Finite deformation finite element calculations are carried out to analyze nonuniform plane strain tensile deformation of single crystals using an elastic-viscoplastic crystal plasticity constitutive relation. The planar crystals considered have two potentially active slip systems with the driving force for slip including a non-Schmid stress. The non-Schmid stress on a slip system is taken to be the Schmid resolved shear stress on the other slip system. In the calculations, the parameters characterizing the non-Schmid term, including its magnitude and the slip interval over which it acts, are varied. The effects of non-Schmid governed slip on the overall stress-strain response and on the strain at which a rapid drop in overall stress occurs is investigated. Additionally, the dissipation rate associated with non-Schmid governed slip is calculated both pointwise and integrated over the entire domain. Results show that one consequence of non-Schmid governed slip is the change in the slip mode and another consequence is that a non-negative dissipation rate for all possible deformation histories is not guaranteed. However, stable calculations up to the rapid drop in overall stress can be carried out even with the dissipation rate being negative at one or more points in the domain over some strain range and, in some cases, even when the integrated dissipation rate is negative. Implications for possible constitutive restrictions are briefly considered.

The role of void nucleation and void growth in triggering localized necking in biaxially stretched sheets is investigated using a rate independent porous plastic constitutive relation with two porosity parameters; one associated with the void volume fraction and the other associated with the weakening effect of void shape changes in shear dominated stress states. Proportional straining plane stress calculations are carried out for ratios of imposed in-plane principal strain rates ranging from approximate to-1 (shear dominated) to 1 (equal biaxial tension). The framework for the localized necking calculations is that in which an imperfection band triggers the onset of localized necking as defined by a loss of ellipticity of the governing equations in the imperfection band. The imperfection band is taken to be either an increase in initial void volume fraction or an increase in the volume fraction of void nucleating particles. The predicted forming limit curves are compared with predictions for a localized necking bifurcation of a rigid-plastic solid. For negative values of the imposed strain ratio, except for near in-plane shear where the second porosity reduces the critical strains, the predicted critical localization strains and the predicted critical localization band orientations differ little from the corresponding critical values predicted by a rigid plastic bifurcation analysis. For biaxial tensile states the critical localization strains are sensitive to the nature and magnitude of the imperfection. When void nucleation occurs over a very narrow range of strain or stress, void nucleation and the onset of localized necking can coincide.

The requirement of a non-negative dissipation rate for all possible deformation histories is generally imposed on plastic constitutive relations. This is a constraint analogous to the Coleman-Noll [1] postulate that the Clausius-Duhem inequality needs to be satisfied for all possible deformation histories. The physical basis for the Clausius-Duhem inequality is as a statistical limit for a large number of discrete events for a long time and is not a fundamental physical requirement for small systems for a short time. The relation between the requirement of a non-negative dissipation rate and the Clausius-Duhem inequality is considered. The consequences of imposing a non-negative dissipation rate for all possible deformation histories are illustrated for: (i) a single crystal plasticity framework that accounts for elastic lattice curvature changes as well as elastic lattice straining; and (ii) for discrete defect theories of plasticity, with attention specifically on discrete dislocation plasticity for crystalline solids and discrete shear transformation zone (STZ) plasticity for amorphous solids. Possible less restrictive conditions on the evolution of dissipation in plasticity formulations are considered as are implications for stability. The focus is on open questions and issues.

A method for solving general boundary-value problems involving discrete dislocations is introduced. Plastic flow emerges from the motion of dislocations in an incremental fashion. At each increment, the displacement, strain and stress fields in the body are obtained by superposition of the infinite medium fields associated with individual dislocations and an image field that enforces boundary conditions. Dislocations are represented as monopoles and dislocation events are treated as a transportation map problem. Long-range interactions are accounted for through linear elasticity with a core regularization procedure. At the current state of development of the method, no ad hoc short-range interactions are included. An approximate loop nucleation model is used for large-scale computations. The image problem is solved using a finite elementFinite elements formulation with the following features: (i) a single Cholesky decompositionDecomposition of the global stiffness matrix, (ii) a consistent enforcement of traction and displacement boundary conditions, and (iii) image force interpolation using an efficient BB-tree algorithm. To ensure accuracy, we explore stable time steps and employ monopole splitting techniques. Special attention is given to the interaction of curved dislocations with arbitrary domain boundaries and free surfacesFree surfaces. The capabilities of the framework are illustrated through a wire torsion problem.

