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Constructing free energies for materials with memory
1. | School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland |
  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.
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), 1037-1090.
doi: 10.1142/S0218202513500760. |
[2] |
G. Amendola, M. Fabrizio and M. Golden, Thermodynamics of Materials with Memory: Theory and Applications, Springer, New York, 2012.
doi: 10.1007/978-1-4614-1692-0. |
[3] |
G. Amendola, M. Fabrizio and J. M. Golden, Algebraic and numerical exploration of free energies for materials with memory, submitted for publication. |
[4] |
V. Berti and G. Gentili, The minimum free energy for isothermal dielectrics with memory, J. Non-Equil. Thermodyn., 24 (1999), 154-176. |
[5] |
B. D. Coleman, Thermodynamics of materials with memory, Arch. Rational Mech. Anal., 17 (1964), 1-46.
doi: 10.1007/BF00283864. |
[6] |
W. A. Day, The thermodynamics of materials with memory, in Materials with Memory, (ed. D. Graffi), Liguori, Naples, (1979). |
[7] |
G. Del Piero and L. Deseri, On the analytic expression of the free energy in linear viscoelasticity, J. Elasticity, 43 (1996), 247-278.
doi: 10.1007/BF00042503. |
[8] |
G. Del Piero and L. Deseri, On the concepts of state and free energy in linear viscoelasticity, Arch. Rational Mech. Anal., 138 (1997), 1-35.
doi: 10.1007/s002050050035. |
[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. |
[10] |
L. Deseri, G. Gentili and J. M. Golden, An explicit formula for the minimum free energy in linear viscoelasticity, J. Elasticity, 54 (1999), 141-185.
doi: 10.1023/A:1007646017347. |
[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), 43-96.
doi: 10.1007/s00205-005-0406-1. |
[12] |
L. Deseri and J. M. Golden, The minimum free energy for continuous spectrum materials, SIAM J. Appl Math., 67 (2007), 869-892.
doi: 10.1137/050639776. |
[13] |
M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM, Philadelphia, 1992.
doi: 10.1137/1.9781611970807. |
[14] |
M. Fabrizio and J. M. Golden, Maximum and minimum free energies for a linear viscoelastic material, Quart. Appl. Math., 60 (2002), 341-381. |
[15] |
M. Fabrizio, G. Gentili and J. M. Golden, Nonisothermal free energies for linear theories with memory, Mathematical and Computer Modeling, 39 (2004), 219-253.
doi: 10.1016/S0895-7177(04)90009-X. |
[16] |
M. Fabrizio, C. Giorgi and V. Pata, A new approach to equations with memory, Arch. Rational Mech. Anal., 198 (2010), 189-232.
doi: 10.1007/s00205-010-0300-3. |
[17] |
J. M. Golden, Free energies in the frequency domain: The scalar case, Quart. Appl. Math., 58 (2000), 127-150. |
[18] |
J. M. Golden, A proposal concerning the physical rate of dissipation in materials with memory, Quart. Appl. Math., 63 (2005), 117-155.
doi: 10.1177/1081286506061450. |
[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), 403-449.
doi: 10.1177/1081286505061450. |
[20] |
I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, Academic Press, New York, 1965. |
[21] |
D. Graffi, Analytic expression of some thermodynamic quantities in materials with memory, Rend. Sem. Mat. Univ. Padova, 68 (1982), 17-29. |
[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), 201-208. |
[23] |
D. Graffi, More on the analytic expression of free energy in materials with memory, Atti Acc. Scienze Torino, 120 (1986), 111-124. |
[24] |
W. Noll, A new mathematical theory of simple materials, Arch. Rational Mech. Anal., 48 (1972), 1-50.
doi: 10.1007/BF00253367. |
show all references
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), 1037-1090.
doi: 10.1142/S0218202513500760. |
[2] |
G. Amendola, M. Fabrizio and M. Golden, Thermodynamics of Materials with Memory: Theory and Applications, Springer, New York, 2012.
doi: 10.1007/978-1-4614-1692-0. |
[3] |
G. Amendola, M. Fabrizio and J. M. Golden, Algebraic and numerical exploration of free energies for materials with memory, submitted for publication. |
[4] |
V. Berti and G. Gentili, The minimum free energy for isothermal dielectrics with memory, J. Non-Equil. Thermodyn., 24 (1999), 154-176. |
[5] |
B. D. Coleman, Thermodynamics of materials with memory, Arch. Rational Mech. Anal., 17 (1964), 1-46.
doi: 10.1007/BF00283864. |
[6] |
W. A. Day, The thermodynamics of materials with memory, in Materials with Memory, (ed. D. Graffi), Liguori, Naples, (1979). |
[7] |
G. Del Piero and L. Deseri, On the analytic expression of the free energy in linear viscoelasticity, J. Elasticity, 43 (1996), 247-278.
doi: 10.1007/BF00042503. |
[8] |
G. Del Piero and L. Deseri, On the concepts of state and free energy in linear viscoelasticity, Arch. Rational Mech. Anal., 138 (1997), 1-35.
doi: 10.1007/s002050050035. |
[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. |
[10] |
L. Deseri, G. Gentili and J. M. Golden, An explicit formula for the minimum free energy in linear viscoelasticity, J. Elasticity, 54 (1999), 141-185.
doi: 10.1023/A:1007646017347. |
[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), 43-96.
doi: 10.1007/s00205-005-0406-1. |
[12] |
L. Deseri and J. M. Golden, The minimum free energy for continuous spectrum materials, SIAM J. Appl Math., 67 (2007), 869-892.
doi: 10.1137/050639776. |
[13] |
M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM, Philadelphia, 1992.
doi: 10.1137/1.9781611970807. |
[14] |
M. Fabrizio and J. M. Golden, Maximum and minimum free energies for a linear viscoelastic material, Quart. Appl. Math., 60 (2002), 341-381. |
[15] |
M. Fabrizio, G. Gentili and J. M. Golden, Nonisothermal free energies for linear theories with memory, Mathematical and Computer Modeling, 39 (2004), 219-253.
doi: 10.1016/S0895-7177(04)90009-X. |
[16] |
M. Fabrizio, C. Giorgi and V. Pata, A new approach to equations with memory, Arch. Rational Mech. Anal., 198 (2010), 189-232.
doi: 10.1007/s00205-010-0300-3. |
[17] |
J. M. Golden, Free energies in the frequency domain: The scalar case, Quart. Appl. Math., 58 (2000), 127-150. |
[18] |
J. M. Golden, A proposal concerning the physical rate of dissipation in materials with memory, Quart. Appl. Math., 63 (2005), 117-155.
doi: 10.1177/1081286506061450. |
[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), 403-449.
doi: 10.1177/1081286505061450. |
[20] |
I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products, Academic Press, New York, 1965. |
[21] |
D. Graffi, Analytic expression of some thermodynamic quantities in materials with memory, Rend. Sem. Mat. Univ. Padova, 68 (1982), 17-29. |
[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), 201-208. |
[23] |
D. Graffi, More on the analytic expression of free energy in materials with memory, Atti Acc. Scienze Torino, 120 (1986), 111-124. |
[24] |
W. Noll, A new mathematical theory of simple materials, Arch. Rational Mech. Anal., 48 (1972), 1-50.
doi: 10.1007/BF00253367. |
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