Reinforcement problems in the calculus of variations. (English) Zbl 0607.73018

Acerbi, Emilio; Buttazzo, Giuseppe

Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3, 273-284 (1986).

The authors investigate the torsion of an elastic bar which is surrounded by an increasingly thin layer with an increasingly hard material. The equations of this model problem may be fully nonlinear. Three different expressions of the limit problem depending on the link between thickness and hardness are obtained. The direct methods of the calculus of variations and \(\Gamma\)-convergence allow to give some answers even in the fully nonlinear case. Paper is important for mathematicians interested in reinforcement, \(\Gamma\)-convergence, integral functionals, and non-equicoercive problems.

Reviewer: H.Bufler

MSC:

74S30	Other numerical methods in solid mechanics (MSC2010)
49J45	Methods involving semicontinuity and convergence; relaxation
74K10	Rods (beams, columns, shafts, arches, rings, etc.)

Keywords:

Gamma convergence; torsion of an elastic bar; limit problem; link between thickness and hardness; fully nonlinear; reinforcement; integral functionals; non-equicoercive problems

References:

[1]	Attouch, H., Variational convergence for functions and operators, Applicable Math. Series (1984), Pitman: Pitman London · Zbl 0561.49012
[2]	Brezis, H.; Caffarelli, L.; Friedman, A., Reinforcement problems for elliptic equations and variational inequalities, Ann. Mat. Pura Appl., t. 123, 4, 219-246 (1980) · Zbl 0434.35079
[3]	Caffarelli, L.; Friedman, A., Reinforcement problems in elastoplasticity, Rocky Mountain J. Math., t. 10, 155-184 (1980) · Zbl 0452.73030
[4]	De Giorgi, E.; Dal Maso, G., Γ-convergence and calculus of variations. Mathematical theories of optimization, (Cecconi, J. P.; Zolezzi, T., Lecture Notes Math, t. 979 (1983), Springer: Springer Berlin), Proceedings, 1981
[5]	Sanchez-Palencia, E., Non-homogeneous media and vibration theory, Lecture Notes Phys., t. 127 (1980), Springer: Springer Berlin · Zbl 0432.70002