The possible emergence of a localized buckled configuration for a simple supported linear elastic beam on an elastic foundation subject to axial compression is analyzed numerically. Beams on a softening foundation and, for comparison purposes, beams on a linear elastic foundation are considered. The critical bifurcation mode involves periodic lateral displacements for both foundations. Calculations are carried out for imperfections that can have a component in the shape of the critical periodic bifurcation mode as well as a component with a localized shape. For beams on a softening foundation, the buckled configuration obtained in the calculations depends on the magnitudes of the periodic and localized imperfection components: (i) for a wide range of imperfections, a rather abrupt growth of localized lateral displacements occurs after a load maximum, even if the imperfection magnitudes are such that the critical periodic mode is suppressed; or (ii) if the magnitude of the periodic imperfection component is sufficiently large, the buckled configuration with a localized imperfection involves lateral displacements that are oscillatory but not periodic. For beams on a linear elastic foundation, a calculation with a sufficiently large magnitude localized imperfection gives a localized buckled configuration under increasing load. For beams on both types of foundation, the main features of the lateral displacements in the localized buckled configuration are similar, for example, involving both positive and negative lateral displacements even if the localized imperfection only involves non-negative deflections.

A method for solving three dimensional discrete dislocation plasticity boundary-value problems using a monopole representation of the dislocations is presented. At each time step, the displacement, strain and stress fields in a finite body are obtained by superposition of infinite body dislocation fields and an image field that enforces the boundary conditions. The three dimensional infinite body fields are obtained by representing dislocations as being comprised of points, termed monopoles, that carry dislocation line and Burgers vector information. The image fields are obtained from a three dimensional linear elastic finite element calculation. The implementation of the coupling of the monopole representation with the finite element method, including the interaction of curved dislocations with free surfaces, is presented in some detail because it differs significantly from an implementation with a line based dislocation representation. Numerical convergence and the modeling of dislocation loop nucleation for large scale computations are investigated. The monopole discrete dislocation plasticity framework is used to investigate the effect of size and initial dislocation density on the torsion of wires with diameters varying over three orders of magnitude. Depending on the initial dislocation source density and the wire diameter, three regimes of torsion-twist response are obtained: (i) for wires with a sufficiently small diameter, plastic deformation is nucleation controlled and is strongly size dependent; (ii) for wires with larger diameters dislocation plasticity is dislocation interaction controlled, with the emergence of geometrically necessary dislocations and dislocation pile-ups playing a key role, and is strongly size dependent; and (iii) for wires with sufficiently large diameters plastic deformation becomes less heterogeneous and the dependence on size is greatly diminished.

The focus is on discrete defects that can be modeled by continuum mechanics, but where the discreteness of the carriers of plastic deformation plays a significant role. The formulations are restricted to small deformation kinematics and the defects considered, dislocations and discrete shear transformation zones (STZs), are described by their linear elastic fields. In discrete defect plasticity both the stress-strain response and the partitioning between defect energy storage and defect dissipation are outcomes of an initial/boundary value problem solution. Discrete dislocation plasticity modeling results are reviewed that illustrate the implications of defect dissipation evolution for friction, fatigue crack growth and thermal softening. Examples are also given of consequences of three modes of the evolution of discrete defects for size dependence and dissipation in constrained shear. The requirement that the dissipation rate is non-negative, which is a specialization of the Clausius-Duhem inequality for a purely mechanical formulation, imposes restrictions on kinetic relations for the evolution of discrete defects. Explicit kinetic relations for discrete dislocation plasticity dissipation and for discrete STZ plasticity dissipation identify conditions that can lead to a negative dissipation rate. The statistical mechanics form of the second law allows the Clausius-Duhem inequality to be violated for a sufficiently small number of discrete events for a short time period. In continuum mechanics, at least in some circumstances, satisfaction of the Clausius-Duhem inequality can be regarded as a requirement for stability. One dimensional continuum calculations illustrate that there can be a negative dissipation rate over a short distance and for a short time period with overall stability maintained. Implications for discrete defect plasticity modeling are briefly discussed.

