In order to evaluate the tan δ value of the complex system consisting of m components with different relaxation time, a new approximation equation has been derived. For the parallel model in which dynamic models representing the viscoelasticity of each component are arranged in parallel, the tan δ value of a whole system can be approximated by the tan δ
e. defined as :
tanδ
e=
mΣ
i=1 V
(i) ·tanδ
(i) For the series model, in which dynamic models are arranged in the series manner, by the tan δ
e, defined as :
tanδ
e=
mΣ
i=1 J
(i) ·tanδ
(i) (i) stands for the value of i-th component. V (i) and J (i) are modulus fraction and compliance fraction of i-th component respectively and are defined by;
V
(i) =E
(i) 1υ
(i) /
mΣ
i=1 E
(i) 1υ
(i) (1)
J
(i) = (υ
(i) /E
(i) 1) /
mΣ
i=1 (υ
(i) /E
(i) 1) (2)
where, E
(i) 1 is the modulus after relaxation and V
(i) is the volume fraction of i-th component. If the peak value of tan δ of the component, (tan δ)
max is same with each other and (tan δ)
max≤0.3, the relative error originated by using the tan δ
e for representing tan δ value of the parallel model is within 5%. This error decreases monotonically with an increase in the distribution width of the relaxation time of its component. This strongly supports the validity of the approximation equationof tan δ_??_tan δ
e in parallel model proposed in our previous paper (this journal, 30, T85 (1977)) for establishing the analyzing procedure of the tan δ-temperature curve.
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