Classical Crystal Plasticity Research Articles

Experimental and digital twinning in ZnAlMg coatings

Twinning is a major deformation mechanism in various materials, especially when few dislocation slip systems are operative. It is the case of zinc-rich coatings in galvanised steel sheets, made of pancake grains on a substrate and where the slip systems with a non-vanishing component along the c-axis present high critical resolved shear stress values. In addition, the abrupt lattice orientation change associated to twinning, the stress relaxation during its propagation and the localised nature of its early stages make it difficult to reproduce this deformation mechanism by using classical crystal plasticity models conceived for dislocation slip. In this sense, this work proposes a hierarchy of three twinning models in combination with a dislocation slip crystal plasticity model, for the case of a ZnAlMg coating. These three models are: a relaxed-Taylor model applied to individual crystal orientations of the coating, a “pseudo-slip” model for twinning and a localised twinning model. The latter incorporates a linear softening in the material law accounting for the unstable twinning initiation and enforces twinning lattice reorientation. A microstructure portion extracted from an in-situ SEM tensile experiment on galvanised steel is used to perform 3D full-field finite element simulations within a finite strain formulation. SEM observations and EBSD acquisitions are used to compare simulation and experimental results during the different steps of the in-situ SEM test, regarding the deformation and damage modes of the zinc-rich coating. The focus is set on twinning evolution inside some individual grains, and the pros and the cons of the three models are finally discussed.

Slip-discreteness-corrected strain gradient crystal plasticity (SDC-SGCP) theory

Strain gradient plasticity theory addresses the plastic strain gradient induced hardening by considering the internal stress and Taylor hardening associated with the geometrically necessary dislocations (GNDs). However, the continuum description of internal stress associated with GNDs is inaccurate due to the coarsening of discrete dislocations. Corrections are thus derived as the difference between the stresses produced by the continuous configuration and the discrete configuration. We further demonstrate the capability of this correction in effectively capturing the internal stress induced strengthening effect associated with GNDs, and elucidate that its role in strengthening is to homogenize the deformation and extend the influence of grain boundaries into the interior of grains within polycrystals. This capability to capture intragranular slip distribution is validated through the simulation of a polycrystalline tensile experiment. This work explains the limitations of classical crystal plasticity theory under high strain gradients and offers a straightforward yet robust slip discreteness correction to crystal plasticity with transparent input from dislocation theory, opening a new perspective for the connections between continuum crystal plasticity theory and dislocation theory.

A crystal plasticity model for multiaxial cyclic deformation of U75V rail steel

The theoretical investigations of cyclic deformation behavior and deformation mechanism of the rail steels under multiaxial loadings are of great significance to the study of rolling contact fatigue. Based on the classical crystal plasticity framework, a new nonproportional parameter at the mesoscale is proposed, and the isotropic and kinematic hardening models are modified. A series of finite element simulations of U75V rail steel under uniaxial and multiaxial cyclic loading are carried out using the proposed model. The simulation results show that the proposed model can reasonably reflect the cyclic softening under strain-controlled uniaxial cyclic loading, cyclic hardening under strain-controlled multiaxial cyclic loadings, and ratcheting behavior under stress-controlled uniaxial and multiaxial cyclic loadings of U75V rail steel. Moreover, the influence of multiaxial loading paths on the plastic strain is explained from a mesoscopic perspective. The values of nonproportional parameters of different slip systems under uniaxial and multiaxial paths are compared, and the difference of nonproportional parameters at the macroscale and mesoscale are discussed. Finally, the simulation results of the proposed model and the existing model with modified kinematic hardening rules are compared. The results show that the proposed model has better predictions on the evolution of ratcheting strain and nonproportionally additional hardening effect caused by nonproportionally multiaxial loadings.

