三次元粘弾性論と Barus 効果の理論

抄録

In previous papers¹⁾ we studied on the phenomenological theory of the non-linear viscoelasticity of three dimensional bodies. Since that time a criticism has been given to it that our stress-strain relation for the three dimensional Maxwell model is not invariant for the rigid rotation of the sample and for the coordinate transformation.²⁾ In this paper a reformed theory is treated for the model. We define the observable displacement (deformation) tensor a and the internal elastic displacement tensor α as
Δr=a·Δr₀andΔξ=α·Δξ₀, (1)
respectively, with the lengths of a line element in the sample at initial state, Δr₀ and at deformation state, Δr, and those of the internal elastic mechanism, Δξ₀ and Δξ.^*** Instead of the fundamental equation of the previous paper for the time derivative of internal elastic displacement tensor, we consider the time derivative of the Cauchy-Green's left internal strain tensor λ=α·α⁺ (+ denotes the transposed tensor). That is,
dλ/dt=da/dt·a^-1·λ+λ·a^+-1·da⁺/dt+(dλ/dt)^* (2)
assuming the dissipation term of the form
(dλ/dt)^*=-β(λ-1). (3)
The stress tensor is written as ∞
σ=Rλ·λ+Qλ+P1=σ_e+P1. (4)
We can get the stress-strain relation from Eqs. (2) and (4):
dσ_e/dt=da/dt·a^-1·σ_e+σ_e·a^+-1·da⁺/dt+Rλ[da/dt·a^-1+a^+-1·da⁺/dt]λ+Rλ·λ+Qλ+(dσ_e/dt)^* (5)
where
(dσ_e/dt)^*=Q(dλ/dt)^*+R[λ·(dλ/dt)^*+(dλ/dt)^*·λ]. (6)
The Cauchy-Green's right internal strain tensor λ'=α⁺·α has a relation similar to Eq. (2) with the modified displacement tensor a'=R⁺·a, where R is the orthogonal tensor representing the rotation of the sample as a whole.
Denoting the additional rigid rotation of the sample by an orthogonal tensor T(t), the displacement tensor and the left internal strain tensor should be transformed to
a=T·a and λ=T·λ·T⁺, (7)
respectively. The relation between a and λ is just similar to Eq. (2), that is, this relation is invariant for the rigid rotation of the sample.
On the other hand, due to the coordinate transformation T(t), the displacement tensor and the left internal strain tensor become
a=T·a·T+and λ=T·λ·T⁺ (8)
respectively. The time derivative of λ has, besides the terms similer to those of Eq. (2), the additional term
-[a·T·dT+/dt·a^-1·λ+λ·a^+-1·dT/dt·T+·a+]. (9)
This term arises from the change of the reference coordinate system for the initial line element of the sample, and if the coordinate transformation is independent of time, this term vanishes.
As an example of our theory, the so-called Barus effect is treated based on the consideration of Metzner et al⁷⁾.