A dislocation density based crystal plasticity finite element model: Application to a two-phase polycrystalline HCP/BCC composites
Introduction
Two-phase metal multilayered composites are gaining much interest due to enhanced properties, such as strength and radiation resistance, compared to their constituents. Many of these exciting results arose from research on small-scale thin film composites made by bottom-up processing such as deposition techniques (Anderson et al., 2003, Beyerlein et al., 2013a, Beyerlein et al., 2013c, Chu and Barnett, 1995, Embury and Hirth, 1994, Ham and Zhang, 2011, Mara et al., 2008, Misra and Kung, 2001, Was and Foecke, 1996). It has also been demonstrated that top-down metal working techniques can make large-scale (bulk) composites of similar composition and architecture as deposited films for a number of material systems: Cu/Ag (Han et al., 1999, Ohsaki et al., 2007), Cu/Nb (Carpenter et al., 2012a, Lee et al., 2012, Segal et al., 1997, Zheng et al., 2013), Al/Zn (Dehsorkhi et al., 2011), Cu/Ni (Liu et al., 2011), Ag/Fe (Yasuna et al., 2000) and Zr/Nb (Knezevic et al., 2013b, Knezevic et al., submitted for publication). Some of these have been tested and found to exhibit high hardness, excellent radiation resistance, good shock resistance and outstanding thermal stability (Beyerlein et al., 2013b, Carpenter et al., 2012b, Carpenter et al., in press, Han et al., 2014, Monclús et al., 2013). However, unlike the deposited multilayers, these bulk composites experience substantial changes in microstructure, such as texture, grain shape, grain boundary properties, bimetal interface and internal grain dislocation storage due to the large-strain processing (Carpenter et al., 2012a, Carpenter et al., in press, Knezevic et al., 2013b, Saito et al., 1999, Valiev and Langdon, 2006, Xue et al., 2007). Thus, it is important to understand and have the ability to predict the microstructure and texture evolution in composites under large strain deformations.
For this purpose, several plasticity models have been developed and applied to two-phase composites. Some works have applied mean-field polycrystalline schemes, by either treating the phases separately or as a homogeneous medium with the appropriate volume fractions of each phase (Beyerlein et al., 2011a, Knezevic et al., submitted for publication). In these codes, however, the effects of co-deformation at the interface on texture evolution were not modeled explicitly. Any possible adjustments in slip activity or lattice reorientation of the two grains on either side of the interface were missed. Spatially resolved polycrystal codes such as crystal plastic finite element models (CPFE) (Kalidindi et al., 1992, Kalidindi et al., 2009, Knezevic et al., 2010, Roters et al., 2010) and Green׳s function fast Fourier transform (FFT) models (Eisenlohr et al., 2013, Lebensohn et al., 2012) can potentially overcome many of the above-mentioned shortcomings. These classes of techniques directly model the grain microstructure and the effects of grain–grain interactions on local granular stress and strain fields. Many of the earlier studies applied CPFE to single-phase materials to capture grain neighbor effects (Diard et al., 2005, Kalidindi et al., 2004, Kalidindi et al., 2006, Roters et al., 2010, Zhao et al., 2008). A few recent studies, however, have used CPFE to study the deformation in two-phase systems (Hansen et al., 2013, Jia et al., 2013, Mayeur et al., 2013). Hansen et al. (2013) showed that CPFE can successfully predict the evolution of texture in roll-bonded Cu–Nb with polycrystalline layers. CPFE has also been used to study the onset of shear banding in Cu/Nb and Cu/Ag bicrystals (Jia et al., 2013). A recent study of Cu/Nb bicrystals (Mayeur et al., 2013) explored the stability of interface character under rolling deformation. Notably, the interfaces deemed “stable” by this CPFE model coincided with those observed experimentally to prevail in nano-layered Cu/Nb composites. A subsequent CPFE study of two-phase composites with polycrystalline layers revealed that the orientational stability of grains attached to the interface deviated from those within the bulk layers (Mayeur et al., 2014).
