American Institute of Mathematical Sciences

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

The state of fractional hereditary materials (FHM)

Luca Deseri 1, , Massiliano Zingales 2, and  Pietro Pollaci 3,

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".
Keywords: microstuctures., power-law, Fractional hereditary materials, functionally graded.
Mathematics Subject Classification: Primary: 35Q74, 74A60; Secondary: 76A1.
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:
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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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G. Scott Blair, The role of psychophysics in rheology, Journal of Colloid Science, 2 (1947), 21-32. Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[16]

M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM Philadelphia, 1992. doi: 10.1137/1.9781611970807.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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A. Gemant, A method of analyzing experimental results obtained from elasto-viscous bodies, Physics, 7 (1936), 311-317. doi: 10.1063/1.1745400.  Google Scholar

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G. Gentili, Maximum recoverable work, minimum free energy and state space in linear viscoelasticity, Quarterly of Applied Mathematics, 60 (2002), 153-182.  Google Scholar

[23]

C. Giorgi and A. Morro, Viscoelastic solids with unbounded relaxation function, Continuum Mechanics and Thermodynamics, 4 (1992), 151-165. doi: 10.1007/BF01125696.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, Prentice Hall, 1970.  Google Scholar

[28]

R. Lakes, Viscoelastic Materials, Cambridge University Press, 2009. doi: 10.1017/CBO9780511626722.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010. doi: 10.1142/9781848163300.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

W. Noll, A new mathematical theory of simple materials, Archive for Rational Mechanics and Analysis, 48 (1972), 1-50.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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F. Riewe, Mechanics with fractional derivatives, Physical Review E, 55 (1997), 3581-3592. doi: 10.1103/PhysRevE.55.3581.  Google Scholar

[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.  Google Scholar

[41]

S. Sakakibara, Relaxation properties of fractional derivative viscoelasticity models, Nonlinear Analysis, 47 (2001), 5449-5454. doi: 10.1016/S0362-546X(01)00649-6.  Google Scholar

[42]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integral and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, 1993.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

G. Scott Blair, The role of psychophysics in rheology, Journal of Colloid Science, 2 (1947), 21-32. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity, SIAM Philadelphia, 1992. doi: 10.1137/1.9781611970807.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

A. Gemant, A method of analyzing experimental results obtained from elasto-viscous bodies, Physics, 7 (1936), 311-317. doi: 10.1063/1.1745400.  Google Scholar

[22]

G. Gentili, Maximum recoverable work, minimum free energy and state space in linear viscoelasticity, Quarterly of Applied Mathematics, 60 (2002), 153-182.  Google Scholar

[23]

C. Giorgi and A. Morro, Viscoelastic solids with unbounded relaxation function, Continuum Mechanics and Thermodynamics, 4 (1992), 151-165. doi: 10.1007/BF01125696.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, Prentice Hall, 1970.  Google Scholar

[28]

R. Lakes, Viscoelastic Materials, Cambridge University Press, 2009. doi: 10.1017/CBO9780511626722.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010. doi: 10.1142/9781848163300.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

W. Noll, A new mathematical theory of simple materials, Archive for Rational Mechanics and Analysis, 48 (1972), 1-50.  Google Scholar

[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.  Google Scholar

[38]

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[39]

F. Riewe, Mechanics with fractional derivatives, Physical Review E, 55 (1997), 3581-3592. doi: 10.1103/PhysRevE.55.3581.  Google Scholar

[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.  Google Scholar

[41]

S. Sakakibara, Relaxation properties of fractional derivative viscoelasticity models, Nonlinear Analysis, 47 (2001), 5449-5454. doi: 10.1016/S0362-546X(01)00649-6.  Google Scholar

[42]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integral and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, 1993.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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