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1.1 MATHMATICS, Applied mathematics

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Pre print of article due for publication in Quarterly of Applied Mathematics, 2018.


A material with memory typically has a set of many free energy functionals associated with it, all members of which yield the same constitutive relations. An alternative interpretation of this set is explored in the present work. Explicit formulae are derived for the free energy and total dissipation of an arbitrary material in the cases of step function and sinusoidal/exponential histories. Expressions for the fraction of stored and dissipated energy are deduced. Also, various formulae are given for discrete spectrum materials. For materials with relaxation function containing one decaying exponential, the associated Day functional is the physical free energy. For more general materials, we seek a best fit of the relaxation function with one decaying expo- nential to that chosen for the general case. The free energy, total dissipation and fractions of stored and dissipated energies relating to the Day material are derived for the various histories. Similar data, in the case of the general mate- rial, are explored for the minimum and maximum free energies and also for a centrally located free energy given in the literature. Various plots of aspects of this data, including comparisons between the behaviour for general and Day materials, are presented and discussed.

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