### Size-Dependent Green’s Function for Bending of Circular Micro Plates Under Eccentric Load

Document Type : Research Paper

Authors

Faculty of Mechanical and Material Engineering, Graduate University of Advanced Technology, Kerman, Iran

Abstract

In this paper, a Green’s function is developed for bending analysis of micro plates under an asymmetric load. In order to consider the length scale effect, the modified couple stress theory is used. This theory can accurately predict the behavior of micro structures. A thin micro plate is considered and therefore the classical plate theory is utilized. The size dependent governing equilibrium equation of a circular micro plate under an eccentric load is obtained by using the minimum total potential energy principle. This equation is a partial differential equation and it is hard to solve it for an arbitrary loading. A transformation of the coordinate system is introduced to obtain the asymmetric exact solution for deflection of circular micro-plates. By using the obtained size dependent Green’s function, the bending behavior of microplates under arbitrary loads can be easily defined. The results are presented for different asymmetric loads. Also, it is concluded that the length scale has a significant effect on bending of micro plates.

Keywords

[1] Lam D.C.C., Yang F., Chong A.C.M., Wang J., Tong P., 2003, Experiments and theory in strain gradient elasticity, Journal of the Mechanics and Physics of Solids 51: 1477-1508.
[2] Yang F., Chong A.C.M., Lam D.C.C., Tong P., 2002, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures 39: 2731-2743.
[3] Mindlin R.D., 1965, Second gradient of strain and surface tension in linear elasticity, International Journal of Solids and Structures 1: 417-438.
[4] Jomehzadeh E., Noori H.R., Saidi A.R., 2011, The size-dependent vibration analysis of micro-plates based on a modiﬁed couple stress theory, Journal of Physica E 43: 877-883.
[5] Reddy J.N., Berry J., 2012, Nonlinear theories of axisymmetric bending of functionally graded circular plates with modiﬁed couple stress, Composite Structures 94: 3664-3668.
[6] Kumar R., Devi Sh., Sharma V., 2017, Axisymmetric problem of thick circular plate with heat sources in modified couple stress theory, Journal of Solid Mechanics 9: 157-171.
[7] Gholami R., Darvizeh A., Ansari R., Pourashraf T., 2017, Analytical treatment of the size-dependent nonlinear post-buckling of functionally graded circular cylindrical micro-/ nano-Shells, Iranian Journal of Science and Technology Transactions of Mechanical Engineering 42: 85-97.
[8] Ansari R., Gholami R., 2016 , Surface effect on the large amplitude periodic forced vibration of first-order shear deformable rectangular nanoplates with various edge supports, Acta Astronautica 118: 72-89.
[9] Gholami R., Ansari R., Darvizeh A., Sahmani S., 2015, Axial buckling and dynamic stability of functionally graded microshells based on the modified couple stress theory, International Journal of Structural Stability and Dynamics 15(04): 1450070.
[10] Ansari R., Gholami R., 2016, Nonlocal free vibration in the pre- and post- buckled states of magneto-electro-thermo elastic rectangular nanoplates with various edge conditions, Smart Materials and Structures 25: 5095033.
[11] Wang Y., Lin W., Zhou C., 2013, Nonlinear bending of size-dependent circular microplates based on the modiﬁed couple stress theory, Archive of Applied Mechanics 84: 391-400.
[12] Stolken J.S., Evans A.G., 1998, A microbend test method for measuring the plasticity length scale, Acta Metallurgica et Materialia 46: 5109-5115.
[13] Saidi A. R., Naderi A., Jomehzadeh E., 2009, A closed form solution for bending/stretching analysis of functionally graded circular plates under as symmetric loading using the Green function, IMECHE Part C Journal of Mechanical Engineering Science 1: 1-3.
[14] Liang K., Yang J., Kitipornchai S., Bradford M.A., 2012, Bending, buckling and vibration of size-dependent functionally graded annular micro-plates, Composite Structures 94: 3250-3257.
[15] Zhang B., He Y., Liu D., Shen L., Lei J., 2015, An efﬁcient size-dependent plate theory for bending, buckling and free vibration analyses of functionally graded microplates resting on elastic foundation, Mathematical Modelling 39: 3814-3845.
[16] Ansari R., Hasrati E., Faghih Shojaei M., Gholami R., Mohammadi V., Shahabodini A., 2016, Size-dependent bending, buckling and free vibration analyses of microscale functionally graded Mindlin plates based on the strain gradient elasticity theory, Latin American Journal of Solids and Structures 13(4): 632-664.
[17] Park S.K., Gao X.L., 2006, Bernoulli-Euler beam model based on a modified couple stress theory, Journal of Micromechanics and Microengineering 16: 2355-2359.
[18] Kong S., Zhou S., Nie, Z., Wang K., 2008, The size-dependent natural frequency of Bernoulli-Euler micro-beams, International Journal of Engineering Science 46: 427-437.
[19] Simsek M., Kocat¨urk T., Akbas S.D., 2013, Static bending of a functionally graded microscale Timoshenko beam based on the modified couple stress theory, Composite Structures 95: 740-747.
[20] Nateghi A., Salamat-talab M., Rezapour J., Daneshian B., 2012, Size dependent buckling analysis of functionally graded micro beams based on modified couple stress theory, Applied Mathematical Modelling 36: 4971-4987.
[21] Roque C.M.C., Fidalgo D.S., Ferreira A.J.M., Reddy J.N., 2013, A study of a microstructure-dependent composite laminated Timoshenko beam using a modified couple stress theory and a meshless method, Composite Structures 96: 532-537.
[22] Ke L.L., Wang Y., Yang J., Kitipornchai S., 2012, Nonlinear free vibration of size-dependent functionally graded micro-beams, International Journal of Engineering Science 50: 256-267.
[23] Ansari R., Gholami R., Faghih Shojaei M., Mohammadi V., Sahmani S., 2014, Bending, buckling and free vibration analysis of size-dependent functionally graded circular/annular micro-plates based on the modified strain gradient elasticity theory, European Journal of Mechanics 49: 251-267.
[24] Baghani M., MohammadSalehi M., Dabaghian P.H., 2016, Analytical couple-stress solution for size-dependent large-amplitude vibrations of FG tapered-nanobeams, Latin American Journal of Solids and Structures 13(1): 95-118.
[25] Karimipour I., Tadi Beni Y., Taheri N., 2017, Influence of electrical double-layer dispersion forces and size dependency on pull-in instability of clamped microplate immersed in ionic liquid electrolytes, Indian Journal of Physics 91(10): 1179-1195.
[26] Karimipour I., Tadi Beni Y., Zeighampour H., 2017, Nonlinear size-dependent pull-in instability and stress analysis of thin plate actuator based on enhanced continuum theories including nonlinear effects and surface energy, Microsystem Technologies 24: 1811-1839.