in the bottom and decreasing towards the top. In Cu [22], the difference between the tilts in the textures measured in the middle and near the top was 5–7°. For the analytical flow models of Section 5.6, this secondorder inhomogeneity can be modeled by making the profile parameter (i.e., b or n) a function of initial material position, as was done in [22,89]. 5.5. Route A: heterogeneity and end-effects Compared to the other routes, route A is prone to increasing end effects with each pass. As shown by Bowen et al. [200], in this route, material from the top layer is being drawn towards and rotated around the head-end and folded there after several passes. (It is incorrectly stated there that route A imposes a constant strain path. Route A (N P 2) imposes a strain path change (see Section 2.1.5).) The tail-end is also expected to grow, and consequently the uniform central region to shrink, with pass number. Texture modeling studies show that route A will be particularly vulnerable to increasing heterogeneity in deformation and texture with N [33]. The problem can be explained as follows. After one pass, the length of the non-uniformly deformed head region ranges from 1 to 2 times the width of the billet,

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increasing with W [196]. In route A, there is no exchange of the top, middle, bottom, and side layers with each pass unlike in routes C, Bc, and Ba. Consequently, end effects from both ends tend to build with each pass at a faster rate in route A than the other routes. The material affected by end effects increases with N, propagating from the ends into the central portion. The resulting texture after several passes would therefore not be a result of shearing but of folding. Consequently, when analyzing textures and texture strengths in route A, it is important to be mindful of the billet length, in particular, texture measurements of route A samples subject to large numbers of passes and relatively short billets (L < 2Nw, where w is the width of the sample). Because of end effects, sample textures after many passes may reflect the complex deformation experienced at the ends and not the severe shear deformation expected from ECAE-shear. This may explain inconsistent trends in reported strengths of textures produced by route A. Billet lengths vary across studies. De Messemaeker et al. [25] find that for low N 6 4, IF steel maintains a monoclinic symmetry and its texture strengthens with N; however, after eight passes the intensity drops and monoclinic symmetry is lost. The billet lengths were short (4–5 cm). Using 8 cm billets in both U = 90° and 120°, Li et al. [31,32] report no definitive weakening or strengthening of the texture in IF steel with N up to four passes. Using a 10 cm billet (Al), Gholinia et al. [192], report a strengthening up to eight passes and a weakening in texture from eight to 12 passes. Using 15 cm long samples (ZK60 Mg alloy), Agnew et al. [118] see a strengthening in route A up to eight passes (from 6 mr to 8 mr). Any sudden weakening in texture at large N is commonly attributed to substructure formation, grain refinement, or dynamic recrystallization. Such hypotheses need to be tested but they are certainly plausible. However, is that it is also possible that such an abrupt change in texture strength after large N in route A could be due to propagating end effects. The shorter the billet length, the earlier such end effects will likely take over and texture weakening will be observed. To determine unambiguously if the texture development in ECAE will eventually ‘randomized’ (or substantially weakened) at large N, appropriate billet lengths need to be processed. The higher N and the larger W, the longer the length that is required. 5.6. Analytical flow models The first analytical model was proposed by Segal [11] for a sharp corner die (W = 0) and non-zero friction surfaces. Using slip-line field analysis (applicable for a rigid plastic isotropic material), it was found that the PDZ broadened into the shape of a fan that was symmetric about the intersection plane. At the corner is a dead metal zone (non-moving material), which is to be distinguished from the free surface corner gap predicted in FE models. A slip field consists of two orthogonal sets of lines parallel to the direction of maximum shear strain rates. Maximum values vary from one slip line to another. Within the fan, one family of slip lines radiates from the inner corner, representing flow by simple shearing on continuously rotating shear planes about the inner corner. The angle of the ‘shear fan’ b was related to the ratio of die surface friction to yield stress and to the die angle U. Later Segal applied the slip field analysis to rounded corner dies with W > 0 [236]. As before, the analysis assumes a rigid plastic material and uniform friction. It was found that the rounded outer corner tends to broaden the PDZ. As calculated in FE simulations (see Sections 5.1 and 5.3–5.4) the PDZ had two distinct parts. The upper part was a shear fan and the lower part, a region of small shear strain and mostly rigid body rotation [22]. The predicted deformation modes are consistent with many FE calculations (see Section 5.1). In this case, the fan angle b depends in a complex way on yield stress, friction, W, and U. Because the shear fan is predicted to be a major component in the two processing situations described above, it is worth describing its deformation in more detail. As shown in Fig. 24, the fanshaped plastic zone is equidistant by an angle a from both sides. For a given die angle U, a is related to the fan angle b through a = (p  U  b)/2. Accordingly, the maximum value of a is amax = (p  U)/2, which is achieved when b = 0 and deformation is simple shearing concentrated on the intersection plane. On the other side, a = 0 corresponds to the largest fan angle possible, bmax = pU. Consider a material point progressing through the die in time t. The angular position of a material point in the  in Fig. 24. Further, bðtÞ  is defined such that at t = t1, when the material fan at time t is denoted by bðtÞ  2 Þ ¼ b. When it enters the fan at  1 Þ ¼ 0; and at t = t2, when it leaves the fan, bðt point enters the fan, bðt

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Fig. 24. The parameters of the ideal fan model for the plastic deformation zone.

time t1, it experiences simple shearing along a plane inclined at a from the x-axis. As it flows through the fan, it continues to experience simple shearing but along planes which are rotating about O in time  and in sync with bðtÞ. When it exits the fan, it experiences its final simple shear deformation along a plane inclined at a + b from the x-axis. (Prior to entering the fan, t < t1, one may choose to model the deformation as uniaxial compression [5] or as no deformation at all.) Because shear fan deformation begins at t = t1, we define a shifted time T = t  t1, and then, to remove any unnecessary dependencies on parameters, normalize it by a time constant tn = (d  x0)/V, where V is the pressing velocity, d is the width of the channel, and x0 is the location of the flow line as indicated in Fig. 24. The angular position  0 Þ is related to this normalized time T’ = (t  t1)/tn via bðT  0 Þ ¼ T 0 cos2 a. Following the prediction bðT made by Segal’s slip field analysis [11], the shape of the flow line within the fan is taken to be circular. For the purposes of polycrystal modeling, the velocity gradient corresponding to the deformation imposed by a shear fan with angle b as described above was determined in Beyerlein and Tomé [28]. The velocity gradient for any die angle U is:

2

 sin 2ðaþbÞ

2 6 L ¼ t n L ¼ 6 2  cos2 a 4 sin ða þ bÞ 0

 0  cos2 ða þ bÞ 

 sin 2ð2aþbÞ 0

3

7 pUb 7 : 05 a ¼ 2

ð17Þ

0

Here the actual velocity gradient L is presented in normalized form  L. The normalization of L is advantageous because it factors out those parameters that are not significant for texture calculation and it reveals that the velocity gradient for an ideal shear fan with constant b only depends on b. As long as b is constant throughout the fan, the deformation field generated by this ideal fan is homogeneous. b can be calculated analytically [11,236] or left as an unknown to be estimated by comparisons with measurement or FE simulation. The latter approach will be demonstrated in the next section. Earlier material deformation in the fan was described as simple shearing on continually rotating planes and L in Eq. (17) expresses the same deformation but in a fixed x–y–z coordinate system, as

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is the case with all forms of velocity gradients used and presented in this article, which is much more  ¼ 0 for all time, and the model recovers the simple convenient for texture modeling. When b = 0, bðtÞ shear model. The velocity gradient in Eq. (17) reduces to the velocity gradient in Eq. (7) for simple shearing along the intersection plane in the x–y–z coordinate system. In this case, L does not change in time. However, because the amount of shear experienced by the material is large within this plane, it is common practice to increment the shear strain in small steps. When b > 0, the components of L  0 Þ is incre 0 Þ and Eq. (17) should be used instead of Eq. (7). In this case, bðT vary continuously with bðT 0  mented starting from 0 to b. In each increment in bðT Þ, L is calculated according to Eq. (17), applied to the polycrystal model, and the texture updated. Like in the case of ideal simple shearing (b = 0), the resolved shear strain that is suddenly experienced by the material point is large at the fan boundary  0 Þ ¼ 0 [28] and as before, it needs to be accumulated in small steps. A similar procedure needs to be bðT  0 Þ ¼ b as well. Between the boundaries, however, this is not performed at the exit boundary at bðT  0 Þ is incremented in small steps. needed; the change in shear will be small as long as bðT As an alternative to the simple shearing description above, one can envision the same deformation  to proceed as follows based on Eq. (17): as material flows through the fan in time or bðtÞ, there is a compression component in the y-direction which decays and a tensile component in the x-direction which increases. There are also non-zero shear components that change sign when the material crosses the intersection plane. This type of evolution has been observed in FE calculations [22,74].  0 Þ, the components of the L decrease in distance from the inner corner. In other words, For fixed bðT the intensity of the deformation, such as the strain rate, decreases for material points further away from the inner corner. The rate of evolution of each L component and the maximum value achieved during deformation are the highest for simple shearing (b = 0) and decrease with increasing b. The distortion of an element (e.g., cube or sphere) after passing through a shear fan will depend on b. The inclination angle of a distorted cube and distorted sphere after one pass will be:

hr ¼ tan1 ð1=KÞ and

ð18Þ

! N2  ðK2 þ 1Þ ; K

1

hs ¼ tan

ð19Þ

where K = 2tana + b/cos2a. According to Eqs. (18) and (19), hr and hs are not very sensitive to b; therefore, they would not be good features for characterizing b. As b increases from zero (simple shearing) to bmax, these angles increase slightly. Iwahashi et al. [237] associated the shear c accumulated in a rounded corner die (W > 0) with the tangent of the final inclination of a cube element in the exit die.