We note that most of the foregoing two-phase studies involve combinations of cubic metals: either two immiscible fcc metals or an fcc and bcc metal. Similar modeling studies on hcp materials (Knezevic et al., 2010) or two-phase combinations involving hcp metals are limited (Venkataramani et al., 2008). The plasticity of hcp metals is more complex than that of cubic materials. In HCP metals, there are several slip and twin modes, each corresponding to planes of different atomic density. The main slip modes are basal 〈a〉 slip, 1/3[112̄0](0001), prismatic 〈a〉 slip, 1/3, and pyramidal 〈c+a〉 slip, of which there are two types: 1/3〈112̄3̄〉{112̄2} or 1/3 (Partridge, 1967, Yoo, 1981, Yoo et al., 2002). The dislocations associated with these different slip modes possess different atomic core structures (Bacon and Vitek, 2002, Hirth and Lothe, 1968), and their mobilities have their own individual dependencies on temperature and strain rate (Beyerlein et al., 2011b). The development of spatially resolved, polycrystalline models, such as CPFE, for two-phase hcp-based systems, presents a new challenge in modeling of multilayered composites to large plastic strains.
The goal of this work is to develop a 3D multiscale code for bi-phase polycrystalline metals, where the crystal structures may include combinations of fcc, bcc, and hcp. Unlike prior 3D versions, this code employs at the meso-scale, a crystal plasticity finite element homogenization, and at the sub-grain scale, a constitutive law governed by the evolution of dislocations on individual slip systems. As a full field model satisfying both stress equilibrium and strain compatibility, CPFE is expected to provide better predictions of local and overall behavior and microstructure evolution of materials containing co-deformation of multiple phases. In this framework, polycrystalline phases are discretized into finite elements and a single crystal hardening law operates at each FE integration point. To describe the resistance to slip for individual slip systems in the individual crystals as a function of strain, temperature, and strain rate, we employ a dislocation density (DD) based hardening law developed in Beyerlein and Tomé (2008). The model includes two populations of DD: forest dislocations mainly responsible for hardening up to and including stage III and substructure dislocations mainly responsible for hardening stage IV. To model extremely large strains, we present a technique to pass key state variables, the crystal orientation and dislocation density, from the deformed (old) to the undeformed (new) mesh.
In prior work, the DD hardening model has been successfully applied to several metals, differing in crystal structure, such as Haynes 25 (Knezevic et al., 2014), Nb (Knezevic et al., in press), Mg (Beyerlein et al., 2011c), Zr (Beyerlein and Tomé, 2008, Capolungo et al., 2009b, Knezevic et al., 2013b), Be (Knezevic et al., 2013a), and even uranium (Knezevic et al., 2012, Knezevic et al., 2013d, Knezevic et al., 2013e). However, in these prior studies, the DD hardening model was incorporated into a visco-plastic self-consistent scheme (VPSC) in order to relate single crystal deformation to the polycrystalline aggregate (e.g., Beyerlein et al., 2011b, Knezevic et al., 2013c, Lebensohn et al., 2007). The VPSC model belongs to a class of more advanced mean-field polycrystal schemes compared to Taylor-type schemes (Al-Harbi et al., 2010, Fast et al., 2008, Knezevic et al., 2008, Knezevic et al., 2009, Knezevic and Kalidindi, 2007, Knezevic and Savage, 2014, Shaffer et al., 2010, Taylor, 1938, Wu et al., 2007) but still cannot capture the effects that grain-to-grain interactions may have on mechanical response and texture evolution. It also cannot model the development of heterogeneities in stress and strain within grains and at grain boundaries and the bimetal interface. To overcome the limitations of mean-field models, the DD model is implemented into a 3D CPFE framework. The combined 3D DD-CPFE model enables predictions of spatially resolved stress and strain fields based on dislocation density and grain reorientation. Earlier studies found that while alterations in these fields near grain boundaries for some grain orientations can be calculated as a function of certain crystallographic relationships (i.e., the Schmid factor across the interfaces) and applied deformation conditions (temperature and strain rate) using simpler Voce type (Voce, 1948) hardening laws and CPFE kinematics (Acharya and Beaudoin, 2000, Beaudoin et al., 1995, Beaudoin et al., 1996, Raabe et al., 2001, Raabe et al., 2002), for many orientations the kinematic treatment of grain boundaries is not sufficient to reveal these micromechanical fields and therefore requires more physically based constitutive descriptions (Ma et al., 2006a, Ma et al., 2006b, Raabe et al., 2004). The impact of these local changes on bulk texture, grain structure, and stress–strain evolution can then be predicted.
