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2.6.2 Continuous Variables
In principle, the objective of the finite element analysis is to solve for the three unknowns, , and , using the partial differential equations that enforce mixture momentum balance in (2.6.1-1), mixture mass balance in (2.6.1-5), and solute mass balance in (2.6.1-6). The remaining solvent and solute momentum balances in (2.6.1-8), and solid mass balance in (2.6.1-7), have been reduced to relations that may be substituted into the three partial differential equations as needed. Solving these equations requires the application of suitable boundary conditions that are consistent with mass, momentum and energy balances across boundary surfaces or interfaces. When defining boundaries or interfaces on the solid matrix (the conventional approach in solid mechanics), whose outward unit normal is , mass and momentum balance relations demonstrate that the mixture traction and normal flux components and must be continuous across the interface [9, 30]. Therefore, , and may be prescribed as boundary conditions.
Combining momentum and energy balances across an interface also demonstrates that and must be continuous [9, 44], implying that these mechano-chemical potentials may be prescribed as boundary conditions. However, because of the arbitrariness of the reference states , , and , and the ill-conditioning of the logarithm function in the limit of small solute concentration, the mechano-chemical potentials do not represent practical choices for primary variables in a finite element implementation. An examination of (2.6.1-3) also shows that continuity of these potentials across an interface does not imply continuity of the fluid pressure or solute concentration . Therefore, pressure and concentration are also unsuitable as nodal variables in a finite element analysis and they must be replaced by alternative choices. Based on the similar reasoning presented by Sun et al. , an examination of the expressions in (2.6.1-3) shows that continuity may be enforced by using where is the effective fluid pressure and is the effective solute concentration in the mixture. Note that represents that part of the fluid pressure which does not result from osmotic effects (since the term may be viewed as the osmotic pressure contribution to , and is a straightforward measure of the solute activity, since . Therefore these alternative variables have clear physical meanings.
Since the unknowns are now given by , and , the governing partial differential equations may be rewritten in the form where Constitutive equations are needed to relate , , , , and to the solid matrix strain and effective solute concentration.