c ¼ 2 cot



U 2

þ

W 2



  U W þ Wcosec þ ¼ tanðhr Þ: 2 2

ð20Þ

The proof is this equation is straightforward: simply equate b to W in the expression for the inclination hr of a distorted grid element Eq. (18) and take the tangent to get Eq. (20). It must be emphasized, however, that generally b and W are not equal. To represent the effect of a rounded corner on deformation for texture modeling, c in Eq. (20) has been used to represent the amount of simple shear concentrated along the intersection plane when W > 0. This method serves, at best, as a rough approximation. In a texture calculation, it only captures the reduction in accumulated strain with increasing W, but not the concomitant change in deformation mode. When the outer corner is rounded, i.e., W > 0, the deformation mode becomes more complex and simple shear does not take place on a single plane. The fan model has some extensions not reviewed here. Skewing of the fan due to strain hardening, as observed in Fig. 23 has been treated in Mahesh et al. [87]. A two-part PDZ with an upper shear fan and lower primarily rotating region was modeled in [28]. The velocity gradient associated with the lower deformation region produced texture predictions that compared favorably with FE calculations and texture measurements in [22].

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Fig. 25. (a) Examples of flow lines in the flow line model of Tóth et al. [74] and (b) comparison of the strain rate components with FE results for n = 6.

A continuous description of the deformation along a flow line is possible with the analytical flowline model of Tóth et al. [26,74] for a 90° die and of Arruffat-Massion et al. [30] for a 120° die. This approach expresses L along a flow line that is defined by its x0 distance from the inner corner (see Fig. 25 for x0). For a 90° sharp corner die, the flow line function is:

/ ¼ ðd  xÞn þ ðd  yÞn ¼ ðd  x0 Þn :

ð21Þ

Note that Eq. (21) is simply the trajectory of the material element. A velocity field is first obtained from Eq. (21) by the following partial derivation:

vx ¼ k

@/ @/ ; v y ¼ k ; @y @x

ð22Þ

where the k(x,y) function is defined from the boundary conditions known for the entering velocity (the same velocity v0 at any point x0). Using Eqs. (21) and (22), L is readily obtained from the velocity field:

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@v x ¼ v 0 ð1  nÞðd  xÞn1 ðd  yÞn1 ðd  x0 Þ12n ; @x @v x Lxy ¼ ¼ v 0 ð1  nÞðd  xÞn ðd  yÞn2 ðd  x0 Þ12n ; @y @v y Lyx ¼ ¼ v 0 ð1  nÞðd  yÞn ðd  xÞn2 ðd  x0 Þ12n ; Lyy ¼ Lxx : @x

483

Lxx ¼

ð23Þ

The curvature of the flow line is governed by the parameter n, which equals 2 for circular flow lines and 1 for the simple shear model. Reasonable n-values can be deduced from FE simulations; see Fig. 25 for examples of some flow lines. Values of n < 1 lead to deviations of the texture components from their expected symmetry positions in agreement with experiments [26,74]. The shape-factor n is found to vary within the die; it depends on the position of the flow line, which is given by x0. Linear dependencies were identified in [89,234]. This dependence of n on x0 was neglected in the derivations defined in Eqs. (22) and (23). However, when it is taken into account, the resulting velocity gradient differs little from (23) [238]. The flow line approach was recently generalized for any die angle, including asymmetric flow [205,238]. 5.7. Texture calculations with analytical flow models When ECAE deformation is non-ideal (i.e. when the deformation mode is not simple shear), polycrystal texture predictions using analytical flow models lead to improvements over the simple shear model [22,73,74,87,88] and predictions comparable to FE [22,65,73]. The latter point is significant because compared to numerical models, analytical formulations are clearly more computationally efficient for polycrystal calculations of ECAE deformation under non-ideal conditions. Whether deformation can be described by simple shearing (b = 0) or spread over a fan-shaped zone PDZ (b > 0), first-pass textures of a variety of cubic materials can be related to classical shear components. Agreement improves over the simple shear model because these models shift the texture components from the ideal positions, proportional to the size of the PDZ (i.e., b) or to the amount that the deformation deviates from ideal simple shearing. Analytical flow models described in the previous Section 5.6 have a so called PDZ ‘profile parameter’ which controls the shift from the ideal ECAE texture component positions (e.g., either b in the ideal fan model or n in the flow line model). Figs. 26 and 27 show the one-pass ECAE textures of an fcc metal in the form of (1 1 1) pole figures in the TD view for both models assuming different b and n. Fig. 26 directly compares the two models and presents the values of b and n where they provide similar results. In the TD-view, an increase in b or decrease in n translates to a larger CCW rotation (or ‘tilt’) about the TD. Note that the strengths of the textures only depend slightly on b or n. Fig. 27 further considers U = 90°, 120°, and 135°, and a relatively large b = 45° (p/4). As expected, the intensities are generally lower as U increases. The textures are tilted Dh (about the 3-axis or z-axis) with respect to one another primarily due to the change in U, i.e., the Dh = DU/2. Accordingly, the U = 90° and 120° textures and 120° and 135° are expected to be tilted 15° and 7.5° from one another. Some guidance is needed for determining practical values of b that represent the deformation over most of the billet cross section. In the case of uniform friction in a sharp corner die, Segal [11] provides an analytical expression for b, i.e., b = p  acos(s/k)  U, where s is the friction stress and k is the yield stress (for a rigid plastic material). For general processing conditions, however, b is unknown. Fortunately, the positions in the ideal ECAE texture components are found to be sensitive to b. If this prediction is accurate, then it suggests that b can be characterized directly from comparisons with texture measurements [22,73]. An example can be found in [22] where comparisons of polycrystal model predictions using the fan model with neutron diffraction measurements, OIM-measurements, and FE predictions for the same ECAE Cu independently estimated the same b value of p/10. Generally estimates for b obtained through comparisons with bulk texture alone should generally be regarded as effective values. These values will largely be influenced by U and W. They may be smaller than the actual size of the PDZ since the low deformation intensities in the outer edges of the PDZ will have little impact on texture evolution. The effective b value reflects only the portion of the PDZ which influences texture. Last, spatially resolved texture measurements or refined FE simulations may suggest slight dispersions in b.

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β=0

β = π/10

β = π/6

Fan model:

Flow line model:

Fig. 26. (1 1 1) Pole figures demonstrating the rotations from ideal positions (b = 0 or n = 1) predicted by the ideal fan model [28] and flow line model [26] for an fcc metal after the first ECAE pass for a 90° die. Axis x is horizontal and directed to the right and axis y is vertical. The broken line indicates the direction normal to the shear plane. Isolines for fan model: 1.0/1.4/2.0/2.8/ 4.0 mr and for flow line model: 0.8/1/1.3/1.6/2/2.5/3.2/4/5/6.4 mr.

Analytical flow models can be used in every pass in multi-pass ECAE processes as long as a broad PDZ persists in every pass. (Otherwise, the SS model with b = 0 suffices.) In [73], the fan model (with b = p/10, a = p/5 in Eqs. (15) and (17)) enabled texture predictions in good agreement with measurements on Cu and Al samples processed up to four passes of route Bc, using a rounded corner die. In the event that material work hardening changes with straining, the size and deformation characteristics of the PDZ will change from one pass to another. Accordingly, the profile parameter may need to be varied. For instance, b may be larger or less symmetric in the first pass than in the second and subsequent passes because of a decrease in work hardening after the first pass. This result was found to be the case in Cu processed by two passes of route C in a rounded corner die [87]. To predict the texture evolution in both passes in agreement with measurement, the shear fan was skewed in the first pass but left symmetric in the second to capture the effect of strain hardening in the first pass and no strain hardening in the second. (Incidentally, the ‘skewed’ fan did not greatly influence the predicted first-pass texture from those of a symmetric fan. Likewise, the first-pass textures calculated from FE-generated deformation histories in Fig. 23 assuming either (a) perfectly-plastic metal or (b) a strain-hardened were also similar [22].) Note that changes in material hardening, which result from changes in microstructure, mechanisms, and texture, can depend on route. Therefore, what values of the profile parameter used in subsequent passes of route Bc may not work for those in route C. 6. Extended texture calculations including grain-scale and subgrain-scale effects Some deficiencies in the standard forms of the FC and VPSC models can affect texture predictions, particularly under large strains and strain path changes. First, correlations between the deformation of a given grain and that of its neighbors are missing and hence grains of the same orientation but

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2

1

Φ = 90º die and β = π/4.

