Grain boundary interface mechanics in strain gradient crystal plasticity

Interactions between dislocations and grain boundaries play an important role in the plastic deformation of polycrystalline metals. Capturing accurately the behaviour of these internal interfaces is particularly important for applications where the relative grain boundary fraction is significant, such as ultra fine-grained metals, thin films and micro-devices. Incorporating these micro-scale interactions (which are sensitive to a number of dislocation, interface and crystallographic parameters) within a macro-scale crystal plasticity model poses a challenge. The innovative features in the present paper include (i) the formulation of a thermodynamically consistent grain boundary interface model within a microstructurally motivated strain gradient crystal plasticity framework, (ii) the presence of intra-grain slip system coupling through a microstructurally derived internal stress, (iii) the incorporation of inter-grain slip system coupling via an interface energy accounting for both the magnitude and direction of contributions to the residual defect from all slip systems in the two neighbouring grains, and (iv) the numerical implementation of the grain boundary model to directly investigate the influence of the interface constitutive parameters on plastic deformation. The model problem of a bicrystal deforming in plane strain is analysed. The influence of dissipative and energetic interface hardening, grain misorientation, asymmetry in the grain orientations and the grain size are systematically investigated. In each case, the crystal response is compared with reference calculations with grain boundaries that are either ‘microhard’ (impenetrable to dislocations) or ‘microfree’ (an infinite dislocation sink).     

Assessment of plastic heterogeneity in grain interaction models using crystal plasticity finite element method

Micromechanical models aimed at simulating deformation textures and resulting plastic anisotropy need to incorporate local plastic strain heterogeneities arising from grain interactions for better predictions. The ALAMEL model [Van Houtte, P., Li, S., Seefeldt, M., Delannay, L. 2005. Deformation texture prediction: from the Taylor model to the advanced Lamel model. Int. J. Plasticity 21, 589–624], is one of the models in which the heterogeneous nature of plastic deformation in metals is introduced by accounting for the influence of a grain boundary on the cooperative deformation of adjacent grains. This is achieved by assuming that neighbouring grains undergo heterogeneous shear rates parallel to the grain boundary. The present article focuses on understanding the plastic deformation fields near the grain boundaries and the influence of grain interaction on intra-grain deformations. Crystal Plasticity Finite Element Method (CPFEM) is employed on a periodic unit cell consisting of four grains discretised into a large number of elements. A refined study of the local variation of strain rates, both along and perpendicular to the grain boundaries permits an assessment of the assumptions made in the ALAMEL model. It is shown that the ALAMEL model imbibes the nature of plastic deformation at the grain boundaries very well. However, near triple junctions, the influence of a third grain induces severe oscillations of the stress tensor, reflecting a singularity. According to CPFEM, such singularity can lead to grain subdivision by the formation of new boundaries originating at the triple junction.     

Defect redistribution within a continuum grain boundary plasticity model

The mechanical response of polycrystalline metals is significantly affected by the behaviour of grain boundaries, in particular when these interfaces constitute a relatively large fraction of the material volume. One of the current challenges in the modelling of grain boundaries at a continuum (polycrystalline) scale is the incorporation of the many different interaction mechanisms between dislocations and grain boundaries, as identified from fine-scale experiments and simulations. In this paper, the objective is to develop a model that accounts for the redistribution of the defects along the grain boundary in the context of gradient crystal plasticity. The proposed model incorporates the nonlocal relaxation of the grain boundary net defect density. A numerical study on a bicrystal specimen in simple shear is carried out, showing that the spreading of the defect content has a clear influence on the macroscopic response, as well as on the microscopic fields. This work provides a basis that enables a more thorough analysis of the plasticity of polycrystalline metals at the continuum level, where the plasticity at grain boundaries matters.     

Predicting the Hall–Petch effect in fcc metals using non-local crystal plasticity

Many conventional continuum approaches to solid mechanics do not address the size sensitivity of deformation to microstructural features like grain boundaries, and are therefore unable to capture much of the experimentally observed behavior of polycrystal deformation. We propose a non-local crystal plasticity model, in which the geometrically necessary dislocation (GND) density is calculated using a non-local integral approach. The model is based on augmented F^eF^p kinematics, which account for the initial microstructure (primarily grain boundaries) present in the polycrystal. With the augmented kinematics, the initial GND and the evolving GND state are determined in a consistent manner. The expanded kinematics and the non-local crystal plasticity model are used to simulate the tensile behavior in copper polycrystals with different grain sizes ranging from 14 μm to 244 μm. The simulation results show a grain size dependence on the polycrystal’s yield strength, which are in good agreement with the experimental data.     

