Maximum and minimum free energies for a linear viscoelastic material

Abstract

Abstract.Certain results about free energies of materials with memory are proved, using the abstract formulation of thermodynamics, both in the general case and as applied within the theory of linear viscoelasticity.In particular, an integral equation for the strain continuation associated with the maximum recoverable work from a given linear viscoelastic state is shown to have a unique solution and is solved directly, using the Wiener-Hopf technique.This leads to an expression for the minimum free energy, previously derived by means of a variational technique in the frequency domain. A new variational method is developed in both the time and frequency domains. In the former case, this approach yields integral equations for both the minimum and maximum free energies associated with a given viscoelastic state. In the latter case, explicit forms of a family of free energies, associated with a given state of a discrete spectrum viscoelastic material, are derived. This includes both maximum and minimum free energies.

Free Energies and Minimal States for Scalar Linear Viscoelasticity

“…The factorization (5.11) corresponds to f = 1 and yields the minimum free energy ψ m (t). Each exchange of zeros, starting from these factors, can be shown to yield a free energy which is greater than or equal to the previous quantity ( [15], [2], p. 363).…”

“…8 is now briefly summarized. It is shown in [15,18] that, for materials with only isolated singularities, the quantity H (ω) is a rational function and has various factorizations other than (5.11), denoted by…”

“…Explicit algebraic representations of the minimum and related free energies have been given for materials with memory which have constitutive equations for stress 1 given by linear functionals of the strain; see for example [1,2,4,10,12,[15][16][17][18][19][20][21].…”

“…The concept of a minimal state was introduced in recent decades ( [2,7,8,10,15,25], for example) based on earlier work by Noll [27]. Recent work on this topic in the context of fractional hereditary materials may be found in [9,13].…”

“…Recent work on this topic in the context of fractional hereditary materials may be found in [9,13]. The term minimal state was first used in [15]. The property that a given quantity is a FMS is of central interest in the present work.…”