Several versions of DD CPFE models currently exist in literature. Among the most promising are the versions that include not only statistical but also geometrically necessary dislocations and therefore rendering the models size sensitive (Evers et al., 2004, Ma et al., 2006a, Ma et al., 2006b, Ma et al., 2006c) and those that consider individual kinetics of screw and edge dislocations and therefore enabling understanding of phenomena associated with the spread of screw dislocation cores (Alankar et al., 2011, Alankar et al., 2014, Ma et al., 2007). A unique feature of the present model is the consideration of hardening by accumulation of substructure dislocation density, which is desirable for simulating severe plastic strains.
The focus application of the present multiscale 3D DD-CPFE model is the large-strain rolling deformation of a Zr/Nb lamellar composite at room temperature recently reported in (Knezevic et al., in press). As individual single-phase metals, the hcp metal Zr and the bcc metal Nb deform using multiple slip families: , prismatic slip; , basal slip; and , pyramidal slip for Zr and {110}〈111〉 and {112}〈111〉 slip in Nb. In particular, a large amount of research has recently been dedicated to understanding deformation mechanisms in Zr, such as the activation of multiple slip and twinning during large strain processing (Jiang et al., 2008, Knezevic et al., 2013b, Long et al., 2013, Yapici et al., 2009, Zhilyaev et al., 2010), twin nucleation (Beyerlein et al., 2011b, Niezgoda et al.,), detwinning (Proust et al., 2010), and slip–twin interactions in abrupt path changes (Capolungo et al., 2009a). These studies established that while the easiest slip mode is prismatic slip, other slip modes, such as pyramidal slip and even basal slip can be activated in large strain deformation (Knezevic et al., 2013b, Long et al., 2013, Yapici et al., 2009). They also show that twinning becomes more prominent below room temperature (Capolungo et al., 2009a) and thus is likely not to be prevalent in the present application. Likewise in Nb, the active slip systems and dislocation dynamics have also been of interest due to its complexity, such as non-Schmid effects (Gröger et al., 2008, Lim et al., 2013, Wang and Beyerlein, 2011), kinematics of glide associated with screw dislocation core structure (Ma et al., 2007), and twinning under shock compression (Gray, 2012, Zhang et al., 2011). Based on the above studies, it is clear that the large-strain deformation behavior and dislocation-based mechanisms operating within a composite of these two relatively complex metals are expected to be both unusual and interesting.
