Modeling transients in the mechanical response of copper due to strain path changes

International Journal of Plasticity 23 (2007) 640–664 www.elsevier.com/locate/ijplas

Modeling transients in the mechanical response of copper due to strain path changes Irene J. Beyerlein

a,*

, Carlos N. Tome´

b

a

b

Theoretical Division, Los Alamos National Laboratory, MS B216, Los Alamos, NM 87545, USA Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Received 28 May 2006 Available online 13 November 2006

Abstract When copper is deformed to large strains its texture and microstructure change drastically, leading to plastic anisotropy and extended transients when it is reloaded along a different strain path. For predicting these transients, we develop a constitutive model for polycrystalline metals that incorporates texture and grain microstructure. The directional anisotropy in the single crystals is considered to be induced by variable latent hardening associated with cross-slip, cut-through of planar dislocation walls, and dislocation-based reversal mechanisms. These effects are introduced in a crystallographic hardening model which is, in turn, implemented into a polycrystal model. This approach successfully explains the flow response of OFHC Cu pre-loaded in tension (compression) and reloaded in tension (compression), and the response of OFHC Cu severely strained in shear by equal channel angular extrusion and subsequently compressed in each of the three orthogonal directions. This new theoretical framework applies to arbitrary strain path changes, and is fully anisotropic.  2006 Elsevier Ltd. All rights reserved. Keywords: A. Strengthening mechanisms; B. Anisotropic material; B. Polycrystalline material; B. Constitutive behavior; Bauschinger effect

*

Corresponding author. Tel.: +1 505 665 2231; fax: +1 505 665 5926. E-mail address: [email protected] (I.J. Beyerlein).

0749-6419/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2006.08.001

I.J. Beyerlein, C.N. Tome´ / International Journal of Plasticity 23 (2007) 640–664

641

1. Introduction When a polycrystalline metal is reloaded after having been deformed plastically, it usually exhibits a plastic anisotropic response. Here ‘plastic anisotropy’ means that the yield stress and hardening evolution depend on the strain mode and direction imposed by the reload. Such anisotropy, which has its basis on texture and microstructure, depends much on prior straining history. With small pre-strains, anisotropy is mostly due to grain microstructure (mainly dislocation cells with very low misorientations) affecting slip activity, as the texture has not yet developed. Upon reloading, the material quickly evolves towards what would be its response in the absence of pre-loading. In contrast, with large prestrains, texture and subgrain microstructures change significantly. They both have a profound impact on anisotropy through their effects on the slip activity within the grains. The crystallographic orientation of the grain can affect the morphology of the substructure (Huang et al., 2001; Mahesh et al., 2004; Xue et al., in press) and substructures can affect directional hardening, slip activity and in turn, grain re-orientation. Changes in strain path can be broadly classified as reversal, pseudo-continuous, or cross. Reversal reloads initially exhibit a so-called Bauschinger effect, a transient drop in yield stress from that attained at the end of pre-straining rf. Further reverse straining may lead to an extended transient region with a hardening behavior different than the monotonic load curve. A continuous or quasi-continuous reload, on the other hand, will start at about the same yield stress rf and exhibit little to no change in hardening. A cross reload leads to an initial spike in the flow stress higher than rf, followed by work-softening and, only later, hardening. When transients of work-softening or zero hardening last over long periods of straining, they can lead to premature localization. Explanations for these effects based solely on crystallographic or morphological texture in the polycrystal are insufficient. Modeling the plastic response after strain path changes, particularly after large pre-strains, requires describing the concurrent evolution of texture and also dislocation structures inside the grains. The key feature of the hardening model that we develop here is the way it is related to slip activity. Our formulation measures the directionality and magnitude of shear in each slip plane of each grain, and couples previous shear activity with new shear activity induced by changes in strain path. The final product is a crystallographically based hardening model for the anisotropic constitutive response of an fcc polycrystal under general strain path changes. 1.1. Stress versus strength evolution Many hardening models which have proven successful in predicting monotonic loading in cubic materials invoke a constitutive law for the strain rate in the form of a power law of the ratio of stress to strength. (e.g., Kocks and Mecking, 2003; Kok et al., 2002; Cheong and Busso, 2004; Hart, 1976; Estrin et al., 1998; Toth et al., 2002; Tome´ et al., 1984). The models differ though, in whether a stress, or a strength term is evolved. Evolving the stress term better reflects the physical microscopic situation, where dislocations only experience the combined stress fields of other dislocations and the applied stress. Some models emphasize internal stress evolution as the sole driver for plasticity (Hart, 1976; ?), while others adjust the applied stress (Cheong and Busso, 2004). Regardless, when the internal stresses that oppose dislocation motion increase, higher applied

642

I.J. Beyerlein, C.N. Tome´ / International Journal of Plasticity 23 (2007) 640–664

stresses are required to move the dislocations, leading to macroscopic strain hardening. Relatively more fundamental and detailed descriptions of dislocation interactions are the subject of dislocation dynamic models (Devincre et al., 2006; Wang et al., in press; Roos et al., 2001; Ghoniem et al., 2000). Although DD alone would be impractical for directly simulating the hardening of polycrystals from zero up to large strains, it can be used to improve strain-hardening laws. As an alternative, another empirical approach for describing strain hardening evolves the ‘strength’ term (also called critical resolved shear stress, slip resistance, hardness, threshold stress). This strength represents an evolving resistance to slip, associated with dislocation multiplication, tangles, clustering, and domain boundary formation. Although these two approaches are conceptually different, hardening in both is related to an increasing difficulty or ease for dislocation motion as straining proceeds. Within the models, internal stress or strength evolution is linked to strain either empirically (e.g., Hart, 1976; Tome´ et al., 1984; Follansbee and Kocks, 1988; Teodosiu, 1996; Garmestani et al., 2001) or related to dislocation-density evolution (Kocks and Mecking, 2003; Toth et al., 2002; Cheong and Busso, 2004; Kok et al., 2002; Peeters et al., 2000, 2001). Implementation is done either at the continuum or grain-scale level, usually relying on effective scalar variables for representing stress and strain rate. 1.2. Modeling transients in flow response Although the approaches reviewed above often neglect the fact that strength and internal stress development are directional, good predictions of monotonic behavior or exact reversals are still achieved. When the strain path changes arbitrarily the slip resistance and internal stresses both change as the microstructure rearranges to reflect the new shear conditions, and accounting for crystallography and directionality is crucial for modeling strain-path-change responses. For this case, hardening models originally developed for monotonic responses need to be modified. Continuum-level models of this type either introduce kinematic hardening or an anisotropic yield function (e.g., Miller et al., 1999; Yoon et al., 2005; Barlat et al., 2005; Banabic et al., 2005), or a set of evolution laws for strengths (e.g., Christodoulou et al., 1986; Rauch, 1992), or a set of evolution laws for both internal stresses and strength (Garmestani et al., 2001; Teodosiu, 1996). Crystallographic models implement modifications at the microscopic-level (grain or subgrain), an approach which holds more promise for modeling transients in flow response associated with arbitrary strain path changes. Modifications generally involve changing single crystal hardening from isotropic to anisotropic to reflect subgrain microstructural heterogeneities or local slip activity. When combined with a polycrystalline plasticity model, the slip activity in each grain is then determined by the crystallographic and morphological orientation of the grain, its current anisotropic strength, interactions with surrounding grains (depending on the model), and the applied stress. Under a macro-strain path change, the local change in slip activity is calculated to be different in different grains. Some grains in the polycrystal can experience a reversal while others, a cross path change, which is unknown a priori. Approaches that combine anisotropic single crystal hardening model with polycrystalline modeling enable texture and subgrain microstructural hetereogeneities to contribute to the mechanical response (e.g., Barlat et al., 2003). Without this important micro–meso scale connection, neither texture evolution nor the macroscopic transients associated with arbitrary strain path changes (e.g., a rolled sheet being tested in tension

I.J. Beyerlein, C.N. Tome´ / International Journal of Plasticity 23 (2007) 640–664

