Constitutive Modeling in Crystal Plasticity 

Constitutive formulations used in crystal plasticity describe plastic flow and strain hardening  at the elementary shear system level. The constitutive flow laws that were suggested during the last decades have gradually developed from empirical viscoplastic formulations into physics-based multiscale internal-variable models of plasticity, including a variety of underlying shear carrieres (dislocations, martensite, twinning), size-dependent effects and interface mechanisms. 

While the kinematic formalism describes the geometrical aspects of the anisotropy of crystal mechanics without considering stresses, the constitutive equations capture the physics of the material behavior, in particular of the dynamics of those lattice defects that act as the elementary carriers of plastic shear. How the microstructural state of the material is defined and how it evolves during loading in a crystal plasticty simulation depends on the kind of constitutive model used. 
Below are some results obtained by using two classes of constitutive models, namely phenomenological models and physics-based models.
Phenomenological constitutive models mostly use a critical resolved shear stress as state variable for each slip system. Therefore, the shear rate is formulated as a function of the resolved shear stress. In contrast to the phenomenological constitutive models, the physically based ones rely on internal variables. In the case of plasticity the most important microstructural state variable certainly is the dislocation density as the dislocations are the carriers of plastic deformation
but twins as well as martensite can be considered as well.

Here we review 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, we also present 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.

We study the link between the indentation size effect and the density of geometrically necessary dislocations (GNDs) through the following approach: four indents of different depth and hardness were placed in a Cu single crystal using a conical indenter with a spherical tip. The deformation-induced lattice rotations below the indents were monitored via a three-dimensional electron backscattering diffraction method with a step size of 50 nm. From these data we calculated the first-order gradients of strain and the GND densities below the indents. This approach allowed us to quantify both the mechanical parameters (depth, hardness) and the lattice defects (GNDs) that are believed to be
responsible for the indentation size effect.

Here a work-hardening model for homogeneous and heterogeneous dislcoation cell-forming alloys is introduced. It distinguishes three internal state variables in terms of three categories of dislocations: mobile dislocations, immobile dislocations in the cell interiors and immobile dislocations in the cell walls. For each dislocation population an evolution law is derived taking into account dislocation generation, annihilation and storage by dipole and lock formation. In particular, these rate equations take into account the number of active glide systems and, thus, introduce texture in the model in addition to the Taylor factor. Microstructure is represented by the dislocation cell structure as well as second-phase particles, which may undergo changes by precipitation and Ostwald ripening. Interaction of mobile dislocations with the microstructure is taken into account through an effective slip length of the mobile dislocations. For the same set of parameters, the predictions are in excellent agreement with measured stress–strain curves of both a precipitation-hardened aluminium alloy (Al–4.16 wt% Cu–1.37 wt% Mg, AlCuMg2) and a precipitation-free model alloy (Al–0.35 wt% Cu–0.25 wt% Mg), the composition of which corresponds to the matrix of the two-phase alloy.

We present a single crystal plasticity model based on edge and screw dislocation densities for body centered cubic (bcc) crystals. In a bcc crystal screw dislocations experience high lattice friction due to their non-planar core. Hence, they have much slower velocity compared to edge dislocations. This phenomenon is modeled by accounting for the motion of screw dislocations via nucleation and expansion of kink-pairs. The model, embedded as a constitutive law into a crystal plasticity framework, is able to predict the crystallographic
texture of a bcc polycrystal subjected to 70%, 80% and 90% thickness reduction. We perform a parametric study based on the velocities of edge and screw dislocations to analyze the effect on plastic anisotropy of electro-deposited pure iron with long needle-shaped grains having sharp crystallographic <111>//ND texture (ND: normal direction). The model shows a large change in the r-value (Lankford value, planar anisotropy ratio) for pure iron when the texture changes from random to <111>//ND. For different simulated cases where the crystallites have an orientation deviation of 1°, 3° and 5°, respectively, from the ideal
<111>//ND axis, the simulations predict r-values between 4.0 and 7.0 which is in excellent agreement with data observed in experiments by Yoshinaga et al. (ISIJ Intern., 48 (2008) 667–670). For these specific orientations of grains, we also model the effect of long needle shaped grains via a procedure that excludes dislocation annihilation.

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.

