In the present paper, an equation of nonlinear superposition principle proposed by Schapery is simplified so that the dependences of equilibrium term of relaxation modulus on strain are identical with that of transient term, and this equation is named general model (model G). From this equation, the several nonlinear constitutive equations can be derived as follows.
(1) Time-strain factorized model
(a) Perfect factorized model (model I)……nonlinearity depends on present strain.
(b) Imperfect factorized model (model II)……nonlinearity depends on strain history.
(c) Mixed factorized model (model M)……to combine model I with model II.
(2) Time-strain reduced model (model III)……the strain to change time scale.
(3) General model (model G)……to combine model M with model III.
It is difficult to assign nonlinear viscoelastic property of a given material to one of the above five models by single step stress relaxation tests. In particular, it is emphasized for power law's materials.
By two step stress relaxation tests, nonlinear characteristics of materials are classified into five models as follows:
The time
tN, when effects of the first strain ε
1, decay and hence two step relaxation stress
σ2(
t′) under ε
2(=ε
1+
Δε) meets single step relaxation stress
σ1(
t′) at ε
2, depends on only
t1 (input time of
Δε) for the time-strain factorized models. For other models,
tN depends on ε
1, ε
2 and
t1. In each case of model I, II and M, |_??_σ
1/_??_ε
2| is linear to ε
1, independent of ε
1, and nonlinear to ε
1, respectively. The another useful properties are that
Δσ/ε
1 (
Δσ=σ
2-σ
1) decrease with ε
2, ε
1, and both ε
1 and ε
2, respectively, for model I, 11 and M. The stress increment δσ at
t1 has only linear relation to
Δε for the model III, while δσ is nonlinear to all ε
1, ε
2 and
t1 for the model G.
The theory is tested for some polyethylenes at room temperature. The samples have nearly power law's properties in single step stress relaxation behavior. The measured values of
tN have properties of the model III or G. Only when
Δε>0 and ε
1_??_2%, the experimental curves of
o2(
t′) agree well with the calculated curves from model III. When ε
1=3% and
Δε>0, the calculated values are less than the experimental values. When
Δε<0, the calculated values are greater than experimental values. In this case, if nonlinearized function
aε (ε) is provided with property of hysteresis, the theory of model III may be improved to approach the experimental results.
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