Homogenization of nonconvex integral functionals and cellular elastic materials. (English) Zbl 0629.73009

Müller, Stefan

Arch. Ration. Mech. Anal. 99, 189-212 (1987).

In this paper the author considers a family of integral functionals depending on a parameter \(\epsilon >0\). The integrand W is a function of the variable space x and of the gradient of the desplacement u(x). u is a vector valued function. The assumptions on W are that it is measurable and periodic in x and it is nonconvex (in general) in the gradient of u, but with a polynomial growth. In the framework of \(\Gamma\)-convergence theory the asymptotic behaviour is studied, in order to obtain an explicit formulation of the limit problem. The author remarks that the convergence result can also be obtained as consequence of a recent theorem of A. Braides [Rend. Accad. Naz. Sci. Detta XL, V. Ser., Mem. Mat. 9, 313-322 (1985; Zbl 0582.49014)] using an approach based on abstract representation theorems from the theory of \(\Gamma\)-convergence.
In the next section convex integrands, which grow faster than a polynomial, are considered. A section is dedicated to the comparison with P. Marcellini’s formula to convex function and with polynomial growth [Ann. Mat. Pura Appl., IV. Ser. 117, 139-152 (1978; Zbl 0395.49007)].

Reviewer: M.Codegone

MSC:

74E05	Inhomogeneity in solid mechanics
74B20	Nonlinear elasticity
49J45	Methods involving semicontinuity and convergence; relaxation
74E10	Anisotropy in solid mechanics

Keywords:

gamma-convergence; family of integral functionals; nonconvex; polynomial growth; asymptotic behaviour; limit problem

Citations:

Zbl 0582.49014; Zbl 0395.49007

References:

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