3-D simulation of spatial stress distribution in an AZ31 Mg alloy sheet under in-plane compression

Abstract

A complete 3-D crystal plasticity finite element method (CPFEM) that considered both crystallographic slip and deformation twinning was applied to simulate the spatial distribution of the relative amount of slip and twin activities in a polycrystalline AZ31 Mg alloy during in-plane compression. A microstructure mapping technique that considered the grain size distribution and microtexture measured by electron backscatter diffraction (EBSD) technique was used to create a statistically representative 3-D microstructure for the initial configuration. Using a 3-D Monte Carlo method, a 3-D digital microstructure that matched the experimentally measured grain size distribution was constructed. Crystallographic orientations obtained from the EBSD data were assigned on the 3-D digital microstructure to match the experimentally measured misorientation distribution. CPFEM captured the heterogeneity of the stress concentration as well as the slip and twin activities of a polycrystalline AZ31 Mg alloy during in-plane compression.

Introduction

Mg alloys exhibit excellent strength-to-weight and stiffness-to-weight ratios compared with those of several other structural materials. In particular, wrought Mg alloys have been applied to electric and lightweight structural parts during recent decades (Kainer et al., 2007, Luo et al., 2007). However, because of their inferior ductility at temperatures near room temperature (RT), their application is limited with parts that depend on warm-forming technology (Yin et al., 2005, Agnew and Duygulu, 2005). It is well known that the critical resolved shear stress (CRSS) of non-basal slip systems, which are favorable to the enhancement of ductility, is much higher than that of a basal slip system at temperatures near RT (Agnew and Duygulu, 2005, Kelly and Hosford, 1968, Agnew et al., 2001, Jain and Agnew, 2007). Therefore, only a limited number of slip systems can be activated to accommodate the external deformation during plastic deformation at temperatures near RT. The various deformation modes, such as basal 〈a〉 slip, prismatic 〈a〉 slip, pyramidal 〈a〉, pyramidal 〈c + a〉 slip, and tensile twinning, mean that the deformation behavior of Mg alloys is complicated compared to, for example, cubic metals. When the c/a ratio of a hexagonal metal (Mg: 1.624) is less than $\sqrt{3}$ (≅1.732), a tensile twin is easily activated by c-axis tension (Yoo, 1981, Yoo and Lee, 1991). Deformation twinning affects both strain hardening behavior and texture evolution (e.g. Reed-Hill, 1960, Wonsiewicz and Backofen, 1967, Kelly and Hosford, 1968, Hartt and Reed-Hill, 1968, Nave and Barnett, 2004, Jiang et al., 2006, Wang and Huang, 2007, Lou et al., 2007, Khan et al., 2011). A number of simulation studies have been conducted to understand texture evolution in Mg alloys during plastic deformation. In particular, many studies have investigated the effect of twin reorientation on texture evolution, macroscopic stress–strain response and macroscopic shape changes during plastic deformation. The key point is that, when a volume of material undergoes twinning, its crystal orientation jumps to the new orientation dictated by the twinning geometry, rather than the gradual rotation that occurs during slip. Several computational schemes that incorporate crystallographic reorientation due to deformation twinning have been proposed (Houtte, 1978, Tomé et al., 1991, Kalidindi, 1998). The approach described by Van Houtte (1978) requires a large number of grain orientations to satisfy the statistical criterion used in the scheme. A predominant twin reorientation (PTR) scheme (Tomé et al., 1991) has been suggested to overcome this disadvantage. The PTR scheme readily tracks the new orientation created by deformation twinning. In the PTR scheme, twinning is considered to be a pseudo-slip mechanism, and a grain is allowed to reorient if an accumulated value reaches a specified threshold. The PTR scheme has been successfully implemented in visco-plastic self-consistent (VPSC) models (Molinari et al., 1987, Lebensohn and Tomé, 1993, Choi et al., 2000) to capture grain reorientation due to deformation twinning (e.g. Agnew et al., 2001, Brown et al., 2005, Yi et al., 2006, Choi et al., 2007, Proust et al., 2007, Proust et al., 2009, Neil and Agnew, 2009). The VPSC model uses a homogenization scheme that can successfully predict strain hardening and texture evolution during plastic deformation. An advantage of the scheme is that the total number of grain aggregates does not change during VPSC simulation. However, a remaining disadvantage of the scheme is that a grain is reoriented only with respect to its dominant twin variant and more than one twin variant in the parent grain cannot be considered simultaneously. Microtexture analysis (Choi et al., 2007, Beyerlein et al., 2010) revealed that the number of active twin variants and the types of dominant twin variants are also dependent on the orientation of their parent grains. Crystal plasticity finite element methods (CPFEM) that are based on an inhomogenization scheme were developed to simulate heterogeneous plastic deformation of hexagonal close-packed (HCP) polycrystalline materials (e.g. Balasubramanian and Anand, 2002, Staroselsky and Anand, 2003, Graff et al., 2007, Wu et al., 2007a, Wu et al., 2007b, Hama and Takuda, 2010, Mayama et al., 