Gradient Plasticity Model Research Articles

Power-law defect energy in a single-crystal gradient plasticity framework: a computational study

A single-crystal gradient plasticity model is presented that includes a power-law type defect energy depending on the gradient of an equivalent plastic strain. Numerical regularization for the case of vanishing gradients is employed in the finite element discretization of the theory. Three exemplary choices of the defect energy exponent are compared in finite element simulations of elastic-plastic tricrystals under tensile loading. The influence of the power-law exponent is discussed related to the distribution of gradients and in regard to size effects. In addition, an analytical solution is presented for the single slip case supporting the numerical results. The influence of the power-law exponent is contrasted to the influence of the normalization constant.

Second order stress gradient plasticity with an application to thin foil bending

The continuum theory of dislocations is applied to formulate the problem of a double ended dislocation pileup under quadratic applied stress. Accordingly, a second order stress gradient plasticity model is presented to address the contribution of the first and the second stress gradients in the effect interpretation. The model is employed to predict the initial strengthening and subsequent hardening in curved and straight thin foils under pure bending within the continuum framework. It is shown that the so-called stress gradient plasticity model that ignores the second stress gradient may not give sound interpretations of the size effects. The plastic response of thin foils is affected by both the first and second stress gradients, yet their interaction strongly depends upon the length scale parameter. The larger the length scale parameter, the quadratic term contribution would be important and the predictions of the first and second order models deviate significantly from each other.

Localization analysis of an energy-based fourth-order gradient plasticity model

Abstract The purpose of this paper is to provide analytical and numerical solutions of the formation and evolution of the localized plastic zone in a uniaxially loaded bar with variable cross-sectional area. An energy-based variational approach is employed and the governing equations with appropriate physical boundary conditions, jump conditions, and regularity conditions at evolving elasto–plastic interface are derived for a fourth-order explicit gradient plasticity model with linear isotropic softening. Four examples that differ by regularity of the yield stress and stress distributions are presented. Results for the load level, size of the plastic zone, distribution of plastic strain and its spatial derivatives, plastic elongation, and energy balance are constructed and compared to another, previously discussed non-variational gradient formulation.

The plastic rotation effect in an isotropic gradient plasticity model for applications at the meso scale

Although formulated to represent a large system of polycrystals at the macroscopic level, isotropic gradient plasticity models have routinely been adopted at the meso scale. For such purposes, it is crucial to incorporate the plastic rotation effect in order to obtain a reasonable approximation of the underlying granular mechanics. This paper focuses on the isotropic plasticity model by Gurtin(2004), which departs from most existing isotropic gradient plasticity theories by incorporating the plastic rotation into the formulation. To elucidate on the role of plastic rotation, we compare and contrast the Gurtin (2004) model with two analogous formulations: the gradient crystal plasticity model by Gurtin and Needleman (2005), which we take as a reference since it retains information on the underlying crystallography; and an isotropic, plastically irrotational gradient plasticity model by Gurtin and Anand (2005). Through a constrained shear problem, it is shown analytically that the plastic rotation is essential for an isotropic model to capture the crystallographic effect predicted by the reference crystal plasticity model with multiple slip systems. The role of plastic rotation is further illustrated numerically with a composite unit cell problem, for which the isotropic model by Gurtin and Anand (2005) predicts a hardening response that differs significantly from the reference solution. Finally, we study the grain boundary effect on a single crystal with the Gurtin (2004) model. By introducing an interfacial resistance at the grain boundaries, the model predicts a Hall–Petch effect where the flow stress scales inversely with the grain size.

Numerical implementation and validation of a consistently homogenized higher order plasticity model

SummaryA homogenization theory was developed earlier that starts with a meso‐scale gradient plasticity model at the bottom and recovers a macroscopically continuous micromorphic model at the top. Through the scale transition framework, the granular mechanics are smeared consistently over a single grain such that the fine scale properties, microstructural length scale (l), and grain size (L) manifest themselves naturally in the homogenized relations, a point of departure from many continuous higher order theories that assume the constitutive relations a priori. This paper elaborates on the numerical implementation of the homogenized model and benchmark its performance through a plane indentation example against detailed numerical simulations. For the two limiting cases of microfree and microhard condition at grain boundaries, the excellent predictive capability of the homogenized model is demonstrated for a range of l/L ratios, at a significantly lower computational cost. It is furthermore highlighted that a more realistic response at grain boundaries can be achieved easily in the homogenized model, by changing the interfacial resistance parameter. In contrast, a non‐trivial numerical treatment is required at the grain boundaries with meso‐scale models. Finally, the homogenized model is shown to perform well even in the absence of a strong scale separation between meso and macro. Copyright © 2015 John Wiley & Sons, Ltd.

