论文标题
宏观微弹性扩散系统建模吸收吸收引起的橡胶泡沫中的肿胀 - 强大的溶解度证明
A macro-micro elasticity-diffusion system modeling absorption-induced swelling in rubber foams -- Proof of the strong solvability
论文作者
论文摘要
在本文中，我们提出了一个宏观微微粒（两尺度）数学模型，用于描述由某些液体的微观吸收引起的橡胶泡沫的宏观肿胀。在我们的建模方法中，我们假设该材料占据了一个一维域，该结构域膨胀如标准梁方程所描述的那样，包括由液压确定的额外项。作为我们模型的特殊特征，吸收是通过较低的长度刻度在橡胶泡沫内部进行的，该尺度假定在这种结构化材料中固有地存在。材料内部液体的吸收和运输是通过源自宏观变形（梁方程的解决方案）定义的非圆柱形结构域中的非线性抛物线方程来建模的。在适当的假设下，我们建立了适合的溶液的存在和独特性，以将在微观非圆柱形域上构成的非线性抛物线方程与在宏观圆柱域上构成的光束方程相连。为了确保非圆柱形结构域的规律性，我们对出现在梁方程中出现的弹性响应函数施加了奇异性。
In this article, we propose a macro-micro (two-scale) mathematical model for describing the macroscopic swelling of a rubber foam caused by the microscopic absorption of some liquid. In our modeling approach, we suppose that the material occupies a one-dimensional domain which swells as described by the standard beam equation including an additional term determined by the liquid pressure. As special feature of our model, the absorption takes place inside the rubber foam via a lower length scale, which is assumed to be inherently present in such a structured material. The liquid's absorption and transport inside the material is modeled by means of a nonlinear parabolic equation derived from Darcy's law posed in a non-cylindrical domain defined by the macroscopic deformation (which is a solution of the beam equation). Under suitable assumptions, we establish the existence and uniqueness of a suitable class of solutions to our evolution system coupling the nonlinear parabolic equation posed on the microscopic non-cylindrical domain with the beam equation posed on the macroscopic cylindrical domain. In order to guarantee the regularity of the non-cylindrical domain, we impose a singularity to the elastic response function appearing in the beam equation.
