A sponge is an elastic solid with connected pores. When immersed in water, the sponge absorbs water. When a saturated sponge is squeezed, water will come out. More generally, the subject is known as diffusion in elastic solids, or elasticity of fluid-infiltrated porous solids, or poroelasticity. The theory has been applied to diverse phenomena. Here are a few examples.

Consolidation of soils. A soil is a composite of solid particles and fluids (mainly water). Particles in the soil are more or less bound together and constitute an elastic skeleton. The interstices of the skeleton are filled with water. When a load is applied to the soil, water will flow out gradually, so that the soil will deform over some time. This process is known as consolidation.

• M.A. Biot, General theory of three-dimensional consolidation, Journal of Applied Physics 12, 155-164 (1941).
• J. Bear, Dynamics of fluids in porous media. Dover reprint, 1988.
• E. Detournay and A.H.-D. Cheng, Fundamentals of Poroelasticity, Chap. 5 in Comprehensive Rock Engineering: Principles, Practice and Projects, Vol. II, Analysis and Design Method, ed. C. Fairhurst, Pergamon, 113-171, 1993.
• J.R. Rice, Elasticity of fluid-infiltrated porous solids, notes for teaching on hydrology and environmental geomechanics.
• J.R. Rice and M.P. Cleary, Some basic stress-diffusion solutions for fluid-saturated elastic porous media with compressible constituents. Reviews of Geophysics and Space Physics 14, 227-241 (1976).
• H. F. Wang, Theory of linear poroelasticity. Princeton University Press, 2000.

Swelling of gels. A gel is a composite of a network of crosslinked molecules, and a solvent consisting of all other molecules that permeate in the interstices of the network, but are not linked to the network. The network is elastic, while the solvent can migrate through the interstices of the network. The elasticity of the network and migration of the solvent are coupled: the network swells where the solvent accumulates, and the solvent migrates in response to the deformation of the network. The gel is called a hydrogel when the solvent is water, or an aerogel when the solvent is a gas.

• J. Dolbow, E. Fried, H. Ji, Chemically induced swelling of hydrogels. Journal of the Mechanics and Physics of Solids 52, 51-84 (2004).
• A. Sidorenko, T. Krupenkin, A. Taylor, P. Fratzl, and J. Aizenberg, Reversible switching of hydrogel-actuated nanostructures into complex micropatterns. Science 315, 487-490 (2007).

Fluid migration in tissues. Nearly all living tissues are porous and elastic, with fluid migrating in the pores inside the tissues to transport nutrients and wastes.

• S.C. Cowin and S.B. Doty, Tissue mechanics. Springer, 2007.

Diffusion in crystals. Metals and ceramics are often in the form of alloys, consisting of dissimilar atoms. Some atoms diffuse much faster than other atoms, so that the slow diffusers may serve the role of an elastic network. For example, some materials can absorb and release large amounts of hydrogen, making them candidates for hydrogen storage technology.

• F.C. Larche and J.W. Cahn, The interactions of composition and stress in crystalline solids, Acta Metallurgica 33, 331-357 (1985).
• P.W. Voorhees and W.C. Johnson, The thermodynamics of elastically stressed crystals, Solid State Physics 59, 1-201 (2004).

However, for most alloys, diffusion is coupled with inelastic deformation, so that the theory of diffusion in elastic crystals is not applicable. See discussions in

• Z. Suo, A continuum theory that couples creep and self-diffusion. Journal of Applied Mechanics 71, 646-651 (2004).

These notes will focus on diffusion in elastic solids. Historically, the theory coupling diffusion and elasticity has caused a great deal of confusion. It might be helpful if we start with elementary ideas.

• Migration of matter in an elastic solid
• Thermodynamics of a fluid of single species of molecules
• Chemical potential of an incompressible liquid
• Chemical potential of an ideal gas
• Equilibrating a liquid and a vapor of the same species of molecules
• Humidity
• Equilibrate a gel with a weight and a moist environment
• A homogeneous field of stress and water concentration
• Alternative free-energy functions
• Inhomogeneous, equilibrium field
• Invariance under rigid-body rotation
• Isotropic material
• Diffusion in a rigid network
• Fluid infiltrating a rigid network
• Thermodynamics of nonequilibrium processes
• Nonlinear poroelasticity
• Summary of equations
• Linear poroelasticity
• Stress in a thin film due to change of the humidity in the environment
• Stress induced by drying
• Analysis of a soil test (Biot, 1941)
• A stationary long crack
• A crack extending at a constant velocity

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