Abstract
The concept of a minimal state was introduced in recent decades, based on earlier work by Noll. The property that a given quantity is a functional of the minimal state is of central interest in the present work. Using a standard representation of a free energy associated with a linear memory constitutive relation, a new condition, involving linear functionals, is derived which, if satisfied, ensures that the free energy is a functional of the minimal state. Using this result and recent work on constructing free energy functionals, it is shown that if the kernel of the rate of dissipation functional is given by sums of products, the associated free energy functional is a functional of the minimal state.
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Notes
We consider for definiteness here isothermal mechanical problems, indeed those for solid viscoelastic materials. Also, only the scalar case is considered, which simplifies the algebra and allows us to focus on the essential structure of the arguments. It must be emphasized however that similar results have been given, with little extra difficulty, for viscoelastic fluids, non-isothermal problems, electromagnetic and non-simple materials with memory, and also for the general tensor theories relating to all of these. Specifically, the fundamental result given by Proposition 1 in Sect. 4 can be shown to hold, with some rephrasing of the argument, for the full tensor case. However, the developments of Sects. 9 and 10 generalize similarly only subject to factorization restrictions, which were first discussed for the simplest case in [10] (see also [2], p. 256).
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Amendola, G., Fabrizio, M. & Golden, J.M. Free Energies and Minimal States for Scalar Linear Viscoelasticity. J Elast 123, 97–123 (2016). https://doi.org/10.1007/s10659-015-9549-y
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DOI: https://doi.org/10.1007/s10659-015-9549-y