A Multiplicative Version of Quasiconvexity for Hyperelasticity

Abstract

As a way to better deal with the fundamental constraint $\det \nabla {\boldsymbol{u}}>0$ for deformations ${\boldsymbol{u}}$ in elastostatics, we propose a multiplicative form of quasiconvexity and of feasible variations. It is based on the classic concept of inner-variations. In general, inner variations are more restrictive than outer general (additive) variations, but they turn out to be equivalent precisely in the situation of hyper-elasticity under the positivity of the determinant. We focus on the weak lower semicontinuity property, and the validity of the weak form of the Euler-Lagrange equations.

Appendix

We gather here, for the convenience of readers, a few fundamental and delicate results that have been used or invoked earlier.

Lemma 6.3

Let $\Omega \subset \mathbb{R}^{N}$ be open and bounded, and ${\boldsymbol{u}}({\boldsymbol{x}})\in W^{1, N}_{\mathit{loc}}(\Omega ; \mathbb{R}^{N})$ such that $\det \nabla {\boldsymbol{u}}({\boldsymbol{x}})>0$ for a.e. ${\boldsymbol{x}}\in \Omega$. Then ${\boldsymbol{u}}$ is continuous, and monotonic in $\Omega$.

The definition of a monotonic mapping is technical and irrelevant for our purposes here. Check [11] or [13] for its precise definition. It is only required for the next result.

Lemma 6.4

Let $\Omega \subset \mathbb{R}^{N}$ be open and bounded. If either ${\boldsymbol{u}}({\boldsymbol{x}})\in W^{1, p}(\Omega ; \mathbb{R}^{N})$ for $p>N$, or ${\boldsymbol{u}}({\boldsymbol{x}})\in W^{1, N}(\Omega ; \mathbb{R}^{N})$ and it is monotonic, then ${\boldsymbol{u}}$ is a.e. differentiable in $\Omega$.

Global injectivity is a very delicate issue in non-linear elasticity. There are several fundamental and elaborate results under varying sets of assumptions. Some classical theorems can be found in [1], [7], [25]. These achieve the injectivity of mappings based on Dirichlet boundary conditions provided by fixed mappings that are one-to-one globally in all of the domain. The smooth, Lipschitz and Sobolev case (with growth exponent larger than dimension) are covered in those references. They require very precise sets of assumptions though.

The following is also a relevant result for us, as it yields a criterion of local invertibility for functions ${\boldsymbol{u}}$ in the Sobolev class $W^{1, N}(\Omega ; \mathbb{R}^{N})$ with $\det \nabla {\boldsymbol{u}}>0$ a.e. in $\Omega$. It is taken from [11]. It has been improved by lowering the exponent $p$ up to $N-1$ in papers like [5].

Theorem 6.5

Let $\Omega \subset \mathbb{R}^{N}$ be bounded and open, and let ${\boldsymbol{u}}\in W^{1, N}(\Omega ; \mathbb{R}^{N})$ be such that $\det \nabla {\boldsymbol{u}}({\boldsymbol{x}})>0$ for a.e. ${\boldsymbol{x}}\in \Omega$. Then ${\boldsymbol{u}}$ is locally invertible in the following sense: for a.e. ${\boldsymbol{x}}_{0}\in \Omega$, there is $r\equiv r({\boldsymbol{x}}_{0})>0$, an open set $\mathbf{D}\equiv \mathbf{D}({\boldsymbol{x}}_{0})\subset \subset \Omega$, and a function

$$ {\boldsymbol{v}}({\boldsymbol{y}}):\mathbf {B}_{r}({\boldsymbol{y}}_{0}) \to \mathbf{D},\quad {\boldsymbol{y}}_{0}={\boldsymbol{u}}({ \boldsymbol{x}}_{0}), $$

such that

$$\begin{gathered} {\boldsymbol{v}}\in W^{1, 1}(\mathbf {B}_{r}({\boldsymbol{y}}_{0})); \mathbb{R}^{N}), \\ {\boldsymbol{v}}\circ {\boldsymbol{u}}({\boldsymbol{x}})={ \boldsymbol{x}}\textit{ for a.e. }{\boldsymbol{x}}\in \mathbf{D}, \\ {\boldsymbol{u}}\circ {\boldsymbol{v}}({\boldsymbol{y}})={ \boldsymbol{y}}\textit{ for a.e. }{\boldsymbol{y}}\in \mathbf {B}_{r}({ \boldsymbol{y}}_{0}), \\ \nabla {\boldsymbol{v}}({\boldsymbol{y}})=\nabla {\boldsymbol{u}}({ \boldsymbol{v}}({\boldsymbol{y}}))^{-1}\textit{ for a.e. }{ \boldsymbol{y}}\in \mathbf {B}_{r}({\boldsymbol{y}}_{0}). \end{gathered}$$