A new3D elasticity numerical solution for nonlinear thermo-mechanical bending of orthotropic annular/circular micro/nanoplates using re-modified couple stress & SAPM
Subject Areas : Mechanical EngineeringAmirreza Golkarian 1 , Mehrdad Jabbarzadeh 2
1 - Ph. D. Student, Science & Research branch, Islamic Azad University, Tehran, Iran
2 - Department of Mechanical Engineering, Mashhad branch, Islamic Azad University, Mashhad, Iran
Keywords: Three dimensional elasticity theory, Re-modified couple stress theory, Semi-analytical polynomial method, Annular nanoplate, Nonlinear bending, Orthotropic material,
Abstract :
In this research, a new 3D elasticity numerical solution based on the Semi-Analytical Polynomial Method (SAPM) is presented for the nonlinear bending analysis of orthotropic annular/circular micro/nanoplates resting on a Winkler-Pasternak elastic foundation, utilizing re-modified couple stress theory. This is the first report of a 3D elasticity numerical solution specifically based on re-modified couple stress theory. Moreover, while previous 3D analytical solutions based on couple stress theory have primarily focused on the bending of rectangular plates with simply supported edges, this study investigates the nonlinear bending of circular plates under various boundary conditions. Additionally, the variation in thickness under different types of loading (mechanical, thermal, and thermo-mechanical) is reported for the first time. The effects of boundary conditions, couple stress scale parameters, aspect ratio, thickness, loading, and the elastic foundation are examined. Both the increasing and decreasing impacts of scale parameters on deflection are observed in this study.
[1] Rafiee M.A., Rafiee J., 2009, Enhanced mechanical properties of nano composites at low graphene contents, ACS Nano3: 3884-90.
[2] Golkarian A.R., Jabbarzadeh M., 2012, The Attitude of Variation of Elastic Modules in Single Wall Carbon Nanotubes: Nonlinear Mass-Spring Model, J of Solid Mech 4: 106-113.
[3] Golkarian A,R., Jabbarzadeh M., 2013, The density effect of van der Waals forces on the elastic modules in graphite layers, Comput Mater Sci 74: 138-142.
[4] Shariati A.A., Golkarian A.R., Jabbarzadeh M., 2014, Investigation of Vibrational Behavior of Perfect and Defective Carbon Nanotubes Using Non–Linear Mass–Spring Model, J of Solid Mech 6: 255-264.
[5] Golkarian A.R., Jabbarzadeh M., 2015, The influence of interlayer interactions on the mechanical properties of polymeric nanocomposites, JSCS–4810 80(11): 1449-1459.
[6] Bao W.X., Zhu C.C., 2004, Simulation of young's modulus of single-walled carbon nanotubes by molecular dynamics, Physica B 352: 156-63.
[7]Zhao H., Min K., 2009, Size and chirality dependent elastic properties of graphene nanoribbons under uniaxial tension, Nano Letter 9: 3012-15.
[8] Toupin R.A., 1962, Elastic materials with couple stresses, Arch Ration Mech Anal 11: 385-414.
[9] Mindlin R.D., Tiersten H.F., 1962, Effects of couple-stresses in linear elasticity, Arch Ration Mech Anal 11: 415-448.
[10] Mindlin R.D., 1963, Influence of couple-stresses on stress concentrations, Exp Mech 3: 1-7.
[11] Koiter W.T., 1964, Couple-stresses in the theory of elasticity: I and II. Proc K Ned Akad Wet B 67: 17-44.
[12] Aifantis E.C., 1999, Strain gradient interpretation of size effects, Int J Fract. 95: 1-4.
[13] Eringen A.C., 1972, Nonlocal polar elastic continua, Int J Eng Sci 10: 1-16.
[14] Gurtin M.E., Weissmuller J., Larche F., 1998, The general theory of curved deformable interfaces in solids at equilibrium, Philos Mag A: 1093-1109.
[15] Yang F., Chong A.C.M., Lam D.C.C., Tong P., 2002, Couple stress based strain gradient theory for elasticity, Int J Solids Struct 39: 2731-43.
[16] Reddy J.N., Berry J., 2012, Nonlinear theories of axisymmetric bending of functionally graded circular plates with modified couple stress, Compos Struct 94 (12): 3664-68.
[17] Reddy J.N., Kim J., 2012, A nonlinear modified couple stress-based third-order theory of functionally graded plates, Compos Struct 94: 1128-1143.
[18] Dastjerdi S., Jabbarzadeh M., 2016, Nonlinear bending analysis of bilayer orthotropic graphene sheets resting on Winkler-Pasternak elastic foundation based on Nonlocal Continuum Mechanics, Compos Part B 87: 161-175.
[19] Dastjerdi S., Jabbarzadeh M., Aliabadi S., 2016, Nonlinear static analysis of single layer annular/circular graphene sheets embedded in Winkler–Pasternak elastic matrix based on non-local theory of Eringen, Ain Shams Eng J 7: 873-884.
[20] Dastjerdi S., Lotfi M., Jabbarzadeh M., 2016, The effect of vacant defect on bending analysis of graphene sheets based on the Mindlin nonlocal elasticity theory, Compos Part B 98: 78-87.
[21] Dastjerdi S., Jabbarzadeh M., 2017, Non-linear bending analysis of multi-layer orthotropic annular/circular graphene sheets embedded in elastic matrix in thermal environment based on non-local elasticity theory, App Math Model 41: 83-101.
