25 research outputs found
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
Artificial boundary conditions on polyhedral truncation surfaces for three-dimensional elasticity systems
Artificial boundary conditions for the Stokes and Navier-Stokes equations in domains that are layer-like at infinity
QUIET FLOWS FOR STOKES AND NAVIER-STOKES PROBLEMS IN DOMAINS WITH CYLINDRICAL OUTLETS TO INFINITY
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