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.

Indentation size effects in pack carbo-nitrided AISI 8620 steels

This paper explores the indentation size effect (ISE) in pack carbo-nitrided AISI 8620 steel surfaces. The surfaces of the AISI 8620 steel samples were pack carbo-nitrided at 900°C using cyanide-containing dried cassava leaves (bio-processed waste). This was achieved by quenching in different pH levels of cyanide-based bio-processed solution (BPS). Nanoindentation was carried out (using a Berkovich tip) on the surface modified steels. This was used to measure the hardness and reduced elastic moduli of the quenched-carbonitrided and the as-received steel surfaces. The surface-modified steel was shown to have higher hardness than the as-received steel. The hardness was also found to depend strongly on the indentation size. The paper also considers the potential contributions from microstructure, geometrically necessary dislocations (GNDs) and C/N diffusion on the measured hardness values. However, the measured ISEs are attributed largely to the role of GNDs that are modeled using the Nix and Gao mechanism-based strain gradient plasticity model. The implications of the results are then discussed for the design of hard carbo-nitrided steel surfaces.

Size effects and temperature dependence on strain-hardening mechanisms in some face centered cubic materials

The mechanical behavior of face centered cubic metals is deeply affected when specimen dimensions decrease from a few millimeters to a few micrometers. At room temperature, a critical thickness (t) to grain size (d) ratio (t/d)c was previously highlighted, under which the softening of mechanical properties became very pronounced both in terms of Hall–Petch relation and work hardening mechanisms. In this work, new experimental results are provided concerning the influence of temperature on this size effect for copper, nickel and Ni–20wt.%Cr, representative of a wide range of deformation mechanisms (i.e. dislocation slip character). It is shown that multicrystalline samples (t/d<(t/d)c) are not deeply affected by an increase in temperature, independently of the planar or wavy character of dislocation glide. For pronounced wavy slip character metals, surface effects in polycrystals (t/d>(t/d)c) are not significant enough to reduce the gap between polycrystal and multicrystal mechanical behavior when the temperature increases. However, a transition from wavy slip to planar glide mechanisms induces a modification of the polycrystalline behavior which tends toward multicrystalline one with a moderate increase in temperature. This work demonstrates that surface effects and grain size influence can be successfully disassociated for the three studied materials using an analysis supported by the Kocks–Mecking formalism. All these results are supported by microscopic investigations of dislocation substructures and compared to numerical simulations using a strain gradient plasticity model.

A stabilization approach for mesh-free simulations of systems developing shocks or extreme strain localizations

A new stabilization scheme, based on a stochastic representation of the discretized field variables, is proposed with a view to reduce or even eliminate unphysical oscillations in the mesh-free numerical simulations of systems developing shocks or exhibiting localized bands of extreme deformation in the response. The origin of the stabilization scheme may be traced to nonlinear stochastic filtering and, consistent with a class of such filters, gain-based additive correction terms are applied to the simulated solution of the system, herein achieved through the element-free Galerkin method, in order to impose a set of constraints that help arresting the spurious oscillations. The method is numerically illustrated through its applications to inviscid Burgers' equations, wherein shocks may develop as a result of intersections of the characteristics, and to a gradient plasticity model whose response is often characterized by a developing shear band as the external load is gradually increased. The potential of the method in stabilized yet accurate numerical simulations of such systems involving extreme gradient variations in the response is thus brought forth.

Microstructural pattern formation in finite-deformation single-slip crystal plasticity under cyclic loading: Relaxation vs. gradient plasticity

We investigate microstructure formation and evolution during cyclic loading in rate-dependent crystal plasticity at finite strains. The non-quasiconvex free energy density in multiplicative single-slip crystal plasticity leads to fine-scale microstructure whose characteristics and resulting effective stress–strain response are studied by two independent approaches: (i) using an incremental formulation based on variational constitutive updates we approximate the quasiconvex hull by lamination, i.e. by constructing an energy-minimizing first-order laminate microstructure, and (ii) a strain-gradient plasticity model applied to a representative unit cell whose effective properties are obtained from homogenization. In the lamination model, three different formulations for updating the accumulated plastic strains are compared and discussed with a specific focus on identifying a suitable description to account for hardening due to changes of the laminate volume fractions. The gradient-plasticity model also predicts a first-order laminate microstructure to form at a comparable stress level upon microstructure initiation. However, the energy associated with the dislocation network is shown to affect the microstructure evolution, leading to considerably higher strain levels at laminate initiation and a stress overshoot. In both models, cyclic loading leads to a degeneration of the stress–strain hysteresis which ultimately experiences elastic shakedown. The amount of work hardening significantly depends on how fast the degeneration occurs. To allow for a comparison, we consider cyclic loading after pre-deformation in the gradient model which delays the degeneration of the stress–strain hysteresis. For low hardening, the two models predict differences in the stress–strain hysteresis, mainly owing to laminate migration in the gradient-plasticity model. As work hardening increases, this phenomenon is restricted and the agreement of the effective stress–strain response between the two models is excellent. Accounting for the energy stored in the domain walls leads to a delayed lamination which is in agreement with the gradient plasticity model.