Crystal plasticity finite element modelling of low cycle fatigue in fcc metals

A new dislocation-based model for low cycle fatigue in fcc metals at a length scale smaller than the feature size of the dislocation structures is presented. It uses the crystal plasticity finite element method and dislocation densities as internal variables. Equations for the dipole distance distribution, for the double cross slip mechanism and a new dislocation multiplication law are introduced, which can predict the emergence of vein and channel structures starting from a randomly perturbed dislocation distribution. The characteristics of these structures in copper and aluminium, as well as the mechanical properties, are compared with experiments. Compared with existing density-based theories, the capability to reproduce dislocation patterning is a significant step forward.     

Multiscale modeling of the plasticity in an aluminum single crystal

This paper describes a numerical, hierarchical multiscale modeling methodology involving two distinct bridges over three different length scales that predicts the work hardening of face centered cubic crystals in the absence of physical experiments. This methodology builds a clear bridging approach connecting nano-, micro- and meso-scales. In this methodology, molecular dynamics simulations (nanoscale) are performed to generate mobilities for dislocations. A discrete dislocations numerical tool (microscale) then uses the mobility data obtained from the molecular dynamics simulations to determine the work hardening. The second bridge occurs as the material parameters in a slip system hardening law employed in crystal plasticity models (mesoscale) are determined by the dislocation dynamics simulation results. The material parameters are computed using a correlation procedure based on both the functional form of the hardening law and the internal elastic stress/plastic shear strain fields computed from discrete dislocations. This multiscale bridging methodology was validated by using a crystal plasticity model to predict the mechanical response of an aluminum single crystal deformed under uniaxial compressive loading along the [4 2 1] direction. The computed strain-stress response agrees well with the experimental data.     

A crystal plasticity theory for latent hardening by glide twinning through dislocation transmutation and twin accommodation effects

Imagine a residual glide twin interface advancing in a grain under the action of a monotonic stress. Close to the grain boundary, the shape change caused by the twin is partly accommodated by kinks and partly by slip emissions in the parent; the process is known as accommodation effects. When reached by the twin interface, slip dislocations in the parent undergo twinning shear. The twinning shear extracts from the parent dislocation a twinning disconnection, and thereby releases a transmuted dislocation in the twin. Transmutation populates the twin with dislocations of diverse modes. If the twin deforms by double twinning, double-transmutation occurs even if the twin retwins by the same mode or detwins by a stress reversal. If the twin deforms only by slip, transmutation is single. Whether single or double, dislocation transmutation is irreversible. The multiplicity of dislocation modes increases upon strain, since the twin finds more dislocations to transmute upon further slip of the parent and further growth of the twin. Thus, the process induces an increasing latent hardening rate in the twin. Under profuse twinning conditions, typical of double-lattice structures, this rate-increasing latent hardening combined with crystal rotation to hard orientations by twinning is consistent with a regime of increasing hardening rate, known as Regime II or Regime B. In this paper, we formulate governing equation of the above transmutation and accommodation effects in a crystal plasticity framework. We use the dislocation density based model originally proposed by Beyerlein and Tomé (2008) to derive the effect of latent hardening in a transmuting twin. The theory is expected to contribute to surmounting the difficulty that current models have to simultaneously predict under profuse twinning, the stress-strain curves, intermediate deformation textures, and intermediate twin volume fractions.     

Crystal plasticity analysis of texture development in magnesium alloy during extrusion

The texture development mechanism during the extrusion of magnesium alloy is investigated by experimental observation and numerical analysis. First, we perform a finite element analysis of a full extrusion process using a phenomenological constitutive equation, and it is confirmed that the loading condition of the extrusion process near the central axis of the billet is approximated by an equi-biaxial compression mode. Then, the equi-biaxial compression problem is adopted as a simplified boundary value problem to be solved using a crystal plasticity model to clarify the detailed texture development mechanism during the extrusion process. The crystal plasticity analysis of equi-biaxial compression successfully reproduces the texture development from an initial random texture to the final experimentally observed texture. The effects of the deformation modes (i.e. slip and twinning systems) implemented in the calculation and the reference stress ratio of basal to nonbasal slip systems on texture development are studied in detail. Finally, the mechanism of texture development during the extrusion process is discussed in terms of the lattice rotation caused by the activated slip systems.     

A homogenization theory of strain gradient single crystal plasticity and its finite element discretization

In this study, a homogenization theory based on the Gurtin strain gradient formulation and its finite element discretization are developed for investigating the size effects on macroscopic responses of periodic materials. To derive the homogenization equations consisting of the relation of macroscopic stress, the weak form of stress balance, and the weak form of microforce balance, the Y-periodicity is used as additional, as well as standard, boundary conditions at the boundary of a unit cell. Then, by applying a tangent modulus method, a set of finite element equations is obtained from the homogenization equations. The computational stability and efficiency of this finite element discretization are verified by analyzing a model composite. Furthermore, a model polycrystal is analyzed for investigating the grain size dependence of polycrystal plasticity. In this analysis, the micro-clamped, micro-free, and defect-free conditions are considered as the additional boundary conditions at grain boundaries, and their effects are discussed.     

