Phase transitions with interfacial energy: interface null lagrangians, polyconvexity, and existence

For interfacial interactions of “separable type” the existence is proved of stable multiphase equilibrium states minimizing the total energy which includes a sharp interface contribution along interfaces separating the phases. The second gradients of deformation do not occur; the theory is based on interfacial null lagrangians as determined in [11–12]. The interfacial interaction is always of separable type if the number of phases does not exceed 3Û for the number of phases 3 4Ù the separable nature of the interface interaction is an assumption. 1 The interfacial energies We consider a body that can exist in states consisting of r inhomogeneous solid phases indexed by a ̈ 1ÙÜ Ù rØWe identify the body with the reference configuration represented by a bounded open set © ⊂ R with lipschitzian boundary. The states are pairs yÙP where y Ú © r Rn is a deformation function and P ̈ E1ÙÜ ÙEr is a partition of© into subsets Ea of © where Ea is the region occupied by phase aØ That one or several of the sets Ea is empty is not excluded. The total energy E yÙP of the state yÙP is given by E yÙP ̈ Eb yÙP + E if yÙP (1.1) where Eb yÙP and Eif yÙP are the bulk and interfacial energies defined as follows. The bulk energy is Eb yÙP ̈ r € a ̈1 ˆ Ea ta ∇y dL (1.2) where ta Ú Lin+ r R is the bulk free energy density of phase a expressed as a function of the deformation gradient Preprint, Institute of Mathematics, AS CR, Prague. 2009-6-30 I N T IT U TE of M ATH TICS A ca d em y o f Sc ie n ce s C ze ch R ep u b lic