Slant edge cracked effect considering the inherent relation between surface energy and mixed-mode crack propagations on the free transverse vibrations of nanobeams with surface effect is investigated. First, the slant edge cracked effect, which considers residual surface stress effect on the crack tip fields of a mode-I and mode-II surface edge crack, is developed and the corresponding stress intensity factors (SIFs) and local flexibility coefficients are derived. Moreover, a refined continuum model of slant cracked nanobeams is established by considering both slant edge cracked effect and surface effect. The effects of fracture angles, crack depth, surface elasticity, surface stress, and surface density on the local flexibility and free transverse vibration characteristics of cracked nanobeams are, respectively, analyzed. The results show that the flexibility coefficients distribute symmetrically about residual surface stress. Fracture angles have a profound influence on both the symmetries of the mode shapes and the natural frequencies of nanobeams, and the influence becomes more pronounced as crack depth ratios increase. Furthermore, the natural frequencies will first decrease and then increase with fracture angles when the slant edge cracked effect is considered. The results demonstrate that the inherent relation between surface energy and crack propagations should be considered for both the stress distributions at the crack tip and the dynamic behavior of cracked nanobeams.

References

References
1.
Abdi
,
H.
,
Nayeb-Hashemi
,
H.
,
Hamouda
,
A.
, and
Vaziri
,
A.
,
2014
, “
Torsional Dynamic Response of a Shaft With Longitudinal and Circumferential Cracks
,”
ASME J. Vib. Acoust.
,
136
(
6
), p.
061011
.
2.
Chatterjee
,
A.
,
2011
, “
Nonlinear Dynamics and Damage Assessment of a Cantilever Beam With Breathing Edge Crack
,”
ASME J. Vib. Acoust.
,
133
(
5
), p.
051004
.
3.
Lin
,
Y.
, and
Chu
,
F.
,
2010
, “
The Dynamic Behavior of a Rotor System With a Slant Crack on the Shaft
,”
Mech. Syst. Signal Process.
,
24
(
2
), pp.
522
545
.
4.
Lin
,
H.-P.
,
Chang
,
S.-C.
, and
Wu
,
J.-D.
,
2002
, “
Beam Vibrations With an Arbitrary Number of Cracks
,”
J. Sound Vib.
,
258
(
5
), pp.
987
999
.
5.
Carneiro
,
S. H.
, and
Inman
,
D. J.
,
2002
, “
Continuous Model for the Transverse Vibration of Cracked Timoshenko Beams
,”
ASME J. Vib. Acoust.
,
124
(
2
), pp.
310
320
.
6.
Chondros
,
T.
, and
Dimarogonas
,
A.
,
1998
, “
Vibration of a Cracked Cantilever Beam
,”
ASME J. Vib. Acoust.
,
120
(
3
), pp.
742
746
.
7.
Feng
,
X.-Q.
,
Wang
,
T.-J.
, and
Gao
,
W.
,
2008
, “
Surface Effects on the Near-Tip Stresses for Mode-I and Mode-III Cracks
,”
ASME J. Appl. Mech.
,
75
(
1
), p.
011001
.
8.
Fu
,
X.
,
Wang
,
G.
, and
Feng
,
X.
,
2008
, “
Surface Effects on the Near-Tip Stress Fields of a Mode-II Crack
,”
Int. J. Fract.
,
151
(
2
), pp.
95
106
.
9.
Fu
,
X.
,
Wang
,
G.
, and
Feng
,
X.
,
2010
, “
Surface Effects on Mode-I Crack Tip Fields: A Numerical Study
,”
Eng. Fract. Mech.
,
77
(
7
), pp.
1048
1057
.
10.
Belytschko
,
T.
,
Xiao
,
S.
,
Schatz
,
G.
, and
Ruoff
,
R.
,
2002
, “
Atomistic Simulations of Nanotube Fracture
,”
Phys. Rev. B
,
65
(
23
), p.
235430
.
11.
Dewapriya
,
M.
, and
Rajapakse
,
R.
,
2014
, “
Molecular Dynamics Simulations and Continuum Modeling of Temperature and Strain Rate Dependent Fracture Strength of Graphene With Vacancy Defects
,”
ASME J. Appl. Mech.
,
81
(
8
), p.
081010
.
12.
Joshi
,
A. Y.
,
Sharma
,
S. C.
, and
Harsha
,
S.
,
2010
, “
Analysis of Crack Propagation in Fixed-Free Single-Walled Carbon Nanotube Under Tensile Loading Using XFEM
,”
ASME J. Nanotechnol. Eng. Med.
