# American Institute of Mathematical Sciences

September  2014, 3(3): 447-483. doi: 10.3934/eect.2014.3.447

## Constructing free energies for materials with memory

 1 School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland

Received  January 2013 Revised  February 2014 Published  August 2014

The free energy for most materials with memory is not unique. There is a convex set of free energy functionals with a minimum and a maximum element. Various functionals have been shown to have the properties of a free energy for materials with particular types of relaxation behaviour. Also, over the last decade or more, forms have been given for the minimum and related free energies. These are all quadratic functionals which yield linear memory terms in the constitutive equations for the stress.
A difficulty in constructing free energy functionals arises in making choices that ensure a non-negative quadratic form both for the free energy and for the rate of dissipation. We propose a technique which renders this task more straightforward. Instead of constructing the free energy and determining from this the rate of dissipation, which may not have the required non-negativity, the procedure is reversed, which guarantees a satisfactory free energy functional.
Certain results for quadratic functionals in the time and frequency domains are derived, providing a platform for this alternative approach, which produces new free energies, including a family of functionals that are generalizations of the minimum and related free energies.
Citation: John Murrough Golden. Constructing free energies for materials with memory. Evolution Equations & Control Theory, 2014, 3 (3) : 447-483. doi: 10.3934/eect.2014.3.447
##### References:
 [1] G. Amendola, M. Fabrizio and J. M. Golden, Free energies in a general non-local theory of a material with memory,, Mathematical Models and Methods in Applied Sciences, 24 (2014), 1037.  doi: 10.1142/S0218202513500760.  Google Scholar [2] G. Amendola, M. Fabrizio and M. Golden, Thermodynamics of Materials with Memory: Theory and Applications,, Springer, (2012).  doi: 10.1007/978-1-4614-1692-0.  Google Scholar [3] G. Amendola, M. Fabrizio and J. M. Golden, Algebraic and numerical exploration of free energies for materials with memory,, submitted for publication., ().   Google Scholar [4] V. Berti and G. Gentili, The minimum free energy for isothermal dielectrics with memory,, J. Non-Equil. Thermodyn., 24 (1999), 154.   Google Scholar [5] B. D. Coleman, Thermodynamics of materials with memory,, Arch. Rational Mech. Anal., 17 (1964), 1.  doi: 10.1007/BF00283864.  Google Scholar [6] W. A. Day, The thermodynamics of materials with memory,, in Materials with Memory, (1979).   Google Scholar [7] G. Del Piero and L. Deseri, On the analytic expression of the free energy in linear viscoelasticity,, J. Elasticity, 43 (1996), 247.  doi: 10.1007/BF00042503.  Google Scholar [8] G. Del Piero and L. Deseri, On the concepts of state and free energy in linear viscoelasticity,, Arch. Rational Mech. Anal., 138 (1997), 1.  doi: 10.1007/s002050050035.  Google Scholar [9] L. Deseri, M. Di Paola, P. Pollaci and M. Zingales, The state of fractional hereditary materials (FHM),, Discrete and Continuous Dynamical Systems - B to appear., ().   Google Scholar [10] L. Deseri, G. Gentili and J. M. Golden, An explicit formula for the minimum free energy in linear viscoelasticity,, J. Elasticity, 54 (1999), 141.  doi: 10.1023/A:1007646017347.  Google Scholar [11] L. Deseri, M. Fabrizio and J. M. Golden, On the concept of a minimal state in viscoelasticity: New free energies and applications to $PDE_S$,, Arch. Rational Mech. Anal., 181 (2006), 43.  doi: 10.1007/s00205-005-0406-1.  Google Scholar [12] L. Deseri and J. M. Golden, The minimum free energy for continuous spectrum materials,, SIAM J. Appl Math., 67 (2007), 869.  doi: 10.1137/050639776.  Google Scholar [13] M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity,, SIAM, (1992).  doi: 10.1137/1.9781611970807.  Google Scholar [14] M. Fabrizio and J. M. Golden, Maximum and minimum free energies for a linear viscoelastic material,, Quart. Appl. Math., 60 (2002), 341.   Google Scholar [15] M. Fabrizio, G. Gentili and J. M. Golden, Nonisothermal free energies for linear theories with memory,, Mathematical and Computer Modeling, 39 (2004), 219.  doi: 10.1016/S0895-7177(04)90009-X.  Google Scholar [16] M. Fabrizio, C. Giorgi and V. Pata, A new approach to equations with memory,, Arch. Rational Mech. Anal., 198 (2010), 189.  doi: 10.1007/s00205-010-0300-3.  Google Scholar [17] J. M. Golden, Free energies in the frequency domain: The scalar case,, Quart. Appl. Math., 58 (2000), 127.   Google Scholar [18] J. M. Golden, A proposal concerning the physical rate of dissipation in materials with memory,, Quart. Appl. Math., 63 (2005), 117.  doi: 10.1177/1081286506061450.  Google Scholar [19] J. M. Golden, A proposal concerning the physical dissipation of materials with memory: the non-isothermal case,, Mathematics and Mechanics of Solids, 12 (2007), 403.  doi: 10.1177/1081286505061450.  Google Scholar [20] I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products,, Academic Press, (1965).   Google Scholar [21] D. Graffi, Analytic expression of some thermodynamic quantities in materials with memory,, Rend. Sem. Mat. Univ. Padova, 68 (1982), 17.   Google Scholar [22] D. Graffi and M. Fabrizio, On the notion of state for viscoelastic materials of "rate'' type,, Atti della Accademia Nazionale dei Lincei, 83 (1990), 201.   Google Scholar [23] D. Graffi, More on the analytic expression of free energy in materials with memory,, Atti Acc. Scienze Torino, 120 (1986), 111.   Google Scholar [24] W. Noll, A new mathematical theory of simple materials,, Arch. Rational Mech. Anal., 48 (1972), 1.  doi: 10.1007/BF00253367.  Google Scholar

