Single Crystal Plasticity Model Research Articles

Void growth and coalescence in triaxial stress fields in irradiated FCC single crystals

Void growth and coalescence, known as main mechanisms of ductile fracture, are investigated for irradiated FCC single crystals. Finite element simulations of voided unit cells are performed with a single crystal plasticity model accounting for strain hardening and softening associated with irradiation-induced defects. The simulations predict a rather brittle overall behavior for the voided irradiated single crystal at high stress triaxiality, with a large amount of local plastic deformation, which is consistent with experimental observations reported in the literature for stainless steels irradiated in fast reactors. Compared with unirradiated single crystals, irradiated crystals exhibit a higher void growth rate leading to an earlier void coalescence, which is caused by a stronger plastic slip localization in the region near the voids.

Simulation of micro‐cutting considering finite deformation crystal plasticity

AbstractMicro‐machining processes on metalic microstructures are influenced by the crystal structure, i. e. the grain orientation. Furthermore, the chip formation underlies large deformations. To perform finite element simulations of micro‐cutting processes, a large deformation material model is necessary in order to model the hyperelastic and finite plastic material behaviour. In the case of cp‐titanium material with hcp‐crystal structure the anisotropic behaviour must be considered by an appropriate set of slip planes and slip directions. In the present work the impact of the grain orientation on the plastic deformation is demonstrated by means of finite element simulations of a finite deformation single slip crystal plasticity model. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Deformation and strain localization in polycrystals with plastically heterogeneous grains

A model of a polycrystalline material is studied, where each grain consists of several zones with different plastic properties. The size and configuration of the zones as well as their properties are mimicking the situation inside the crystals of artificially aged Al alloys with precipitate free zones (PFZs). The properties of the different zones are conjectured based on micromechanical models and experimental observations of the AA6000 series of Al alloys. A periodic patch of such a composite material, subjected to plane-strain tension, is modelled using the finite element method. The material behaviour is described by a single crystal plasticity model. The results of the simulations show a number of characteristic features emerging in composite material systems, including sliding and distortion of grain boundaries and shear band formation at grain boundary triple points. The assumptions made about the properties of the different zones in the crystals are evaluated. The distributions of plastic strain and deviatoric and hydrostatic stresses in the composite crystalline systems are discussed.

Extraction of Anisotropic Mechanical Properties From Nanoindentation of SiC-6H Single Crystals

Because brittle solids fail catastrophically during normal tension and compression testing, nanoindentation is often a useful alternative technique for measuring their mechanical properties and assessing their deformation characteristics. One practical question to be addressed in such studies is the relationship between the anisotropy in the uniaxial mechanical behavior to that in the indentation response. To this end, a systematic study of the mechanical behavior the 6H polytype of a hexagonal silicon carbide single crystal (SiC-6H) was performed using standard nanoindentation methods. The indentation elastic modulus and hardness measured using a Berkovich indenter at a peak load of 500 mN varied over a wide range of crystal orientation by only a few percent. The variation in modulus is shown to be consistent with an anisotropic elastic contact analysis based on the known single crystal elastic constants of the material. The variation in hardness is examined using a single crystal plasticity model that considers the anisotropy of slip in hexagonal crystals. When compared to experimental measurements, the analysis confirms that plasticity in SiC-6H is dominated by basal slip. An anisotropic elastic contact analysis provides insights into the relationship between the pop-in load, which characterizes the transition from elasticity to plasticity during nanoindentation testing, and the theoretical strength of the material. The observations and analyses lay the foundations for further examination of the deformation and failure mechanisms in anisotropic materials by nanoindentation techniques.

