Athermal Nonlinear Elastic Constants of Amorphous Solids

TL;DR

This paper derives the lowest nonlinear elastic constants for amorphous solids at zero temperature, revealing their importance in understanding instabilities like plastic flow and fracture, and connecting thermal and athermal elastic theories.

Contribution

It provides explicit expressions for third-order nonlinear elastic constants of amorphous solids and links thermal and athermal elastic theories, with applications to elasto-plasticity.

Findings

Derived expressions for nonlinear elastic constants up to third order.

Demonstrated the role of these constants in predicting mechanical instabilities.

Connected thermal nonlinear elastic theory to athermal conditions.

Abstract

We derive expressions for the lowest nonlinear elastic constants of amorphous solids in athermal conditions (up to third order), in terms of the interaction potential between the constituent particles. The effect of these constants cannot be disregarded when amorphous solids undergo instabilities like plastic flow or fracture in the athermal limit; in such situations the elastic response increases enormously, bringing the system much beyond the linear regime. We demonstrate that the existing theory of thermal nonlinear elastic constants converges to our expressions in the limit of zero temperature. We motivate the calculation by discussing two examples in which these nonlinear elastic constants play a crucial role in the context of elasto-plasticity of amorphous solids. The first example is the plasticity-induced memory that is typical to amorphous solids (giving rise to the Bauschinger effect). The second example is how to predict the next plastic event from knowledge of the nonlinear elastic constants. Using the results of this paper we derive a simple differential equation for the lowest eigenvalue of the Hessian matrix in the external strain near mechanical instabilities; this equation predicts how the eigenvalue vanishes at the mechanical instability and the value of the strain where the mechanical instability takes place.