Elias C. Aifantis

The physics of plastic deformation

Abstract A simplified physical picture is extracted from the many complicated processes occuring during plastic deformation. It is based upon a set of continuously distributed straight edge dislocations, the carriers of plastic deformation, moving along their slip plane, interacting with each other and the lattice, multiplying and annihilating. The principles of continuum physics, that is the conservation laws of mass and momentum, and results from discrete dislocation modelling are then employed to analyze the situation and deduce a closed set of relations describing the evolution of deformation and the associated forces that bring it about. A simple method is suggested for extending these relations to macroscales. This way, current phenomenological models of plasticity are physically substantiated. Moreover, a framework is provided for rigorously constructing small and large deformation theories of plasticity. Finally, a new possibility is made available for capturing the salient features of the heteogeneity of plastic flow including the wavelength of persistent slip bands, the width of shear bands, and the velocity of Portevin-Le Chatelier bands.

On the role of gradients in the localization of deformation and fracture

Abstract A brief account on the role of higher order strain gradients in the localization of plastic flow, the formation and propagation of deformation bands, and the determination of the structure of the crack tip is given. In the case of shear localization, the higher order gradients provide a mechanism for capturing the evolution of plastic flow in the “material softening” regime, deriving estimates for the width or spacing of the localized zones, and obtaining expressions for the velocity of propagating deformation bands. In the case of determination of the structure of the crack tip, the higher order gradients provide a mechanism for eliminating the strain singularity and deriving a displacement profile which satisfies the smooth closure condition and substantiates Barenblatts cohesive zone concept.

Strain gradient interpretation of size effects

It is shown that the phenomenon of strength dependence on size, for otherwise geometrically similar specimens, may be interpreted on the basis of gradient elasticity and gradient plasticity arguments. This is illustrated by adopting a simple strength of materials approach for considering torsion and bending of solid bars and subsequent comparison with available experimental data. Solutions of boundary value problems based on gradient elasticity and gradient plasticity including those of fracture and shear banding can also be used to interpret size effects in more complex situations. This is not discussed here, however, in order to maintain simplicity and clarity when illustrating the theoretical predictions in comparison with the experimental trends.

Update on a class of gradient theories

This article, written in honor of Professor Nemat-Nasser, provides an update of the standard theories of dislocation dynamics, plasticity and elasticity properly modified to include scale effects through the introduction of higher order spatial gradients of constitutive variables in the governing equations of material description. Only a special class of gradient models, namely those developed by the author and his co-workers, are considered. After a brief review of the basic mathematical structure of the theory and certain gradient elasticity solutions for dislocation fields, the physical origin and form of the gradient terms (for all classes of elastic, plastic, and dislocation dynamics behavior), along with the nature of the associated phenomenological coefficients are discussed. Applications to the interpretation of deformation patterning and size effects are given. Two new features are noted: the role of wavelet analysis and stochasticity in interpreting deformation heterogeneity measurements and serrations of the stress–strain graph. 2002 Elsevier Science Ltd. All rights reserved.

A gradient approach to localization of deformation. I. Hyperelastic materials

By utilizing methods recently developed in the theory of fluid interfaces, we provide a new framework for considering the localization of deformation and illustrate it for the case of hyperelastic materials. The approach overcomes one of the major shortcomings in constitutive equations for solids admitting localization of deformation at finite strains, i.e. their inability to provide physically acceptable solutions to boundary value problems in the post-localization range due to loss of ellipticity of the governing equations. Specifically, strain-induced localized deformation patterns are accounted for by adding a second deformation gradient-dependent term to the expression for the strain energy density. The modified strain energy function leads to equilibrium equations which remain always elliptic. Explicit solutions of these equations can be found for certain classes of deformations. They suggest not only the direction but also the width of the deformation bands providing for the first time a predictive unifying method for the study of pre- and post-localization behavior. The results derived here are a three-dimensional extension of certain one-dimensional findings reported earlier by the second author for the problem of simple shear.

