Single Crystal Plasticity Model Research Articles

Effect of core energy on mobility in a continuum dislocation model

We present a first-principles-based, multiscale single-crystal plasticity model for fcc metals and apply it to nickel. The model consists of a phase field approach to dislocation dynamics (PFDD) with all its input parameters obtained from equilibrium and nonequilibrium molecular-dynamics (MD) simulations. The atomistic information used to inform the phase field model includes elastic constants, dislocation core energy, crystal disregistry energy (gamma surface), and dislocation mobility. We show that the PFDD model can be simplified to the Frenkel-Kontorowa equations for straight dislocations, and under these conditions an analytical time-dependent solution enables a direct connection to nonequilibrium MD simulations. This time-dependent analytical solution provides a relationship between dislocation mobility (ratio between dislocation velocity and applied stress) and fundamental atomic-scale materials properties that arise from the atomistics: unstable stacking fault energy and dislocation core energy and width. We find that the dislocation mobility increases linearly with the ratio between the core energy and unstable stacking fault energy in the PFDD theory.

A micro mechanical study on failure initiation of dual phase steels under tension using single crystal plasticity model

Both experimental and numerical methods were employed to investigate the mechanism of failure in dual phase steels. The tensile test was interrupted in different steps to capture the mechanism of void initiation and void growth during material failure. The results can be considered as a first report for the commercial DP800 steel. Numerical simulations, which were carried out using the real micro-structure, are able to predict the void initiation in the material. In addition, through the numerical simulation a new understanding of the deformation localization was gained. Deformation localization, which causes severely deformed regions in the material, is most probably the main source of rupture in the final stages of the failure. In the SEM micrographs of the material after failure some voids are observable which can validate the results obtained by the simulation.

Void growth and coalescence in f.c.c. single crystals

In this study, we investigate deformation behavior of f.c.c. single crystals containing microvoids by using three-dimensional finite element methods. The unit cell analysis has been conducted to study the effect of stress triaxialities, crystallographic orientations and initial void volume fractions on the growth and coalescence of voids in f.c.c. single crystals. The locally homogeneous constitutive model for the rate-dependent single crystal plasticity is implemented into a finite element program (ABAQUS) by means of the user-defined subroutine (UMAT). To identify the effect of stress triaxiality and the crystallographic orientation on the void evolution, the stress triaxiality was kept constant during deformation. The numerical results showed that the stress triaxiality and the deformation mode specified by the crystallographic orientation have a competitive effect on the evolution of voids. For the low level of stress triaxiality, the deformation mode is mainly determined by the crystallographic orientation. For high stress triaxiality, however, the deviation from the specific deformation mode is large even for incipient void growth and the void growth rate is mainly determined by stress triaxiality and the initial void volume fraction. For the small initial void volume fraction, the growth rate of a void is rapid compared with the large one and the effect of the initial crystallographic orientation is significant.

Deformation distribution maps of β-HMX molecular crystals

β-HMX, extensively used as an energetic material, exists as monoclinic molecular crystals which exhibit highly nonlinear anisotropic mechanical behaviour. To explain the unique features of the deformation behaviour of β-HMX, we have developed the so-called ‘deformation distribution maps’ (DDMs) based on a single crystal plasticity model based on empirical data. These DDMs reveal that under uniaxial compression there are ten deformation fibres along which β-HMX exhibits maximum plastic deformation and six other deformation fibres along which the deformation response is mostly elastic. These DDMs are applicable to a wider class of molecular crystals for not only understanding their complex mechanical response, but also to serve as important design tools.