Heterogeneous deformation and spall of an extruded tungsten alloy: plate impact experiments and crystal plasticity modeling

The role of microstructure in the dynamic deformation and fracture of a dual phase, polycrystalline tungsten alloy under high-rate impact loading is investigated via experiments and modeling. The material studied consists of pure tungsten crystals embedded in a ductile binder alloy comprised of tungsten, nickel, and iron. The tungsten crystals are elongated in a preferred direction of extrusion during processing. Plate impact tests were conducted on samples oriented either perpendicular or parallel to the extrusion direction. Spatially resolved interferometric data from these tests were used to extract wave propagation behavior and spall strength dependent upon position in the sample microstructure. Finite element simulations of impact and spall in digitally reproduced microstructural geometries were conducted in parallel with the experiments. Finite deformation crystal plasticity theory describes the behavior of the pure tungsten and binder phases, and a stress- and temperature-based cohesive zone model captures fracture at grain and phase boundaries in the microstructure. In results from both experiments and modeling, the grain orientations affect the free-surface velocity profile and spall behavior. Some aspects of distributions of free-surface velocity and spall strength among different microstructure configurations are qualitatively similar between experimental and numerical results, while others are not as a result of differing scales of resolution and modeling assumptions. Following a comparison of experimental and numerical results for different microstructures, intergranular fracture is identified as an important mechanism underlying the spall event.     

Incorporation of twinning into a crystal plasticity finite element model: Evolution of lattice strains and texture in Zircaloy-2

A crystal plasticity finite element code is developed to model lattice strains and texture evolution of HCP crystals. The code is implemented to model elastic and plastic deformation considering slip and twinning based plastic deformation. The model accounts for twinning reorientation and growth. Twinning, as well as slip, is considered to follow a rate dependent formulation. The results of the simulations are compared to previously published in situ neutron diffraction data. Experimental results of the evolution of the texture and lattice strains under uniaxial tension/compression loading along the rolling, transverse, and normal direction of a piece of rolled Zircaloy-2 are compared with model predictions. The rate dependent formulation introduced is capable of correctly capturing the influence of slip and twinning deformation on lattice strains as well as texture evolution.     

Constitutive modeling for nanocrystalline metals based on cooperative grain boundary mechanisms

In nanocrystalline metals, the plastic deformation is accommodated primarily at the grain boundaries. Yang and Wang (J. Mech. Phys. Solids, 2003) suggested a deformation model based on clusters consisting of nine grains and incorporating both the Ashby-Verrall mechanism and a 30° rotation of closely linked pairs of grains. In the present article, the insertion and rotation processes are considered together as a cooperative deformation mechanisms, and the degree to which each process contributes is determined by the application of the principle of maximum plastic work. Plane strain and three-dimensional constitutive relations based on this concept are derived for which a general stress state drives the orientation evolution of various grain clusters under the Reuss assumption. Detailed calculation shows that the strain rate depends linearly on the stress, with the values of the coefficients in this linear relationship dictated by the microscopic energy dissipation. The deformation contributed to the overall response by the grain boundary mechanism is discussed in the spirit of the Hashin-Shtrikman bounds.     

Grain-boundary sliding and separation in polycrystalline metals: application to nanocrystalline fcc metals

In order to model the effects of grain boundaries in polycrystalline materials we have coupled a crystal-plasticity model for the grain interiors with a new elastic-plastic grain-boundary interface model which accounts for both reversible elastic, as well irreversible inelastic sliding-separation deformations at the grain boundaries prior to failure. We have used this new computational capability to study the deformation and fracture response of nanocrystalline nickel. The results from the simulations reflect the macroscopic experimentally observed tensile stress-strain curves, and the dominant microstructural fracture mechanisms in this material. The macroscopically observed nonlinearity in the stress-strain response is mainly due to the inelastic response of the grain boundaries. Plastic deformation in the interior of the grains prior to the formation of grain-boundary cracks was rarely observed. The stress concentrations at the tips of the distributed grain-boundary cracks, and at grain-boundary triple junctions, cause a limited amount of plastic deformation in the high-strength grain interiors. The competition of grain-boundary deformation with that in the grain interiors determines the observed macroscopic stress-strain response, and the overall ductility. In nanocrystalline nickel, the high-yield strength of the grain interiors and relatively weaker grain-boundary interfaces account for the low ductility of this material in tension.     

