WIAS Preprint No. 1311, (2008)

A study on the eigenstrain problem in solid mixtures



Authors

  • Dreyer, Wolfgang
  • Duderstadt, Frank
  • Kimmerle, Sven-Joachim

2010 Mathematics Subject Classification

  • 74-99 74A10 74E05 74F05 74F10 74F20 74N25

Keywords

  • misfit, inclusions, St. Venant-Kirchhoff law, elasticity, inelastic deformation, change of reference configuration, intermediate configuration, thermal, expansion, diffusion, phase transition

Abstract

We introduce a framework that is capable to model the appearance of mechanical stresses due to inelastic deformations. Among these we consider in particular thermal expansions, diffusion and phase transitions. Among the quantities of central importance are the eigenstrain and the misfit strain. They describe the phenomenon that different material volumes of a compact body may not be compatible to each other in a stress-free reference configuration, so that here a compact body may not exist. We shall show that it is possible to find a further reference configuration, where the body is compact but not free of stress. A typical example where misfit appears concerns a body whose local parts differently transform their phase. This might be a change of the crystal lattice from the ferrite to the austenite symmetry in steel, or the formation of liquid droplets in crystalline gallium arsenide. In both cases the new interior phase has with respect to the parent phase different volume or shape in its state that is free of stress. In this study we consider the eigenstrain problem for pure substances as well as for mixtures. In the latter case subtle arguments are needed for an appropriate description. Special focus is given to the equivalence of interface boundaries with discontinues and continues displacement vectors.