等方性粘弾性体の非線型構成方程式

抄録

Based upon the phenomenological theory of non-linear responses of many variable systems (K. Okano and O. Nakada, J. Phys. Soc. Japan, 16, 2071 (1961)) the non-linear constitutive equations for isotropic viscoelastic materials are presented.
Let σ(t) denote the stress tensor at an instant of time t, and D(t) be the displacement gradient ∇S (S being the displacement vector) or velocity gradient ∇V (V being the velocity vector) according as the material concerned is a viscoelastic solid or a viscoelastic liquid, then we have (K. Okono, Japanese J. Appl. Phys., 1, 302 (1961)) the following non-linear constitutive equation (up to the second order terms in D) for an isotropic viscoelastic material:
σ(t)=∫^t_-∞[a⁽¹⁾(t-τ)D⁽⁰⁾(τ)+b⁽¹⁾(t-τ)D⁽²⁾(τ)]dτ+∫^t_-∞∫^t_-∞[a⁽²⁾(t-τ₁, t-τ₂)D⁽⁰⁾(τ₁): D⁽⁰⁾(τ₂)I
+b⁽²⁾(t-τ₁, t-τ₂)D⁽²⁾(τ₁): D⁽²⁾(τ₂)I+c⁽²⁾(t-τ₁, t-τ₂)D⁽⁰⁾(τ₁)·D⁽²⁾(τ₂)
+d⁽²⁾(t-τ₁, t-τ₂){1/2D⁽²⁾(τ₁)·D⁽²⁾(τ₂)+1/2D⁽²⁾(τ₂)·D⁽²⁾(τ₁)-1/3D⁽²⁾(τ₁): D⁽²⁾(τ₂)I}
+e⁽²⁾(t-τ₁, t-τ₂){1/2D⁽¹⁾(τ₁)·D⁽²⁾(τ₂)-1/2D⁽²⁾(τ₂)·D⁽¹⁾(τ₁)}]dτ₁dτ₂
+higher order terms.
In the above equation I is the unit tensor and
D⁽⁰⁾≡1/3∇·SI or 1/3∇·VI
D⁽¹⁾≡1/2(D-D)
D⁽²⁾≡1/2(D+D)-D⁽⁰⁾
and a⁽¹⁾(t), b⁽¹⁾(t), a⁽²⁾(t₁, t₂), ect. are the scalar material functions characterizing the viscoelastic response of the system concerned. The third order terms in D are given in the text. (eq. 2.5).
If the material concerned is incompressible the terms on the right hand of the above constitutive equation which are connected with a dilatational deformation (the terms containing a⁽¹⁾, a⁽²⁾, b⁽²⁾, c⁽²⁾) should be replaced by an indeterminate hydrostatic pressure: -pI