The investigation of the variation problem in mechanical systems has two purposes. One of them is to find out the variation principle as a fundamental principle of mechanical system, and the other is to build up a technique of obtaining an approximate solution of the equation of motion. In the case of the conservation system, these two viewpoints fortunately stand together. In the case of viscoelastic material, on the other hand, it is not possible to make the variation problem as the basis of mechanics. In fact, since the stress in such a system is not the conserving force, we cannot get the variation function in a closed form. However, making use of a suitable technique for the variation of the parameters, we have been enabled to establish the variation method, on the basis of the so-called Hamilton's principle of mechanics for the purpose of finding the approximate solution of the equation of motion.
Hamilton's principle in our case is written as
∫t∫v[1/2ρδr2-Sp(σE·δeE)+ρKE·δr]dtdV+∫t∫SFE·δrdtdS=0
in the Euler form, and
∫t∫V0[1/2ρ0δr2-Sp(σL·δeL)+ρ0KL·δr]dtdV0+∫t∫S0FL·δrdtdS0=0
in the Lagrange form. Here σ is the stress tensor, e the strain tensor (=a+·a/2 in Lagrange system, a: the displacement tensor), ρ the density, K the volumic force acting on the unit mass of the sample, and F the surface force acting on the boundary of the sample.
In further treatment, we must use a suitable model describing the mechanical behavior of the sample. In the viscoelastic material it is supposed that there exists energy-storing mechanism as well as energy-dissipative mechanism. For each mechanism we suppose the displacement tensor and the stress tensor, a1 and σ1, and a2 and σ2, respectively, in addition to the observable ones, a and σ. As the viscoelastic model, we define the three dimensional Voigt model by the relation (in E-system)1)
a=a1=a2 and σ=σ1+σ2,
and the Maxwell model by
δa·a-1=δa1·a1-1+δa2·a2-1 and σ=σ1=σ2.
Introducing the stored energy density w(e1), we have the following variation functions:
a) The Voigt model.
IE(V)=∫t∫V[1/2ρr2-WE(e)-Sp[(a-1·σE, 2)c·a]+ρKE·r]dtdV-∫t∫SFE·r dtdS
in E-system, and
IL(V)=∫t∫V0[1/2ρ0r2-WL(e)-Sp[(σL, 2)c·eL]+ρ0KL·r]dtdV0-∫t∫S0FL·r dtdS0
in L-system.
b) The Maxwell model.
IE(M)=∫t∫V[1/2ρr2-WE(e1)-Sp[(a2-1·σE)c·a2]+ρKE·r]dtdV-∫t∫SFE·r dtdS