A simple approach to solve boundary-value problems in gradient elasticity

SummaryWe outline a procedure for obtaining solutions of certain boundary value problems of a recently proposed theory of gradient elasticity in terms of solutions of classical elasticity. The method is applied to illustrate, among other things, how the gradient theory can remove the strain singularity from some typical examples of the classical theory.

On the Problem of Diffusion in Solids

We rationalize diffusion in solids on the basis of a differential equation of balance expressing conservation of momentum for the diffusing species. The balance equation contains a tensor, modelling the stress supported by the diffusing species, and a diffusive force vector, modelling the exchange of momentum between the diffusing species and the species of the solid matrix. These two quantities, which are not identified in classical diffusion interpretations, are the basic ingredients of the theory. The effect of state and constitution of interdiffusing materials is reflected in the form of the constitutive equations for the stress and the diffusive force. Within our framework, the main results of classical theories are rigorously derived in a unified manner. New interesting findings are also deduced and their implications are discussed. The applicability of the theory to a variety of problems, ranging from metallurgy to polymer physics and geophysics, is illustrated.ZusammenfassungWir beschreiben die Diffusion in festen Körpern auf der Basis von Bilanzdifferentialgleichungen, die die Erhaltung des Impulses der diffundierenden Teilchen ausdrücken. Die Bilanzgleichung enthält einen Tensor, der die durch das diffundierende Material hervorgerufenen Spannungen beschreibt und den Vektor einer Diffusionskraft, die den Impulsaustausch zwischen dem diffundierenden Medium und der Festkörpermatrix beschreibt. Diese zwei Größen, die in den klassischen Behandlungen des Diffusionsproblems nicht festgestellt wurden, sind die tragenden Säulen für diese Theorie. Zustand und Beschaffenheit der austauschenden Materialien werden durch konstitutive Gleichungen für die Spannungen und die diffusive Kraft beschrieben. Innerhalb dieses Rahmens werden die grundlegenden Ergebnisse der klassischen Theorie streng auf einheitliche Weise hergeleitet. Neue, interessante Ergebnisse werden abgeleitet und ihre Folgerungen besprochen.Die Anwendbarkeit der Theorie auf verschiedene Problemstellungen, von der Metallurgie über Polymerphysik bis zur Geophysik, wird aufgezeigt.

A gradient flow theory of plasticity for granular materials

SummaryA flow theory of plasticity for pressure-sensitive, dilatant materials incorporating second order gradients into the flow-rule and yield condition is suggested. The appropriate extra boundary conditions are obtained with the aid of the principle of virtual work. The implications of the theory into shear-band analysis are examined. The determination of the shear-band thickness and the persistence of ellipticity in the governing equations are discussed.

On Some Aspects in the Special Theory of Gradient Elasticity

In this paper a special form of gradient-dependent elasticity is considered. The motivation for considering higher-order gradients of strains in elasticity is discussed. Equilibrium equations and boundary conditions are discussed. The relationship between the special form of gradient elasticity adopted in this study and mixture or nonlocal theories is considered. Solutions to certain problems including the propagation of harmonic waves, the longitudinal vibrations of a beam, and the displacement field in an infinite medium weakened by a line crack are given.

Observation and measurement of grain rotation and plastic strain in nanostructured metal thin films

Abstract The deformation behavior of nanostructured gold thin films, with grain diameters of 10 nm and film thicknesses of 10–20 nm, has been studied by means of in situ high resolution transmission electron microscopy. Grain rotation was observed by measuring the changes in the angular relationships between the lattice fringes of different grains during deformation at low strain rates. The strain tensor was calculated by measuring the relative displacements of three material points, and using an analysis similar to that for strain gage rosettes. Relative grain rotations of up to 15 degrees, along with effective plastic strains on the order of 30%, were measured. No evidence of dislocation activity was detected during or after straining. Identical experiments on coarser-grained silver thin films, with grain diameters around 110 nm, yielded clear evidence of dislocation activity. These results indicate that grain rotation and grain boundary sliding can make significant contributions to the deformation of nanostructured thin films at low homologous temperatures.