A previously proposed strain gradient plasticity theory is extended to incorporate a non-quadratic power law function of the plastic strain gradient in the free energy expression with an exponent of N + 1. The values of N are taken to vary from N = 1 to N = 0. A simple shear problem of a metal layer between rigid boundaries is analyzed. Two stages of plastic deformation are considered. In stage I, the plastic strain is taken to be zero at the boundaries. Stage I ends when a specified magnitude of the plastic strain gradient is attained at the boundaries. In stage II, the magnitude of the plastic strain gradient at the boundaries is fixed at the specified value. With N = 0, a critical plastic strain gradient cannot be specified at the boundaries because the plastic strain gradient is infinite at the boundaries. The theory with N = 0 predicts a constant plateau stress immediately after initial yield, and the dependence of the plateau stress on the layer thickness can fit experimentally observed plateau stress values. However, with N = 0, a stress gap occurs between the initial yield stress and the plateau stress. The theory with 0 < N = 1 and with stage II also can reproduce the experimentally observed dependence of the plateau stress on the layer thickness for any value of N in that range, with an appropriate value of critical plastic strain gradient at the boundaries. The solution for 0 < N = 1 includes that for N = 0 as a limiting case.

Ductile materials subject to impact loading conditions can undergo spall fracture when an incoming compressive stress wave reflects off interfaces and free surfaces as a tensile stress wave. Experimental observations suggest that in ductile materials, spall fracture is driven by the evolution of porosity. However, the presence of initial porosity in ductile materials also introduces plastic compressibility, which can attenuate the incoming compressive stress wave and, as a result, reduce the amplitude of the reflected tensile stress wave. This, in turn, can mitigate or delay spall fracture. In this work, we report on finite deformation finite element calculations that analyze the response of porous ductile materials subjected to impact loading conditions. Two sets of calculations are carried out, in the first set the material contains initial porosity values ranging from 0% to 5% while in the second set the material also undergoes stress-controlled porosity nucleation. Both sets of calculations are carried out for a wide range of imposed impact velocities. Our results show that porosity in ductile materials can, under certain circumstances, mitigate spall fracture by attenuating stress waves. Results correlating the effects of impact velocity, initial porosity, and porosity nucleation on spall fracture are presented and the underlying mechanisms are discussed.

The most widely used phenomenological constitutive relation to model ductile failure of structural metals at room temperature is based on the work of Gurson (1977) and has the advantage of having both a micromechanics basis and sufficient simplicity to enable complex engineering calculations to be carried out. The predictions of this framework have been most successful in circumstances where the stress triaxiality, the ratio of mean normal stress to Mises effective stress, is relatively large. For stress states with smaller values of stress triaxiality, the predictive capability has typically been much reduced. We present a shear modified enhanced Gurson constitutive relation that combines the shear modification of Nahshon and Hutchinson (2008) with the second porosity concept of Gologanu et al. (1994). This maintains both the connection with micromechanics and the computational simplicity. The predictive capability of the framework is illustrated by the good agreement with cell model calculations for localization of deformation over a wide range of stress states. Strain localization analyses with an initial porosity imperfection predict that for axisymmetric stress states with a superposed hydrostatic tension, the minimum critical strain for the onset of localization follows the onset of coalescence, as defined within the context of the constitutive model here, if the value of the stress triaxiality is sufficiently small. For an imposed shear stress state with a superposed hydrostatic tension, the minimum critical strain for the onset of localization is predicted to precede the onset of coalescence. A strong sensitivity of the critical strain for localization of deformation to the strain or stress range over which void nucleation occurs is also predicted, with void nucleation and localization of deformation essentially coinciding for sufficiently abrupt nucleation.