Calibration of Nanocrystal Grain Boundary Model Based on Polycrystal Plasticity Using Molecular Dynamics Simulations

Decohesion parameters are computed for the tilt grain boundaries through molecular simulations and the parameters are employed in a elastoplastic deformation model of a face centered cubic (FCC) nanocrystal. The calibrated continuum grain boundary model accounts for reversible elastic and irreversible inelastic separation sliding deformations. The intragranular plasticity was modeled using a rate independent single crystal plasticity model. Atomistic calculations are presented for a planar, copper grain boundary interface with a tilt lattice misorientation for cases of loading and unloading. The interface models are deformed to full separation and then relaxed to study inelastic behavior. Plots of stress versus displacement show a distinctly different deformation response between normal and tangential interface loading conditions. Two dimensional microstructures uniaxially loaded using the calibrated cohesive model indicate that the macroscopically observed nonlinearity in the mechanical response is mainly due to the inelastic response of the grain boundaries. Plastic deformation in the interior of the grains prior to the initiation of grain boundary cracks was not observed. Although key features of the molecular simulation results have been introduced in the cohesive model, a few discrepancies between the behavior of cohesive model when compared to molecular simulations are noted.

Numerical exploration of the Dang Van high cycle fatigue criterion: application to gradient effects

The objective of this paper is to show that a number of key features of the Dang Van high cycle fatigue criterion can be observed using simple polycrystalline computational models. This paper presents a series of numerical computations for an inclusion consisting of 156 grains embedded in a homogeneous matrix. The grains are modeled using a polycrystalline single slip elasto-plastic model, whilst the matrix is considered as elastic. As expected the numerical simulations confirm the theoretical prediction on which the Dang Van fatigue criterion is based, that if a large enough number of grains is considered under uniform loading, a grain with the least favourable lattice orientation will always be present. This grain will constitute the weakest link in the assembly and thus its fatigue life largely determines the fatigue life of the bulk material. Next the question of stress-gradients in the high cycle fatigue regime is addressed. An example of stress gradients appears around notches as they create stress concentrations in structures. It is a well known problem that fatigue criteria have to be locally arranged using stress-factors or critical distances in order to give satisfactory predictions. The work presented here shows that an analysis of the problem at the grain scale explains the apparent discrepancy when using classical fatigue criteria. The discussion is based on a numerical model of single slip crystal plasticity and the Dang Van fatigue criterion.

Constitutive modelling of the anisotropic creep behaviour of nickel-base single crystal superalloys

This paper focuses on the modelling of primary, secondary and tertiary creep of nickel-base single crystal superalloys at high temperatures. In particular, we propose an extension of the Cailletaud single crystal plasticity model [Méric L, Poubanne P, Cailletaud G. Single crystal modeling for structural calculations: part I—model presentation. Transactions of the ASME 1991;133:162–170] to include tertiary creep. This is achieved by introducing an additional evolution equation for a scalar damage variable per slip system. In addition, a methodology for the calibration of the material parameters of the model to fit the results from experiments has been implemented. The parameter identification rests upon a two-membered evolution strategy. The comparison with uniaxial and multiaxial test data shows a good agreement between model and experiment. The structural simulations have been performed by means of a special element technology which enables efficient and accurate finite element computations.

Crystallographically based model for transformation-induced plasticity in multiphase carbon steels

The microstructure of multiphase steels assisted by transformation-induced plasticity consists of grains of retained austenite embedded in a ferrite-based matrix. Upon mechanical loading, retained austenite may transform into martensite, as a result of which plastic deformations are induced in the surrounding phases, i.e., the ferrite-based matrix and the untransformed austenite. In the present work, a crystallographically based model is developed to describe the elastoplastic transformation process in the austenitic region. The model is formulated within a large-deformation framework where the transformation kinematics is connected to the crystallographic theory of martensitic transformations. The effective elastic stiffness accounts for anisotropy arising from crystallographic orientations as well as for dilation effects due to the transformation. The transformation model is coupled to a single-crystal plasticity model for a face-centered cubic lattice to quantify the plastic deformations in the untransformed austenite. The driving forces for transformation and plasticity are derived from thermodynamical principles and include lower-length-scale contributions from surface and defect energies associated to, respectively, habit planes and dislocations. In order to demonstrate the essential features of the model, simulations are carried out for austenitic single crystals subjected to basic loading modes. To describe the elastoplastic response of the ferritic matrix in a multiphase steel, a crystal plasticity model for a body-centered cubic lattice is adopted. This model includes the effect of nonglide stresses in order to reproduce the asymmetry of slips in the twinning and antitwinning directions that characterizes the behavior of this type of lattices. The models for austenite and ferrite are combined to simulate the microstructural behavior of a multiphase steel. The results of the simulations show the relevance of including plastic deformations in the austenite in order to predict a more realistic evolution of the transformation process.