Quasi-static response and texture evolution of α- and γ-RDX: a comparative study

In this paper, we undertake a comparative study of the stress–strain response and slip activity of α- and γ-polymorph of hexahydro-1,3,5-trinitro-1,3,5-triazine (RDX) under pressure loading using a rate-dependent single-crystal plasticity model. Texture evolution studies are performed to further understand the effects of the dominant slip systems. The simulations indicate that the difference in elastic moduli and lattice parameters for α- and γ-RDX lead to different elastic–plastic constitutive response in the two polymorphs. γ-RDX exhibits more plastic slip compared to α-RDX for loading on (1 1 1) plane and the two polymorphs have different sets of dominant slip systems. We observe that the high-pressure slip system (0 0 1)[0 1 0] that is determined using molecular dynamics simulations is the most dominant slip system for this orientation. Whereas, for loading on (2 1 0) plane, α-RDX has marginally higher plastic slip than γ-RDX, though the same slip system is dominant for both the polymorphs. The texture evolution for loading on (1 1 1) and (2 1 0) planes follow the path towards the most dominant slip systems for both the polymorphs. We predict that the larger plastic slip in γ-RDX for loading on (1 1 1) plane might play an important role in understanding the reduced sensitivity for shock loading on (1 1 1) plane, when compared to (2 1 0) for which γ-RDX has lesser plastic slip, and (1 0 0) which is purely elastic.

Numerical implementation of a crystal plasticity model with dislocation transport for high strain rate applications

This paper details a numerical implementation of a single crystal plasticity model with dislocation transport for high strain rate applications. Our primary motivation for developing the model is to study the influence of dislocation transport and conservation on the mesoscale response of metallic crystals under extreme thermo-mechanical loading conditions (e.g. shocks). To this end we have developed a single crystal plasticity theory (Luscher et al (2015)) that incorporates finite deformation kinematics, internal stress fields caused by the presence of geometrically necessary dislocation gradients, advection equations to model dislocation density transport and conservation, and constitutive equations appropriate for shock loading (equation of state, drag-limited dislocation velocity, etc). In the following, we outline a coupled finite element–finite volume framework for implementing the model physics, and demonstrate its capabilities in simulating the response of a [1 0 0] copper single crystal during a plate impact test. Additionally, we explore the effect of varying certain model parameters (e.g. mesh density, finite volume update scheme) on the simulation results. Our results demonstrate that the model performs as intended and establishes a baseline of understanding that can be leveraged as we extend the model to incorporate additional and/or refined physics and move toward a multi-dimensional implementation.

Modeling the evolution of microstructures in finite plasticity

AbstractKochmann and Hackl introduced in [1] a micromechanical model for finite single crystal plasticity. Based on thermodynamic variational principles this model leads to a non‐convex variational problem. Employing the Lagrange functional, an incremental strategy was outlined to model the time‐continuous evolution of a first order laminate microstructure. Although this model provides interesting results on the material point level, due to the global minimization in the evolution equations, the calculation time and numerical instabilities may cause problems when applying this model to macroscopic specimens.In order to avoid these problems, a smooth transition zone between the laminates is introduced to avoid global minimization, which makes the numerical calculations cumbersome compared to the model in [1]. By introducing a smooth viscous transition zone, the dissipation potential and its numerical treatment have to be adapted. We obtain rate‐dependent time‐evolution equations for the internal variables based on variational techniques and show as an example single slip shear. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A variational viscosity-limit approach to the evolution of microstructures in finite crystal plasticity

A micromechanical model for finite single crystal plasticity was introduced by Kochmann & Hackl (2011 Contin. Mech. Thermodyn. 23, 63–85 ( doi:10.1007/s00161-010-0714-5 )). This model is based on thermodynamic variational principles and leads to a non-convex variational problem. Based on the Lagrange functional, an incremental strategy was outlined to model the time-continuous evolution of a first-order laminate microstructure. Although this model provides interesting results on the material point level, owing to the global minimization in the evolution equations, the calculation time and numerical instabilities may cause problems when applying this model to macroscopic specimens. In this paper, a smooth transition zone between the laminates is introduced to avoid global minimization, which makes the numerical calculations cumbersome compared with the model in Kochmann & Hackl. By introducing a smooth viscous transition zone, the dissipation potential and its numerical treatment have to be adapted. We outline rate-dependent time-evolution equations for the internal variables based on variational techniques and show as first examples single-slip shear and tension/compression tests.

