Strain Gradient Crystal Plasticity Research Articles

Attaining the rate-independent limit of a rate-dependent strain gradient plasticity theory

The existence of characteristic strain rates in rate-dependent material models, corresponding to rate-independent model behavior, is studied within a back stress based rate-dependent higher order strain gradient crystal plasticity model. Such characteristic rates have recently been observed for steady-state processes, and the present study aims to demonstrate that the observations in fact unearth a more widespread phenomenon. In this work, two newly proposed back stress formulations are adopted to account for the strain gradient effects in the single slip simple shear case, and characteristic rates for a selected quantity are identified through numerical analysis. Evidently, the concept of a characteristic rate, within the rate-dependent material models, may help unlock an otherwise inaccessible parameter space.

On modeling micro-structural evolution using a higher order strain gradient continuum theory

Published experimental measurements on deformed metal crystals show distinct pattern formation, in which dislocations are arranged in wall and cell structures. The distribution of dislocations is highly non-uniform, which produces discontinuities in the lattice rotations. Modeling the experimentally observed micro-structural behavior, within a framework based on continuous field quantities, poses obvious challenges, since the evolution of dislocation structures is inherently a discrete and discontinuous process. This challenge, in particular, motivates the present study, and the aim is to improve the micro-structural response predicted using strain gradient crystal plasticity within a continuum mechanics framework. One approach to modeling the dislocation structures observed is through a back stress formulation, which can be related directly to the strain gradient energy. The present work offers an investigation of constitutive equations for the back stress based on both considerations of the gradient energy, but also includes results obtained from a purely phenomenological starting point. The influence of model parameters is brought out in a parametric study, and it is demonstrated how a proper treatment of the back stress enables dislocation wall and cell structure type response in the adopted framework.

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.

Lattice Rotation Patterns and Strain Gradient Effects in Face-Centered-Cubic Single Crystals Under Spherical Indentation

Strain gradient effects are commonly modeled as the origin of the size dependence of material strength, such as the dependence of indentation hardness on contact depth and spherical indenter radius. However, studies on the microstructural comparisons of experiments and theories are limited. First, we have extended a strain gradient Mises-plasticity model to its crystal plasticity version and implemented a finite element method to simulate the load–displacement response and the lattice rotation field of Cu single crystals under spherical indentation. The strain gradient simulations demonstrate that the forming of distinct sectors of positive and negative angles in the lattice rotation field is governed primarily by the slip geometry and crystallographic orientations, depending only weakly on strain gradient effects, although hardness depends strongly on strain gradients. Second, the lattice rotation simulations are compared quantitatively with micron resolution, three-dimensional X-ray microscopy (3DXM) measurements of the lattice rotation fields under 100 mN force, 100 μm radius spherical indentations in 〈111〉, 〈110〉, and 〈001〉 oriented Cu single crystals. Third, noting the limitation of continuum strain gradient crystal plasticity models, two-dimensional discrete dislocation simulation results suggest that the hardness in the nanocontact regime is governed synergistically by a combination of strain gradients and source-limited plasticity. However, the lattice rotation field in the discrete dislocation simulations is found to be insensitive to these two factors but to depend critically on dislocation obstacle densities and strengths.

Modeling the Hall-Petch Relation of Ni-Base Polycrystalline Superalloys Using Strain-Gradient Crystal Plasticity Finite Element Method

Abstract A strain-gradient crystal plasticity constitutive model was developed in order to predict the Hall-Petch behavior ofa Ni-base polycrystalline superalloy. The constitutive model involves statistically stored dislocation and geometrically necessarydislocation densities, which were incorporated into the Bailey-Hirsch type flow stress equation with six strength interactioncoefficients. A strain-gradient term (called slip-system lattice incompatibility) developed by Acharya was used to calculate thegeometrically necessary dislocation density. The description of Kocks-Argon-Ashby type thermally activated strain rate was alsoused to represent the shear rate of an individual slip system. The constitutive model was implemented in a user materialsubroutine for crystal plasticity finite element method simulations. The grain size dependence of the flow stress (viz., the Hall-Petch behavior) was predicted for a Ni-base polycrystalline superalloy NIMONIC PE16. Simulation results showed that thepresent constitutive model fairly reasonably predicts 0.2 %-offset yield stresses in a limited range of the grain size.Key wordscrystal plasticity, strain gradient plasticity, finite element method, hall-petch relation, polycrystal.