Three-dimensional finite element simulation of a polycrystalline copper specimen

The application of crystal plasticity in finite element codes provides a virtual copy of a real grain structure, including stress–strain state and slip system activity. This paper presents, first, finite element computations of an oxygen-free, high-conductivity copper multicrystal under monotonic tension. A series of polishing operations are used to reveal the real three-dimensional (3-D) microstructure, so that the mesh is a full 3-D mesh. A single crystal plasticity model is used to represent the elasto-plastic behavior for each of the approximately 100 grains. The results obtained with a full 3-D mesh and an extended 2-D mesh with columnar grains are compared in order to check the bias introduced by a simplified mesh generation. The paper goes on to compare the computational results to experimental data obtained by orientation image microscopy and local strain field measurements. Local strain and lattice rotation fields are shown. Finally, the numerical results for the slip system activity are analyzed.

A meshless grain element for micromechanical analysis with crystal plasticity

This study concerns the development of a 2-D meshless grain element for elasto-plastic deformation and intergranular damage initiation and propagation in polycrystalline fcc metals under static loading. The crystallographic material behaviour of the grains is represented by a rate-independent single-crystal plasticity model while including material orthotropy. The two slip planes are arbitrarily located with respect to the crystallographic axis of the grain. A non-linear constitutive model known as the cohesive zone model is employed to represent the inelastic interaction between the grain boundaries, thus permitting grain boundary opening and sliding. The cohesive model describes the deformation characteristics of the grain boundaries through a non-linear relation between the effective grain boundary tractions and displacements. Because of the presence of non-linear material behaviour both inside the grain and along the cohesive grain boundaries, the method utilizes the principle of virtual work in conjunction with the meshless formulation in the derivation of the system of non-linear incremental equilibrium equations. The solution is obtained via an incremental procedure based on the Taylor series expansion about the current equilibrium configuration. The fidelity of the present approach is verified by considering simple polycrystalline metals of only a few grains. Copyright © 2006 John Wiley & Sons, Ltd.

Creep of Single Crystals – Modelling and Numerical Aspects

AbstractA number of constitutive models, utilizing both microstructural and/or phenomenological considerations, have been developed for the simulation of the creep behaviour of nickel‐base single crystal superalloys at elevated temperatures. In this work, emphasis is placed on the rate‐dependent single crystal plasticity model [1]. A strategy for the identification of the material parameters of the model to fit the results from experiments has been implemented. The parameter fitting methodology rests upon a two‐membered evolution strategy. In addition, a proposal is made for the extension of the Cailletaud model [1] by means of an evolution equation for a damage variable which enables the modelling of the tertiary creep stage. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Two-scale analysis for deformation-induced anisotropy of polycrystalline metals

The anisotropic macroscopic mechanical behavior of polycrystalline metals is characterized by incorporating the microscopic constitutive model of single crystal plasticity into the two-scale modeling based on the mathematical homogenization theory, which enables us to derive both micro- and macro-scale governing equations. The two-scale simulations are conducted to evaluate the macroscopic anisotropy induced by microscopic plastic deformation histories of the polycrystalline aggregate. In the simulations, the representative volume element (RVE) composed of several crystal grains is uniformly loaded in one direction, unloaded to macroscopically zero stress in a certain stage of deformation and then re-loaded in the different directions. The last re-loading calculations provide different macroscopic responses of the RVE, which can be the appearance of material anisotropy. We then try to examine the effects of the intergranular and intragranular behaviors on the anisotropy by means of various illustrations of microscopic plastic deformation process without referring to the change of crystallographic orientations.

