Variation Method in Viscoelasticity II

Abstract

In a previous paper the variation method was developed to find the solution for the equation of motion of three dimensional viscoelastic materials, on the basis of Hamilton's principle of mechanics. Since the stress in that case was not conservative force, we could not get the variation function in a closed form. In the present paper it is shown that if the suitable representation is taken for the variation of strain-rate tensor and a somewhat different form is assumed for Hamilton's principle, we can obtain the variation function in a closed form as a matter of form. The Hamilton's principle is here applied to the variation of displacement-rate instead of that of displacement of the material point. In our case, the variation corresponds to the extremum of work done by the external force.
With attention to the equation of motion in the Euler-system (E-system), we put
ρξ=Divσ_E+ρK_E (1)
The Hamilton's principle is then assumed to be given in the form
∫_t∫_V[ρξ-Divσ_E-ρK_E]·δξdVdt=0 (2)
where ξ is the coordinate of the mass point in the material which is placed at x in the natural state, σ_E is stress tensor referring to E-system, K_E is external force acting on the unit mass of the sample and ρ is the material density. Making use of the displacement tensor a: Δξ=a·Δx where Δ denotes the difference of the coordinates of neighboring two points in the sample, we define the strain tensor e_L=a⁺·a/2 as well as the strain-rate tensor e_L=(a⁺·a+a⁺·a)/2 in the Lagrange system (L-system). The variations of these tensors are assumed to be δe_L=(δa⁺·a+a⁺·δa)/2 and δe_L=(δa⁺·a+a⁺·δa)/2, and these tensors are transformed to those in E-system by A_E=a^+-1·A_L·a^-1. In viscoelastic materials there exist two mechanisms; one of them is elastic, stored mechanism (1) and the other is viscous, dissipative mechanism (2). It is assumed that each of them is characterized by the corresponding stress tensor and strain tensor σ_i and e_i, i=1, 2. The elastic stored strain energy density and the viscous dissipative energy density are considered to be the functions of e₁ and e₂, respectively: w=w(e₁) and ε=ε(e₂). After suitable mathematical calculation, we can obtain the variation functions in both E-system and L-system as
I_E=∫_V1/2ρξ²dV+∫_Vw_E(a₁)dV+∫_V∫_tε_E(e_E2)dtdV+∫_V∫_ξρK_E·dξdV-_??__S∫_ξF_E·dξdS (3)
I_L=∫_V01/2ρ₀ξ²dV₀+∫_V0w_L(e_L1)dV₀+∫_V0∫_tε_L(e_L2)dtdV₀+∫_V0∫_ξρ₀K_L·dξdV₀-∫_S0∫_ξF_L·dξdS₀ (4)