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 80

wall. There is no displacement of the gel at this point. Perpendicular to the cross sec-
tion, presented on Figure 1, the gel is considered of an infinite width. This assumption
means that there is no deformation in this dimension.

FIGURE 1 The geometry of the model system .

Let us assume that the friction between the gel and the adjacent plates is zero and
Poisson ratio is also zero. Consider the deformation of the gel while the upper plate
moves down by

d . Then the process of this deformation can be seen as superposition
of the two deformations:
(1) The deformation with no volume change (which means that the volume of
the gel redistributes in such a way that the free surface moves to the right by
Δ

Δ

h Δ

d
d

x = ).

(2) The compression of the layer adjacent to the surface to compensate to this
movement.
As a result, the boundary remains at the same point x

h , but the solid phase near
this surface is compressed. The volume of À uid that À ows out to the bath is equal to
the volume by which the plate moved. Additionally, stresses and strains do not depend
on z because of any friction assumption [2, 16]. Hence, as we will see, all functions in
the governing equation depend only on x .
Let us introduce the virtual displacement u of solid phase that at each time mo-
ment means the displacement relative to the state #1 in x -direction. It is called vir-
tual because it is the displacement from the virtual state that has never existed. The
displacement at the state #1 relative to the undeformed state is

=

x , where ε ⊥ is the
applied strain in z -direction. Hence, the real displacement u x ( x , t ) in x -direction is a
sum of the two described virtual displacements: u x

−ε ⊥ ⋅

u

= − ε ⊥ ⋅ x .