This paper describes the development of a fractional calculus approach to model the hysteretic behavior shown by the variation in electrical resistances with strain in conductive polymers. Experiments have been carried out on a conductive polymer nanocomposite sample to study its resistance-strain variation under strain varying with time in a triangular manner. A combined fractional derivative and integer order integral model and a fractional integral model (with two submodels) have been developed to simulate this behavior. The efficiency of these models has been discussed by comparing the results, obtained using these models, with the experimental data. It has been shown that by using only a few parameters, the hysteretic behavior of such materials can be modeled using the fractional calculus with some modifications.

1.
Hammerand
,
D. C.
, and
Kapania
,
R. K.
, 1999, “
Thermo-Viscoelastic Analysis of Composite Structures Using a Triangular Flat Shell Element
,”
AIAA J.
0001-1452,
37
(
2
), pp.
238
247
.
2.
Hammerand
,
D. C.
, and
Kapania
,
R. K.
, 2000, “
Geometrically Nonlinear Shell Element for Hygrothermorheologically Simple Linear Viscoelastic Composites
,”
AIAA J.
0001-1452,
38
(
12
), pp.
2305
2319
.
3.
Heymans
,
N.
, 2003, “
Constitutive Equations for Polymer Viscoelasticity Derived From Hierarchical Models in Cases of Failure of Time-Temperature Superposition
,”
Signal Process.
0165-1684,
83
(
11
), pp.
2345
2357
.
4.
Flandin
,
L.
,
Brechet
,
Y.
, and
Cavaille
,
J. -Y.
, 2001, “
Electrically Conductive Polymer Nanocomposites as Deformation Sensors
,”
Compos. Sci. Technol.
0266-3538,
61
(
6
), pp.
895
901
.
5.
Knite
,
M.
,
Teteris
,
V.
,
Kiploka
,
A.
, and
Kaupuzs
,
J.
, 2004, “
Polyisoprene-Carbon Black Nanocomposites as Tensile Strain and Pressure Sensor Materials
,”
Sens. Actuators, A
0924-4247,
110
(
1–3
), pp.
142
149
.
6.
Rekhviashvili
,
S. S.
, 2007, “
Non-Steady-State Electrical Conduction of Polymers in the Model With Fractional Integro-Differentiation
,”
Phys. Solid State
1063-7834,
49
(
8
), pp.
1598
1602
.
7.
Viswamurthy
,
S. R.
, and
Ganguli
,
R.
, 2007, “
Modeling and Compensation of Piezoceramic Actuator Hysteresis for Helicopter Vibration Control
,”
Sens. Actuators, A
0924-4247,
135
(
2
), pp.
801
810
.
8.
Preisach
,
F.
, 1935, “
Uber Die Magnetische Nachwirkung
,”
Z. Phys.
0044-3328,
94
, pp.
277
302
.
9.
Everett
,
D. H.
, and
Whitton
,
W. I.
, 1952, “
A General Approach to Hysteresis
,”
Trans. Faraday Soc.
0014-7672,
48
, pp.
749
757
.
10.
Cornejo
,
D. R.
, and
Missell
,
F. P.
, 1999, “
Application of the Preisach Model to Nanocrystalline Magnets
,”
J. Magn. Magn. Mater.
0304-8853,
203
, pp.
41
45
.
11.
Roshko
,
R. M.
, and
Huo
,
D. L.
, 2001, “
A Preisach Characterization of the Barkhausen Spectrum of a Canonical Ferromagnet SrRuO3
,”
Physica B
0921-4526,
306
(
1–4
), pp.
246
250
.
12.
Vandenbossche
,
L.
,
Dupré
,
L.
, and
Melkebeek
,
J.
, 2005, “
Preisach-Based Magnetic Evaluation of Fatigue Damage Progression
,”
J. Magn. Magn. Mater.
0304-8853,
290–291
(
1
), pp.
486
489
.
13.
Schiffer
,
A.
, and
Ivanyi
,
A.
, 2006, “
Preisach Distribution Function Approximation With Wavelet Interpolation Technique
,”
Physica B
0921-4526,
372
(
1–2
), pp.
101
105
.
14.
Yu
,
Y.
,
Xiao
,
Z.
,
Lin
,
E. -B.
, and
Naganathan
,
N.
