Rheological Equations of Viscoelastic Liquid

Abstract

Assuming that the elastic displacement of a material particle is expressible by the displacement from θ_eⁱ to Θ^j in general curvilinear coordinate system or that from x_eⁱ to Xⁱ in Cartesian coordinate system, the contravariant component of the elastic strain tensor will be defined by
ε^ij=1/2(∂Θⁱ/∂x_e^α∂Θ^j/∂x_e^α-∂Θⁱ/∂Xβ∂Θ^j/∂Xβ) (1)
The contravariant component of the actual strain rate tensor will be given by
f^ij=1/2(V^i/j+V^j/i) (2)
where Vⁱ represents the contravariant component of velocity vector and V^i/j denotes the contravariant differentiation of Vⁱ by Θ^j.
We assume that the stress tensor τ is a function of both the elastic strain tensor ε and the actual strain rate tensor f, and may be expressible as a polynomial in ε and f. If the material is isotropic in its rest state, this relation must be invariant to the transformation of the coordinate system. Then, considering the symmetric property of τ, the relation will be written as follows:
τ=(-P+a₀)1+α₁ε+α₂f+α₃ε²+α₄f²+α₅(εf+fε)+α₆(ε²f+fε²)+α₇(εf²+f²ε)+α₈(ε²f²+f²ε²) (3)
where P is the hydrostatic pressure, 1 is the metric tensor and α₀, α₁, …… α₈ are polynomials of ten invariants such as t_rε, t_rε², t_rε³, t_rf, t_rf², t_rf³, t_rεf, t_rε²f, trεf² and t_rε²f². Considering that the principal axes of ε are equivalent to those of τ but not to f, Eq. (3) will be reduced to
τ=(-p+α₀)1+α₁ε (4)
This equation assures the coincidence of the principal axes of τ with those of ε in every case, but in some cases it may happen that the principal axes of the other term of Eq. (3) become equal to those of τ. In simple shearing flow, such term is α₇(εf²+f²ε). Then, Eq. (3) will be
τ=(-p+α₀)1+α₁ε+α₇(εf²+f²ε) (5)
which may be expressed in terms of the contravariant components of tensors as follows:
τ^ij=(-P+α₀)G^ij+α₁ε^ij+α₇(εⁱ_αf_βαfβj+fⁱ_αf_β;αεβj) (6)
The applications of our theory to simple shearing flow and steady flow through pipe have been made with satisfactory results; especially it has been shown that two normal stresses in the direction perpendicular to the stream are equivalent to each other.