SIAM Journal on Applied Mathematics, Volume 80, Issue 5, Page 2170-2193, January 2020.
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 nondimensionalization, leads to a scaled model involving a small parameter $\epsilon>0$, which is suitable for periodic-homogenization techniques. The problem is formally homogenized using the method of two-scale asymptotic expansions, which leads to a model of distributed-microstructure type in the limit. Finally, numerical simulations based on finite elements showcase the model behavior of the distributed-microstructure model.

中文翻译：

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具有微米级相分离的Cahn-Larché系统的多尺度建模和仿真

SIAM应用数学杂志，第80卷，第5期，第2170-2193页，2020年1月。
我们考虑了机械应力影响下的二元体系的相分离过程，并推导了一个数学多尺度模型，该模型描述了考虑到所涉及材料的弹性特性的不断发展的微观结构。在膜平衡实验中，通过在脂质单层中观察到的相分离过程，该模型的起点是Cahn-Hilliard方程和线性弹性方程，即所谓的Cahn-Larché系统。由于机械变形发生在宏观尺度上，而相分离发生在微观层面上，因此必须采用多尺度方法。我们假设正在演变的微观结构的模式具有与之相关的固有长度尺度，在进行无量纲化之后，导致包含小参数$ \ epsilon> 0 $的缩放模型，适用于周期均化技术。该问题使用两步渐近展开法在形式上均一化，这导致了极限范围内的分布微结构类型模型。最后，基于有限元的数值模拟显示了分布式微结构模型的模型行为。