Constitutive modeling of creep behavior in single crystal superalloys: Effects of rafting at high temperatures

Rafting and creep modeling of single crystal superalloys at high temperatures are important for the safety assessment and life prediction in practice. In this research, a new model has been developed to describe the rafting evolution and incorporated into the Cailletaud single crystal plasticity model to simulate the creep behavior. The driving force of rafting is assumed to be the relaxation of the strain energy, and it is calculated with the local stress state, a superposition of the external and misfit stress tensors. In addition, the isotropic coarsening is introduced by the cube root dependence of the microstructure periodicity on creep time based on Ostwal ripening. Then the influence of rafting on creep deformation is taken into account as the Orowan stress in the single crystal plasticity model. The capability of the proposed model is validated with creep experiments of CMSX-4 at 950°C and 1050°C. It is able to predict the rafting direction at complex loading conditions and evaluate the channel width during rafting. For [001] tensile creep tests, good agreement has been shown between the model predictions and experimental results at different temperatures and stress levels. The creep acceleration can be captured with this model and is attributed to the microstructure degradation caused by the precipitate coarsening.

Modeling dislocation absorption by surfaces within the framework of strain gradient crystal plasticity

Surfaces play important roles in plastic deformation of crystals at submicron scale by emitting/absorbing dislocations and hence may contribute to the complicated size effect. To take into account the dislocation absorption on the continuum level, an energetic surface model has been proposed by considering the change of surface energy contributed by the formation of surface steps via dislocation absorption. In combination with the conventional higher-order strain gradient theory for grain interior, a new model for single crystal plasticity has been developed. Within the model, there exists a critical threshold for the onset of dislocation absorption by surfaces, alternatively, an independent yield criterion for surfaces and the microscopic boundary conditions can be obtained naturally. As an example of application, plastic behavior of a thin film under plane constrained shear has been studied. Results demonstrate that surface yield strength strongly depends on the film thickness and the orientation of the activated slip system. Comparison with the strain gradient plasticity models under the conventional microscopic boundary conditions, i.e., the microhard and microfree conditions has also been made. It has been found that the plastic behavior is sensitive to the surface boundary conditions.

Multiscale modeling of irradiated polycrystalline FCC metals

We propose a set of models for the post-irradiation deformation response of polycrystalline FCC metals. First, a defect- and dislocation-density based evolution model is developed to capture the features of irradiation-induced hardening as well as intra-granular softening. The proposed hardening model is incorporated within a rate-independent single crystal plasticity model. The result is a non-homogeneous deformation model that accounts for defect absorption on the active slip planes during plastic loading. The macroscopic non-linear constitutive response of the polycrystalline aggregate of the single crystal grains is then obtained using a micro–macro transition scheme, which is realized within a Jacobian-free multiscale method (JFMM). The Jacobian-free approach circumvents explicit computation of the tangent matrix at the macroscale by using a Newton–Krylov process. This has a major advantage in terms of storage requirements and computational cost over existing approaches based on homogenized material coefficients in which explicit Jacobian computation is required at every Newton step. The mechanical response of neutron-irradiated single and polycrystalline OFHC copper is studied and it is shown to capture experimentally observed grain-level phenomena.

On the strain hardening and texture evolution in high manganese steels: Experiments and numerical investigation

We present a systematic investigation on the strain hardening and texture evolution in high manganese steels where twinning induced plasticity (TWIP) plays a significant role for the materials' plastic deformation. Motivated by the stress–strain behavior of typical TWIP steels with compositions of Fe, Mn, and C, we develop a mechanistic model to explain the strain-hardening in crystals where deformation twinning dominates the plastic deformation. The classical single crystal plasticity model accounting for both dislocation slip and deformation twinning are then employed to simulate the plastic deformation in polycrystalline TWIP steels. While only deformation twinning is activated for plasticity, the simulations with samples composed of voronoi grains cannot fully capture the texture evolution of the TWIP steel. By including both twinning deformation and dislocation slip, the model is able to capture both the stress–strain behaviors and the texture evolution in Fe–Mn–C TWIP steel in different boundary-value problems. Further analysis on the strain contributions by both mechanisms suggests that deformation twinning plays the dominant role at the initial stage of plasticity in TWIP steels, and dislocation slip becomes increasingly important at large strains.

