# Crystal Plasticity Finite Element simulations & experiments

**Overview of crystal plasticity FEM simulations**

Review of continuum-based variational formulations for describing the elastic–plastic deformation of anisotropic heterogeneous crystalline matter. Crystal plasticity finite-element models, are impor

Crystal Plasticity Finite Element Modeli[...]

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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.

**Shear banding in crystal plasticity FEM simulations: Texture evolution under consideration of shear banding**

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 band

Acta Materialia 60 (2012) 1099 Jia Shear[...]

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**Simulation of shear banding in heterophase co-deformation: Example of plane strain compressed Cu–Ag and Cu–Nb metal matrix composites**

CPFEM shear band hetero Acta 2013.pdf

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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.

**Acta Materialia 60 (2012) 3415: Orientation dependence of shear banding in face-centered-cubic single crystals**

Acta Materialia 60 (2012) 3415 orientati[...]

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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.

**Crystal plasticity modeling of a metal–ceramic composite**

Crystal plasticity polycrystal model of the structure evolution and the mechanical behavior of a Mo–TiC30 vol.% metal–ceramic composite material using a three-dimensional microstructure map obtain

Acta Materialia 60 (2012) 1623 3D crysta[...]

PDF-Dokument [638.3 KB]

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.

**Crystal plasticity prediction In-grain orientation gradients**

Acta Materialia 50 (2002) 421 theory ori[...]

PDF-Dokument [728.5 KB]

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 dislocation density-based crystal plasticity constitutive model for prismatic slip in alpha-titanium**

Acta Materialia 59 (2011) 7003 titanium [...]

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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.

**Combination of ab initio models with polycrystal models**

epjp1100181-offprints.pdf

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**Plastic anisotropy of titanium aluminides**

Analysis of plastic anisotropy and pre-yielding of titanium-aluminide microstructures

Zambaldi2011 plastic anisotropy 2-phase [...]

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**Dislocation constitutive model for alpha-Titanium**

titanium-dislocation-model-Acta Material[...]

PDF-Dokument [363.3 KB]

**Overview on microstructure based modeling**

2011-overview-microstructure-models-JOM.[...]

PDF-Dokument [1.5 MB]

**Lamination theory in plasticity**

Acta Materialia 57 (2009) 3439-3449

Acta Materialia 57 (2009) 3439 laminatio[...]

PDF-Dokument [863.1 KB]

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.

**Virtual lab for metal forming based on using crystal plasticty FEM**

Computational Materials Science 46 (2009) 383-392

Computational Materials Science 46 (2009[...]

PDF-Dokument [2.5 MB]

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.

**Bauschinger effect at small scale in Ci single crystals**

Acta Materialia 58 (2010) 6055_Bausching[...]

PDF-Dokument [940.0 KB]

**On the dependence of in-grain subdivision and deformation texture of aluminum on grain interaction**

raabe_interaction.pdf

PDF-Dokument [281.0 KB]

**Experimental investigation of plastic grain interaction**

Mater Sc Engin A336 (2002) 81 interactio[...]

PDF-Dokument [682.0 KB]

**On the influence of the grain boundary misorientation on the plastic deformation of aluminum bicrystals**

Bicrystal_Al_Acta Materialia 51 (2003) 4[...]

PDF-Dokument [647.1 KB]

**A dislocation density based constitutive model for crystal plasticity FEM including geometrically necessary dislocations**

Acta Materialia 54 (2006) 2169 GND in CP[...]

PDF-Dokument [535.3 KB]

**On the consideration of interactions between dislocations and grain boundaries in crystal plasticity finite element modeling**

Acta Materialia 54 (2006) 2181 CPFEM bic[...]

PDF-Dokument [2.3 MB]

**Effects of orientation in single crystal microcompression: A crystal plasticity finite element**

Acta Materialia 55 (2007) 4567 CPFEM Pil[...]

PDF-Dokument [2.2 MB]

**MICROMECHANICAL AND MACROMECHANICAL EFFECTS IN GRAIN SCALE POLYCRYSTAL PLASTICITY EXPERIMENTATION AND SIMULATION**

Raabe_micro_acta_2001.pdf

PDF-Dokument [471.6 KB]

**The mechanical size effect as a mean-field breakdown phenomenon: Example of microscale single crystal beam bending**

Acta Materialia 58 (2010) 1876_size_effe[...]

PDF-Dokument [1'000.4 KB]

**Investigation of three-dimensional aspects of grain-scale plastic surface deformation of an aluminum oligocrystal**

Int J Plast 24 (2008) 2278 CPFEM dogbone[...]

PDF-Dokument [2.7 MB]

**WORK HARDENING IN HETEROGENEOUS ALLOYS—A MICROSTRUCTURAL APPROACH BASED ON THREE INTERNAL STATE VARIABLES**

actav48p4181.pdf

PDF-Dokument [151.9 KB]

**Grain-scale micromechanics of polycrystal surfaces during plastic straining**

raabe_rough.pdf

PDF-Dokument [1.9 MB]

**Smaller is stronger: The effect of strain hardening**

Acta Materialia 57 (2009) 5996–smaller-i[...]

PDF-Dokument [803.6 KB]

**Plastic anisotropy of gamma-TiAl revealed by axisymmetric indentation**

Acta Materialia 58 (2010) 3516-3530

Acta Materialia 58 (2010) 3516 Zambaldi [...]

PDF-Dokument [939.4 KB]

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.

**Simulation of earing and anisotropy by crystal plasticity FEM simulations**

bcc_earing.pdf

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**Integrated experimental–simulation analysis of stress and strain partitioning in multiphase alloys**

Acta Materialia 81 (2014) 386-400

Acta Mater 81 (2014) 386 stress strain p[...]

PDF-Dokument [1.6 MB]

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.

**Strain localization and damage in dual phase steels investigated by coupled in-situ deformation experiments and crystal plasticity simulations**

International Journal of Plasticity 63 (2014) 198-210

Intern Journ Plasticity 63 (2014) 198 In[...]

PDF-Dokument [964.9 KB]

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.