Utilizing a novel lattice orientation based stress characterization method to study stress fields of shear bands

Journal of the Mechanics and Physics of Solids 128 (2019) 105–116

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Utilizing a novel lattice orientation based stress characterization method to study stress fields of shear bands Darren C. Pagan a,∗, Armand J. Beaudoin a,b a b

Cornell High Energy Synchrotron Source, Cornell University, Ithaca, NY 14853, USA Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

a r t i c l e

i n f o

Article history: Received 17 August 2018 Revised 9 January 2019 Accepted 1 April 2019 Available online 1 April 2019

a b s t r a c t A new framework for evaluating stress fields from distributions of crystallographic orientation is presented. The framework employs the kinematics of crystal plasticity and field dislocation mechanics to connect measured lattice orientation, crystallographic slip, and geometrically necessary dislocations to recover the elastic deformation field present, and subsequently determine the stress using a finite-element scheme. As a demonstration of the framework’s utility, stress fields generated by the formation of shear bands in a copper single crystal are studied, including how these bands’ stress fields drive secondary slip. The results indicate that small amounts of secondary slip shielded stresses produced by the shear bands and that the bands were oriented in a low-energy configuration, stabilizing their structure. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction At the macroscopic scale, heterogeneity of plastic deformation is known to produce distributions of strain (eigen strains) and stress as a body seeks to maintain compatibility. These stress fields can lead to premature failure when superimposed upon stresses created by applied loading. Similarly, at the microscale stresses are produced due to heterogeneity (and localization) of crystallographic slip across a crystal. These stresses may subsequently influence the onset of yield, the direction of plastic deformation, and the formation of voids and microcracks. As these stresses are of significant importance, methods to quantify them using experimental data are of great use to failure prediction efforts. In this paper, we present a new method to use experimentally measured lattice misorientation data, which are increasingly available and can now be rapidly gathered from large, mm-sized 3-D volumes (even in-situ), to characterize stresses arising from slip heterogeneity. Rather than attempt to directly quantify elastic strains (as done by most methods), crystal plasticity and continuum dislocation kinematics are leveraged to recover ‘missing’ elastic strain information. With this information, stresses are then solved for using a finite element based linear elasticity and field dislocation mechanics formulation which has the ability to enforce known boundary conditions and static equilibrium across a body. The framework is used to analyze stresses associated with the formation of shear bands and provide insights into long-standing issues related to the driving forces of crystallographic slip within the shear bands.

∗

Corresponding author. E-mail address: [email protected] (D.C. Pagan).

https://doi.org/10.1016/j.jmps.2019.04.003 0022-5096/© 2019 Elsevier Ltd. All rights reserved.