Understanding and improving the mechanical properties of tungsten is a critical task for the materials fusion energy program. The plastic behavior in body-centered cubic (bcc) metals like tungsten is governed primarily by screw dislocations on the atomic scale and by ensembles and interactions of dislocations at larger scales. Modeling this behavior requires the application of methods capable of resolving each relevant scale. At the small scale, atomistic
methods are used to study single dislocation properties, while at the coarse-scale, continuum models are used to cover the interactions between dislocations. In this work we present a multiscale model that comprises atomistic, kinetic Monte Carlo (kMC) and continuum-level crystal plasticity (CP) calculations. The function relating dislocation velocity to applied stress
and temperature is obtained from the kMC model and it is used as the main source of constitutive information into a dislocation-based CP framework. The complete model is used to perform material point simulations of single-crystal tungsten strength. We explore the entire crystallographic orientation space of the standard triangle. Non-Schmid effects are inlcuded in the model by considering the twinning-antitwinning asymmetry in the kMC calculations. We consider the importance of 1 1 1{1 1 0} and 1 1 1{1 1 2} slip systems in the homologous temperature range from 0.08Tm to 0.33Tm, where Tm=3680 K is the melting
point in tungsten.

Here we present a new physics-based constitutive model for low-SFE fcc metals that exhibit deformation twinning. The model is a combination and extension of the 3IVM of Roters and the twin nucleation model of Mahajan and Chin. Dislocation cells, grain size and twin volume fraction evolution are included.

Very good agreement with experimental compression data (Fe–22Mn–0.6C TWIP steel) was found between 293 and 873 K using a single set of physically motivated parameters. The model reveals that the intermediate strainhardening
regime that is responsible for the high formability of TWIP steels is due to the dynamic increase of the twin-related interface density and its interaction with the
dislocations. In addition, due to the good prediction of hardening behavior over a temperature range spanning almost 600 K, the door is now open to the inclusion of adiabatic heating effects caused by shear banding, the implementation of temperature-sensitive forming simulations and the improvement of failure simulations. The twin nucleation model introduced follows twinning at the mechanistic level and considers both the dislocation activity necessary to create twin nuclei and the stress state responsible for the expansion of the nuclei into twins. The nucleation rate of twins is linked directly to the dislocation density, the size of the twin nucleus and the SFE through the critical twin stress and the probability of formation of the twin nucleus.
The simulated temperature of the sample evolves during deformation owing to dissipation, a phenomenon that has been ignored in models until now, but is vital to include. The predicted changes in sample temperature during RT compression or tensile testing can exceed 100 K in the case studied here.  The SFE of an alloy, which is the key parameter for twinning, can nowadays be calculated ab initio by density functional theory. In combination with the presented model, it is therefore possible to tailor the SFE (i.e. the alloy composition) to achieve desired macroscopic properties. This is a big step forward in predictive hierarchical
materials modeling.

In this project a martensitic phase transformation model has been developed
within a crystal plasticity framework to simulate the gamma -  ε transformation in high Mn and high Ni base steels. The novelty of this model is that martensitic phase transformation is incorporated in addition to dislocation glide and twinning. By conducting FFT-based crystal plasticity simulations using the spectral solver
within DAMASK, quantitative comparisons are conducted between the experiments and simulations on the stress-strain curves, strain hardening curves, twin volume fraction and ε-martensite for a Fe- 22Mn-0.6C steel.
The model is able to predict, based on the temperature, which in turn changes the stacking fault energy of the material, the activation of martensitic phase transformation at low temperatures, the predominance of twinning at intermediate temperatures, and plastic deformation solely by pure slip at high temperatures. It is observed from the experiments that gamma-ε transformation occurs at low temperatures and increases as the temperature decreases.
However, twinning occurs starting at room temperature and increases as the temperature increases.

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 plas-
ticity 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.

A dislocation density based constitutive model for the face centered cubic crystal structure has been implemented into a crystal-plasticity finite element framework and extended to consider the mechanical interaction between mobile dislocations and grain boundaries by the authors [Ma, A., Roters, F., Raabe, D., 2006a. A dislocation density based constitutive model for crystal-plasticity FEM including geometrically necessary dislocations. Acta Materialia 54, 2169–2179; Ma, A., Roters, F., Raabe, D., 2006b. On the consideration of interactions between dislocations and grain boundaries in crystalplasticity finite element modeling – theory, experiments, and simulations. Acta Materialia 54, 2181–2194]. The approach to model the grain boundary resistance against slip is based on the introduction of an additional activation energy into the rate equation for mobile dislocations in the vicinity of internal interfaces. This energy barrier is derived from the assumption of thermally activated dislocation penetration events through grain boundaries. The model takes full account of the geometry of the grain boundaries and of the Schmid factors of the critically stressed incoming and outgoing slip systems. In this study we focus on the influence of the one remaining model parameter which can be used to scale the obstacle strength of the grain boundary.