643

or shear along an arbitrary direction, Wilson et al., 1990; Mahesh et al., 2004; Wu et al., 2005) can be predicted. 1.3. Single crystal anisotropic hardening Anisotropic hardening of the slip resistance (‘strength’) within a single crystal is typically associated with dislocation patterning or special interactions between different slip systems (Kocks, 1964; Bassani and Wu, 1991; Devincre et al., 2006). Theoretically it has been introduced by means of latent hardening models, Hall–Petch hardening terms, or dislocation-density evolution laws. Below we restrict our discussion to their applications in modeling flow responses after strain path changes. Latent hardening is the hardening of inactive (latent) slip systems by active ones. Deviations from unity of the ratio of latent to active hardening h changes hardening from isotropic to anisotropic. Assuming h = 1, Lopes et al. (2003) show that the degree of anisotropy in rolled pure Al was underestimated. Assuming h = 1.4, Kalidindi and Anand (1994) successfully predicted the post-compression response of Cu after plane strain compression of 0.34. On the other hand, Beyerlein et al. (2005a) found that after severe plastic shearing by ECAE (100% simple shear), the compression anisotropy of Cu was only reasonably predicted by ratios less than one (h = 0.2) and not h = 1.0 or h = 1.4. In any case, while one can be successful in modeling monotonic straining and texture evolution with any reasonable latent hardening scheme (e.g. 0 < h < 2), whether or not the evolution of latent hardening during straining is correctly captured is not apparent until there is a strain path change. In addition, latent hardening does not reproduce transients, such as the work-softening seen in a cross-test or the extended low hardening plateau seen in a reverse test (Beyerlein et al., 2005a). Hall–Petch effects and evolution of specific dislocation populations have been used to model the role played by planar dislocation boundaries on directional hardening in Al (Hansen and Juul Jensen, 1992; Winther et al., 1997; Winther, 2005), bcc steels (Peeters et al., 2000, 2001), Cu (Mahesh et al., 2004) and single crystal fcc steels (Canadinc et al., 2005). In these works, dislocation sheets or cell block boundaries are assumed to form either parallel to the active crystallographic planes or approximately parallel to the maximum stress direction in the grains, both of which have been experimentally observed. Generally these authors find that treating these planar defects as directional barriers to the propagation of dislocations results in anisotropy of the yield stress that agrees with experimental measurement. A model proposed by Peeters et al. (2000, 2001), based on the evolution of three different dislocation populations in each grain, accurately captures the reload response of bcc steel. However, when the model is adapted for Cu, it is found that it does not perform well in predicting the reload response of rolled fcc Cu (Mahesh et al., 2004). We attribute this result to the lack of a stage IV hardening in the model, a wider variety of substructure morphologies in Cu compared to steel, and the rather unruly nature of the various parameters that control dislocation-density evolution (Mahesh et al., 2004). 1.4. Current approach Our objective is to develop a directional formulation for polycrystalline constitutive response which operates at the single crystal level and evolves a strength (or slip resistance)

644

I.J. Beyerlein, C.N. Tome´ / International Journal of Plasticity 23 (2007) 640–664

for each slip plane in each grain. This strength evolution will be directional, depending explicitly on strain vectors, strain rate vectors, and strain-path-change matrix indicators, which characterize and quantify strain path changes (e.g., reversal, continuous, cross) in individual grains. Such measures of crystallographic shearing activity can be calculated directly from a polycrystal model. Also, texture evolution, a dominant factor in plastic anisotropy is naturally incorporated into the constitutive calculation. An advantage of this approach is that no a priori assumptions are required on the type of strain path change, amount of pre-strain, or amount of post-strain. At this stage in model development, the strength components will not be explicit functions of dislocation densities and dislocation-density evolution will not be calculated. Rather the functions that relate strength to strain will be empirical. However, as we will discuss, the functional forms can be supported by dislocation-based arguments. Moreover, the modeling framework presented here would allow for incorporation of evolving dislocation densities (which are state variables unlike strain). As we show here, this approach is effective for capturing how the subgrain microstructure (either dislocation cells or extended planar dislocation walls) impose directional barriers to slip. First we propose a latent hardening model which evolves with straining. Regarding strain path changes, within the model, strain reversals on crystallographic planes are favored because dislocations can move when the applied stress and internal stress act in the same sense, while new slip across previously active planes is impeded because it requires dislocations to overcome the existing structure. Next we present criteria for the activation of these transient deformation mechanisms in each grain, which are based on strain-path-change indicators which relate current to previous slip activity. Following this, new microscale models for the alterations in the strength reflecting reversal mechanisms and a mechanism of localized channeling and dissolution of microstructure (cross-effect) are presented. We incorporate this new hardening model into a visco-plastic self-consistent (VPSC) polycrystal model (Lebensohn and Tome´, 1993; Tome´ and Lebensohn, 2004) and predict the flow response of pure polycrystalline Cu under strain path changes. This multi-scale constitutive model aims to predict not only the yield strength, but the complete flow response up to large strains. In order to test the reversal mechanism in isolation, we compare our predictions with results from Christodoulou et al. (1986) for the stress reversal response following tensile or compressive pre-strains. The advantage of such tests is that the loading history is relatively simple. Finally, we test our microscale model against compression reload tests on a sample processed by equal channel angular extrusion (ECAE). The load history for ECAE is more complex, involving continuously varying shearing over large strains (Beyerlein and Tome´, 2004). The challenge in this case is to predict the anisotropy in the processed sample using the same parameters to simulate ECAE deformation and the three subsequent compression tests, which can activate any combination of reversal and cut-through mechanisms. 2. Modeling approach We use the rate-dependent VPSC polycrystal plasticity model (Lebensohn and Tome´, 1993; Tome´ and Lebensohn, 2004), to relate the macroscopic polycrystal deformation to that of the single crystals. The polycrystal is modeled as a collection of grains each with a distinct crystallographic orientation and volume fraction. A representative grain is

I.J. Beyerlein, C.N. Tome´ / International Journal of Plasticity 23 (2007) 640–664

645

modeled as an ellipsoid embedded in an anisotropic homogeneous effective medium with average properties of all the grains. At each increment the visco-plastic compliance of the polycrystal is determined self-consistently. Each grain deforms depending on its interaction with the homogeneous medium, the applied deformation, its shape evolution, and a visco-plastic flow rule which can account for the directional anisotropy in slip resistance. Consequently polycrystal hardening will have two contributions, one from grain orientations (texture) and the other from the evolution of the hardness with strain inside each grain. Useful outputs are the grain stress, grain strain rate, slip activity, shape change and re-orientation for each deformation step. Note that in self-consistent schemes, it is not necessary for each grain to activate a minimum of five slip systems, like in the Taylor model. In general, for fcc materials the number is found to be on the order of 2–4 per grain, most often distributed on one to two slip planes, depending on grain orientation and loading. VPSC neglects the elastic contribution, so that deformation behavior associated with extremely small applied strains (<0.001), anelasticity, unloading, or stress relaxation, are not modeled.

2.1. Crystallographic shear vectors A common measure of accumulated shear in a grain is C, calculated by the sum of the accumulated shears on all slip systems: Z tX C¼ j_cs j dt: ð1Þ 0

s

s

Here c_ is the shear rate on s, a scalar quantity which can be positive or negative. Rather than considering slip systems individually, we will associate the slip activity with its slip plane. For fcc Cu we consider the three slip systems on each of the four {1 1 1} planes as a group. The {1 1 1} planes will be designated as a, b = 1, 2, 3, 4. Modeling the alterations in flow response due to changes in strain path requires knowledge of the magnitude and direction of crystallographic strain and strain rates within each grain. To such effect we define directional shear vectors which measure the direction and magnitude of shearing on a slip plane. The shear rate vector v_ a and shear vector va on slip plane a are defined as v_ a ¼

X

 bs c_ s

and va ¼

Z

t

v_ a dt;

ð2Þ

0

s2a

where  bs is the normalized Burgers vector of slip system s expressed in crystal axes, such that the vectors are independent of crystallographic rotations. The bar above the symbol indicates that the quantity is a vector. The shear vectors in Eq. (2) are contained in plane a and, unlike a straightforward sum of c_ s , such as C in Eq. (1), they weight the relative contributions of the three slip systems on the plane a. For instance if slip system s dominates then va will be nearly parallel to  bs , whereas if two slip systems s and s 0 dominate, then va s s0   will lie in-between b and b . ‘Plane specific’ scalar shear rate and shear are defined by the norms of v_ a and va : v_ a ¼ kv_ a k

and

va ¼ kva k:

ð3Þ

I.J. Beyerlein, C.N. Tome´ / International Journal of Plasticity 23 (2007) 640–664

646

Total scalar shear rates and shear are defined as sums over the plane specific ones: v_ ¼

4 X

v_ a

and

v¼

4 X

a¼1

va :

ð4Þ

a¼1

2.2. Deformation mechanisms and local strengths The local strength enters into the polycrystal model through a visco-plastic law (e.g., Kocks et al., 1998), where the shear rate of active slip systems, c_ s , varies as a power of the ratio between the resolved shear stress and the strength of the system sac :  s n m : r ð5Þ c_ s ¼ c_ 0  a  signðms : rÞ ðs 2 aÞ; sc where c_ 0 is a normalizing strain rate, r is the grain stress, ms is the Schmid tensor for slip system s, and n is a model parameter.1 Here, the three slip systems s acting on plane a, all have the same sac . Under monotonic loading, dislocation accumulation and microstructure development are prevalent. These mechanisms are sense-insensitive and hence in our model, the strength associated with these mechanisms sah will be related to sense-independent shears, like C in Eq. (1). When the externally applied strain path changes, three outcomes are possible within each grain: (a) slip activity can continue ‘monotonically’ on the same planes and in the same direction as during pre-straining; (b) slip activity can take place on different planes than those used before. This new slip activity may have to locally ‘cut-through’ previously generated microstructure and may lead to an immediate increase in flow stress followed by a softening response; (c) slip activity can operate on the same planes as during pre-strain, but in the reverse sense. This path change involves deformation mechanisms in which small populations of dislocations that are immobile in forward glide become mobile in reverse glide. These reversal mechanisms are responsible for the macroscopic Bauschinger effect and transient changes in hardening observed during stress reversal tests. In our model, the differences in strength contributed by the cut-through and reversal mechanisms, (b) Dsacut and (c) Dsarev , depend on plane specific and sense-dependent shears Eqs. (2) and (3), in order to model their directionality. Dislocation storage and arrangement into substructure leads to permanent alterations in hardening. Reversal-related and cut-through mechanisms, in contrast, lead to transient alterations in grain hardening. (To emphasize their transitory nature, we use the D symbol to denote their contribution to strength.) All components, whether permanent or transient, can affect texture evolution. Therefore it is possible that even transient changes in grain hardening can lead to permanent changes in macroscopic hardening. In summary, the strength sac is the result of three contributions: sac ¼ sah þ Dsacut þ Dsarev ;

ð6Þ

where each components is updated incrementally with strain. Under monotonic straining the strength sah is the result of obstacles determined by dislocation multiplication and 1

The exponent n is set equal to 20 and c_ 0 is set equal to the macroscopic strain rate. As a consequence strain rate effects induced by n are removed. They can be introduced into the strength term if desired.

