Crystal Plasticity Finite Element simulations & experiments

Overview of continuum-based variational formulations for describing the elastic–plastic deformation of anisotropic heterogeneous crystalline matter. These approaches, commonly referred to as crystal plasticity finite-element models, are important both for basic microstructure-based mechanical predictions as well as for engineering design and performance simulations involving anisotropic media.
Besides the discussion of the constitutive laws, kinematics, homogenization schemes and multiscale approaches behind these methods, the paper also presents some examples, including, in particular, comparisons of the predictions with experiments. The applications stem from such diverse fields as orientation stability, microbeam bending, single-crystal and bicrystal deformation, nanoindentation, recrystallization, multiphase steel (TRIP) deformation, and damage prediction for the microscopic and mesoscopic scales and multiscale predictions of rolling textures, cup drawing, Lankfort (r) values and stamping simulations for the macroscopic scale.

The co-deformation and shear localization in heterophase alloys is studied using two-dimensional crystal plasticity finite element simulations on plane strain compressed Cu–Ag and Cu–Nb metal matrix composites. The aim is to study the fundamentals of micromechanics, co-deformation and shear banding in materials with heterophase interfaces. It is observed that, depending on the initial orientations of the crystals, co-deformation of the constituent heterophases often proceeds via collective mechanisms, i.e. by pronounced shear banding triggered by stress concentration at the interfaces. This phenomenon leads to highly localized strains within the bands, exceeding the average strain in part by two orders of magnitude. Shear band development is related to the inherent mechanical properties of each crystal and also to the properties of the abutting crystals. The predicted topology and nature of the cross-phase shear bands, i.e. the extreme local strains, significant bending of the interface regions, and sharp strain localization that propagates across the interfaces, agree well with experimental observations in cold-rolled composites. The simulations reveal that cross-phase shear banding leads to large and highly localized values of stress and strain at heterophase interfaces. Such information is essential for a better understanding of the micromechanical boundary conditions inside co-deformed composites and the associated shear-induced chemical mixing.

We present crystal plasticity finite element simulations of plane strain compression of a-Brass single crystals with different initial orientations. The aim is to study the fundamentals of mesoscale structure and texture development in face-centered-cubic (fcc) metals with low stacking fault energy (SFE). Shear banding depends on the initial orientation of the crystals. In Copper and Brass-R-oriented crystals which show the largest tendency to form shear bands, an inhomogeneous texture distribution induced by shear banding is observed. To also understand the influence of the micromechanical boundary conditions on shear band formation, simulations on Copper-oriented single crystals with varying sample geometry and loading conditions are performed. We find that shear banding can be understood in terms of a mesoscopic softening mechanism. The predicted local textures and the shear banding patterns agree well with experimental observations in low SFE fcc crystals.

The mechanical behavior of a Mo–TiC30 vol.% ceramic–metal composite was investigated over a wide temperature range (25–700 °C). High-energy X-ray tomography was used to reveal percolation of the hard titanium carbide phase through the composite. Using a polycrystal approach for a two-phase material, finite-element simulations were performed on a real three-dimensional (3-D) aggregate of the material. The 3-D microstructure, used as the starting configuration for the predictions, was obtained by serial sectioning in a dual beam focused ion beam scanning electron microscope coupled to an electron backscattered diffraction system. The 3-D aggregate consists of a molybdenum matrix and a percolating TiC skeleton. As for most body-centered cubic (bcc) metals, the molybdenum matrix phase is characterized by a change in plasticity mechanism with temperature. We used a polycrystal model for bcc materials which was extended to two phases (TiC and Mo). The model parameters of the matrix were determined from experiments on pure molydenum. For all tem-
peratures investigated the TiC particles were considered to be brittle. Gradual damage to the TiC particles was treated, based on an accumulative failure law that is approximated by evolution of the apparent particle elastic stiffness. The model enabled us to determine the evolution of the local mechanical fields with deformation and temperature. We showed that a 3-D aggregate representing the actual microstructure of the composite is required to understand the local and global mechanical properties of the composite studied.