[22] Dastjerdi S., Jabbarzadeh M., 2016, Nonlocal Bending Analysis of Bilayer Annular/Circular Nano Plates Based on First Order Shear Deformation Theory, J of Solid Mech 8: 645-661.
[23] Dastjerdi S., Jabbarzadeh M., 2016, Non-Local Thermo-Elastic Buckling Analysis of Multi- Layer Annular/Circular Nano-Plates Based on First and Third Order Shear Deformation Theories Using DQ Method, J of Solid Mech 8: 859-874.
[24] Thai H.T., Choi D.H., 2013, Size-dependent functionally graded Kirchhoff and Midline plate models based on a modified couple stress theory, Compos Struct 95: 142-153.
[25] Thai H.T., Kim Se., 2013, A size-dependent functionally graded Reddy plate model based on a modified couple stress theory, Compos Part B 50: 1636-1645.
[26] Thai H.T., Vo T.P., 2013, A size-dependent functionally graded sinusoidal plate model based on a modified couple stress theory, Compos Struct 96: 376-383.
[27] Sahmani S., Ansari R., 2013, On the free vibration response of functionally graded higher-order shear deformable microplates based on the strain gradient elasticity theory, Compos Struct 95: 430-442.
[28] Daneshmehr A., Rajabpoor A., pourdavood M., 2014, Stability of size dependent functionally graded nanoplate based on nonlocal elasticity and higher order plate theories and different boundary conditions, Int J Eng Sci 82: 84-100.
[29] Kashtalyan M., 2004, Three-dimensional elasticity solution for bending of functionally graded rectangular plates, Eur J MechA Solids 23: 853-864.
[30] Kashtalyan M., Menshykova M., 2009, Effect of a functionally graded interlayer on three-dimensional elastic deformation of coated plates subjected to transverse loading, Compos Struct 89: 167-176.
[31] Huang Z.Y., Lu C.F., Chen W.Q., 2008, Benchmark solutions for functionally graded thick plates resting on Winkler–Pasternak elastic foundations, Compos Struct 85: 95-104.
[32] Woodward B., Menshykova M., 2011, Three-dimensional elasticity solution for bending of transversely isotropic functionally graded plates, Eur J Mech A Solids 30: 705-718.
[33] Salehipour H., Nahvi H., Shahidi A., Mirdamadi H.R., 2017, 3D elasticity analytical solution for bending of FG micro/nanoplates resting on elastic foundation using modified couple stress theory, App Math Model 47: 174-188.
[34] Reddy J.N., Romanoff J., Loya J.A., 2016, Nonlinear finite element analysis of functionally graded circular plates with modified couple stress theory, Euro J of Mech A/Solids 56: 92-104.
[35] Yang F., Chong A.C.M., Lam D.C.C., Tong P., 2002, Couple stress based strain gradient theory for elasticity, Int J Solids Struct 39: 2731-43.
[36] Chen W., Li X., 2014, A new modified couple stress theory for anisotropic elasticity and microscale laminated Kirchhoff plate model, Arch Appl Mech 84 (3): 323-41.
[37]Hosseini Kordkheili S.A., Moshrefzadeh-Sani H., 2013, Mechanical properties of double-layered graphene sheets, Comput Mat Sci 69: 335-343.
[38] Altekin M., Yu¨ kseler R.F., 2011, London, U.K. WCE. In: Proof the World Cong on Eng.
[39] Timoshenko S., Woinowsky-Krieger S., 1959, Theories of Plates and Shells. New York: McGraw-Book Company, p. 410-412.
[40] Szilard R., 1974, Theory and Analysis of Plates, Englewood Cliffs: Prentice-Hall Inc. p. 622.
[41] Reddy J.N., Wang C.M., Kitipornchai S., 1999, Axisymmetric bending of functionally grade circular and annular plates, Euro J MechA/Solids18: 185-99.
[42] Golmakani M.E., 2013, Non-linear bending analysis of ring-stiffened functionally graded circular plates under mechanical and thermal loadings, Int J Mech Sci 79: 130-42.
[43] Sobhy M., 2014, Thermo mechanical bending and free vibration of single-layered graphene sheets embedded in an elastic medium, Physica E 56: 400-409.
[44] Panyatong M., Chinnaboon B., Chucheepsakul S., 2018, Nonlinear bending analysis of nonlocal nanoplates with general shapes and boundary conditions by the boundary-only method, Eng Anal with Bound Elem 87: 90-110.
[45] Kim J., Reddy J.N., 2013, Analytical solutions for bending, vibration, and buckling of FGM plates using a couple stress-based third-order theory, Compos Struct 103: 86-98
[46] Shaat M., Mahmoud F.F., Gao X.L., Faheem A.F., 2014, Size-dependent bending analysis of Kirchhoff nano-plates based on a modified couple-stress theory including surface effects, Int J of Mech Sci 79: 31-37.
[47] Sahmania S., Mohammadi Aghdama M., Rabczuk T., 2018, Nonlinear bending of functionally graded porous micro/nano-beams reinforced with graphene platelets based upon nonlocal strain gradient theory, Compos Struct 186: 68-78.
[48] Sobhy M., 2015, Levy-type solution for bending of single-layered graphene sheets in thermal environment using the two-variable plate theory. Int J of Mech Sci 90: 171-178.
[49] 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, Compos Struct 125: 304-313.
[50] Kolahchi R., 2017, A comparative study on the bending, vibration and buckling of viscoelastic sandwich nano-plates based on different nonlocal theories using DC, HDQ and DQ methods, Aero Sci & Tech 66: 235-248.
