Multiscale Modeling &Simulation, Volume 18, Issue 1, Page 163-197, January 2020.
Swelling media (e.g., gels, tumors) are usually described by mechanical constitutive laws (e.g., Hooke or Darcy laws). However, constitutive relations of real swelling media are not well-known. Here, we take an opposite route and consider a simple packing heuristics, i.e., the particles can't overlap. We deduce a formula for the equilibrium density under a confining potential. We then consider its evolution when the average particle volume and confining potential depend on time under two additional heuristics: (i) any two particles can't swap their position; (ii) motion should obey some energy minimization principle. These heuristics determine the medium velocity consistently with the continuity equation. In the direction normal to the potential level sets the velocity is related with that of the level sets, while in the parallel direction, it is determined by a Laplace--Beltrami operator on these sets. This complex geometrical feature cannot be recovered using a simple Darcy law.

中文翻译：

---

不可压缩膨胀材料的新连续谱理论

多尺度建模与仿真，第18卷，第1期，第163-197页，2020年1月。
溶胀介质（例如凝胶，肿瘤）通常由机械本构定律（例如胡克或达西定律）描述。但是，真正的膨胀介质的本构关系并不为人所知。在这里，我们采取相反的路线，并考虑简单的填充启发法，即粒子不能重叠。我们推导了一个约束电位下的平衡密度公式。然后，在另外两种启发式方法下，当平均粒子体积和约束潜力取决于时间时，我们考虑其演化：（i）任何两个粒子都不能交换位置；（ii）运动应遵循一些能量最小化原则。这些试探法与连续性方程一致地确定了介质速度。在垂直于电位水平集的方向上，速度与水平集的速度相关，而在平行方向上，则由这些集合上的Laplace-Beltrami运算符确定。使用简单的达西定律无法恢复这种复杂的几何特征。