Bending and vibration analysis of delaminated Bernoulli–Euler micro-beams using the modified couple stress theory

Document Type : Article

Authors

1 Department of Mechanical Engineering, Babol Noshirvani University of Technology, Babol 47148-71167, Iran

2 Department of Mechanical Engineering, Babol Noshirvani University of Technology, Babol 47148 - 71167, Iran

3 Department of Mechanical Engineering, The City College of the City University of New York, NY 10031, USA

Abstract

In this paper, we study static bending and free vibration behavior of Bernoulli–Euler micro-beams with a single delamination using the modified couple stress theory. The delaminated beam is modeled by four interconnected sub-beams using the delamination zone as their boundaries. The free and constrained mode theories have been utilized to model the interaction of delamination surfaces in the damaged area. The continuity as well as compatibility conditions are satisfied between the neighboring sub-beams. After verification of the results for some case studies with available solutions, the effect of various parameters such as spanwise and thicknesswise locations of the delamination, material length scale parameter, and boundary conditions on the static bending and free vibration characteristics of the size-dependent micro-beam have been investigated in detail.

Keywords

Main Subjects


References
1. Hung, E.S. and Senturia, S.D. Extending the travel
range of analog-tuned electrostatic actuators", J. Microelectromech.
S., 8(4), pp. 497-505 (1999).
2. Pei, J., Tian, F., and Thundat, T. Novel glucose
biosensor based on the microcantilever", Anal. Chem.,
76, pp. 292-297 (2004).
3. Pereira, R.D.S. Atomic force microscopy as a novel
pharmacological tool", Biochem. Pharmacol., 62, pp.
975-983 (2001).
4. Jafari-Talookolaei, R.A., Abedi, M., Simsek, M., and
Attar, M. Dynamics of a micro scale Timoshenko
beam subjected to a moving micro particle based on
the modi ed couple stress theory", J. Vib. Control,
24(3), pp. 527-548 (2018).
5. Mindlin, R. and Tiersten, H. E ects of couple-stresses
in linear elasticity", Arch. Ration. Mech. An., 11, pp.
415-448 (1962).
6. Toupin, R.A. Elastic materials with couple-stresses",
Arch. Ration. Mech. An., 11, pp. 385-414 (1962).
7. Yang, F., Chong, A., Lam, D., and Tong, P. Couple
stress based strain gradient theory for elasticity", Int.
J. Solids Struct., 39, pp. 2731-2743 (2002).
8. Park, S.K. and Gao, X.L. Bernoulli-Euler beam
model based on a modi ed couple stress theory", J.
Micromech. Microeng., 16, pp. 2355-2359 (2006).
9. Ma, H.M., Gao, X.L., and Reddy, J. A microstructure-
dependent Timoshenko beam model based
on a modi ed couple stress theory", J. Mech. Phys.
Solids, 56, pp. 3379-3391 (2008).
10. Dehrouyeh-Semnani, A.M. and Nikkhah-Bahrami, M.
A discussion on incorporating the Poisson e ect in
micro-beam models based on modi ed couple stress
theory", Int. J. Eng. Sci., 86, pp. 20-25 (2015).
11. Reddy, J.N. and El-Borgi, S. Eringen's nonlocal
theories of beams accounting for moderate rotations",
Int. J. Eng. Sci., 82, pp. 159-177 (2014).
12. Kahrobaiyan, M.H., Asghari, M., Rahaeifard, M., and
Ahmadian, M.T. Investigation of the size-dependent
dynamic characteristics of atomic force microscope
microcantilevers based on the modi ed couple stress
theory", Int. J. Eng. Sci., 48, pp. 1985-1994 (2010).
13. Asghari, M., Kahrobaiyan, M.H., and Ahmadian, M.T.
A nonlinear Timoshenko beam formulation based on
the modi ed couple stress theory", Int. J. Eng. Sci.,
48, pp. 1749-1761 (2010).
14. Akgoz, B. and Civalek,  O. Strain gradient elasticity
and modi ed couple stress models for buckling analysis
of axially loaded micro-scaled beams", Int. J. Eng.
Sci., 49, pp. 1268-1280 (2011).