We suggest a theory of in-grain orientation gradients in plastically strained metals. It is an approach to explain why initially uniformly oriented crystals can -under gradient-free external loadings -build up in-grain orientation gradients
during plastic deformation and how this phenomenon depends on intrinsic factors (crystal orientation) and extrinsic factors (neighbor grains).
The intrinsic origin (orientation dependence) of in-grain orientation gradients is investigated by quantifying the change in crystal reorientation upon small changes in initial orientation. This part of the approach is formulated by
applying a divergence operator to reorientation rate vector fields (in the present paper calculated by using strain-rate homogenization Taylor–Bishop–Hill theory). The obtained scalar divergence function (but not the reorientation vector
field itself) quantifies the kinematic stability of grains under homogeneous boundary conditions as a function of their orientation. Positive divergence (source in the reorientation rate vector field) characterizes orientations with diverging non-zero reorientation rates which are kinematically unstable and prone to build up orientation gradients. Zero divergence indicates orientations with reorientation rate identity with the surrounding orientations which are not prone to build up orientation gradients. Negative divergence (sink in the reorientation rate vector field) characterizes orientations with converging non-zero reorientation rates which are kinematically stable and not prone to build up orientation gradients. Corresponding results obtained by use of a crystal plasticity finite element formulation are in good agreement with the reorientation field divergence function derived by homogenization theory. The extrinsic origin of in-grain orientation gradients (influence of grain–neighbor interaction) is addressed using a crystal plasticity finite element bicrystal model. The simulations show that a significant dependence of orientation gradients on the neighbor crystals occurs for grains with high positive divergence. The build-up of orientation gradients in grains with close to zero or negative divergence is in body centered cubic crystals less sensitive to the presence of neighbor orientations than in face centered cubic crystals (Goss and cube orientation).

A new constitutive plasticity model for prismatic slip in hexagonal a-titanium is developed. In the concept pure edge and screw dislocation densities evolve on the {1 0 1 0}<1 2 1 0> slip systems. The model considers that the screw dislocation segments have a spread out core, leading to a much higher velocity of edge compared with screw dislocations. This enables the model to describe the observed transition in strain hardening from stage I to stage II in single crystals oriented for prismatic slip. Good agreement is found between the
experimentally observed and simulated stress–strain behavior.

We investigate the formation of microscopic patterns in a copper single crystal deformed in a shear experiment. Using high-resolution electron backscatter diffraction imaging, we find a band-like microstructure consisting of confined areas in the sample with rotated lattice. Digital image correlation allows us to exactly determine the macroscopic state of deformation of the sample. This data can be used as a side condition to calculate the lamination parameters from the theory of kinematically compatible lamination of separate material regions, each deforming in single slip. The parameters given by the theory agree with the measured properties, i.e. a lattice rotation of 3 and a lamination normal rotated 7 counterclockwise from a <111> direction.

In the modern practice of stamping simulation of complex industrial parts the prediction of springback still lacks accuracy. In commercial software packages various empirical constitutive laws for stamping are available. Limited to simple empirical models for material anisotropy they do not take into account in a full manner the effects of microstructure and its evolution during the deformation process. The crystal plasticity finite element method bridges the gap between the polycrystalline texture and macroscopic mechanical properties that opens the way for more profound consideration of metal anisotropy in the stamping process simulation. In this paper the application of crystal plasticity FEM within the concept of virtual material testing with a representative volume element (RVE) is demonstrated. Using virtual tests it becomes possible, for example, to determine the actual shape of the yield locus and Lankford parameters and to use this information to calibrate empirical constitutive models. Along with standard uniaxial tensile tests other strain paths can be investigated like biaxial tensile, compressive or shear tests. The application of the crystal plasticity FEM for the virtual testing is demonstrated for DC04 and H320LA steel grades. The parameters of the Vegter yield locus are calibrated and the use case demonstration is completed by simulation of a typical industrial part in PAMSTAMP 2G.

