Transient and Temperature Response of a Distributed, Thermally Activated System

Abstract

After a brief survey of methods of representing the transient response of a distributed linear system, a general analysis is presented of the temperature dependence of relaxation times of a thermally activated system. It is assumed that either the pre-exponential factor τd (or its inverse, the vibrational frequency vd) or the activation energy E may be separately distributed or that they both depend linearly on a structure factor S and so are similarly distributed. Further, τd, E, and parameters of their distributions are taken temperature independent. A two-parameter, generalized, truncated exponential probability density function is chosen for the distribution of S and transformed to yield a power-law distribution for G(τ), the over-all relaxation-time distribution function. Expressions yielding the time and temperature response of the system are then derived from G(τ). The specific transient response considered is that following the imposition of a constant forcing function at t=0 and may represent the charge and current response of a dielectric system after application or removal of a constant voltage, the strain and rate-of-strain response of a mechanical system after constant stress is applied or removed, or the stress and rate-of-stress response on application and maintenance of a constant strain. Representative results are found to agree quantitatively or qualitatively with available transient and temperature responses for a wide variety of materials under electrical or mechanical stimulation. In particular, the current transient response or stress relaxation response is constant for measuring times less than the shortest relaxation time of the system, then exhibits one or two regions with t−(1+ρi) behavior, and, finally, decreases very rapidly when the measuring time exceeds the longest time constant of the system. The theory leads to specific possibilities for the temperature behavior of ρi and to reasons why it is usually found to be less than unity in magnitude. Finally, conditions are analyzed for which the time-temperature superposition law of rheology applies.