Gradient Plasticity Model Research Articles

Localization properties of strain-softening gradient plasticity models. Part II: Theories with gradients of internal variables

The second part of this paper compares and evaluates enhancements of the conventional plasticity theory by gradients of internal variables. Attention is focused on their performance as localization limiters. Both explicit and implicit gradient formulations are considered. It is shown that certain models suffer by serious mathematical deficiencies that would complicate their numerical implementation. Some other models are appropriate only at early stages of the softening process but later exhibit locking accompanied by a spurious expansion of the localized plastic zone. The comparative study indicates that a convenient and robust tool for regularized modeling of the entire localization process is provided by the implicit gradient approach combined with a suitable form of the hardening/softening law.

Localization properties of strain-softening gradient plasticity models. Part I: Strain-gradient theories

This paper compares and evaluates strain-gradient extensions of the conventional plasticity theory. Attention is focused on the ability of individual formulations to act as localization limiters, i.e., to regularize the boundary value problem in the presence of softening and to prevent localization of plastic strain increments into a set of zero measure. To keep the presentation simple and to highlight the essential properties of the investigated models, only the static, rate-independent response in the small-strain range and in the one-dimensional setting is considered. These restrictions permit an analytical or semi-analytical treatment of the problem, while the basic characteristics of the solutions remain valid in the general, multi-dimensional case. The onset of localization is characterized as a bifurcation from a uniform state. The subsequent evolution of the localized process zone and of the shape of the strain profile is studied numerically. It is shown that certain pathologies, e.g., expansion of the plastic region accompanied by stress locking, may arise at later stages of localization. A similar analysis of models with gradients of internal variables is presented in a companion paper.

Formulation of strain gradient plasticity with interface energy in a consistent thermodynamic framework

In this work, the strain gradient formulation is used within the context of the thermodynamic principle, internal state variables, and thermodynamic and dissipation potentials. These in turn provide balance of momentum, boundary conditions, yield condition and flow rule, and free energy and dissipative energies. This new formulation contributes to the following important related issues: (i) the effects of interface energy that are incorporated into the formulation to address various boundary conditions, strengthening and formation of the boundary layers, (ii) nonlocal temperature effects that are crucial, for instance, for simulating the behavior of high speed machining for metallic materials using the strain gradient plasticity models, (iii) a new form of the nonlocal flow rule, (iv) physical bases of the length scale parameter and its identification using nano-indentation experiments and (v) a wide range of applications of the theory. Applications to thin films on thick substrates for various loading conditions and torsion of thin wires are investigated here along with the appropriate length scale effect on the behavior of these structures. Numerical issues of the theory are discussed and results are obtained using Matlab and Mathematica for the nonlinear ordinary differential equations (NODE) which constitute the boundary value problem. This study reveals that the micro-stress term has an important effect on the development of the boundary layers and hardening of the material at both hard and soft interface boundary conditions in thin films. The interface boundary conditions are described by the interfacial length scale and interfacial strength parameters. These parameters are important to control the size effect and hardening of the material. For more complex geometries the generalized form of the boundary value problem using the nonlocal finite element formulation is required to address the problems involved.

Yielding and tensile behavior of nanocrystalline copper

Two theoretical frameworks are calibrated to capture the inverse Hall–Petch phenomenon, as well as the stress–strain response of nanocrystalline Cu during plastic flow. The first framework employed is gradient plasticity enhanced with an interface energy term, the use of which is dictated by the fact that at the nanoscale interfaces play a dominant role in the mechanical behavior of nanocrystalline materials. The second formulation involves a simplified gradient plasticity model without an interface energy term coupled with wavelet analysis.

Plastic failure of nonlocal beams

This paper questions the mode of collapse of some simple softening nonlocal structural systems comprising the classical cantilever beam. Nanobeams can be concerned by such an elementary model. The homogeneous cantilever beam loaded by a concentrated force at its extremity is first considered as a structural paradigm. A nonlocal plasticity model is developed in order to control the localization process induced by microcracking phenomena. An implicit gradient plasticity model equivalent to a nonlocal integral plasticity model is used in this paper. It is shown that the regularized problem is well posed. Closed-form solutions of the elastoplastic deflection are finally derived. The length of the plastic zone grows during the softening process until an asymptotic limited value, which depends on the characteristic length of the material. Scale effects are clearly obtained for these static bending tests. Other structural cases are also presented, including the simply supported beam under uniform transverse loading. It is concluded that the mode of collapse is firmly a nonlocal phenomenon.