Plastic deformation of crystalline metals mainly takes place by the nucleation and motion of dislocations, line defects in the atomic lattice. At a length scales of micrometers, dislocations can conveniently be idealized as discrete line defects in an elastic continuum and plastic deformation can be represented by the evolution of the dislocation structure. This mesoscopic description is commonly referred to as discrete dislocation plasticity. The focus in the present article is on the solution of initial/boundary value problems using discrete dislocation plasticity. A brief summary of the underlying dislocation dynamics is given, methods used to describe plasticity in terms of evolving dislocation structures are described, some results are presented that illustrate unique capabilities of discrete dislocation plasticity, and needs as well as opportunities for future development are indicated.

Load and hold conical indentation responses calculated for materials having creep stress exponents of 1.15, 3.59 and 6.60 are regarded as input ‘experimental’ responses. A Bayesian-type statistical approach (Zhang et al. 2019 J. Appl. Mech. 86, 011002 (doi:10.1115/1.4041352)) is used to infer power-law creep parameters, the creep exponent and the associated pre-exponential factor, from noise-free as well as noise-contaminated indentation data. A database for the Bayesian-type analysis is created using finite-element calculations for a coarse set of parameter values with interpolation used to create the refined database used for parameter identification. Uniaxial creep and stress relaxation responses using the identified creep parameters provide a very good approximation to those of the ‘experimental’ materials with stress exponents of 1.15 and 3.59. The sensitivity to noise increases with increasing stress exponent. The uniaxial creep response is more sensitive to the accuracy of the predictions than the uniaxial stress relaxation response. Good agreement with the indentation response does not guarantee good agreement with the uniaxial response. If the noise level is sufficiently small, the model of Bower et al. (1993 Proc. R. Soc. Lond. A 441, 97–124 ()) provides a good fit to the ‘experimental’ data for all values of creep stress exponent considered, while the model of Ginder et al. (2018 J. Mech. Phys. Solids 112, 552–562 ()) provides a good fit for a creep stress exponent of 1.15.

A three-dimensional finite deformation study of necking and failure in rectangular tensile bars is carried out using a constitutive relation for porous material plasticity. The fully dynamic formulation accounts for void nucleation and growth along with thermal and rate effects, but here focus is on quasi-static response with a specified initial void volume fraction. The constitutive relation takes into account void shape changes and associated void rotations for three-dimensional voids. The constitutive update is carried out using a generalized rate tangent scheme for an elastic-viscoplastic solid. The sensitivity of necking and failure patterns to the aspect ratio of the rectangular bar is investigated with focus on the plane strain limit and a square tensile bar. The calculations predict the well-known slant fracture in plane strain tension and the emergence of a cup-cone like failure region for a square cross-section. Details are provided for the development of porosity in the bar with a square cross-section, including void shape changes and void rotations. The numerical examples show the capability of a constitutive relation for porous plasticity that can model details of void evolution, thus paving the way for advanced analyses of ductile failure under arbitrary loadings.

Plane strain finite element analyses are used to model the experiments of Mu et al. (2014, 2016) for a thin metal layer confined between elastic solids. The thin metal layer undergoes elastic–plastic deformations, here modeled by strain gradient plasticity, while the rest of the solid deforms only elastically. Plane strain is assumed and finite strains are accounted for. The microscopic boundary condition at the band interface is initially taken to be zero plastic strain increment, but plastic straining at the interface is allowed when the plastic strain gradient at the interface has exceeded a critical value. When the narrow metal band is inclined 45° relative to the direction of compression of the solid, simple shear develops in the band and predictions are rather similar to those of a one-dimensional analysis for an infinite band. For a band at 90° the elastic parts give significant constraint on plastic flow, the stress fields are non-uniform along the band and the average nominal compressive stress shows strong sensitivity to the layer thickness and to the width of the specimen. Remarkably good qualitative, and in some cases quantitative, agreement is found with the experimental observations of Mu et al. (2014, 2016). Considering also other band angles, both the average nominal shear and compressive stress show variations with the angle.

