Levy (stable) probability densities and mechanical relaxation in solid polymers

Abstract

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 Kohlrauschφ(t)= exp[−(t/τ α )] 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/τ α )] is interpreted as the survival probability of a frozen segment in a swarm of hopping defects with a stable waiting-time distributionAt −α for defect motion. The exponent a is the fractal dimension of a hierarchical scaling set of defect hopping times. Integral transforms ofφ(t) needed for data analysis are evaluated; the cosine and inverse-Laplace transforms are stable probability densities. The reciprocal kernel for short-time compliance is discussed.