Numerical approximation of incremental infinitesimal gradient plasticity

AbstractWe investigate a representative model of continuum infinitesimal gradient plasticity. The formulation is an extension of classical rate‐independent infinitesimal plasticity based on the additive decomposition of the symmetric strain tensor into elastic and plastic parts. It is assumed that dislocation processes contribute to the storage of energy in the material whereby the curl of the plastic distortion appears in the thermodynamic potential and leads to an additional nonlocal backstress tensor. The formulation is cast into a numerical framework by a saddle point approximation of the corresponding minimization problem in each incremental loading step. This allows one to reformulate the (nonlocal) dissipation inequality to a point‐wise flow rule and yields a solution scheme, which is a direct extension of the standard approach in classical plasticity. Our numerical results show the regularizing effects of the additional physically motivated terms. Copyright © 2008 John Wiley & Sons, Ltd.

Application of Strain Gradient Plasticity Modelling to High Pressure Torsion

An analytical model describing the deformation behaviour of copper during the high-pressure torsion (HPT) processing is presented. The model was developed on the microstructural basis where the material is partitioned in two ‘phases’, the dislocation densities in cell walls and the dislocation densities cell interior, entering the model as scalar internal variables. The resulting ’phase mixture’ model is combined with strain gradient theory to account for strain non-uniformity inherent in SPD. It was demonstrated that gradient plasticity model is capable of describing the experimentally observed trends and accounting for a homogenisation of the accumulated shear strain across the HPT sample. The predictions of the model with respect to the ultrafine grain size produced by HPT and evolution of dislocation densities are in good agreement with experimental results reported by other research groups.

A regularization study of some softening beam problems with an implicit gradient plasticity model

The modelling of plastic beams experiencing softening is studied. The homogeneous cantilever beam loaded by a concentrated force at its extremity is considered. This simple structural problem with gradient bending moment allows an analytical treatment of the evolution problem. A gradient plasticity model is developed in order to overcome Wood’s paradox. Surprisingly, explicit gradient plasticity models do not eliminate this paradox, since the beam response is found to be not continuous with respect to the loading parameter. A new implicit gradient plasticity model is used in this paper. It is shown that the new regularized problem is well-posed. Closed-form solutions of the elastoplastic deflection are finally derived. These results are valid for the beam bending problem, but also for the simple analogy of the bar subjected to distributed axial force.

Micro-bending tests: A comparison between three-dimensional discrete dislocation dynamics simulations and experiments

Discrete dislocation dynamics simulations in three dimensions are performed on micro-sized bending beams and the results are compared with experiments. A strong size dependence of the flow stress σf (or bending moment) is found. The flow stress scales approximately inversely with the beam thickness t. The simulations show that the dislocation structure exhibits pronounced pile-ups around the neutral plane of the beam. The back stress from these pile-ups on the dislocation sources is analyzed by means of an analytical pile-up model. It is shown that the scaling behavior σf∝t-1 can be explained by a combination of pile-up and source size limitation. Subsequently, the applicability of strain gradient plasticity models on micro-bending is discussed.

Finite element formulation of a new nonlocal damage model

It is now widely recognized in literature that the results of the local damage model are mesh dependent. Regularization methods which use gradients of equivalent plastic strain in the material yield function are also popularly known as gradient plasticity models. Many times, these nonlocal forms of equivalent plastic strain have been used as a measure of material damage in the integral and gradient enhanced formulations. Regularization methods based on nonlocal forms of ductile void volume fraction are limited in literature, especially for the Rousselier's model. Moreover, the mesh independent nature of the nonlocal solutions are usually demonstrated with the help of 1D bar, shear band and notched tensile specimens, etc. Comparisons of the nonlocal solutions with those of experiment are hardly done, especially, for problems involving prediction of the fracture resistance behaviour of cracked specimens. In this work, the Rousselier's damage model has been extended to a nonlocal form using the nonlocal damage parameter ‘ d’ as an additional degree of freedom of the finite element (FE) model. The FE equations have been derived using the weak forms of the governing equations for both mechanical stress equilibrium and the damage equilibrium. The mesh independent nature of the model has been demonstrated through various examples, such as an axisymmetric tensile specimen and a standard fracture mechanics specimen (for which predicted results have been compared with those of experiment) using different mesh sizes. The ability of the new model to predict the effect of crack tip constraint on the fracture resistance behaviour has been demonstrated by analysing two different types of standard fracture mechanics specimens.

