Compatibility condition for the Eulerian left Cauchy-Green deformation tensor field

TL;DR

This paper derives an explicit compatibility condition for the Eulerian left Cauchy-Green deformation tensor, establishing its role as the finite-strain equivalent of Saint-Venant's condition and clarifying differences from the Lagrangian framework.

Contribution

It provides the first explicit compatibility condition formula for the Eulerian left Cauchy-Green tensor, linking finite-strain theory with classical compatibility conditions.

Findings

The compatibility condition is explicitly formulated for the Eulerian tensor.

It demonstrates the tensor's role as the finite-strain counterpart of Saint-Venant's condition.

The paper highlights key differences between Eulerian and Lagrangian compatibility problems.

Abstract

An explicit compatibility condition formula is presented for the Eulerian left Cauchy-Green deformation tensor field. It is shown to be the appropriate finite-strain counterpart of Saint-Venant's compatibility condition. The difference between the Eulerian problem and the Lagrangian one is highlighted.