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8.6.1 Initial State of Swelling
Under traction-free conditions, a multiphasic material is usually in a state of swelling due to the osmotic pressure difference between the interstitial fluid and the external environment. This osmotic pressure arises from the difference in interstitial versus external concentrations of cations and anions, caused by the charged solid matrix and the requirement to satisfy electroneutrality. An osmotic pressure difference arising from such electrostatic interactions is known as the Donnan osmotic pressure. When the Donnan pressure is non-zero, traction-free conditions do not produce stress-free conditions for the solid matrix, since the matrix must expand until its stressed state resists the swelling pressure.
The Donnan pressure reduces to zero when the fixed charged density is zero, or when the external environment is infinitely hypertonic (having ion concentrations infinitely greater than the interstitial fixed charge density). Since these two conditions represent special cases, it is generally necessary to devise methods for achieving the desired initial state of swelling, in an analysis where loads or displacements need to be prescribed over and above this swollen state. Swelling occurs as a result of the influx of solvent into the porous solid matrix. This influx is a time-dependent process that could require extensive analysis time. Therefore, it is computationally efficacious to achieve the initial state of swelling by using a multi-step analysis (Chapter 6↑) where the first step is a steady-state analysis (Section 3.3.1↑). In this steady-state step, the fixed charge density may be ramped up from zero to the desired value using a load curve for that property or, alternatively, the external environmental conditions may be reduced from a very high hypertonic state down to the desired level. In the second step, prescribed displacement boundary conditions (when needed) may be specified to be of type relative (Section 3.10.1↑), so that they become superposed over and above the initial swelling state.
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