##
**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.]

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.]

Reviewer: Hervé Le Dret (Paris)

### MSC:

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 |

### Keywords:

three-dimensional hyperelasticity; minimize an elastic energy functional; generalized diffeomorphisms; subspaces of the space of rectifiable currents; Sobolev spaces; weak sequential closedness results; existence theorems; globally invertible minimizers; coercivity hypotheses; cavitation
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\textit{M. Giaquinta} et al., Arch. Ration. Mech. Anal. 106, No. 2, 97--159 (1989; Zbl 0677.73014)

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