Crystal Instabilities and Elastic Responses of Metals under Extreme Strain Rates

TL;DR

This paper develops a generalized elastic instability criterion for crystals under extreme strain rates, clarifying the physical nature of higher-order elastic instabilities and explaining elastic response singularities in metals.

Contribution

It introduces a unified criterion at continuum and atom levels that encompasses existing instability criteria and clarifies the sign paradox in strain-gradient theories.

Findings

The criterion reproduces known instability criteria like the modified Born and {\Lambda}-criteria.

Modified Born criterion is less precise than the {\Lambda}-criterion under heterogeneous stress.

Application to copper and aluminum explains their elastic response singularities.

Abstract

Despite of some progresses in investigating the roles of the higher-order strain gradients on elastic stabilities of solids, the physical nature on the higher-order elastic instabilities of crystals, especially under extreme strain rates, is still a mystery. In this work, a generalized elastic instability criterion for infinite crystals is consistently established at both continuum and atom level under frameworks of a higher-order phenomenological theory. The established criterion could consistently reproduce the well-known strain-based lattice instability criteria, such as modified Born criterion, {\Lambda}-criterion, as well as a higher-order one proposed by Bardenhagen et al. Our results show that modified Born criterion is not as precise as the {\Lambda}-criterion under heterogeneous stress states. Different from the higher-order criterion, contributions from the third order gradients of displacements are considered so that the well-known sign paradox in the first strain-gradient theory could be reproduced. According to microscopic comprehensions on the higher-order phenomenological theory, the sign paradox is well clarified. Finally, the established criterion is employed to investigate the elastic stabilities of single crystalline copper and aluminum. The obtained results could well explain the singularities of elastic responses of the two metals under ramp compressions.