Single crystals of gamma-TiAl cannot be grown in the near-stoichiometric compositions that are present inside two-phase gamma/a2-microstructures

with attractive mechanical properties. Therefore, the single-crystal constitutive behavior of gamma-TiAl was studied by nanoindentation experiments in single-phase regions of these gamma/a2-microstructures. The experiments were characterized by orientation microscopy and atomic force microscopy to quantify the orientation-dependent mechanical response during nanoindentation. Further, they were analyzed by a three-dimensional crystal plasticity finite element model that incorporated the deformation behavior of gamma-TiAl. The spatially resolved activation of competing deformation mechanisms during indentation was used to assess their relative strengths. A convention was defined to unambiguously relate any indentation axis to a crystallographic orientation. Experiments and simulations were combined
to study the orientation-dependent surface pile-up. The characteristic pile-up topographies were simulated throughout the unit triangle of gamma-TiAl and represented graphically in the newly introduced inverse pole figure of pile-up patterns. Through this approach, easy activation of ordinary dislocation glide in stoichiometric gamma-TiAl was confirmed independently from dislocation observation by transmission electron microscopy.

The mechanical response of multiphase alloys is governed by the microscopic strain and stress partitioning behavior among microstructural constituents. However, due to limitations in the characterization of the partitioning that takes place at the submicron scale, microstructure optimization of such alloys is typically based on evaluating the averaged response, referring to, for example, macroscopic stress–strain curves. Here, a novel experimental–numerical methodology is introduced to strengthen the integrated understanding of the microstructure and mechanical properties of these alloys, enabling joint analyses of deformation-induced evolution of the microstructure, and the strain and stress distribution therein, down to submicron resolution. From the experiments, deformation-induced evolution of (i) the microstructure, and (ii) the local strain distribution are concurrently captured, employing in situ secondary electron imaging and electron backscatter diffraction (EBSD) (for the former), and microscopic-digital image correlation (for the latter). From the simulations, local strain as well as stress distributions are revealed, through 2-D full-field crystal plasticity (CP) simulations conducted with an advanced spectral solver suitable for heterogeneous materials. The simulated model is designed directly from the initial EBSD measurements, and the phase properties are obtained by additional inverse CP simulations of nanoindentation experiments carried out on the original microstructure. The experiments and simulations demonstrate good correlation in the proof-of-principle study conducted here on a martensite–ferrite dual-phase steel, and deviations are discussed in terms of limitations of the techniques involved. Overall, the presented integrated computational materials engineering approach provides a vast amount of well-correlated structural and mechanical
data that enhance our understanding as well as the design capabilities of multiphase alloys.

Ferritic–martensitic dual phase (DP) steels deform spatially in a highly heterogeneous manner, i.e. with strong strain and stress partitioning at the micro-scale. Such heterogeneity in local strain evolution leads in turn to a spatially heterogeneous damage distribution, and thus, plays an important role in the process of damage inheritance and fracture. To understand and improve DP steels, it is important to identify connections between the observed strain and damage heterogeneity and the underlying microstructural parameters,
e.g. ferrite grain size, martensite distribution, martensite fraction, etc. In this work we pursue this aim by conducting in-situ deformation experiments on two different DP steel grades, employing two different microscopic-digital image correlation (lDIC) techniques to achieve microstructural strain maps of representative statistics and high-resolution. The resulting local strain maps are analyzed in connection to the observed damage incidents (identified by image post-processing) and to local stress maps (obtained from crystal plasticity (CP) simulations of the same microstructural area). The results reveal that plasticity
is typically initiated within ‘‘hot zones’’ with larger ferritic grains and lower local
martensite fraction. With increasing global deformation, damage incidents are most often observed in the boundary of such highly plastified zones. High-resolution lDIC and the corresponding CP simulations reveal the importance of martensite dispersion: zones with bulky martensite are more susceptible to macroscopic localization before the full strain hardening capacity of the material is consumed. Overall, the presented joint analysis establishes
an integrated computational materials engineering (ICME) approach for designing advanced DP steels.