Free-Boundary Dynamics in Elasto-plastic Amorphous Solids: The Circular Hole Problem

TL;DR

This paper develops an athermal shear-transformation-zone theory to analyze the boundary dynamics of voids in amorphous solids under stress, capturing plastic deformation and residual stresses with potential for broader applications.

Contribution

It introduces a novel STZ-based framework that tracks internal state variables and models elasto-plastic boundary behavior in amorphous solids with analytical solutions for symmetric cases.

Findings

Computed deformations and residual stresses after stress pulses.

Demonstrated the importance of internal state variables like disorder temperature.

Explored boundary-layer theory for less symmetric geometries.

Abstract

We develop an athermal shear-transformation-zone (STZ) theory of plastic deformation in spatially inhomogeneous, amorphous solids. Our ultimate goal is to describe the dynamics of the boundaries of voids or cracks in such systems when they are subjected to remote, time-dependent tractions. The theory is illustrated here for the case of a circular hole in an infinite two-dimensional plate, a highly symmetric situation that allows us to solve much of the problem analytically. In spite of its special symmetry, this example contains many general features of systems in which stress is concentrated near free boundaries and deforms them irreversibly. We depart from conventional treatments of such problems in two ways. First, the STZ analysis allows us to keep track of spatially heterogeneous, internal state variables such as the effective disorder temperature, which determines plastic response to subsequent loading. Second, we subject the system to stress pulses of finite duration, and therefore are able to observe elasto-plastic response during both loading and unloading. We compute the final deformations and residual stresses produced by these stress pulses. Looking toward more general applications of these results, we examine the possibility of constructing a boundary-layer theory that might be useful in less symmetric situations.