197 research outputs found

    Geometry of polycrystals and microstructure

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    We investigate the geometry of polycrystals, showing that for polycrystals formed of convex grains the interior grains are polyhedral, while for polycrystals with general grain geometry the set of triple points is small. Then we investigate possible martensitic morphologies resulting from intergrain contact. For cubic-to-tetragonal transformations we show that homogeneous zero-energy microstructures matching a pure dilatation on a grain boundary necessarily involve more than four deformation gradients. We discuss the relevance of this result for observations of microstructures involving second and third-order laminates in various materials. Finally we consider the more specialized situation of bicrystals formed from materials having two martensitic energy wells (such as for orthorhombic to monoclinic transformations), but without any restrictions on the possible microstructure, showing how a generalization of the Hadamard jump condition can be applied at the intergrain boundary to show that a pure phase in either grain is impossible at minimum energy.Comment: ESOMAT 2015 Proceedings, to appea

    Comparison results for the Stokes equations

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    This paper enfolds a medius analysis for the Stokes equations and compares different finite element methods (FEMs). A first result is a best approximation result for a P1 non-conforming FEM. The main comparison result is that the error of the P2-P0-FEM is a lower bound to the error of the Bernardi-Raugel (or reduced P2-P0) FEM, which is a lower bound to the error of the P1 non-conforming FEM, and this is a lower bound to the error of the MINI-FEM. The paper discusses the converse direction, as well as other methods such as the discontinuous Galerkin and pseudostress FEMs. Furthermore this paper provides counterexamples for equivalent convergence when different pressure approximations are considered. The mathematical arguments are various conforming companions as well as the discrete inf-sup condition