Plastic anisotropy of electro-deposited pure α-iron with sharp crystallographic // texture in normal direction: Analysis by an explicitly dislocation-based crystal plasticity model

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 dislocation-based multi-rate single crystal plasticity model

The goal of this work is to formulate a constitutive model for the deformation of metallic single crystals over a wide range of strain rates, which is integral to computing reliable stress states of metallic polycrystals under shock loading. An elastic-viscoplastic, slip-based single crystal model that accounts for crystallographic orientation, temperature, and strain rate dependence has been formulated based on dislocation dynamics simulations and existing experimental data. The plastic model transitions from the low-rate, thermally-activated regime, to the high-rate, drag-dominated regime, by use of a distribution of dislocation velocities including kinetic effects. It has been compared favorably with experimental and computational results of copper. The transition to drag-dominated dislocation motion is predicted rather than empirically fit to experimental data.

Simulation of yield surfaces for aluminium sheets with rolling and recrystallization textures

The influence of crystallographic texture and homogenization scheme on the yield surface for aluminium sheets has been studied numerically. A rate-dependent single crystal plasticity model was adopted to describe the behaviour of the grains, whereas the behaviour of the polycrystal was obtained either by the crystal plasticity finite element method (CP-FEM) or by the full-constraint Taylor approach. A representative volume element (RVE) of the material including 800 grains was considered. The grain orientations were defined according to typical rolling and recrystallization textures exhibited in rolled aluminium sheets. With CP-FEM, the RVE was discretized with finite elements, using in average 30 cubical solid elements per grain, whereas in the full-constraint Taylor approach, all grains are assumed to be subjected to the same deformation. Stress states located on the yield surface were determined by subjecting the RVE to proportional deformation in different directions, assuming yielding to occur at a given plastic work per unit volume. The yield surface was then described analytically by fitting the Yld2004-18p yield function to the obtained stress states at yield, and further employed to calculate stress ratios and Lankford coefficients in uniaxial tension. The resulting yield surfaces for aluminium with rolling and recrystallization textures are markedly different. The influence of homogenization scheme is moderate, but significant effects are found on the predicted Lankford coefficients.

Surface effects and the size-dependent hardening and strengthening of nickel micropillars

We evaluate the extent to which two mechanisms contribute to the observed size effect of the ultimate yield strength of micropillars of diameters in the range of 1–30μm: dislocation pile-ups, modeled by means of a physically based non-local single-crystal plasticity model; and the short-range interaction of dislocations with the free surface of the micropillars, e.g., through the formation of surface steps. To this end, we formulate a crystal-plasticity model that accounts for the self-energy of geometrically necessary dislocations and the formation energy of dislocation steps at the boundary of the solid. These two additional sources of energy have the effect of rendering the internal energy of the solid non-local, thereby introducing the possibility of size effects. By way of validation of the model, we simulate the uniaxial compression tests on [269] nickel micropillars of Dimiduk et al. (2005). The calculated dependence of the ultimate strength of the micropillars exhibits strong power-law behavior, and is in good agreement with observation. Our analysis suggests that non-local hardening due to the self-energy of geometrically necessary dislocations does not suffice to account for the observed size effect of the ultimate yield strength of micropillars, and that surface effects, such as resulting from the formation energy of dislocation steps, contribute significantly to that size effect.

Near Grain Boundary Behavior of Aluminum Bicrystals Deformed in Plane Strain Conditions

The paper describes the mechanism of deformation at 77 K of pure aluminum bicrystals of different grain orientations. The following orientations were selected: {100}/{110} (cube/Goss) and - {100}/{100} (cube/shear) to represent the unstable vs. stable and the unstable vs. unstable behaviours, respectively. The bicrystalline samples were deformed in the plane strain conditions with the use of a channel-die immersed inside a reservoir with liquid nitrogen. The low temperature deformation increases the tendency to form plain strain inhomogeneities of the deformation in the grains with an unstable orientation. In both sets of crystallite compositions, the grain boundary was situated perpendicularly to the compression plane. A particular interest was paid to the analysis of the tendencies of the crystal lattice rotations near the grain boundary and the description of the deformation behaviour of the material in the macro- scale (hardening behaviour). A detailed analysis of the crystal lattice rotations was possible with the application of the local orientation measurements by means of scanning and transmission electron microscopes, equipped with the electron backscattered diffraction and convergent beam electron diffraction facility, respectively. The experimental results of the local orientation measurements were used to evaluate the accuracy of the numerical prediction of the macro-scale behaviour of bi-crystalline samples by a single Cristal Plasticity Model. The investigation shows that the crystallites behave essentially as single crystals in the same deformation conditions. Due to the similar hardening behaviour of the investigated crystallites (similar values of the Taylor factors) the grain boundary remains unchanged. The calculated lattice rotations are similar to those observed experimentally. Key words: aluminium bi-crystals, texture, microstructure, single crystal plasticity model