2

1

Φ = 120º die and β = π/4.

2

1

Φ = 135º die and β = π/4. Fig. 27. One-pass fcc textures predicted by the VPSC approach using the ideal fan model for a relatively large value of b as a function of the die angle U. Contour levels are 1/2/2.8/4.0 mr.

different local environments deform the same. Also, the evolution of individual grain shapes [27], but not of the grain size, is simulated. Last, these models assume that the grain is homogeneous, so that the lattice of the entire grain rotates under deformation. Therefore, effects of some microstructural features, from macroscopic shear bands down to microscopic twins or subgrains, are missing. The latter, subgrain-scale structures, are of particular interest. Their effects on measured textures were discussed in Sections 3.1.10, 3.1.11 and 3.2.5. Subgrain and twin boundaries affect texture evolution in a direct way, by reorienting a substantial volume fraction of the grains, or in an indirect way, by creating directional barriers for slip, e.g., resisting slip in planes normal to the twin interface and promoting slip in planes parallel to the twin interface. While these substructure elements do not deform independently, they do operate using different slip systems than their surroundings, which can lead to a further growth in misorientation during deformation. Standard PX models with homogeneous grains often predict faster texture evolution rates and intensities than measured, a discrepancy which is commonly attributed to neglect of grain–grain interactions and substructural evolution in their formulations.

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In this final section of the article, we first examine modeling attempts to address these lower scale microstructural features, which can affect texture. Next, as a special application, we consider texture evolution in route C, which is found to be particularly sensitive to behavior at all three scales. We follow with a discussion on the relationship between texture and substructural evolution. Finally texture and microstructural evolution together affect the anisotropy in the mechanical properties of ECAE processed metals and we conclude with a discussion on this important topic. 6.1. Grain–grain interactions In a polycrystalline metal, grains do not deform independently of their neighbors. Interactions between a grain and its local neighborhood can affect its distortion (stretch plus rotation), choice of slip systems, intragranular heterogeneities, and possibly its propensity to twin. Grains of similar orientation in an aggregate may, therefore, deform differently due to their location, resulting in spatial deformation heterogeneities. Furthermore, grain neighborhoods have the effect of constraining grain deformation, and hence, slowing down texture evolution. As described earlier, the polycrystal models FC-Taylor and VPSC treat the effect of neighboring grains in an approximate way. Some recent works have been performed to alleviate this assumption. To simulate grain–grain interactions in a polycrystalline setting, Tomé et al. [150] proposed a co-rotational (CR) scheme to force paired grains to co-rotate during deformation. In this scheme, grains with the same initial orientation but having different neighbors can follow different orientation paths during deformation, and as a consequence, texture evolution slows down. Some recent ECAE studies using VPSC with the CR scheme [65,71,76,227] have shown that the texture intensities are lowered to values consistent with measurement starting with first pass and up to multiple passes of route A, Bc, and C. The main components and texture features, however, did not change, only the dispersion of orientations about them. A more accurate way to model grain–grain interactions is via three-dimensional finite element models employing a crystal plasticity formulation, such as CPFE [239]. Only a few studies thus far have applied CPFE to simulate the ECAE process [221,240]. The authors in [221] showed that grain–grain interactions lowered texture intensities when multiple elements are assigned to one grain. For one pass, it was shown that the predictions were more accurate than FC-Taylor and similar to those of VPSC with the CR scheme. 6.2. Grain shape effects Grain shape, in principle, can affect texture evolution. Interactions with the homogeneous medium (representing an average neighborhood) and the stress-state within the grains will be affected by grain shape. Grain shape can influence texture by changing the number and distribution of active slip systems from that of a spherical grain. The grain shape effect could be important in SPD because for some processes and routes, the grains can deform to extreme shapes (see for example Ni in route A after three passes in Fig. 28 taken from [10]). Extreme grain shapes or neighbors with extreme shapes may affect a grain’s ability to reorient in deformation. With regard to numerical considerations, when grains become severely distorted they will not maintain relatively uniform internal stress states as assumed by polycrystal models, such as FC-Taylor and VPSC. To correct this, one model that can be employed in self-consistent schemes, which represent grains as ellipsoidal spherical inclusions, is grain splitting (Fig. 29 [27]). In this scheme, subdivision of the ellipsoidal grains is determined according to the length ratios of the long (L), medium (M) and short (S) axes of the ellipsoidal grain in comparison with a critical value R. As illustrated in Fig. 29, a grain is divided into two grains when L/S P R and M/S < R/2 or into four grains when L/S P R and M/S P R/2. The first case corresponds to a cigar-shaped grain and the second to a pancake-shaped grain. The crystallographic orientation immediately before and after the split remains the same. R = 5 was shown to be a reasonable choice [27]. Two reasons to invoke grain-splitting routine are to reduce the error introduced in calculating the stress-state in an extremely distorted inclusion and to best represent the morphology of the crystal elements (volumes of near homogeneous orientation) under high strains. It is not intended to model

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Fig. 28. Optical micrographs of Ni of the (a) original material and after (b) three passes of route A. Taken from Segal (1995) [10].

L L/2

S

(a) original grain, Vf

Vf/2 new grains Vf/4

M/2 M

(b) Fig. 29. Illustration of the grain splitting criteria for an (a) elongated, cigar shaped grain and a (b) flat, disk-like grain [27].

grain refinement by formation of subgrain dislocation structures and the evolution of misorientations between them. The grain-splitting routine was found useful in VPSC simulations of large strain deformation. To demonstrate the effect of splitting, VPSC calculations of large strain monotonic shearing are performed in an initially random textured polycrystalline fcc metal (1000 orientations) with initially spherical grains. In these simulations, each orientation is allowed to deform according to its individual orientation, shape, hardening, and interaction with a homogeneous medium. Therefore, each grain will reach R and split at different moments in deformation. Fig. 30 shows the effect of R ( = 2.5, 5, 20) on shear textures under positive s12 shear at shear strains c of 2, 4, and 6. When R = 2.5, there is virtually no

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2

2

2

1

1

1

(a) R = 2.5

2

2

2

1

1

1

(b) R = 5.0 2

2

1

2

1

1

(c) R = 20.0 Fig. 30. The effect of critical splitting aspect ratio R on texture evolution (displayed as (1 1 1) pole figures) in simple shear for c = 2, 4, and 6. Contour levels are 1.0/2.0/4.0/8.0/11.3/16.0 mr.

change in grain shape throughout straining, whereas when R = 20, grains are allowed to distort and align close to the shear direction, without causing error in calculation. As shown R affects texture very little at c = 2. Grain morphology effects become pronounced between c = 2 and 4 and are maintained after c = 4. For all values of R, the shear textures up to c = 2 contain A, B, and C shear components. As the grain aspect ratio is allowed to deform to larger values, the main components change from A to C with straining. As shown in Fig. 30, for R = 2.5, the A components strengthen and the C components weaken with straining. For R = 5, both A and C strengthen. Last, for R = 20, the C component dominates after large strain. Texture evolution in these cases R = 2.5 to R = 20 seem to cover the variety of shear textures found in most metals after pffiffiffilarge strain torsion. Ni containing high Co alloy content (Ni–60%Co) exhibits textures (at evM ¼ c= 3  4) resembling the R = 2.5 or R = 5.0 cases, whereas pure Ni exhibits textures resembling the R = 20 case [241]. (Fig. 27 clearly shows that pure Ni can deform to extreme shapes.) At high temperatures (537 K), both metals at evM = 2.1 (c = 3.63) developed the same texture as those predicted with R = 5 for c = 4. Severely sheared (evM  4–6) Cu and Al exhibited textures similar to the R = 5 and R = 20 cases, respectively [58]. However, under high temperatures, Cu under similar large strain torsion developed a texture reminiscent of the R = 20-like texture. It seems that, from these comparisons, high stacking fault energy materials, like Ni and Al, or metals under high temperatures, are more likely to allow grains to severely deform, than medium and low stacking fault energy materials, like Ni–60%Co and Cu, or metals under low temperatures. (This certainly makes sense as the former conditions promote dynamic recovery and ductility.) Some of the differences observed in their large strain shear textures, therefore, could be attributed to differences in grain shape. Accordingly,

