A theory of filler-reinforcement more precise than the previous one
^{1)4)}is presented here, based upon the same model and by means of more accurate internal deformation
^{1)}. Suppose just the same dispersed system with the one of the previous theory where
M spherical rigid particles with a radius
d are uniformly dispersed in a vulcanized rubber-like substance whose volume is
V_{r}. Then the volume
V of the system, the volume ratio
X of the filler to the rubbery medium and the volume fraction
Y of the filler are given by
V=V
_{r}+M(4π/3)d
^{3}, X=M(4π/3)d
^{3}/V
_{r} and Y=M(4π/3)d
^{3}/V,
respectively.
Let the system be considered as a double network system constructed both from the ordinary network chains spatially distributed in the rubbery medium with volume density
g_{r} and from the adhered network chains over the particle surfaces with surface density
g_{f}, and let it be considered as our model for the filled vulcanized rubber-like specimens.
The following assumptions used in the previous theory are used here again:
(a) Spherical rigid particles are uniformly disperesed in the rubbery medium.
(b) The movements of the center of each particle under the external deformation accords to the requirment of proportionality (: an affine deformation).
(c) The one-body approximation in terms of“
D-sphere”is adopted.
(d) The adhesion state is represented by the“mixing”approximation.
(e) The shape off the surface of cavity assumes the ellipsoid of revolution.
(f) The volume of rubbery medium is kept constant under the external deformation.
(g) The network chains accord to the“deformation theorem”derived from the“internal deformation”.
(h) The assumptions in the ordinary theory of rubber elasticity
^{3)} are adopted here except the requirement of proportionality; especially
(α) The network chains are ideal Gaussian.
(β) The free energy of the system consists of two parts: the energy due to entropy of individual chains and the liquid like interaction energy
U among the chain segments.
Against the external deformation α which is uniformly in appearance the internal deformation α
_{R} is defined as transformation matrix in formally by which any point
P of the rubbery medium with a position vector
R with respect to the center in the
D-sphere transforms to the point
P' with a vector
R':
R'=α
_{R}R. (0)
It is clear that the internal deformation α
_{R} must satisfy at least the following conditions:
i)α
_{R}=α over the
D-sphere,
ii)α
_{R}=γ over the
d-sphere(: particle is sometimes called the
d-sphere),
iii)α
_{R}→ α as
Y→0,
iv)α
R minimizes the deformation energy of the system for the given α.
Here γ represents the deformation of the surface of the cavity occurred around each particle and is assumed as the ellipsoid of revolution of such a shape thatγ
_{1}=γ
_{2}=1, γ
_{3}=γ, when the external deformation α is the simple elongation such that α
_{1}=α
_{2}=β, α
_{3}=α(cf. Eq.(6)in the main discourse), and γ is an undecided function of α. Taking differential Δ
R' of.
R' in Eq. (0) for increment Δ
R in
R and rewriting asΔ
R=
r, Δ
R'=
r', the“deformation theorem”of the network chain with end-to-end vector
r located at
P is derived. as in Eq.(4).
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