A finite deformation theory of higher-order gradient crystal plasticity

For higher-order gradient crystal plasticity, a finite deformation formulation is presented. The theory does not deviate much from the conventional crystal plasticity theory. Only a back stress effect and additional differential equations for evolution of the geometrically necessary dislocation (GND) densities supplement the conventional theory within a non-work-conjugate framework in which there is no need to introduce higher-order microscopic stresses that would be work-conjugate to slip rate gradients. We discuss its connection to a work-conjugate type of finite deformation gradient crystal plasticity that is based on an assumption of the existence of higher-order stresses. Furthermore, a boundary-value problem for simple shear of a constrained thin strip is studied numerically, and some characteristic features of finite deformation are demonstrated through a comparison to a solution for the small deformation theory. As in a previous formulation for small deformation, the present formulation applies to the context of multiple and three-dimensional slip deformations.     

Modeling texture evolution in equal channel angular extrusion using crystal plasticity finite element models

A new approach to modeling crystallographic texture evolution in Equal Channel Angular Extrusion (ECAE) is presented in this paper. The proposed approach utilizes an elastic–viscoplastic single crystal constitutive model implemented in a finite element framework. A representative volume element of the polycrystal is subjected to boundary conditions that simulate the approximate deformation history experienced by different regions of the sample (at different through-thickness depths) in both Route A and Route C processing. The proposed approach aims to capture the influence of the complex interactions that ensue among the constituent individual crystals of a polycrystal in controlling the texture evolution in the sample, while capturing the boundary conditions inherent to ECAE deformation. The predictions from the proposed approach are compared against previously reported experimental measurements in ECAE of copper. It is observed that the proposed approach provides significantly better agreement with the measurements when compared against previously reported model predictions.     

Microstructure-based crystal plasticity modeling of cyclic deformation of Ti–6Al–4V

Ti–6Al–4V is a dual phase material with range of possible complex microstructures. It is well known that mechanical behavior of Ti–6Al–4V is significantly affected by its texture and microstructure morphology. A three-dimensional microstructure-based constitutive model for monotonic and cyclic deformation of duplex Ti–6Al–4V is developed and implemented. The model includes length scale effects associated with dislocation interactions with different microstructure features, and is calibrated using polycrystalline finite element simulations to fit the measured macroscopic responses (overall stress–strain behavior) of a duplex heat treated Ti–6Al–4V alloy subjected to a complex cyclic loading history. Representative microstructures are simulated using a three-dimensional finite element mesh with periodic boundary conditions imposed in all directions. The measured orientation and misorientation distributions of grains of this duplex Ti–6Al–4V are considered, and similar probability density distributions of the crystallographic orientations are assigned to the finite element mesh. The misorientation distributions are then fit using the simulated annealing method. Effects of microstructural features are examined and compared with the experimental data in terms of their influence on the material yield strength. The results are shown to be in good agreement with the experimental observations.     

Multiscale modelling of the induced plastic anisotropy in bcc metals

This paper presents a new framework to predict the qualitative and quantitative variation in local plastic anisotropy due to crystallographic texture in body-centered cubic polycrystals. A multiscale model was developed to examine the contribution of mesoscopic and local microscopic behaviour to the macroscopic constitutive response of bcc metals during deformation. The model integrated a dislocation-based hardening scheme and a Taylor-based crystal plasticity formulation into the subroutine of an explicit dynamic FEM code (LS-DYNA). Numerical analyses using this model were able to predict not only correct grain rotation during deformation, but variations in plastic anisotropy due to initial crystallographic orientation. Optimal results were obtained when {1 1 0}〈1 1 1〉, {1 1 2}〈1 1 1〉, and {1 2 3}〈1 1 1〉 slip systems were considered to be potentially active. The predicted material heterogeneity can be utilised for research involving any texture-dependent work hardening behaviour, such as surface roughening.     

A numerical modelling of 3D polycrystal-to-polycrystal diffusive phase transformations involving crystal plasticity

A FE modelling of the elastoplastic interactions occurring within a 3D polycrystal subjected to diffusive phase transformation is proposed. The parent polycrystal is represented by a Voronoi tessellation, where grains differ in shape, size and crystallographic orientation. Grains of the new phase nucleate at favourable sites of the parent polycrystal then grow isotropically, following specific kinetics. This process can result in various product polycrystal morphologies where grains are distinguished by their morphologies and their crystallographic orientations, and have crystalline properties different from those of the parent grains. Application is performed on the austenite-to-ferrite transformation of a low carbon steel, by analysing different basic cases of transformation history with different constitutive modellings. Microplasticity and its related internal stresses are shown to develop during the phase transformations and to affect significantly the elastoplasticity of the product medium.