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the critical aspect ratio R is a material-dependent parameter, as some materials are more likely to contain highly distorted grains than others after severe plastic deformation. For the ECAE process, a few studies have theoretically explored the impact of grain shape with pass number. The result that grain shape takes effect around accumulated strains of three to four (Fig. 30) was first shown in [27]. Signorelli et al. [227] applied a (decoupled) FE-VPSC model with this grain splitting routine, as well as with co-rotation (see Section 6.1), for the first three passes of route A (U = 120°). They found that the model predicts texture strengths that do not monotonically weaken or strengthen, and also that it does not grossly overestimate texture intensities. Noticeable impact and improvement in agreement between polycrystal calculations with grain splitting and texture measurements (neutron diffraction) was shown to occur after the fourth pass (route Bc) [65]. A sophisticated subdivision routine, which redefines both the shape and misorientations of the subgrains was introduced in [242,243] and applied to route C [242]. These texture results suggest that the grain shape is expected to affect deformation behavior after large strains. It is tempting to relate model grain shapes to subgrain morphologies rather than grain morphologies found in real metals; however, the correlations are not strong. Both Ni and Ni–60% Co exhibit elongated subgrains [241]. Al usually has more grains with a cellular substructure than Cu. Similarly, metals under high temperature torsion contain more grains with cellular substructure than those under room temperature torsion. Thus, the correlations are stronger with grain shape than subgrain shape. In summary, the grain-splitting criterion is necessary in large strain deformation to properly model the evolution of the grain shape and not necessarily the evolution of the subgrain shape, which is dictated by intragranular strain heterogeneities and dislocation kinetics. The analysis in this section shows that grain shape affects texture evolution after large strains in a manner that depends on how much the grains can deform under severe strains (material ductility). Further study is needed to better clarify the relationship between grain ductility, cross-slip, and substructure evolution. Two ways of grain subdivision are considered, see Fig. 29. For some materials, it may be more suitable to postpone splitting until an appreciable amount of strain has been imposed (R = 20) and then drop R thereafter (R = 5) in further straining. Also, the effect of initial grain shape was not addressed. The calculations in this section assumed that the grains were initially spherical (modeling a more or less equiaxed grain microstructure), but due to fabrication or pre-processing, the initial grain shapes could be elongated, oddly shaped, or heavily distorted. 6.3. Substructure models Subgrain evolution models can be divided into two categories: those that predict the evolution of misoriented subgrains but without possible hardening from subgrain boundaries and those that predict the hardening from substructure but not the misorientation. While for dislocation density based hardening laws misorientations could be calculated indirectly from the dislocation density, the reorientation of the lattice associated with the misorientation is not directly modeled. The few models in the first category are sequential lamination [244], deformation banding [87,245], and disclination boundary models [243]. They predict the formation of subgrain volume elements and the growth in the misorientations across their boundaries due to differences in slip activities in the neighboring elements. The first two treat the formation of bands and have been implemented into the FC-Taylor polycrystal model. In both, the primary bands are allowed to form secondary bands, which can, in turn, form tertiary bands (and so on). The third model, the disclination-based substructure evolution model by Nazarov et al. [243,246], predicts the subdivision of grains into smaller grains, which can subdivide further into even smaller grains, and so on. This model has been incorporated into VPSC. The orientation of the deformation bands, lamina or substructure boundaries in these models are found to be [245] (or assumed to be [242,243]) non-crystallographic. Experimentally, however, lamina boundaries in one-pass Cu and Al are observed to lie along {1 1 1} slip planes [107,247]. The last two models [87,242] predict more diffuse textures which are in better agreement than the standard polycrystal models after one pass. They also predict nearly shear-like textures after two passes of route C, unlike a standard polycrystal model which would predict near recovery of the initial texture. Full field numerical calculations using crystal plasticity finite element (CPFE) [240] suggest that the impact of substructure evolution on texture evolution could increase for

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larger N. In [240], each of the 100 grains was subdivided into 36 subgrains with the same initial orientation as the parent. The misorientations between the subgrains increased with pass number in route C. Extending the polycrystal based substructure evolution models [87,242–245] to larger N and to model grain refinement will require treating the interactions between previously generated and newly formed substructure from pass to pass. Models in the second category include single crystal hardening laws. Baik et al. [82] developed a multi-scale model using a dislocation density-based hardening model [229] incorporated into the full-constraints (FC) Taylor model. A deformation history calculated from a separate 2D FEM simulation (assuming an isotropic continuum hardening law), was then introduced to the FC-Taylor model. They compared and achieved good agreement with measured textures for routes A and C. However, because single crystal hardening was isotropic, we suspect that the good agreement is attributed to the more realistic deformation from the FE simulation (see Sections 5.1 and 5.3). It can be demonstrated that for either FC-Taylor or VPSC and slip-deforming fcc metals, the texture simulated with constant CRSS values are almost indistinguishable from that simulated with CRSS values8 that harden equally (isotropic hardening), regardless if they are active or not. In another work, [248] proposed a constitutive model for fcc metals which introduces directional anisotropy in singlecrystal hardening. The single crystal anisotropy in their model was considered to result from the buildup and disintegration of dislocation sheets on crystallographic planes, as observed experimentally [107], and the interaction of slip systems with these dislocation sheets.9 The model evolves latent hardening with strain according to the relative slip activities within a grain, with higher hardening rates for the more active systems [249]. Beyerlein et al. [90,249] and Li et al. [23] implemented this latent hardening model and found that the effect on texture evolution was most noticeable for route C (see Section 6.5) but otherwise, it was minor for the other routes. The model used for anisotropic single crystal hardening does have, however, a significant impact on the calculated post-ECAE deformation response. As will be discussed in Section 6.5.3, good models for anisotropic single crystal hardening and texture evolution are needed to predict the transients in hardening rate and plastic anisotropy observed in ECAE-processed fcc metals [3,4,248]. 6.4. Deformation twinning To account for deformation twinning, polycrystal models need to specify when and where they grow (nucleation), the rate of growth, how slip and twins interact, and the crystallographic reorientation within the twinned domains. The latter is required to capture the effect of twinning on texture evolution. The processes underlying twin nucleation, growth, and interactions with slip dislocations are not well understood, particularly for metals with non-cubic crystal structures. Very few SPD studies have attempted to model materials that deform by both slip and twinning [5,95]. In these studies, twin initiation and growth were treated like those for slip and slip–twin interactions were modeled via latent hardening. For twin reorientation, the predominant twin reorientation (PTR) scheme was employed. The PTR scheme [250,251] fully reorients the orientation of the grain to its predominant twin system (PTS) when the volume fraction associated with the activity of this twin exceeds a threshold value Vthres. In other words, a ‘twinned’ grain fully adopts the orientation of its predominant twin and remains homogeneous. The threshold volume fraction Vthres is related to the bulk aggregate properties as follows:

V thres ¼ A0 þ A1

V reor V shear

;

ð24Þ

8 ‘Isotropic’ hardening in slip implies that all slip systems harden equally regardless of their activity. The CRSS values for each system are equal in all strain increments. The same is also true for the non-hardening case. As a result, the activation of slip systems and thus the texture do not change between the two cases. 9 Dislocation sheets or walls were considered those dense dislocation structures which span the width of the grain and may contain one to several dislocation cells across their thickness.

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where Vshear is the volume fraction of twins in the aggregate, and Vreor is the volume fraction of twinned grains in the aggregate. Vshear is based on the contribution of shearing on all twin systems to the accumulated strain in the polycrystal. Initially Vreor = 0, and hence, Vthres = A0. A0 represents an ‘incubation’ volume that the PTS must accumulate in the grain before it is considered for reorientation. As twinning accumulates in the polycrystal, Vreor approaches Vshear, and Vthres approaches A0 + A1, the ‘saturation’ value for Vthres. Within this model, ‘re-twinning’ is allowed; a twinned grain can twin again, such as in the event of a strain path change when going from one pass to another. Within a previously twinned grain, a new PTS is determined and if the new PTS reaches Vthres, then the grain is reoriented to the orientation of the new PTS. Texture development in Ag was found to be distinct from another fcc metal, Cu (see Section 3.1.10). As described there, there were many plausible reasons for the texture differences: differences in SFE, initial texture effects (initially Ag had a random texture and Cu, a strong fiber texture), and deformation twinning. EBSD and transmission electron microscopy (TEM) confirmed the presence of twins in each pass [95]. To determine their role in texture evolution, polycrystal model simulations employing the PTR scheme for twinning were performed. It was confirmed that twinning caused the texture differences. Both microscopy and VPSC calculations showed that grains near the A1U orientation twin readily under simple shear, explaining why this component is consistently weak in Ag in all passes (Fig. 11). At the same time, they found that  U components are the least likely to twin, which explained why these compothe A2U and the BU =B nents are the strongest. This mechanism alone, however, does not explain why the material twins in each pass. The explanation lies in the strain path changes associated with route A and a U = 90° die. In the first pass, the A1U-oriented grains twin into the A2U orientation. The rotation of the billet between passes causes grains and twinned domains in the A2U orientation to be placed near the unstable A1U position prior to second-pass shearing. During the second pass, these preferentially re-positioned elements re-orient again into the A2U orientation. This process repeats itself in the third pass. This work strongly implies that route and U can influence twinning in each pass. It would be of interest to conduct similar studies using different routes and U. As discussed in Section 3.2.5 for hcp metals, deformation twinning is more prevalent, especially when they are processed at low temperatures and in suitably oriented crystals. Polycrystal modeling using VPSC and the PTR scheme was applied recently to pure Zr processed by a single ECAE pass at room temperature [5]. In [5], texture analysis (see Figs. 18 and 19) and model interpretations sug 2g tensile twinning occurred in the beginning of the extrusion, as well as during subgested that f1 0 1 sequent compression testing depending on the direction of loading. The onset of twinning could be linked directly to the orientation relationship between texture evolution and the imposed deformation field. More research studies on the effect of twinning on texture evolution, involving complementary polycrystalline modeling, is needed. As demonstrated in this section, one of the advantages of coupling texture modeling with measurement is to elucidate the mechanisms involved in deformation. More sophisticated models than discussed here have been developed for laminated twinned grains, twin nucleation and growth, and slip–twin interactions, e.g., see [252–255]. These models have been tested in small strain deformation. Determining if their formulations hold at larger strains requires more investigation. 6.5. Application: Modeling texture evolution in route C Predicting texture evolution in route C has proven to be quite challenging. Because of the reverse path change during two passes of route C, a straightforward application of most polycrystalline models predict near return of the initial texture. However, as reviewed in Sections 3.1.2–3.1.5,3.2.3, recovery of the initial texture after two consecutive passes of route C is rarely observed. A variety of materials, Cu, Al and Al alloys, Fe, IF-steel, Ti, Be, and Zr processed under various conditions produce this same result. These experimental observations in ECAE are not unlike the unaltered shear textures measured after reverse torsion tests in Fe, brass, and Cu, see [256] and references therein. In this section, the possible reasons why grain reorientation would not follow the reverse path back to its original position are discussed in, arguably, increasing order of importance. They involve

