On the assembly of $C^1-$stationary points of a polyconvex functional and finite BOP-theory

TL;DR

This paper constructs and analyzes radially symmetric stationary points of a polyconvex energy functional in finite elasticity, extending BOP-theory to arbitrary covering maps and revealing complex behaviors of these points.

Contribution

It generalizes BOP-theory to arbitrary covering maps in finite elasticity, constructing smooth stationary points with detailed properties.

Findings

Constructed radially symmetric $M$-covering stationary points.

Extended BOP-theory beyond the $M=2$ case.

Revealed richer behaviors of stationary points in finite elasticity.

Abstract

In this work the following energy is considered $I(u)=\int\limits_B{\frac{1}{2}|\nabla u|^2+\rho(\det\nabla u)\;dx},$ where $B\subset\mathbb{R}^2$ denotes the unit ball, $u\in W^{1,2}(B,\mathbb{R}^2),$ and $\rho:\mathbb{R}\rightarrow\mathbb{R}_0^+$ smooth and convex with $\rho(s)=0$ for all $s\le0$ and $\rho$ becomes affine when $s$ exceeds some value $s_0>0.$ Additionally, we may impose $(M\in \mathbb{N}\setminus\{0\})-$covering maps as boundary conditions in a suitable fashion. For such situations we then construct radially symmetric $M-$covering stationary points of the energy, which are at least $C^1$ (in some circumstances even $C^\infty),$ and verify more refined properties, which these stationary points need to satisfy. We do so by following the strategy first and foremost developed by P. Bauman, N. C. Owen, and D. Phillips (BOP) confirming and generalising that the method remains valid beyond the $M=2-$case for an arbitrary $M.$ Furthermore, as far as we know, this is the first treatise of BOP-theory in finite elasticity. The finiteness, not imposing such strict conditions, allows for a richer class of possible behaviours of the stationary points, making it more difficult to completely determine them.