Abstract
In this paper the thermo-mechanical response of functionally graded plates and shells is studied using a continuum shell finite element model with high-order spectral/hp basis functions. The shell element is based on the seven-parameter first-order shear deformation theory, and it does not utilize reduced integration or stabilization ideas and yet exhibits no locking. The static and dynamic response of functionally graded shells, with power-law variation of the constituents, under mechanical and thermal loads is investigated by varying the volume fraction of the constituents. Numerical results for deflections and stresses are presented and compared with available analytical and finite element results from the literature. The performance of the shell element for transient thermal problems is found to be excellent.
Acknowledgements
The first author wishes to acknowledge the financial support of the Mexican government through the Consejo Nacional de Ciencia y Tecnologia (CONACYT) and the Secretaria de Educacion Publica (SEP). The second author gratefully acknowledges the support of the Oscar S. Wyatt Endowed Chair. The numerical simulations presented were made using the commercial codes available in the High Performance Research Computing at Texas A&M University; their support is gratefully acknowledged.
[1] Shen HS. Functionally graded materials: nonlinear analysis of plates and shells. Boca Raton, FL: CRC Press, 2009 .Search in Google Scholar
[2] Reddy JN. Mechanics of laminated composite plates and shells: theory and analysis, 2nd Boca Raton, FL: CRC Press, 2004.10.1201/b12409Search in Google Scholar
[3] Yamanoushi M, Koizumi M, Hiraii T, Shiota I.. Proceedings of the First International Symposium on Functionally Gradient Materials.Japan, 1990.Search in Google Scholar
[4] Koizumi M. The concept of FGM. Ceram Trans Funct Gradient Mater. 1993;34:3–10.Search in Google Scholar
[5] Thai HT, Kim SE. A review of theories for the modeling and analysis of functionally graded plates and shells. Compos Struct. 2015;128:70–86.10.1016/j.compstruct.2015.03.010Search in Google Scholar
[6] Ma LS, Wang TJ. Nonlinear bending and post-buckling of a functionally graded circular plate under mechanical and thermal loadings. Int J Solids Struct. 2003;40:3311–3330.10.1016/S0020-7683(03)00118-5Search in Google Scholar
[7] Woo J, Meguid SA, Ong LS. Nonlinear free vibration behavior of functionally graded plates. J Sound Vib. 2006;289:595–611.10.1016/j.jsv.2005.02.031Search in Google Scholar
[8] Praveen GN, Reddy JN. Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates. Int J Solids Struct. 1998;35:4457–4476.10.1016/S0020-7683(97)00253-9Search in Google Scholar
[9] Alijani F, Bakhtiari-Nejad F, Amabili M. Nonlinear vibrations of FGM rectangular plates in thermal environments. Nonlinear Dyn. 2011;66:251–270.10.1007/s11071-011-0049-8Search in Google Scholar
[10] Reddy JN. A simple higher-order theory for laminated composite plates. J Appl Mech. 1984;51:745–752.10.1115/1.3167719Search in Google Scholar
[11] Shen HS. Nonlinear bending response of functionally graded plates subjected to transverse loads and in thermal environments. Int J Mech Sci. 2002;44:561–584.10.1016/S0020-7403(01)00103-5Search in Google Scholar
[12] Yang J, Shen HS. Nonlinear bending analysis of shear deformable functionally graded plates subjected to thermo-mechanical loads under various boundary conditions. Compos B Eng. 2003;34:103–115.10.1016/S1359-8368(02)00083-5Search in Google Scholar
[13] Huang XL, Shen HS. Nonlinear vibration and dynamic response of functionally graded plates in thermal environments. Int J Solids Struct. 2004;41:2403–2427.10.1016/j.ijsolstr.2003.11.012Search in Google Scholar
[14] Aliaga W, Reddy JN. Nonlinear thermoelastic response of functionally graded plates using the third-order plate theory. Int J Comput Eng Sci. 2004;54:753–780.Search in Google Scholar
[15] Duc ND, Cong PH. Nonlinear postbuckling of symmetric S-FGM plates resting on elastic foundations using higher order shear deformation plate theory in thermal environments. Compos Struct. 2013;100:566–574.10.1016/j.compstruct.2013.01.006Search in Google Scholar
[16] Alijani F, Amabili M. Effect of thickness deformation on large-amplitude vibrations of functionally graded rectangular plates. Compos Struct. 2014;113:89–107.10.1016/j.compstruct.2014.03.006Search in Google Scholar
[17] Na KS, Kim JH. Nonlinear bending response of functionally graded plates under thermal loads. J Therm Stresses. 2006;29:245–261.10.1080/01495730500360427Search in Google Scholar
[18] Woo J, Meguid SA. Nonlinear analysis of functionally graded plates and shallow shells. Int J Solids Struct. 2001;38:7409–7421.10.1016/S0020-7683(01)00048-8Search in Google Scholar
