A hydrogel is a cross-linked polymer network with water as solvent. Industrially widely used superabsorbent polymers (SAP) are partially neutralized sodium polyacrylate hydrogels. The extremely large degree of swelling is one of the most distinctive characteristics of such hydrogels, as the volume increase can be about 30 times its original volume when exposed to physiological solution. The large deformation resulting from the swelling demands careful numerical treatment. In this work, we present a biphasic continuum-level swelling model using the mixed hybrid finite element method (MHFEM) in three dimensions. The hydraulic permeability is highly dependent on the swelling ratio, resulting in values that are orders of magnitude apart from each other. The property of the local mass conservation of MHFEM contributes to a more accurate calculation of the deformation as the permeability across the swelling gel in a transient state is highly non-uniform. We show that the proposed model is able to simulate the free swelling of a random-shaped gel and the squeezing of fluid out of a swollen gel. Finally, we make use of the proposed numerical model to study the onset of surface instability in transient swelling.

Ionized hydrogels, as osmoelastic media, swell enormously (1000 times its original volume in unionized water) due to the osmotic pressure difference caused by the presence of the negatively charged ion groups attached to the solid matrix (polymer chains). The coupling between the extremely large deformations (induced by swelling) and fluid permeation is a field of application that regular poroelasticity formulations cannot handle. In this work, we present a mixed hybrid finite element (MHFE) computational framework featuring a three-field (deformation-chemical potential-flux) formulation. This formulation guarantees that mass conservation is preserved both locally and globally. The impact of such a property on the swelling simulations is demonstrated by four numerical examples in 2D. This paper focuses on the implementation aspects of the MHFE model and shows that it stays robust and accurate for a volume increase of more than 3000%.

The triphasic theory on soft charged hydrated tissues (Lai, W. M., Hou, J. S., and Mow, V. C., 1991, "A Triphasic Theory for the Swelling and Deformation Behaviors of Articular Cartilage," ASME J. Biomech. Eng., 113, pp. 245-258) attributes the swelling propensity of articular cartilage to three different mechanisms: Donnan osmosis, excluded volume effect, and chemical expansion stress. The aim of this study is to evaluate the thermodynamic plausibility of the triphasic theory. The free energy of a sample of articular cartilage subjected to a closed cycle of mechanical and chemical loading is calculated using the triphasic theory. It is shown that the chemical expansion stress term induces an unphysiological generation of free energy during each closed cycle of loading and unloading. As the cycle of loading and unloading can be repeated an indefinite number of times, any amount of free energy can be drawn from a sample of articular cartilage, if the triphasic theory were true. The formulation for the chemical expansion stress as used in the triphasic theory conflicts with the second law of thermodynamics.

This theory results in a coupled system of nonlinear parabolic differential equations together with an algebraic constraint for electroneutrality. In this model, it is desirable to obtain accurate approximations of the fluid flow and ions flow. Such accurate approximations can be determined by the mixed finite element method. The solid displacement, fluid and ions flow and electro-chemical potentials are taken as degrees of freedom. In this article the lowest-order mixed method is discussed. This results into a first-order nonlinear algebraic equation with an indefinite coefficient matrix. The hybridization technique is then used to reduce the list of degrees of freedom and to speed up the numerical computation. The mixed hybrid finite element method is then validated for small deformations using the analytical solutions for one-dimensional confined consolidation and swelling. Two-dimensional results are shown in a swelling cylindrical hydrogel sample. Mathematics

Abstract.The swelling and shrinkage of biological tissues are modelled by a four-component mixture theory in which a deformable and charged porous medium is saturated with a fluid with dissolved ions. Four components are defined: solid, liquid, cations and anions. The aim of this paper is the construction of the Lagrangian model of the four-component system. It is shown that, with the choice of Lagrangian description of the solid skeleton, the motion of the other components can be described in terms of Lagrangian initial system of the solid skeleton as well. Such an approach has a particularly important bearing on computer-aided calculations. Balance laws are derived for each component and for the whole mixture. In cooperation of the second law of thermodynamics, the constitutive equations are given. This theory results in a coupled system of nonlinear parabolic differential equations together with an algebraic constraint for electroneutrality. In this model, it is desirable to obtain an accurate approximation of the fluid flow and ions flow. Such an accurate approximation can be determined by the mixed finite element method. Part II is devoted to this task. Mathematics