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The stored-energy for some discontinuous deformations in nonlinear elasticity. (English) Zbl 0679.73006
Partial differential equations and the calculus of variations. Essays in Honor of Ennio De Giorgi, 767-786 (1989).
[For the entire collection see Zbl 0671.00007.]
In this work an idea, which Ennio De Giorgi demonstrated to be fruitful in various contexts, for example in the study of minimal surfaces or of $$\Gamma$$-convergence [Boll. Un. Mat. Ital., IV. Ser. 8, Suppl. to 2, 80-88 (1973; Zbl 0289.49042); Celocisl. Potoki minimal. Poverhn. 113-120 (1973; Zbl 0276.49031)], is applied to nonlinear elastic deformations which are subsets of $$R^ n$$, $$n\geq 2$$ (sic!). It concerns “estensione dello spazio ambiente”: just as it is efficacious first to seek surfaces of minimum area in the class BV of functions of bounded variation, rather than among the functions of class $$C^ 1$$, so in problems of $$\Gamma$$-convergence or existence of a minimizer in the calculus of variations it is efficacious to extend the function space $$C^ 1$$ to the Sobolev space $$H^{1,p}$$, or even to $$L^ p$$. (From author’s summary in Italian.)
The specific problem treated is that of cavitation produced at the centre of a unit ball due to its dilatation. Care is given to the extension of the domain of a specific stored-energy function defined on $$C^ 1$$ to the larger space.
Reviewer: J.Dunwoody

##### MSC:
 74B20 Nonlinear elasticity 74S30 Other numerical methods in solid mechanics (MSC2010) 49Q05 Minimal surfaces and optimization
##### Keywords:
cavitation; unit ball; dilatation; stored-energy function
##### Citations:
Zbl 0671.00007; Zbl 0289.49042; Zbl 0276.49031