In the present work, a nonlocal model based on the conformal strain energy, utilizing the conformable derivative definition, has been obtained. The model has two additional free parameters compared to the classical (local) mechanical formulations. The first one specifies the amount of the integer and the noninteger gradient of strain in the strain energy relation, and the second one controls the order of the strain derivatives in the conformable energy relation. The obtained governing (nonlinear) equation has been solved by the Galerkin method and the effects of both free parameters have been shown. As a case study, the bending and buckling of nanobeam structures has been studied.

References

References
1.
Pradhan
,
S. C.
, and
Phadikar
,
J. K.
,
2009
, “
Nonlocal Elasticity Theory for Vibration of Nanoplates
,”
J. Sound Vib.
,
325
(
1–2
), pp.
206
223
.
2.
Aksencer
,
T.
, and
Aydogdu
,
M.
,
2011
, “
Levy Type Solution Method for Vibration and Buckling of Nanoplates Using Nonlocal Elasticity Theory
,”
Phys. E
,
43
(
4
), pp.
954
959
.
3.
Ruud
,
J. A.
,
Jervis
,
T. R.
, and
Spaepen
,
F.
,
1994
, “
Nanoindention of Ag/Ni Multilayered Thin Films
,”
J. Appl. Phys.
,
75
(
10
), pp.
4969
4974
.
4.
Chowdhury
,
R.
,
Adhikari
,
S.
,
Wang
,
C. W.
, and
Scarpa
,
F.
,
2010
, “
A Molecular Mechanics Approach for the Vibration of Single Walled Carbon Nanotubes
,”
Comput. Mater. Sci.
,
48
(
4
), pp.
730
735
.
5.
Eringen
,
A. C.
,
2010
,
Nonlocal Continuum Field Theories
,
Springer
,
New York
.
6.
Peddieson
,
J.
,
Buchanan
,
G. R.
, and
McNitt
,
R. P.
,
2003
, “
The Role of Strain Gradients in the Grain Size Effect for Polycrystals
,”
Int. J. Eng. Sci.
,
41
(
3–5
), pp.
305
312
.
7.
Toupin
,
R. A.
,
1962
, “
Elastic Materials With Couple-Stress
,”
Arch. Ration. Mech. Anal.
,
11
(
1
), pp.
385
414
.
8.
Mindlin
,
R. D.
, and
Eshel
,
N. N.
,
1968
, “
On First Strain-Gradient Theories in Linear Elasticity
,”
Int. J. Solids Struct.
,
4
(
1
), pp.
109
124
.
9.
Nowacki
,
W.
,
1972
,
Theory of Micropolar Elasticity
,
CISM
,
Udine, Italy
.
10.
Eringen
,
A. C.
,
1966
, “
Linear Theory of Micropolar Elasticity
,”
J. Math. Mech.
,
15
(
6
), pp.
909
923
.https://www.jstor.org/stable/24901442
11.
Gurtin
,
M. E.
, and
Murdoch
,
A. I.
,
1975
, “
A Continuum Theory of Elastic Material Surfaces
,”
Arch. Ration. Mech. Anal.
,
57
(
4
), pp.
291
323
.
12.
Davis
,
G. B.
,
Kohandel
,
M.
,
Sivaloganathan
,
S.
, and
Tenti
,
G.
,
2006
, “
The Constitutive Properties of the Brain Parenchyma—Part 2: Fractional Derivative Approach
,”
Med. Eng. Phys.
,
28
(
5
), pp.
455
459
.
13.
Bagley
,
R. L.
, and
Torvik
,
P. J.
,
1983
, “
A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity
,”
J. Rheol.
,
27
(
3
), p.
201
.
14.
Cajic
,
M. S.
,
Mihailo
,
P. L.
, and
Tomislav
,
B. S.
,
2014
, “
Robotic System With Viscoelastic Element Modeled Via Fractional Zener Model
,”
International Conference on Fractional Differentiation and Its Applications (ICFDA'14)
, Catania, Italy, June 23–25, pp.
1
6
.
15.
Belyaev
,
A. K.
,
2014
, “
Fractional Derivatives Appearing in Some Dynamic Problems
,”
Mechanics and Model-Based Control of Advanced Engineering Systems
,
Springer
,
Vienna, Austria
, pp.
41
48
.
16.
Beda
,
P.
,
2017
, “
Dynamical Systems Approach of Internal Length in Fractional Calculus
,”
Eng. Trans.
,
65
(
1
), pp.
209
215
.http://et.ippt.gov.pl/index.php/et/article/view/703
17.
Ahmad
,
W. M.
, and
El-Khazali
,
R.
,
2007
, “
Fractional-Order Dynamical Models of Love
,”
Chaos, Solitons Fractals
,
33
(
4
), pp.
1367
1375
.
18.
Lima
,
M. F.
,
Machado
,
J. A. T.
, and
Crisóstomo
,
M. M.
,
2007
, “
Experimental Signal Analysis of Robot Impacts in a Fractional Calculus Perspective
,”
JACIII
,
11
(
9
), pp.
1079
1085
.
19.
Torvik
,
P. J.
, and
Bagley
,
R. L.
,
1984
, “
On the Appearance of the Fractional Derivative in the Behavior of Real Materials
,”
ASME J. Appl. Mech.
,
51
(
2
), pp.
294
298
.
20.
De Espındola
,
J. J.
,
da Silva Neto
,
J. M.
, and
Lopes
,
E. M.
,
2005
, “
A Generalised Fractional Derivative Approach to Viscoelastic Material Properties Measurement
,”
Appl. Math. Comput.
,
164
(
2
), pp.
493
506
.
21.
Podlubny
,
I.
,
Petraš
,
I.
,
Vinagre
,
B. M.
,
O'lear
,
P.