Application of Strain Gradient Plasticity to Micro‐torsion Experiments

AbstractThe applicability of a strain gradient crystal plasticity model to size effects observed on microwires in torsion experiments is discussed. Finite Element simulations of simplified cylindrical grain aggregations are presented and the resulting overall mechanical response is compared to experimental data for gold from the literature. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Crystal Plasticity Analysis of Scale Effect on Tensile Properties of Ferrite/Cementite Fine Lamellar Structure under Lattice Strain

Tensile properties of ferrite lamella in pearlite under lattice strain are examined by a strain gradient crystal plasticity analysis. Tensile direction is made to be parallel to the lamella. Obtained results of macroscopic stress-strain relation of the lamella show significant increase of yield stress and strain hardening rate with the reduction of the lamella thickness and further increase of the yield stress with positive normal lattice strain parallel to the tensile direction in the ferrite layer. Whereas normal lattice strain perpendicular to the tensile direction contributes little to the tensile properties.

Simulation of Torsion of Thin FCC Single Crystal Wires by Mechanism-Based Strain Gradient Crystal Plasticity

The size effects observed in the torsion of thin FCC single crystal wires is modelled by the employment of mechanism-based strain gradient crystal plasticity (MSG-CP). In the formulation the total slip resistance in each active slip system is assumed to be due to a mixed population of forest obstacles arising from both statistically stored and geometrically necessary dislocations. The MSG-CP constitutive model is implemented into the Abaqus/Standard FE platform by developing the User MATerial subroutine UMAT. By implementing the formulation, the relationship between the non-dimensional torque and the surface strain of the thin copper single crystal wires of different diameters is obtained with the [001] direction along that of the wire axis. The simulation results of torsion reveal size effects, which is in a qualitative agreement with those reported in existing literatures. An appreciable axial elongation is also found in the torsion of single crystal wires.Key words Size effect, MSG-CP model, Torsion

Strain-gradient modelling of grain size effects on fatigue of CoCr alloy

A strain-gradient crystal plasticity framework based on physical dislocation mechanisms is developed for simulation of the experimentally observed grain size effect on the low cycle fatigue of a CoCr alloy. Finite-element models of the measured microstructure are presented for both as-received and heat-treated CoCr material, with significantly different grain sizes. Candidate crystallographic slip-based parameters are implemented for prediction of fatigue crack initiation. The measured beneficial effects of fine grain size on both cyclic stress–strain response and crack initiation life are predicted. The build-up of geometrically necessary dislocations as a result of strain-gradients, leading to grain-size-dependent material hardening, is shown to play a key role.

Numerical modelling of micro-machining of f.c.c. single crystal: Influence of strain gradients

A micro-machining process becomes increasingly important with the continuous miniaturization of components used in various fields from military to civilian applications. To characterise underlying micromechanics, a 3D finite-element model of orthogonal micro-machining of f.c.c. single crystal copper was developed. The model was implemented in a commercial software ABAQUS/Explicit employing a user-defined subroutine VUMAT. Strain-gradient crystal-plasticity and conventional crystal-plasticity theories were used to demonstrate the influence of pre-existing and evolved strain gradients on the cutting process for different combinations of crystal orientations and cutting directions.

Coupled glide-climb diffusion-enhanced crystal plasticity

This paper presents a fully coupled glide-climb crystal plasticity model, whereby climb is controlled by the diffusion of vacancies. An extended strain gradient crystal plasticity model is therefore proposed, which incorporates the climbing of dislocations in the governing transport equations. A global–local approach is adopted to separate the scales and assess the influence of local diffusion on the global plasticity problem. The kinematics of the crystal plasticity model is enriched by incorporating the climb kinematics in the crystallographic split of the plastic strain rate tensor. The potential of the fully coupled theory is illustrated by means of two single slip examples that illustrate the interaction between glide and climb in either bypassing a precipitate or destroying a dislocation pile-up.