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1.1. Characterization of microscale stress using X-rays The quantification of stress using diffraction-based methods generally relies on the measurement of elastic strains and then calculating local stress using a constitutive relationship (Hooke’s Law). Laboratory X-ray techniques have long been used to probe stress states at the surface of materials through the measurement of lattice strain (Noyan and Cohen, 2013), but new high-energy X-ray techniques performed at synchrotron light sources enable the probing of lattice strain fields with enhanced spatial resolution or deep within deforming samples. Laue X-ray techniques have achieved sub μm spatial resolution (Ice et al., 2005; Larson and Levine, 2013; Phan et al., 2016) and high-energy ‘scanning 3D-XRD’ techniques are able to penetrate much further into specimens and achieve excellent spatial resolution ( ≈ μm) (Hayashi et al., 2017). However, while these techniques are proving to be very valuable and continue to advance, in practice, the volumes that may be probed restrict study to mechanical gradients at scales much shorter than 1 mm. To attempt to probe larger volumes, other avenues for quantifying microscale stress fields using measured orientation or grain averaged elastic strains have been developed which solve for stress within full mechanics frameworks (Chatterjee et al., 2017; Demir et al., 2013; McNelis et al., 2013; Park et al., 2013; Pokharel and Lebensohn, 2017). Indeed, recent efforts to measure lattice strains to determine microscale stress fields are of considerable value, but the orientation of a crystal (or volume of a crystal) is generally more straight forward to determine that its full strain state: diffraction peak transients due to lattice rotations are generally much larger than shifts due to lattice strain, and only three independent pieces of information are needed to evaluate an orientation (as opposed to six for strain). Therefore, lower signal-to-noise ratios are necessary to probe lattice orientation fields and subsequently larger volumes can be probed quickly. As a consequence, experimental techniques to evaluate lattice orientation in large volumes at the sub-grain scale are becoming increasingly widespread. Specifically, monochromatic high-energy X-ray techniques used at synchrotron light sources are enabling researchers to non-destructively probe full 3-D microscale lattice orientation fields deep within deforming crystals over macroscopically relevant volumes of material (Jakobsen et al., 2006; Li et al., 2012; Lind et al., 2014; Ludwig et al., 2009; Suter et al., 2006). Similarly, tri-beam microscopy techniques can determine the orientation field of similarly sized volumes with increased accuracy (Echlin et al., 2012) in comparison to high-energy X-ray techniques (though the sample is destroyed in the process). Analysis techniques to further leverage these new data to understand stress fields will be of great value. At the core of the stress evaluation framework presented herein is the relationship between geometrically necessary dislocation (GND) density fields and elastic deformation used to evaluate the stress fields they generate. Procedures for evaluating a stress field from a GND distribution of dislocations have long been established (Eshelby, 1956), but rarely used. This is because direct quantification of geometrically necessary dislocations outside of small volumes using transmission electron microscopy is still not possible, so it generally must be calculated from an elastic deformation. Unfortunately, this points to the problem, the elastic deformation must already be characterized to quantify GND content, so calculating GND content becomes an unnecessary step for finding stress. Researchers have tried to get around this problem when evaluating GND dislocation content by neglecting elastic strains (Pantleon, 2008), minimizing strain energy (Kysar et al., 2010; Ruggles et al., 2016), or minimization of dislocation density on specific slip systems (Jiang et al., 2013; Littlewood et al., 2011). In this work, the missing elastic strain information necessary to evaluate GND content is recovered by first analyzing the plastic distortion undergone by a crystal. The plastic distortion is found by constraining the plastic deformation to be consistent with crystal plasticity kinematics and measured lattice orientation data. Recent work has demonstrated that orientation gradients can be used to determine the amounts of plastic shear strain on varying slip systems and gain information about the distributions of shear strains in the deforming crystals (Pagan and Miller, 2016). With this information, full GND content can be found to link the measurable orientation data to stress distributions. Thus, we develop a framework with which experimentally measured orientation fields are used to determine plastic distortion fields, and the stresses resulting as a consequence heterogeneous plastic distortion at the sub-grain scale. 1.2. Slip and the formation of shear bands in single crystals One of the most common forms of slip heterogeneity observed at the microscale is the localization of slip into bands (Kuhlmann-Wilsdorf, 1999; Kuhlmann-Wilsdorf et al., 1999), referred to as shear or deformation bands. The study of the ubiquitous formation of shear bands in single crystals of fcc metals and alloys under uniaxial loading, specifically when slip is primarily on a single system, has received long-standing attention (Cahn, 1951; Honeycombe, 1951). In these tests, the presence of small amounts of secondary slip is often detected and a great deal of research has gone into understanding how secondary slip may play a role in the stability of these shear bands. A review is provided by J. A. Wert and Inoko (2003), who points out the classic treatment of formation of a shear band by Mott (1951). In this model, tilt boundaries result from polygonalization, creating a shear band lying perpendicular to the plane of slip; the formation of dislocation pile-ups from the interactions of dislocation loops on the primary slip system are the foundation of the band localization. In later works, it was also shown that finely distributed secondary slip serves to relax stresses from primary edge dislocations (Basinski and Basinski, 2004). Generally, these studies focused on local interactions of dislocations; the actual driving forces responsible for generating the secondary slip received less attention. At the sample scale, it has been proposed that secondary slip may develop as a consequence of kinematic constraints imposed by gripping of the specimen (see (J. A. Wert and Inoko, 2003) and references therein). However, compression of a copper single crystal in an apparatus specially designed for unconstrained

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uniaxial compression (Florando et al., 2007) demonstrates that secondary slip can still occur free from boundary conditions and be quite dramatic (Magid et al., 2009). These results indicate that secondary slip does not arise solely from loading conditions, and may be a direct result of primary slip activity. While shear bands have been linked to material failure, systematic studies of the 3-D stress fields they produce have been limited due to a lack of experimental methods to quantify them. The framework presented will be used to explore driving forces for secondary slip as a shear band forms with its ability to characterize stress fields due to GNDs. The framework is used to evaluate stress fields produced by shear bands in a copper single crystal oriented for single slip that have been experimentally characterized using lattice orientation data from high-energy X-ray diffraction. The utility of the presented stress characterization will be demonstrated with analysis of stresses produced by shear bands. The next section will begin by summarizing previous results analyzing heterogeneous slip in a deforming copper single crystal that provide context and further motivation for stress analysis subsequently described. Next, the methodology with which fields of lattice orientation are used to determine a stress field in a deforming crystal is presented. Following this, distributions of plastic shear strain informed by lattice orientation measurements made in a deforming single crystal will be used to understand stresses produced by the formation of shear bands. Lastly, how the distributions of stress further influence the subsequent deformation of the crystal, specifically secondary slip, will be discussed. Bold characters will indicate vectors (a) and tensors (A). The vectors ex , ey , and ez will be reserved for basis vectors of rectangular Cartesian coordinate systems. Super-scripts will denote the coordinate systems in which vectors or tensors are expressed (aS ). Sub-scripts will generally identify specific scalars, vectors or tensors  (a0 , a0 , A0 ). The magnitude of a vector √ is ||a|| is defined as ai ai and the Froebenius norm of a tensor ||A|| is defined as Ai j Ai j . 2. Previous experimental results For convenience, previous results described in Pagan et al. (2018) will be summarized, as they serve as a starting point for this work and is referenced throughout the paper. A copper single crystal oriented for single slip ([123] crystallographic direction aligned with the loading direction) was deformed in uniaxial compression. The copper specimen had a 1 mm2 cross sectional area and a length of 2.4 mm in the loading direction ey S . The copper specimen was deformed to approximately 0.6% strain and the final applied load was -60MPa. The Schmid factors of different slip systems when loaded in the [123] direction along with the naming convention used in this paper (Basinski and Basinski, 1979) are given in Table 1 for reference. Using this convention, the slip system with the highest Schmid factor and that was primarily active is labeled B4. Diffraction measurements were performed in-situ as the crystal was compressed using a high-energy very far-field diffraction instrument at Sector-1ID-E at the Advanced Photon Source (Lienert et al., 2011). With this instrument, measurements were made of the evolution of the distribution of lattice plane orientation through a 250 μm tall volume that extended across the entire cross section of the specimen. During the test, two crossed shear bands formed and were tracked simultaneously to the diffraction measurements using digital image correlation on a single sample surface. These bands can be observed in Fig. 1A which shows the distribution of the 2-D equivalent strain on the sample surface. Using the combination of the diffraction measurements (probing orientation distribution evolution the volume in the sample interior) and the DIC measurements (providing the distribution of strain on a sample surface), the spatial distribution of slip throughout the entire specimen was reconstructed. Throughout this paper we will refer to a primary and secondary band that have been labeled in Fig. 1A. In the analysis of the data, it was determined that the distributions of lattice plane orientation (plotted on pole figures in Fig. 1B) that the formation of shear bands split distributions of lattice plane orientation and provided a diffraction ‘signature’ of shear band formation. Analysis of the directions that the distributions of lattice plane orientation

