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Description:
We consider the process of phase separation of a binary system under the influence of mechanical stress and we derive a mathematical multiscale model, which describes an evolving microstructure taking into account the elastic properties of the involved materials. Motivated by phase-separation processes observed in lipid monolayers in film-balance experiments, the starting point of the model is the Cahn–Hilliard equation coupled with the equations of linear elasticity, the so-called Cahn–Larché system. Owing to the fact that the mechanical deformation takes place on a macrosopic scale whereas the phase separation happens on a microscopic level, a multiscale approach is imperative. We assume the pattern of the evolving microstructure to have an intrinsic length scale associated with it, which, after non-dimensionalisation, leads to a scaled model involving a small parameter ɛ > 0, which is suitable for periodic-homogenisation techniques. Furthermore, we present a linearised Cahn–Larché system. For the associated ɛ-dependent problem, we proof the existence and uniqueness of a weak solution by a Galerkin approach, for every ɛ > 0. As discretisation in space leads to a linear differential–algebraic system of equations, we apply solution theory for such equations in a weak setting. A-priori estimates enable us to homogenise the linear system rigorously using the concept of two-scale convergence. The full nonlinear problem is formally homogenised using the method of two-scale asymptotic expansion. Both systems leads to models of distributed-microstructure type in the limit. Properties of the limit models are discussed. Finally, numerical simulations based on a finite-element approach are considered to showcase the model behaviour of the nonlinear distributed-microstructure model. ; Wir betrachten den Prozess der Phasenseparation eines binären Systems unter Einfluss von mechanischer Spannung und leiten ein mathematisches Multiskalenmodell her, das die Entwicklung einer Mikrostruktur unter Berücksichtigung der ...