Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden
Abstract:
We study the forms of a range of free energy functionals for materials with
memory for two types of strain history, namely sinusoidal and exponential
behaviours. The work deals with discrete spectrum materials, which are those
with relaxation functions given by sums of decaying exponentials.
Various standard free energy functionals are explored, including the minimum
free energy and related quantities. It is shown that quite different formulae
are obtained, depending on the manner in which these functionals are evaluated,
particularly for those related to the minimum free energy. Such differences
are resolved. Various numerical plots of free energies are presented and discussed,
along with the rates of dissipation associated with them. This is the first
comprehensive exploration of quantitative comparisons between the various free
energy functionals. The results both illustrate certain properties that have
been deduced theoretically and expose an interesting proximity between apparently
unconnected functionals.
Submitted October 8, 2014. Published March 24, 2015.
Math Subject Classifications: 80A20, 74F05.
Key Words: Thermodynamics; memory effects; free energy functional;
rate of dissipation; numerical examples.
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Giovambattista Amendola Dipartimento di Matematica, Universita di Pisa Largo Bruno Pontecorvo 5 56127-Pisa, Italy email: amendola@dma.unipi.it |
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Mauro Fabrizio Dipartimento di Matematica, Universita di Bologna Piazza di Porta S. Donato 5 40127-Bologna, Italy email: fabrizio@dm.unibo.it |
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John Murrough Golden School of Mathematical Sciences Dublin Institute of Technology Kevin Street, Dublin 8, Ireland email: murrough.golden@dit.ie |
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