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Energy change due to the appearance of cavities in elastic solids. (English) Zbl 1035.74009

Summary: The paper presents an overview of the problem of assessing the increment of strain energy due to the appearance of small cavities in elastic solids. The following approaches are discussed: the compound asymptotic Mazja method, the Eshelby-like method used in classical works on mechanics of composites, the homogenization method, and the topological derivative method. The increment of energy is expressed by a quadratic form with respect to strains referring to the virgin solid. All the methods lead to the same formula for the increment of energy. It is expressed by a quadratic form with respect to strains referring to the virgin solid. This quadratic form turns out to be unconditionally positive definite. Explicit formulae are derived for an elliptical hole and for a spherical cavity. The results derived determine the characteristic function of bubble method of optimal shape design for elastic two- and three-dimensional structures.

MSC:

74B99 Elastic materials
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