Evaluation of finite element based analysis of 3D multicrystalline aggregates plasticity: Application to crystal plasticity model identification and the study of stress and strain fields near grain boundaries

Plastic heterogeneities of hexagonal close-packed (HCP) materials are numerically investigated at the grain level. Intensive use of parallel Finite Elements computations enables us to study micro-plasticity of realistic 3D multicrystalline aggregates, including, macroscopic mechanical responses but also average responses in each grain and particularly local stress and strain fields. This paper focuses on three applications of this simulation method. The first part of this paper is devoted to a fine analysis of micro-plasticity of HCP materials. Intergranular but also intragranular stress and strain heterogeneities are described and micro-plasticity patterns are displayed throughout the 3D microstructures. A special attention is paid to the sensitivity of simulations with respect to the mesh discretization, the element interpolation and the geometrical representation of grain boundaries, in terms of macroscopic and local responses. Later, a simplified homogenization method is evaluated, regarding results of the first part. Afterwards, this method is applied with a zirconium alloy to identify a set of coefficients for a single crystal plasticity model. Finally, in order to provide critical information for intergranular damage phenomena (reported in literature for zirconium alloys), the third part provides a statistical analysis of over-stresses at grain boundaries.

Evaluation of Induced Anisotropy in Polycrystalline Metals Based on Homogenization Method

Macro-scale anisotropic mechanical behavior of polycrystalline meals is characterized by incorporating the micro-scale constitutive model of single crystal plasticity into the two-scale modeling based on the mathematical homogenization theory. The two-scale simulations are conducted to characterize the macro-scale anisotropy induced by micro-scale plastic deformation of the polycrystalline aggreagate. In the simulations, the micro-scale representative volume element of a polycrystalline aggregate is uniformly loaded in one direction, unloaded to macroscopially zero stress in a certain stage of deformation and then re-loaded in the different directions. Then the effect of the intergranular and intragranular behaviors are examined by several ways of illustrations of plastic deformation process. In particular, we are concerned with the concentrations of accumulated plastic strain and effective stress in the neighborhood of grain boundaries. In addition, the principal slip directions in each loading state are shown to examine the effect of self and latent hardenig on the deformation-induced anisotropy.

Anisotropic finite elastoplastic analysis of shells: simulation of earing in deep-drawing of single- and polycrystalline sheets by Taylor-type micro-to-macro transitions

The paper presents new aspects of anisotropic elastoplastic analyses of shells at finite strains. On the side of the computational shell analysis we design a surface-orientated brick-type mixed finite shell element in a setting relative to the parameter space of the shell equipped with shell-typical assumed strain modifications and an additive definition of an enhanced current metric in the Lagrangian representation. For this element we outline an interface to strain-driven constitutive stress update algorithms of multiplicative plasticity for single crystals and polycrystals at finite strains. On the side of computational plasticity we outline details of a constitutive model for finite single crystal plasticity. Here, we consider a distinct incremental variational formulation of the local constitutive elasto–visco-plastic response of single crystals in a multisurface format, where a quasi-hyperelastic stress potential is obtained from a local minimization problem with respect to the internal variables. It is shown that this local minimization problem determines the internal state of the material for finite increments of time. The proposed nonlinear shell formulation is applied to the simulation of earing in deep-drawing of anisotropic sheets. We present finite element simulations of cup-drawings of f.c.c. single crystal sheets and previously rolled polycrystalline f.c.c. sheets. The anisotropic polycrystal is modeled by a Taylor-type texture analysis based on an assumed microstructure of single crystal grains. A comparison with experiments underlines the performance of the simulation.

