Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Analytical solutions for static bending of edge cracked micro beams

  • Received : 2015.08.24
  • Accepted : 2016.06.10
  • Published : 2016.08.10
⟨ Previous Next ⟩

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

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. https://doi.org/10.12989/sem.2013.48.2.195
  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. https://doi.org/10.1016/j.apm.2014.11.029
  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. https://doi.org/10.1016/j.matdes.2009.12.006
  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. https://doi.org/10.1007/s12206-016-0337-9
  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. https://doi.org/10.1142/S0219519415500372
  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. https://doi.org/10.1080/15376494.2015.1029167
  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. https://doi.org/10.1016/j.ijengsci.2015.05.007
  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. https://doi.org/10.4028/www.scientific.net/AMM.332.331
  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. https://doi.org/10.1016/j.ijmecsci.2014.08.019
  16. Eringen, A.C. (1972), "Nonlocal polar elastic continua", Int J. Eng. Sci., 10(1), 1-16. https://doi.org/10.1016/0020-7225(72)90070-5
  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. https://doi.org/10.1016/S0030-3992(03)00004-5
  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. https://doi.org/10.1088/0953-8984/15/49/005
  19. Farokhi, H. and Ghayesh, M.H. (2015), "Thermo-mechanical dynamics of perfect and imperfect Timoshenko microbeams", Int. J. Eng. Sci., 91, 12-33. https://doi.org/10.1016/j.ijengsci.2015.02.005
  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. https://doi.org/10.1016/0022-5096(93)90072-N
  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. https://doi.org/10.1016/j.ijengsci.2013.04.003
  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. https://doi.org/10.1016/j.tsf.2010.12.143
  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. https://doi.org/10.1016/j.ijengsci.2010.06.003
  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. https://doi.org/10.1016/j.physe.2011.03.009
  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. https://doi.org/10.12989/sem.2013.46.3.417
  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. https://doi.org/10.1016/j.ijengsci.2007.10.002
  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. https://doi.org/10.4028/www.scientific.net/AMR.694-697.221
  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. https://doi.org/10.1016/S0022-5096(03)00053-X
  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. https://doi.org/10.1063/1.3068370
  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. https://doi.org/10.1016/j.jmps.2008.09.007
  32. Mindlin, R.D. (1963), "Influence of couple-stresses on stress concentrations", Exp. Mech., 3, 1-7. https://doi.org/10.1007/BF02327219
  33. Mindlin, R.D. and Tiersten, H.F. (1962), "Effects of couple-stresses in linear elasticity", Arch. Rat. Mech. Anal., 11, 415-48. https://doi.org/10.1007/BF00253946
  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. https://doi.org/10.1088/0960-1317/16/11/015
  35. Pei, J., Tian, F. and Thundat, T. (2004), "Glucose biosensor based on the microcantilever", Anal. Chem., 76, 292-297. https://doi.org/10.1021/ac035048k
  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. https://doi.org/10.1007/s00542-006-0245-5
  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. https://doi.org/10.1590/S1679-78252014000500005
  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. https://doi.org/10.1016/j.sna.2015.04.025
  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. https://doi.org/10.1016/j.actaastro.2015.09.015
  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. https://doi.org/10.1016/j.ijengsci.2010.09.027
  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. https://doi.org/10.1016/j.compstruct.2012.08.036
  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. https://doi.org/10.1016/j.ijengsci.2014.07.006
  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. https://doi.org/10.1016/j.tsf.2012.06.063
  47. Toupin, R.A. (1962), "Elastic materials with couple stresses", Arch. Rat. Mech. Anal., 11, 385-414. https://doi.org/10.1007/BF00253945
  48. Wang, L. (2010), "Size-dependent vibration characteristics of fluid-conveying microtubes", J. Fluid. Struct., 26, 675-684. https://doi.org/10.1016/j.jfluidstructs.2010.02.005
  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. https://doi.org/10.1016/j.ijengsci.2013.03.004
  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. https://doi.org/10.1016/j.ijengsci.2010.04.010
  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. https://doi.org/10.1016/S0020-7683(02)00152-X
  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. https://doi.org/10.1016/j.ijengsci.2014.04.008
  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. https://doi.org/10.1007/s00419-014-0929-8
  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. https://doi.org/10.1007/s00707-015-1318-2
  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.