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phenomena at multiple length scales. Modeling efforts associated with different scales are presented. It will become apparent that modeling texture evolution in route C is an excellent example of how modeling serves as a powerful tool in testing hypotheses. Also, the reader will come to appreciate that inaccurate assumptions made in the model at any lengthscale can result in failure to predict texture evolution in route C observed in measurement. 6.5.1. Grain–grain interactions Interactions between neighboring grains can prevent individual grains from exactly retracting their deformation path in reverse shearing. However, this is a weak constraint, especially relative to the severe strain deformations imposed in each ECAE pass, and therefore, cannot be responsible for the shear texture components evident in second-pass route C textures. Polycrystal models which make some account for grain–grain interactions cannot reproduce the second-pass measurements [27,65]. 6.5.2. Inhomogeneous deformation Another reason is concerned with texture measurements made over large areas of the ECAE sample that encompass substantial texture gradients. Li et al. [65] demonstrated through FE/polycrystal modeling how texture inhomogeneities across the billet can alone contribute to shear-like textures after two passes of route C. They argued that the inhomogeneity that develops when the outer corner of the die is rounded would lead to shear textures in the top and bottom layers of the billet after two passes of route C. The central portion of the billet, in contrast, is likely to experience homogeneous deformation over the two-pass sequence, and only there, deformation is reversed and the texture recovers the initial one. They showed that, in this event, the ‘average’ texture across the entire thickness would be a shear-like texture. Later, OIM measurements were made on a relatively small area (in which the scan height was 1/20 of the thickness) in the center of the same two-pass Cu sample and displayed a relatively weak shear texture [23]. Therefore, even the material in the dead center of the billet did not recover the initial texture and formed a shear texture instead. In other words, in no part of the sample was the initial texture recovered. The results of this local OIM measurement are shown in Fig. 31 [23]. It is evident from the observed grain shapes that the deformation was reversed in going from the first to the second pass. The Cu material started with equiaxed grains, developed elongated grains after the first pass, and then recovered its nearly spherical grain shape after two passes of route C. A similar grain shape evolution is seen in Be (see Fig. 32 [88]), Ni [10] and reverse torsion of Cu [256]. 6.5.3. Grain hardening and substructure evolution The OIM results naturally lead one to look at the lower length scales to seek explanations for shear textures after even passes of route C. Comparing the microstructures of the original with that of the second pass in Fig. 31 one can see that within the grains there was an appreciable increase in dislocation density. Concomitantly, the resistance to slip (or so-called critical resolved shear stresse sV) must have increased during the two-pass sequence. Changes in hardening behavior at the single crystal level could have prevented the grain from rotating back to its original state when the applied strain was reversed. The most straightforward way to model the effects of accumulated dislocation density on texture evolution is to affect single crystal hardening behavior in the polycrystal model. To test this idea, we perform VPSC polycrystal simulations of two-pass route C and account for single crystal hardening with both material and geometric contributions. Material hardening of each fcc grain is made to follow an extended Voce law [257] below. According to this law, the critical resolved shear stress sV for each of the {1 1 1}h1 1 0i slip systems evolves with the total accumulated strain in the grain C according to

sV ðCÞ ¼ s0 þ ðs1 þ hs1 CÞ½1  expðh0 C=s1 Þ;

ð25Þ

where the parameters are the same for each slip system, s0, s1, h0, and h1 = 20, 80, 300, and 10. Geometric hardening evolves differently for each grain according to its crystallographic orientation. Randomly paired grains were made to have the same rotation in order to model grain–grain interactions (see CR scheme in Section 6.1) [27,76,150].

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Second Pass

First Pass

Initial

493

(a) First Pass

Second Pass

2

2

1

1

(b) Fig. 31. (a) EBSD/OIM images of the microstructure of pure Cu before and after the first and second passes of ECAE route C [23] (b) (1 1 1) pole figures of pure Cu after the first and second passes of route C as measured by EBSD/OIM. Contour levels are in integer units of mr starting with 1.

Fig. 33 shows the predicted texture evolution in route C assuming ideal simple shearing (U = 90°) for both ECAE extrusions. The initially random texture (1000 orientations) is nearly recovered, in spite of irreversible hardening at the single crystal level and use of the CR scheme for grain–grain interactions. The disagreement between the predictions in Fig. 33 and measurement may be attributed to the neglect of the formation of misorientations and boundaries associated with substructure evolution in the calculations. Evident in Fig. 31, subgrain boundaries formed after each pass. The distribution of misorientations across these subgrain boundaries was found to be wide [23,258] but, on average, misorientations increased from 3° after the first pass to 6–8° after the second [23] and fourth passes [81]). These smaller misoriented regions within the grain will use different slip systems in reverse deformation than the unfragmented parent grain used in forward deformation. Substructure development may, therefore, explain the development of shear textures after the reversal in second pass of route C. Models for substructure evolution that consider the development of regions (lamina, bands, subgrains) misoriented from the parent grain were developed in [87,242–245]. Implementing these subgrain models into either a Taylor [87] or VPSC [242] formulation generated second-pass route C textures different than the initial one with some shear components, although their intensity was still much weaker than the measurement. Substructure formation in these models diffused the texture about the main components, but was not sufficient to form all the shear texture components after a reversal. It is likely that substructure/subgrain development would be a more significant contributor to shear texture development in reversals if the strain levels per pass were much higher than 1.0, such that the amount of microstructural change per pass is greater. Such is the case in reverse torsion

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(a)

(b)

ND

ND

ND

ED

First Pass

Second Pass Route A

Second Pass Route C

Fig. 32. (a) Grain shapes before deformation and after two passes in route C in pure Be. (b) XRD measured basal pole figures of the texture after the first [116] and second pass [88] of routes A and C using a sharp corner U = 90° die. Contour levels are 1/2/ 2.83/4.0 mr.

experiments [256]: unaltered shear textures were observed when the strains imposed by the forward and reversal passes were each much larger than a true strain of 1 and not when they were less. Substructure evolution can also cause grain hardening to change from isotropic to anisotropic; meaning each slip system in the grain hardens at a different rate, depending on its relative activity. For the calculations in Fig. 33 the single crystal hardening was isotropic; that is, the CRSS for all slip systems harden at the same rate, whether they are active or not. Li et al. [23] tested the effect of anisotropic hardening by considering latent hardening. When the active planes were made to harden faster or slower than the inactive (latent) ones, texture development in the second pass was altered. Yet still, the shear textures were not as pronounced as those of the measurement. Although latent hardening between slip systems can inhibit perfectly reversed deformation, its effect is weak. (The effect of latent hardening may not be weak for other materials which exhibit plastic anisotropy because they deform by multiple deformation mechanisms, such as low SFE cubic metals or low symmetry metals that activate several slip and twinning modes. Similarly, texture evolution during large strain reversals will be different in metals that contain solutes, precipitates, or other inclusions that exhibit strong Bauschinger effects and other reversal-related transient behaviors [259].) 6.5.4. Imperfect reversal Thus far, no irreversibilities in the microstructure can fully explain the generation of nearly complete shear textures in the second pass of route C. Although they should not be disregarded, the theoretical textures resulting from these effects alone do not contain as many shear texture components as the measurements. The reason their effects have been minimal in the calculations is because the