Table 1 Components of the slip directions ss and slip plane normals ns of an FCC crystal expressed in the crystal coordinate system. Also given are the Schmid factors (SF) of the twelve slip systems of an FCC single crystal specimen with a 123 crystallographic direction along the loading axis. ss B4 B5 B2 C1 C5 C3 D4 D1 D6 A3 A6 A2

√1 2 √1 2 1 √ 2 √1 2 √1 2 1 √ 2 √1 2 1 √ 2 √1 2 √1 2 1 √ 2 √1 2

ns

{101} {011} {11¯ 0} {110} {011} {101¯ } {101} {110} {011} {101¯ } {01¯ 1} {11¯ 0}

√1 3 √1 3 1 √ 3 √1 3 √1 3 1 √ 3 √1 3 1 √ 3 √1 3 √1 3 1 √ 3 √1 3

SF

{111¯ } {111¯ } {111¯ } {11¯ 1} {11¯ 1} {11¯ 1} {11¯ 1¯ } {11¯ 1¯ } {11¯ 1¯ } {111} {111} {111}

0.47 0.35 0.12 0.29 0.18 0.12 0 0 0 0.35 0.18 0.18

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Fig. 1. A) Distribution of effective strain on the surface of a copper single crystal deformed in uniaxial compression measured using digital image correlation. B) Distributions of lattice plane orientation from sets of lattice planes associated with the primary slip system B4: sB4 (4¯ 04¯ ), nB4 (3¯ 3¯ 3), and mB4 (24¯ 2¯ ). Red arrows plotted on the distributions of lattice planes indicate predicted directions that the distributions will spread during single slip.

spread on the (4¯ 04¯ ) and (3¯ 3¯ 3 ) pole figures (in addition to minimal spread of the (24¯ 2¯ ) pole figure) indicated that both of the shear bands were due primarily to slip on the B4 slip system. The expected pole figure extension directions used to make this conclusion are shown in Fig. 1B with red arrows and can be seen to be closely aligned with the data. With the slip mode established, the magnitude of plastic shear on the primary slip system that was necessary to spread the lattice plane distributions was then determined using a kinematic model. While it was found that the deformation was primarily accommodated by slip on the B4 slip system, minor amounts of secondary slip were detected in the distributions of lattice orientation. This secondary slip was believed to cause the small angular deviations between predicted spreading directions (red arrows) and the lattice orientation distributions on the (4¯ 04¯ ) and (3¯ 3¯ 3 ) pole figures in Fig. 1B. From analysis of the spreading directions on the pole figures, it was determined that the angular deviations were most likely due to slip activity on the B5 slip system. However, it was never determined why activation of B5 slip system was favored over the A3 slip system, even though these slip systems had the same Schmid factor (Table 1). The diffraction data and kinematic analysis provided insight into the deformation occurring inside the bulk of the crystal, but questions still remained about the underlying mechanics of the shear band formation (overlapping with previously identified issues regarding driving forces for secondary slip): 1. What are the effects of stresses (if any) produced by the localization of deformation into shear bands leading to deviations from homogeneous uniaxial deformation? 2. Why did the small observed amounts of secondary slip occur and which slip system was it associated with? A goal of this paper is to apply the presented stress characterization framework to answering these questions. 3. Framework for evaluating microscale stress from lattice orientation fields In this section we lay out a framework from which to find an incompatibility induced stress field from an orientation field with crystal plasticity and continuum dislocation kinematics. In the development, the displacement gradient ∇ u will be decomposed into elastic (UE ) and plastic (UP ) parts. The elastic and plastic parts are then further decomposed into two different decompositions. The first is an often-used decomposition into symmetric and skew parts

U_ = (U_ )SYM + (U_ )SKEW = ε_ + _

(1)

where the symmetric and skew parts are associated with strain ε and rotation  respectively (underscores can be replaced with E or P in Eq. 1 for decomposing elastic or plastic parts). The second is a less familiar decomposition into compatible and incompatible parts

U_ = U_C + U_I .

(2)

In the second decomposition, the compatible part is that which exists inside the null space of the curl operator

∇ × U_C = 0,

(3)

implying that the total curl is equal to the curl of the incompatible part

∇ × U_ = ∇ × U_I .

