Analytical solutions for static bending of edge cracked micro beams

- Journal title : Structural Engineering and Mechanics
- Volume 59, Issue 3, 2016, pp.579-599
- Publisher : Techno-Press
- DOI : 10.12989/sem.2016.59.3.579

Title & Authors

Analytical solutions for static bending of edge cracked micro beams

Akbas, Seref Doguscan;

Akbas, Seref Doguscan;

Abstract

In this study, static bending of edge cracked micro beams is studied analytically under uniformly distributed transverse loading based on modified couple stress theory. The cracked beam is modelled using a proper modification of the classical cracked-beam theory consisting of two sub-beams connected through a massless elastic rotational spring. The deflection curve expressions of the edge cracked microbeam segments separated by the rotational spring are determined by the Integration method. The elastic curve functions of the edge cracked micro beams are obtained in explicit form for cantilever and simply supported beams. In order to establish the accuracy of the present formulation and results, the deflections are obtained, and compared with the published results available in the literature. Good agreement is observed. In the numerical study, the elastic deflections of the edge cracked micro beams are calculated and discussed for different crack positions, different lengths of the beam, different length scale parameter, different crack depths, and some typical boundary conditions. Also, the difference between the classical beam theory and modified couple stress theory is investigated for static bending of edge cracked microbeams. It is believed that the tabulated results will be a reference with which other researchers can compare their results.

Keywords

open edge crack;modified couple stress theory;cracked microbeam;

Language

English

Cited by

1.

Forced vibration analysis of viscoelastic nanobeams embedded in an elastic medium,;

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2.

3.

References

1.

Akbas, S.D. (2016), "Free vibration of edge cracked functionally graded microscale beams based on the modified couple stress theory", Int. J. Struct. Stab. Dyn., 1750033.

2.

Akgoz, B. and Civalek, O. (2012), "Analysis of microtubules based on strain gradient elasticity and modified couple stress theories", Adv. Vib. Eng., 11(4), 385-400.

3.

Akgoz, B. and Civalek, O. (2013), "Buckling analysis of linearly tapered micro-columns based on strain gradient elasticity", Struct Eng. Mech., 48(2), 195-205.

4.

Alashti, R.A. and Abolghasemi, A.H. (2013), "A size-dependent Bernoulli-Euler beam formulation based on a new model of couple stress theory", Int J. Eng. Tran. C: Aspect., 27(6), 951.

5.

Ansari, R., Ashrafi, M.A. and Arjangpay, A. (2015), "An exact solution for vibrations of postbuckled microscale beams based on the modified couple stress theory", Appl. Math. Model., 39(10-11), 3050-3062.

6.

Ansari, R., Gholami, R. and Darabi, M.A. (2012), "A nonlinear Timoshenko beam formulation based on strain gradient theory", J. Mech. Mater. Struct., 7(2), 95-211.

7.

Asghari, M., Ahmadian, M.T., Kahrobaiyan, M.H. and Rahaeifard, M. (2010), "On the size dependent behavior of functionally graded micro-beams", Mater. Des., 31, 2324-3249.

8.

Ataei, H., Beni, Y.T. and Shojaeian, M. (2016), "The effect of small scale and intermolecular forces on the pull-in instability and free vibration of functionally graded nano-switches", J. Mech. Sci. Tech., 30(4), 1799-1816.

9.

Beni, Y.T. and Zeverdejani, M.K. (2015), "Free vibration of microtubules as elastic shell model based on modified couple stress theory", J. Mech. Med BioI., 15(03), 1550037.

10.

Beni, Y.T., Jafari, A. and Razavi, H. (2015), "Size effect on free transverse vibration of cracked nano-beams using couple stress theory", Int. J. Eng., 28(2), 296-304.

11.

Beni, Y.T., Mehralian, F. and Zeighampour, H. (2016), "The modified couple stress functional graded cylindrical thin shell formulation", Mech. Adv. Mater. Struct., 23(7), 791-801.

12.

Broek, D. (1986), Elementary engineering fracture mechanics, Martinus Nijhoff Publisher, Dordrecht.

13.

