Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity. (English) Zbl 0677.73014

The authors consider several new classes of generalized (or weak) diffeomorphisms of an open set \(\Omega\) in \(\mathbb{R}^n\). Their main goal, in the particular case of three-dimensional hyperelasticity, is to minimize an elastic energy functional of the form \[ F(u;\Omega)=\int_{\Omega}W(x,Du)\,dx, \] where \(W\) is a polyconvex function satisfying appropriate coercivity conditions, in such a way that the minimizers thus found be mappings that are globally invertible in some sense, i.e. deformations that are admissible from the viewpoint of mechanics.
The generalized diffeomorphisms are introduced as special subspaces of the space of rectifiable currents in the sense of geometric measure theory [cf. H. Federer, Geometric measure theory. Berlin etc.: Springer Verlag (1969; Zbl 0176.00801)]. Usual diffeomorphisms are identified with the current \(T_u\) that is integration on the graph of \(u\). This point of view allows u and its inverse \(u^{-1}\) to play a symmetrical role. The first part of the paper is devoted to the description and properties of various classes of such currents that generalize the usual Sobolev spaces \(W^{1/p}(\Omega;\mathbb{R}^n)\) and still have diffeomorphism-like properties. Keeping in mind applications to the calculus of variations, the authors prove several weak sequential closedness results in these classes. It should be noted that some of these results of weak continuity of determinants are proved in a more elementary fashion by S. Müller [C.R. Acad. Sci. Paris, Sér. I 307, No. 9, 501–506 (1988; Zbl 0679.34051)].
Application of these results to three-dimensional elasticity yields existence theorems with globally invertible minimizers under coercivity hypotheses such as \[ | M(F)| =^{def}\{1+| F|^ 2+| \operatorname{Adj} F|^ 2+| \det F|^ 2\}^{1/2}, \]
\[ W(x,F)\geq | M(F)|^p+(| M(F)|^q/| \det F|^{q-1}). \] Furthermore, the equilibrium equations in the deformed configuration are shown to be satisfied as well as conservation of energy. Finally, cavitation in elasticity is studied, cf. (*) J. M. Ball [Philos. Trans. R. Soc., A 306, 557–611 (1982; Zbl 0513.73020)]. It is shown that it cannot happen in the present framework, and the connection with (*) is discussed in detail.
[Note that the authors have recently written an Erratum-Addendum to the present paper.]


74B20 Nonlinear elasticity
74S30 Other numerical methods in solid mechanics (MSC2010)
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
49Q99 Manifolds and measure-geometric topics
58A15 Exterior differential systems (Cartan theory)
49Q20 Variational problems in a geometric measure-theoretic setting
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