I.J. Beyerlein, C.N. Tome´ / International Journal of Plasticity 23 (2007) 640–664

647

recovery and patterning (stages I–IV in monotonic loading). Under a cross-path change, new slip on plane a could be activated and if so, it must overcome the substructure left behind by prior slip activity on the other planes, leading to an additional resistance to slip Dsacut . When slip on plane a is reversed, a fraction of previously immobile dislocations are set in motion. Reverse glide encounters a lower slip resistance compared to the previously forward glide. This difference Dsarev , is decomposed into two contributions, each related to a different mechanism: Dsarev ¼ DsaB þ DsaR :

ð7Þ

DsaB and DsaR are the reductions in glide resistance for dislocations previously immobilized by elastic backstresses and those tangled or locked up by deformation-induced microstructure, respectively. Another way to interpret Eqs. (6) and (7) is that these ‘reversible’ dislocations represent a surplus, and therefore the grain does not harden until they are used up (annihilated or stopped again) Eq. (6) assumes that processes behind each term are de-coupled. In Sections 2.3–2.5 that follow we present evolution models for sah , Dsacut , DsaB , and DsaR , and define slip-activity-dependent criteria for when they are invoked during deformation. The criteria are designed so that these strengths are not ‘switched on’ or ‘switched off’ arbitrarily by the user, usually when the macroscopic stress state is changed. 2.3. Latent hardening model First we describe the strength component sah in Eq. (6). Under multiple slip, the strength of slip system s contained on plane a (s 2 a) evolves with the strain rate on its plane and that of the other slip planes b by (Hill, 1966) oss ab b s_ ah ¼ V  h v_ ; s 2 a: ð8Þ oC The hardness for each slip system ssV (C) evolves with the accumulated strain in each grain C according to the extended Voce law (Tome´ et al., 1984) ssV ðCÞ ¼ ss0 þ ðss1 þ hs1 CÞ½1  expðhs0 C=ss1 Þ

ð9Þ

This law is meant to model hardening stages II–IV, usually associated with dislocation multiplication, recovery, debris formation, and dislocation patterning. In this work, the material parameters s0, s1, h0, and h1 are equal for the 12 {1 1 1}Æ1 1 0æ systems. For fcc crystals, the common approach is to consider isotropic hardening, hab = 1 in Eq. (8), or ‘fast’ and constant latent hardening, e.g., haa = 1 for coplanar slip and hab = 1.2–1.4 for non-coplanar slip (Asaro and Needleman, 1985; Kalidindi and Anand,  1994). These values are supported by small strain latent hardening experiments on single crystals varying widely in SFE, e.g., a-brass, silver, copper and aluminum (Franciosi et al., 1980; Kocks, 1964; Phillips and Robertson, 1958). At the other extreme, some authors have used a nonlatent hardening regime, where  hab = 0 for a 6¼ b or even coplanar slip systems (Bassani and Wu, 1991; Wu et al., 1996) and coupling between slip systems was incorporated into self-hardening. As for the ‘slow’ latent hardening regime 0 < hab < 1.0, in which active systems harden faster than passive ones, evidence of it can be found in some experiments (Phillips, 1962; Hosford, 1993; Kocks, 1964; Bassani and Wu, 1991; Wu et al., 1991) and dislocation dynamic simulations (Devincre et al., 2006) in medium to high SFE single crystals, such

648

I.J. Beyerlein, C.N. Tome´ / International Journal of Plasticity 23 (2007) 640–664

as Cu and Al. Recently Devincre et al. (2006) considered the hardening expression due to a pffiffiffiffiffiffiffiffiffiffi forest of dislocations qf by Franciosi et al., ssc ¼ bl asr qrf , and characterized the interaction parameters asr between slip systems s and r using 3-D dislocation dynamics. Although this forest hardening expression applies to small strain (not beyond stage III), it can be considered an appropriate dislocation-based analogy to the hardening law in Eq. (8). The interaction parameter a between cross-slip systems was found to be the strongest with 1=2 a value of 0.625 ± 0.044 and increasing towards unity with increasing qf (i.e., as strain decreases). In a polycrystal, the slow latent hardening regime merits examination in connection with large strain deformation. First this regime reflects the development of planar boundaries in fcc crystals, which form nearly parallel to active slip planes under large strains (Miyamoto et al., 2004; Nes et al., 1986; Huang et al., 2001; McCabe et al., 2004; Mahesh et al., 2004; Canadinc et al., 2005; Xue et al., in press). (After small strains, this may not be the case; certain orientations can generate walls perpendicular to slip planes (Xue et al., in press).) Therefore the dislocations of these systems contribute the most to building the walls, collect the most debris along their slip planes and consequently harden at a faster rate. Second, less hardening on the less active systems builds in a feedback mechanism, promoting non-coplanar slip and possibly evolving hab back towards unity. Evidence of such a transition in slip activity was found in TEM analyses by Xue et al. (in press), who tracked the microstructural evolution in several Cu grains up to large strains in simple shear. To sum, in a polycrystal, although  hab P 1 seems to be more likely at low strains, ab the ‘slow’ latent hardening  h < 1 may prevail at large strains, since it tends to favor crossslip, a dominant mechanism in large strain deformation of polycrystals with medium to high SFE. Based on the evidence discussed above we introduced a latent hardening function that captures the influence of microstructural evolution on slip interactions by evolving hab during the course of straining (Beyerlein and Tome´, 2006). There we proposed hab in Eq. (8) to evolve with deformation according to ab

ab ¼ ð1  hsat Þed þ hsat ; h

ð10Þ

where dab ¼ gjva  vb j depends on the difference in the accumulated shear on planes a and b (Eq. (3)) and g characterizes the rate at which  hab changes from 1.0 (isotropic) to a saturation value hsat. For Cu, we assume a slow latent hardening regime, characterized by 0 < hsat < 1 in Eq. (10). The following cases illustrate how this model operates: (a) For small strains jva  vbj  1/g and  hab  1. In this case the grain substructure is relatively weak, consisting mainly of a three-dimensional network of cells formed by statistical trapping and with low misorientations (<1). This structure will lead to isotropic hardening as all systems are likely to encounter resistance from them. (b) After large strains in plane a, i.e., jva  vbj > 1/g, hsat < hab < 1, the hardening turns from isotropic to anisotropic. Planar dislocation boundaries tend to form along the most active slip planes (Winther et al., 2004; Xue et al., in press; Huang and Liu, 1998; Mahesh et al., 2004; Miyamoto et al., 2004), leading to an anisotropic hardening condition in which the active planes harden faster than the latent ones. (c) An exception occurs when two planes are equally active, va  vb, and a 6¼ b. In this case,  hab  1, meaning non-coplanar slip systems with nearly equivalent activities harden equally.

I.J. Beyerlein, C.N. Tome´ / International Journal of Plasticity 23 (2007) 640–664

649

Both hsat and g are material parameters. The value hsat will be related to the propensity for cross-slip: for hsat below unity (high SFE), cross-slip is favored, whereas values of hsat greater than unity should apply to low SFE materials, where cross-slip is difficult. The parameter g governs how quickly stable planar substructure develops with strain, which also may be faster as the SFE increases. Results are not very sensitive to the values of hsat (within 0.2–0.9) and g (>10) and we do not consider them as adjustable parameters for fitting. Based on previous discussion, for Cu, we assign g = 100 and hsat = 0.6, meaning that the ssh for systems on the most active planes harden at a higher rate than the latent ones. 2.4. Cut-through model Dominant slip activity on one or two slip planes is primarily responsible for the build up of dislocation walls. Single crystal and polycrystal TEM observations suggest that, after large strains, these extended planar dislocation walls are more or less aligned along the most (or more) active plane(s), albeit with some slight 5 deviation (Wu et al., 2004; Mahesh et al., 2004; Xue et al., in press). As a consequence, the slip planes that were latent near the end of large pre-straining are likely to be non-coplanar to these walls. Therefore, if these latent systems are suddenly activated in the event of a strain-path change, they would encounter resistance in attempting to channel through dislocation boundaries and build a microstructure of their own. In what follows we describe a model that accounts for such a mechanism. 2.4.1. Cut-through (cross-effect) matrix Xab When there is a change in activity in plane a, the difference in instantaneous shear rates ðv_ a;nþ1  v_ a;n Þ (see Eq. (2)) between strain increment n and n + 1, will deviate from zero. However, for determining if new activity on a will cut-through previously formed microstructure, detecting an activity change is not a sufficient criterion. We need to further consider the shearing history on planes non-coplanar to a. Suppose that prior to the activity change, plane b has accumulated much shear, i.e., vb is large, and slip plane b has also contributed significantly to the deformation of the grain, i.e., vb/v is also large (see Eq. (3)). In such a case, most likely (but not necessarily) dislocation walls are closely aligned with b and so slip plane b will act as a barrier to the propagation of dislocations on other slip planes in the same grain. Accordingly, the channeling through and dissolution of these b barriers by new dislocations gliding on a, will depend on the magnitude and alignment of the current shear rate in a, v_ a;nþ1 , with respect to the accumulated shear vector in b, vb . An appropriate measure of cut-through is their crossproduct · because the larger their relative orientation, the greater is the severity of the cutthrough mechanism. We define a ‘cut-through’ matrix Xab, coupling shear activity in different slip planes:  a;nþ1   a;n  v_ v_ vb  vb  ab    : ð11Þ X ¼     v v v_ v_ Normalization by scalars v_ and v means higher values of Xab are obtained when the current slip activity occurs mostly on a and was historically predominant on b. It also means 1 < Xab < 1. The following cases have to be considered:

