M.Sc Student Trapper Pavel A Simple Theory of Gradient Plasticity with Two Characteristic Lengths Department of Civil and Environmental Engineering Professor Konstantin Volokh Mr. Dimitry Val Abstract

A new strain gradient plasticity theory is formulated to accommodate more than one material length parameter. The theory is an extension of the classical J2 flow theory of metal plasticity to the micron scale. Distinctive features of the proposed theory as compared to other existing theories are the simplicities of mathematical formulation, numerical implementation and physical interpretation. The proposed theory is based on the physical idea that the effective plastic strain is a measure of the defect density. (Point, line dislocations, and surface defects underlying plastic flow are considered. These defects should not be confused with microcracks, voids etc underlying damage accumulation. The latter is out of the scope of the present work.) The latter allows interpreting the yield/hardening condition as an equation describing (quasi-static) evolution of defects. At the macroscopic scale, the defect diffusion is negligible and the yield condition is a purely algebraic equation. At the scale of microns, however, the defect diffusion is sound and it can be included in the yield condition through the divergence of the vector of the defect flux. We propose that the defect flux is linearly proportional to the gradient of the defect density, i.e. the accumulated plastic strain. This proportionality is provided by a second-order tensor of the defect conductivity. Such tensor is the main source of the material characteristic lengths. If, for example, this tensor is equivalent to the length-scaled identity tensor then we arrive at the earlier Aifantis theory of strain-gradient plasticity where the defect diffusion is isotropic. It is reasonable, however, to assume that the defect diffusion is not purely isotropic and stresses induce anisotropy. The latter means that the defect conductivity tensor should depend on the scaled stress tensor. The corresponding formulation of strain gradient plasticity is developed in the present work. It appears that the proposed theory is, probably, the simplest one among the theories, which include more than one characteristic length. It is also important that in contrast to the majority of strain gradient theories no direct use of higher-order stresses, which can hardly be interpreted in simple physical terms, has been made in the present work.