2011). Numerical formulation and verification of CPFEM has been performed for FCC and BCC materials that deform by crystallographic slip (e.g. Becker, 1991, Becker and Panchanadeeswaran, 1995, Sarma et al., 1998, Raabe et al., 2002, Choi, 2003). Unlike FCC and BCC materials, crystallographic slip and deformation twinning are important deformation modes for HCP materials. An awkward aspect of the current CPFEM theoretical framework is the incorporation of a reorientation scheme for deformation twinning into the constitutive equations. Staroselsky and Anand (2003) used a probabilistic approach to simulate both the texture evolution and the stress–strain response of a polycrystalline Mg alloy. In this approach, the grain orientations were replaced with twin-related orientations only if the twinned volume fraction exceeded a certain random number. Kalidindi (1998) proposed a total Lagrangian approach to simulate rolling textures in a polycrystalline Zr alloy. The same theoretical framework was used to simulate both the texture evolution and the macroscopic stress–strain response in high purity α-Ti (Salem et al., 2005, Wu et al., 2007a, Wu et al., 2007b). Forming limit diagrams for AM30 Mg alloy tubes were simulated by combining this approach with M–K analysis (Lévesque et al., 2010). Walde and Riedel (2007) used the PTR model to simulate earing profiles after deep drawing and calculated the effects of the initial texture components on earing profile evolution. The authors did not, however, compare their theoretical results with experiments. Graff et al. (2007) conducted a simulation of channel die compression tests that revealed both strong anisotropy and asymmetric yield behavior. This model did not take the crystallographic reorientation of the twinned volume into account. The effect of deformation twinning on the heterogeneous stress concentration at either the grain boundaries or twin boundaries must be understood theoretically to improve the mechanical properties of AZ31 Mg alloys. The CPFEM-based theoretical studies that have been conducted to date are insufficient to explain how the deformation twinning affects the stress concentration at grain or twin boundaries in AZ31 Mg alloys during plastic deformation. In a previous paper (Choi et al., 2010a), CPFEM was used to simulate the spatial stress concentration in a deformed AZ31 Mg alloy under in-plane compression. Electron backscatter diffraction (EBSD) data were directly mapped (Erieau and Rey, 2004, Héripré et al., 2007) onto quasi 3-D finite element meshes to consider a real microstructure as the initial configuration. A PTR scheme was modified to simultaneously consider activation of more than one twin variant in the parent grains. This modified PTR scheme was successfully implemented to capture grain reorientation due to deformation twinning in twin-dominated deformation. The CPFEM successfully simulated more than one twin orientation in the parent grains. It also successfully modeled the heterogeneous stress concentration at the grain level during in-plane compression. However, since the quasi 3-D mesh of the model could not be used to consider the interactions with neighboring grains above and below the model grains, a complete 3-D mesh was required to capture the interactions. However, polycrystalline AZ31 Mg alloys are known to have inhomogeneous size and shape distributions, and can also exhibit significant morphological variation. It is expected that the artificial grain geometry constructed from a uniform array of cubes or a simple polyhedron (Choi, 2003, Delannay et al., 2006) may have difficulty in capturing inhomogeneous plastic deformation in a realistic microstructure. Several techniques have been suggested to create a 3-D digital microstructure that fills the 3-D space of polycrystalline materials. The Voronoi tessellation or 3-D Monte Carlo (MC) method can be used to construct a statistically representative microstructure (Kanit et al., 2003, Buchheit et al., 2005, Diard et al., 2005, Delannay et al., 2009). More rigorous approaches have been suggested to consider the statistical distribution of grain size and shape obtained from experimental observation of the 2-D orthogonal sections (Saylor et al., 2004, Brahme et al., 2006, St-Pierre et al., 2008). 3-D XRD microscopy (Ludwig et al., 2009) is an emerging non-destructive technique by which real 3-D microstructures can be constructed without destroying the specimen as occurs with serial sectioning by a focused ion beam (Xu et al., 2007, Groeber et al., 2008).

In the present work, CPFEM was used for a complete 3-D simulation of the spatial distribution of the stress concentration in a deformed AZ31 Mg alloy under in-plane compression. A microstructure mapping technique that considered both the average grain size and microtexture measured by an EBSD technique was used to create a statistically representative 3-D digital microstructure for the initial configuration. A modified PTR scheme was implemented in the CPFEM to simulate the activation of more than one twin variant in the parent grains during twin-dominated deformation. A new implicit time integration scheme was adopted to achieve a more stable and accurate time integration of the constitutive relationships. CPFEM captured both the heterogeneity of the stress concentration as well as the slip and twin activities through the thickness direction in a polycrystalline AZ31 Mg alloy during in-plane compression.