Variational constitutive updates for strain gradient isotropic plasticity

Abstract

In the past decades, various strain gradient isotropic plasticity theories have been developed to describe the size-dependence plastic deformation mechanisms observed experimentally in micron-indentation, torsion, bending and thin-film bulge tests in metallic materials. Strain gradient plasticity theories also constitute a convenient device to introduce ellipticity in the differential equations governing plastic deformation in the presence of softening. The main challenge to the numerical formulations is that the effective plastic strain, a local internal variable in the classic isotropic plasticity theory, is now governed by the partial differential equation which includes spatial derivatives. Most of the current numerical formulations are based on Aifantis' one-parameter model with a Laplacian term [Aifantis and Muhlhaus, ijss, 28:845-857, 1991]. As indicated in the paper [Fleck and Hutchinson, jmps, 49:2245-2271, 2001], one parameter is not sufficient to match the experimental data. Therefore a robust and efficient computational framework that can deal with more parameters is still in need. In this thesis, a numerical formulation based on the framework of variational constitutive updates is presented to solve the initial boundary value problem in strain gradient isotropic plasticity. One advantage of this approach compared to the mixed methods is that it avoids the need to solve for both the displacement and the effective plastic strain fields simultaneously. Another advantage of this approach is, as has been amply established for many other material models, that the solution of the problem follows a minimum principle, thus providing a convenient basis for error estimation and adaptive remeshing.

(cont.) The advantages of the framework of variational constitutive updates have already been verified in a wide class of material models including visco-elasticity, visco-plasticity, crystal plasticity and soil, however this approach has not been implemented in the strain gradient plasticity models. In this thesis, a three-parameter strain gradient isotropic plasticity model is formulated within the variational framework, which is then taken as a basis for finite element discretization. The resulting model is implemented in a computer code and exercised on the benchmark problems to demonstrate the robustness and versatility of the proposed method.