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Injectivity and self-contact in nonlinear elasticity. (English) Zbl 0628.73043
Self-contact may occur in any traction problem within the theory of large deformations of (hyperelastic) bodies; in fact, self-contact may lead to non-trivial equilibrium placements even when the body is totally unloaded and under no condition of place.
The authors were the first to adress the problem of self-contact in hyperelasticity; their results, announced in C. R. Acad. Sci., Paris, Sér. I 301, 621-624 (1985; Zbl 0585.73018) are proved here in full. Variational methods are used and the solution of the mixed problem under dead body load and zero surface traction is taken to be the global minimizer of the total energy (thus alternative solutions corresponding to local minima are not considered). The stored-energy function w is supposed to satisfy Ball’s conditions of polyconvexity, coerciveness and singularity \((W\to +\infty\) as det \(F\to 0+\); \(F=\nabla \phi\) being the displacement gradient). Interpenetration is denied by the condition \(\int_{\Omega}\det F dx\leq vol \phi (\Omega)\) imposed upon all allowable functions (here \(\Omega\) is the set occupied by the body in the reference placement); in fact, by a classical formula, \(\int_{\phi (\Omega)}card \phi^{-1}(x')dx'=\int_{\Omega}\det Fdx\). Actually, as the negation of interpenetration introduces anyway a sort of unilateral constraint, the authors impose on the body also a condition of confinement without frictin (\(\phi\) (\({\bar \Omega}\)) must belong to a closed set with smooth boundary).
The main theorem assures the existence of at least one minimizer; other theorems describe properties of the minimizer under the presumption that it be sufficiently smooth (for instance card \(\phi\) \({}^{-1}(x')=1\) except when x’\(\in \phi (\partial \Omega)\cap int B\), and then card \(\phi\) \({}^{-1}(x')=1\) or 2; in the latter case there are two opposite outer normal vectors to \(\phi\) (\(\partial \Omega)\) at x’).
Reviewer: G.Capriz

MSC:
74B20 Nonlinear elasticity
74S30 Other numerical methods in solid mechanics (MSC2010)
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