The mathematical foundations of anelasticity: Existence of smooth global intermediate configurations

Author: 

Goodbrake, C
Goriely, A
Yavari, A

Publication Date: 

6 January 2021

Journal: 

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Last Updated: 

2021-10-19T13:24:04.563+01:00

Issue: 

2245

Volume: 

477

DOI: 

10.1098/rspa.2020.0462

abstract: 

A central tool of nonlinear anelasticity is the multiplicative decomposition of the deformation tensor that assumes that the deformation gradient can be decomposed as a product of an elastic and an anelastic tensor. It is usually justified by the existence of an intermediate configuration. Yet, this configuration cannot exist in Euclidean space, in general, and the mathematical basis for this assumption is on unsatisfactory ground. Here, we derive a sufficient condition for the existence of global intermediate configurations, starting from a multiplicative decomposition of the deformation gradient. We show that these global configurations are unique up to isometry. We examine the result of isometrically embedding these configurations in higher-dimensional Euclidean space, and construct multiplicative decompositions of the deformation gradient reflecting these embeddings. As an example, for a family of radially symmetric deformations, we construct isometric embeddings of the resulting intermediate configurations, and compute the residual stress fields explicitly.

Symplectic id: 

1147509

Submitted to ORA: 

Submitted

Publication Type: 

Journal Article