Handbook of Solvents - George Wypych - ChemTech - Ventech!

306 E. Ya. Denisyuk, V. V. Tereshatov
possible mechano-diffusion modes which differ qualitatively. Self-similar solutions obtained
for these modes describe asymptotic properties at the initial stage of swelling. These
modes are related to thermodynamical material properties. The theoretical predictions have
been verified in the experiments on real elastomers.
6.1.2 FORMULATION OF SWELLING FOR A PLANE ELASTOMER LAYER
Consider an infinite plane elastomer layer of thickness 2h embedded in a low-molecular liquid.
Suppose that the elastomer initially does not contain liquid and is unstrained. This state
is taken as a reference configuration. Let us introduce the Cartesian coordinates (x,y,z) with
the origin placed in the layer center and relate them to a polymer matrix. In the examined
problem, the Cartesian coordinates will be used as the material coordinates. With reference
to the layer, the x axis has a transverse direction and the other axes have longitudinal directions.
In our approach, we define the problem under consideration as a one-dimensional
problem, in which all quantities characterizing the elastomer state depend only on the x-coordinate.
On swelling, the layer experiences transversal and longitudinal deformations which
can be written as
X= X(,) xt Y= ν() ty Z= ν () tz
[6.1.1]
where (X,Y,Z) are the spatial Cartesian coordinates specifying the actual configuration of
the polymeric matrix. From this it follows that the relative longitudinal stretch of the layer is
λ2 = λ3 = ν(t)
and the relative transversal stretch is λ1 = λ(x,t)
= ∂X/ ∂x.
The quantity
J = λλ λ = λν
1 2 3
2
[6.1.2]
characterizes a local relative change in the material volume due to liquid absorption.
The boundary conditions and the relations describing free swelling of the plane layer
in the reference configuration are represented in 5 as
where:
∂N1
∂ ⎛ ∂N1
⎞
= ⎜D
⎟ , N1 = N1( x, t)
[6.1.3]
∂t
∂x
⎝ ∂x
⎠
∂N / ∂t
= 0
[6.1.4]
2
∂σ / ∂x
= 0
[6.1.5]
1
( )
N1x, 0 = 0
[6.1.6]
( , ) / , ( , )
∂N 0t ∂x
= 0 X 0t = 0
[6.1.7]
1
( ht , ) , ( ht , )
μ = 0 σ = 0
( xt , ) ( xt , )
1
[6.1.8]
σ = σ = [6.1.9]
2 3 0
N1,N2 the molar concentrations of the liquid and the chains of polymeric network of elastomer,
respectively,
μ
σk the chemical potential of the liquid dissolved in material
(k = 1,2,3) are the principal values of the Piola stress tensors.