粘弾性における変分法 : 分散形のレオロジー, その他
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概要
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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[1/2ρδr^^・^2-Sp (σ_E・δe_E)+ρK_E・δr)]dtdV+∫_t∫_s・F_EδrdtdS=0 in the Euler form, and ∫_t∫_<v0>[1/2ρ_0δr^^・^2-Sp (σ_L・δe_L)+ρ_0K_L・δr]dtdV_0+∫_t∫_<so>F_L・δrdtdS_0=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, a_1 and σ_1, and a_2 and σ_2, 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) a=a_1=a_2 and σ=σ_1=σ_2, and the Maxwell model by δa・a^<-1>+δa_2・a_2^%lt;-1> and σ=σ_1=σ_2. Introducing the stored energy density w(e_1), we have the following variation functions : a) The Voigt model. I_E^<(V)>=∫_t∫_v[1/2ρr^^・^2-W_E (e)-Sp[(a^<-1>・σ_<E,2>)_c・a]+ρK_E・r]dtdV-∫_t∫_sF_E・rdtdS in E-system. and I_L^<(V)>=∫_t∫_<v0>[1/2ρ_0r^^・^2-W_L (e)-Sp[(σ_<L,2)_c・e_L]+ρ_0K_L・r]dtdV_0-∫_t∫_<s0>F_L・r dtdS_0 in L-system. b) The Maxwell model. I_E^<(M)>=∫_t∫_v[1/2ρ_0r^^・^2-W_E (e_1)-Sp[(a_2^<-1>・σ_E)_c・a_2]+ρK_E・r]dtdV-∫_t∫_sF_E・r dtdS in E-system, and I_L^<(M)>=∫_t∫_<v0>[1/2ρ_0r^^・^2-W_L(e_1)-Sp[{(a_2^<-1>・a)・a_L・(a_2^<-1>・a)^<-1>}_c・e_2]+ρ_0K_L・r]dtdV_o-∫_t∫_<so>F_L・r dtdS_0
- 社団法人日本材料学会の論文
- 1964-05-15
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