Viscoelasticity of Three Dimensional Bodies
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概要
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A phenomenological theory of the non-linear viscoelasticity of three dimensional bodies is treated. The two elementary models of the classical theory of linear viscoelasticity, that is the Maxwell and the Voigt model, are extended to three dimensional non-linear cases. In addition to the observable (external) deformation tensor a and the stress tensor σ, we define the two internal deformation tensors and the stress tensors: The internal elastic deformation tensor, αE and stress tensor σE correspond to the energy stored mechanism while the internal viscous deformation αη and stress ση represent the state of the energy dissipative mechanism. They are just the three dimensional analogues of these of the spring and the dashpot in the mechanical model of the classical theory of viscoelasticity.The three dimensional Maxwell model is defined by the conditionσ=σE=ση. (1)On the other hand, the three dimensional Voigt model is characterized by the restrictiona=αE=αη. (2)From the energy consideration, we find the fundamental equations of the three dimensional Maxwell model in such the formsdα/dt=da/dt·a-1·α+(dα/dt)*, (3)σ=P1+Qλ+Rλ·λ, (4)λ=α·α+, (5)(dα/dt)*=-β(α-1), (6)where α is the rewriting of the internal elastic deformation αE, and (dα/dt)* corresponds to the deformation production due to the dissipative mechanism. Equation (6) is merely a assumption for the dissipative term, and β represents the reciprocal of the so-called relaxation time, τ=1/β. P, Q and R may be the scalar functions of the three invariants associating to the internal (elastic) deformation, α.On the other hand, the fundamental equations of the three dimensional Voigt model are as follows:σ≡σE+ση (7)σE=P'1+Q'ε+R'ε·ε (8)ση=p1+qε+rε·ε (9)ε=a·a+ (10)ε=da/dt·a-1+a+-1·da+/dt (11)where P', Q' and R' and p, q and r are the scalar functions of the three invariants associating to the strain tensor ε and the strain-rate tensor ε, respectively.Our models are shown to be successful in analysing the so-called normal stress effects, i.e., the Weissenberg effect for elasticoviscous liquids (in the case of Maxwell model) and the Poynting effect for viscoelastic solids (Voigt model).Non-linearities in viscoelastic properties are classified into three cases: elastic, viscous or relaxational, and geometrical ones. Elastic non-linearity corresponds to the non-ideality of the stress-deformation relation of the energy stored mechanism, and viscous non-linearity represents nothing but the kinetic character of the energy dissipative mechanism. An essential feature of the three dimensional bodies is the geometrical non-linearity. The typical examples of this are the said normal stress effects which arises from the cross effect between the shearing deformation and the normal stress components.
- 社団法人 日本材料学会の論文
社団法人 日本材料学会 | 論文
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