ALEX 2018 Workshop: Abstracts

Mixed Lagrangian-Eulerian formulations

Ulisse Stefanelli

University of Vienna, Faculty of Mathematics (Austria)

In presence of finite strains, reference and actual configurations need to be distinguished. This is particularly relevant when mechanics is combined with other effects, requiring to simultaneously deal with both Lagrangian and Eulerian variables. Magnetoelastic materials are a first example in this direction, for the energy is defined in terms of deformation (Lagrangian) and magnetization (Eulerian). Other examples are nematic polymers, where the Eulerian variable is the nematic orientation, and piezoelectrics, which involve the Eulerian polarization instead. In fact, an interplay of Lagrangian and Eulerian effects occurs already in case of space-dependent forcings, as well as in some specific models of finite plasticity, where plastic deformations compose with the elastic ones. Mixed Lagrangian-Eulerian formulations arise in fluid-structure interaction, where the deformed body defines the (complement of the) fluid domain, and in solid-solid phase change, in case actual phase interphases are considered. I will present some classical and recent results on the topic.