A dislocation density based constitutive model for face-centred cubic crystals is introduced and implemented into a crystal plasticity finite element framework. The approach assumes a homogeneous dislocation structure and tracks the dislocation evolution on each slip system. In addition to the statistically stored dislocations, the geometrically necessary dislocation density is introduced in order to consider strain gradients and thus render the model size sensitive. Furthermore, we develop a consistent algorithm for the updating of the
geometrically necessary dislocation density. A simple shear experiment of an aluminium single crystal is used to calibrate the material parameters of the model and demonstrate its size sensitivity.

We introduce a crystal plasticity constitutive model for BCC materials which is build on dislocation movement and uses dislocation density variables as internal state variables. Besides the statistically stored dislocations geometrically necessary dislocations are used to consider nonlocal effects as recently proposed by Ma, Roters and Raabe for the FCC crystal structure. In this paper the model will be adopted to the BCC crystal structure. Due to the special core structure of screw dislocations formed at low temperatures, the mechanical
behavior of BCC crystals is controlled by the movement of screw dislocations rather than edge dislocations. For this reason, the Peierls mechanism has to be considered and several modifications have been introduced which include a new scaling relation between the mobile and immobile dislocations, and new flow rules for bulk and grain boundary elements. A pure Nb bicrystal is studied experimentally and numerically under channel die compression boundary conditions, to demonstrate the applicability of the new model variant.

We suggest a dislocation based constitutive model to incorporate the mechanical interaction between mobile dislocations and grain boundaries into a crystal plasticity finite element framework. The approach is based on the introduction of an additional activation energy into the rate equation for mobile dislocations in the vicinity of grain boundaries. The energy barrier is derived by using a geometrical model for thermally activated dislocation penetration events through grain boundaries. The model takes full account of the geometry of the grain boundaries and of the Schmid factors of the critically stressed incoming and outgoing slip systems and is formulated as a vectorial conservation law. The new model is applied to the case of 50% (frictionless) simple shear deformation of Al bicrystals with either a small, medium, or large angle grain boundary parallel to the shear plane. The simulations are in excellent agreement with the experiments in terms of the von Mises equivalent strain distributions and textures. The study reveals that the incorporation of the misorientation alone is not sufficient to describe the influence of grain boundaries on polycrystal micro-mechanics. We observe three mechanisms which jointly entail pronounced local hardening in front of grain boundaries (and other interfaces) beyond the classical kinematic hardening effect which is automatically included in all crystal plasticity finite element models owing to the change in the Schmid factor across grain boundaries. These are the accumulation of geometrically necessary dislocations (dynamic effect; see [Ma A, Roters F, Raabe D. A dislocation density based constitutive model for crystal plasticity FEM including geometrically necessary dislocations. Acta Mater 2006;58:2169–79]), the resistance against slip penetration (dynamic effect; this paper), and the change in the orientation spread
(kinematic effect; this paper) in the vicinity of grain boundaries.

We present crystal plasticity finite element simulations of the texture evolution in a-brass polycrystals under plane strain compression. The novelty is a non-crystallographic shear band mechanism [Anand L, Su C. J Mech Phys Solids 2005;53:1362] that is incorporated into the constitutive model in addition to dislocation and twinning. Non-crystallographic deformation associated with shear banding leads to weaker copper and S texture components and to a stronger brass texture compared to simulations enabling slip and twinning only. The
lattice rotation rates are reduced when shear banding occurs. This effect leads to a weaker copper component. Also, the initiation of shear banding promotes brass-type components. In summary the occurrence of non-crystallographic deformation through shear bands shifts face-centered-cubic deformation textures from the copper type to the brass type.

We present the orientation dependent indentation response of pure magnesium during single grain indentation. A conical indenter and maximum loads between 50 mN and 900 mN were employed. Indent topographies were acquired by confocal microscopy. The indents were also characterized by electron backscatter orientation microscopy for their microstructures. Pronounced activation of specific twinning systems was observed around the impressions. The resulting data were compiled into the inverse pole figure presentation of indent microstructures and topographies after Zambaldi and Raabe, Acta Mater. (2010). Three-dimensional crystal plasticity finite element simulation of the indentation deformation supports the interpretation of the orientation dependent slip and twinning patterns around the indents. The match between the activation of observed and simulated twinning variants is discussed with respect to the conditions for nucleation and growth of extension twins. Furthermore, the compatibility of the twinning strains with the imposed deformation is discussed based on the expanding cavity model of indentation. The orientation dependent
response of magnesium during indentation is compared to the literature data for indentation of alpha-titanium and beryllium. Recommendations are given on how to exploit the characteristic nature of the observed indentation patterns to rapidly assess the relative activity of deformation mechanisms and their critical shear stresses during alloy development.