,
1
(
4
), p.
041008
.
13.
Tsai
,
J.-L.
,
Tzeng
,
S.-H.
, and
Tzou
,
Y.-J.
,
2010
, “
Characterizing the Fracture Parameters of a Graphene Sheet Using Atomistic Simulation and Continuum Mechanics
,”
Int. J. Solids Struct.
,
47
(
3
), pp.
503
509
.
14.
He
,
L.
,
Lim
,
C.
, and
Wu
,
B.
,
2004
, “
A Continuum Model for Size-Dependent Deformation of Elastic Films of Nano-Scale Thickness
,”
Int. J. Solids Struct.
,
41
(
3
), pp.
847
857
.
15.
Kim
,
C.
,
Schiavone
,
P.
, and
Ru
,
C.-Q.
,
2010
, “
The Effects of Surface Elasticity on an Elastic Solid With Mode-III Crack: Complete Solution
,”
ASME J. Appl. Mech.
,
77
(
2
), p.
021011
.
16.
Kim
,
C.
,
Schiavone
,
P.
, and
Ru
,
C.-Q.
,
2011
, “
The Effect of Surface Elasticity on a Mode-III Interface Crack
,”
Arch. Mech.
,
63
(
3
), pp.
267
286
.
17.
Nan
,
H.
, and
Wang
,
B.
,
2013
, “
Effect of Crack Face Residual Surface Stress on Nanoscale Fracture of Piezoelectric Materials
,”
Eng. Fract. Mech.
,
110
(
1
), pp.
68
80
.
18.
Nan
,
H.
, and
Wang
,
B.
,
2012
, “
Effect of Residual Surface Stress on the Fracture of Nanoscale Materials
,”
Mech. Res. Commun.
,
44
(
1
), pp.
30
34
.
19.
Nan
,
H.
, and
Wang
,
B.
,
2014
, “
Effect of Interface Stress on the Fracture Behavior of a Nanoscale Linear Inclusion Along the Interface of Bimaterials
,”
Int. J. Solids Struct.
,
51
(
23
), pp.
4094
4100
.
20.
Hasheminejad
,
B. S. M.
,
Gheshlaghi
,
B.
,
Mirzaei
,
Y.
, and
Abbasion
,
S.
,
2011
, “
Free Transverse Vibrations of Cracked Nanobeams With Surface Effects
,”
Thin Solid Films
,
519
(
8
), pp.
2477
2482
.
21.
Wang
,
G.-F.
, and
Feng
,
X.-Q.
,
2009
, “
Surface Effects on Buckling of Nanowires Under Uniaxial Compression
,”
Appl. Phys. Lett.
,
94
(
14
), p.
141913
.
22.
Wang
,
K.
, and
Wang
,
B.
,
2013
, “
Timoshenko Beam Model for the Vibration Analysis of a Cracked Nanobeam With Surface Energy
,”
J. Vib. Control
,
21
(
12
), pp.
2452
2464
.
23.
Hosseini-Hashemi
,
S.
,
Fakher
,
M.
,
Nazemnezhad
,
R.
, and
Sotoude Haghighi
,
M. H.
,
2014
, “
Dynamic Behavior of Thin and Thick Cracked Nanobeams Incorporating Surface Effects
,”
Composites, Part B
,
61
(
1
), pp.
66
72
.
24.
Lu
,
P.
,
He
,
L.
,
Lee
,
H.
, and
Lu
,
C.
,
2006
, “
Thin Plate Theory Including Surface Effects
,”
Int. J. Solids Struct.
,
43
(
16
), pp.
4631
4647
.
25.
Loya
,
J.
,
López-Puente
,
J.
,
Zaera
,
R.
, and
Fernández-Sáez
,
J.
,
2009
, “
Free Transverse Vibrations of Cracked Nanobeams Using a Nonlocal Elasticity Model
,”
J. Appl. Phys.
,
105
(
4
), p.
044309
.
26.
Torabi
,
K.
, and
Nafar Dastgerdi
,
J.
,
2012
, “
An Analytical Method for Free Vibration Analysis of Timoshenko Beam Theory Applied to Cracked Nanobeams Using a Nonlocal Elasticity Model
,”
Thin Solid Films
,
520
(
21
), pp.
6595
6602
.
27.
Hsu
,
J.-C.
,
Lee
,
H.-L.
, and
Chang
,
W.-J.
