6533b85bfe1ef96bd12baa8e

RESEARCH PRODUCT

Generalized inverse of the compliance tensor, and behaviour of incompressible anisotropic materials - Application to damage

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Before the final rupture, most structural materials exhibit an import damage in the form of microvoids. The overall behaviour of a damaged elastic material depends on the void volume fraction f. Undamaged polymers are generally considered as incompressible elastic. Metals at high temperature may be considered as linearly viscoplastic. Thus the undamaged material may be described by an incompressible behaviour, while the overall behaviour of the damaged material is compressible depending on the void volume fraction. The transition from a compressible to an incompressible behaviour leads to a singular compliance matrix and an undefined rigidity matrix. The generalized inverse of the compliance matrix of the damaged and undamaged material shows to be of primary interest to study both incompressible and compressible materials with the same formulation. We show that incompressible behaviour of anisotropic elastic materials can be formulated in terms of generalized inverse of the compliance matrix. When incompressibility conditions are required, the compliance matrix of a material with known moduli becomes singular. The Moore-Penrose generalized inverse can help us to formulate Hooke’s law for such materials. Examples of such relations are given for isotropic, transversally isotropic and orthotropic materials.