(4)

Also, it is important to remember that in this framework, stresses arise from defects left behind by an incompatible deformation; a corresponding elastic deformation must be present that simultaneously makes the total deformation compatible and produces a residual stress field that satisfies static equilibrium.

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3.1. Calculating a stress field from a misorientation distribution We employ a linearized decomposition of the deformation gradient F and displacement gradient contains both elastic and plastic parts

F = I + ∇ u = I + UE + UP .

(5)

We will refer to UE as the elastic deformation and UP as the plastic distortion. The instantiated plastic distortion that is inferred from measurement, is composed of a combination of shearing motions on different slip systems following most crystal plasticity formulations

U˜P =

SS 

γ˜s (ss  ns )

(6)

s=1

where ss is a slip direction, ns is a slip plane normal, and γ˜s is the total shear strain on a slip system, to be discussed below. The tilde denotes measured quantities used to instantiate the stress calculation. Importantly, only the elastic portion of the deformation, which stretches and rotates the crystal lattice, can be probed using diffraction techniques. In previous work, it has been demonstrated that at the early stages of plasticity during different types of constrained deformation (e.g. tension, compression, plane strain compression) (Pagan and Miller, 2016), the total rotation of the material ˜ E is equal and can be assumed to be negligible. If this is the case, the measurable skew portion of the elastic deformation  opposite to the skew portion of the instantiated plastic distortion

˜ E = − ˜P =− 

SS 

γ˜s (ss  ns )SKEW ,

(7)

s=1

˜ E to the plastic distortion that has occurred. We note that if the total rotation of material is known linking the measurable  (such as during a torsion test), other deformations can be analyzed. Using Eq. 7 and readily measured slip directions and slip plane normal orientations, plastic shear strains γ˜s that have occurred on different slip systems can be determined. From these plastic shear strains, a total plastic distortion U˜P can be reconstructed using Eq. 6. With the plastic distortion determined, the rest of the framework is concerned with determining the stress state. As mentioned previously, the stress arises from the incompatible portion of the plastic deformation which must now be determined. We take UP = U˜P and note that in order for the total deformation to be compatible, the sum of the incompatible portions of both the elastic deformation (UEI ) and plastic distortion (UPI ) must be equal to 0

UEI + UPI = 0.

(8)

From the continuum theory of dislocations, it is found that the GND tensor α is equal to the curl of the incompatible portion of the elastic deformation

∇ × UEI = α.

(9)

Through use of Eq. 8 and noting that by definition the curl of the incompatible portion of the plastic distortion and total plastic distortion are equivalent (Eq. 4), we find that:

α = −∇ × UPI = −∇ × UP .

(10)

Using Eq. 10, the GND tensor is evaluated across the body. This still leaves the incompatible portion of the plastic distortion unknown. Eq. 10 is reformulated into a least squares residual that can be solved using the finite element method

1 1 (α − ∇ × UPI )2 + (∇ · UPI )2 = 0. 2 2

(11)

The first term in 11 ensures that the portions inside the curl of UP and UPI are equivalent while the second term ensures that UPI contains no compatible portions. Eq. 11 can be solved using the least squares finite element method (LSQFEM) described in more detail in Roy and Acharya (2005). The final portion of the procedure is to determine the loads generated by the incompatible portion of the plastic deformation and then solve for a complete stress field σ across the body that satisfies both static equilibrium and the appropriate boundary conditions. This procedure closely follows those outlined by Eshelby (1956) and Acharya (2001); Roy and Acharya (2005). From compatibility (Eq. 8), we determine that the incompatible plastic deformation UPI (determined using LSQFEM, Eq. 11) is equal and opposite to the incompatible elastic deformation UEI

UPI = −UEI .

(12)

We then calculate a field of body forces fI generated by the incompatible elastic deformation using Hooke’s Law:

fI = ∇ · (C : εEI )

(13)

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where εEI is the symmetric portion of the incompatible elastic deformation and C is the fourth-order elastic stiffness tensor. In this formulation, equilibrium is given as:

∇ · σ = − fI .

(14)

We then solve for a compatible strain field εEC using Hooke’s Law such that equilibrium

∇ · (C : εEC ) = − fI .

(15)

and boundary conditions are satisfied. Eq. 15 is solved using a standard linear finite element formulation that finds a compatible strain field by construction. From the starting point of an orientation field, assumptions about the kinematics and constraints of the deformation are used to instantiate a total plastic distortion U˜P . This plastic distortion is then decomposed into compatible and incompatible portions using Eq. 11. The incompatible portions of the elastic deformation and plastic distortion are assumed to be equivalent so that the total deformation is compatible. Lastly, the compatible portion of the elastic deformation is determined using linear elastic finite elements with stresses arising from the incompatible portion of the deformation serving as body forces. The interpretation of this procedure is that incompatibilities arising from heterogeneous plastic deformation are counteracted by an equal and opposite incompatible elastic deformation that in turn produces a stress field throughout the body. A secondary compatible portion of the elastic deformation is necessary for the body to maintain equilibrium. Following intuition, the compatible portion of the plastic deformation plays no role in the stress field in the body. 3.2. Stress relaxation The framework in section 3.1 only employs Hooke’s Law as a constitutive relationship and stresses can be calculated that are aphysical and far outside the yield surface. Evaluating an evolution problem is necessary to relax stresses arising from the instantiated GND density indicated by Pagan and Miller (2014); Pagan et al. (2018) and determine valid stress fields. The mesoscale field dislocation mechanics approach detailed in Acharya et al. (2008); Acharya and Roy (2006); Roy and Acharya (2006), as implemented in Varadhan (2007); Varadhan et al. (2006), is applied to carry out the viscoplastic relaxation. Of relevance to the present effort, the evolution problem provides insight into the origins of secondary slip. The treatment of stress relaxation builds upon the procedure described in 3.1, while finding the solution for a relaxed incompatible part of the plastic distortion, UPI , follows Eqs. 10 and 11. However, the relaxation problem differs from the previous section in that a compatible plastic deformation must be found to relax the stress field. Solving for the compatible plastic distortion, UPC involves evaluating an auxiliary problem for the gradient of a vector field z such that

