On monotonicity of the relaxation functions of viscoelastic materials

Abstract

This note is concerned with one-dimensional viscoelastic materials. Experiments show that for linear materials the relaxation functions are monotone decreasing functions of the time. This monotonic property has not, as far as I am aware, been characterized and in this note I provide a characterization formulated in terms of an assertion about the work done on the material over certain closed paths in strain space. More explicitly, consider any path in strain space which starts from equilibrium and arrives at the strain e at time t. We can now take the material around a closed path by either retracing the given path immediately or we can do it by holding the strain fixed at value e for a time r and then retracing the given path. Let us call the total work done on the material on the first closed path w(0) and on the second w(r). In general, because of the memory of the material, w(t) ≠ w(0). If it is true that w(r) ≥ w(0), whatever value r ≥ 0 has and whatever the initial strain path starting from equilibrium, we say that work is always increased by delay on retraced paths. The characterization proved here is that if G(·) is the relaxation function, with equilibrium elastic modulus G(∞), then G(·) −G(∞) is completely monotone if and only if work is always increased by delay on retraced paths.†