650

I.J. Beyerlein, C.N. Tome´ / International Journal of Plasticity 23 (2007) 640–664

(a) Xab = 0 (no cut-through) implies either no new activity on a or historically no activity on b or both. (b) Xab < 0 (no cut-through) implies a reduction in activity of a. Only positive values of Xab are of interest. (c) Xab > 0 implies increased activity on a and non-zero activity on b. Xab > 0 is therefore a necessary (but not sufficient) indicator that localized cutting mechanism operates in a grain. (d) Xaa 6¼ 0 (no cut-through) can happen because the instantaneous rate v_ a may experience a finite change or a reversal under a strain path change, or even between successive deformation increments. This case, however, is not relevant because cutthrough is restricted to occur between non-coplanar slip planes. Following case (c) above, we now establish a necessary and sufficient criterion for when the cut-through mechanism is likely to be relevant. Suppose   at some instance a previously latent plane a becomes active, meaning v_ a;nþ1  > 0 and v_ a;n  ¼ 0. The cut-through mechanism most effectively operates when slip activity is mostly localized in one plane a (say:  v_ a;nþ1  > 1=2Þ. Formation of one set of boundaries along b is possible when b was one of v_ the most active planes in the grain (say: vb/v > 1/2). Finally the cut-through mechanism is most effective when the orientation relationship between v_ a and vb is nearly normal (say:  pffiffiffi  a;nþ1    b hab > 45; sin hab > 1= 2Þ. Combining these circumstances in Eq. (11), X ab  v_ v_  vv     sin hab > 12 12 p1ffiffi2 > 0:15, which represents an empirical threshold for the cut-through mechanism. In other words, the condition Xab > Xcut = 0.15 amounts to localized plastic cut-through caused by newly activated planar slip a channeling through dislocation boundaries b non-coplanar to it. 2.4.2. Cut-through strength New slip activity a will encounter an additional resistance to slip Dsacut when attempting to permeate boundary b, when Xab > Xcut. This resistance will be proportional to the ‘strength’ of the boundary, which depends on boundary characteristics like its dislocation density q, dislocation spacing d or misorientation Dh. Using dislocation dynamics, Madec et al. (2002) determined that the bow-out stress sd for dislocations propagating through a boundary containing an array of dislocations with spacing d follows   b d sd ¼ kl ln ; ð12Þ d b where b/d is the ratio of the Burgers vector to the dislocation spacing in the boundary, k is a material-independent constant determined to be k = 0.086, and l is the shear modulus, which is 42 GPa for Cu. If we let sbd be the strength associated with boundary b, then the initial value of the cutthrough strength sacut;0 will be  X ab sbd when Xab > Xcut or sacut;0 ¼ 0 otherwise. In the likely event that a grain contains more than one family of dislocation boundaries, sacut;0 must account for the possibility that planar slip on a must channel through boundaries lying on several sets of planes. X X ab sbd ; ð13Þ sacut;0 ¼ b6¼a

I.J. Beyerlein, C.N. Tome´ / International Journal of Plasticity 23 (2007) 640–664

651

where only Xab > Xcut are considered in Eq. (13). As straining proceeds, gliding dislocations on a can either permeate through or react with the boundary b dislocations, inducing untangling, annihilation, dissociation, or recombination. As a consequence, channels are created and d within the boundary increases locally, resulting in a reduction in Dsacut needed for dislocations to pass through. Thus it will become easier for new slip activity on a to propagate as straining progresses. The reduction rate of Dsacut will be proportional to v_ a ; a higher v_ a indicates a higher flux of dislocations permeating and dissolving the boundaries. This process is appropriately represented by the following decay rate equation D_sacut ¼ xDsacut v_ a ;

ð14Þ

where x characterizes the rate at which newly activated glide locally dissolves microstructure. A similar form for the evolution of a continuum-level strength component associated with softening of pre-existing boundaries was presented by Teodosiu (1996) and Rauch (1992). Solving Eq. (14) leads to an exponential decay law for Dsacut Dsacut ¼ sacut;0 expðxvanew Þ;

ð15Þ

vanew

where is the shear on the plane a accumulated since the instant when the condition Xab > Xcut is met. The material parameter x in Eqs. (14) and (15) reflects how rapidly the slip activities vanew in newly activated planes can breakdown dislocation barriers created during pre-load. The critical stresses required for such processes depend sensitively on the dislocation mechanics of the particular material and the temperature. We expect that x will be proportional to temperature and SFE as these two factors aid dislocation reactions and cross-slip. At room temperature and for Cu, we estimate, by comparison to experimental data, that x = 30 provides a good quantitative description of the cut-through process. In order to calculate sbd in Eqs. (12) and (13) we need to relate it to its associated shear vector vb. b/d is expected to have a non-linear relationship with the shear vb, illustrated schematically in Fig. 1. At small strains, v < v1, dislocations boundaries have not yet formed and b/d is negligibly small. As boundaries are forming, b/d rapidly increases as walls assume low-energy configurations with the aid of straining. Once stabilized, at

1.0

(b /d)/kc

0.8 0.6 0.4 0.2

v1

v2

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

vβ Fig. 1. Relationship between the boundary quantity b/d (related to dislocation density) along plane b and the accumulated shear on plane b. The meaning of v1 and v2 are also shown. See Eq. (16).

652

I.J. Beyerlein, C.N. Tome´ / International Journal of Plasticity 23 (2007) 640–664

v = v2, b/d saturates with deformation. The following sigmoidal dependence of b/d with the shear vb represents well the functional form in Fig. 1,  ðbÞ b kc  where vb ¼ ð2vb  v2  v1 Þ=ðv2  v1 Þ; v2 > v1 : ¼ ð16Þ  d 1 þ expð2mb Þ Eq. (16) connects evolution of inverse mean free path (dislocation density) in plane b with shear strain v. The dislocation-density coefficient kc is found to be 0.023 for Cu. The material parameters v1 and v2 are assigned physically reasonable values (0.3 and 0.7, respectively) based on TEM observations by Xue et al. (in press) on polycrystalline Cu and are not adjusted in characterization. For Al, in which dislocation walls form earlier, v1 and v2 will be lower (e.g., Beyerlein et al., in press). 2.5. Model for strain reversal effects The widely accepted definition of the Bauschinger effect is the reduction in the yield stress upon reversal of loading direction. At the same time, it is also well-known that the reverse flow response can involve more than a simple drop in yield stress, but also a transient change in hardening rate over a long straining interval (Vincze et al., 2005; Gracio et al., 2004) and permanent changes in flow stress (Stout and Rollett, 1990; Wilson et al., 1990). When dealing with simple tests (such as tension–compression, or shearreverse shear), one may characterize the transient reversal by comparison with the extrapolated forward stress–strain response (tension or shear in this case). More complex reloads also exhibit reversal-type transient effects (Wilson et al., 1990; Beyerlein and Tome´, 2006), although it may not be possible to characterize them by comparison with a hypothetical ‘forward response’. In this section we describe the reversal effect as a multi-stage transient response associated with reloading conditions that induces total or partial shear reversal inside the grains. The stages are schematically illustrated in Fig. 2. The primary stage (A–B) covers the first 1–3% of reverse straining and is commonly associated with the Bauschinger effect. It consists of the initial reduction in the yield stress upon reloading, followed by a high hardening rate. The secondary stage (B–C) is a subsequent softening response, which may or may not be present, and may affect the 3–10% reload strain interval. The tertiary stage (C–D) is

monotonic forward B

|σ| A

D

C

reverse reload

strain Fig. 2. Multiple stages of a macroscopic flow response after a reversal reload (solid thick line).