Global existence in rate‐independent infinitesimal strain gradient plasticity

AbstractWe propose a model of infinitesimal strain gradient plasticity including phenomenological Prager type linear kinematical hardening and nonlocal kinematical hardening due to dislocation interaction. The model is a thermodynamically admissible model of infinitesimal plasticity involving only the Curl of the non‐symmetric plastic distortion p. Linearized spatial and material covariance under constant infinitesimal rotations is satisfied. Uniqueness of strong solutions of the infinitesimal model is obtained if two non‐classical boundary conditions on the plastic distortion p are introduced: ṗ.τ = 0 on the microscopically hard boundary ΓD ⊂ ∂Ω and [Curl p ].τ = 0 on the microscopically free boundary ∂Ω\ΓD, where τ are the tangential vectors at the boundary ∂Ω. Moreover, a weak reformulation of the infinitesimal model allows for a global in‐time solution of the corresponding rate‐independent initial boundary value problem. The method of choice are a formulation as a quasi‐variational inequality with symmetric and coercive bilinear form. Use is made of new Hilbert‐space suitable for dislocation density dependent plasticity. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Computations of fatigue crack growth with strain gradient plasticity and an irreversible cohesive zone model

Computations of fatigue crack growth with a first-order strain gradient plasticity (SGP) model and an irreversible cohesive zone model are reported. SGP plays a significant role in the model predictions and leads to increased fatigue crack growth rates relative to predictions with classical plasticity. Increased magnitudes of tractions and material separation at the crack tip together with reduced crack closure appear as the cause for accelerated crack growth in SGP. Under plane strain conditions SGP appears as an essential feature of the development of the crack closure zone. Size effects are explored relative to changes in internal material length scale as well as to structural length scales.

A discontinuous Galerkin formulation for classical and gradient plasticity. Part 2: Algorithms and numerical analysis

This work is the second of a two-part investigation into the use of discontinuous Galerkin methods for obtaining approximate solutions to problems of classical and gradient plasticity. Part I [J.K. Djoko, F. Ebobisse, A.T. McBride, B.D. Reddy, A discontinuous Galerkin formulation for classical and gradient plasticity. Part 1: Formulation and analysis, Comput. Methods Appl. Mech. Engrg., 196 (2007) 3881–3897] presented the formulation and analysis of such problems. This part focusses on algorithmic and computational aspects of the problem. In particular, it is shown that the predictor–corrector algorithms of classical plasticity are readily extended to the case of gradient plasticity, and to discontinuous Galerkin formulations. Conditions for convergence of the algorithms are presented, for the elastic, secant, and consistent tangent predictors. The form of the consistent tangent modulus is established for the case of gradient plasticity. A selection of numerical examples is presented and discussed with a view to illustrating aspects of the approximation scheme and algorithms, as well as features of the model of gradient plasticity adopted here.

Modification of composite hardness models to incorporate indentation size effects in thin films

The indentation hardness of copper films of thickness 25 to 1400 nm on oxidized silicon substrates is determined by nanoindentation. The load-displacement curves are analysed using the Oliver and Pharr method. An indentation size effect is observed at low relative indentation depths for the copper films. Composite hardness models are applied to the data without an indentation size effect and the Korsunksy and Puchi–Cabrera models are found to give very good fits to this data. The Nix and Gao mechanism based strain gradient plasticity (MSG) model is used to account for the indentation size effect. An attempt is made to modify the existing composite hardness models using the MSG model to incorporate indentation size effect. By modification of both the Korsunsky and Puchi-Cabrera models a good fit to the entire range of data is obtained.

Adiabatic Shear Localization for Steels Based on Johnson-Cook Model and Second- and Fourth-Order Gradient Plasticity Models

To consider the effects of the interactions and interplay among microstructures, gradient-dependent models of second- and fourth-order are included in the widely used phenomenological Johnson-Cook model where the effects of strain-hardening, strain rate sensitivity, and thermal-softening are successfully described. The various parameters for 1006 steel, 4340 steel and S-7 tool steel are assigned. The distributions and evolutions of the local plastic shear strain and deformation in adiabatic shear band (ASB) are predicted. The calculated results of the second- and fourth-order gradient plasticity models are compared. S-7 tool steel possesses the steepest profile of local plastic shear strain in ASB, whereas 1006 steel has the least profile. The peak local plastic shear strain in ASB for S-7 tool steel is slightly higher than that for 4340 steel and is higher than that for 1006 steel. The extent of the nonlinear distribution of the local plastic shear deformation in ASB is more apparent for the S-7 tool steel, whereas it is the least apparent for 1006 steel. In fourth-order gradient plasticity model, the profile of the local plastic shear strain in the middle of ASB has a pronounced plateau whose width decreases with increasing average plastic shear strain, leading to a shrink of the portion of linear distribution of the profile of the local plastic shear deformation. When compared with the second-order gradient plasticity model, the fourth-order gradient plasticity model shows a lower peak local plastic shear strain in ASB and a higher magnitude of plastic shear deformation at the top or base of ASB, which is due to wider ASB. The present numerical results of the second- and fourth-order gradient plasticity models are consistent with the previous numerical and experimental results at least qualitatively.

