A methodology to derive fractional derivative constitutive models for finite deformation of viscoelastic materials is proposed in a continuum mechanics treatment. Fractional derivative models are generalizations of the models given by the objective rates. The method of generalization is applied to the case in which the objective rate of the Cauchy stress is given by the Truesdell rate. Then, a fractional derivative model is obtained in terms of the second Piola–Kirchhoff stress tensor and the right Cauchy-Green strain tensor. Under the assumption that the dynamical behavior of the viscoelastic materials comes from a complex combination of elastic and viscous elements, it is shown that the strain energy of the elastic elements plays a fundamental role in determining the fractional derivative constitutive equation. As another example of the methodology, a fractional constitutive model is derived in terms of the Biot stress tensor. The constitutive models derived in this paper are compared and discussed with already existing models. From the above studies, it has been proved that the methodology proposed in this paper is fully applicable and effective.

References

References
1.
Gemant
,
A.
,
1936
, “
A Method of Analyzing Experimental Results Obtained From Elasto-Viscous Bodies
,”
J. Appl. Phys.
,
7
(
8
), pp.
311
317
.
2.
Blair
,
S. G. W.
,
1944
, “
Analytical and Integrative Aspects of the Stress–Strain–Time Problems
,”
J. Sci. Instrum.
,
21
(
5
), pp.
80
84
.10.1088/0950-7671/21/5/302
3.
Rabotnov
,
Y. N.
,
1980
,
Elements of Hereditary Solid Mechanics
,
Mir Publishers
,
Moscow, Russia
.
4.
Bagley
,
R. L.
, and
Torvik
,
P. J.
,
1983
, “
A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity
,”
J. Rheol.
,
27
(
3
), pp.
201
210
.10.1122/1.549724
5.
Bagley
,
R. L.
, and
Torvik
,
P. J.
,
1983
, “
Fractional Calculus—A Different Approach to the Analysis of Viscoelasticity Damped Structures
,”
AIAA J.
,
21
(
5
), pp.
741
748
.10.2514/3.8142
6.
Koeller
,
R. C.
,
1984
, “
Application of Fractional Calculus to the Theory of Viscoelasticity
,”
ASME J. Appl. Math.
,
51
(
2
), pp.
299
307
.10.1115/1.3167616
7.
Oldham
,
K. H.
, and
Spanier
,
J.
,
1974
,
The Fractional Calculus
,
Mineola, NY
.
8.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic Press
,
NY
.
9.
Kilbas
,
A. A.
,
Srivastava
,
H. M.
, and
Trujillo
,
J. J.
,
2006
,
Theory and Applications of Fractional Differential Equations
,
Elsevier
,
Amsterdam
.
10.
Caputo
,
M.
,
1967
, “
Linear Model of Dissipation Whose q is Almost Frequency Independent—II
,”
Geophys. J. R. Astron. Soc.
,
13
(
5
), pp.
529
539
.10.1111/j.1365-246X.1967.tb02303.x
11.
Lodge
,
A. S.
,
1964
,
Elastic Liquids
,
Academic Press
,
London
.
12.
Freed
,
A. D.
, and
Diethelm
,
K.
,
2006
, “
Fractional Calculus in Biomechanics: A 3D Viscoelastic Model Using Regularlized Fractional Derivative Kernels With Application to the Human Calcaneal Fat Pad
,”
Biomech. Modell. Mechanobiol.
,
5
(
4
), pp.
203
215
.10.1007/s10237-005-0011-0
13.
Fukunaga
,
M.
, and
Shimizu
,
N.
,
2009
, “
Analysis of Impulse Response of a Gel by Nonlinear Fractional Derivative Model
,”
ASME DETC 2009
, San Diego, CA, Aug. 30–Sept. 2,
ASME
Paper No. DETC2009/86803.10.1115/DETC2009/86803
14.
Bird
,
R. B.
,
Armstrong
,
R. C.
, and
Hassager
,
O.
,
1987
,
Dynamics Elastic Liquids
, Vol.
1
,
Wiley
,
NY
.
15.
Bird
,
R. B.
,
Curtiss
,
C. F.
,
Armstrong
,
R. C.
, and
Hassager
,
O.
,
1987
,
Dynamics Elastic Liquids
, Vol.
2
,
Wiley
,
NY
.
16.
Drozdov
,
A. D.
,
1997
, “
Fractional Differential Models in Finite Viscoelasticity
,”
Acta Mech.
,
124
(
1–4
), pp.
155
180
.10.1007/BF01213023
17.
Haupt
,
P.
, and
Lion
,
A.
,
2002
, “
On Finite Linear Viscoelasticity of Incompressible Isotropic Materials
,”
Acta Mech.