Computational modeling of intrinsically induced strain gradients during compression of c-axis-oriented magnesium single crystal

A finite-deformation strain gradient crystal plasticity model is implemented in a three-dimensional finite-element framework in order to analyze the deformation behavior and the stress–strain response of magnesium single crystals under c-axis orientation. The potential-based and thermodynamically consistent material model is formulated in a non-local and non-linear inelastic context in which dislocation densities are introduced via plastic strain gradients. Experiments have shown that the internal length scale of the microstructure starts to affect the overall stress–strain response when the sample size decreases to the micron scale. As a consequence, strain gradients develop, leading to an additional energetic-like hardening effect which results in an increase of the macroscopic strength with decreasing crystal size. In the case of uniaxial compression of c-axis-oriented single-crystal micropillars, the model is able to predict the discrete dislocation glide in terms of a band-shaped slip zone. Two different pillar sample sizes are taken into account in order to investigate the intrinsic size effect during plastic deformation where the crystallographic orientation leads to the activation of pyramidal {112¯2}〈112¯3〉 slip systems as reported in various experimental studies. The interaction of those slip systems is expressed in terms of latent hardening and excess dislocation development. A comparison between numerical results and corresponding experimental data is presented.

Gradient crystal plasticity modelling of anelastic effects in particle strengthened metallic thin films

It is now a well known phenomenon that thin films are susceptible to size effects, which can be captured adequately by gradient plasticity theories. Besides the scale dependency, metal thin films also exhibit time dependent behavior: anelasticity (deformation recovery over time following elastic spring back upon load removal) and creep (permanent deformation developed over time at constant loads). This work focuses on the extension of a strain gradient crystal plasticity (SGCP) model (Int J Solids Struct 43:7268–7286, 2006; Phil Mag 87:1361–1378, 2007; J Mech Phys Solids 52:2379–2401, 2004; Int J Solids Struct 41:5209–5230, 2004), previously developed for the scale dependent behavior of pure fcc metals, so that it can be exploited for the description of the scale and time dependent mechanical behavior of thin films that are made of metal alloys with second phase particles. For this purpose, an extended physically based slip law is developed for crystallographic slip in fcc metals by considering the deformation mechanisms that are active within the grains. In doing so, the interaction of dislocations with other dislocations and with second phase particles is taken into account. Three types of dislocation–particle interactions are considered: (i) the Orowan mechanism, (ii) the Friedel mechanism, and (iii) dislocation climb. Finite element simulations of the bending of a single crystalline beam show that at low stress levels, the plastic slip rate is controlled by dislocation climb within the presented model. Provided that a considerable lattice diffusion occurs and sufficiently large back stresses exist in the material, the extended SGCP model predicts a noticeable time dependent recovery, reducing the residual deformation after unloading. The magnitude and the characteristic time scale of the anelastic recovery are controlled by dislocation glide limited by climb.

Modeling of dislocation–grain boundary interactions in a strain gradient crystal plasticity framework

This paper focuses on the continuum scale modeling of dislocation---grain boundary interactions and enriches a particular strain gradient crystal plasticity formulation (convex counter-part of Yalcinkaya et al., J Mech Phys Solids 59:1---17, 2011; Int J Solids Struct 49:2625---2636, 2012) by incorporating explicitly the effect of grain boundaries on the plastic slip evolution. Within the framework of continuum thermodynamics, a consistent extension of the model is presented and a potential type non-dissipative grain boundary description in terms of grain boundary Burgers tensor (see e.g. Gurtin, J Mech Phys Solids 56:640---662, 2008) is proposed. A fully coupled finite element solution algorithm is built-up in which both the displacement $${{\varvec{u}}}$$ u and plastic slips $$\gamma ^{\alpha }$$ ? ? are considered as primary variables. For the treatment of grain boundaries within the solution algorithm, an interface element is formulated. The proposed formulation is capable of capturing the effect of misorientation of neighboring grains and the orientation of the grain boundaries on slip evolution in a natural way, as demonstrated by bicrystal specimen examples.

Microscopic mechanism of plastic deformation in a polycrystalline Co–Cr–Mo alloy with a single hcp phase

A Co–Cr–Mo alloy with a single ε (hexagonal close-packed, hcp) phase exhibits excellent tensile properties with a 0.2% proof stress of 630MPa, an ultimate tensile stress of 1072MPa and an elongation to fracture of 38.3%. The dominant deformation modes are basal 〈a〉 slip and prismatic 〈a〉 slip, and the apparent respective critical resolved shear stresses at room temperature are calculated to be 184 and 211MPa. This simultaneous activation of both 〈a〉 slips can be explained in terms of the lattice constant ratio c/a of 1.610. There is a tendency for the geometrically necessary dislocations (GNDs) to accumulate at grain boundaries, and the magnitude of this GND accumulation at a particular boundary is dependent on its character. Numerical analysis using a dislocation-model-based strain gradient crystal plasticity calculation makes it possible to characterize the distributions of dislocation density, local stress and local strain in the polycrystalline ε Co–Cr–Mo alloy, and the calculation is largely consistent with the experimental results. This simulation reveals that the activity of the prismatic 〈a〉 slip in addition to the basal 〈a〉 slip contributes to the stress relaxation at the boundary. For this reason, excellent tensile ductility is obtained in the polycrystalline ε Co–Cr–Mo alloy.