Simulation of crystal plasticity in irradiated metals: A case study on Zircaloy-4

A classical crystal plasticity formulation based on dislocation slip was extended to include the mechanisms of dislocation channelling, with associated strain softening which is observed in many alloys post irradiation. The performance of the model was evaluated against experimental data on Zircaloy-4, which included engineering stress-strain response and high-resolution digital image correlation strain mapping. Variants of the model were developed to evaluate the influence on the strain hardening law used, comparing hardening based on a linear relationship for effective plastic strain with that based on the evolution of geometrically necessary dislocations. In addition, governing equations for simulating the interaction between gliding dislocations with various types of irradiation defect were investigated; this included the comparison of isotropic and anisotropic interactions based on the resultant reaction segments for each interaction. It was shown that the engineering stress-strain response measured by experiment could be captured by many of the model variants, but the simulation of characteristic strain heterogeneity was more sensitive to the model used. Direct modelling of the HRDIC experiments indicated that the model successfully predicted the activation of slip systems in many cases and exhibited localised strain distribution as observed in the experiment. In all models localised kink band formation was predicted, which is not observed experimentally, which highlights limitations in modelling of softening materials with a local crystal plasticity approach and a required area of development going forward.

Modeling and analysis of grain morphology effects on deformation response based on crystal plasticity finite element method

AbstractIn order to investigate the grain morphology effects on deformation response at macro‐ and micro‐scale in polycrystalline metal, a method considering grain size and shape effects is proposed based on the classical crystal plasticity finite element method in this paper. A grain morphology controlled polycrystalline geometry model consisting of grain shape models and grain size transition models is developed based on Voronoi diagram. Grain size effect is introduced into the classical crystal plasticity constitutive model quantitatively on the basis of Hall‐Petch type relationship. With the aid of the method, grain aggregate finite element meshes with different mean grain sizes are established to simulate the deformation response of polycrystalline pure copper. The predicted values of flow stresses show good agreement with experimental ones. Finally, a set of grain aggregate finite element meshes with same mean grain size but different grain distributions are developed to simulate the uniaxial tension processes of polycrystalline pure copper. The results shown that the effects of grain morphology on the macro‐scale deformation response are insignificant under a large number of grains with random morphology and orientation. When the grain distributions changes regularly along the x direction, the maximum decrease of flow stress data is 10 MPa.

Experimental and numerical investigation of differential hardening of cold rolled steel sheet under non-proportional loading using biaxial tensile test

Strain path change is common phenomenon during industrial sheet metal forming processes. These strain paths can be proportional and non-proportional in nature. To enable strain path change in a single experiment is key challenge faced by many researchers. To predict deformation behavior accurately during such strain paths, it is required to represent initial yielding, the evolution of subsequent yield surface and hardening behavior precisely. This work deals with the mechanical characterization of Interstitial Free High Strength Steel (IFHS) during multi-axial proportional and non-proportional loading for the determination of yield surface evolution and strain hardening behavior. A set of tensile tests at various orientations with respect to the rolling direction and in-plane biaxial tests using cruciform specimens for various linear and bi-linear stress ratios were performed on as-received materials. It was found that at the onset of stress path change, the stress strain curve shows a change in the hardening rate for rolling and transverse directions of the biaxial specimen due to the change in active slip systems. To predict this behaviour various yield criterion like von Mises, Hill's 48 and Yld2000-2d were used. The experimental results show that the mechanical behavior of the subjected specimen deviates from isotropic behavior predicted by the von Mises and anisotropic behavior predicted by the Hill's 48 yield criterion. The initial yield loci measured by biaxial testing machine is in good agreement with what is predicted by Yld2000-2d yield criterion. However, it was observed that such phenomenological models were unable to capture the observed deformation behaviour. It is also shown that classical Crystal Plasticity Finite Element Method (CPFEM) can capture change in strain hardening rate during non-proportional deformation for few cases reasonably well while in few cases it could not predict.