Super-convergent second derivative recovery for lower-order strain gradient plasticity

An element patch based super-convergent second derivative recovery technique is developed in this paper for evaluating the first and second strain derivatives for a lower-order strain gradient plasticity model. The element based patches are defined and the patch least square fitting process is introduced. A classical shear band simulation is then conducted to test the new technique with various unstructured meshes with mixed quadratic quadrilateral and triangular C0 elements. The result shows that a direct recovery from the patch least square fitting process is more reliable than that from the nodal averaging process usually necessary for the super-convergent second derivative recovery methods. The proposed technique is proven to be effective for recovering both first and second order strain derivatives and has the potential for even higher order derivative recovery.

A dislocation-based, strain–gradient–plasticity strengthening model for deformation processed metal–metal composites

Deformation processed metal–metal composites (DMMCs) are high-strength, high-electrical conductivity composites developed by severe plastic deformation of two ductile metal phases. The extraordinarily high strength of DMMCs is underestimated using the rule of mixture (or volumetric weighted average) of conventionally work-hardened metals. In this article, a dislocation-density-based, strain–gradient–plasticity model is proposed to relate the strain-gradient effect with the geometrically necessary dislocations emanating from the interface to better predict the strength of DMMCs. The model prediction was compared with the experimental findings of Cu–Nb, Cu–Ta, and Al–Ti DMMC systems to verify the applicability of the new model. The results show that this model predicts the strength of DMMCs better than the rule-of-mixture model. The strain-gradient effect, responsible for the exceptionally high strength of heavily cold worked DMMCs, is dominant at large deformation strain since its characteristic microstructure length is comparable with the intrinsic material length.

Some Remarks on the Numerical Solution of a Strain Gradient Plasticity Theory

AbstractA one‐dimensional strain gradient plasticity model is reviewed in the context of Finite Element Method (FEM) solvability. It is found that under certain conditions specified the resulting system of two coupled partial differential equations in space and time leads to a symmetric global stiffness matrix. This is achieved upon standard weak form approach and yields a favorable system for an efficient numerical solution. (© 2013 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Questioning size effects as predicted by strain gradient plasticity

AbstractThe analytical solution of the elastic-plastic response of a two-phase laminate microstructure subjected to periodic simple shear loading conditions is derived considering strain gradient and micromorphic plasticity models successively. One phase remains purely elastic, whereas the second one displays an isotropic elastic-plastic behavior. Although no classic hardening is introduced at the individual phase level, the laminate is shown to exhibit an overall linear hardening scaling with the inverse of the square of the cell size. The micromorphic model leads to a saturation of the hardening at small length scales in contrast to Aifantis strain gradient plasticity model displaying unlimited hardening. The models deliver qualitatively relevant size effects from the physical metallurgical point of view, but fundamental quantitative discrepancy is pointed out and discussed, thus requiring the development of more realistic nonlinear equations in strain gradient plasticity.

Interpreting the stress–strain response of Al micropillars through gradient plasticity

Micropillar compression has fascinated the materials and mechanics communities for over a decade, due to the unique stochastic effects and slip zones that dictate their stress–strain curves and microstructure. Although plethora studies exist that capture experimentally the mechanical response of various types of micropillars, limited theoretical models can interpret the observed behavior. Particularly, single crystal micropillars exhibit multiple serrations in their stress–strain response, indicating the activation of slip zones, while bi-crystal pillars, in which the grain boundary lies parallel to the pillar axis, do not display such serrations, but rather a distinct “knee”, which indicates dislocation pileups at the grain boundary. In-situ synchrotron microdiffraction experiments have illustrated that not only dislocations, but also significant plastic strain gradients develop during micropillar compression. In the present study, therefore, appropriate gradient plasticity models that can account for the pillar microstructure, are successfully used to capture the stress–strain response of single- and bi-crystal Al pillars.