, 2000, “
Analytic and Experimental Studies of a Wavelet Identification of Preisach Model of Hysteresis
,”
J. Magn. Magn. Mater.
0304-8853,
208
(
3
), pp.
255
263
.
15.
Oldham
,
K. B.
, and
Spanier
,
J.
, 1974,
The Fractional Calculus
,
Academic
,
New York
.
16.
Chang
,
T.
, 2002, “
Seismic Response of Structures With Added Viscoelastic Dampers
,” Ph.D. thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA.
17.
Euler
,
L.
, 1738, “
De Progressionibus Transcendentibus, Sev Quarum Termini Generales Algebraice Dari Nequent
,”
Commentarii Academiae Scientiarum Imperialis Scientiarum Petropolitanae
,
5
, pp.
38
57
.
18.
Lacroix
,
S. L.
, 1819,
Traite du Calcul differentiel et du Calcul Integral
,
2nd ed.
,
Courcier
,
Paris
, pp.
409
410
.
19.
Hardy
,
G. H.
, 1917, “
On Some Properties of Integrals of Fractional Order
,”
Messenger Math.
,
47
, pp.
145
150
.
20.
Osler
,
T. J.
, 1970, “
Leibniz Rule for Fractional Derivatives Generalized and an Application to Infinite Series
,”
SIAM J. Appl. Math.
0036-1399,
18
(
3
), pp.
658
674
.
21.
Nutting
,
P. G.
, 1921, “
A New General Law of Deformation
,”
J. Franklin Inst.
0016-0032,
191
, pp.
679
685
.
22.
Bagley
,
R. L.
, and
Torvik
,
P. J.
, 1983, “
A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity
,”
J. Rheol.
0148-6055,
27
(
3
), pp.
201
210
.
23.
Gemant
,
A.
, 1936, “
A Method of Analyzing Experimental Results Obtained From Elasto-Viscous Bodies
,”
Physica (Amsterdam)
0031-8914,
7
, pp.
311
317
.
24.
Scott-Blair
,
G. W.
, and
Reiner
,
M.
, 1950, “
The Rheological Law Underlying the Nutting Equation
,”
Appl. Sci. Res.
0003-6994,
A2
, pp.
225
234
.
25.
Belavine
,
V. A.
,
Nigmatullin
,
R. S.
,
Miroshnikov
,
A. I.
, and
Lutskaya
,
N. K.
, 1964, “
Fractional Differentiation of Oscillographic Polarograms by Means of an Electrochemical Two-Terminal Network
,”
Tr. Kazan. Aviacion. Inst.
,
5
, pp.
144
145
.
26.
Bagley
,
R. L.
, and
Torvik
,
P. J.
, 1983, “
Fractional Calculus—A Different Approach to the Analysis of Viscoelastically Damped Structures
,”
AIAA J.
0001-1452,
21
(
5
), pp.
741
748
.
27.
Bagley
,
R. L.
, and
Torvik
,
P. J.
, 1986, “
On the Fractional Calculus Model of Viscoelastic Behavior
,”
J. Rheol.
0148-6055,
30
(
1
), pp.
133
155
.
28.
de Espindola
,
J. J.
,
da Silva Neto
,
J. M.
, and
Lopes
,
E. M. O.
, 2005, “
A Generalised Fractional Derivative Approach to Viscoelastic Material Properties Measurement
,”
Appl. Math. Comput.
0096-3003,
164
(
2
), pp.
493
506
.
29.
Horr
,
A. M.
, and
Schmidt
,
L. C.
, 1996, “
A Fractional-Spectral Method for Vibration of Damped Space Structures
,”
Eng. Struct.
0141-0296,
18
(
12
), pp.
947
956
.
30.
Davis
,
G.
,
Kohandel
,
M.
,
Sivaloganathan
,
S.
, and
Tenti
,
G.
, 2006, “
The Constitutive Properties of the Brain Paraenchyma: Part 2. Fractional Derivative Approach
,”
Med. Eng. Phys.
1350-4533,
28
(
5
), pp.
455
459
.
31.
Schmidt
,
A.
, and
Gaul
,
L.
, 2006, “
On the Numerical Evaluation of Fractional Derivatives in Multi-Degree-of-Freedom Systems
,”
Signal Process.
0165-1684,
86
(
10
), pp.
2592
2601
.
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