    ∇ · z˙ = ∇ · U˙ PC = ∇ · (α × vα + LP )

(16)

where vα is the velocity of the local GND content and LP is the plastic portion of the velocity gradient. The initial condition of UPC follows from the solution of Eq. 15 (Acharya, 2001; Roy and Acharya, 2005). The first term on the right of Eq. 16 α × vα , describes the transport of the GND content and the second term LP describes shearing motions due to statistically stored dislocations (SSDs) and is defined as:

LP =

SS 

γ˙ s (ss  ns ).

(17)

s=1

The shearing rates on different slip systems γ˙ s are

γ˙ s = ρm b vs

(18)

where ρ m is the fixed mobile dislocation density and b is the Burgers vector. The velocity vs follows a power law relation

vs = v0

 τ  τ  m1 −1 s  s   τ τ

(19)

where τ s is the resolved shear stress on a slip system ((ss  ns ): σ ), and τ is an isotropic slip system strength strength. The constant v0 is taken such that a reference strain rate γ˙ 0 = ρm b v0 is unity. The slip system strength τ has an isothermal component τ 0 and a contribution from a population of statistically stored forest dislocations

 τ = τ0 + a b C44 ρ f

(20)

where a is a fit parameter. The hardening rate of the strength τ is controlled by the evolution of forest SSD density ρ f described as

ρ˙ f =

c0   α · ns b s

 ˙ +

c  1 b

 . ρ f − c2 ρ f  .

(21)

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The parameters c0 , c1 and c2 are selected so as to match the measured work hardening response. The plastic strain rate, ˙ takes into account both the motion of the GND and mobile SSD densities and is defined as

˙ = ||α × vα || +



|γ˙ s |.

(22)

s

The transport of GND density α with velocity vα is

α˙ = −∇ × (α × vα + LP ).

(23)

Following Acharya and Roy (2006), the GND velocity field vα acts in a direction d which is guided by the local stresses acting on the GND field. The GND velocity field renders positive dissipation and pressure independence,

vα = v

d

||d|| 

d= f−

f·

n ||n||

i jk σ jr αrk ni ≡ i jk α jk

 n ||n||

fi ≡

(24)

where σ is the deviatoric stress, f are forces acting on the GND distribution due the deviatoric stress, n is the active slip plane normal, and the velocity of the GND density follows from the slip system activity, as

v=

SS 

|vs |.

(25)

s=1

A critical feature of these evolution equations is the coupling between the parts of the plastic deformation due to SSDs and GNDs. The plastic velocity gradient LP affects the transport of the GND content α˙ and in turn α can produce a stress field that affects the motion of SSDs. This coupling can give rise to slip modes that differ from what would be predicted by standard crystal plasticity formulations. 4. Results To evaluate the stress field produced by the localized slip in the shear bands, a virtual specimen with distributions of slip informed by both the optical DIC measurements of the surface strains and the distribution of misorientation present within the crystal (see Fig. 1) was generated. The stress evaluation method described in section 3 was then used to evaluate the stress field produced by the distributions of heterogeneous slip in the specimen. 4.1. Stress fields due to slip bands in a deforming single crystal As described in section 2, as the copper single crystal was compressed along the [123] crystallographic direction, two crossed shear bands formed in the sample. These two shear bands (labeled as primary and secondary in Fig. 1A) were attributed to slip primarily on a single slip system, B4, through analysis of distributions of lattice orientation (misorientation) that developed. To analyze the distribution of stress in the crystal, the full distribution of slip (and related plastic distortion) was instantiated. To generate shear bands consistent with the data, an analytic function was used to simulate a slip gradient along a direction p. The direction p is in the plane created by the slip direction sB4 and the slip plane normal nB4 and can have a variable orientation in the plane by adjusting the angle θ between p and nB4 . For each shear band, the distribution of slip along the direction pˆ on the B4 slip system is given by:

  ⎧ −( p+t/2 )2 , for p ≤ −t/2 ⎪ ⎨γM exp 2l 2 γB4 = γM ,   for − t/2 < p < t/2 ⎪ ⎩γ exp −( p−t/2)2 , for p ≥ t/2 M 2l 2

(26)

where p is coordinate along pˆ, γ M is the maximum shear strain within the crystal, t controls the thickness of the shear band, and l controls the length of the gradient region. This form of the slip distribution produces slab shaped shear bands within the crystal. By adjusting the variable band parameters, the physical extent of a shear band can be adjusted. This shear band model was first introduced and validated in Pagan et al. (2018), but the ability to control the angle of the slip band with respect to the slip plane normal using the variable θ is new. Importantly, for a fixed maximum slip γ M , reducing l will shrink the gradient region separating inside and outside the band and generally increase the dislocation content in the crystal as the dislocation content is directly related to the magnitude of the derivatives of the slip distribution. We note that Eq. 26 produces a distribution of slip satisfying continuity requirements necessary for a valid distribution of α.

