A set of universal equations on the reduced stress relaxation modulus with K-W-W stretched exponential function has been derived from the dynamics of α and β structural relaxation processes. In the present work, the K-W-W decay function is used to define the three types of relaxations (single α, single β relaxation and α-β co-relaxation), then their average times of relaxation are theoretically calculated from the reduced shear stress relaxation modulus and the relaxation time spectrum function H(τ). When the average time of co-relaxation, the reference temperatures (ficitive Tf and glass transition Tg) and the isostructural parameter achieved from the conditions of isostructural glass state are introduced into the reduced shear stress relaxation modulus (GT) under the equilibrium state, a set of correlations between isochoric fragility index (mvα, mvβ and mvαβ) and the coupling strength (α and β) under the reference temperatures are derived from the exact definition of isochoric fragility. So the theory of dynamic fragility for glass substances at isochoric state is developed. The theory can predict the following main features of structural relaxations and behavior of isochoric fragility: the temperature dependence of peak relaxation frequency exhibits a bifurcation with a pair of single α and single β relaxations; the temperature dependence of Stickel equation on 1/T exhibits two crossovers with VFTH(1) and VFTH(2) at the temperatures of Tf and Tg regime; there are two linear correlations between isochoric fragility index (mvα and mvβ) and the coupling strength. Fine agreements between the theoretical calculation and experimental results are obtained.
Based on the structure of glass （or liquid） polymers consisting of α-domain, β-co-domain, and entanglement constituent chain networks, and the nonexponentially viscoelastic behavior, a “heterophase fluctuation” model was proposed. It was found that the dynamics of cooperative rearrangement on the “fluidized domain” has a great shear rate, domain size, and temperature dependences. When the shear rate, domain size, and temperature dependences were taken account into the cooperatively localized rearrangement on the fluidized domain by the degradation of primary α-domain and the reformation of secondary β-co-domain constituent chains. A new dynamic theory of cooperatively localized rearrangement on the fluidized domain constituent chains with different size and different network chain length during physical and mechanical aging was established. The total viscoelastic free en-ergy of deformation resulting from the change in conformations of α-domain, β-co-domain, crytallite, crosslinked, and trapped entanglement constituent chains during aging processes was calculated by the combining method of kinetics and statistical mechanics. The constitu- tive equations and reduced stress relaxation modulus and creep compliances for three types of polymers were also derived. Finally, two reduced universal equations on creep compliance and stress relaxation modulus with a non-linear and two nonexponential parameters α and β were theoretically derived from the dynamic theory and a statistically extended mode coupling theory for double aging effects of polymers was developed. Results show that the two reduced universal equations have the same form as Kohlraush-Williams-Watts （K-W-W） stretched exponential function. The nonlinearity and the nonexponentiality are, respectively, originated from the memory effects of nonthermal and thermal history. The correlation of nonlinearity, α and β to the aging time, aging temperature, and the mesomorphic structure of fluidized domains was also established.
Three kinds of polymeric materials are taken as example for the verification of linear ex-trapolation method from unified master lines with reduced universal equations on creep and stress relaxation tests. The theoretical values of long-term mechanical behavior and lifetime for a cured epoxide, polypropylene, poly（methyl-methacrylate）, and SBR rubber are directly evaluated with the universal equations on reduced creep compliance and reduced stress relax-ation modulus and are compared with their predicted values by the linear extrapolation from the unified master lines of creep and stress relaxation. The results show that the theoretical values of dimensional stability, bearing ability and lifetime are in an excellent agreement with the predicted values, it shows that the linear extrapolation method is more simple and reliable. The dependences of long-term mechanical behaviors and lifetime on the different aging times are discussed.