A crystal plasticity model that accounts for grain size effects and slip system interactions on the deformation of austenitic stainless steels

To ensure a more representative simulation of meso‑scale and macro-scale deformation, it is important the underlying constitutive relations are advanced to more effectively incorporate the micro-mechanisms such as those operative near the grain boundaries. Critical features necessary for accurate predictions include the influence of grain size, morphology, and misorientation on the local deformation. In this study, a length scale dependence is incorporated into classical crystal plasticity simulations. The implementation is based on the influence of dislocation slip pile-up at grain boundaries based on the Hall-Petch theory in materials. This was achieved by applying a purpose-built algorithm to extract the slip distance in adjacent grains along each slip direction. This extracted data was used to adjust the critical resolved shear stress within the grain. The adjusted slip law was then implemented in constitutive models to predict the yield stress and hardening evolution of material for different grain sizes. The interaction of the slip systems between grains was also considered, which resulted in the adjustment in the extent of slip transfer permitted between grains based on misorientation. The proposed approach can be applied in both finite element and spectral methods for solving the continuum differential equations. The accuracy of the proposed model was investigated by considering the meso‑scale predictions using two-dimensional simulations, in addition to macro-scale predictions using three-dimensional models. Furthermore, the validity of the method was experimentally supported by tracking the development of intragranular residual elastic stresses, measured via high-resolution electron backscatter diffraction. The discussed results highlight a new potential of enhancing the way in which complex grain boundary interactions contribute to local deformation in crystal plasticity simulations.

A method to extract slip system dependent information for crystal plasticity models

A tool to implement a length scale dependency to classical crystal plasticity simulations is presented. Classical crystal plasticity models do not include a size effect; therefore, the size of the grain does not influence the simulated deformation. Classical crystal plasticity advancements have been through the inclusion of stress or strain gradient based constitutive models to improve the simulation of length scale dependent deformation. However, this tool presents an alternative to implementing a length scale, where the influence of slip pile-up in the form of dislocations at grain boundaries as a potential to explaining the Hall-Petch effect in materials. This is achieved by calculating the slip distance in adjacent grains for each slip system, by assuming the total slip length spans the grain in the slip direction. These calculations can occur in two ways. The first is the analysis occurs at the start of the simulation, therefore, only occurs once. If this approach is used, the computational cost of this tool is minute. However, if the simulations consider large deformations, during which it is expected that the grains are going to undergo large rotations, then it would be advantageous to the have the tool recalculate the information during the analysis. Consequently, the computational cost would depend on the resolution of the modelled geometry, the number of grains, and the number of slip systems. The tool also provides a capability to develop constitutive models based on complex grain boundary features which can be implemented in classical crystal plasticity models and gradient based crystal plasticity models. The described calculation process is implemented through a Fortran subroutine, which has been designed to be easily used in crystal plasticity simulations. The presented tool also includes Python code designed to link with microstructures built using DREAM.3D to extract the required input data to the Fortran subroutine.The proposed tool is not limited to classical crystal plasticity formulations, instead the data extracted and outputted from the Fortran subroutine can be used to serve alternative purposes in both stress and strain gradient crystal plasticity models.The proposed tool can be modified to extract additional data to that presented.The slip distance in the adjacent grain, the distance from the grain boundary of the current calculation point, and the interaction between slip systems between grains can be used in any crystal plasticity constitutive models.

Polycrystal plasticity with grain boundary evolution: a numerically efficient dislocation-based diffuse-interface model

Grain structure plays a key role in the mechanical properties of alloy materials. Engineering the grain structure requires a comprehensive understanding of the evolution of grain boundaries (GBs) when a material is subjected to various manufacturing processes. To this end, we present a computationally efficient framework to describe the co-evolution of bulk plasticity and GBs. We represent GBs as diffused geometrically necessary dislocations, whose evolution describes GB plasticity. Under this representation, the evolution of GBs and bulk plasticity is described in unison using the evolution equation for the plastic deformation gradient, an equation central to classical crystal plasticity theories. To reduce the number of degrees of freedom, we present a procedure which combines the governing equations for each slip rates into a set of governing equations for the plastic deformation gradient. Finally, we outline a method to introduce a synthetic potential to drive migration of a flat GB. Three numerical examples are presented to demonstrate the model. First, a scaling test is used to demonstrate the computational efficiency of our framework. Second, we study the evolution of a tricrystal, formed by embedding a circular grain into a bicrystal, and demonstrate qualitative agreement between the predictions of our model and those of molecular dynamics simulations by Trautt and Mishin (2014 Acta Mater. 65 19–31). Finally, we demonstrate the effect of applied loading in texture evolution by simulating the evolution of a synthetic polycrystal under applied displacements.