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D.C. Pagan and A.J. Beaudoin / Journal of the Mechanics and Physics of Solids 128 (2019) 105–116 Table 2 Parameters describing the instantiated shear bands in the simulated copper crystal.

γM

Band

−2

–1.250×10 –0.625×10−2

Primary Secondary

t (mm)

l (mm)

θ (◦ )

0.1 0.3

0.025 0.025

1 89

Fig. 2. A) Instantiated distributions of slip on the primary slip system B4 associated with the two experimentally observed slip bands. B) Corresponding distribution of the only non-zero component of the dislocation density tensor α xz expressed in a coordinate system associated with the primary slip system. C) Non-zero regions of the absolute value α xz showing the locations of GND content associated with the two slip bands. Table 3 Single crystal elastic moduli used for calculating stress in GPa (Kelly and Knowles, 2012). C11

C12

C44

164

122

75

A set of shear band parameters were selected to match the spatial distributions of slip observed using DIC in Fig. 1A, and the magnitude of lattice misorientation observed on the pole figures in Fig. 1B. These parameters are listed in Table 2 and the distribution of slip on the primary slip system in the crystal can be seen in Fig. 2. From the instantiated distribution of slip and Eq. 6, the plastic distortion U˜P is evaluated across the crystal, and next the GND density tensor α can be determined using Eq. 10. The distribution of α generated by the shear bands in the simulated crystal are shown in Fig. 2B. In a coordinate B4 B4 system associated with the primary slip system (eB4 x parallel to sB4 , ey parallel to nB4 , and ez parallel to mB4 ), the only non-zero component of α is the xz component. Two sharp slabs of GND content with opposite sign associated with the secondary slip band are prominent in Fig. 2B. Notably, the dislocation content of the primary band is less evident and is nearly an order of magnitude smaller – even though the amount of slip in the primary band is significantly higher (γ M is larger) and the gradient region length term l is smaller, both being expected to produced elevated dislocation content. To B4 is taken and the dislocation content was thresholded at a aid the viewing of the primary band, the absolute value of αxz −8 −1 value of 10 nm . In Fig. 2C, the distribution of GND content throughout the crystal is more clearly visible. The two shear bands produce four slabs that cross near the center of the crystal. Next, to evaluate the stress distribution in the virtual specimen created using the above sample instantiation procedure, values of U˜P were calculated at nodal points of a 128,0 0 0 element mesh (dimensions 40 × 80 × 40 along ex S , ey S , and ez S ). Using the procedure outlined in section 3 the incompatible portion of the elastic deformation UEI , serving to balance incompatibilities generated during the slip process, was calculated using the least squares finite element method. The elastic incompatibility multiplied by the single crystal elastic moduli for pure copper are applied as a body force in the linear elasticity formulation (Eq. 15). The single crystal copper moduli used are listed in Table 3 (Kelly and Knowles, 2012). The results from evaluating the resulting stress field in the virtual specimen with traction free boundary conditions are shown in Fig. 3. Fig. 3A presents the distribution of normal stress along the loading direction generated by the shear S ), while Fig. 3B presents the distribution of von Mises stress (σ bands (σyy vM ). In the specimen in Fig. 3A, alternating bands of tensile and compressive stress can be seen to correlate with the bounds of the shear bands and the regions of high dislocation density (Fig. 2). These normal stresses are on the order of the applied load during the experiment (30 MPa). The highest normal stress along the loading direction exists at the intersection of the shear bands and the sample edges. Tensile stresses generated by the dislocations can serve as ‘back stresses’ which would lower regions of the crystal off the yield surface. The effects of the compressive stresses will be further explored in the discussion. In addition, the intersection of the two shear bands in the center of the specimen produce a sharp stress gradient as evidenced by the relatively high von Mises stress near the center of the sample.

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Fig. 3. A) Distribution of stress along the loading direction σ yy produced by the heterogeneity of slip in the two slip bands. B) Distribution of von Mises stress σ vm produced by the heterogeneity of slip in the two slip bands.

Fig. 4. A) Distribution of stress along the loading direction σ yy produced by the heterogeneity of slip in the two slip bands after application of a -60 MPa traction along the loading direction and relaxation of the dislocation structure. B) Distribution of the GND tensor component α xz expressed in a coordinate system associated with the primary slip system after application of the external traction and subsequent relaxation. C) Distribution of the non-zero GND tensor component α xx that developed during the relaxation of the dislocation structure. Table 4 Plasticity parameters used during the stress relaxation process. b (nm) 0.256

