Hussein M. Zbib

On plastic deformation and the dynamics of 3D dislocations

Abstract A three-dimensional (3D) mesoscopic model to simulate the collective dynamic behavior of a large number of curved dislocations of finite lengths has been developed for the purpose of analyzing deformation patterns and instabilities, including the formation of dislocation cell structures. Each curved dislocation is approximated by a piecewise continuous array of straight line segments. The interactions among the segments, including line-tension and self-interactions, are treated explicitly. For longer-range interactions, the space is divided into a regular cellular array and the elastic fields of the dislocations in a remote cell approximated by a multipolar expansion, leading to an order N algorithm for the description of a cell containing N dislocations. For large arrays, the simulation volume is divided into cubical cells. A discrete random starting array is selected for the master cell and its nearest neighbors, which constitute an order 2 cell. Reflection boundary conditions are imposed for near-neighbor order 2 cells and so forth, creating an NaCl-type lattice array. The boundaries between the cells are considered to be relaxed grain boundaries. That is, recovery within the boundaries and rotation across them are considered to occur so that the boundaries have no associated elastic fields. This cell hierarchy, coupled with the multipole expansion, is suitable for the use of massively parallel computation, with individual cells assigned to separate processors.

A multiscale model of plasticity

Abstract A framework for investigating size-dependent small-scale plasticity phenomena and related material instabilities at various length scales ranging from the nano-microscale to the mesoscale is presented. The model is based on fundamental physical laws that govern dislocation motion and their interaction with various defects and interfaces. Particularly, the multi-scale framework merges two scales, the nano-microscale where plasticity is determined by explicit three-dimensional dislocation dynamics analysis providing the material length-scale, and the continuum scale where energy transport is based on basic continuum mechanics laws. The result is a hybrid elasto-viscoplastic simulation model coupling discrete dislocation dynamics with finite element analyses. With this hybrid approach, one can address complex size-dependent problems including, dislocation boundaries, dislocations in heterogeneous structures, dislocation interaction with interfaces and associated shape changes and lattice rotations, as well as deformation in nano-structured materials, localized deformation and shear bands.

On the gradient-dependent theory of plasticity and shear banding

SummaryAfter a brief review of a recently developed gradient-dependent theory of plasticity various questions related to the yield function and the loading-unloading condition in the presence of higher order strain gradients and the determination of the corresponding phenomenological coefficients are addressed. For rate-independent materials, we construct as before an analytical solution for the strain profile in the postlocalization regime providing the shear band thickness and strain within it but we now compare these results to recently obtained experimental data by assigning appropriate values to the gradient coefficients. We also address some questions recently raised in the literature regarding our nonlinear shear band analysis. For rate-dependent materials, the resulting spatio-temporal differential equation for the strain is solved numerically using the finite difference method. It is shown that the band width does not depend on the grid size, as long as the the grid size is smaller than a certain characteristic length. Various initial imperfections of different amplitudes and sizes are examined, and the possibility of simultaneous development of two shear bands and their interaction is investigated.

A Gradient-Dependent Flow Theory of Plasticity: Application to Metal and Soil Instabilities

We propose a gradient-dependent flow theory of plasticity for metals and granular soils and apply it to the problems of shear banding and liquefaction. We incorporate higher order strain gradients either into the constitutive equation for the flow stress or into the dilantancy condition. We examine the effect of these gradients on the onset of instabilities in the form of shear banding in metals or shear banding and liquefaction in soils under both quasi-static and dynamic conditions. It is shown that the higher order gradients affect the critical conditions and allow for a wavelength selection analysis leading to estimates for the width or spacing of shear bands and liquefying strips. Finally, a nonlinear analysis is given for the evolution of shear bands in soils deformed in the post-localization regime.

3D dislocation dynamics : Stress-strain behavior and hardening mechanisms in fcc and bcc metals

A dislocation dynamics (DD) model for plastic deformation, connecting the macroscopic mechanical properties to basic physical laws governing dislocation mobility and related interaction mechanisms, has been under development. In this model there is a set of critical reactions that determine the overall results of the simulations, such as the stress-strain curve. These reactions are, annihilation, formation of jogs, junctions, and dipoles, and cross-slip. In this paper we discuss these reactions and the manner in which they influence the simulated stress- strain behavior in fcc and bcc metals. In particular, we examine the formation (zipping) and strength of dipoles and junctions, and effect of jogs, using the dislocation dynamics model. We show that the strengths (unzipping) of these reactions for various configurations can be determined by direct evaluation of the elastic interactions. Next, we investigate the phenomenon of hardening in metals subjected to cascade damage dislocations. The microstructure investigated consists of small dislocation loops decorating the mobile dislocations. Preliminary results reveal that these loops act as hardening agents, trapping the dislocations and resulting in increased hardening.