Micromechanical Simulation of the Hall‐Petch Effect with a Crystal Gradient Theory including a Grain Boundary Yield Criterion

AbstractA viscoplastic strain gradient crystal plasticity theory based on the gradient of the equivalent plastic strain ∇γeq is proposed. A grain boundary yield condition is introduced. The microstructural explanation of the Hall‐Petch effect, accounting for notch‐like stress concentrations at the grain boundary as a result of discrete slip bands, is reviewed. Periodic tensile test FEM simulation results illustrate the prediction of the numerical model. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Numerical analysis of the indentation size effect using a strain gradient crystal plasticity model

This work presents a finite element analysis of the indentation size effect (ISE) experimentally observed in tests performed at submicron scale. A 3D model of a conical rigid surface indenting on a Nb single crystal at different depths has been developed. The bcc Nb material has been characterized within a finite-strain framework through a crystal plasticity model incorporating strain-gradient hardening. The hardness evolution for different material orientations and for different initial dislocation densities has been studied. The numerical results are compared with predictions of existing analytical models and with experimental results.

Equivalent plastic strain gradient crystal plasticity – Enhanced power law subroutine

AbstractA gradient crystal plasticity theory is presented including a defect energy based on the gradient of an equivalent plastic strain measure. Preserving the single crystal slip kinematics, the gradient hardening contribution models dislocation long range interactions adding a back‐stress term to the flow rule, similar to other gradient crystal plasticity theories. Owing to a reduced number of four nodal degrees of freedom the finite element implementation can handle systems consisting of an increased number of grains (compared to other theories) without elaborate or costly computer systems. Emphasis is put on the enhancement of the power law material subroutine. The associated implicit Euler scheme is optimized based on an improved starting value for the Newton scheme. Three‐dimensional simulations illustrate that the proposed algorithm facilitates significantly larger time steps compared to the standard Newton scheme. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Crystal plasticity based modeling of time and scale dependent behavior of thin films

AbstractThe micro and sub‐micro scale dimensions of the components of modern high‐tech products pose challenging engineering problems that require advanced tools to tackle them. An example hereof is time dependent strain recovery, here referred to as anelasticity, which is observed in metallic thin film components of RF‐MEMS switches. Moreover, it is now well known that the properties of a thin film material strongly depend on its geometrical dimensions through so‐called size effects. A strain gradient crystal plasticity formulation (SGCP) was recently proposed [1–4], involving a back stress in terms of strain gradients capturing the lattice curvature effect. In the present work, the SGCP model is used in a realistic simulation of electrostatic bending of a free standing thin film beam made of either a pure fcc metal or a particle strengthened Al‐Cu alloy. The model capabilities to describe the anelastic and plastic behavior of metallic thin films in comparison with experimentally available data are thereby assessed. Simulation results show that the SGCP model is able to predict a macroscopic strain recovery over time following the load removal. The amount of the anelastic relaxation and the accompanying relaxation times result from the rate dependent modeling approach, the basis of which is phenomenological only. The SGCP model is not fully capable of describing the permanent deformations in an alloy thin beam as observed in electrostatic experiments. Hence, to incorporate realistic time constants and the influence of the microstructure into the mechanical behavior of the thin film material, an improved constitutive law for crystallographic slip is necessary within the SGCP formulation. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Computational strain gradient crystal plasticity

A numerical method for viscous strain gradient crystal plasticity theory is presented, which incorporates both energetic and dissipative gradient effects. The underlying minimum principles are discussed as well as convergence properties of the proposed finite element procedure. Three problems of plane crystal plasticity are studied: pure shear of a single crystal between rigid platens as well as plastic deformation around cylindrical voids in hexagonal close packed and face centered cubic crystals. Effective in-plane constitutive slip parameters for plane strain deformation of specifically oriented face centered cubic crystals are developed in terms of the crystallographic slip parameters. The effect on geometrically necessary dislocation structures introduced by plastic deformation is investigated as a function of the ratio of void radius to plasticity length scale.