a 0.333

τ 0 (MPa) 25.0

c0 3.0

c1 0.00256

c2 1.86

m 0.025

ρ m (m−2 ) 10

13

v0 (ms−1 ) 3.91 × 10−4

4.2. Stress distributions under applied load with relaxation The stress distributions caused by slip heterogeneity presented in the previous section act in tandem to the externally applied load. To better reflect the stress distributions during loading, a -60 MPa traction was applied to the top and bottom surfaces of the virtual specimen. The dislocation content and the stresses in the specimen were allowed to relax following the evolution equations in section 3.2. Fig. 4A shows the distribution of stress in the specimen after the application of the external traction and following relaxation of stress. The materials parameters used in the evolution process are listed in Table 4 and adapted from parameters found in Pagan (2016). As expected, the stress field generated by the GND content causes the stress in the crystal to deviate significantly from a constant uniaxial stress. As mentioned previously, the tensile stresses serve as a back stress and move regions of the crystal off of the yield surface. The regions of elevated compressive stress however are significantly lower than what would be expected from simply adding the applied uniaxial and dislocation-generated stress fields ( ≈-70 MPa after relaxation versus ≈-90 MPa expected). As described, part of the relaxation process includes the production and evolution of the GND content in the crystal. B4 after relaxation. In the figure, there is a slight spreading of the dislocation content in comparison to Fig. 4B, shows αxz the instantiated GND content (Fig. 2B), but these differences are minimal, indicating that the instantiated GND structure B4 in the primary slip system was stable. However, during the relaxation process, a new non-zero GND tensor component, αxx coordinate system, developed that is shown in Fig. 4C. The increase in this dislocation density tensor component corresponds to either an increase in the number of screw dislocations on the primary slip system or edge dislocations on the B5 slip system that share the same slip plane as B4 and have a relatively high Schmid factor. The attribution of this GND component

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to these dislocation types is done by projecting the GND tensor on to these slip modes and finding that both are non-zero: α · (sB4  sB4 ) for screw dislocations on B4 and α · (mB5  sB5 ) for edge dislocation on B5 where m is the dislocation line direction. Importantly, these secondary dislocation arise from the transport of α˙ in Eq. 22 and the stresses generated by the incompatibilities of slip on the primary slip system. 5. Discussion In this paper, we have presented a new framework for determining spatial distributions of stress from lattice misorientation data. The analysis differs from more typical stress measurements using diffraction processes (whether by electrons, X-rays, or neutrons) which rely on measuring elastic strains from changes in lattice plane spacing and calculating stresses directly using Hooke’s Law. Instead, lattice misorientation data is analyzed using crystal plasticity and continuum dislocation kinematics to calculate slip system activity and build the plastic distortion U˜P . From this analysis, GND content α is calculated, and the incompatibilities of the plastic distortion field are used to generate stress. The methods for solving for a stress distribution from a given distribution of GNDs are not new Eshelby (1956), but application of these methods has been hindered by limited capabilities to attain full characterization of the GND content. The current work closes this gap by connecting previously developed kinematic analyses linking misorientation to slip heterogeneity (Pagan, 2016; Pagan et al., 2018) with a finite element solution scheme that can calculate stresses from GND content and relax these solutions back to the yield surface with field dislocation mechanics (Acharya and Roy, 2006; Roy and Acharya, 2005, 2006). This advance enables new insights to be gleaned about stress distributions in plastically deforming and deformed materials, in addition to providing new means of virtual sample instantiation for finite element modeling. Early in this work, two open questions were posed regarding previously reported experimental results from a plastically deforming copper single crystal including: (1) effects of stress produced by slip heterogeneity in the shear band and (2) causes of small amounts of detected secondary slip. The results presented appear to provide answers to these questions and will be discussed shortly in section 5.1. We believe it is worth highlighting that the proposed framework can be used to characterize residual stress. Upon unloading the crystal, the stress distribution produced by the underlying dislocation content is the residual stress distribution in the specimen. A wide variety of X-ray techniques have been developed that are capable of providing 3-D distributions of lattice orientation during in-situ tensile deformation with μm scale resolution. The framework that has been presented and developed can be readily translated to these techniques, creating a new class of analyses for the 3-D characterization of microscale residual stress fields. 5.1. Shear band stress distributions and band stability The stresses produced by the shear band and the appearance of secondary slip seem to be closely intertwined and aid stabilization of the shear band, preventing its expansion in the sample. In section 4, it was shown that the intersection of the shear bands with each other and the bounds of the sample were found to produce stress concentrations and also that the shear bands form alternating tensile and compressive stresses in regions of high gradients of slip and high dislocation content (Fig. 3). Interestingly, the distributions of shear used to produce the bands are symmetric about p = 0 (Eq. 26) while the resulting dislocation content and stress distributions are anti-symmetric p = 0 (see Fig. 2B). This change of symmetry is related to the curl relationship between UP and α reversing the sign of the GND content at the opposing sides of the shear band and also the stress. The asymmetry of the stress around the bands influenced the subsequent relaxation within the virtual specimen after the traction was applied. The tensile stresses at one face of the shear band reduced the total compressive stresses in these regions and moved these volumes of crystal off of the yield surface, stabilizing one bound of the shear band preventing its expansion. The response of the compressive region of the shear band was more complicated under applied load. Combining the compressive applied stress to the stress generated by the shear band was expected to greatly accelerate slip on the primary slip system. However, instead during the application of the traction, small amounts of dislocation content associated with secondary slip on the same slip plane as the primary slip system naturally arose (B5 system) and no ‘run-away’ slip activity was observed. The stresses generated by the heterogeneity of slip on the B4 system within the shear band appears to drive secondary slip on the B5 system, explaining the previous observations of B5 slip in the original experiment (Section 2). In turn the dislocation content created by B5 slip activity ‘shielded’ the compressive stresses produced by slip gradient on the B4 system and significantly lowered the total stress in the specimen. From this observation, we infer that secondary slip can serve to stabilize the bounds of the shear band. This hypothesis is consistent with other researchers that have contended that small amounts of secondary slip are necessary to stabilize shear bands (Basinski and Basinski, 2004; Brown, 2012). We also note B4 (see Figs. 2B and 4B), pointing to the stability of that during the relaxation process, there was minimal evolution of αxz the shear band structure. In total, both the stress distributions and secondary slip develop in manners that allow slip to concentrate within the bounds of the shear band, serving to maintain the geometric structure. A question related to the stability of the shear bands is why do they appear nearly parallel and perpendicular to the primary slip plane? The reason remains unclear from the simulations that were conducted on the virtual specimen. However, the framework presented provides the opportunity to further test different shear band configurations with the hope that