Cited by

  1. Forced vibration analysis of viscoelastic nanobeams embedded in an elastic medium vol.18, pp.6, 2016, https://doi.org/10.12989/sss.2016.18.6.1125
  2. A semi-continuum-based bending analysis for extreme-thin micro/nano-beams and new proposal for nonlocal differential constitution vol.172, 2017, https://doi.org/10.1016/j.compstruct.2017.03.070
  3. Forced Vibration Analysis of Functionally Graded Nanobeams vol.09, pp.07, 2017, https://doi.org/10.1142/S1758825117501009
  4. Dynamic responses of a beam with breathing cracks by precise integration method vol.60, pp.5, 2016, https://doi.org/10.12989/sem.2016.60.5.891
  5. Forced vibration analysis of cracked nanobeams vol.40, pp.8, 2018, https://doi.org/10.1007/s40430-018-1315-1
  6. Nonlocal thermo-electro-mechanical vibration analysis of smart curved FG piezoelectric Timoshenko nanobeam vol.20, pp.3, 2016, https://doi.org/10.12989/sss.2017.20.3.351
  7. A simple quasi-3D sinusoidal shear deformation theory with stretching effect for carbon nanotube-reinforced composite beams resting on elastic foundation vol.13, pp.5, 2016, https://doi.org/10.12989/eas.2017.13.5.509
  8. Forced vibration analysis of cracked functionally graded microbeams vol.6, pp.1, 2016, https://doi.org/10.12989/anr.2018.6.1.039
  9. The discrete element method simulation and experimental study of determining the mode I stress-intensity factor vol.66, pp.3, 2018, https://doi.org/10.12989/sem.2018.66.3.379
  10. Large deflection analysis of a fiber reinforced composite beam vol.27, pp.5, 2016, https://doi.org/10.12989/scs.2018.27.5.567
  11. Bending of a cracked functionally graded nanobeam vol.6, pp.3, 2016, https://doi.org/10.12989/anr.2018.6.3.219
  12. Numerical analysis of the receding contact problem of two bonded layers resting on an elastic half plane vol.72, pp.6, 2016, https://doi.org/10.12989/sem.2019.72.6.775
  13. A GN-based modified model for size-dependent coupled thermoelasticity analysis in nano scale, considering nonlocality in heat conduction and elasticity: An analytical solution for a nano beam with ene vol.73, pp.3, 2016, https://doi.org/10.12989/sem.2020.73.3.287
  14. Simulating vibration of single-walled carbon nanotube using Rayleigh-Ritz's method vol.8, pp.3, 2016, https://doi.org/10.12989/anr.2020.8.3.215
  15. Modal analysis of viscoelastic nanorods under an axially harmonic load vol.8, pp.4, 2020, https://doi.org/10.12989/anr.2020.8.4.277
  16. Static analysis of cutout microstructures incorporating the microstructure and surface effects vol.38, pp.5, 2016, https://doi.org/10.12989/scs.2021.38.5.583
  17. Effect of nonlinear FG-CNT distribution on mechanical properties of functionally graded nano-composite beam vol.78, pp.2, 2021, https://doi.org/10.12989/sem.2021.78.2.117
  18. A Force-Based Rectangular Cracked Element vol.13, pp.4, 2016, https://doi.org/10.1142/s1758825121500472
  19. Size-dependent dynamics of double-microbeam systems with various boundary conditions via modified couple stress theory vol.27, pp.8, 2016, https://doi.org/10.1007/s00542-020-05183-z
  20. Free vibration analysis of carbon nanotube RC nanobeams with variational approaches vol.11, pp.2, 2021, https://doi.org/10.12989/anr.2021.11.2.157
  21. Propagation of waves with nonlocal effects for vibration response of armchair double-walled CNTs vol.11, pp.2, 2016, https://doi.org/10.12989/anr.2021.11.2.183
  22. An investigation of mechanical properties of kidney tissues by using mechanical bidomain model vol.11, pp.2, 2016, https://doi.org/10.12989/anr.2021.11.2.193