Equilibrium for Multiphase Solids with Eulerian Interfaces

Abstract

We describe a general phase-field model for hyperelastic multiphase materials. The model features an elastic energy functional that depends on the phase-field variable and a surface energy term that depends in turn on the elastic deformation, as it measures interfaces in the deformed configuration. We prove existence of energy minimizing equilibrium states and -convergence of diffuse-interface approximations to the sharp-interface limit.

Bifurcation of elastic curves with modulated stiffness

“…As the volume constraint (14) passes to the limit under convergences ( 17) and (18), from ( 17), (19), and (20) we get that y ∈ Y(φ). Hence, owing to inequality (21) and the convergence (22) we conclude that (y, φ, V ) solves the topology optimization problem (16).…”

“…Remark 8 (Multiple phases). Although the model above deals only with two phases, an extension to a general multiphase material is possible in a similar way as in [4], or [18,28]. More precisely, one can describe the case of m ∈ N distinct phases by redefining…”

“…In particular, it his model the perimeters of interfaces in the reference and deformed configurations, as well as the deformations of lines in the referential interfaces are penalized. A more explicit characterization of interface polyconvexity can be found in [17,18], discussing the case of materials with more than two phases as well. Again, let us mention that the mathematical treatment of multiphase materials without surface-energy penalization typically leads to ill-posed problems where the existence of a solution is not necessarily guaranteed, and some relaxation is needed, cf.…”

Two phase models for elastic membranes with soft inclusions

“…We finally would like to point out that in the present contribution we focus on a situation with one of the phases being degenerate and Lagrangian surface energies that are given in the reference configuration. By way of contrast, models with non-degenerate bulk elasticity and jump contributions measured in the deformed configuration have been proposed in [Š10, Š11] and analyzed in [GKMS20]. Lagrangian models appropriately measure surface contributions, which on a molecular level result from boundary layers forming within the elastic matrix in the vicinity of voids even for large deformations.…”

Equilibria of charged hyperelastic solids

“…The membrane and Von Kármán regimes are the subject of [11] and [6], respectively. For energy functionals featuring both bulk and surface terms, as well as for refined phase-field models, we refer to [28], [35], and to the two recent contributions [19,20].…”