A new finite element approach to model microscale strain localization within olivine aggregates

Abstract. This paper presents a new mesoscopic full field approach for the modeling of microstructural evolutions and mechanical behavior of olivine aggregates. The mechanical framework is based on a reduced crystal plasticity (CP) formulation which is adapted to account for non-dislocation glide strain-accommodating mechanisms in olivine polycrystals. This mechanical description is coupled with a mixed velocity–pressure finite element (FE) formulation through a classical crystal plasticity finite element method (CPFEM) approach. The microstructural evolutions, such as grain boundary migration and dynamic recrystallization, are also computed within a FE framework using an implicit description of the polycrystal through the level-set approach. This numerical framework is used to study the strain localization, at the polycrystal scale, on different types of pre-existing shear zones for thermomechanical conditions relevant to laboratory experiments. We show that both fine-grained and crystallographic textured pre-existing bands favor strain localization at the sample scale. The combination of both processes has a large effect on strain localization, which emphasizes the importance of these two microstructural characteristics (texture and grain size) on the mechanical behavior of the aggregate. Table 1 summarizes the list of the acronyms used in the following.

Modeling plasticity of cubic crystals using a nonlocal lattice particle method

In this paper, a novel nonlocal crystal plasticity lattice particle method (CPLPM) is proposed and verified for modeling plasticity of cubic crystals. The proposed CPLPM is fundamentally different from the classical crystal plasticity finite element method, in which the underlying lattice structures are explicitly incorporated in the model formulation and lattice rotation is used to represent the material anisotropy of crystals. As a first attempt to develop a comprehensive CPLPM framework, constitutive relations based on the Schmid’s law and rate-dependent crystal plasticity for infinitesimal deformations are formulated for single crystals of face-centered and body-centered cubic lattice structures. Slip-system-dependent plastic deformation is automatically handled using the underlying crystal topology incorporated in LPM. Numerical solution algorithms for global iteration and plasticity state variables update are discussed in detail. The validity and prediction accuracy of CPLPM are established using several numerical examples, e.g., perfectly plastic deformation of single crystals, plastic slip near crack tip, and the calibration and prediction of experimental results. The performance of the CPLPM algorithm is also discussed considering different viscous properties.

FFT-based simulations of slip and kink bands formation in 3D polycrystals: Influence of strain gradient crystal plasticity

The ability of softening strain gradient plasticity to simulate intragranular plastic slip localization modes called slip and kink bands, at is investigated at incipient plasticity. A strain gradient crystal plasticity model based on a quadratic energy contribution of the Nye tensor has been implemented within the massively parallel FFT-based solver AMITEX_FFTP. It is shown that a classical evaluation of the double curl operator does not properly account for the backstress induced by GNDs in slip and kink bands. Consequently, a better suited differentiation operator is proposed for this purpose. Besides, four practical implementations of interface conditions at grain boundaries are implemented and discussed. It is demonstrated that these conditions have a strong influence on GND induced hardening but weakly affect the formation of slip localization patterns.A theoretical and numerical study of an idealized kink band shows that this modeling framework, contrary to classical crystal plasticity models, is able to capture their essential physical features. A quantitative analysis of high resolution 3D polycrystalline simulations, based on imaged processing and a peak detection on plastic slip profiles, is applied to show that gradient effects strongly influence the mechanisms of selection between slip and kink banding, to the favor of the first. In particular, some kink bands are shown to decompose into an homogeneous distribution of parallel slip bands of finite length. Finally, the competition between structural effects and energetic cost of lattice curvature in the formation of localized slip bands networks in polycrystals is discussed. The present results demonstrate the relevance of softening crystal plasticity modeling including an energetic contribution of GND density to accurately capture slip localization in polycrystals.