,
159
(
1–4
), pp.
87
124
.10.1007/BF01171450
18.
Adolfsson
,
K.
, and
Enelund
,
M.
,
2003
, “
Fractional Derivative Viscoelasticity at Large Deformations
,”
Nonlinear Dyn.
,
33
(
3
), pp.
301
321
.10.1023/A:1026003130033
19.
Adolfsson
,
K.
,
2004
, “
Nonlinear Fractional Order Viscoelasticity at Large Deformations
,”
Nonlinear Dyn.
,
38
(
1–4
), pp.
233
246
.10.1007/s11071-004-3758-4
20.
Freed
,
A. D.
, and
Diethelm
,
K.
,
2007
, “
Caputo Derivatives in Viscoelasticity: A Non-Linear Finite-Deformation Theory for Tissue
,”
Fractional Calculus Appl. Anal.
,
10
(
3
), pp.
219
248
.
21.
Nasuno
,
H.
,
2009
, “
Nonlinear Viscoelastic Finite Element Analysis by Means of Fractional Calculus
,” Ph.D. dissertation, Graduate Course of Science and Engineering, Iwaki Meisei University, Iwaki, Japan.
22.
Xiao
,
H.
,
Bruhns
,
O. T.
, and
Meyers
,
A.
,
1997
, “
Hypo-Elasticity Model Based Upon the Logarithmic Stress Rate
,”
J. Elasticity
,
47
(
1
), pp.
51
68
.10.1023/A:1007356925912
23.
Schiessel
,
H.
, and
Blumen
,
A.
,
1993
, “
Hierarchical Analogues to Fractional Relaxation Equations
,”
J. Phys. A
,
26
(
19
), pp.
5057
5069
.10.1088/0305-4470/26/19/034
24.
Heymans
,
N.
, and
Bauwens
,
J. C.
,
1994
, “
Fractional Rheological Models and Fractional Differential Equations for Viscoelastic Behavior
,”
Rheol. Acta
,
33
(
3
), pp.
210
210
.10.1007/BF00437306
25.
Heymans
,
N.
,
1996
, “
Hierarchical Models for Viscoelasticity: Dynamic behavior in the Linear Range
,”
Rheol. Acta
,
35
(
5
), pp.
508
519
.10.1007/BF00369000
26.
Rouse
,
P. E.
, Jr.
,
1953
, “
A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers
,”
J. Chem. Phys.
,
21
(
7
), pp.
1272
1280
.10.1063/1.1699180
27.
Zimm
,
B. H.
,
1956
, “
Dynamics of Polymer Molecules in Dilute Solution: Viscoelasticity, Flow Birefringence and Dielectric Loss
,”
J. Chem. Phys.
,
24
(
2
), pp.
269
278
.10.1063/1.1742462
28.
Coleman
,
B. D.
, and
Noll
,
W.
,
1961
, “
Foundation of Linear Viscoelasticity
,”
Rev. Mod. Phys.
,
33
(
2
), pp.
239
249
.10.1103/RevModPhys.33.239
29.
Caputo
,
M.
, and
Mainardi
,
F.
,
1971
, “
Linear Models of Dissipation in Anelastic Solids
,”
Riv. del Nuovo Cimento
,
1
(
2
), pp.
161
198
.10.1007/BF02820620
30.
Strobl
,
G. R.
,
1996
,
The Physics of Polymers
,
Springer
,
Berlin, Germany
.
31.
Mainardi
,
F.
,
2010
,
Fractional Calculus and Waves in Linear Viscoelasticity
,
Imperial College Press
,
London, UK
.
32.
Lu
,
S. C. H.
, and
Pister
,
K. S.
,
1975
, “
Decomposition of Deformation and Representation of the Free Energy Function for Isotropic Thermoelastic Solids
,”
Int. J. Solids Struct.
,
11
(
7,8
), pp.
927
934
.10.1016/0020-7683(75)90015-3
33.
Lubliner
,
J.
,
1985
, “
A Model of Rubber Viscoelasticity
,”
Mech. Res. Commun.
,
12
(
2
), pp.
93
99
.10.1016/0093-6413(85)90075-8
34.
Simo
,
J. C.
,
1987
, “
On a Fully Three-Dimensional Finite-Strain Viscoelastic Damage Model: Formulation and Computational Aspects
,”
Comput. Method Appl. Mech. Eng.
,
60
(
2
), pp.
153
173
.10.1016/0045-7825(87)90107-1
35.
Govindjee
,
S.
, and
Simo
,
C.
,
1992
, “
Mullins' Effect and the Strain Amplitude Dependence of the Storage Modus
,”
Int. J. Solids Struct.
,
29
(
14,15
), pp.
