## Fachbereich Mathematik

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#### Year of publication

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#### Keywords

- Continuum mechanics (1)
- Elastoplasticity (1)
- Elastoplastizität (1)
- Geostrophic flow (1)
- Homotopiehochhebungen (1)
- Homotopy lifting (1)
- Hysterese (1)
- Injectivity of mappings (1)
- Injektivität von Abbildungen (1)
- Jiang's constitutive model (1)
- Jiangsches konstitutives Gesetz (1)
- Kontinuumsmechanik (1)
- Nichtlineare/große Verformungen (1)
- Nonlinear/large deformations (1)
- Ratenunabhängigkeit (1)
- Variationsungleichungen (1)
- hysteresis (1)
- local approximation of sea surface topography (1)
- locally supported (Green’s) vector wavelets (1)
- rate-independency (1)
- variational inequalities (1)

In this article, we present an analytic solution for Jiang's constitutive model of elastoplasticity. It is considered in its stress controlled form for proportional stress loading under the assumptions that the one-to-one coupling of the yield surface radius and the memory surface radius is switched off, that the transient hardening is neglected and that the ratchetting exponents are constant.

In this article we give a sufficient condition that a simply connected flexible body does not penetrate itself, if it is subjected to a continuous deformation. It is shown that the deformation map is automatically injective, if it is just locally injective and injective on the boundary of the body. Thereby, it is very remarkable that no higher regularity assumption than continuity for the deformation map is required. The proof exclusively relies on homotopy methods and the Jordan-Brouwer separation theorem.

This paper deals with the problem of determining the sea surface topography from geostrophic flow of ocean currents on local domains of the spherical Earth. In mathematical context the problem amounts to the solution of a spherical differential equation relating the surface curl gradient of a scalar field (sea surface topography) to a surface divergence-free vector field(geostrophic ocean flow). At first, a continuous solution theory is presented in the framework of an integral formula involving Green’s function of the spherical Beltrami operator. Different criteria derived from spherical vector analysis are given to investigate uniqueness. Second, for practical applications Green’s function is replaced by a regularized counterpart. The solution is obtained by a convolution of the flow field with a scaled version of the regularized Green function. Calculating locally without boundary correction would lead to errors near the boundary. To avoid these Gibbs phenomenona we additionally consider the boundary integral of the corresponding region on the sphere which occurs in the integral formula of the solution. For reasons of simplicity we discuss a spherical cap first, that means we consider a continuously differentiable (regular) boundary curve. In a second step we concentrate on a more complicated domain with a non continuously differentiable boundary curve, namely a rectangular region. It will turn out that the boundary integral provides a major part for stabilizing and reconstructing the approximation of the solution in our multiscale procedure.