Maria Specovius-Neugebauer

Polarization matrices in anisotropic heterogeneous elasticity

Polarization matrices (or tensors) are generalizations of mathematical objects like the harmonic capacity or the virtual mass tensor. They participate in many asymptotic formulae with broad applications to problems of structural mechanics. In the present paper polarization matrices for anisotropic heterogeneous elastic inclusions are investigated, the ambient anisotropic elastic space is allowed to be inhomogeneous near the inclusion as well. By variational arguments the existence of unique solutions to the corresponding transmission problems is proved. Using results about elliptic problems in domains with a compact complement, polarization matrices can be properly defined in terms of certain coefficients in the asymptotic expansion at infinity of the solution to the homogeneous transmission problem. Representation formulae are derived from which properties like positivity or negativity can be read of directly. Further the behavior of the polarization matrix is investigated under small changes of the interface.

Optimal Control of Inclusion and Crack Shapes in Elastic Bodies

The paper is concerned with the control of the shape of rigid and elastic inclusions and crack paths in elastic bodies. We provide the corresponding problem formulations and analyze the shape sensitivity of such inclusions and cracks with respect to different perturbations. Inequality type boundary conditions are imposed at the crack faces to provide a mutual nonpenetration between crack faces. Inclusion and crack shapes are considered as control functions and control objectives, respectively. The cost functional, which is based on the Griffith rupture criterion, characterizes the energy release rate and provides the shape sensitivity with respect to a change of the geometry. We prove an existence of optimal solutions.

Artificial Boundary Conditions for the Stokes and Navier–Stokes Equations in Domains that are Layer-Like at Infinity

Artificial boundary conditions are presented to approximate solutions to Stokesand Navier-Stokes problems in domains that are layer-like at infinity. Based on results about existence and asymptotics of the solutions v∞, p∞ to the problems in the unbounded domain Ω the error v∞−vR, p∞−pR is estimated in H(ΩR) and L(ΩR), respectively. Here v, p are the approximating solutions on the truncated domain ΩR, the parameter R controls the exhausting of Ω. The artificial boundary conditions involve the Steklov-Poincare operator on a circle together with its inverse and thus turn out to be a combination of local and nonlocal boundary operators. Depending on the asymptotic decay of the data of the problems, in the linear case the error vanishes of order O(R−N ), where N can be arbitrarily large.

Crack propagation in anisotropic composite structures

A crack approaching a material interface between two elastic materials may stop or may advance by either penetrating the interface or deflecting into the interface (cf. N.Y. He and J.W. Hutchinson, Int. J. Solids Struct. 25 (1989), 1053-1067). Mathematical models for crack path prediction are based on the asymptotic behavior near the crack tip. In this work, an idea to compute the asymptotic expansion of the displacement field for structures composed of two dissimilar elastic anisotropic materials is shown, if the crack impinges the material interface. In contrast to the well-known case of isotropic materials, logarithmic terms can appear in the asymptotic decomposition of the displacement field in anisotropic composites. Based on this results, the energy release rate is calculated for different scenarios of crack propagation in composite structures.

Hölder continuous Young measure solutions to coercive non-monotone parabolic systems in two space dimensions

We consider parabolic systems u t  − div(a(∇u)) = f in two space dimensions where the elliptic part is derived from a potential and is coercive, but not monotone. With natural assumptions on the data we obtain the existence of a long-time Hölder continuous solution in the sense of Young measures.