Computational Modeling of Dislocation Slip Mechanisms in Crystal Plasticity: A Short Review

The bridge between classical continuum plasticity and crystal plasticity is becoming narrower with continuously improved computational power and with engineers’ desire to obtain more information and better accuracy from their simulations, incorporating at the same time more effects about the microstructure of the material. This paper presents a short overview of the main current techniques employed in crystal plasticity formulations for finite element analysis, as to serve as a point of departure for researchers willing to incorporate microstructure effects in elastoplastic simulations. We include both classical and novel crystal plasticity formulations, as well as the different approaches to model dislocations in crystals.

A unification of finite deformation J2 Von-Mises plasticity and quantitative dislocation mechanics

We present a framework which unifies classical phenomenological J2 and crystal plasticity theories with quantitative dislocation mechanics. The theory allows the computation of stress fields of arbitrary dislocation distributions and, coupled with minimally modified classical (J2 and crystal plasticity) models for the plastic strain rate of statistical dislocations, results in a versatile model of finite deformation mesoscale plasticity. We demonstrate some capabilities of the framework by solving two outstanding challenge problems in mesoscale plasticity: 1) recover the experimentally observed power-law scaling of stress-strain behavior in constrained simple shear of thin metallic films inferred from micropillar experiments which all strain gradient plasticity models overestimate and fail to predict; 2) predict the finite deformation stress and energy density fields of a sequence of dislocation distributions representing a progressively dense dislocation wall in a finite body, as might arise in the process of polygonization when viewed macroscopically, with one consequence being the demonstration of the inapplicability of current mathematical results based on Γ-convergence for this physically relevant situation. Our calculations in this case expose a possible ‘phase transition’ - like behavior for further theoretical study. We also provide a quantitative solution to the fundamental question of the volume change induced by dislocations in a finite deformation theory, as well as show the massive non-uniqueness in the solution for the (inverse) deformation map of a body inherent in a model of finite strain dislocation mechanics, when approached as a problem in classical finite elasticity.

A quantitative study of stress fields ahead of a slip band blocked by a grain boundary in unalloyed magnesium

Stress localization ahead of a slip band blocked by a grain boundary is measured for three different grain boundaries in unalloyed Mg using high-resolution electron backscatter diffraction (HR-EBSD). The results are compared with a theoretical dislocation pile-up model, from which slip system resistance and micro-Hall–Petch coefficients for different grain boundary types are deduced. The results indicate that grain boundary character plays a crucial role in determining micro-Hall–Petch coefficients, which can be used to strengthen classical crystal plasticity constitutive models to make predictions linked to the effect of grain boundary strengthening.

On the implementation of rate‐independent gradient‐enhanced crystal plasticity theory

AbstractIn the present work, classical local crystal plasticity is reformulated using incremental energy minimization. The resulting constrained optimization problem is efficiently solved by a nonlinear complementary problem, which additionally solves the problem of determination of the set of active slip systems. The model is extended to a nonlocal, gradient‐enhanced, formulation by employing and adapting the micromorphic approach. By doing so, the implementation of the gradient‐enhanced model is almost identical to that of the original local one, with the exception of additional balance equations.