Phenomenological crystal plasticity modeling and detailed micromechanical investigations of pure magnesium

We present a single crystal plasticity model for pure Mg incorporating slip and deformation twinning. The model uses the basic framework of Kalidindi (1998), but proposes constitutive descriptions for the slip and twin evolution and their interactions that are motivated by experimental observations. Based on compelling experimental evidences, we distinguish between the constitutive descriptions of the tension and compression twinning to better represent their roles in the overall hardening of Mg single crystals. With these improved phenomenological descriptions, we first calibrate material parameters for the different slip and twin modes by performing three-dimensional simulations mimicking the plane-strain compression experiments by Kelley and Hosford (1967, 1968) on single crystal pure Mg. In doing so, these computational responses are critically compared with their corresponding orientation-dependent microscopic (slip and twin activities) and macroscopic (stress–strain responses) experimental observations. Then, the calibrated parameters are used to predict several other experimental results on pure single- and poly-crystal Mg under different loading conditions. We also investigate the role of pre-existing heterogeneities such as initial twin population and stiff, elastic inclusions on the single crystal macroscopic and microscopic responses. Microstructural characteristics show that such heterogeneities strongly influence the local and global evolution of the slip and twin activities, and in some cases modulate the strength anisotropy that is commonly observed in monolithic single crystals. These results may provide useful indicators toward designing novel composite Mg microstructures.

Simulation of texture evolution during plastic deformation of FCC, BCC and HCP structured crystals with crystal plasticity based finite element method

Two alternative formulations of single crystal plasticity model were introduced respectively and two schemes were implemented in the explicit FE code with software ABAQUS/Explicit by writing the user subroutine VUMAT. Meshes containing material data were created with solid elements. Each element represented an individual grain, and the grain orientations were explicitly stored and updated at each increment. Tangential modulus method was employed to calculate the plastic shear strain increment of deformation systems in combination with a hardening law to describe the hardening responses. Both two developed subroutines were applied to simulate the texture evolution during the uniaxial tension of copper (FCC), cold rolling of IF steel (BCC) and uniaxial compression of AZ31 magnesium alloy (HCP). The predicted texture distributions are in qualitative agreement with the experimental results.

Bending of single crystal thin films modeled with micropolar crystal plasticity

The scale-dependent mechanical response of single crystal thin films subjected to pure bending is investigated using a dislocation-based model of micropolar single crystal plasticity via finite element simulations. Due to the presence of couple stresses, the driving force for plastic slip in a micropolar crystal contains an intrinsic back stress component that is related to gradients in lattice torsion-curvature. Strain gradient-dependent back stresses are a common feature of various types of generalized crystal plasticity theories; however, it is often introduced either in a phenomenological manner without additional kinematics or by designating the plastic slips as generalized degrees-of-freedom. The treatment of lattice rotations as fundamental degrees-of-freedom instead of plastic slips greatly reduces the complexity (computational expense) of the single crystal model, and leads to the incorporation of additional elastoplastic kinematics since the lattice torsion-curvature is taken as a work-conjugate continuum deformation measure. A recently proposed single criterion micropolar framework is employed in which the evolution of both the plastic strains and torsion-curvatures are coupled through the use of a unified flow rule. The deformation behavior is characterized by the moment-rotation response and the dislocation substructure evolution for various slip configurations and specimen thicknesses. The results are compared to analogous simulations carried out using a model of discrete dislocation dynamics as well as a statistical-mechanics inspired, flux-based model of nonlocal crystal plasticity. The micropolar model demonstrates good qualitative and quantitative agreement with the previous results up to certain inherent limitations of the current formulation.