Scale transition of a higher order plasticity model – A consistent homogenization theory from meso to macro

Standard plasticity models cannot capture the microstructural size effect associated with grain sizes, as well as structural size effects induced by external boundaries and overall gradients. Many higher-order plasticity models introduce a length scale parameter to resolve the latter limitation – microstructural influences are not explicitly account for. This paper adopts two distinct length scales in the formulation, i.e. an intrinsic length scale (l) governing micro-processes such as dislocation pile-up at internal boundaries, as well as the characteristic grain size (L), and aims to unravel the interaction between these two length scales and the characteristic specimen size (H) at the macro level. At the meso-scale, we adopt the strain gradient plasticity model developed in Gurtin (2004) [Gurtin, M.E., 2004. A gradient theory of small-deformation isotropic plasticity that accounts for the Burgers vector and for dissipation due to plastic spin. J. Mech. Phys. Solids 52, 2545–2568] which accounts for the direct influence of grain boundaries. Through a novel homogenization theory, the plasticity model is translated consistently from meso to macro. The two length scale parameters (l and L) manifest themselves naturally at the macro scale, hence capturing both types of size effects in an average sense. The resulting (macro) higher-order model is thermodynamically consistent to the meso model, and has the same structure as a micromorphic continuum. Finally, we consider a bending example for the two limiting cases – microhard and microfree conditions at grain boundaries – and illustrate the excellent match between the meso and homogenized solutions.

Material size effects on crack growth along patterned wafer-level Cu–Cu bonds

The role of micron-scale patterning on the interface toughness of bonded Cu-to-Cu nanometer-scale films is analyzed, motivated by experimental studies of Tadepalli, Turner and Thompson. In the experiments 400nm Cu films were deposited in various patterns on Si wafer substrates and then bonded together. Crack growth along the bond interface is here studied numerically using finite element analyses. The experiments have shown that plasticity in the Cu films makes a major contribution to the macroscopic interface toughness. To account for the size dependence of the plastic flow a strain gradient plasticity model is applied here for the metal. A cohesive zone model is applied to represent the crack growth along the bond between the two Cu films. This cohesive zone model incorporates the effect of higher order stresses in the continuum, such that the higher order tractions on the crack faces decay to zero values when the crack separation process takes place.The analyses focus on a pattern of Cu lines orthogonal to the crack growth direction, and the analyses are carried out for plane strain conditions with the assumption of small scale yielding under remote mode I loading. When crack growth over a Cu line initiates at the 90° edge on the Cu substrate the resistance curve shows a high peak before the toughness decays to a rather constant plateau. Later, when the crack tip approaches the 90° edge at the end of the Cu line, the fracture toughness decays below the plateau. It is found that both the toughness peak and the subsequent plateau level are highly sensitive to the value of the characteristic material length. A small material length, relative to the thickness of the Cu film, gives high toughness whereas a length comparable to the film thickness gives much reduced crack growth resistance.

On the Fleck and Willis homogenization procedure in strain gradient plasticity

We revisit the homogenization process for a heterogeneous small strain gradient plasticity model considered in [5]. We derive a precise homogenized behavior, independently of any kind of periodicity assumption and demonstrate that it reduces to a model studied in [8] when periodicity is re-introduced.

Localization analysis of variationally based gradient plasticity model

The paper presents analytical or semi-analytical solutions for the formation and evolution of localized plastic zone in a uniaxially loaded bar with variable cross-sectional area. A variationally based formulation of explicit gradient plasticity with linear softening is used, and the ensuing jump conditions and boundary conditions are discussed. Three cases with different regularity of the stress distribution are considered, and the problem is converted to a dimensionless form. Relations linking the load level, size of the plastic zone, distribution of plastic strain and plastic elongation of the bar are derived and compared to another, previously analyzed gradient formulation.

Effect of alternating electric current on the nanoindentation of copper

Electromechanical interaction determines the structural reliability of electronic interconnects. Using the nanoindentation technique, the effect of alternating electric current on the indentation deformation of copper strips was studied for the indentation load in a range of 100 to 1600 μN at room temperature. During the test, an alternating electric current of the electric current density in a range of 1.25 to 4.88 kA/cm2 was passed through the copper strips. The indentation results showed that the reduced contact modulus decreased linearly with increasing the electric current density. The indentation hardness decreased with increasing the indentation deformation, demonstrating the normal indentation size effect. Using the model of strain gradient plasticity, we found that the strain gradient underneath the indentation decreased slightly with increasing the electric current density for the same indentation depth.