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##### References:
 [1] G. Amendola, M. Fabrizio and J. M. Golden, Free energies in a general non-local theory of a material with memory,, Mathematical Models and Methods in Applied Sciences, 24 (2014), 1037.  doi: 10.1142/S0218202513500760.  Google Scholar [2] G. Amendola, M. Fabrizio and M. Golden, Thermodynamics of Materials with Memory: Theory and Applications,, Springer, (2012).  doi: 10.1007/978-1-4614-1692-0.  Google Scholar [3] G. Amendola, M. Fabrizio and J. M. Golden, Algebraic and numerical exploration of free energies for materials with memory,, submitted for publication., ().   Google Scholar [4] V. Berti and G. Gentili, The minimum free energy for isothermal dielectrics with memory,, J. Non-Equil. Thermodyn., 24 (1999), 154.   Google Scholar [5] B. D. Coleman, Thermodynamics of materials with memory,, Arch. Rational Mech. Anal., 17 (1964), 1.  doi: 10.1007/BF00283864.  Google Scholar [6] W. A. Day, The thermodynamics of materials with memory,, in Materials with Memory, (1979).   Google Scholar [7] G. Del Piero and L. Deseri, On the analytic expression of the free energy in linear viscoelasticity,, J. Elasticity, 43 (1996), 247.  doi: 10.1007/BF00042503.  Google Scholar [8] G. Del Piero and L. Deseri, On the concepts of state and free energy in linear viscoelasticity,, Arch. Rational Mech. Anal., 138 (1997), 1.  doi: 10.1007/s002050050035.  Google Scholar [9] L. Deseri, M. Di Paola, P. Pollaci and M. Zingales, The state of fractional hereditary materials (FHM),, Discrete and Continuous Dynamical Systems - B to appear., ().   Google Scholar [10] L. Deseri, G. Gentili and J. M. Golden, An explicit formula for the minimum free energy in linear viscoelasticity,, J. Elasticity, 54 (1999), 141.  doi: 10.1023/A:1007646017347.  Google Scholar [11] L. Deseri, M. Fabrizio and J. M. Golden, On the concept of a minimal state in viscoelasticity: New free energies and applications to $PDE_S$,, Arch. Rational Mech. Anal., 181 (2006), 43.  doi: 10.1007/s00205-005-0406-1.  Google Scholar [12] L. Deseri and J. M. Golden, The minimum free energy for continuous spectrum materials,, SIAM J. Appl Math., 67 (2007), 869.  doi: 10.1137/050639776.  Google Scholar [13] M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity,, SIAM, (1992).  doi: 10.1137/1.9781611970807.  Google Scholar [14] M. Fabrizio and J. M. Golden, Maximum and minimum free energies for a linear viscoelastic material,, Quart. Appl. Math., 60 (2002), 341.   Google Scholar [15] M. Fabrizio, G. Gentili and J. M. Golden, Nonisothermal free energies for linear theories with memory,, Mathematical and Computer Modeling, 39 (2004), 219.  doi: 10.1016/S0895-7177(04)90009-X.  Google Scholar [16] M. Fabrizio, C. Giorgi and V. Pata, A new approach to equations with memory,, Arch. Rational Mech. Anal., 198 (2010), 189.  doi: 10.1007/s00205-010-0300-3.  Google Scholar [17] J. M. Golden, Free energies in the frequency domain: The scalar case,, Quart. Appl. Math., 58 (2000), 127.   Google Scholar [18] J. M. Golden, A proposal concerning the physical rate of dissipation in materials with memory,, Quart. Appl. Math., 63 (2005), 117.  doi: 10.1177/1081286506061450.  Google Scholar [19] J. M. Golden, A proposal concerning the physical dissipation of materials with memory: the non-isothermal case,, Mathematics and Mechanics of Solids, 12 (2007), 403.  doi: 10.1177/1081286505061450.  Google Scholar [20] I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products,, Academic Press, (1965).   Google Scholar [21] D. Graffi, Analytic expression of some thermodynamic quantities in materials with memory,, Rend. Sem. Mat. Univ. Padova, 68 (1982), 17.   Google Scholar [22] D. Graffi and M. Fabrizio, On the notion of state for viscoelastic materials of "rate'' type,, Atti della Accademia Nazionale dei Lincei, 83 (1990), 201.   Google Scholar [23] D. Graffi, More on the analytic expression of free energy in materials with memory,, Atti Acc. Scienze Torino, 120 (1986), 111.   Google Scholar [24] W. Noll, A new mathematical theory of simple materials,, Arch. Rational Mech. Anal., 48 (1972), 1.  doi: 10.1007/BF00253367.  Google Scholar
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