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The state of fractional hereditary materials (FHM)
1. | Department of Civil, Environmental and Mechanical Engineering, Via Mesiano 77, 38123 Trento, Italy |
2. | Department of Civil, Environmental and Aerospace Engineering, Viale delle Scienze - Build. 8 - 90128 Palermo, Italy |
3. | Department of Civil and Environmental Engineering, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, United States |
References:
[1] |
G. Amendola, M. Fabrizio and M. J. Golden, Thermodynamics of Materials with Memory: Theory and Applications, Springer, 2012.
doi: 10.1007/978-1-4614-1692-0. |
[2] |
R. L. Bagley and P. J. Torvik, Fractional calculus - A different approach to analysis of viscoelastically damped structures, The American Institute of Aeronautics and Astronautics, 21 (1983), 741-748.
doi: 10.2514/3.8142. |
[3] |
R. L. Bagley and P. J. Torvik, On the fractional calculus model of viscoelastic behavior, Journal of Rheology, 30 (1986), 133-155.
doi: 10.1122/1.549887. |
[4] |
M. Baumgaertel and H. H. Winter, Interrelation between continuous and discrete relaxation time spectra, Journal of Non-Newtonian Fluid Mechanics, 44 (1992), 15-36.
doi: 10.1016/0377-0257(92)80043-W. |
[5] |
G. Scott Blair, The role of psychophysics in rheology, Journal of Colloid Science, 2 (1947), 21-32. |
[6] |
B. D. Coleman and D. C. Newman, On the rheology of cold drawing II. Viscoelastic materials, Journal of Polymer Science: Part B: Polymer Physics, 30 (1992), 25-47.
doi: 10.1002/polb.1992.090300104. |
[7] |
G. Del Piero and L. Deseri, On the Concepts of state and free energy in linear viscoelasticity, Archive for Rational Mechanics and Analysis, 138 (1997), 1-35.
doi: 10.1007/s002050050035. |
[8] |
L. Deseri and G. Zurlo, The stretching elasticity of biomembranes determines their line tension and bending rigidity, Biomechanics and Modeling in Mechanobiology, 12 (2013), 1233-1242.
doi: 10.1007/s10237-013-0478-z. |
[9] |
L. Deseri, M. D. Piccioni and G. Zurlo, Derivation of a new free energy for biological membranes, Continuum Mechanics and Thermodynamics, 20 (2008), 255-273.
doi: 10.1007/s00161-008-0081-1. |
[10] |
L. Deseri, M. Di Paola, M. Zingales and P. Pollaci, Power-law hereditariness of hierarchical fractal bones, International Journal of Numerical Methods in Biomedical Engineering, 29 (2013), 1338-1360.
doi: 10.1002/cnm.2572. |
[11] |
L. Deseri, G. Gentili and M. J. Golden, An expression for the minimal free energy in linear viscoelasticity, Journal of Elasticity, 54 (1999), 141-185.
doi: 10.1023/A:1007646017347. |
[12] |
L. Deseri, M. Fabrizio and M. J. Golden, The concept of a minimal state in viscoelasticity: New free energies and applications to PDEs, Archive for Rational Mechanics and Analysis, 181 (2006), 43-96.
doi: 10.1007/s00205-005-0406-1. |
[13] |
L. Deseri and M. J. Golden, The minimum free energy for continuous spectrum materials, SIAM Journal on Applied Mathematics, 67 (2007), 869-892.
doi: 10.1137/050639776. |
[14] |
M. Di Paola, F. P. Pinnola and M. Zingales, A discrete mechanical model of fractional hereditary materials, Meccanica, 48 (2013), 1573-1586.
doi: 10.1007/s11012-012-9685-4. |
[15] |
M. Di Paola and M. Zingales, Exact mechanical models of fractional hereditary materials, Journal of Rheology, 56 (2012), 983-1004.
doi: 10.1122/1.4717492. |
[16] |
M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM Philadelphia, 1992.
doi: 10.1137/1.9781611970807. |
[17] |
M. Fabrizio and M. J. Golden, Minimum free energies for materials with finite memory, Journal of Elasticity, 72 (2003), 121-143.
