Minimum Free Energies for Materials with Finite Memory

Abstract

Finite memory viscoelastic materials are of interest because (a) they are not necessarily experimentally distinguishable from materials with infinite memory; and (b) the assumption of infinite memory can, in certain contexts, lead to results that run counter to physical intuition. An example of this - the quasi-static viscoelastic membrane in a frictional medium - is discussed. It is shown that, for a finite memory material, the singularity structure of the Fourier transform of the relaxation function derivative is quite different from the infinite memory case in the sense that it is an entire function with all its singularities being essential singularities at infinity. The formula for the minimum free energy [1] is still valid in this case. In contrast to the work function, this quantity, and all other functions of the minimal state, depend only on the values of the history over the period when the relaxation function derivative is nonzero. The factorization required to determine the form of the minimum free energy can be carried out explicitly for simple step-function choices of the relaxation function derivative. The two simplest cases are fully worked through and explicit formulae are given for all relevant quantities.