Lévy (stable) probability densities and mechanical relaxation in solid polymers (Q1069719)

Early investigations by Weber, R. and F. Kohlrausch, Maxwell, and Boltzmann of relaxation in viscoelastic solids are reviewed. A two-state model stress-tensor describing strain coupling to internal conformations of a polymer chain is used to derive a linear response version of the Boltzmann superposition principle for shear stress relaxation. The relaxation function of Kohlrausch \(\phi (t)=\exp [-(t/\tau)^{\alpha}]\) is identical to the Williams-Watts empirical dielectric relaxation function and in the model corresponds to the autocorrelation function of a segment's differential shape anisotropy tensor. By analogy with the dielectric problem, \(\exp [-t/\tau)^{\alpha}]\) is interpreted as the survival probability of a frozen segment in a swarm of hopping defects with a stable waiting-time distribution \(At^{-\alpha}\) for defect motion. The exponent \(\alpha\) is the fractal dimension of a hierarchical scaling set of defect hopping times. Integral transforms of \(\phi\) (t) needed for data analsis are evaluated; the cosine and inverse-Laplace transforms are stable probability densities. The reciprocal kernel for short-time compliance is discussed.