doi: 10.1023/B:ELAS.0000018771.71385.05. |
[18] |
M. Fabrizio, G. Gentili and M. J. Golden, The minimum free energy for a class of compressible viscoelastic fluids, Differential and Integral Equations, 7 (2002), 319-342. |
[19] |
L. Galuppi and G. Royer Carfagni, Laminated beams with viscoelastic interlayer, International Journal of Solids and Structures, 49 (2012), 2637-2645.
doi: 10.1016/j.ijsolstr.2012.05.028. |
[20] |
L. Galuppi and G. Royer Carfagni, The design of laminated glass under time-dependent loading, International Journal of Mechanical Sciences, 68 (2013), 67-75.
doi: 10.1016/j.ijmecsci.2012.12.019. |
[21] |
A. Gemant, A method of analyzing experimental results obtained from elasto-viscous bodies, Physics, 7 (1936), 311-317.
doi: 10.1063/1.1745400. |
[22] |
G. Gentili, Maximum recoverable work, minimum free energy and state space in linear viscoelasticity, Quarterly of Applied Mathematics, 60 (2002), 153-182. |
[23] |
C. Giorgi and A. Morro, Viscoelastic solids with unbounded relaxation function, Continuum Mechanics and Thermodynamics, 4 (1992), 151-165.
doi: 10.1007/BF01125696. |
[24] |
D. Graffi and M. Fabrizio, Sulla nozione di stato per materiali viscoelastici di tipo "rate'', Atti. Accademia dei Lincei, Rendiconti di Fisica, 83 (1989), 201-208. |
[25] |
R. Guedes, J. Simoes and J. Morais, Viscoelastic behaviour and failure of bovine cancellous bone under constant strain rate, Journal of Biomechanics, 39 (2006), 49-60.
doi: 10.1016/j.jbiomech.2004.11.005. |
[26] |
A. Jaishankar and G. H. McKinley, Power-law rheology in the bulk and at the interface: Quasi-properties and fractional constitutive equations, Proceedings of the Royal Society A, 469 (2013).
doi: 10.1098/rspa.2012.0284. |
[27] |
A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, Prentice Hall, 1970. |
[28] |
R. Lakes, Viscoelastic Materials, Cambridge University Press, 2009.
doi: 10.1017/CBO9780511626722. |
[29] |
R. L. Magin and T. J. Royston, Fractional-order elastic models of cartilage: A multi-scale approach, Communications in Nonlinear Science and Numerical Simulation, 15 (2010), 657-664.
doi: 10.1016/j.cnsns.2009.05.008. |
[30] |
F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, in Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, Wien, 1997, 291-348. arXiv:1201.0863.
doi: 10.1007/978-3-7091-2664-6_7. |
[31] |
F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010.
doi: 10.1142/9781848163300. |
[32] |
F. Mainardi and G. Spada, Creep, relaxation and viscosity properties for basic fractional models in rheology, The European Physical Journal, 193 (2011), 133-160. arXiv:1110.3400.
doi: 10.1140/epjst/e2011-01387-1. |
[33] |
G. Mishuris, A. Piccolroaz and E. Radi, Steady-state propagation of a Mode III crack in couple stress elastic materials, International Journal of Engineering Science, 61 (2012), 112-128.
doi: 10.1016/j.ijengsci.2012.06.015. |
[34] |
L. Morini, A. Piccolroaz, G. Mishuris and E. Radi, On fracture criteria for a crack propagating in couple stress elastic materials, International Journal of Engineering Science, 71 (2013), 45-61.
doi: 10.1016/j.ijengsci.2013.05.005. |
[35] |
S. Nawaz, P. Sanchez, K. Bodensiek, S. Li, M. Simons and I. A. T. Schaap, Cell viscoelasticity measured with AFM and vertical optical trapping at sub-micrometer deformations, PLoS ONE, 7 (2012), e45297.
doi: 10.1371/journal.pone.0045297. |
[36] |
W. Noll, A new mathematical theory of simple materials, Archive for Rational Mechanics and Analysis, 48 (1972), 1-50. |
[37] |
P. Nutting, A new general law of deformation, Journal of The Franklin Institute, 191 (1921), 679-685.
doi: 10.1016/S0016-0032(21)90171-6. |
[38] |
V. Quaglini, V. La Russa and S. Corneo, Nonlinear stress relaxation of trabecular bone, Mechanics Research Communications, 36 (2009), 275-283.
