Ioannis Tsagrakis

Recent Developments in Gradient Plasticity¿Part I: Formulation and Size Effects

The purpose of this two-purt article, is first to give an update of recent developments of gradient plasticity as this was advanced by Aifantis and co-workers in the early eighties to address dislocation patterning and chear band problems, and then to elaborate on two specific issues of current interest: size effects and plastic heterogeneity. In Part I, a brief review of gradient dislocation dynamics as providing a direct motivation for the simplest form of gradient plasticity is given. Then, a more general phenomenological formulation of gradient plasticity is given and used to interpret size effects. In Part H, wavelet analysis is used as a potential tool to describe plastic heterogeneity at very fine scales for which experimental results are not available, as well as for providing another means to interpret size effects through the derivation of scale-dependent constitutive equations.

Strain gradient and wavelet interpretation of size effects in yield and strength

Abstract The problem of yield stress and ultimate strength dependence on the specimen size is considered. This dependence, in otherwise geometrically similar specimens, is herein interpreted on the basis of gradient plasticity and wavelet analysis which introduce an internal length scale in the stress vs. strain relation. In particular, gradient plasticity within a simple strength of materials approach and wavelets within a simple scale-dependent constitutive equations approach, are used to derive expressions for the yield stress and the ultimate strength of solid bars subjected to torsion and bending. It is shown that these expressions depend on the size of the specimen. When the dimensions of the specimen become comparable to the internal length scale, size effects become important, while for larger specimens such dependence becomes negligible. Comparisons of the theoretical predictions to available experimental data are given.

On the Effect of Strain Gradient on Adiabatic Shear Banding

Most of the work on adiabatic shear banding is based on the effect of temperature gradients on shear band nucleation and evolution. In contrast, the present work considers the coupling between temperature and strain gradients. The competition of thermal and strain gradient terms on the onset of instability and its dependence on specimen size is illustrated. It is shown that heat conduction promotes the instability initiation in the hardening part of the homogeneous stress–strain, while the strain gradient term favors the occurrence of this initiation in the softening regime. This behavior is size dependent, i.e., small specimens can support stable homogeneous deformations even in the softening regime. The spacing of adiabatic shear bands is also evaluated by considering the dominant instability mode during the primary stages of the localization process and it is found that it is an increasing function of the strain gradient coefficient.

Size Effects in Tension: Gradient Internal Variable and Wavelet Models

A gradient internal variable model and a gradient-based wavelet analysis are employed to interpret size effects in uniaxial tension o f smooth un-notched specimens. This task could not be addressed within the standard strain gradient theory because macroscopic strain gradients are absent for this type o f loading. It turns out that both approaches can be used to model such size effects quite satisfactorily.

Element-Free Galerkin Implementation of Gradient Plasticity. Part II: Applications to 2D Strain Localization and Size Effects

This is the second part of a two-part article that focuses on strain gradient plasticity implementation within the element-free Galerkin (EFG) framework. In the first part, a generalized f low theory of gradient plasticity has been presented along with its EFG formulation for the solution of incremental boundary value problems. The one-dimensional tensile bar test has also been employed to study the properties of this formulation for an elastic-softening plastic material. In the present part, the aforementioned EFG implementation is employed to solve two-dimensional boundary value problems under plane strain or plane stress conditions. In particular, the following examples are considered: a) Plastic strain localization and size effects in compression of an elastic gradient plastic material with linear isotropic softening, b) Size effect in uniaxial tension of a perforated strip made of an elastic gradient plastic material with linear isotropic hardening. As in the one-dimensional problem, the non-pointwise satisfaction of the yield condition, combined with the long-range nodal interaction in the EFG method, induces stress oscillations within the plastic region and non-proper, sub-quadratic convergence. Upon discretization refinement, however, robust applied stress (or normalized load) vs. nominal strain graphs, equivalent plastic strain patterns and traction distributions are obtained. 1. I N T R O D U C T I O N The interest in higher-order strain gradient theories has been recently revived because of their ability to model phenomena that cannot be captured by standard continuum theories of elasticity and plasticity, which do not involve any internal length scale in their constitutive equations. Most of the relevant work pertains to

Size Effects in Thick-Walled Hollow Cylinders: Deformation versus Flow Theory of Gradient Plasticity

The purpose of this article is to employ a deformation, as well as a flow theory of gradient plasticity, to investigate the deformation behavior of elastoplastic, thick-walled, hollow cylinders subjected to a uniform internal pressure. Deformation theory is formulated within an approximate strength of materials approach which provides an analytical solution, while flow theory is formulated within a finite element framework since a respective analytical solution is not feasible. The obtained results are compared on the basis of size effects in geometrically similar cylinders. Despite the general qualitative agreement in the predictions of the two models, some considerable differences, related to the sign of the gradient coefficient, the size influence on the onset of plastic yielding, and the position of the elastic-plastic interface, are observed.

Gradient Elasticity Effects on the Two-Phase Lithiation of LIB Anodes

A coupled gradient chemoelasticity theory is employed to model the two-phase mechanism that occurs during lithiation of silicon nanoparticles used to fabricate next generation Li-ion battery (LIB) anodes. It is shown that the strain gradient length scale is able to predict the propagation of an interface front of nonzero thickness advancing from the lithiated to unlithiated region without necessarily including higher-order concentration gradients of the Li ions. Larger strain gradient coefficients (elastic internal lengths) induce more diffused interfaces and faster lithiation, which affect both internal strain and stress distributions in a similar way. Estimates for the migration velocity of the phase boundary are obtained and a range of values of the strain gradient length scale is shown to simulate the observed experimental results.