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2

2

1

Simple shear model

First Pass

1

Second Pass, Route C

Fig. 33. VPSC simulated texture evolution in an fcc metal assuming ideal uniform deformation (far left) in both passes of route C for a 90° die. Contour levels are in integer units of mr starting with 1.

model for the applied deformation is decoupled from the single crystal and polycrystalline hardening behavior. In reality, microstructural evolution will impart a change in flow stress and work hardening, which, in turn, changes the nature of plastic flow throughout the extrusion. In the first pass, the metal work-hardens through stages II to IV, whereas throughout the second pass, its work-hardening rate is either mild and constant (stage IV) or has saturated. The first pass deformation characteristics will, therefore, be different than those of the second pass. Therefore, after the second pass of route C, shear textures develop because the second pass is not a pure reversal of the first pass. Consider, for instance, a sharp corner die (W = 0). When the material is strain hardening, a corner gap forms and a fan-like plastic deformation zone (PDZ) is generated as in Fig. 34a. When the material is perfectly-plastic, however, it fills the die, and the deformation is close to simple shearing [11] (Fig. 34b). For the same fcc material as in Fig. 33, Fig. 34c,d shows the corresponding VPSC calculations for the first two passes of route C for a sharp corner U = 90° die where in the first pass (Fig. 34a), the deformation is a symmetric PDZ (b = 18° in Eq. (17)) and in the second pass (Fig. 34b), it is ideal simple shearing. (The effect of the route C 180° rotation between passes is modeled but not shown.) Development of a second-pass shear texture is evident in Fig. 34c,d. For a rounded corner die (W > 0), it is less likely that a corner gap forms and a PDZ develops regardless of material behavior. For this geometry, as demonstrated by the FE simulations in Fig. 23, the PDZ for the strain hardening material is skewed slightly CCW (Fig. 35a), whereas for the perfectly-plastic material it is symmetric (Fig. 35b). In Fig. 35a we repeat the calculation for a rounded corner die where the first-pass deformation is an asymmetric PDZ skewed h = p/30 (b = 18°) due to strain hardening (Fig. 35a) and the second-pass deformation is a symmetric PDZ due to perfect-plastic behavior (Fig. 35b). As in the case of the sharp corner die (Fig. 34), a shear-texture is generated after the second pass. Mahesh et al. [87] first demonstrated that accounting for this skew effect (h = p/50) was necessary in order to predict texture evolution in agreement with OIM measurements made in the center of a two-pass route C Cu sample [23] processed using a rounded corner die. Well-developed shear texture components after two and four passes of route C are also observed in other crystal structures. Fig. 36 considers hcp Be processed using a rounded corner U = 120° die (Eq. (17), (b = p  2p/3  2a)). Be deforms primarily by basal slip, more so than prismatic or pyramidal hc + ai slip. A different set of Voce parameters (Eq. (25)) [88] are estimated for each of these three slip modes by fitting to mechanical test data under monotonic loading and 425 °C [260–262]. The parameters corresponding to the fit in Fig. 36a are shown in Table 5. When the deformation follows a symmetric PDZ (b = 10°) for both the first and second passes, the initial random texture (1000 orientations) is nearly recovered (Fig. 36b). In this case, the fan-deformation of the second pass is still a perfect reversal of the fan-deformation of the first (when the 180° twist is made between passes). However, when the PDZ of the first pass is slightly skewed (h = p/30) while that of the second pass remains symmetric, a second-pass shear texture develops (Fig. 36c). In this case, the deformation is

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α βm

α

(a)

(b)

2

2

1

(c)

1

(d)

Fig. 34. Hypothetical sequence of deformation in two passes of ECAE for a 90° die with a sharp outer corner angle. (a) In the first pass, deformation spreads out over a fan-shaped zone due to strain hardening and formation of a corner gap. (b) Second-pass deformation is close to ideal simple shearing because work hardening has saturated. (c) The corresponding VPSC simulated texture evolution in an fcc metal after the first pass and (d) second pass of route C. Contour levels are in integer units of mr starting with 1.

an imperfect reversal. Notably the first-pass texture agrees well with the measurement shown in Fig. 13a. The two-pass deformation sequences considered above can be used for any route, not just route C. Fig. 37 considers the same Be, but processed by a sharp corner U = 90° die [88] via routes A and C. The corresponding measurements are shown in Fig. 32b (and also in Fig. 13b). In this example, the two deformation sequences shown in Figs. 33 and 34 were considered. In the first sequence (perfect reversal), ideal simple shearing occurs in both the first and second passes Fig. 37a. In the second (imperfect reversal), an ideal fan develops in the first pass due to strain hardening and ideal simple shearing in the second once strain hardening has saturated. The corresponding texture predictions are shown in Fig. 37b for both routes A and C. As shown, with the imperfect reversal deformation sequence, the agreement with the measurement for route C in Fig. 13b improves markedly. We find that the grain shapes returned to nearly spherical after the two-pass route C sequence (not shown). For route A, on the other hand, texture evolution is less sensitive to the differences between the two deformation sequences. Notably the predictions using the imperfect reversal sequence capture well the location of the maxima in these two routes (see [88] for more discussion10).

10 In these simulations for Be, basal slip was the main deformation mode followed by prismatic and pyramidal hc + ai [88]. The basal slip activity increased in the second passes for both routes [88]. The relative activities and texture results in Figs. 36 and 37 changed slightly when latent hardening between the slip modes was raised from one to two.

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α

α+θ

β

β

α−θ

(a)

α

(b)

2

2

1

(c)

1

(d)

Fig. 35. Hypothetical sequence of deformation in the first two passes of ECAE for a 90° die with a rounded outer corner angle. (a) In the first pass, the deformation spreads out over a fan-shaped zone which is skewed counter-clockwise by an angle h due to strain hardening. (b) Second-pass deformation is close to a symmetric fan because work hardening in the material has saturated and the material fills the rounded outer corner. (c)–(d) The corresponding texture evolution after the (c) first pass and (d) second pass. Contour levels are in integer units of mr starting with 1.

In the above examples, shear textures develop in second pass route C due to the deformation imposed by the second pass; they are not retained shear textures from the first pass. Because of changes in material hardening, the largest difference in deformation characteristics is expected to occur between the first and second passes. As a result, the second-pass texture will be a shear texture. Because material hardening nearly saturates at large strains, differences between the second and subsequent passes are likely to be negligible. The deformation between even-number passes greater than two will, therefore, be nearly perfect reversals. The fourth-pass texture will recover this second-pass shear texture under a perfectly reversible deformation and so on for the higher N. Consequently, if the second pass is a shear texture, the subsequent even-numbered passes will likely be so as well. In summary, while route C, for the most part, imposes a reverse deformation sequence, it is not a perfect reversal. Even the slightest imperfection can lead to a shear texture after the reversal sequence. The calculations presented in this section considered both fcc and hcp metals, and rounded and sharp corner dies. In all cases, a slightly imperfect reversal deformation path was shown to lead to shear textures. Because the deformation is still by nature a reversal, albeit imperfect, the grain shapes nearly return to their original shape. Imperfect reversals are the result of changes in the microstructure, substructure, and single crystal hardening behavior from one to two passes. The relatively simple simulations conducted in these examples demonstrate the principle and motivate the need for more sophisticated multi-scale ECAE simulations, such as those that directly couple

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500

Compression Stress (MPa)

450

Through Thickness Compression Rolled Be

400 350

Hot Pressed Be No Initial Texture

300 250 200

DATA VPSC

150 100 -0.05

0.00

0.05

0.10

(a)

0.15

0.20

0.25

0.30

0.45

1

1

Second Pass, Rt. C

First Pass

2

2

1

(c)

0.40

2

2

(b)

0.35

Compression Strain

First Pass

1

Second Pass, Rt. C

Fig. 36. (a) Comparison of prediction using parameters in Table 5 with mechanical test data on pure Be at 425 °C [260–262]. (b) Texture evolution for the first two passes of route C for Be using a 120° ECAE die (with a rounded corner) assuming PDZ b = 10° in both passes or assuming (c) a skewed PDZ (few degrees) in the first pass and symmetric PDZ in the second pass, both with b = 10°. The first-pass textures in (b) and (c) compare well with that in Fig. 11(a). Only the second-pass texture in (c) agrees with the measurement. Contour levels are in integer units of mr starting with 1.