Dai, H.L., Wang, Y.K. and Wang, L. (2015), "Nonlinear dynamics of cantilevered microbeams based on modified couple stress theory", Int. J. Eng. Sci., 94, 103-112.

14.

Daneshmehr, A.R., Abadi, M.M. and Rajabpoor, A. (2013), "Thermal effect on static bending, vibration and buckling of reddy beam based on modified couple stress theory", Appl. Mech. Mater., 332,331-338.

15.

Darijani, H. and Mohammadabadi, H. (2014), "A new deformation beam theory for static and dynamic analysis of micro beams", Int. J. Mech. Sci., 89, 31-39.

17.

Fang, T.H. and Chang, W.J. (2003b), "Sensitivity analysis of scanning near-field optical microscope probe", Opt. Laser Tech., 35(4), 267-271.

18.

Fang, T.H., Chang, W.J. and Liao, S.C. (2003a), "Simulated nanojet ejection process by spreading droplets on a solid surface", J. Phys. Condens. Matter., 15(49), 8263-8271.

19.

Farokhi, H. and Ghayesh, M.H. (2015), "Thermo-mechanical dynamics of perfect and imperfect Timoshenko microbeams", Int. J. Eng. Sci., 91, 12-33.

20.

Fleck, H.A. and Hutchinson, J.W. (1993), "A phenomenological theory for strain gradient effects in plasticity", J. Mech. Phys. Solid, 41, 1825-57.

21.

Ghayesh, M.H., Amabili, M. and Farokhi, H. (2013), "Three-dimensional nonlinear size-dependent behaviour of Timoshenko microbeams", Int. J. Eng. Sci., 71, 1-14.

22.

Hasheminejad, B.S.M., Gheshlaghi, B., Mirzaei, Y. and Abbasion, S. (2011), "Free transverse vibrations of cracked nanobeams with surface effects", Thin Solid. Film., 519(8), 2477-2482.

23.

Kahrobaiyan, M.H., Asghari, M., Rahaeifard, M. and Ahmadian, M.T. (2010), "Investigation of the size dependent dynamic characteristics of atomic force microscope microcantilevers based on the modified couple stress theory", Int. J. Eng. Sci., 48, 1985-1994.

24.

Ke, L.L., Wang, Y.S. and Wang, Z.D. (2011), "Thermal effect on free vibration and buckling of size-dependent microbeams", Physica E: Low Dimens. Syst. Nanostroct, 43(7), 1387-1393.

25.

Kocaturk, T. and Akbas, S.D. (2013), "Wave propagation in a microbeam based on the modified couple stress theory", Struct. Eng. Mech., 46, 417-431.

26.

Kong, S., Zhou, S., Nie, Z. and Wang, K. (2008), "The size-dependent natural frequency of Bernoulli-Euler micro-beams", Int. J. Eng. Sci., 46, 427-437.

27.

Kong, S.L. (2013), "Size effect on natural frequency of cantilever micro-beams based on a modified couple stress theory", Adv. Mater. Res., 694-697, 221-224.

28.

Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J. and Tong, P. (2003), "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Solid., 51(8), 1477-508.

29.

Liu, S.T., Qi, S.H. and Zhang, W.M. (2013), "Vibration behavior of a cracked micro-cantilever beam under electrostatic excitation", J. Vib. Shock., 32(17), 41-45.

30.

Loya, J., Lopez-Puente, J., Zaera, R. and Fernandez-Saez, J. (2009), "Free transverse vibrations of cracked nanobeams using a nonlocal elasticity model", J. Appl. Phys., 105(4), 044309.

31.

Ma, H.M., Gao, X.L. and Reddy, J.N. (2008), "A microstructure-dependent Timoshenko beam model based on a modified couple stress theory", J. Mech. Phys. Solid, 56, 3379-3391.

32.

33.

Mindlin, R.D. and Tiersten, H.F. (1962), "Effects of couple-stresses in linear elasticity", Arch. Rat. Mech. Anal., 11, 415-48.

34.

Park, S.K. and Gao, X.L. (2006), "Bernoulli-Euler beam model based on a modified couple stress theory", J. Micromech. Microeng, 16, 2355-2359.