,
2011
, “
Longitudinal Vibration of Cracked Nanobeams Using Nonlocal Elasticity Theory
,”
Curr. Appl. Phys.
,
11
(
6
), pp.
1384
1388
.
28.
Loya
,
J.
,
Aranda-Ruiz
,
J.
, and
Fernández-Sáez
,
J.
,
2014
, “
Torsion of Cracked Nanorods Using a Nonlocal Elasticity Model
,”
J. Phys. D: Appl. Phys.
,
47
(
11
), p.
115304
.
29.
Roostai
,
H.
, and
Haghpanahi
,
M.
,
2014
, “
Vibration of Nanobeams of Different Boundary Conditions With Multiple Cracks Based on Nonlocal Elasticity Theory
,”
Appl. Math. Modell.
,
38
(
3
), pp.
1159
1169
.
30.
Liu
,
C.
, and
Rajapakse
,
R.
,
2010
, “
Continuum Models Incorporating Surface Energy for Static and Dynamic Response of Nanoscale Beams
,”
IEEE Trans. Nanotechnol.
,
9
(
4
), pp.
422
431
.
31.
Gurtin
,
M. E.
, and
Murdoch
,
A. I.
,
1975
, “
A Continuum Theory of Elastic Material Surfaces
,”
Arch. Ration. Mech. Anal.
,
57
(
4
), pp.
291
323
.
32.
He
,
J.
, and
Lilley
,
C. M.
,
2008
, “
Surface Effect on the Elastic Behavior of Static Bending Nanowires
,”
Nano Lett.
,
8
(
7
), pp.
1798
1802
.
33.
Park
,
H. S.
,
Klein
,
P. A.
, and
Wagner
,
G. J.
,
2006
, “
A Surface Cauchy–Born Model for Nanoscale Materials
,”
Int. J. Numer. Methods Eng.
,
68
(
10
), pp.
1072
1095
.
34.
Shenoy
,
V. B.
,
2005
, “
Atomistic Calculations of Elastic Properties of Metallic FCC Crystal Surfaces
,”
Phys. Rev. B
,
71
(
9
), p.
094104
.
35.
De Pasquale
,
G.
, and
Soma
,
A.
,
2011
, “
MEMS Mechanical Fatigue: Effect of Mean Stress on Gold Microbeams
,”
J. Microelectromech. Syst.
,
20
(
4
), pp.
1054
1063
.
36.
Kisa
,
M.
, and
Brandon
,
J.
,
2000
, “
The Effects of Closure of Cracks on the Dynamics of a Cracked Cantilever Beam
,”
J. Sound Vib.
,
238
(
1
), pp.
1
18
.
37.
Ou
,
Z.
,
Wang
,
G.
, and
Wang
,
T.
,
2008
, “
Effect of Residual Surface Tension on the Stress Concentration Around a Nanosized Spheroidal Cavity
,”
Int. J. Eng. Sci.
,
46
(
5
), pp.
475
485
.
38.
Wang
,
G.
, and
Wang
,
T.
,
2006
, “
Deformation Around a Nanosized Elliptical Hole With Surface Effect
,”
Appl. Phys. Lett.
,
89
(
16
), p.
161901
.
39.
Wang
,
B.
,
Han
,
J.
, and
Du
,
S.
,
2000
, “
Cracks Problem for Non-Homogeneous Composite Material Subjected to Dynamic Loading
,”
Int. J. Solids Struct.
,
37
(
9
), pp.
1251
1274
.
40.
Rubio
,
L.
, and
Fernández-Sáez
,
J.
,
2010
, “
A Note on the Use of Approximate Solutions for the Bending Vibrations of Simply Supported Cracked Beams
,”
ASME J. Vib. Acoust.
,
132
(
2
), p.
024504
.
41.
Timoshenko
,
S.
, and
Goodier
,
J.
,
1970
,
Theory of Elasticity
,
McGraw-Hill
,
New York
.
42.
Rao
,
S. S.
,
2007
,
Vibration of Continuous Systems
,
Wiley
,
Hoboken, NJ
.
43.
Aydin
,
K.
,
2007
, “
Vibratory Characteristics of Axially-Loaded Timoshenko Beams With Arbitrary Number of Cracks
,”
ASME J. Vib. Acoust.
,
129
(
3
), pp.
341
354
.
44.
Miller
,
R. E.
, and
Shenoy
,
V. B.
,
2000
, “
Size-Dependent Elastic Properties of Nanosized Structural Elements
,”
Nanotechnology
,
11
(
3
), pp.
139
–147.
You do not currently have access to this content.