Non-Linear Constitutive Equations for Isotropic Viscoelastic Materials

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 andD(0)≡1/3∇·SI or 1/3∇·VID(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