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Table 5 Hardening parameters of the Voce single crystal hardening law (Eq. (25)) used in the simulations displayed in Figs. 36 and 37. The latent hardening between all modes is 1.0. Deformation mode

s0 (MPa)

s1 (Mpa)

h0 (MPa)

h1 (MPa)

Basal slip Prismatic slip  2gh1 1 2  3i Pyramidalhc þ aiSlipf1 1 2

100 150 420

130 150 1100

3500 4500 1E+05

0 70 0

(a)

ND

ND

ND

ED

First pass (b)

ND

Second pass, Rt. A ND

Second pass, Rt. C ND

ED

First pass

Second pass, Rt. A

Second pass, Rt. C

Fig. 37. (a) Predicted texture evolution in Be for the first two passes of routes A and C (a) assuming the deformation sequence in Fig. 34 with PDZ b = 10° in the first pass and ideal simple shear in the second pass versus (b) assuming ideal uniform deformation (Fig. 33) in both passes. Predictions in (a) agree well with those in Fig. 30. Contour levels are 1/2/2.83/4.0 mr. Simulation results taking from [88].

deformation fields with anisotropic polycrystalline hardening behavior and those that account for grain refinement. 6.6. Sensitivity of texture to microstructural evolution (grain refinement) Observations in fcc materials after large N (four or larger) find that texture evolution is still dependent on processing route, but microstructure development is not. The dependencies of microstructural features on route have been slight. In pure Al, McNelley et al. [263] find that there is very little

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difference in the misorientation distribution and in the average grain size among routes. Huang et al. [264] find that Cu processed by route Bc and C have the same microstructure, with only slight differences in HABs (20% in route C vs. 30% in route Bc) and in the percentage of recrystallized grains, (0% C vs. 1.5% Bc) after eight passes. In IF steel [31,32], the evolution of cell size and laminar widths do not vary significantly with route, being only slightly smaller for route Bc, than A, Ba, or C. Therefore, at least for cubic metals that deform by slip only, the sensitivity of texture to microstructural evolution is minimal. As we have mentioned in Sections 6.3 and 6.5, results from extended modeling efforts to incorporate single crystal hardening have demonstrated that it has subtle effects on texture. In a polycrystal model, slip resistance in each grain is represented by the CRSS prescribed for each slip system. For a cubic metal that plastically deforms by slip only, details of CRSS evolution, which depend in microstructure, are found not to be important for texture evolution. In fact, texture calculations which have neglected the evolution of the CRSS still have been able to achieve reasonable agreement for Ni, Al, IF steel, and Cu [31,32,65,73,71,76]. The apparent insensitivity of texture evolution to details of CRSS evolution results from the assignment of equal CRSS or isotropic hardening for all slip families used. It can be demonstrated that texture predictions that use (i) equal and constant CRSS values (no hardening) and (ii) equal CRSS and isotropic hardening are indistinguishable. Single crystal models which assume, instead, anisotropic (or latent) hardening, on the other hand, are found to have some effect on texture development [23,90]. As further support, recent modeling work incorporating substructure [87,242–246] in fcc materials shows that substructure disperses orientations around the same preferred shear components, but otherwise does not alter first-pass textures significantly. Some of these studies also considered the first two passes of route C, for which they predict weak shear-like textures and increases in subgrain misorientations after the second pass [87,242]. These efforts, however, were performed for no more than two passes. Further work is needed to see if substructure evolution or grain refinement has any affect at higher N. The biggest issue faced by these models is treatment of the substructure rearrangement associated with the strain path change between passes. While it is important to incorporate microstructural evolution in ECAE simulations, the texture predictions will likely not be sensitive to many microstructural details. The predictability of a combined hardening evolution model and texture evolution model needs to be assessed by comparisons with other measurements such as plastic behavior and anisotropy in addition to texture. Examples will be presented in the next section. Last, when attempting to determine sources of certain texture features and differences across textures, it is best to first investigate the applied deformation, initial texture, and slip (or twinning) activities before attributing them to microstructural features, such as grain shape, grain size, subgrain morphologies (laminar vs. cellular), and submicron-thick bands or twins. 6.7. Influence of texture on post deformation of SPD materials Texture analyses of this chapter have concentrated both on the overall trends in texture evolution during SPD, the factors that affect texture evolution, detailed characterization, and texture model performance. Consequences of texture, specifically how texture can significantly impact mechanical properties, have not been addressed. The role of texture in elastic and plastic behaviors is, arguably, one of the primary motives researchers attempt to understand and examine material texture. A review of texture-property relations deserves a separate article of its own and is beyond the scope of this work. For completeness, we only briefly touch on this topic below. The most commonly studied properties of the SPD processed metal are yield strength, peak flow stress, hardness, ductility, work hardening, fatigue, superplasticity, and creep behavior. It is common to expect that microstructural features, such as grain size, dislocation density, and twinning can have a profound effect on these mechanical properties. Unfortunately, similar considerations on the effect of SPD textures on mechanical properties have been largely overlooked. As discussed below, some progress has been made in linking plastic behavior with texture in post-SPD metals, yet more studies are needed. Relating texture to the properties listed above certainly deserves study, particularly as scaling-up to industrial size samples, forming parts from SPD-processed metals, and using the final products in service, rely on knowledge of these properties.

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As a result of texture development, the relative proportion of ‘soft’ and ‘hard’ grains changes, and in this way, texture changes the mechanical properties. In a conventional metal, what makes a grain hard or soft depends on the availability and activation of the slip or twinning systems. Geometric effects like Schmid tensors and crystal structure and material effects like dislocation motion and substructure are some of the factors that govern this selection. In SPD-processed metals, however, some of the less common ones, such as individual grain history; grain neighborhood and misorientations; ultra-fine to nano grain sizes; extreme grain shapes; and excess dislocations, defects, or excess volume in the grain boundary, may play an increasingly important role in determining slip activity and hence, texture, especially after large strains and under strain path changes. Current predictive capability is a long way off from understanding the impact of these material factors. The microstructure and texture generated after the SPD process can introduce plastic anisotropy in an otherwise nearly isotropic starting metal. When the SPD metal is plastically anisotropic, its plastic behavior (flow stress, work hardening, ductility) can depend on both the direction of loading and sense of loading. The material may be strong or ductile in one direction, yet disappointingly weak or brittle in another. Only a few of studies to date have experimentally evaluated the plastic anisotropy of ECAE materials by testing them in several directions. They have shown that ECAE-processed Ti [265,266], Mg alloys [133,267], Al [3,268], Zr [5] and Cu [1,3,4,269] exhibit an anisotropic response. For ECAE-processed Cu the compression flow stress was higher in the TD than ND and both were higher than in the ED. This considerable anisotropy (TD > ND > ED) persisted up to 16 passes of route Bc and four passes of route C [3]. A specific case of anisotropy is tension–compression asymmetry. Along the ED, the material can be stronger in compression, yet weaker in tension as seen in Cu [1] or vice versa, as seen in Fe [2] and Cu [4]. Moreover in [4] it was found that the tension–compression asymmetry was different in the ND and TD than ED. This difference is related to the strain path change induced by each test direction when going from ECAE to post axial testing. It was shown that each test direction, in turn, promoted different dislocation rearrangements, processes, and substructure evolution. Compared to cubics, in hcp or in other low symmetry crystals, texture can have an even bigger impact on the post-deformation response. If the texture-favored mode (highest Schmid factor) is not necessarily the easiest (lowest CRSS), then its activation could lead to strengthening. Conversely, a large concentration of orientations favoring an easy slip mode under a given loading direction can lead to softening. Good examples of studies linking texture and mechanical properties have been performed for Mg alloys processed by ECAE. In two studies [123,267], two different Mg alloys (AZ61 and Mg–Li alloy) processed by routes A and Bc were studied. Not surprisingly after four passes, these two routes developed different textures. Although they both had axisymmetric initial textures, one had basal poles concentrated close to the entry axis (8.2 mr) [123] and the other a weaker one away from the billet axis (7 mr) [267]. Despite the difference in initial texture, both studies find that the route Bc sample has a lower yield stress (after four passes in [123] and eight passes in [267]), but a larger elongation to failure than route A. Both studies speculate that route A is stronger and less ductile because its texture promoted more non-basal activity, unlike route Bc which develops a texture which promotes mostly basal activity, the easiest slip mode. Liu et al. [123] show a strengthening in the UTS after four passes of both routes, while Kim et al. [267] observe a decrease after four (and eight passes) of route Bc. The authors in the latter attribute the decrease to the development of a texture that favorably oriented most of the grains for basal slip, which is the easiest slip mode in Mg alloys. This reduction occurs in spite of the refined grain size, emphasizing the importance of texture effects. In a few of these studies mechanical properties were predicted accounting for the texture development that took place either during ECAE or post-testing [3–5,90,248,269]. Assuming simple shearing, it was shown in [27] that some routes invoke geometric (texture) hardening and some geometric softening during the ECAE extrusion. Considering additionally material hardening based on a dislocation cell-hardening model, Toth pointed out [270] that the strain rate variations during the passage through a broad PDZ may even lead to some strain softening of the material when it leaves the PDZ. Most studies are, however, concerned with the effect of ECAE generated textures on post mechanical properties as these are possible to measure. In a series of works, the ECAE-induced plastic anisotropy in Cu and Al [3,4,248] was modeled using VPSC in combination with a single crystal

