FFT-based investigation of the shear stress distribution in face-centered cubic polycrystals
Introduction
Due to their polycrystalline structure, metals display anisotropic behavior at the grain level inducing a heterogeneous local stress distribution. This distribution has a major influence on nonlinear effects and failure mechanism, such as micro-plastic deformations, e.g. grain boundary pop-ins (Javaid et al., 2020), or incipient damage, e.g. fatigue fracture (Blochwitz et al., 1996). Therefore, investigating the local stress distribution in polycrystalline materials has been the subject to numerous experimental, analytical and numerical studies.
In experiments, techniques such as neutron diffraction (Clausen et al., 1999, Pang et al., 2000), X-ray diffraction (Tamura et al., 2003, Lienert et al., 2004), electron back-scattering diffraction (EBSD) (Blochwitz et al., 1996, Raabe et al., 2001) and digital image correlation (DIC) (Tasan et al., 2014, Raabe et al., 2001) were used for investigating the mechanical fields on the microscale. For instance, in their recent article Berger et al. (2022) use the latter two techniques for a detailed examination of the spatial strain distribution in -FE polycrystals. As experiments are associated with considerable cost and effort, analytical and numerical approaches appear attractive for predicting the stress distribution, owing to their higher flexibility. However, in this context, experimental data is still crucial for validating analytical (Clausen et al., 1999, Raabe et al., 2001) and numerical results (Lienert et al., 2004, Demir and Gutierrez-Urrutia, 2021). Therefore, according to Berger et al. (2022), the dialogue between experiments and simulations should be sought.
Early analytical homogenization approaches assumed a homogeneous strain Voigt, 1928, Taylor, 1938, Bishop and Hill, 1951a, Bishop and Hill, 1951b or stress field (Reuss, 1929, Sachs, 1928), enabling a (crude) estimate for the localization of the conjugate stress and strain field, respectively. More sophisticated models, using a statistical description of the microstructure, were developed to arrive at more accurate predictions. For instance, the self consistent approach for linear (Hershey, 1954, Kröner, 1958) and nonlinear material behavior (Hill, 1965, Hutchinson, 1976, Molinari et al., 1987, Lebensohn and Tomé, 1993) has been exploited for predicting the first and second statistical moment of the stress field in polycrystals (Lebensohn et al., 2004, Brenner et al., 2009). A different analytical technique is the Maximum Entropy Method (MEM) pioneered by Kreher and Pompe (1989), see Krause and Böhlke (2020) for a review and an in-depth study on the capabilities of the method.
In numerical studies, full-field simulations are carried out on discretized microstructures. As solvers, commercial finite element software (Hashimoto and Margolin, 1983, Kumar et al., 1996, Barbe et al., 2001, Sauzay, 2007, Wong and Dawson, 2010) and dedicated FFT-based micromechanics codes (Lebensohn et al., 2004, Brenner et al., 2009, Lavergne et al., 2013, Castelnau et al., 2006) are most commonly used. In particular, the computational efficiency of FFT-based methods (Moulinec and Suquet, 1994, Moulinec and Suquet, 1998) and improvements in solver technology (Michel et al., 2001, Gélébart and Mondon-Cancel, 2013, Wicht et al., 2020b) have enabled the study of larger samples (Brenner et al., 2009, Kasemer et al., 2020). Due to the flexibility of the full-field approach, various studies have focused on different aspects of the stress localization in polycrystals. For instance, it was shown that the neighborhood effect – comprising the impact of the neighboring grains’ stiffness, their relative position with respect to loading axis and their crystallographic orientation (Bretin et al., 2019) – leads to a large scatter of the mean stress in a grain Sauzay, 2007, Guilhem et al., 2010, Bretin et al., 2019, Castelnau et al., 2020, Gélébart, 2021. This influence is suspected to be greater than the scatter resulting from the grain’s own orientation (Sauzay, 2007). However, a grain’s orientation has a major impact on the stress distribution in the plastified regime (Wong and Dawson, 2010, Bieler et al., 2009), whereas, in the elastic regime and during the elastic–plastic transition, the single crystal anisotropy is more crucial (Sauzay, 2007, Kumar et al., 1996, Wong and Dawson, 2010, Lebensohn et al., 2012). In addition to the grain’s orientation, the hardening behavior tends to influence the stress distribution in fully plastified polycrystals (Kasemer et al., 2020). A detailed study on the impact of phenomenological and physical hardening laws can be found in Demir and Gutierrez-Urrutia (2021). At the grain boundaries, on average, larger disorientations (Kühbach and Roters, 2020), a higher number of active slip systems and an increased sum of plastic slips (Cailletaud et al., 2003) can be found. The resulting local stress and strain concentrations are suspected to be the precursor of crack initiation and growth (Cailletaud et al., 2003, Guilhem et al., 2010) although Bieler et al. (2009) state that the first microcracks do not occur at the exact location of the stress or strain peaks but in close proximity. Therefore, the spatial distribution of the stresses has attracted increased research interest (Cailletaud et al., 2003, Rollett et al., 2010, Gonzalez et al., 2014, Hure et al., 2016, El Shawish et al., 2020, Kühbach and Roters, 2020).