A discontinuous Galerkin formulation for classical and gradient plasticity – Part 1: Formulation and analysis

A discontinuous Galerkin formulation is developed and analyzed for the cases of classical and gradient plasticity. The model of gradient plasticity is based on the von Mises yield function, in which dependence is on the isotropic hardening parameter and its Laplacian. The problem takes the form of a variational inequality of the second kind. The discontinuous Galerkin formulation is shown to be consistent and convergent. Error estimates are obtained for the cases of semi- and fully discrete formulations; these mimic the error estimates obtained for classical plasticity with the conventional Galerkin formulation.

Well-posedness of a model of strain gradient plasticity for plastically irrotational materials

The initial boundary value problem corresponding to a model of strain gradient plasticity due to [Gurtin, M., Anand, L., 2005. A theory of strain gradient plasticity for isotropic, plastically irrotational materials. Part I: Small deformations. J. Mech. Phys. Solids 53, 1624–1649] is formulated as a variational inequality, and analysed. The formulation is a primal one, in that the unknown variables are the displacement, plastic strain, and the hardening parameter. The focus of the analysis is on those properties of the problem that would ensure existence of a unique solution. It is shown that this is the case when hardening takes place. A similar property does not hold for the case of softening. The model is therefore extended by adding to it terms involving the divergence of plastic strain. For this extended model the desired property of coercivity holds, albeit only on the boundary of the set of admissible functions.

Size-effects in porous metals

The intrinsic size-effect for porous metals is investigated. The analyses are carried out numerically using a finite strain generalization of a higher order strain gradient plasticity model. Results for plane strain growth of cylindrical voids are presented in terms of response curves and curves of relative void growth. The influence of void size compared with a constitutive length parameter is analysed and it is shown that strain gradient hardening suppresses void growth on the micron scale. This increased resistance to void growth is accompanied by an increase in the overall strength of the material. For porous materials with small void volume fractions under highly triaxial tension, void growth is analysed through cavitation instabilities using a finite element Rayleigh–Ritz procedure. Cavitation instabilities are found to be delayed for small voids, so that higher stress levels are needed in order to obtain unstable growth. Cavitation diagrams for cylindrical voids are compared with cavitation diagrams for axi-symmetric void growth of initially spherical voids.

Effects of constitutive parameters on adiabatic shear localization for ductile metal based on JOHNSON-COOK and gradient plasticity models

By using the widely used JOHNSON-COOK model and the gradient-dependent plasticity to consider microstructural effect beyond the occurrence of shear strain localization, the distributions of local plastic shear strain and deformation in adiabatic shear band(ASB) were analyzed. The peak local plastic shear strain is proportional to the average plastic shear strain, while it is inversely proportional to the critical plastic shear strain corresponding to the peak flow shear stress. The relative plastic shear deformation between the top and base of ASB depends on the thickness of ASB and the average plastic shear strain. A parametric study was carried out to study the influence of constitutive parameters on shear strain localization. Higher values of static shear strength and work to heat conversion factor lead to lower critical plastic shear strain so that the shear localization is more apparent at the same average plastic shear strain. Higher values of strain-hardening exponent, strain rate sensitive coefficient, melting point, thermal capacity and mass density result in higher critical plastic shear strain, leading to less apparent shear localization at the same average plastic shear strain. The strain rate sensitive coefficient has a minor influence on the critical plastic shear strain, the distributions of local plastic shear strain and deformation in ASB. The effect of strain-hardening modulus on the critical plastic shear strain is not monotonous. When the maximum critical plastic shear strain is reached, the least apparent shear localization occurs.

Cavity expansion of a gradient-dependent solid cylinder

This paper presents an elasto-plastic analysis for cavity expansion in a solid cylinder. The solid is modelled using a strain gradient plasticity model to account for the influence of microstructures on the macroscopic mechanical behaviour. A numerical shooting method, together with Broyden’s iteration procedure, is developed to solve the resulting fourth-order ordinary differential equation with two-point boundary conditions for the gradient-dependent problem. Fully elastic-plastic solutions to the cavity expansion are obtained and they are compared with conventional results for a number of examples. The effects of microstructure on macroscopic behaviour for the cavity expansion problem are analysed. It is demonstrated that, with consideration of microstructural effects, the deformation and stress distributions in the cylinder are highly inhomogeneous during both the initial loading and the subsequent elastic and plastic expansion stages. The gradient effects can result in a stiffer response in the elastic regime (as compared with the corresponding conventional prediction), but a weaker response in the plastic regime. As expected, the overall elasto-plastic behaviour of the gradient-dependent cylinder depends on the material parameters as well as the cylinder thickness. It is shown that the strain gradient theory solutions reduce to the conventional ones as a special case when the dimension of the microstructures is negligible compared with the cylinder size. The results in this paper can be used as a benchmark for further numerical investigations of the cavity expansion problem.