On the change of energy caused by crack propagation in 3-dimensional anisotropic solids

summary:Crack propagation in anisotropic materials is a persistent problem. A general concept to predict crack growth is the energy principle: A crack can only grow, if energy is released. We study the change of potential energy caused by a propagating crack in a fully three-dimensional solid consisting of an anisotropic material. Based on methods of asymptotic analysis (method of matched asymptotic expansions) we give a formula for the decrease in potential energy if a smooth inner crack grows along a small crack extension

Hodge-Helmholtz Decompositions of Weighted Sobolev Spaces in Irregular Exterior Domains with Inhomogeneous and Anisotropic Media

We study in detail Hodge-Helmholtz decompositions in non-smooth exterior domains filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms belonging to weighted Sobolev spaces into irrotational and solenoidal forms. These decompositions are essential tools, for example, in electro-magnetic theory for exterior domains. In the appendix we translate our results to the classical framework of vector analysis.Comment: Key Words: Hodge-Helmholtz decompositions, Maxwell's equations, electro-magnetic theory, weighted Sobolev space

ASYMPTOTIC ANALYSIS AND POLARIZATION MATRICES

Abstract. Polarization matrices are considered for the elasticity boundary value problems in two and three spatial dimensions. The matrices are introduced in the framework of asymptotic analysis for boundary value problems depending on small geometrical parameter, it is the size of an elastic inclusion or a defect (cavity, crack) in an elastic body. Our analysis is performed for some representative classes of boundary value problems, however the method is general and can be applied to the modelling and optimization in structural mechanics or for coupled models like piezoelectricity. The explicit properties obtained for polarization matrices are useful for mathematical analysis and for numerical solution of control, inverse and shape optimization problems with mathematical models derived by the asymptotic analysis in singularly perturbed geometrical domains. The analysis is performed by some different techniques including asymptotics in unbounded domains, singular perturbations and shape sensitivity. In particular, since the polarization matrices can be identified for some classes of shapes, we provide the formulae for numerical evaluation of such matrices for nearby shapes by means of the shape sensitivity analysis. 1