The present work is devoted to investigating the local shear stress distribution using FFT-based solvers in face-centered cubic (fcc) polycrystals (Lebensohn et al., 2012, Sauzay, 2007, Lim et al., 2019). In particular, we study the stress distribution with respect to grain orientations, focusing on the , and grains. For the elastic regime, we follow the strategy of Brenner et al. (2009) and Sauzay (2007), studying the characteristics of the stress distribution based on a large ensemble. In particular, we focus on the impact of elastic anisotropy and the specific shape of the stress distribution. The results are supplemented by spatial information, taking the distance with respect to grain boundaries into account (Rollett et al., 2010, Castelnau et al., 2020). Furthermore, we compare our results to predictions of the Maximum Entropy Method (MEM), which has proven to be a powerful analytical tool for polycrystals (Krause and Böhlke, 2020). In the plastic regime, studies are often limited to a single realization (Rollett et al., 2010, Gonzalez et al., 2014) or a small ensemble (e.g. 30 realizations (Vincent et al., 2011)), due to the associated computational cost. Thus, an analysis based on a large ensemble is of interest to obtain representative results. Building upon the elastic results, we investigate the onset of plastification. More precisely, we study the characteristics of the stress distribution during the elastic–plastic transition and correlate the grain boundary distance to incipient plasticity.
The outline is as follows. After a short introduction of the methods in Section 2, a preliminary study follows in Section 3 in order to determine the required resolution and ensemble size. In the first part of the actual study, the influence of anisotropy on the stress distribution in the linear elastic range is examined (see Section 4.1). An analysis of the specific shape of the stress distribution is given in Section 4.2, followed by a comparison to results of the Maximum Entropy Method (MEM) in Section 4.3 and an investigation of the stress distribution with respect to grain boundary distance in Section 4.4. Secondly, the stress distribution is examined during the elastic–plastic transition in Section 5.1, taking a closer look at the location of the plastified regions in Section 5.2.
Section snippets
Crystal plasticity
For the present study, we rely on a small-strain single-crystal elasto-viscoplasticity model. The elastic behavior is governed by Hooke’s law where the total strain is decomposed additively into an elastic part and a plastic part . As we restrict to crystals with cubic crystal symmetry, the stiffness tensor is specified by three elastic constants (, , ) and the orientations of the lattice vectors. The Zener parameter serves as
Resolution study for a single microstructure
Computing the stress and strain field on a microstructure is associated with considerable computational effort, especially for more involved material laws such as the crystal plasticity model in Section 2.1. Thus, before committing to computations on a large ensemble, we seek a suitable resolution which limits the computational cost but is still fine enough to permit a meaningful investigation of the stress field. More precisely, for a single microstructure discretized with various
Influence of anisotropy
In the following, we investigate the stress distribution of fcc polycrystals in the linear elastic range, focusing on the effect of anisotropy. To this end, we consider a set of 4 materials with increasingly anisotropic stiffness, see Table 2, with aluminum being nearly isotropic () and -plutonium as the most anisotropic example.
For illustrating the different degrees of anisotropy, the Young’s modulus as a function of load direction is visualized in Fig. 5 in the crystal coordinate
Statistical moments
In the following, we expand our investigation of the stress distribution into the elastic–plastic range. To keep computational costs manageable, we restrict to a single set of material parameters corresponding to copper, see Table 5, taken from Eghtesad et al. (2018) and adapted to the material model in Section 2.1 by Wicht et al. (2020a). For the investigation at hand, we focus on the initial stages of plastification, i.e. the transition from the elastic to the plastic regime, building upon
Conclusions
In the present work, the local shear stress distribution in polycrystalline metals with fcc crystal structure and anisotropic stiffness is investigated numerically using FFT-based solvers. In particular, we focus on characteristically oriented grains in the linear elastic range and at incipient elasto-viscoplastic deformation. All studies are carried out on a data set of 200 microstructures.
Regarding the influence of anisotropy on the stress distribution in the linear elastic range, an increase
CRediT authorship contribution statement
Flavia Gehrig: Methodology, Software, Formal analysis, Investigation, Data curation, Visualization, Writing – original draft, Writing – review & editing. Daniel Wicht: Conceptualization, Methodology, Software, Formal analysis, Investigation, Writing – review & editing. Maximilian Krause: Software, Formal analysis, Writing – review & editing. Thomas Böhlke: Supervision, Project administration, Funding acquisition, Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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2023, Mechanics Research CommunicationsCitation Excerpt :There is a large literature on geometric modelling of polycrystalline metals and foams using Laguerre tessellations and weighted Voronoi diagrams; see for example the following recent papers and their references: [1–7]. Applications include generating representative volume elements (RVEs) for computational homogenisation [8,9], fitting Laguerre tessellations to imaging data of polycrystalline microstructures [10,11], and modelling grain growth [12]. This paper builds on the research programme initiated in [2], where recent results from optimal transport theory [13] were exploited to develop fast algorithms for generating Laguerre tessellations with grains of given volumes.