An intrinsic geometric formulation of Hyper-elasticity, pressure potential and non-holonomic constraints

TL;DR

This paper develops an intrinsic geometric framework for isotropic hyper-elasticity, pressure potential, and non-holonomic constraints directly on the body manifold, offering a novel geometric perspective on elasticity theory.

Contribution

It introduces a geometric formulation of hyper-elasticity and pressure potential on manifolds, extending Poincaré's formula to infinite dimensions for non-holonomic constraints.

Findings

Intrinsic geometric formulation of hyper-elasticity and pressure potential

Extension of Poincaré's formula to infinite-dimensional settings

Identification of optimal non-holonomic constraints for pressure potentials

Abstract

Isotropic hyper-elasticity, altogether with the equilibrium equation and the usual boundary conditions, are formulated directly on the body B, a three-dimensional compact and orientable manifold with boundary equipped with a mass measure. Pearson-Sewell-Beatty pressure potential is formulated in an intrinsic geometric manner. It is shown that Poincar{\'e}'s formula extended to infinite dimension, provides, in a straightforward manner, the optimal (non-holonomic) constraints for such a pressure potential to exist.