doi: 10.1016/j.mechrescom.2008.10.012. |
[39] |
F. Riewe, Mechanics with fractional derivatives, Physical Review E, 55 (1997), 3581-3592.
doi: 10.1103/PhysRevE.55.3581. |
[40] |
S. Saha Ray, B. P. Poddar and R. K. Bera, Analytical solution of a dynamic system containing fractional derivative of order one-half by Adomian decomposition method, Journal of Applied Mechanics, 72 (2005), 290-295.
doi: 10.1115/1.1839184. |
[41] |
S. Sakakibara, Relaxation properties of fractional derivative viscoelasticity models, Nonlinear Analysis, 47 (2001), 5449-5454.
doi: 10.1016/S0362-546X(01)00649-6. |
[42] |
S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integral and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, 1993. |
[43] |
N. Sarkar and A. Lahiri, Effect of fractional parameter on plane waves in a rotating elastic medium under fractional order generalized thermoelasticity, International Journal of Applied Mechanics, 4 (2012).
doi: 10.1142/S1758825112500305. |
[44] |
V. Sharma, A. Jaishankar, Y. C. Wang and G. H. McKinley, Rheology of globular proteins: Apparent yield stress, high shear rate viscosity and interfacial viscoelasticity of bovine serum albumin solutions, Soft Matter, 7 (2011), 5150-5160.
doi: 10.1039/c0sm01312a. |
[45] |
M. K. Suchorsky and R. H. Rand, A pair of van der Pol oscillators coupled by fractional derivatives, Nonlinear Dynamics, 69 (2012), 313-324.
doi: 10.1007/s11071-011-0266-1. |
[46] |
M. R. Sunny, R. K. Kapania, R. D. Moffitt, A. Mishra and N. Goulbourne, A modified fractional calculus approach to model hysteresis, Journal of Applied Mechanics, 77 (2010).
doi: 10.1115/1.4000413. |
[47] |
P. Yang, Y. C. Lam and K. Q. Zhu, Constitutive equation with fractional derivatives for generalized UCM model, Journal of Non-Newtonian Fluid Mechanics, 165 (2010), 88-97.
doi: 10.1016/j.jnnfm.2009.10.002. |
[48] |
show all references
References:
[1] |
G. Amendola, M. Fabrizio and M. J. Golden, Thermodynamics of Materials with Memory: Theory and Applications, Springer, 2012.
doi: 10.1007/978-1-4614-1692-0. |
[2] |
R. L. Bagley and P. J. Torvik, Fractional calculus - A different approach to analysis of viscoelastically damped structures, The American Institute of Aeronautics and Astronautics, 21 (1983), 741-748.
doi: 10.2514/3.8142. |
[3] |
R. L. Bagley and P. J. Torvik, On the fractional calculus model of viscoelastic behavior, Journal of Rheology, 30 (1986), 133-155.
doi: 10.1122/1.549887. |
[4] |
M. Baumgaertel and H. H. Winter, Interrelation between continuous and discrete relaxation time spectra, Journal of Non-Newtonian Fluid Mechanics, 44 (1992), 15-36.
doi: 10.1016/0377-0257(92)80043-W. |
[5] |
G. Scott Blair, The role of psychophysics in rheology, Journal of Colloid Science, 2 (1947), 21-32. |
[6] |
B. D. Coleman and D. C. Newman, On the rheology of cold drawing II. Viscoelastic materials, Journal of Polymer Science: Part B: Polymer Physics, 30 (1992), 25-47.
doi: 10.1002/polb.1992.090300104. |
[7] |
G. Del Piero and L. Deseri, On the Concepts of state and free energy in linear viscoelasticity, Archive for Rational Mechanics and Analysis, 138 (1997), 1-35.
doi: 10.1007/s002050050035. |
[8] |
L. Deseri and G. Zurlo, The stretching elasticity of biomembranes determines their line tension and bending rigidity, Biomechanics and Modeling in Mechanobiology, 12 (2013), 1233-1242.
doi: 10.1007/s10237-013-0478-z. |
[9] |
L. Deseri, M. D. Piccioni and G. Zurlo, Derivation of a new free energy for biological membranes, Continuum Mechanics and Thermodynamics, 20 (2008), 255-273.
