Effects of Cyclic Loading and Binding to Extracellular Matrix on Macromolecular Transport in Soft Biological Tissue - Progress in Organic and Physical Chemistry: Structures and Mechanisms - page 81

7.3 MATHEMATICAL MODEL
7.3.1 The Governing Equation for Fluid and Matrix Motion
The mathematical model, that describes fluid motion, solid phase deformation, and
solute transport in the model system, is presented. The model is based on biphasic
theory, developed by Mow et al. in 1980 for mechanics of cartilage [16]. The gel is
considered as porous media, consisting of fluid phase (interstitial fluid), solid phase
(polymeric network), and solute.
Porosity of the gel is de¿ ned as follows:

V f
V

,

φ =

where V f is the volume of fluid phase, and V is the total volume of the gel. The vol-
ume of the solute phase can be assumed to be zero. Consequently, the volume fraction
of solid phase is equal to 1

V s
V

.

− φ =

The governing equations for mechanics of the gel will be derived under the fol-
lowing assumptions:
(1) Solid and fluid phases are incompressible.
(2) Solid phase is linearly elastic (this assumption is valid for polymeric gels if
strains are not more than 5% [17], and this will be maximum strain amplitude
in the present study) and elastic modulus is independent of strain.
(3) Body forces are not considered.
(4) Poisson ratio of the gel is zero that is the whole volume change during defor-
mation is due to fluid outflow from the pores or reverses inflow. There is no
friction between the gel and the adjacent impermeable plates.
(5) Permeability k is constant (it has been shown (data not presented) using theo-
retical formula [15] that k differs negligibly for strain <=5%).
(6) Diffusivity D is constant (it has been shown (data not presented) using theo-
retical formula [15] that D differs negligibly for strain <=5%).
According to the biphasic theory [16, 18] under the assumption (4) the balance of
momentum for the whole system reads [2]:

∂σ
∂

0

(1)

=

x

( x , t ) is the total stress of the gel, that, taking into account assumption (1), can
be expressed as the sum of stresses in fluid and solid phases[16, 18]:

σ

where

σ

s

=− (1 − φ ) ⋅ p + σ

e

f

σ =−φ ⋅ p

f s e

σ = σ + σ =− p + σ

(2)

where p is the hydrostatic pore pressure,

e is an effective stress of the solid matrix,
that describes elastic stress of polymeric network:

σ