I.J. Beyerlein, C.N. Tome´ / International Journal of Plasticity 23 (2007) 640–664

653

a plateau region of null or low strain hardening. In some metals, this stage can cover a wide strain range, on the order of the pre-strain (Vincze et al., 2005; Stout and Rollett, 1990). The final stage starts at D and is the resumption of positive ‘normal’ work hardening, which may or may not lead to rejoining the ‘forward’ response. 2.5.1. Reversal indicators Paa Even if the macroscopic strain is reversed, locally each slip system in each grain may not experience an exact reverse shear. To measure the reversibility of slip in a given plane under arbitrary loading, we use the projection of the instantaneous shear rate vector on the accumulated shear vector (Beyerlein and Tome´, 2006): P aa ¼

va v_ a ; v v_

1 6 P aa 6 1:

ð17Þ

The reversibility in slip plane a is characterized by the sign and magnitude of Paa. When Paa P 0, there is no shear reversibility, while 1 6 Paa < 0 indicates that there is a reverse loading in the plane, and Paa = 1 for a full shear reversal. According to Eq. (17) Paa evolves with straining. When and if Paa becomes negative, it will remain negative until the grain has accumulated sufficient strain in reverse, at which point Paa becomes positive again even though the direction of straining continues to be reversed. The multiple inflection points in Fig. 2, which are often observed in soft metals or after large pre-strains, cannot be explained by one mechanism alone. As introduced in Section 2.2, Eq. (7), the reduction in strength Dsarev associated with a reversal event (1 6 Paa < 0) has two components, DsaB and DsaR . Each one is linked to a separate mechanism. As developed below, both components will depend on the reversal indicator when Paa < 0. They are zero when Paa > 0. 2.5.2. Bauschinger effect DsaB In connection with Fig. 2, the first component DsaB , is responsible for the so-called Bauschinger effect (portion A–B of the reversal response). During forward loading internal stresses are generated by dislocation configurations (such as cell walls and sheet-like laminar boundaries), and dislocation interactions with grain boundaries and other defects. Part of this internal stress field works against the applied stress and tends to immobilize a fraction of the dislocations. This field is commonly categorized as a backstress. The backstress is often related to elastic strains and therefore it is limited by plastic relaxation. It includes contributions from nearby dislocation arrays or line tension of bowed-out dislocations, or low energy (stable) boundaries (such as twin or grain boundaries). When the applied stress is reversed and becomes aligned with the backstresses, these qaB dislocations become mobile again and glide in reverse. As reverse straining continues, the backstress, however, decays exponentially (within 1–2% of straining) (Wilson and Bate, 1986; Wilson et al., 1990) and is replaced by a backstress opposing the new shear direction. The maximum value of the backstress, therefore, is the initial value when the strain is first reversed. It is found to increase slightly with pre-strain level for small pre-strains while the decay rate under reloading does not (Wilson et al., 1990). The reverse motion of these dislocations under the combined action of the applied stress and backstress manifests as the initial macroscopic yield drop commonly known as the Bauschinger effect. The rapid hardening that immediately follows is due to the re-orientation of the backstress.

654

I.J. Beyerlein, C.N. Tome´ / International Journal of Plasticity 23 (2007) 640–664

Microscopically, we represent the reduction in strength due to these dislocations by DsaB . Its initial value at the instance of reversal (1 6 Paa < 0) is saB;0 and it is made to decay exponentially with reverse straining. saB;0 will depend on the degree of reversibility aa through P aa at the instance of reversal, and the dislocations qaB collected 0 , the value of P on the slip plane a during the pre-strain. The latter is represented by varev , which evolves with the previously accumulated shear va, but in a manner analogous qaB . As suggested by measurements made by (Wilson and Bate, 1986), the nature of qaB dislocations will be to increase monotonically with va (pre-strain) for small values but quickly saturate for large va. Accordingly, varev ¼ vB;sat ð1  expð#B va ÞÞ;

ð18Þ

where vB,sat is the maximum amount of shear that may be available. Because the amount of qaB frozen by elastic backstresses will be limited, the maximum vB,sat will be small, on the order of 103, and # will be large, i.e., 5 < # < 20. (These material parameters are not expected to vary outside of these narrow regions.) With these values, for pre-strains larger than 5–10% saturation is achieved and varev ¼ vB;sat . Finally, saB;0 is related to the product a of P aa 0 and vrev by a saB;0 ¼ lP aa 0 vrev ;

DsaB

ðno sumÞ;

P aa < 0;

then decays exponentially with further straining along the new path  a  v DsaB ¼ saB;0 exp  new ; vB

ð19Þ vanew : ð20Þ

where vanew is the shear on the plane a accumulated since reversal. The parameter vB is related to how fast the backstress is nullified with vanew , typically within 0.5–2% strain for Cu at room temperature, according to our study. It is a material parameter that is insensitive to pre-strain level va. It tends to be larger for heavily alloyed metals than single-phase pure metals like Cu (Wilson et al., 1990). a There are a few important aspects of Eqs. (18)–(20) to note. P aa 0 provides to DsB an orientation dependence, which will vary from grain to grain even when the macroscopic a strain is a precise reversal. If P aa 0 is close to zero then there are not going to be many qB aa a a dislocations in motion and hence DsB will be low. P 0 also provides to DsB a history dependence: if the plane a was not historically active (i.e., when va in Eq. (17) is low), then there are not going to be many qaB dislocations available and again DsaB will be low. varev in Eq. (18) provides to DsaB a dependence on pre-strain only when va is low (va < 5%). 2.5.3. Extended reversal response DsaR The component of Dsarev in Eq. (7), DsaR , is responsible for the extended transient portion (B–D) of the reversal response. The reverse motion of a group of dislocations, called qaR , reduces the strength by an amount DsaR . Under forward motion, these dislocations become trapped in dislocation tangle, cell walls, and lamellar dislocation sheets. When the stress is reversed, qaR dislocations find it easier to glide in reverse. Unlike the Bauschinger qaB , qaR can grow with pre-straining to be relatively large and finite amounts of reverse straining are needed to untangle them. The ones that are loosely trapped require small amounts of strain and are released first. Those that are more tightly locked up require higher reverse strain levels. Therefore, the rate at which qaR are released from entrapment f 0 first increases, but then decreases with straining as the remaining qaR become harder to unlock and are

I.J. Beyerlein, C.N. Tome´ / International Journal of Plasticity 23 (2007) 640–664

655

depleted. The total strain required for complete depletion increases with the amount of qaR . Formally, f 0 is defined as Z v Dqa ðvÞ f ðvÞ ¼ f 0 ðxÞdx ¼ aR ; where f 0 P 0; qtotal 0 where f(v) is the fraction of qaR released by reverse shear v, DqaR , with respect to the total dislocation density on slip plane a, qatotal . We suppose that DsaR is proportional to f 0 , according to 0 a DsaR ¼ lP aa 0 f ðvnew Þ:

ð21Þ

a As before, P aa 0 is included to provide an orientation and history dependence to DsR . 0 Fig. 3 illustrates the general form proposed for f as a function of reverse straining vanew . At the instant of reversal, vanew ¼ 0, f 0 = 0, and DsaR ð0Þ ¼ 0. For small vanew , f 0 and DsaR remain close to zero. This is in contrast to the Bauschinger-dislocations qaB , where DsaB ð0Þ ¼ saB;0 > 0, which are readily available and contribute immediately to reversal. The qaR dislocations, on the other hand, need to untangle first from the ‘core’ microstructure and create their own channels, as suggested by Hasegawa et al. (1976). Once this process takes place f 0 and DsaR increase rapidly. The initial rise in f 0 shown in Fig. 3 reflects this behavior. Eventually, with further straining, this population of dislocations is depleted via attrition and their contribution to reverse shear becomes less significant. Therefore, f 0 0 decays after reaching fpeak . This exhaustion process is captured by the asymptotic tail of f 0 in Fig. 3, which slows down as the pre-strain increases. Increasing the pre-strain has the effect of increasing the total density qaR , which in turn increases the amount of reverse straining required to completely deplete them. If we define a shear vR as a measure of prestrain and assume that vR only affects the decay rate of f 0 , an empirical function which captures the behavior discussed above is

g ¼ expðvanew =vR Þlognormalðvanew jlR ; rR Þ;

ð22Þ

where lR and rR are the parameters of the lognormal distribution; lR and rR depend on material but are independent of pre-strain or loading mode. The expðvanew =mR Þ term

f' 0.0010

peak

vR =

f' 0.0005

0.4 0.2 0.0000 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Reverse straining, vnew Fig. 3. Illustration of the release function f 0 and how it changes with vR, which is roughly proportional to vR. The 0 parameters used in this example are rR = 0.85, lR = 0.25, fpeak ¼ 0:001, vR = 0.2 and 0.4.

656

I.J. Beyerlein, C.N. Tome´ / International Journal of Plasticity 23 (2007) 640–664

extends the tail of the lognormal as vR increases, as shown in Fig. 3 (mR = 0.2 versus 0 mR = 0.4). Note also that in Fig. 3 the peak value, fpeak does not change with vR. The peak 0 value fpeak is made independent of vR by defining f 0 as g 0 f 0 ¼ fpeak : ð23Þ gpeak 0 , rR and lR are listed in Table 2, together with the rest. The material parameters fpeak

3. Model applications The hardening model for single crystals, including the evolution of latent hardening, cut-through and reversal mechanisms, is applied to two sets of tests found in the literature for pure polycrystalline Cu. All the model parameters are, in principle, material dependent ones and therefore we attempt to use the same set of parameters for these two applications (see Table 1). However because the starting Cu for each case came from different sources, and had different pre-processing conditions and grain sizes, two of the Voce parameters 0 associated with Eq. (9) were slightly different, as were vB and fpeak (see Table 2), associated with the reversal model. For each application, the same set of parameters is used when simulating pre-straining and reloading. 3.1. Application to reversal tests on copper For the first case we use tension–compression test results for Cu (initial grain size 20 lm) reported by Christodoulou et al. (1986). The tests are 8% tension followed by compression, and 18% or 28% compression followed by tension (Fig. 4). The VPSC polycrystal is represented by 1000 randomly oriented and initially spherical grains. The Voce parameters (Table 2) were estimated by comparing the simulated and measured responses under the pre-strain loading up to large strain (100% or higher). Although available, during the monotonic loading the cut-through and reversal mechanisms did not activate. To best compare the reversal response with the monotonic response, the absolute value of the tensile and compressive stress and strain is plotted in Fig. 4. As shown, the model predicts well several features of the reversal response: the initial 10–20% drop in yield at A, Table 1 List of the material-dependent parameters common to both test applications of pure Cu Symbol

Value

Description

Method of characterization

kc x lR rR v1 v2 #B vB,sat g hsat

2.3 · 102 30 0.25 0.85 0.3 0.7 15 2.8 · 103 100 0.6

Dislocation-density coefficient Rate of cut-through Lognormal scale parameter Lognormal shape parameter Strain at which wall formation begins Strain at which wall formation is completed Rate of saturation for reverse shear Maximum shear available for Bauschinger effect Rate of saturation for latent hardening Saturation value for latent hardening

Adjusted to ECAE data Adjusted to ECAE data Fixed Adjusted to reversal data Fixed (0.1–0.3) Fixed (0.7–1.0) Fixed (5–30) Fixed in reasonable range Fixed (10–103) Fixed (0.2–1.4)

Only kc, x and rR are estimated by agreement with the measurements; while the rest are fixed. A ‘fixed’ parameter is not adjusted freely to fit the data. They will vary if the material is changed and therefore a recommended range of values is provided.