Intragranular localization induced by softening crystal plasticity: Analysis of slip and kink bands localization modes from high resolution FFT-simulations results

We investigate the ability of local continuum crystal plasticity theory to simulate intense slip localization at incipient plasticity observed experimentally in metals exhibiting softening mechanisms. A generic strain softening model is implemented within a massively parallel FFT solver framework to study intragranular strain localization throughout high resolution polycrystalline simulations. It is coupled to a systematic analysis strain localization modes: Equivalent plastic strain and lattice rotation fields are processed to create binary maps of slip and kink bands populations, estimate their volume fraction and mean strain level. High resolution simulations show the formation of an intragranular localization band network. The associated localization maps are used to identify accurately slip and kink bands populations and highlight the distinct evolution of kink bands, influenced by lattice rotation. Results highlight that the analysis of the nature of localization bands in numerical studies is fundamental to asses the validity of polycrystalline simulations. Indeed, it is evidenced that selection between slip or kink localization modes is only due to grain to grain incompatibilities as these two localization modes are equivalent in classical crystal plasticity models. As a result they predict the formation of a large amount of kink bands in contradiction with experimental observations of softening metals. We show that this holds for complex physics based models too. Hence, the use of classical crystal plasticity for strain localization simulation should be reconsidered in order to predict realistic localization modes.

In situ EBSD investigation of deformation processes and strain partitioning in bi-modal Ti-6Al-4V using lattice rotations

Identification of operating deformation processes and assessment of the resulting strain partitioning are critical concerns for mechanical properties prediction and microstructure optimization in complex alloys such as α/β titanium alloys. Lattice rotation relative to the initial orientation was presently used as a marker of slip activity. A Ti-6Al-4V specimen with a bi-modal microstructure was tested in tension in a scanning electron microscope. Crystallographic orientations were characterized in situ using electron back-scattered diffraction (EBSD). A successful prediction of activated slip systems was achieved using the rotation axis associated with plastic activity. The combination of this procedure and slip traces analysis offers an insight into the determination of both slip plane and slip direction of active slip systems. Based on classical crystal plasticity formulations, the magnitude of the rotation relative to the initial orientation was interpreted in terms of plastic shear magnitude. A quantitative assessment of plastic strain at the microstructure scale was then carried out using lattice rotation data. This approach enabled to discuss strain partitioning in Ti-6Al-4V considering the influence of microstructural features and active slip modes.

Dislocation pattern formation in finite deformation crystal plasticity

Stressed dislocation pattern formation in crystal plasticity at finite deformation is demonstrated for the first time. Size effects are also demonstrated within the same mathematical model. The model involves two extra material parameters beyond the requirements of standard classical crystal plasticity theory. The dislocation microstructures shown are decoupled from deformation microstructures, and emerge without any consideration of latent hardening or constitutive assumptions related to cross-slip. Crystal orientation effects on the pattern formation and mechanical response are also demonstrated. The manifest irrelevance of the necessity of a multiplicative decomposition of the deformation gradient, a plastic distortion tensor, and the choice of a reference configuration in our model to describe the micromechanics of plasticity as it arises from the existence and motion of dislocations is demonstrated.

Modelling nano-indentation of ion-irradiated FCC single crystals by strain-gradient crystal plasticity theory

In this work, a strain-gradient crystal plasticity theory with irradiation effect is proposed for the surface nano-indentation of ion-irradiated FCC metals. In order to characterize the indentation size effect and irradiation hardening of ion-irradiated materials indented by the Berkovich indenter, the hardening contributions of geometrically necessary dislocations and non-uniformly distributed defects are incorporated into the classical crystal plasticity theory. The constitutive equations are implemented into ABAQUS through the user material subroutine VUMAT to numerically simulate the surface nano-indentation of ion-irradiated single crystal copper. The theoretical model is calibrated by comparing the numerical results with experimental data. Both the simulated force-depth relationship without irradiation effect and hardness-depth relationship with/without irradiation effect can match well with corresponding experimental data. The dominant features as observed in the nano-indentation of ion-irradiated materials can be effectively characterized by the proposed theoretical framework, which include the indentation size effect, depth-dependent irradiation hardening, and unirradiated substrate softening effect. Moreover, the evolution of different hardening mechanisms during the surface nano-indentation is systematically analyzed to help understand the macroscopic deformation behavior.