OxPDE Lunchtime Seminar
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Wed, 27/02/2008 12:00 |
Benson Muite |
OxPDE Lunchtime Seminar  |
Gibson 1st Floor SR |
| We describe how to perform high resolution simulations of viscoelastic continuum mechanical models for martensitic transformations with diffuse interfaces. The computational methods described may also be of use in performing high resolution simulations of time dependent partial differential equations where solutions are sufficiently smooth. | |||
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Thu, 28/02/2008 10:00 |
Patrizio Neff |
OxPDE Lunchtime Seminar |
Gibson 1st Floor SR |
| We are concerned with the derivation of the Γ-limit to a three dimensional geometrically exact Cosserat model as the relative thickness h > 0 of a at domain tends to zero. The Cosserat bulk model involves already exact rotations as a second independent field and this model is meant to describe defective elastic crystals liable to fracture under shear. It is shown that the Γ-limit based on a natural scaling assumption con- sists of a membrane like energy contribution and a homogenized transverse shear energy both scaling with h, augmented by an additional curvature stiffness due to the underlying Cosserat bulk formulation, also scaling with h. No specific bending term appears in the dimensional homogenization process. The formulation exhibits an internal length scale Lc which sur- vives the homogenization process. A major technical difficulty, which we encounter in applying the Γ-convergence arguments, is to establish equi- coercivity of the sequence of functionals as the relative thickness h tends to zero. Usually, equi-coercivity follows from a local coerciveness assump- tion. While the three-dimensional problem is well-posed for the Cosserat couple modulus μc ≥ 0, equi-coercivity forces us to assume a strictly pos- itive Cosserat couple modulus μc > 0. The Γ-limit model determines the midsurface deformation m ∈ H1,2(ω;R3). For the case of zero Cosserat couple modulus μc= 0 we obtain an estimate of the Γ - lim inf and Γ - lim sup, without equi-coercivity which is then strenghtened to a Γ- convergence result for zero Cosserat couple modulus. The classical linear Reissner-Mindlin model is "almost" the linearization of the Γ-limit for μc = 0 apart from a stabilizing shear energy term. | |||
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Thu, 06/03/2008 12:00 |
Sorin Mardare |
OxPDE Lunchtime Seminar |
Gibson 1st Floor SR |
| As all analysts know, solving a problem which contains some kind of nonlinearity is generally far from being obvious. This is of course the case when we deal with nonlinear PDEs, but also when we have linear systems with some nonlinear compatibility conditions. One example is constituted by the systems of first order linear partial differential equations. If the coefficients are regular, there is a classical way to solve this systems. However, if the coefficients are only of class Lp, the classical methods cannot be applied. We will show how this problem can be solved by using a method of regularization which allows to preserve the nonlinear compatibility conditions. Then we will present some possible applications to the theory of elasticity. In the end some open problems related to similar aspects will be discussed. Example of such a problem: the rigidity of deformations of class H1. | |||