Simulations on the growth of dislocation density during Stage 0 deformation in BCC metals

This study focuses on modelling the behaviour of single crystal tantalum in Stage 0 characterized by the large activity of edge dislocations and relative inactivity of screw dislocations. The multiplication of dislocation density is investigated using dislocation dynamics (DD) simulations and a dislocation density based continuum model of single crystal plasticity. The DD simulations are used to guide the constitutive development of the continuum model and to determine its material specific parameters. While not all of the material constants needed by the continuum model can be determined, due to the limited strain histories considered in the simulations, interpreting the DD simulations through a dislocation mechanics based continuum plasticity model allows for the efficient extraction of scaling laws controlling the growth of dislocation density.

A Continuum Model for Single Crystal Cyclic Plasticity

AbstractA constitutive model was developed to bridge the cyclic plasticity behavior of single crystals and the corresponding characteristic dislocation structures. Yield and flow were built on the individual slip systems. The Armstrong-Frederick kinematic hardening rule was invoked to capture the Bauschinger effect. A material memory parameter was introduced to consider the amplitude dependence of cyclic hardening. Latent hardening considering the interactions among the slip systems was used to describe the anisotropic cyclic behavior. The experimental results of copper single crystals were used to validate the model developed. It was found that the model was able to adequately describe the well-known three distinctive regions in the cyclic stress-strain curve of the FCC single crystal oriented for single slip and the associated dislocation substructures. The model was capable of capturing the enhanced hardening observed in copper single crystals in multi-slip orientations. For a given loading history, the model can predict not only the saturated stress-strain response but also the detailed evolution of the transient cyclic behavior. The characteristic dislocation structures can be featured with the slip evolution.

Crystal-plasticity based modeling and simulation of geometrically-necessary dislocations at a crack front in ductile single crystals

This work is concerned with the formulation and numerical implementation of a constitutive model for ductile single crystals containing cracks or other geometric imperfections. In particular, attention is focused here on the modeling of the emergence of geometrically-necessary dislocations (GNDs) at such imperfections due to inhomogeneous deformation there. To this end, we formulate a glide-system model for GND density consistent with the model of Ashby and use it to extend the Cailletaud model for single crystal plasticity to include the effects of GNDs on the material behaviour of such crystals. This extended model is then implemented in an FE-based 3D-simulation of the material behaviour at a crack front in round bars consisting of single crystals of the nickel superalloy SC16. In particular, preliminary simulation results are presented and discussed for GND development at the crack front for two crystal-crack orientations. In these, the GND density is maximal generally at the crack front in the middle and decreases toward its edges.

Capturing the influence of surface constraints in small and thin samples using polycrystalline plasticity theory

A rate-dependent, single-crystal plasticity model for face-centred cubic crystal structures has been implemented into a large strain elastic - plastic, finite-element code to examine the mechanical influence of the reduced surface constraints of relatively small polycrystalline aggregates. The implemented model simulates deformation of a polycrystal composed of cubic grains where each grain is a single finite element. Mechanical constraint is varied by changing (a) specimen thickness and (b) specimen volume, relative to grain size. Numerical uniaxial tensile tests have been performed to a strain level of 0.01. Direct and statistical examination of the model results revealed the reduced flow stress of grains at specimen surfaces, edges and corners. The results of these simulations are in good agreement with previous experimental studies which suggest that 5 - 10 grains across the minimum dimension of a structure are necessary to approximate true continuum polycrystalline response.

Instability phenomena in plasticity: Modelling and computation

We presented aspects and results related to the broad field of strain localization with special focus on large strain elastoplastic response. Therefore, we first re-examined issues related to the classification of discontinuities and the classical description of localization with a particular emphasis on an Eulerian geometric representation. We touched the problem of mesh objectivity and discussed results of a particular regularization method, namely the micropolar approach. Generally, regularization has to preserve ellipticity and to reflect the underlying physics. For example ductile materials have to be modelled including viscous effects whereas geomaterials are adequately described by the micropolar approach. Then we considered localization phenomena within solids undergoing large strain elastoplastic deformations. Here, we documented the influence of isotropic damage on the failure analysis. Next, the interesting influence of an orthotropic yield condition on the spatial orientation of localized zones has been studied. Finally, we investigated the localization condition for an algorithmic model of finite strain single crystal plasticity.