I.J. Beyerlein, C.N. Tome´ / International Journal of Plasticity 23 (2007) 640–664

657

Table 2 The material parameters used in the model which were different between the two batches of Cu Test

Grain size (lm)

s0 (MPa)

s1 (MPa)

h0 (MPa)

h1 (MPa)

vB

vR

0 fpeak

Axial reversal (Christodoulou et al., 1986) ECAE reload (Alexander and Beyerlein, 2005)

20

20

150

510

26

0.02

0.2

1 · 103

50

20

175

440

26

0.01

0.4

0.74 · 103

The Voce parameters, s0, s1, h0, and h1 in Eq. (9) were estimated by fitting axial compression or tension response of the measured copper, assuming hsat = 0.6 and g = 100 in Eq. (10). vB is the decay rate of the reversal strength 0 reduction DsaB (Eq. (20)). fpeak is the peak value of the release function f 0 (Eq. (23)).

a 400

b

18% Compression Pre-strain

350

300

300

250

250

Stress (MPa)

Stress (MPa)

400

8% Tension Pre-strain

350

200 150

Experiment Predicted Monotonic Predicted Reverse

100 50

200 150

Experiment Predicted Monotonic Predicted Reverse

100 50

0

0

0.0

0.2

0.4

0.6

0.8

0.0

1.0

0.2

0.4

Axial strain

c

0.6

0.8

1.0

Axial strain

400

28% Compression Pre-strain

350

Stress (MPa)

300 250 200 150

Experiment Predicted Monotonic Predicted Reverse

100 50 0 0.0

0.2

0.4

0.6

0.8

1.0

Axial strain

Fig. 4. Comparison between the model predictions and experimental results of Christodoulou et al. (1986) on axial reversal tests on pure copper. (a) 0.08 pre-strain in tension followed by compression and (b and c) 0.18 and 0.28 pre-strain in compression followed by tension.

the region of very high hardening rate, noticeable work-softening for the larger pre-strains (18% and 28%), and increasing tertiary stage (C–D) with increasing pre-strain. In 96% of the grains the most active plane in reverse loading coincided with the most active in forward loading.

658

I.J. Beyerlein, C.N. Tome´ / International Journal of Plasticity 23 (2007) 640–664

The predicted reverse response is extended beyond the data to see if and when it will rejoin the monotonic response. For this Cu, the monotonic response is recovered roughly at 0.45, 0.8, and 0.9 total strain for the 0.08, 0.18, and 0.28 pre-strain cases, respectively. Such an extended transient response has been observed in pure Cu in large strain shear reversal tests (Stout and Rollett, 1990). Although the cut-through mechanism is allowed to be active, it did not operate during reverse deformation. For the largest pre-strain (0.28) new systems were activated in a small number of grains (in 0.3% grains, Xab > Xcut). However, the corresponding cut-through strengths were low (5 MPa or less). 3.2. Application to ECAE-processed copper The ECAE process involves extruding a metal sample through a die consisting of two channels, intersecting at an angle /, usually 90 (Segal, 1995). As the metal is pushed around the corner, it experiences simple shearing along the intersection plane. In a set of systematic studies (Alexander and Beyerlein, 2005), Cu samples (initial grain size 50 lm), extruded by one ECAE pass, / = 90, were compressed in three orthogonal directions, one parallel to the extrusion direction (ED), another parallel to the entry channel (ND), and the third normal to the plane of the die (TD). As shown in Fig. 5, the ECAE-processed Cu exhibits marked plastic anisotropy and transient response upon being reloaded in compression. The compressive strength in the TD direction is significantly higher than in the ND and ED directions, with the ED direction being the weakest for the strain range tested (up to 20–25%). A subtle amount of softening is also seen in ND. After the initial transient work-softening response in TD, finite strain hardening occurs in all three directions and the anisotropy decreases. The hardening rate, however, is slightly different for each direction. In practice, the ECAE process typically involves multiple extrusions through the die, not just a single one. Regardless of the number of prior extrusions, anisotropy in the compression strength was always present; the out-of-plane response (in TD) was always higher

425

Stress (MPa)

400 375 350 325 ED TD ND Model

300 275 250 0.00

0.05

0.10

0.15

0.20

0.25

0.30

Strain Fig. 5. Comparison between the measured compression reload responses after one ECAE pass (Alexander and Beyerlein, 2005) and model predictions (solid lines).

I.J. Beyerlein, C.N. Tome´ / International Journal of Plasticity 23 (2007) 640–664

659

than the in-plane responses (ND and ED), (i.e. TD > ND > ED) (Alexander and Beyerlein, 2005). As in the experiment, a complete simulation involves a single pass of ECAE immediately followed by compression testing in one of the three orthogonal directions. To model ECAE deformation we use an analytical flow model (Beyerlein and Tome´, 2004) to describe the sequential shearing deformation that occurs over a fan-shaped plastic deformation zone generated at the intersection corner (Segal, 1995), instead of a simple localized shear at the intersection plane, inclined along the 45. We show in previous work that this flow model agrees well with FE simulations (Li et al., 2005) better than the simple shear model. When combined with VPSC, the resulting texture predictions agree well with separate measurements made by neutron diffraction and OIM in a variety of materials (Li et al., 2005; Beyerlein et al., 2005b). For this work, the Cu polycrystal is represented by 1516 distinct orientations distributed according to an OIM measurement of the initial texture (Li et al., 2004), consisting in a weak fiber about the billet long axis. The polycrystal initially has spherical grains. The Voce parameters for this Cu (Table 2) are estimated by fits with the tension and compression responses of the annealed copper. With the macroscopic and microscopic input listed in Tables 1 and 2, VPSC reproduces the texture and grain shape of the material after extrusion. For subsequent compression testing we assume traction-free surfaces perpendicular to the direction of axial compression. The experimental and predicted compression curves for Cu pre-strained by ECAE are compared in Fig. 5. In the TD response, there is an initial jump in stress followed immediately by a high degree of work-softening over the first 15–20% of compression straining and afterwards the material begins to work harden at a normal rate. In the ED response, the initial concavity followed by an extended plateau region of low hardening are indicative of stages A–D of the Bauschinger effect. Using the same parameters during ECAE and in all reload directions, the model predicts well the flow anisotropy of the severely processed material and the transient characteristics of the flow response. The strain path change sequences associated with reloading ECAE material in compression involved a mix of both cut-through and reversal mechanisms. Each direction operated a different combination. As an indication of local reverse activity, the number of grains in which the most active plane during pre-loading was also the most active during reloading (but in the reverse sense) was counted for each test. A larger number of grains in ED (76.1%) than in ND (0%), and TD (12%), underwent reversal. Because of this, all stages of the Bauschinger effect are present in the ED response. Unlike the reversal mechanisms, all directions activated cut-through mechanisms with the largest percentage in the TD (79.2%) followed by ND (39.4%) and ED (31.4%). Because compression along the TD includes more cross mechanisms, the work-softening shortly after reloading in TD is more pronounced than in the other directions. A small fraction of grains in the ED and TD (2.9% and 7.3%, respectively) had both mechanisms operative. These ‘mixed’ grains had at least one of its planes satisfy, 1=3 < sacut =ðsarev þ sacut Þ < 2=3. Clearly for general strain path changes, such as this case, it is important to allow for the possibility of either mechanism to operate in each grain. Geometric effects play a large role in the strain hardening following the path change. When texture evolution, cut-through, and reversal mechanisms are suppressed during the simulation of subsequent compression (not shown), the ND and ED responses have a significantly lower hardening rate (nearly flat) and the TD response has a higher one compared to their respective responses in Fig. 5. This result implies that texture evolution

