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Almost-everywhere injectivity in nonlinear elasticity. (English) Zbl 0656.73010
This article presents an extension of the result of P. G. Ciarlet and J. Nečas [Arch. Ration. Mech. Anal. 97, 171-188 (1987; Zbl 0628.73043)] on global invertibility in nonlinear elasticity to the case when the deformation u: \(\Omega\) \(\subset {\mathbb{R}}^ n\to {\mathbb{R}}^ n\) is less regular than \(W^{1,n}\). The author deals with deformations that lie in the sets \(A^+_{p,q}(\Omega)=\{v\in W^{1,p}(\Omega;{\mathbb{R}}^ n)\), adj\(\nabla v\in L^ q(\Omega;{\mathbb{R}}^{n^ 2})\), \(\det \nabla v>0\) a.e. in \(\Omega\) \(\}\) where \(p>n-1\) and \(q>p/(p-1)\). He gives a generalization of Ciarlet and Nečas’ condition, \[ (1)\quad Vol(u(\Omega))\geq \int_{\Omega}\det \nabla u(x)dx, \] which is valid for such deformations. The derivation of this generalized condition relies heavily upon the work of V. Šverák [ibid. 100, No.2, 105-127 (1988)] on the regularity properties of elements of \(A^+_{p,q}(\Omega)\), which allows in particular for an adequate definition of the “image” of u, Ê(u,\(\Omega)\). Ciarlet and Nečas’ condition becomes: \[ (2)\quad {\mathcal L}^ n(\hat E(u(\Omega))\geq \int_{\Omega}\det \nabla u(x)dx. \] The author then studies several properties implied by (2). He shows, among other things, that (2) is preserved under weak convergence in \(A^+_{p,q}(\Omega)\). The existence of an almost-everywhere defined inverse (in an appropriate sense) to u is then shown to follow from (2), again by building on Šverák’s results, and its regularity is studied. In a final section, the author applies his results to the minimization of the elastic energy of a body with given loads and boundary conditions, with a polyconvex stored energy function W(F) whose coerciveness agrees with the definition of \(A^+_{p,q}(\Omega)\). He proves the existence of a minimizer in \(A^+_{p,q}(\Omega)\) subjected to the constraints (2) and Ê(u(\(\Omega)\))\(\subset B\) (a confinement condition). This minimizer is thus almost-everywhere injective and formally satisfies noninterpenetration of matter while allowing self-contact of the boundary without friction.
Reviewer: H.Le Dret

MSC:
74B20 Nonlinear elasticity
74S30 Other numerical methods in solid mechanics (MSC2010)
49J27 Existence theories for problems in abstract spaces
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