zbMATH — the first resource for mathematics

Homogenization of nonconvex integral functionals and cellular elastic materials. (English) Zbl 0629.73009
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
Full Text:
References:
 [1] H. Attouch [1981]. Sur la {$$\Gamma$$}-convergence, in: H. Brézis and J. L. Lions (Eds.), Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. I, p. 7–41, Pitman, London. [2] H. Attouch [1984]. Variational convergence for functions and operators, Pitman, London. · Zbl 0561.49012 [3] J. M. Ball [1977]. Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal., 63, 337–403. · Zbl 0368.73040 [4] J. M. Ball & F. Murat [1984]. W1,p-quasiconvexity and variational problems for multiple integrals, J. Funct. Anal., 58, 225–253. · Zbl 0549.46019 [5] A. Bensoussan, J. L. Lions & G. Papanicolaou [1978]. Asymptotic analysis for periodic structures, North-Holland, Amsterdam. · Zbl 0404.35001 [6] A. Braides [1985]. Homogenization of some almost periodic coercive functional, Rend. Accad. Naz. Sci. XL, 103, 313–322. · Zbl 0582.49014 [7] L. Carbone & C. Sbordone [1979]. Some properties of {$$\Gamma$$}-limits of integral functionals, Annali Mat. Pura Appl., 122, 1–60. · Zbl 0474.49016 [8] D. Cioranescu & J. Saint Jean Paulin [1979]. Homogenization in open sets with holes, J. Math. Anal. Appl., 71, 590–607. · Zbl 0427.35073 [9] G. Dal Maso & L. Modica [1981]. A general theory of variational functionals, in: Topics in functional analysis 1980–81, by F. Strocchi, E. Zarantonello, E. De Giorgi, G. Dal Maso, L. Modica, Scuola Normale Superiore, Pisa, p. 149–221. [10] I. Ekeland & R. Temam [1974]. Analyse convexe et problèmes variationnels, Dunod, Gauthier-Villars. · Zbl 0281.49001 [11] N. Fusco [1983]. On the convergence of integral functionals depending on vector-valued functions, preprint no. 102, Univ. Napoli. · Zbl 0563.49007 [12] L. J. Gibson & M. F. Ashby [1982]. The mechanics of three-dimensional cellular materials, Proc. R. Soc. Lond., A 382, 43–59. [13] E. De Giorgi [1975]. Sulla convergenza di alcune successioni d’integrali del tipo dell’ area, Rend. Matsmatica, 8, 277–294. · Zbl 0316.35036 [14] E. De Giorgi [1979]. Convergence problems for functions and operators, Proc. Int. Meeting on ”Recent methods in nonlinear analysis”, Rene 1978, ed. E. De Giorgi, E. Magenes, Mosco Pitagora, Bologna, p. 131–188. [15] E. De Giorgi & G. Dal Maso [1983]. {$$\Gamma$$}-convergence and calculus of variations. In: J. P. Cecconi and T. Zolezzi (Eds.), Mathematical theories of optimization, Lecture Notes Maths. 979, Springer, Berlin, Heidelberg, New York. · Zbl 0511.49007 [16] E. De Giorgi & T. Franzoni [1975]. Su un tipo di convergenza variazionale, Atti. Accad. Naz. Lincei, Rend. Cl. Sc. Mat. (8), 58, 842–850. · Zbl 0339.49005 [17] A. N. Gent & A.G. Thomas [1959]. The deformation of foamed elastic materials, J. Appl. Polym. Sci., 1, 107–113. [18] D. Gilbarg & N. S. Trudinger [1983]. Elliptic partial differential equations of second order, 2nd edition, Springer, Berlin, Heidelberg, New York. · Zbl 0562.35001 [19] N. C. Hilyard (Ed.) [1982]. Mechanics of cellular plastics, Applied Science Publ., London. [20] P. Marcellini [1978]. Periodic solutions and homogenization of nonlinear variational problems, Annali Mat. Pura Appl. 117, 139–152. · Zbl 0395.49007 [21] Yu. G. Reshetnyak [1967]. On the stability of conformal mappings in multidimensional spaces, Siberian Math. J., 8, 69–85. · Zbl 0172.37801 [22] E. Sanchez-Palencia [1980]. Nonhomogeneous media and vibration theory, Lecture Notes in Physics, 127, Springer, Berlin, Heidelberg, New York. · Zbl 0432.70002 [23] I. Tartar [1977]. Cours Peccot au Collège de France, Paris.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.