This paper is structured as follows. We first present the multiscale DD-CPFE modeling methodology, highlighting new features such as the methods needed to incorporate dislocation evolution at the sub-crystalline scale into the CPFE code and to enable modeling severe plastic strains (>1). Next we use the model to study and predict microstructural evolution of a two-phase hcp Zr/bcc Nb composite in rolling reported previously (Knezevic et al., in press). For completeness, we review the experimental results and then describe our independent procedure for characterizing the DD model parameters for Zr and for Nb using uniaxial compression tests on these materials in monolithic form. The DD-CPFE is then applied to predict texture evolution of the Zr and Nb phases within the composite during large-strain plane strain compression. We show good agreement with the experimental data on texture. Next, we use the model to predict the associated slip activities within the grains and to identify possible gradients in texture due to the bimetal interface. We show that even at large strains, the predicted textures agreed well with the measured deformation textures for each phase in the composite. Also, in spite of large strain development, the texture within the fine lamellar layer thickness (4 µm) was the same as the texture in close proximity of the interfaces. The model predicts that the observed textures developed by a combination of prismatic 〈a〉 slip, pyramidal 〈c+a〉 slip, and anomalously basal 〈a〉 slip in Zr and primarily {110}〈111〉 slip and secondly {112}〈111〉 slip in Nb.
Section snippets
Multiscale modeling approach
The multiscale model developed here couples self-consistently a 3D crystal plasticity finite element model of a meshed polycrystal, an anisotropic elasticity and rate- and temperature-dependent plasticity formulation for the plasticity of the single crystals, and a dislocation density model for the behavior of individual slip systems within a crystal. Fig. 1 presents a flow chart, which illustrates how the different modeling components are connected. To model very large plastic strains
Materials and experiments
To demonstrate the capabilities of the multiscale model described above, we examine texture and microstructural evolution within a two-phase Zr/Nb composite deformed in rolling. In prior work (Knezevic et al., in press), this layered composite was fabricated by accumulative roll bonding (ARB) to a final nominal layer thickness of ~4 μm, requiring strains of ~4.0. In this section, we briefly review the material and manufacturing details and the experimental results on texture evolution that we
Model characterization
Before predicting the behavior of the Zr/Nb composite in rolling deformation, the material parameters associated with the DD model of Zr and Nb had to be characterized. To do this, separate simulations and experiments were carried out for uniaxial compression of monolithic Zr and Nb. In simulation, the initial model microstructure consisted of a single-phase polycrystal with 15,000 grain orientations representing the measured initial texture (Fig. 3), and spherical shaped grain. With respect to
Model set up
Fig. 8 shows the 3D bi-phase FE models in the global 1–2–3 (RD–TD–ND) orthogonal coordinate system. The top polycrystalline layer is Nb and the bottom one is Zr. In developing the initial model microstructure, our objective was to have as many elements as necessary to realistically capture the measured initial texture. The elements were hexahedral (brick) elements, containing eight integration points representing eight grains with distinct orientations. Each phase has 16×16×8 elements yielding
von Mises and equivalent strain distributions
Fig. 8 shows the predicted von Mises stress fields in the deformed bi-phase composite at different stages of straining. It can be seen that the Zr phase experiences higher stress than the Nb phase during co-deformation. This can be expected since in monolithic form, the flow stress of Zr is higher than Nb, as seen in Fig. 7. Fig. 9 shows the corresponding equivalent strain fields. Some degree of localization can be seen, starting at the end of the second pass. This strain is approximately the
Discussion
In the experimental community, much progress has been made towards 3D microstructural characterizations of grain size, grain orientation, grain boundary character, and triple junctions (Groeber et al., 2006, Hefferan et al., 2012, Saylor et al., 2004a, Saylor et al., 2004b). In the modeling community, advancements have also been made in establishing relationships between these microscopic features and macroscopic response. However, a complete 4D (3D plus time) understanding of
Conclusions
In summary, we present a 3D multiscale model for polycrystalline metals. The model uniquely incorporates a dislocation density-based hardening law for multiple slip modes into a single-crystal plasticity model, which in turn is linked to the polycrystalline response using finite elements. With these components, the model is capable of predicting the coupled evolution of the local stress and strain fields, dislocation density and texture. The model is applied to the challenging problem of a
Acknowledgments
M.K. and M.A. were supported by the University of New Hampshire faculty startup funds. I.J.B. would like to acknowledge support through a Los Alamos National Laboratory Directed Research and Development (LDRD) project ER20140348.
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