doi: 10.1007/s00161-008-0081-1. |
[10] |
L. Deseri, M. Di Paola, M. Zingales and P. Pollaci, Power-law hereditariness of hierarchical fractal bones, International Journal of Numerical Methods in Biomedical Engineering, 29 (2013), 1338-1360.
doi: 10.1002/cnm.2572. |
[11] |
L. Deseri, G. Gentili and M. J. Golden, An expression for the minimal free energy in linear viscoelasticity, Journal of Elasticity, 54 (1999), 141-185.
doi: 10.1023/A:1007646017347. |
[12] |
L. Deseri, M. Fabrizio and M. J. Golden, The concept of a minimal state in viscoelasticity: New free energies and applications to PDEs, Archive for Rational Mechanics and Analysis, 181 (2006), 43-96.
doi: 10.1007/s00205-005-0406-1. |
[13] |
L. Deseri and M. J. Golden, The minimum free energy for continuous spectrum materials, SIAM Journal on Applied Mathematics, 67 (2007), 869-892.
doi: 10.1137/050639776. |
[14] |
M. Di Paola, F. P. Pinnola and M. Zingales, A discrete mechanical model of fractional hereditary materials, Meccanica, 48 (2013), 1573-1586.
doi: 10.1007/s11012-012-9685-4. |
[15] |
M. Di Paola and M. Zingales, Exact mechanical models of fractional hereditary materials, Journal of Rheology, 56 (2012), 983-1004.
doi: 10.1122/1.4717492. |
[16] |
M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM Philadelphia, 1992.
doi: 10.1137/1.9781611970807. |
[17] |
M. Fabrizio and M. J. Golden, Minimum free energies for materials with finite memory, Journal of Elasticity, 72 (2003), 121-143.
doi: 10.1023/B:ELAS.0000018771.71385.05. |
[18] |
M. Fabrizio, G. Gentili and M. J. Golden, The minimum free energy for a class of compressible viscoelastic fluids, Differential and Integral Equations, 7 (2002), 319-342. |
[19] |
L. Galuppi and G. Royer Carfagni, Laminated beams with viscoelastic interlayer, International Journal of Solids and Structures, 49 (2012), 2637-2645.
doi: 10.1016/j.ijsolstr.2012.05.028. |
[20] |
L. Galuppi and G. Royer Carfagni, The design of laminated glass under time-dependent loading, International Journal of Mechanical Sciences, 68 (2013), 67-75.
doi: 10.1016/j.ijmecsci.2012.12.019. |
[21] |
A. Gemant, A method of analyzing experimental results obtained from elasto-viscous bodies, Physics, 7 (1936), 311-317.
doi: 10.1063/1.1745400. |
[22] |
G. Gentili, Maximum recoverable work, minimum free energy and state space in linear viscoelasticity, Quarterly of Applied Mathematics, 60 (2002), 153-182. |
[23] |
C. Giorgi and A. Morro, Viscoelastic solids with unbounded relaxation function, Continuum Mechanics and Thermodynamics, 4 (1992), 151-165.
doi: 10.1007/BF01125696. |
[24] |
D. Graffi and M. Fabrizio, Sulla nozione di stato per materiali viscoelastici di tipo "rate'', Atti. Accademia dei Lincei, Rendiconti di Fisica, 83 (1989), 201-208. |
[25] |
R. Guedes, J. Simoes and J. Morais, Viscoelastic behaviour and failure of bovine cancellous bone under constant strain rate, Journal of Biomechanics, 39 (2006), 49-60.
doi: 10.1016/j.jbiomech.2004.11.005. |
[26] |
A. Jaishankar and G. H. McKinley, Power-law rheology in the bulk and at the interface: Quasi-properties and fractional constitutive equations, Proceedings of the Royal Society A, 469 (2013).
doi: 10.1098/rspa.2012.0284. |
[27] |
A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, Prentice Hall, 1970. |
[28] |
R. Lakes, Viscoelastic Materials, Cambridge University Press, 2009.
doi: 10.1017/CBO9780511626722. |
[29] |
R. L. Magin and T. J. Royston, Fractional-order elastic models of cartilage: A multi-scale approach, Communications in Nonlinear Science and Numerical Simulation, 15 (2010), 657-664.
