Shim Jay J, Ateshian Gerard A
Department of Mechanical Engineering, Columbia University, New York, NY 10027.
Arch Appl Mech. 2022 Feb;92(2):491-511. doi: 10.1007/s00419-020-01851-8. Epub 2021 Jan 7.
The primary aim of this study is to establish the theoretical foundations for a solid-fluid biphasic mixture domain that can accommodate inertial effects and a viscous interstitial fluid, which can interface with a dynamic viscous fluid domain. Most mixture formulations consist of constituents that are either all intrinsically incompressible or compressible, thereby introducing inherent limitations. In particular, mixtures with intrinsically incompressible constituents can only model wave propagation in the porous solid matrix, whereas those with compressible constituents require internal variables, and related evolution equations, to distinguish the compressibility of the solid and fluid under hydrostatic pressure. In this study, we propose a hybrid framework for a biphasic mixture where the skeleton of the porous solid is intrinsically incompressible but the interstitial fluid is compressible. We define a state variable as a measure of the fluid volumetric strain. Within an isothermal framework, the Clausius-Duhem inequality shows that a function of state arises for the fluid pressure as a function of this strain measure. We derive jump conditions across hybrid biphasic interfaces, which are suitable for modeling hydrated biological tissues. We then illustrate this framework using confined compression and dilatational wave propagation analyses. The governing equations for this hybrid biphasic framework reduce to those of the classical biphasic theory whenever the bulk modulus of the fluid is set to infinity and inertia terms and viscous fluid effects are neglected. The availability of this novel framework facilitates the implementation of finite element solvers for fluid-structure interactions at interfaces between viscous fluids and porous-deformable biphasic domains, which can include fluid exchanges across those interfaces.
本研究的主要目的是为一个能考虑惯性效应和粘性孔隙流体的固液双相混合域建立理论基础，该混合域可与动态粘性流体域相衔接。大多数混合公式所包含的组分要么本质上都是不可压缩的，要么都是可压缩的，从而带来了固有的局限性。特别是，由本质上不可压缩的组分构成的混合物只能模拟多孔固体基质中的波传播，而由可压缩组分构成的混合物则需要内部变量及相关的演化方程来区分固体和流体在静水压力下的可压缩性。在本研究中，我们提出了一种双相混合物的混合框架，其中多孔固体的骨架本质上是不可压缩的，但孔隙流体是可压缩的。我们定义一个状态变量来衡量流体的体积应变。在等温框架内，克劳修斯 - 杜亥姆不等式表明，流体压力作为该应变度量的函数会产生一个状态函数。我们推导了适用于模拟水合生物组织的混合双相界面处的跳跃条件。然后，我们通过受限压缩和膨胀波传播分析来说明这个框架。只要将流体的体积模量设为无穷大，并忽略惯性项和粘性流体效应，这个混合双相框架的控制方程就会简化为经典双相理论的方程。这种新颖框架的可用性有助于在粘性流体与多孔可变形双相域之间的界面处实现用于流固相互作用的有限元求解器，其中可以包括跨这些界面的流体交换。