1737
1751
.10.1016/0020-7683(92)90167-R
36.
Holzapfel
,
G. A.
, and
Simo
,
J. C.
,
1996
, “
A New Viscoelastic Constitutive Model for Continuous Media at Finite Thermomechanical Changes
,”
Int. J. Solids Struct.
,
33
(
20–22
), pp.
3019
3034
.10.1016/0020-7683(95)00263-4
37.
Simo
,
J. C.
, and
Hughes
,
T. J. R.
,
1998
,
Computational Inelasticity
.
Springer
,
NY
.
38.
Lion
,
A.
,
1998
, “
Thixotropic Behavior of Rubber Under Dynamic Loading Histories: Experimental Results and Theory
,”
J. Mech. Phys. Solids
,
46
(
5
), pp.
895
930
.10.1016/S0022-5096(97)00097-5
39.
Holzapfel
,
G. A.
,
2000
,
Nonlinear Solid Mechanics
,
Wiley
,
Chichester, NY
.
40.
Haupt
,
P.
, and
Seldan
,
K.
,
2002
, “
Viscoelasticity of Elastic Materials: Experimental Facts and Constitutive Modelling
,”
Acta Mech.
,
159
(
1–4
), pp.
87
124
.10.1007/BF01171450
41.
de Gennes
,
P.
,
1979
,
Scaling Concepts in Polymer Physics
,
Cornell University Press
,
Ithaca, NY
.
42.
Treloar
,
L. R. G.
,
1975
,
The Physics of Rubber Elasticity, Third Edition, Oxford Classic Texts in the Physical Sciences
,
Clarendon Press
,
Oxford, UK
.
43.
Fukunaga
,
M.
, and
Shimizu
,
N.
,
2011
, “
Nonlinear Fractional Derivative Models of Viscoelastic Impact Dynamics Based on Viscoelasticity And Generalized Maxwell Law
,”
ASME J. Comput. Nonlinear Dyn.
,
6
(
2
), p.
021005
.10.1115/1.4002383
44.
Fukunaga
,
M.
,
Shimizu
,
N.
, and
Nasuno
,
H.
,
2009
, “
A Nonlinear Fractional Derivative Models of Impulse Motion for Viscoelastic Materials
,”
Phys. Scr.
,
T136
, p.
014010
.10.1088/0031-8949/2009/T136/014010
45.
Lubliner
,
J.
,
1969
, “
On Fading Memory in Materials of Evolutionary Type
,”
Acta Mech.
,
8
(
1,2
), pp.
75
81
.10.1007/BF01178535
46.
Valanis
,
K. C.
,
1971
, “
A Theory of Viscoelasticity With a Yield Surface Part I—General Theory
,”
Arch. Mech.
,
23
(
4
), pp.
517
533
.
47.
Truesdell
,
C.
, and
Noll
,
W.
,
2004
,
The Non-Linear Field Theories of Mechanics
,
3rd ed.
,
Springer
,
Heidelberg, Germany
.
48.
Fukunaga
,
M.
, and
Shimizu
,
N.
,
2011
, “
Three-Dimensional Fractional Derivative Models for Finite Deformation
,” ASME 2011 IDETC/CIE 2011, Washington, DC, Aug. 28–31,
ASME
Paper No. IDETC2011/47552.10.1115/IDETC2011/47552
49.
Bonet
,
J.
, and
Wood
,
R. D.
,
1997
,
Nonlinear Continuum Mechanics for Finite Element Analysis
,
Cambridge University Press
,
NY
.
50.
Biot
,
M. A.
,
1965
,
Mechanics of Incremental Deformations
,
Wiley
,
NY
.
51.
Dienes
,
J. K.
,
1979
, “
On the Analysis of Rotation and Stress Rate in Deforming Bodies
,”
Acta Mech.
,
32
(
4
), pp.
217
232
.10.1007/BF01379008
52.
Kleiber
,
M.
,
1986
, “
On Errors Inherent in Commonly Accepted Rate Forms of Elastic Constitutive Laws
,”
Arch. Mech.
,
38
(
3
), pp.
271
277
.
53.
Meyers
,
A.
,
Xiao
,
H.
, and
Bruhns
,
O. T.
,
2006
, “
Choice of Objective Rate in Single Parameter Hypoelastic Deformation Cycles
,”
Comput. Struct.
,
84
(
17,18
), pp.
1134
1140
.10.1016/j.compstruc.2006.01.012
54.
Fukunaga
,
M.
, and
Shimizu
,
N.
,
2014
, “
Comparison of Fractional Derivative Models for Finite Deformation With Experiments of Impulse Response
,”
J. Vib. Control
,
20
(
7
), pp.
1033
1041
.10.1177/1077546313481051
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