We develop a continuum mechanics model of blastocyst hatching. The blastocyst and the zona pellucida are modeled as concentric thick-walled initially spherical shells embedded in a viscous medium. Each shell is characterized by a nonlinear elastic–viscous–constitutive relation. The stiffer outer shell (the zona pellucida) contains an opening. The softer inner shell (the blastocyst) is subject to a continually increasing pressure, which can eventually drive the escape of the inner shell from the outer shell ("hatching"). The focus is on the continuum mechanics modeling framework and illustrating the sort of quantitative predictions that can be made. Numerical examples are presented for the predicted dependence of the evolution of the escape process on values of parameters characterizing the constitutive response of the shells, on the viscosity of the external medium and on the size of the opening in the zona pellucida.

The response in spherical indentation is used to identify material properties for plastically compressible materials with presumed known elastic properties. The materials are assumed to be characterized by a Deshpande-Fleck constitutive relation (Deshpande and Fleck, 2000). The indentation force versus indentation depth responses and the residual surface profiles (the surface profiles after unloading) are calculated for two sets of material parameters. These are regarded as the "experimental'' input materials. The constitutive parameters for these two materials are identified using the Bayesian-type statistical approach of Zhang et al. (2019) both for noise-free and for noise-contaminated data. The uniaxial stress-strain responses obtained from the spherical indentation responses are good approximations of those of the "experimental'' input materials, particularly if surface profile data is used, but the quality of the approximation decreases with increasing noise amplitude. Plastic compressibility is found to have a relatively small effect on the correction factor. beta in the Oliver-Pharr relation (Oliver and Pharr, 2004) between the unloading slope and the effective (or reduced) elastic modulus. The indentation response of the plastically compressible materials can be well-represented by a nearly incompressible plastic constitutive relation but the inferred uniaxial stress-strain response is a poor representation of the ``experimental'' material uniaxial stress-strain response. The predicted residual surface profile is less dependent on the assumed elastic constant values than is the indentation force versus indentation depth response. The indentation force versus indentation depth responses in spherical indentation for three materials with very different uniaxial stress-strain curves are found to be indistinguishable if the indentation depth is sufficiently small but are distinguishable if the indentation depth is sufficiently large.

Dynamic frictional slip along an interface between plastically compressible solids is analyzed. The plane strain, small deformation initial/boundary value problem formulation and the numerical method are identical to those in Shi et al. (Int J Fract 162:51, 2010) except that here the material constitutive relation allows for plastic compressibility. The interface is characterized by a rate and state dependent friction law. The specimens have an initial compressive stress and are subject to shear loading by edge impact near the interface. Two loading conditions are analyzed, one giving rise to a crack-like mode of slip propagation and the other to a pulse-like mode of slip propagation. In both cases, the initial compressive stress is taken to vary with plastic compressibility such that the associated initial effective stress is the same for all values of plastic compressibility. The volume change for the crack-like slip mode is mainly plastic while the elastic volume change plays a larger role for the pulse-like mode. For the crack-like slip mode, the proportion of plastic dissipation in the material increases with the increasing plastic compressibility, but the effect of plastic compressibility on the energy partitioning for the pulse-like slip mode is much smaller. The predicted propagation speeds approach a speed about the dilational wave speed for both the crack-like and pulse-like slip modes and this speed is not sensitive to the value of the plastic compressibility parameter. Plastic dissipation is found to be mainly associated with the deformation induced by the loading wave rather than with the deformation arising from slip propagation. The amplitude of the slip rate in the slip pulses is found to be largely governed by the value of the initial compressive stress regardless of the value of plastic compressibility.