Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part I: Modelling of incompressible charged porous media

Abstract

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

Extended isogeometric analysis of a progressively fracturing fluid‐saturated porous medium

“…Fluid flow in a deforming porous medium is a topic of major attention due to its numerous applications in, for instance, petroleum and geotechnical engineering, 1,2 biology and medical sciences, [3][4][5] and three or four-phase media. [6][7][8] Initially, the theory was limited to intact porous materials.…”

Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part II: Mixed-hybrid finite element solution

“…The four-component model in the Lagrangian formulation by assuming infinitesimal deformation for the solid phase is derived [11]. Note that for a finite deformation model it is important that the Lagrangian formulation to be considered.…”

“…The initial and boundary conditions for the displacement and electro-chemical potentials with respect to the steady state initial state t = t 0 are: 11) in which µ β t0 = 0, β = l, +, −, and 12) ∆c out is the change in the external concentration and t + 0 and t − 0 are the time just after and before t 0 when chemical loading is applied.…”

“…The driving mechanism is an interplay of mechanical, chemical and electrical forces. In Part I [11], a finite deformation four component model has been derived to account for osmotic effects. To account for finite deformation the set of equations is written in Lagrangian coordinates.…”

“…To account for finite deformation the set of equations is written in Lagrangian coordinates. This leads to a Also remember the relation for fixed charges density in [11], equation (4. We choose a linear elastic material and therefore the elastic energy part is of the form:…”