[19] Duc ND, Quan TQ. Nonlinear stability analysis of double-curved shallow FGM panels on elastic foundations in thermal environments. Mech Compos Mater. 2012;48:435–448.10.1007/s11029-012-9289-zSearch in Google Scholar
[20] Duc ND, Quan TQ. Nonlinear postbuckling of imperfect eccentrically stiffened P-FGM double curved thin shallow shells on elastic foundations in thermal environments. Compos Struct. 2013;106:590–600.10.1016/j.compstruct.2013.07.010Search in Google Scholar
[21] Duc ND, Quan TQ. Nonlinear dynamic analysis of imperfect functionally graded material double curved thin shallow shells with temperature-dependent properties on elastic foundation. J Vib Control. 2015;21:1340–1362.10.1177/1077546313494114Search in Google Scholar
[22] Du C, Li Y. Nonlinear resonance behavior of functionally graded cylindrical shells in thermal environments. Compos Struct. 2013;102:164–174.10.1016/j.compstruct.2013.02.028Search in Google Scholar
[23] Arciniega RA, Reddy JN. Large deformation analysis of functionally graded shells. Int J Solids Struct. 2007;44:2036–2052.10.1016/j.ijsolstr.2006.08.035Search in Google Scholar
[24] Payette GS, Reddy JN. A seven-parameter spectral/hp finite element formulation for isotropic, laminated composite and functionally graded shell structures. Comput Methods Appl Mech Eng. 2014;278:664–704.10.1016/j.cma.2014.06.021Search in Google Scholar
[25] Gutierrez Rivera M, Reddy JN. Stress analysis of functionally graded shells using a 7-parameter shell element. Mech Res Commun. 2016;78:60–70.10.1016/j.mechrescom.2016.02.009Search in Google Scholar
[26] Zhang W, Hao YX. Nonlinear dynamic of functionally graded cylindrical shells under the thermal mechanical loads. Int Mech Eng Congr Exp, 2009.10.1115/IMECE2009-10044Search in Google Scholar
[27] Sheng GG, Wang X. Non-linear response of functionally graded cylindrical shells under mechanical and thermal loads. J Therm Stresses. 2011;34:1105–1118.10.1080/01495739.2011.606016Search in Google Scholar
[28] Pradyumna S, Nanda N. Geometrically nonlinear transient response of functionally graded shell panels with initial geometric imperfection. Mech Adv Mater Struct. 2013;20:217–226.10.1080/15376494.2011.584148Search in Google Scholar
[29] Tung HV, Duc ND. Nonlinear response of shear deformable FGM curved panels resting on elastic foundations and subjected to mechanical and thermal loading conditions. Appl Math Model. 2014;38:2848–2866.10.1016/j.apm.2013.11.015Search in Google Scholar
[30] Alijani F, Amabili M, Bakhtiari-Nejad F. Thermal effects on nonlinear vibrations of functionally graded doubly curved shells using higher order shear deformation theory. Compos Struct. 2011;93:2541–2553.10.1016/j.compstruct.2011.04.016Search in Google Scholar
[31] Sansour C. A theory and finite element formulation of shells at finite deformations involving thickness change: Circumventing the use of a rotation tensor. Arch Appl Mech. 1995;65:194–216.10.1007/BF00799298Search in Google Scholar
[32] Bischoff M, Ramm E. Shear deformable shell elements for large strains and rotations. Int J Numer Methods Eng. 1997;40:4427–4449.10.1002/(SICI)1097-0207(19971215)40:23<4427::AID-NME268>3.0.CO;2-9Search in Google Scholar
[33] Ahmad S, Irons BM, Zienkiewicz OC. Analysis of thick and thin shell structures by curved finite elements. Int J Numer Methods Eng. 1970;2:419–451.10.1002/nme.1620020310Search in Google Scholar
[34] Arciniega RA, Reddy JN. Tensor-based finite element formulation for geometrically nonlinear analysis of shell structures. Comput Methods Appl Mech Eng. 2007;196:1048–1073.10.1016/j.cma.2006.08.014Search in Google Scholar
[35] Stojanovitch R, Djuritch S, Vujoshevitch L. On finite thermal deformations. Archiwum Mechaniki Stosowanej. 1964;16:103–108.Search in Google Scholar
[36] Javaheri R, Eslami MR. Thermal buckling of functionally graded plates. AIAA J. 2002;40:162–169.10.2514/2.1626Search in Google Scholar
[37] Reddy JN. An introduction to continuum mechanics, 2nd New York: Cambridge University Press, 2013.Search in Google Scholar
[38] Reddy JN. Energy principles and variational methods in applied mechanics, 2nd New York: John Wiley & Sons, 2002 .Search in Google Scholar
[39] Karniadakis GE, Sherwin SJ. Spectral/hp element methods for CFD. Oxford: Oxford University Press, 1999.Search in Google Scholar
[40] Reddy JN. An introduction nonlinear finite element analysis, 2nd Oxford: Oxford University Press, 2015.10.1093/acprof:oso/9780199641758.001.0001Search in Google Scholar
[41] Newmark NM. A method of computation for structural dynamics. J Eng Mech Div. 1959;85:67–94.10.1061/JMCEA3.0000098Search in Google Scholar
[42] Fu Y, Hu S, Mao Y. Nonlinear transient response of functionally graded shallow spherical shells subjected to mechanical load and unsteady temperature field. Acta Mech Solida Sin. 2014;27:496–508.10.1016/S0894-9166(14)60058-6Search in Google Scholar
© 2017 Walter de Gruyter GmbH, Berlin/Boston
