A finite element implementation of finite deformation surface and bulk poroelasticity

Abstract

We present a theoretical and computational model for the behavior of a porous solid undergoing two interdependent processes, the finite deformation of a solid and species migration through the solid, which are distinct in bulk and on surface. Nonlinear theories allow us to systematically study porous solids in a wide range of applications, such as drug delivery, biomaterial design, fundamental study of biomechanics and mechanobiology, and the design of sensors and actuators. As we aim to understand the physical phenomena at a smaller length scale, towards comprehending fundamental biological processes and miniaturization of devices, surface effect becomes more pertinent. Although existing methodologies provide the necessary tools to study coupled bulk effects for deformation and diffusion; however, very little is known about fully coupled bulk and surface poroelasticity at finite strain. Here we develop a thermodynamically consistent formulation for surface and bulk poroelasticity, specialized for soft hydrated solids, along with a corresponding finite element implementation that includes a three-field weak form. Our approach captures the interplay between competing multiphysical processes of finite deformation and species diffusion, accounting for surface kinematics and surface transport, and provides invaluable insight when surface effects are important.

Appendices

Appendix A: Derivation of weak form of species balance

Staring form Eq. (14a),

$$\begin{aligned} \int _{ V } \dot{ C }\,\delta \mu \, \textrm{d}V + \int _{ V } \left( \varvec{\nabla }_{{\textbf{X}}}\cdot {\textbf{J}}\right) \delta \mu \, \textrm{d}V = 0 \end{aligned}$$

(A1)

By applying the product rule and divergence theorem [29], and then substituting Eq. (14b) into Eq. (A1),

$$\begin{aligned}{} & {} \int _{ V } \dot{ C }\,\delta \mu \, \textrm{d}V - \int _{ V } {\textbf{J}}\cdot \varvec{\nabla }_{{\textbf{X}}}\delta \mu \, \textrm{d}V \nonumber \\{} & {} \quad + \int _{ S } \dot{\widetilde{ C }}\,\delta \mu \, \textrm{d}S + \int _{ S } \left( \widetilde{\varvec{\nabla }}_{\widetilde{{{\textbf {X}}}}}\cdot \widetilde{{\textbf{J}}}\right) \delta \mu \, \textrm{d}S = 0 \end{aligned}$$

(A2)

where the last term in Eq. (A2) can be rewritten by product rule and surface divergence theorem,

$$\begin{aligned}{} & {} \int _{ S } \left( \widetilde{\varvec{\nabla }}_{\widetilde{{{\textbf {X}}}}}\cdot \widetilde{{\textbf{J}}}\right) \delta \mu \, \textrm{d}S = - \int _{ S } \widetilde{{\textbf{J}}}\cdot \widetilde{\varvec{\nabla }}_{\widetilde{{{\textbf {X}}}}}\delta \mu \, \textrm{d}S \nonumber \\{} & {} \quad + \int _{ L } \left( \widetilde{{\textbf{J}}}\cdot \widetilde{{\textbf{N}}}\right) \delta \mu \, \textrm{d}L \end{aligned}$$

(A3)

where the second term in the right-hand side of Eq. (A3) is assumed to be zero. By substituting Eq. (A3) into Eq. (A2),

$$\begin{aligned}{} & {} \int _{ V } \dot{ C }\,\delta \mu \, \textrm{d}V - \int _{ V } {\textbf{J}}\cdot \varvec{\nabla }_{{\textbf{X}}}\delta \mu \, \textrm{d}V \nonumber \\{} & {} \quad + \int _{ S } \dot{\widetilde{ C }}\,\delta \mu \, \textrm{d}S - \int _{ S } \widetilde{{\textbf{J}}}\cdot \widetilde{\varvec{\nabla }}_{\widetilde{{{\textbf {X}}}}}\delta \mu \, \textrm{d}S = 0 \end{aligned}$$

(A4)

This equation corresponds to Eq. (34b) in the manuscript.

Appendix B: Convergence study on cube-contracting example

We perform the convergence study to demonstrate the simulation results and mesh resolution (see Fig. 11) on cube-contracting example in Sect. 5.1. During the surface energy ramping phase ($t /\tau < 10^{0}$), the bulk gains species, leading to a rapid increase in normalized diameters. While the surface energy is fixed ($t /\tau \ge 10^{0}$), the species continue to migrate from the surface into the bulk due to the chemical potential, resulting in a gradual increase in normalized diameter. The diameters stop increasing once the chemical potentials reach equilibrium ($t /\tau \sim 1.6\times 10^{5}$). It is important to note that the normalized diameters at multiple normalized times approach the reference values. Here we confirm a convergence of the solution following a global deformation metric.