A Thermodynamical Theory of Gradient Elastoplasticity with Dislocation Density Tensor. I : Fundamentals

Abstract A thermodynamical theory of gradient elastoplasticity, including kinematic hardening, is developed by introducing the concept of dislocation density tensor. The theory is self-consistent and is based on two fundamental principles, the principle of increase of entropy and the maximal entropy production rate. Thermodynamically consistent constitutive equations for plastic stretching, plastic spin and back stress are rigorously derived. Also, an expression for the plastic spin is obtained from the constitutive equation of dislocation drift rate and an expression for the back stress is given as a balance equation expressing equilibrium between internal stress and microstress conjugate to the dislocation density tensor. Moreover, it is shown that the present gradient theory yields a symmetric stress tensor. Some generalities and utility of this theory are discussed and comparisons with other gradient theories are given.

On the concept of relative and plastic spins and its implications to large deformation theories. Part I: Hypoelasticity and vertex-type plasticity

SummaryThe concept of relative spinWD/S is introduced to facilitate the formulation of large deformation stress-rate models of inelasticity. Roughly speaking, it is a measure of non-coaxiality between stress and deformation rate of the formWD/S=W−WS withWS andW signifying the angular velocities or spins of the two material frames corresponding to the stress and strain rate respectively. It is suggested that an objective stress rate be defined with respect toWS for use in the constitutive equations and this requires explicit representations forWD/S reflecting the aforementioned non-coaxiality. It is shown that this practice leads conveniently to an elegant generalization of previous proposals resorting to either use of a variety of different spins or considerably complex constitutive equations, in order to dispense with undesirable oscillatory solutions of simple shear problems.

Strain gradients and continuum modeling of size effect in metal matrix composites

SummaryConstitutive modeling for the particle size effect on the strength of particulate-reinforced metal matrix composites is investigated. The approach is based on a gradient-dependent theory of plasticity that incorporates strain gradients into the expression of the flow stress of matrix materials, and a finite unit cell technique that is used to calculate the overall flow properties of composites. It is shown that the strain gradient term introduces a spatial length scale in the constitutive equations for composites, and the dependence of the flow stress on the particle size/spacing can be obtained. Moreover, a nondimensional analysis along with the numerical result yields an explicit relation for the strain gradient coefficient in terms of particle size, strain, and yield stress. Typical results for aluminum matrix composites with ellipsoidal particles are calculated and compare well with data measured experimentally.

Constitutive modeling of deformation and damage in superplastic materials

Abstract The superplastic deformation and cavitation damage characteristics of a modified aluminum alloy are investigated at a temperature range from 500 to 550°C. The baseline alloy is AA5083. Nominally this alloy contains about 4.5% Mg, 0.8% Mn, 0.2% Cr, 0.037% Si, 0.08% Fe and 0.025% Ti by weight. The experimental program consists of uniaxial tension tests and digital image analysis for measuring cavitation. The experiments reveal that evolution of damage is due to both nucleation and growth of voids. A viscoplastic model for describing deformation and damage in this alloy is developed based on a continuum mechanics framework. The model includes the effect of strain hardening, strain rate sensitivity, dynamic and static recovery, and nucleation and growth of voids. The model predictions compare well with the experimental results.

Simulation of shock-induced plasticity including homogeneous and heterogeneous dislocation nucleations

A model of plasticity that couples discrete dislocation dynamics and finite element analysis is used to investigate shock-induced dislocation nucleation in copper single crystals. Homogeneous nucleation of dislocations is included based on large-scale atomistic shock simulations. The resulting prodigious rate of dislocation production takes the uniaxialy compressed material to a hydrostatically compressed state after a few tens of picoseconds. The density of dislocations produced in a sample with preexisting dislocation sources decreases slightly as shock rise time increases, implying that relatively lower densities would be expected for isentropic loading using extremely long rise times as suggested experimentally.