Large deformations of a soft porous material

Abstract

Compressing a porous material will decrease the volume of the pore space, driving fluid out. Similarly, injecting fluid into a porous material can expand the pore space, distorting the solid skeleton. This poromechanical coupling has applications ranging from tissue mechanics to hydrogeology. The classical theory of linear poroelasticity captures this coupling by combining Darcy's law with Terzaghi's effective stress and linear elasticity in a linearized kinematic framework. Linear poroelasticity is a good model for very small deformations, but it becomes increasingly inappropriate for moderate to large deformations, which are common in the context of phenomena such as swelling and damage, and for soft materials such as gels and tissues. The well-known theory of large-deformation poroelasticity combines Darcy's law with Terzaghi's effective stress and nonlinear elasticity in a rigorous kinematic framework. This theory has been used extensively in biomechanics to model large elastic deformations in soft tissues, and in geomechanics to model large elastoplastic deformations in soils. Here, we first provide an overview and discussion of this theory with an emphasis on the physics of poromechanical coupling. We present the large-deformation theory in an Eulerian framework to minimize the mathematical complexity, and we show how this nonlinear theory simplifies to linear poroelasticity under the assumption of small strain. We then compare the predictions of linear poroelasticity with those of large-deformation poroelasticity in the context of two uniaxial model problems: Fluid outflow driven by an applied mechanical load (the consolidation problem) and compression driven by a steady fluid throughflow. We use these problems to explore the steady and dynamical errors associated with the linear model in both situations, as well as the impact of introducing a deformation-dependent permeability.

Cited by in corpus (22)

• Dynamics of swelling and drying in a spherical gel
• The Biot-Stokes coupling using total pressure: formulation, analysis and application to interfacial flow in the eye
• From arteries to boreholes: Steady-state response of a poroelastic cylinder to fluid injection
• The breakdown of Darcy's law in a soft porous material
• A Locally Conservative Mixed Finite Element Framework for Coupled Hydro-Mechanical-Chemical Processes in Heterogeneous Porous Media
• Surface tension and the Mori-Tanaka theory of non-dilute soft composite solids
• Migration, trapping, and venting of gas in a soft granular material
• From arteries to boreholes: Transient response of a poroelastic cylinder to fluid injection
• Modelling a Hydrodynamic Instability in Freely Settling Colloidal Gels
• Compression and reswelling of microgel particles after an osmotic shock
• Non-intrusive reduced order modeling of poroelasticity of heterogeneous media based on a discontinuous Galerkin approximation
• Collective diffusion coefficient of a charged colloidal dispersion: interferometric measurements in a drying drop
• Fluid-fluid phase separation in a soft porous medium
• Energetic Formulation of Large-Deformation Poroelasticity
• Coupling Stokes flow with inhomogeneous poroelasticity
• Finite element methods for large-strain poroelasticity/chemotaxis models simulating the formation of myocardial oedema
• Solute transport due to Periodic Loading in a Soft Porous Material
• Drying-induced stresses in poroelastic drops on rigid substrates
• Nonlinear plane waves in saturated porous media with incompressible constituents
• Cyclic loading of a heterogeneous non-linear poroelastic material
• Modeling Multiphase Flow Through and Around Multiscale Deformable Porous Materials
• Surfing on chemical waves: a simple yet dynamically rich two-sphere responsive gel swimmer