Roles of texture and latent hardening on plastic anisotropy of face-centered-cubic materials during multi-axial loading
Introduction
Automotive part production demands accurate modeling and simulation of the material response during metal forming processes. It is straightforward to predict the isotropic behavior (a material behaves identically in all loading directions). The von Mises hypothesis is one of the most common approaches for predicting the isotropic response at the continuum scale. The von Mises hypothesis of isotropy assumes that the flow stress in the deviatoric stress space at the continuum scale should follow these conditions:
- (1)
the stress at which the material initially yields is not a function of material orientation with respect to the frame of the test (i.e., isotropic yielding);
- (2)
there exists a multi-axial yield locus that is described by a single value of stress that corresponds to yield in uniaxial tension (i.e., stress equivalency);
- (3)
on hardening, the multi-axial yield locus expands by the same amount in every direction in the π-plane, which is the plane that has its normal parallel to [111] in the deviatoric stress space (i.e., isotropic hardening);
- (4)
there is an associated flow rule, i.e., the strain increment is normal to the yield locus.
Requirements (3) and (4) lead to a consequence that a stress ratio corresponding to any given strain ratio remains constant and is equal to the strain ratio during plastic deformation. In other words, there are specific stress (or strain) ratios for different loading paths. In this present study, “loading” can be either mechanical straining or stressing, and a loading path means a locus of stress (or strain) states that a material is subjected to during deformation. Examples of (monotonic) loading paths in this study include uniaxial tension, plane strain tension and balanced biaxial tension. In addition, isotropic hardening requires that the hardening rate is the same for all loading paths.
In practice, it is challenging to accurately model complex behavior during multi-axial forming along various loading paths because most engineering materials behave differently along different loading directions, i.e., anisotropic behavior. The anisotropic behavior can be considered as the deviation of material response from an ideally isotropic response (e.g., von Mises isotropy). In this paper, the plastic anisotropy will be represented by the deviation in both the initial plastic yield and hardening behavior, while the anisotropic hardening is characterized by the evolution of stress (or strain) and the increment in stress (i.e., hardening rate) during plastic deformation. It is often seen that stress (or strain) paths, in response to uniaxial (or plane strain or equi-biaxial) deformation, do not follow ideal isotropic stress (or strain) paths. The hardening rate is also not the same for all loading paths. During forming, the anisotropic response of engineering materials during plastic deformation is complex and challenging to model accurately. This study aims to understand the plastic anisotropic response during multi-axial loading of face-centered-cubic materials via a dislocation-based approach in order to provide information for improving constitutive models for accurate predictions of the material behavior during forming.
There has been substantial modeling effort devoted to progressing from empirical models to physics-based models, particularly for modeling anisotropy (Taylor, 1938, Asaro and Needleman, 1985, Lebensohn and Tomé, 1993), because these approaches are expected to enhance the accuracy and effectiveness of models. The anisotropy is known to be a function of the crystallographic texture of the material (Kocks et al., 1998). In addition, it is also known that dislocation behaviors, such as dislocation interactions (both with other dislocations and other defects), result in changes in the yielding and hardening behavior (Franciosi, 1985a, Franciosi, 1985b, Kocks et al., 1991, Franciosi et al., 1980). One of most obvious manifestations of plastic anisotropy due to dislocations is referred to as latent hardening (also known as “cross-hardening”). Latent hardening is used to describe yielding and hardening phenomena, where a higher yield strength and higher hardening rate are often observed in polycrystalline materials when the strain path changes (Franciosi, 1985a, Franciosi et al., 1980). Latent hardening results from the fact that the activity on one slip system affects other systems to different degrees because the stored dislocations on one system act as forest dislocations on intersecting slip systems (Franciosi, 1985a, Franciosi et al., 1980). In particular, during sheet metal forming, the multiaxial stress state induces slip activity on multiple systems, resulting in the formation of a spectrum of dislocation junctions and their associated strengths (Franciosi, 1985a, Franciosi et al., 1980, Madec et al., 2003, Hirth, 1961). The junctions act as obstacles to mobile dislocations on intersecting slip systems. Strain path changes can activate new sets of dislocation interactions, resulting in different dislocation junctions and, consequently, changes in hardening behavior (Franciosi, 1985a, Franciosi et al., 1980, Pham et al., 2013). Even during monotonic loading, a specific loading path results in the formation of dislocation junction types different than those along other paths, leading to anisotropic hardening behavior for different monotonic loading paths (Franciosi, 1985a, Franciosi et al., 1980). In other words, dislocation interaction-based latent hardening significantly contributes to plastic anisotropy not only during loading path changes, but also during different monotonic loading paths. Consequently, to understand and then accurately model the mechanical response during forming, it is necessary to study not only the influence of texture on plastic anisotropy, but also (1) those of dislocation interaction-induced latent hardening and (2) the synergistic influences of both texture and latent hardening.