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Fig. 5. A) Example distribution of slip only on a single slip band. The angle θ is the angle between the gradient direction of the band p and the active slip plane normal. B) Maximum von Mises stress in the crystal as a function of θ .

simulating these configurations may explain why they are not observed experimentally. Applying this logic, a series of virtual crystals were generated in which the primary band was generated with fixed width t, gradient length l, and maximum shear stress γ M while θ , the angle between the shear band gradient direction p and nB4 was varied between 0◦ and 90◦ . Element counts in the virtual crystals matched those of the previous specimen (128,0 0 0). Fig. 5A shows the slip distribution on the primary slip system in an example virtual crystal. The stress distributions were calculated in each of the virtual crystals and the maximum von Mises stresses were gathered and plotted in Fig. 5B. In Fig. 5B, we see that stress minimums exist near 0◦ and 90◦ . The maximum stresses rapidly increase away from these two values. This observation implies that forming a shear band away from ≈ 0◦ and ≈ 90◦ (like those observed experimentally) is extremely energetically unfavorable. We note that a gradient at θ = 90◦ is a configuration similar to a low angle tilt boundary which has long been established to be low strain energy configurations (Hull and Bacon, 2001). Recent dislocation dynamics simulations modeling the evolution of dislocation configurations have produced similar results with the emergence of perpendicular edge dislocation configurations during simple shearing (Ispánovity et al., 2017). These simulations appear to help explain why the observed shear bands are stable: bands nearly parallel and perpendicular to the active slip plane are low energy configurations that provide deformation modes enabling single slip to accommodate constrained uniaxial compression. 6. Summary A new framework for calculating stress fields from lattice misorientation data measure using X-ray diffraction has been presented. The framework was used to explore how slip localizing into shear bands can influence the progress of plastic deformation in a deforming crystal. We found that shear bands formed at angles approximately 0◦ and 90◦ from the primary slip plane correspond to low strain energy configurations, and secondary slip at the bounds of the shear band promotes stability of the GND structure. The framework presented can be applied to many state-of-the-art lattice orientation characterization techniques to gather new insights about stress distributions produces by heterogeneous slip, including the development of residual stress. Acknowledgments DCP is supported by the Cornell High Energy Synchrotron Source (CHESS) which is supported by the National Science Foundation and the National Institutes of Health, National Institute of General Medical Sciences under NSF Award No. DMR1332208. AJB is supported by the Office of Naval Research In-Sitμ program (Contract N0 0 014-16-1-3126). We would like to thank Professor Amit Acharya for helpful discussions of this work. References Acharya, A., 2001. A model of crystal plasticity based on the theory of continuously distributed dislocations. J. Mech. Phys. Solids 49 (4), 761–784. doi:10. 1016/S0 022-5096(0 0)0 0 060-0. Acharya, A., Beaudoin, A., Miller, R., 2008. New perspectives in plasticity theory: dislocation nucleation, waves, and partial continuity of plastic strain rate. Math. Mech. Solids 13, 292–315. Acharya, A., Roy, A., 2006. Size effects and idealized dislocation microstructure at small scales: predictions of a phenomenological model of mesoscopic field dislocation mechanics: part i. J. Mech. Phys. Solids 54 (8), 1687–1710. Basinski, S., Basinski, Z., 1979. Chapter 16 Plastic Deformation and Work Hardening. In: Dislocations in Solids, 13. Elsevier, pp. 499–601. Basinski, Z.S., Basinski, S.J., 2004. Quantitative determination of secondary slip in copper single crystals deformed in tension. Philos. Mag. 84, 213–251. Brown, L.M., 2012. Constant intermittent flow of dislocations: central problems in plasticity. Mater. Sci. Technol. 28 (11), 1209–1232. doi:10.1179/ 174328412X13409726212768. Cahn, R., 1951. Slip and polygonization in aluminum. J. Inst. Metals 79, 129–158. Chatterjee, K., Ko, J., Weiss, J., Philipp, H., Becker, J., Purohit, P., Gruner, S., Beaudoin, A., 2017. Study of residual stresses in ti-7al using theory and experiments. J. Mech. Phys. Solids 109, 95–116. doi:10.1016/j.jmps.2017.08.008. Demir, E., Park, J.-S., Miller, M., Dawson, P., 2013. A computational framework for evaluating residual stress distributions from diffraction-based lattice strain data. Comput. Methods Appl. Mech. Eng. 265, 120–135. doi:10.1016/j.cma.2013.06.002.

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