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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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##### References:
 [1] Federer, Geometric Measure Theory (1969) · Zbl 0176.00801 [2] DOI: 10.1016/0022-1236(81)90085-9 · Zbl 0459.35020 [3] DOI: 10.1007/BF00279992 · Zbl 0368.73040 [4] Antman, Arch. Rational Mech. Anal. 61 pp 353– (1976) [5] Neumann, Functional Operators 1 (1950) [6] Temam, Problèmes Mathematiques en Plasticité (1983) [7] Schwartz, Cours d’Analyse (1967) [8] Necas, Les Methodes Directes en Théories des Equations Elliptiques (1967) [9] Morrey, Multiple Integrals in the Calculus of Variations (1966) · Zbl 0142.38701 [10] DOI: 10.1098/rsta.1982.0095 · Zbl 0513.73020 [11] DOI: 10.1090/S0002-9904-1973-13319-1 · Zbl 0275.49041 [12] DOI: 10.1007/BF00281557 · Zbl 0589.73017 [13] Dieudonne, Treatise on Analysis (1972) [14] Ciarlet, C. R. Acad. Sci. Paris, Ser. A 301 pp 621– (1985) [15] DOI: 10.1007/BF00250917 · Zbl 0557.73009 [16] Ciarlet, Elasticite Tridimensionnelle (1985) [17] Ball, Proc. Roy. Soc. Edinburgh Sect. A 88 pp 315– (1981) · Zbl 0478.46032
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