Analysis of Nanoplate with a Central Crack Under Distributed Transverse Load Based on Modified Nonlocal Elasticity Theory
Subject Areas : Mechanics of SolidsM Rajabi 1 , H Lexian 2 , A Rajabi 3
1 - Mechanical Engineering, Malek Ashtar University of Technology (MUT), Tehran, Iran
2 - Faculty of Material & Manufacturing Technology, Malek Ashtar University of Technology (MUT), Tehran, Iran
3 - Mechanical Engineering, Malek Ashtar University of Technology (MUT), Tehran, Iran
Keywords: Nanoplate, Small-scale effect, Crack, Singularity, Nonlocal elasticity theory,
Abstract :
In this paper, using the complete modified nonlocal elasticity theory, the deflection and strain energy equations of rectangular nanoplates, with a central crack, under distributed transverse load were overwritten. First, the deflection of nanoplate was obtained using Levy's solution and consuming it; strain energy of nanoplate was found. As regards nonlocal elasticity theory wasn’t qualified for predicting the static behavior of nanoplates under distributed transverse load, using modified nonlocal elasticity theory, the deflection of nanoplate with a central crack for different values of the small-scale effect parameter was achieved. It was gained with the convergence condition for the complete modified nonlocal elasticity theory. To verify the result, the results for the state of the small-scale effect parameter were placed equal to zero (plate with macro-scale) and then were compared with the numerical results as well as the classical analytical solution results available in the valid references. It was shown that the complete modified nonlocal elasticity theory does not show any singularity at the crack-tip unlike the classical theory; therefore, the method presented is a suitable method for analysis of the nanoplates with a central crack.
[1] Taheri A.H., 2009, The Crack Study and Analysis on Nano-Dimensional Plates Based on the Expanded Finite Element Method, K. N. Toosi University of Technology, Tehran, Iran.
[2] Gao J., 1999, An asymmetric theory of nonlocal elasticity – Part 2: Continuum field, International Journal of Solids And Structures 36(20): 2959-297l.
[3] Zhou Z.G., Han J.C., Du S.Y., 1999, Investigation of a Griffith crack subject to anti-plane shear by using the nonlocal theory, International Journal of Solids and Structures 36: 3891-3901.
[4] Zhou Z.G., Wang B., Du S.Y., 2003, Investigation of anti-plane shear behavior of two collinear permeable cracks in a piezoelectric material by using the nonlocal theory, ASME Journal of Applied Mechanics 69: 388-390.
[5] Zhou Z.G., Shen Y.P., 1999, Investigation of the scattering of harmonic shear waves by two collinear cracks using the nonlocal theory, Acta Mechanica 135: 169-179.
[6] Zhou Z.G., Wang B., 2003, Investigation of anti-plane shear behavior of two collinear impermeable cracks in the piezoelectric materials by using the nonlocal theory, International Journal of Solids and Structures 39:1731-1742.
[7] Sun Y.G., Zhou Z.G., 2004, Stress field near the crack tip in nonlocal anisotropic elasticity, European Journal of Mechanics A/Solids 23(2): 259-269,2004.
[8] Zhou Z.G., Wang B., 2003, Nonlocal theory solution of two collinear cracks in the functionally graded materials, International Journal of Solids and Structures 43: 887-898.
[9] Huang L.Y., Han Q., Liang Y.J., 2012, Calibration of nonlocal scale effect parameter for bending single-layered graphene sheet under molecular dynamics, NANO: Brief Reports and Reviews 5: 1250033-1250041.
[10] Yan J.W., Tong L.H., Li C., Zhu Y., Wang Z.W., 2015, Exact solutions of bending deflections for nano-beams and nano- plates based on nonlocal elasticity theory, Composite Structures 125: 304-313.
[11] Rezatalab J., Golmakani M.E., 2015, Nonlinear bending analyzing of graphen nanoplate in polymeric environment using eringen nonlocal model, 22th Mechanical Engineering Conference, Ahvaz, Iran.
[12] Şeref A., 2016, Static analysis of a nano plate by using generalized differential quadrature method, International Journal of Engineering & Applied Sciences 8(2): 30-39.
[13] Eskandari Shahraki M., Heydari Bani M., Zamani M.R., Eskandari Jam J., 2018, Bending analyzing of graphen kirshoff nanoplate with simply supports using modified couple stress theory, 5th Mechanical Engineering Conference, Aerospace Industries, Mashhad, Iran.
[14] Mousavi Z., Shahidi S.A., Boroomand B., 2017, A new method for bending and buckling analysis of rectangular nano plate: full modified nonlocal theory, Springer 52(11): 2751-2768.
[15] Shvabyuk V., Pasternak I., Sulym H., 2011, Bending of orthotropic plate containing a crack parallel to the median plane, Acta Mechanica et Automatica 5: 94-102.
[16] Chattopadhyay L., 2011, Analytical solution for bending stress intensity factor from reissner’s plate theory, Scientific Research Publishing 3: 517-524.
[17] Yang W.H., 1968, On an integral equation solution for a plate with internal support, The Quarterly Journal of Mechanics and Applied Mathematics 21(4): 503-515.
[18] Keer L.M., Sve C., 1970, On the bending of cracked plates, International Journal of Solids and Structures 6: 1545-1599.