, and
Dorčák
,
Ľ.
,
2002
, “
Analogue Realizations of Fractional-Order Controllers
,”
Nonlinear Dyn
,,
29
(
1/4
), pp.
281
296
.
22.
Xue
,
D.
,
Zhao
,
C.
, and
Chen
,
Y.
,
2006
, “
Fractional Order PID Control of a DC-Motor With Elastic Shaft: A Case Study
,”
American Control Conference
(
ACC
), Minneapolis, MN, June 14–16, pp.
3182
3187
.https://www.semanticscholar.org/paper/Fractional-Order-PID-Controller-Design-for-Speed-of-Mehra-Srivastava/2f41bf87b2264698bc37f48bcf5d985db122c442
23.
Silva
,
M. F.
,
Machado
,
J. T.
, and
Lopes
,
A. M.
,
2004
, “
Fractional Order Control of a Hexapod Robot
,”
Nonlinear Dyn.
,
38
(
1–4
), pp.
417
433
.
24.
Sommacal
,
L.
,
Melchior
,
P.
,
Oustaloup
,
A.
,
Cabelguen
,
J. M.
, and
Ijspeert
,
A. J.
,
2008
, “
Fractional Multi-Models of the Frog Gastrocnemius Muscle
,”
J. Vib. Control
,
14
(
9–10
), pp.
1415
1430
.
25.
Heymans
,
N.
,
2008
, “
Dynamic Measurements in Long-Memory Materials: Fractional Calculus Evaluation of Approach to Steady State
,”
J. Vib. Control
,
14
(
9–10
), pp.
1587
1596
.
26.
Bohannan
,
G. W.
,
2008
, “
Analog Fractional Order Controller in Temperature and Motor Control Applications
,”
J. Vib. Control
,
14
(
9–10
), pp.
1487
1498
.
27.
Lazopoulos
,
K. A.
,
2006
, “
Non-Local Continuum Mechanics and Fractional Calculus
,”
Mech. Res. Commun.
,
33
(
6
), pp.
753
757
.
28.
Drapaca
,
C. S.
, and
Sivaloganathan
,
S.
,
2012
, “
A Fractional Model of Continuum Mechanics
,”
J. Elasticity
,
107
(
2
), pp.
105
123
.
29.
Atanackovic
,
T. M.
, and
Stankovic
,
B.
,
2009
, “
Generalized Wave Equation in Nonlocal Elasticity
,”
Acta Mech.
,
208
(
1–2
), pp.
1
10
.
30.
Carpinteri
,
A.
,
Cornetti
,
P.
, and
Sapora
,
A.
,
2011
, “
A Fractional Calculus Approach to Nonlocal Elasticity
,”
Eur. Phys. J.: Spec. Top.
,
193
(
1
), pp.
193
204
.
31.
Sumelka
,
W.
,
2015
, “
Non-Local Kirchhoff–Love Plates in Terms of Fractional Calculus
,”
Arch. Civ. Mech. Eng.
,
15
(
1
), pp.
231
242
.
32.
Sumelka
,
W.
,
Blaszczyk
,
T.
, and
Liebold
,
C.
,
2015
, “
Fractional Euler-Bernoulli Beams: Theory, Numerical Study and Experimental Validation
,”
Eur. J. Mech. A/Solids
,
54
, pp.
243
251
.
33.
Kilbas
,
A. A.
,
Srivastava
,
H. M.
, and
Trujillo
,
J. J.
,
2006
,
Theory and Applications of Fractional Differential Equations
,
Elsevier Science Limited
, Amsterdam, The Netherlands, p.
204
.
34.
Di Paola
,
M.
, and
Zingales
,
M.
,
2008
, “
Long-Range Cohesive Interactions of Non-Local Continuum Faced by Fractional Calculus
,”
Int. J. Solids Struct.
,
45
(
21
), pp.
5642
5659
.
35.
Eringen
,
A. C.
, and
Edelen
,
D. G. B.
,
1972
, “
Nonlocal Elasticity
,”
Int. J. Eng. Sci.
,
10
(
3
), pp.
233
248
.
36.
Kroner
,
E.
,
1967
, “
Elasticity Theory of Materials With Long Range Cohesive Forces
,”
Int. J. Solids Struct.
,
3
, pp.
731
742
.
37.
Carpinteri
,
A.
,
Cornetti
,
P.
,
Sapora
,
A.
,
Di Paola
,
M.
, and
Zingales
,
M.
,
2009
, “
Fractional Calculus in Solid Mechanics: Local Versus Non-Local Approach
,”
Phys. Scr.
,
2009
, p.
T136
.
38.
Rahimi
,
Z.
,
Shafiei
,
S.
,
Sumelka
,
W.
, and
Rezazadeh
,
G.
,
2018
, “
Fractional Strain Energy and Its Application to the Free Vibration Analysis of a Plate
,”
Microsyst. Technol.
, (epub).
39.
Khalil
,
R.
,
Al Horani
,
M.
,
Yousef
,
A.
, and
Sababheh
,
M.
,
2014
, “
A New Definition of Fractional Derivative
,”
J. Comput. Appl. Math.
,
264
, pp.
65
70
.
40.
Abdeljawad
,
T.
,
2015
, “
On Conformable Fractional Calculus
,”
J. Comput. Appl. Math.
,
279
, pp.
57
66
.
41.
Katugampola
,
U. N.
,
2014
, “
A New Fractional Derivative With Classical Properties
,” preprint
arXiv:1410.6535
.https://arxiv.org/abs/1410.6535
42.
Reddy
,
J. N.
,
2007
, “
Nonlocal Theories for Bending, Buckling and Vibration of Beams
,”
Int. J. Eng. Sci.
,
45
(
2–8
), pp.
288
307
.
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