660

I.J. Beyerlein, C.N. Tome´ / International Journal of Plasticity 23 (2007) 640–664

contributes a positive hardening rate to the total ND and ED responses and a negative one (geometric softening) to the TD response in Fig. 5, as already shown by Alexander and Beyerlein (2004). 4. Discussion In this work we propose a combined hardening law-polycrystal approach for modeling the mechanical response of Cu associated with strain-path changes. The model accounts for texture and its evolution but also, and even more importantly, for dislocation microstructure evolution. The latter is representative of the directionality and sense of slip taking place during the pre-load stage in each individual grain. Upon reloading, cut-through and reversals mechanisms play an important role in determining the directional anisotropy of single grains. Dislocation substructures are found to develop on the more active slip planes in the grain. As a consequence, we describe hardening in terms of the activity on crystallographic slip planes rather than in terms of individual slip systems. Depending on the ability for dislocations to cross-slip, these active planes can be harder (cross-slip is likely) or softer (cross-slip unlikely) than the latent planes. This first role of substructure-induced anisotropy is modeled by the extended Voce law for single slip and the hab model for coupling non-coplanar slip (Beyerlein and Tome´, 2006). The latent hardening hab model evolves the ratio of latent to active hardening with strain. For copper this ratio is allowed to fluctuate between 1 and 0.6, depending on relative plane activity. Within the model, cross-slip is encouraged on latent planes after large straining along active planes. In what concerns shear reversals on crystallographic planes, we consider the reverse glide of previously immobile dislocations, either ‘Bauschinger-type’ dislocations which were immobilized by internal backstresses, or the dislocations tangled in the microstructure to varying degrees but preserving a ‘memory’ of the shear direction that induced them. The first type moves under reversal due to the realignment of the applied stress with the previously generated backstress. Unlocking of the second type is driven by the new stress state and has little to do with internal backstresses. Each contributes to a characteristic domain of the reload curve (see Fig. 2): Bauschinger dislocations affect the first 1–2% of it, and cause a drop in the yield stress, followed by a short period of rapid hardening, while ‘polar’ dislocations may induce softening and a large plateau in the hardening response (of the order of the pre-strain). The pre-strain parameter vR in the release function of the polar dislocations Eq. (22) was found to be 0.2 for all three Christodoulou et al. tests and 0.4 for the ECAE test, suggesting that it may increase slowly for small to medium pre-strain levels and eventually saturate at large levels. Controlled pre-load tests to different strains followed by reversals will be required to characterize the dependence of vR on pre-strain, as well as the other model parameters. Finally, dislocation substructures may affect directional anisotropy when new slip activity attempts to permeate and locally dissolve pre-existing structure. Our cut-through model favors this mechanism when prior slip activity on one plane is large and when new slip, non-coplanar to it, is activated. The channeling process induces a transient work-softening effect on the flow stress in the event of a cross-type strain path change. The parameters introduced in the model are material dependent. Some will be less sensitive than others. Further studies, like in Beyerlein et al. (in press), where this model is applied to two different materials need to be conducted to assess how they change with

I.J. Beyerlein, C.N. Tome´ / International Journal of Plasticity 23 (2007) 640–664

661

0 material. For example there it is found that hsat, rR, fpeak , kc, differed the most between Al and Cu, in a manner consistent with known differences in their stacking fault energies. The directional formulation developed here, with the use of strain vectors and strainpath-change matrices, can be extended to include temperature and rate effects (e.g., Kok et al., 2002) and microstructure (e.g., Cheong et al., 2005), by making a more rigorous connection between dislocation densities and the directional shear vectors v_ a and va . Clearly some parameters, such as v1, v2, vB,sat, and vR, can be linked to certain dislocation populations. However most dislocation-density-based hardening models for single crystals apply up to stage III and do not keep track of specific dislocation densities in different planes. As a consequence, they are not designed for predicting the flow response under strain path changes, particularly with pre-strains greater than 20–30%. The formulation presented here utilizes an intuitive understanding of the connection between dislocations and shear, while keeping to a minimum the parameters that need to be adjusted. Most importantly our formulation tackles the issues above in a manner which fully accounts for the crystallographic nature of slip, the directionality of shear and dislocation obstacles, and texture effects. Such a model provides a very general and stable approach for the simulation and analysis of arbitrary strain path changes in cubic materials, e.g., in rolling, Beyerlein and Tome´ (in press).

Acknowledgements Supports by Los Alamos Laboratory-Directed Research and Development project (No. 20030216) and Office of Basic Energy Sciences Project FWP 06SCPE401 are gratefully acknowledged. The authors are grateful to Prof. Armand Beaudoin for several insightful suggestions and discussions during the course of this work. References Alexander, D.J., Beyerlein, I.J., 2004. Mechanical properties of high-purity copper processed by equal channel angular extrusion. In: Zhu, Y.T. et al. (Eds.), Ultrafine Grained Materials III. TMS, Warrendale, PA, pp. 517–522. Alexander, D.J., Beyerlein, I.J., 2005. Anisotropy in mechanical properties of high-purity copper processed by equal channel angular extrusion. Mater. Sci. Eng., 480–484. Asaro, R.J., Needleman, A., 1985. Texture development and strain hardening in rate dependent polycrystals. Acta Metall. 33, 923–953. Banabic, D., Aretz, H., Comsa, D.S., Paraianu, L., 2005. An improved analytical description of orthotropy in metallic sheets. Int. J. Plast. 21, 493–512. Barlat, F., Aretz, H., Yoon, J.W., Karabin, M.E., Brem, J.C., Dick, R.E., 2005. Linear transformation-based anisotropic yield functions. Int. J. Plast. 21, 1009–1039. Barlat, F., Duarte, J.M.F., Gracio, J.J., Lopes, A.B., Rauch, E.F., 2003. Plastic flow for non-monotonic loading conditions of an aluminum alloy sheet sample. Int. J. Plast. 19, 1215–1244. Bassani, J.L., Wu, T.Y., 1991. Latent hardening in single crystals II. Analytical characterization and predictions. Proc. Roy. Soc. A435, 21–41. Beyerlein, I.J., Tome´, C.N., 2004. Analytical modeling of material flow in equal channel angular extrusion (ECAE). Mater. Sci. Eng. A380, 171–190. Beyerlein, I.J., Tome´, C.N., 2006. Modeling directional anisotropy in grains subjected to large strains and strain path changes. In: Zhu, Y.T., Langdon, T.G., Horita, Z., Zehetbauer, M.J., Semiatin, S.L., Lowe, T.C. (Eds.), Ultrafine Grained Materials, vol. IV. TMS, The Minerals, Metals & Materials Society, pp. 63–71. Beyerlein, I.J., Tome´, C.N., in press. In: Proceedings of THERMEC 2006, Materials Science Forum, published by Trans Tech Publications, Vancouver, Canada, July 3–7, 2006.

662

I.J. Beyerlein, C.N. Tome´ / International Journal of Plasticity 23 (2007) 640–664

Beyerlein, I.J., Li, S., Alexander, D.J., 2005a. Modeling the plastic anisotropy in pure copper after one pass ECAE. Mater. Sci. Eng. A410–A411, 201–206. Beyerlein, I.J., Li, S., Necker, C.T., Alexander, D.J., Tome´, C.N., 2005b. Non-uniform microstructure and texture evolution during equal channel angular extrusion. Philos. Mag. 85 (13), 1359–1394. Beyerlein, I.J., Alexander, D.J., Tome´, C.N., in press. Plastic anisotropy in aluminum and copper pre-strained by equal channel angular extrusion (ECAE). J. Mater. Sci. Canadinc, D., Sehitoglu, H., Maier, H.J., Chumlyakov, Y.I., 2005. Strain hardening behavior of aluminum alloyed Hadfield steel single crystals. Acta Mater. 53, 1831–1842. Cheong, K.-S., Busso, E.P., 2004. Discrete dislocation modelling of single phase FCC polycrystal aggregates. Acta Mater. 52, 5665–5675. Cheong, K.-S., Busso, E.P., Arsenlis, A., 2005. A study of microstructural length scale effects on the behavior of fcc polycrystals using strain gradient concepts. Int. J. Plast. 21, 1797–1814. Christodoulou, N., Woo, O.T., MacEwen, S.R., 1986. Effect of stress reversals on the work hardening behavior of polycrystalline copper. Acta Metall. 34 (8), 1553–1562. Devincre, B., Kubin, L., Hoc, T., 2006. Physical analyses of crystal plasticity by DD simulations. Scripta Mater. 54, 741–746. Estrin, Y., Toth, L.S., Molinari, A., Brechet, Y., 1998. Dislocation-based model for all hardening stages in large strain deformation. Acta Mater. 46 (15), 5509–5522. Follansbee, P.S., Kocks, U.F., 1988. A constitutive description of the deformation of copper based on the use of the mechanical threshold stress as an internal state variable. Acta Metall. 36, 81–93. Franciosi, P., Berveiller, M., Zaoui, A., 1980. Latent hardening in aluminum and copper single crystals. Acta Metall. 28 (3), 273–283. Garmestani, H., Vaghar, M.R., Hart, E.W., 2001. A unified model for inelastic deformation of polycrystalline materials—application to transient behavior in cyclic loading and relaxation. Int. J. Plast. 17, 1367–1391. Gracio, J.J., Barlatt, F., Rauch, E.F., Jones, P.T., Neto, V.F., Lopes, A.B., 2004. Artificial aging and shear deformation behavior of 6022 aluminum alloy. Int. J. Plast. 20, 427–445. Ghoniem, N.M., Tong, S.H., Sun, L.Z., 2000. Parametric dislocation dynamics: a thermodynamcis-based approach to investigations of mesoscopic plastic deformation. Phys. Rev. B B1 (1), 913–927. Hansen, N., Juul Jensen, D., 1992. Flow stress anisotropy caused by geometrically necessary boundaries. Acta Metall. Mater. 40, 3265–3275. Hart, E.W., 1976. Constitutive relations for nonelastic deformation of metals. J. Eng. Mater. Technol. 98, 193– 202. Hasegawa, T., Yakou, T., Karashima, S., 1976. Deformation behavior and dislocation structures upon stress reversal in polycrystalline aluminum. Mater. Sci. Eng. 20, 267–276. Hill, R., 1966. Generalized constitutive relations for incremental deformation of metal crystals by multislip. J. Mech. Phys. Solids 14, 95–102. Huang, X., Borrego, A., Pantleon, W., 2001. Polycrystal deformation and single crystal deformation: dislocation structure and flow stress in copper. Mater. Sci. Eng. A319–321, 237–241. Huang, X., Liu, Q., 1998. Determination of crystallographic and macroscopic orientation of planar structures in TEM. Ultramicroscopy 74 (3), 123–130. Hosford, W.F., 1993. The Mechanics of Crystals and Textured Polycrystals. Oxford University Press, New York, p. 29. Kalidindi, S.R., Anand, L., 1994. Macroscopic shape change and evolution of crystallographic texture in pretextured fcc metals. J. Mech. Phys. Solids 42, 459–490. Kocks, U.F., 1964. Latent hardening and secondary slip in aluminum and silver. Trans. Met. Soc. AIME 230, 1160–1167. Kocks, U.F., Tome´, C.N., Wenk, H.-R., 1998. Texture and Anisotropy. Cambridge University Press. Kocks, U.F., Mecking, H., 2003. Physics and phenomenology of strain hardening: the FCC case. Prog. Mater. Sci. 48 (3), 171–273. Kok, S., Beaudoin, A.J., Tortorelli, D.A., 2002. On the development of stage IV hardening using a model based on the mechanical threshold. Acta Mater. 50, 1653–1667. Lebensohn, R.A., Tome´, C.N., 1993. Self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: application to zirconium alloys. Acta Metall. Mater. 41, 2611–2624. Li, S., Beyerlein, I.J., Necker, C.T., Alexander, D.J., Bourke, M.A.M., 2004. Heterogeneity of deformation texture in equal channel angular extrusion of copper. Acta Mater. 52, 4859–4875.