35.

Pei, J., Tian, F. and Thundat, T. (2004), "Glucose biosensor based on the microcantilever", Anal. Chem., 76, 292-297.

36.

Rezazadeh, G., Tahmasebi, A. and Zubtsov, M. (2006), "Application of piezoelectric layers in electrostatic MEM actuators: controlling of pull-in voltage", J. Microsyst. Tech., 12, 1163-1170.

37.

Sedighi, H.M. and Bozorgmehri, A. (2016), "Dynamic instability analysis of doubly clamped cylindrical nanowires in the presence of Casimir attraction and surface effects using modified couple stress theory", Acta Mechanica, 227(6), 575-1591.

38.

Sedighi, H.M., Changizian, M. and Noghrehabadi, A. (2014), "Dynamic pull-in instability of geometrically nonlinear actuated micro-beams based on the modified couple stress theory", Lat. Am. J. Solid Struct., 11(5), 810-825.

39.

Senturia S.D. (1998), "CAD challenges for microsensors, microactuators, and microsystems", Froc. IEEE, 86, 1611-1626.

40.

Shojaeian, M. and Beni, Y.T. (2015), "Size-dependent electromechanical buckling of functionally graded electrostatic nano-bridges", Sens. Actuat. A: Fhys., 232, 49-62.

41.

Shojaeian, M., Beni, Y.T. and Ataei, H. (2016), "Electromechanical buckling of functionally graded electrostatic nanobridges using strain gradient theory", Acta Astronautica, 118, 62-71.

42.

Simsek, M. (2010), "Dynamic analysis of an embedded microbeam carrying a moving microparticle based on the modified couple stress theory", Int. J. Eng. Sci., 48, 1721-1732.

43.

Simsek, M., Kocaturk, T. and Akbas, S.D. (2013), "Static bending of a functionally graded microscale Timoshenko beam based on the modified couple stress theory", Compos. Struct, 95, 740-747.

44.

Tada, H., Paris, P.C. and Irwin, G.R. (1985), The Stress Analysis of Cracks Handbook, Paris Production Incorporated and Del Research Corporation.

45.

Tang, M., Ni, Q., Wang, L., Luo, Y. and Wang, Y. (2014), "Size-dependent vibration analysis of a microbeam in flow based on modified couple stress theory", Int. J. Eng. Sci., 85, 20-30.

46.

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. Film., 520(21), 6595-6602.

47.

48.

Wang, L. (2010), "Size-dependent vibration characteristics of fluid-conveying microtubes", J. Fluid. Struct., 26, 675-684.

49.

Wang, L., Xu, Y.Y. and Ni, Q. (2013), "Size-dependent vibration analysis of three-dimensional cylindrical microbeams based on modified couple stress theory: a unified treatment", Int. J. Eng. Sci., 68, 1-10.

50.

Xia, W., Wang, L. and Yin, L. (2010), "Nonlinear non-classical microscale beams: static, bending, postbuckling and free vibration", Int. J. Eng. Sci., 48, 2044-053.

51.

Yang, F., Chong, A., Lam, D. and Tong, P. (2002) "Couple stress based strain gradient theory for elasticity", Int. J. Solid. Struct., 39(10), 2731-2743.

52.

Zeighampour, H. and Beni, Y.T. (2014), "Analysis of conical shells in the framework of coupled stresses theory", Int. J. Eng. Sci., 81, 107-122.

53.

Zeighampour, H. and Beni, Y.T. (2015), "A shear deformable cylindrical shell model based on couple stress theory", Arch. Appl. Mech., 85(4), 539-553.

54.

Zeighampour, H., Beni, Y.T. and Mehralian, F. (2015), "A shear deformable conical shell formulation in the framework of couple stress theory", Acta Mechanica, 226(8), 2607-2629.

55.

Zook, J.D., Burns, D.W., Guckel, H., Smegowsky, J.J., Englestad, R.L. and Feng, Z. (1992), "Characteristics of polysilicon resonant microbeams", Sens. Actuat., 35, 31-59.