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Energies in SBV and variational models in fracture mechanics. (English) Zbl 0904.73045
Cioranescu, Doina (ed.) et al., Homogenization and applications to material sciences. Proceedings of the international conference, Nice, France, June 6–10, 1995. Tokyo: Gakkotosho. GAKUTO Int. Ser., Math. Sci. Appl. 9, 1-22 (1995).
One of the fundamental point of fracture mechanics is Griffith’s criterium of fracture related to the surface energy of crack. This criterion has been applied by mechanicians without sufficiently rigorous mathematical background. In the present paper the authors discuss Griffith’s theory as a free surface variational problem. It is natural that the authors discuss a weak version of the problem. In order to overcome many difficulties appearing in this approach, the authors introduce a special class of functions of bounded variation SBV. Theorems of existence based on compactness and semi-continuity of SBV functions are proven. The main attention is paid to fracture of composite materials, Barenblatt-type materials, elastic and Hecky’s materials.
For the entire collection see [Zbl 0873.00028].

74R99 Fracture and damage
74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
26A45 Functions of bounded variation, generalizations