# American Institute of Mathematical Sciences

September  2014, 19(7): 2065-2089. doi: 10.3934/dcdsb.2014.19.2065

## 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

Received  April 2013 Revised  September 2013 Published  August 2014

The widespread interest on the hereditary behavior of biological and bioinspired materials motivates deeper studies on their macroscopic minimal" state. The resulting integral equations for the detected relaxation and creep power-laws, of exponent $\beta$, are characterized by fractional operators. Here strains in $SBV_{loc}$ are considered to account for time-like jumps. Consistently, starting from stresses in $L_{loc}^{r}$, $r\in [1,\beta^{-1}], \, \, \beta\in(0,1)$ we reconstruct the corresponding strain by extending a result in [42]. The minimal" state is explored by showing that different histories delivering the same response are such that the fractional derivative of their difference is zero for all times. This equation is solved through a one-parameter family of strains whose related stresses converge to the response characterizing the original problem. This provides an approximation formula for the state variable, namely the residual stress associated to the difference of the histories above. Very little is known about the microstructural origins of the detected power-laws. Recent rheological models, based on a top-plate adhering and moving on functionally graded microstructures, allow for showing that the resultant of the underlying microstresses" matches the action recorded at the top-plate of such models, yielding a relationship between the macroscopic state and the microstresses".
Citation: Luca Deseri, Massiliano Zingales, Pietro Pollaci. The state of fractional hereditary materials (FHM). Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2065-2089. doi: 10.3934/dcdsb.2014.19.2065
##### 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. 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. 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. 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.

[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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.

[19]

L. Galuppi and G. Royer Carfagni, Laminated beams with viscoelastic interlayer,, International Journal of Solids and Structures, 49 (2012), 2637. 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. 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. 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.

[23]

C. Giorgi and A. Morro, Viscoelastic solids with unbounded relaxation function,, Continuum Mechanics and Thermodynamics, 4 (1992), 151. doi: 10.1007/BF01125696.

[24]

D. Graffi and M. Fabrizio, Sulla nozione di stato per materiali viscoelastici di tipo "rate'',, Atti. Accademia dei Lincei, 83 (1989), 201.

[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. 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. 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, (1997), 291. doi: 10.1007/978-3-7091-2664-6_7.

[31]

F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity,, Imperial College Press, (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. 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. 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. 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). 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.

[37]

P. Nutting, A new general law of deformation,, Journal of The Franklin Institute, 191 (1921), 679. 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. doi: 10.1016/j.mechrescom.2008.10.012.

[39]

F. Riewe, Mechanics with fractional derivatives,, Physical Review E, 55 (1997), 3581. 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. doi: 10.1115/1.1839184.

[41]

S. Sakakibara, Relaxation properties of fractional derivative viscoelasticity models,, Nonlinear Analysis, 47 (2001), 5449. 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. 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. 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. doi: 10.1016/j.jnnfm.2009.10.002.

[48]
##### 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. 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. 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. 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.

[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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.

[19]

L. Galuppi and G. Royer Carfagni, Laminated beams with viscoelastic interlayer,, International Journal of Solids and Structures, 49 (2012), 2637. 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. 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. 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.

[23]

C. Giorgi and A. Morro, Viscoelastic solids with unbounded relaxation function,, Continuum Mechanics and Thermodynamics, 4 (1992), 151. doi: 10.1007/BF01125696.

[24]

D. Graffi and M. Fabrizio, Sulla nozione di stato per materiali viscoelastici di tipo "rate'',, Atti. Accademia dei Lincei, 83 (1989), 201.

[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. 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. 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, (1997), 291. doi: 10.1007/978-3-7091-2664-6_7.

[31]

F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity,, Imperial College Press, (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. 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. 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. 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). 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.

[37]

P. Nutting, A new general law of deformation,, Journal of The Franklin Institute, 191 (1921), 679. 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. doi: 10.1016/j.mechrescom.2008.10.012.

[39]

F. Riewe, Mechanics with fractional derivatives,, Physical Review E, 55 (1997), 3581. 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. doi: 10.1115/1.1839184.

[41]

S. Sakakibara, Relaxation properties of fractional derivative viscoelasticity models,, Nonlinear Analysis, 47 (2001), 5449. 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. 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. 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. doi: 10.1016/j.jnnfm.2009.10.002.

[48]

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