doi: 10.1016/j.cnsns.2009.05.008. |
[30] |
F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, in Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, Wien, 1997, 291-348. arXiv:1201.0863.
doi: 10.1007/978-3-7091-2664-6_7. |
[31] |
F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010.
doi: 10.1142/9781848163300. |
[32] |
F. Mainardi and G. Spada, Creep, relaxation and viscosity properties for basic fractional models in rheology, The European Physical Journal, 193 (2011), 133-160. arXiv:1110.3400.
doi: 10.1140/epjst/e2011-01387-1. |
[33] |
G. Mishuris, A. Piccolroaz and E. Radi, Steady-state propagation of a Mode III crack in couple stress elastic materials, International Journal of Engineering Science, 61 (2012), 112-128.
doi: 10.1016/j.ijengsci.2012.06.015. |
[34] |
L. Morini, A. Piccolroaz, G. Mishuris and E. Radi, On fracture criteria for a crack propagating in couple stress elastic materials, International Journal of Engineering Science, 71 (2013), 45-61.
doi: 10.1016/j.ijengsci.2013.05.005. |
[35] |
S. Nawaz, P. Sanchez, K. Bodensiek, S. Li, M. Simons and I. A. T. Schaap, Cell viscoelasticity measured with AFM and vertical optical trapping at sub-micrometer deformations, PLoS ONE, 7 (2012), e45297.
doi: 10.1371/journal.pone.0045297. |
[36] |
W. Noll, A new mathematical theory of simple materials, Archive for Rational Mechanics and Analysis, 48 (1972), 1-50. |
[37] |
P. Nutting, A new general law of deformation, Journal of The Franklin Institute, 191 (1921), 679-685.
doi: 10.1016/S0016-0032(21)90171-6. |
[38] |
V. Quaglini, V. La Russa and S. Corneo, Nonlinear stress relaxation of trabecular bone, Mechanics Research Communications, 36 (2009), 275-283.
doi: 10.1016/j.mechrescom.2008.10.012. |
[39] |
F. Riewe, Mechanics with fractional derivatives, Physical Review E, 55 (1997), 3581-3592.
doi: 10.1103/PhysRevE.55.3581. |
[40] |
S. Saha Ray, B. P. Poddar and R. K. Bera, Analytical solution of a dynamic system containing fractional derivative of order one-half by Adomian decomposition method, Journal of Applied Mechanics, 72 (2005), 290-295.
doi: 10.1115/1.1839184. |
[41] |
S. Sakakibara, Relaxation properties of fractional derivative viscoelasticity models, Nonlinear Analysis, 47 (2001), 5449-5454.
doi: 10.1016/S0362-546X(01)00649-6. |
[42] |
S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integral and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, 1993. |
[43] |
N. Sarkar and A. Lahiri, Effect of fractional parameter on plane waves in a rotating elastic medium under fractional order generalized thermoelasticity, International Journal of Applied Mechanics, 4 (2012).
doi: 10.1142/S1758825112500305. |
[44] |
V. Sharma, A. Jaishankar, Y. C. Wang and G. H. McKinley, Rheology of globular proteins: Apparent yield stress, high shear rate viscosity and interfacial viscoelasticity of bovine serum albumin solutions, Soft Matter, 7 (2011), 5150-5160.
doi: 10.1039/c0sm01312a. |
[45] |
M. K. Suchorsky and R. H. Rand, A pair of van der Pol oscillators coupled by fractional derivatives, Nonlinear Dynamics, 69 (2012), 313-324.
doi: 10.1007/s11071-011-0266-1. |
[46] |
M. R. Sunny, R. K. Kapania, R. D. Moffitt, A. Mishra and N. Goulbourne, A modified fractional calculus approach to model hysteresis, Journal of Applied Mechanics, 77 (2010).
doi: 10.1115/1.4000413. |
[47] |
P. Yang, Y. C. Lam and K. Q. Zhu, Constitutive equation with fractional derivatives for generalized UCM model, Journal of Non-Newtonian Fluid Mechanics, 165 (2010), 88-97.
doi: 10.1016/j.jnnfm.2009.10.002. |
[48] |
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