Although dislocation-based hardening models have been intensively applied in constitutive models for plastic anisotropy, most models employ a self-hardening law based on self-interaction of dislocations. Recently, there have been efforts to include non-self-interactions of dislocations in constitutive models to account for latent hardening. For example, clear differentiation between self and non-self-interactions (Peirce et al., 1983, Erinosho and Cocks, 2013, Erinosho et al., 2013) (or between coplanar and non-coplanar interactions (Miraglia et al., 2007), or between collinear and non-collinear interactions (Hoc and Forest, 2001)) was used to better account for latent hardening. There have also been efforts to include more detailed dislocation interactions (e.g., (Madec et al., 2003; Liu et al., 2014; Dequiedt et al., in press; Devincre et al., 2006; Pham et al., 2015)). However, it still requires significant effort to understand the plastic anisotropy under the influence of specific latent hardening sources and texture conditions, in particular during multi-axial forming conditions. It is also interesting to note that it is unclear whether latent hardening influences anisotropy and/or texture evolution. Kocks et al. (1991) showed that latent hardening has little effect on texture development, as well as on the plastic anisotropy (except for compression). However, more recent work argued that latent hardening can significantly affect the evolution of texture compared to self-hardening (Miraglia et al., 2007, Toth et al., 1997, Young et al., 2006). Consequently, the present study aims at understanding the plastic anisotropy under the presence of texture with and without latent hardening to provide some insight for the above-mentioned discrepancies in the literature.
In order to explicitly account for dislocation interactions, one could use discrete dislocation dynamics, which, however, are currently limited to single crystals or idealized multi-crystal domains. When one realizes that the problems in metal forming involve complex polycrystals, it is not practical to use discrete dislocation dynamics to systematically study the anisotropy of materials in multi-axial deformation along complex load paths with various initial textures. Our objective is to use a simulation platform that enables a reasonable prediction of texture evolution of an ensemble of grains during deformation while allowing for the inclusion of dislocation interactions inside each grain. This enables us to study the separate and joint effects of textures and latent hardening on the anisotropy of metals. The self-consistent approach is known for its texture prediction capability with the large number of grains (Lebensohn and Tomé, 1993) that is necessary for sheet metal forming while being only moderately computationally expensive. It is readily adapted to include effects of latent hardening (Pham et al., 2015, Hu et al., 2012). In this study, all types of dislocation interactions in pairs in face-centered-cubic (FCC) materials and texture are incorporated in a visco-plastic self-consistent (VPSC) model to understand the relationships between latent hardening, texture and plastic anisotropy. The plastic anisotropy of FCC materials as a function of Random texture along a number of monotonic straining paths with self-hardening is first investigated as a baseline (Section 3.3). Corresponding discussions of these results are given in Section 4.2. Subsequently, the effects of dislocation-based latent hardening without the presence of preferred texture on plastic anisotropy are presented in Section 3.4 and discussed in Section 4.3. Finally, plastic anisotropy under the presence of both preferred textures and latent hardening is investigated to understand the synergistic influences of texture and latent hardening on the overall anisotropic response (3.5 Initial texture: Cube, 3.6 Initial texture: Copper, 3.7 Initial texture: S, 3.8 Brass component, 3.9 Initial texture: Goss, 4.4 Synergistic roles of texture and latent hardening on plastic anisotropy). Based on this analysis, we seek possible explanations (Section 4.4) for the discrepancy in the influence of latent hardening on plastic anisotropy between Kocks and other studies as referred to above (Kocks et al., 1991, Miraglia et al., 2007, Toth et al., 1997).
Section snippets
Stress-strain constitutive model
The stress-strain constitutive relationship is described in this section. The detailed treatment of the deformation slip kinematics is given in (Tomé and Lebensohn, 2012). At the grain level, the plastic strain rate is given by the sum of the shear strain rates from all the active slip systemswhere N is the number of slip systems, is the Schmid tensor of the slip system . The Schmid tensor is defined aswhere is the slip direction and is the slip
Results
The combinations of 5 different paths, 6 hardening models and 6 initial texture components result in 180 different simulations of the flow stress response and texture evolution. In the following sub-sections, we plot stress (or strain) for all 5 different loading paths in a sub-figure. Sub-figures that correspond to the same initial texture are shown in the same row, while those corresponding to a given hardening model are shown in the same column. For plotting the texture development,
Influences of initial texture and latent hardening on strain paths during uniaxial loading
The uniaxial loading in the RD (or TD) direction was done by imposing both velocity gradient in the RD (or TD) direction and the normal stress components of in the TD and ND (or RD and ND) directions (to assure that or for uniaxial loading in the RD or TD, respectively). This imposed loading condition is different to those of PS and EB straining during which the velocity gradient were controlled. Because all considered textures (except the Brass) have an r-value in the RD
Conclusions
The separate and synergistic roles of crystallographic textures (Random, Cube, Copper, S, Brass and Goss) and latent hardening (Collinear, Coplanar, Glissile, Hirth and Lomer-Cottrell) on the anisotropy behavior of FCC polycrystalline materials were comprehensively studied in order to obtain insight into the complex plastic anisotropy during multi-axial loading. The main result is that texture and individual latent hardening types can counteract others to produce effective isotropy or act
Acknowledgements
We would like to thank Dr. Carlos Tomé and Dr. Ricardo Lebensohn at Los Alamos National Laboratory (USA) for providing the VPSC program. This work was supported by the National Institute of Standards and Technology (NIST) Center for Automotive Lightweighting. The financial support of Materials Genome Initiative at NIST is also acknowledged.
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