3D Semi-Analytical Solutions for Functionally Grade Power Law Varied Laminate Subjected to Thermo-Mechanical Loading

Document Type : Original Article


1 Research Scholar, Structural Engineering Department, Veermata Jijabai Technological Institute, Matunga, Mumbai 400 019, India

2 Associate Professor, Structural Engineering Department, Veermata Jijabai Technological Institute, Matunga, Mumbai 400 019, India


This paper describes formulation for calculation of actual through thickness temperature variation followed by stress and displacement analysis of all-around simply supported functionally graded (FG) laminate using a semi-analytical approach. This approach has depended on a two-point boundary value problem (BVP) governed by first-order ordinary differential equations (ODEs). Developed formulation carries the advantage of both elasticity solution as well as ESL or approximate theories. This new model capable of providing accurate results without any approximation along the thickness of FG laminate. Material properties like heat conductivity, modulus of elasticity and thermal expansion coefficient are considered to be varied by a power law. The numerical investigation has been performed to examine thermal loading response on the FGM laminate and transverse loading applied on the laminate's top surface. The results are obtained for two types of thermal loading, obtained by heat conduction formulation received by developed semi-analytical approach and another with assumed power law variation and compared with each other. Leads outcomes from parametric studies, which will be helpful for further research in this area.


Main Subjects

[1]     Vel SS, Batra RC. Three-dimensional analysis of transient thermal stresses in functionally graded plates. Int J Solids Struct 2003;40:7181–96. doi:10.1016/S0020-7683(03)00361-5.
[2]     Javaheri R, Eslami MR. Thermal Buckling of Functionally Graded Plates Based on Higher-Order Theory. J Therm Stress 2002;25:603–25. doi:10.1080/01495730290074333.
[3]     Ghannadpour SAM, Ovesy HR, Nassirnia M. Buckling analysis of functionally graded plates under thermal loadings using the finite strip method. Comput Struct 2012;108–109:93–9. doi:10.1016/j.compstruc.2012.02.011.
[4]     Damanpack AR, Bodaghi M, Ghassemi H, Sayehbani M. Boundary element method applied to the bending analysis of thin functionally graded plates. Lat Am J Solids Struct 2013;10:549–70.
[5]     Shen H-S. Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments. Int J Mech Sci 2002;44:561–84. doi:10.1016/S0020-7403(01)00103-5.
[6]     Lanhe W. Thermal buckling of a simply supported moderately thick rectangular FGM plate. Compos Struct 2004;64:211–8. doi:10.1016/j.compstruct.2003.08.004.
[7]     Dai KY, Liu GR, Han X, Lim KM. Thermomechanical analysis of functionally graded material (FGM) plates using element-free Galerkin method. Comput Struct 2005;83:1487–502. doi:10.1016/j.compstruc.2004.09.020.
[8]     Nguyen T-K, Sab K, Bonnet G. First-order shear deformation plate models for functionally graded materials. Compos Struct 2008;83:25–36. doi:10.1016/j.compstruct.2007.03.004.
[9]     Yang J, Shen H-S. Nonlinear bending analysis of shear deformable functionally graded plates subjected to thermo-mechanical loads under various boundary conditions. Compos Part B Eng 2003;34:103–15. doi:10.1016/S1359-8368(02)00083-5.
[10]   Zenkour AM. Generalized shear deformation theory for bending analysis of functionally graded plates. Appl Math Model 2006;30:67–84. doi:10.1016/j.apm.2005.03.009.
[11]   Saidi AR, Jomehzadeh E. On the analytical approach for the bending/stretching of linearly elastic functionally graded rectangular plates with two opposite edges simply supported. Proc Inst Mech Eng Part C J Mech Eng Sci 2009;223:2009–16. doi:10.1243/09544062JMES1431.
[12]   Kadoli R, Akhtar K, Ganesan N. Static analysis of functionally graded beams using higher order shear deformation theory. Appl Math Model 2008;32:2509–25. doi:10.1016/j.apm.2007.09.015.