I.J. Beyerlein, C.N. Tome´ / International Journal of Plasticity 23 (2007) 640–664

663

Li, S., Beyerlein, I.J., Alexander, D.J., 2005. Texture evolution during equal channel angular extrusion: effect of initial texture from experiment and simulation. Scripta Mater. 52, 1099–1104. Lopes, A.B., Barlat, F., Gracio, J.J., Ferreira Duarte, J.F., Rauch, E.F., 2003. Effect of texture and microstructure on strain hardening anisotropy for aluminum deformed in uniaxial tension and simple shear. Int. J. Plast. 19, 1–22. Mahesh, S., Tome´, C.N., McCabe, R.J., Kaschner, G.C., Beyerlein, I.J., Misra, A., 2004. Application of a substructure-based hardening model to copper under loading path changes. Metall. Mater. Trans. A35, 3763– 3774. Madec, R., Devincre, B., Kubin, L.P., 2002. From dislocation junctions to forest hardening. Phys. Rev. Lett. 89, 255508-1–255508-4. McCabe, R.J., Misra, A., Mitchell, T.E., 2004. Experimentally determined content of a geometrically necessary dislocation boundary in copper. Acta Mater. 52, 705–714. Miller, M.P., Harley, E.J., Bammann, D.J., 1999. Reverse yield experiments and internal variable evolution in polycrystalline metals. Int. J. Plast. 15, 93–117. Miyamoto, H., Erb, U., Koyama, T., Mimaki, T., Vinogradov, A., Hashimoto, S., 2004. Microstructure and texture development of copper single crystals deformed by equal-channel angular pressing. Philos. Mag. Lett. 84, 235–243. Nes, E., Hutchinson, W.B., Ridha, A.A., 1986. On the formation of microbands during plastic straining of metals. In: McQueen (Ed.), Proceedings of the Seventh International Conference on Strength of Metals and Alloys, vol. 1, pp. 57–62. Peeters, B., Kalidindi, S.R., Van Houtte, P., Aernoudt, E., 2000. A crystal plasticity based work hardening/ softening model for BCC metals under changing strain paths. Acta Mater. 48, 2123–2133. Peeters, B., Seefeldt, M., Teodosiu, C., Kalidindi, S.R., Van Houtte, P., Aernoudt, E., 2001. Work-hardening/ softening behaviour of BCC polycrystals during changing strain paths: I. An integrated model based on substructure and texture evolution, and its prediction of the stress–strain behaviour of an IF steel during twostage strain paths. Acta Mater. 49, 1607–1619. Phillips Jr., W.L., 1962. Aluminum and copper tested in direct shear. Trans. Met. Soc. AIME 22, 845– 850. Phillips Jr., W.L., Robertson, W.D., 1958. Work-hardening in the latent slip directions of alpha brass during easy glide. Trans. Met. Soc. AIME, 406–412. Rauch, E.F., 1992. The flow law of mild steel under monotonic or complex strain path. Solid State Phenom. (23/ 24), 317–334. Segal, V.M., 1995. Materials processing by simple shear. Mater. Sci. Eng. A197, 157. Stout, M.G., Rollett, A.D., 1990. Large strain Bauschinger effects in fcc metals and alloys. Metall. Trans. 21A, 3201–3213. Teodosiu, C., 1996. Plastic anisotropy induced by the microstructural evolution at large strains. RIKEN Rev: Focused Comput. Sci. Eng. 14, 35–36. Tome´, C.N., Canova, G.R., Kocks, U.F., Christodoulou, N., Jonas, J.J., 1984. The relation between macroscopic and microscopic strain hardening in FCC polycrystals. Acta Metall. 32 (10), 1637–1653. Tome´, C.N., Lebensohn, R.A., 2004. Self consistent homogenization methods for texture and anisotropy. In: Raabe, D., Roters, F., Barlat, F., Chen, L.Q. (Eds.), Continuum Scale Simulation of Engineering Materials: Fundamentals, Microstructures, Process Applications. Wiley-VCH Verlag GmbH & Co, KGaA, Weinheim, pp. 473–499. Toth, L.S., Molinari, A., Estrin, Y., 2002. Strain hardening at large strains as predicted by dislocation based polycrystal plasticity model. J. Eng. Mater. Technol. 124 (1), 71–77. Roos, A., De Hosson, J.T.M., Van der Giessen, E., 2001. High-speed dislocations in high strain rate deformations. Comput. Mater. Sci. 20, 19–27. Vincze, G., Rauch, E.E., Gracio, J.J., Barlat, F., Lopes, A.B., 2005. A comparison of the mechanical behavior of an AA1050 and a low carbon steel deformed upon strain reversal. Acta Mater. 53, 1005–1013. Wang, Z.Q., Beyerlein, I.J., LeSar, R., in press. Dislocation motion in high-strain-rate deformation. Philos. Mag. Wilson, D.V., Bate, P.S., 1986. Reversibility in the work hardening of spheroidised steels. Acta Metall. 34 (6), 1107–1120. Wilson, D.V., Zandrahimi, M., Roberts, W.T., 1990. Effects of changes in strain path on work-hardening in CP aluminum and an Al–Cu–Mg alloy. Acta Metall. Mater. 38, 215–226. Winther, G., Juul Jensen, D., Hansen, N., 1997. Modelling flow stress anisotropy caused by deformation induced dislocation boundaries. Acta Mater. 45, 2455–2465.

664

I.J. Beyerlein, C.N. Tome´ / International Journal of Plasticity 23 (2007) 640–664

Winther, G., 2005. Effect of grain orientation dependent microstructures on flow stress anisotropy modeling. Scripta Mater. 52, 995–1000. Winther, G., Huang, X., Godfrey, A., Hansen, N., 2004. Critical comparison of dislocation boundary alignment studied by TEM and EBSD: technical issues and theoretical consequences. Acta Mater. 52 (15), 4437. Wu, T.Y., Bassani, J.L., Laird, C., 1991. Latent hardening in single crystals I. Theory and experiment. Proc. Roy. Soc. A435, 1–19. Wu, P.C., Chang, C.P., Kao, P.W., 2004. The distribution of dislocation walls in the early processing stage of equal channel angular extrusion. Mater. Sci. Eng. A374, 196–203. Wu, P.D., Neale, K.W., Van der Giessen, E., 1996. Simulation of the behavior of fcc polycrystals during reversed torsion. Int. J. Plast. 12, 1199–1219. Wu, P.D., MacEwen, S.R., Lloyd, D.J., Jain, M., Tugcu, P., Neale, K.W., 2005. On pre-straining and the evolution of material anisotropy in sheet metals. Int. J. Plast. 21, 723–739. Xue, Q., Beyerlein, I.J., Alexander, D.J., in press. Mechanisms for initial grain refinement in OFHC copper in equal channel angular extrusion. Acta Mater. Yoon, J.W., Barlat, F., Gracio, J.J., Rauch, E.F., 2005. Anisotropic strain hardening behavior in simple shear for cube textured aluminum alloy sheets. Int. J. Plast. 21, 2426–2447.