15. Simsek, M. and Reddy, J.N. Bending and vibration of
functionally graded micro-beams using a new higher
order beam theory and the modi ed couple stress
theory", Int. J. Eng. Sci., 64, pp. 37-53 (2013).
16. Akgoz, B. and Civalek,  O. Free vibration analysis
of axially functionally graded tapered Bernoulli-Euler
micro-beams based on the modi ed couple stress theory",
Compos. Struct., 98, pp. 314-322 (2013).
17. Mohammad Abadi, M. and Daneshmehr, A.R. An investigation
of modi ed couple stress theory in buckling
analysis of micro composite laminated Euler-Bernoulli
and Timoshenko beams", Int. J. Eng. Sci., 75, pp.
40-53 (2014).
18. Darijani, H. and Mohammadabadi, H. A new deformation
beam theory for static and dynamic analysis of
micro-beams", Int. J. Mech. Sci., 89, pp. 31-39 (2014).
19. Dai, H.L., Wang, Y.K., and Wang, L. Nonlinear dynamics
of cantilevered micro-beams based on modi ed
couple stress theory", Int. J. Eng. Sci., 94, pp. 103-112
(2015).
20. Mohammad-Abadi, M. and Daneshmehr, A.R. Size
dependent buckling analysis of micro-beams based on
modi ed couple stress theory with high order theories
and general boundary conditions", Int. J. Eng. Sci.,
74, pp. 1-14 (2014).
21. Simsek, M. Nonlinear static and free vibration analysis
of micro-beams based on the nonlinear elastic
foundation using modi ed couple stress theory and
He's variational method", Compos. Struct., 112, pp.
264-272 (2014).
22. Ghayesh, M.H., Farokhi, H., and Amabili, M. Nonlinear
dynamics of a microscale beam based on the
modi ed couple stress theory", Compos. Eng., 50, pp.
318-324 (2013).
23. Farokhi, H., Ghayesh, M.H., and Amabili, M. Nonlinear
dynamics of a geometrically imperfect micro-beam
based on the modi ed couple stress theory", Int. J.
Eng. Sci., 68, pp. 11-23 (2013).
24. Wang, J.T.S., Liu, Y.Y., and Gibby, J.A. Vibrations
of split beams", J. Sound Vib., 84, pp. 491-502 (1982).
25. Mujumdar, P.M. and Suryanarayan, S. Flexural vibrations
of beams with delaminations", J. Sound Vib.,
125, pp. 441-461 (1988).
26. Shen, M.H. and Grady, J.E. Free vibrations of delaminated
beams", AIAA J., 30, pp. 1361-1370 (1992).
27. Della, C.N., Shu, D., and MSRao, P. Vibrations of
beams with two overlapping delaminations", Compos.
Struct., 66, pp. 101-108 (2004).
28. Manoach, E., Warminski Mitura, J.A., and Samborski,
S. Dynamics of a composite Timoshenko beam with
delamination", Mech. Res. Commun., 46, pp. 47-53
(2012).
29. Kargarnovin, M.H., Ahmadian, M.T., and Jafari-
Talookolaei, R.A. Forced vibration of delaminated
Timoshenko beams subjected to a moving load", Sci.
Eng. Compos. Mater., 19, pp. 145-157 (2012).
688 R.-A. Jafari-Talookolaei et al./Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 675{688
30. Kargarnovin, M.H., Jafari-Talookolaei, R.A., and Ahmadian,
M.T. Vibration analysis of delaminated
Timoshenko beams under the motion of a constant
amplitude point force traveling with uniform velocity",
Int. J. Mech. Sci., 70, pp. 39-49 (2013).
31. Szekrenyes, A. Coupled
exural-longitudinal vibration
of delaminated composite beams with local stability
analysis", J. Sound Vib., 333, pp. 5141-5164
(2014).
32. Szekrenyes, A. A special case of parametrically excited
systems: Free vibration of delaminated composite
beams", Eur. J. Mech. A-Solid, 49, pp. 82-105
(2015).
33. Attar, M., Karrech, A., and Regenauer-Lieb, K. Nonlinear
modal analysis of structural components subjected
to unilateral constraints", J. Sound Vib., 389,
pp. 380-410 (2016).
34. Attar, M., Karrech, A., and Regenauer-Lieb, K. Nonlinear
analysis of beam-like structures on unilateral
foundations: a lattice spring model", Int. J. Solids
Struct., 88, pp. 192-214 (2016).
35. Clive, L.D. and Shames, I.H., Solid Mechanics, A
Variational Approach, Springer (2013).