[13]   Mantari JL, Oktem AS, Guedes Soares C. Bending response of functionally graded plates by using a new higher order shear deformation theory. Compos Struct 2012;94:714–23. doi:10.1016/j.compstruct.2011.09.007.
[14]   Kant T, Jha DK, Singh RK. A higher-order shear and normal deformation functionally graded plate model: some recent results. Acta Mech 2014;225:2865–76. doi:10.1007/s00707-014-1213-2.
[15]   Ferreira AJM, Roque CMC, Jorge RMN, Fasshauer GE, Batra RC. Analysis of Functionally Graded Plates by a Robust Meshless Method. Mech Adv Mater Struct 2007;14:577–87. doi:10.1080/15376490701672732.
[16]   Wu C-P, Li H-Y. An RMVT-based third-order shear deformation theory of multilayered functionally graded material plates. Compos Struct 2010;92:2591–605. doi:10.1016/j.compstruct.2010.01.022.
[17]   Xiang S, Kang G. A nth-order shear deformation theory for the bending analysis on the functionally graded plates. Eur J Mech - A/Solids 2013;37:336–43. doi:10.1016/j.euromechsol.2012.08.005.
[18]   Tounsi A, Al-Dulaijan SU, Al-Osta MA, Chikh A, Al-Zahrani MM, Sharif A, et al. A four variable trigonometric integral plate theory for hygro-thermo-mechanical bending analysis of AFG ceramic-metal plates resting on a two-parameter elastic foundation. Steel Compos Struct 2020;34:511–24.
[19]   Chikr SC, Kaci A, Bousahla AA, Bourada F, Tounsi A, Bedia EA, et al. A novel four-unknown integral model for buckling response of FG sandwich plates resting on elastic foundations under various boundary conditions using Galerkin’s approach. Geomech Eng 2020;21:471–87.
[20]   Refrafi S, Bousahla AA, Bouhadra A, Menasria A, Bourada F, Tounsi A, et al. Effects of hygro-thermo-mechanical conditions on the buckling of FG sandwich plates resting on elastic foundations. Comput Concr 2020;25:311–25.
[21]   Boussoula A, Boucham B, Bourada M, Bourada F, Tounsi A, Bousahla AA, et al. A simple nth-order shear deformation theory for thermomechanical bending analysis of different configurations of FG sandwich plates. Smart Struct Syst 2020;25:197–218.
[22]   Balubaid M, Tounsi A, Dakhel B, Mahmoud SR. Free vibration investigation of FG nanoscale plate using nonlocal two variables integral refined plate theory. Comput Concr 2019;24:579–86.
[23]   Kaddari M, Kaci A, Bousahla AA, Tounsi A, Bourada F, Tounsi A, et al. A study on the structural behaviour of functionally graded porous plates on elastic foundation using a new quasi-3D model: bending and free vibration analysis. Comput Concr 2020;25:37–57.
[24]   Roy AM. Barrierless melt nucleation at solid-solid interface in energetic nitramine octahydro-1, 3, 5, 7-tetranitro-1, 3, 5, 7-tetrazocine. Materialia 2021;15:101000. doi:10.1016/j.mtla.2021.101000.
[25]   Smilowitz L, Henson BF, Asay BW, Dickson PM. The β–δ phase transition in the energetic nitramine-octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine: Kinetics. J Chem Phys 2002;117:3789–98. doi:10.1063/1.1495399.
[26]   Henson BF, Smilowitz L, Asay BW, Dickson PM. The β–δ phase transition in the energetic nitramine octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine: Thermodynamics. J Chem Phys 2002;117:3780–8. doi:10.1063/1.1495398.
[27]   Bowlan P, Henson BF, Smilowitz L, Levitas VI, Suvorova N, Oschwald D. Kinetics of the γ–δ phase transition in energetic nitramine-octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine. J Chem Phys 2019;150:064705. doi:10.1063/1.5080010.
[28]   Levitas VI, Henson BF, Smilowitz LB, Asay BW. Solid-Solid Phase Transformation via Virtual Melting Significantly Below the Melting Temperature. Phys Rev Lett 2004;92:235702. doi:10.1103/PhysRevLett.92.235702.
[29]   Pendhari SS, Kulkarni SP. Cylindrical Bending of Power Law Varied Functionally Graded Laminate Subjected to Thermo-Mechanical Loading. Comput Eng Phys Model 2020;3:22–46. doi:10.22115/cepm.2020.243741.1124.
[30]   Kantorovich LV. Approximate methods of higher analysis. Interscience 1958.