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hardening law that accounted for the development and rearrangement of substructure. The couplings between the grain orientation (texture), deformation history, anisotropic hardening, and slip activity accounted for in the model were necessary to effectively predict the post-deformation responses. The model was able to predict the compression plastic anisotropy in flow stress (TD > ND > ED) [248] and work hardening and tension–compression asymmetry in all three directions [4]. As discussed in Section 5.7, the size of the PDZ affects texture evolution and causes the textures to be tilted with respect to those formed under ideal simple shearing. With multi-scale modeling, it was shown in [90] that a change in the fan angle b used to describe the PDZ size in ECAE deformation resulted in changes in the mechanical anisotropy. Creep rates and texture evolution during tensile creep testing were found to be lower in ECAE processed Al than those in the as-received material [9]. The reduction is not expected because smaller grain sizes (1.3 lm in the ECAE material compared to 1 mm for the initial Al) usually lead to higher creep rates. Modeling suggested that the main deformation mechanism during the creep test was still slip and the texture formed after ECAE and direction of the creep test favored the slow texture evolution. The above examples consider fcc metals and so as a final example we consider an hcp metal, which prior to SPD processing is plastically anisotropic. In [5], pure Zr with two different starting textures was processed by one ECAE pass and subsequently tested in compression in three sample directions. The textures developed after the extrusion depended on the initial texture. This difference in the final ECAE texture leads to differences in the work hardening behavior and plastic anisotropy in the compression stress–strain response. During the 30% post straining, the texture was found to evolve from the ECAE texture depending on the direction of straining. With complementary VPSC polycrystal modeling, it was possible to infer the active slip systems. Not surprisingly, the relative activity of each deformation mechanism during ECAE and during each post-compression test changed depending on the initial texture prior to ECAE. A more subtle relationship between texture and flow stress that has yet to be explored is the effect of texture on subgrain structure evolution. Grain orientations that favor development of cellular substructures would pose barriers equally to all slip systems leading to a more isotropic response, whereas orientations favoring planar sheet-like structures would pose barriers only to some slip systems, yielding directional anisotropy. In summary, plastic behavior, particularly anisotropy, partly depends on texture evolution. It is, therefore, recommended that texture analyses be conducted under the notion that even the smallest of texture features could potentially affect mechanical performance. 7. Concluding remarks and recommendations Measuring, predicting, and controlling texture are important components in materials synthesis, processing and design. With the potential of SPD processes for fabricating new materials comes the opportunity to study texture evolution under large strains and strain path changes. Under such conditions, texture evolution is substantial and cannot be neglected. Texture will have a significant contribution to the anisotropy (directional dependence) in the mechanical and microstructure properties of the processed material. This article examines the progress made in the areas of measurement, characterization, analysis, and prediction, to date. While the reader may leave with the impression that many textures studies have been conducted in the past decade, understanding texture evolution in SPD is still in its infancy. In particular, texture evolution in low symmetry metals needs further study. Compared to cubic materials, modeling of ECAE texture evolution of hcp materials has been slim. Scientific challenges in modeling hcp texture evolution are found in modeling slip and twinning and their interactions. Modeling at several lengthscales is discussed with the emphasis on polycrystal modeling, the lengthscale at which texture is commonly measured. However, it is emphasized that the prediction of texture evolution will not only be influenced by the polycrystal model used at the mesoscale level, but also by the model used for the applied deformation at the macroscale and the models used for slip and/or twinning mechanisms at the microscale. For further improvement, substructure evolution and grain refinement models, which account for misorientations, are needed. The impact of texture on the material behavior of SPD-processed metals, such as anisotropy in plastic

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behavior, creep, fatigue, and fracture, is a largely untouched, but crucial issue. Last, there are many SPD processes, other than ECAE, being developed, and texture evolution during these operations has yet to be measured and understood. Acknowledgements IJB acknowledges with much gratitude collaborations with Dr. Carlos Tomé (Los Alamos), Dr. Ricardo Lebensohn (Los Alamos), and Dr. Saiyi Li (Guangzhou, China) that were very valuable in the development of this work. LST acknowledges intense and fruitful collaborations with Prof. Werner Skrotzki (Dresden), Ass. Prof. Satyam Suwas (Bangalore), Prof. Kenneth Neale (Sherbrooke, Canada), Dr. Roxane Arruffat-Massion (Metz), Dr. Benoit Beausir (Metz), Dr. Jean-Jacques Fundenberger (Metz), Mr. Arman Hasani (Metz) and Miss Mandana Arzaghi (Metz) that provided a great help in the development of this subject. References [1] Alexander DJ, Beyerlein IJ. Anisotropy in mechanical properties of high-purity copper processed by equal channel angular extrusion. Mater Sci Engng A 2005;410–411:480–4. [2] Han BQ, Lavernia EJ, Mohamed FA. Mechanical properties of iron processed by severe plastic deformation. Metall Mater Trans 2003;A34:71–83. [3] Beyerlein IJ, Alexander DJ, Tomé CN. Plastic anisotropy in aluminum and copper pre-strained by equal channel angular extrusion. J Mater Sci 2007;42:1733–50. [4] Yapici GG, Beyerlein IJ, Karaman I, Tomé CN. Tension–compression asymmetry in severely deformed pure copper. Acta Mater 2007;55:4603–13. [5] Yapici GG, Beyerlein IJ, Karaman I, Tomé CN, Vogel S, Liu C. Plastic anisotropy of pure zirconium after severe deformation at room temperature. Acta Mater, submitted for publication. [6] Li S, Beyerlein IJ, Bourke MAM. Texture formation during equal channel angular extrusion of fcc and bcc materials: comparison with simple shear. Mater Sci Engng 2005;A394:66–77. [7] Suwas S, Tóth LS, Fundenberger J-J, Eberhardt A, Skrotzki W. Evolution of crystallographic texture during equal channel angular extrusion of silver. Scr Mater 2003;49:1203–8. [8] Kaschner GC, Tomé CN, Beyerlein IJ, Vogel SC, Brown DW, McCabe RJ. Role of twinning in the hardening response of zirconium during temperature reloads. Acta Mater 2006;54:2887–96. [9] Kawasaki M, Beyerlein IJ, Vogel SC, Langdon TG. Characterization of creep textures in pure Al processed by equal-channel angular pressing. Acta Mater 2008;56:2307–17. [10] Segal VM. Materials processing by simple shear. Mater Sci Engng 1995;A197:157–64. [11] Segal VM. Equal channel angular extrusion: from macromechanics to structure formation. Mater Sci Engng 1999;A271:322–33. [12] Segal VM. Methods of stress–strain analysis in metalforming Minsk, Sc.D. thesis; 1974 [in Russian]. [13] Valiev RZ. Structure and mechanical properties of ultrafine-grained metals. Mater Sci Engng A 1997;234–236:59–66. [14] Valiev RZ, Gunderov DV, Zhilyaev AP, Popov AG, Pushin VG. Nanocrystallization induced by severe plastic deformation of amorphous alloys. J Metastable Nanocryst Mater 2004;22:21–6. [15] Valiev RZ, Langdon TG. Principles of equal-channel angular pressing as a processing tool for grain refinement. Prog Mater Sci 2006;51:881–981. [16] Segal VM, Reznikov VI, Drobyshevski AE, Kopylov VI. Plastic working of metals by simple shear. Russ Metall 1981;1:99–105. [17] Fang DR, Zhang ZF, Wu SD, Huang CX, Zhang H, Li JJ, et al. Effect of equal channel angular pressing on tensile properties and fracture modes of casting Al–Cu alloys. Mater Sci Engng 2006;A426:305–13. [18] Han WZ, Zhang ZF, Wu SD, Li SX, Wang YD. Anisotropic compressive properties of iron subjected to single-pass equalchannel angular pressing. Philos Mag Lett 2006;86:435–41. [19] Han WZ, Zhang ZF, Wu SD, Li SX. Nature of shear flow lines in equal-channel angular-pressed metals and alloys. Philos Mag Lett 2007;87:735–41. [20] Kamachi M, Furukawa M, Horita Z, Langdon TG. A model investigation of the shearing characteristics in equal-channel angular pressing. Mater Sci Engng 2003;A347:223–30. [21] Shan AD, Moon IG, Ko HS, Park JW. Direct observation of shear deformation during equal channel angular pressing of pure aluminium. Scr Mater 1999;41:353–7. [22] Beyerlein IJ, Li S, Necker CT, Alexander DJ, Tomé CN. Non-uniform microstructure and texture evolution during equal channel angular extrusion. Philos Mag 2005;85:1359–94. [23] Li S, Beyerlein IJ, Necker CT. On the development of microstructure and texture heterogeneity in ECAE via route C. Acta Mater 2006;54:1397–408. [24] Shin DH, Kim I, Kim J, Kim YS, Semiatin SL. Microstructure development during equal-channel angular pressing of titanium. Acta Mater 2003;51:983–96. [25] De Messemaeker J, Verlinden B, Van Humbeeck J. Texture of IF steel after equal channel angular pressing (ECAP). Acta Mater 2005;53(15):4245–57. [26] Tóth LS. Texture evolution in severe plastic deformation by equal channel angular extrusion. Adv Engng Mater 2003;5:308–16.

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