Stress Analysis of Transversely Loaded Functionally Graded Plates with a Higher Order Shear and Normal Deformation Theory
Publication: Journal of Engineering Mechanics
Volume 139, Issue 12
Abstract
Static analysis of orthotropic functionally graded (FG) elastic, rectangular, and simply supported (diaphragm) plates under transverse loads is presented based on a higher order shear and normal deformation theory (HOSNT). Although functionally graded materials (FGMs) are highly heterogeneous in nature, they are generally idealized as continua with mechanical properties changing smoothly with respect to the spatial coordinates. The material properties of FG plates are assumed here to be varying through the thickness of the plate in a continuous manner. The Poisson’s ratios of the FG plates are assumed to be constant, but their Young’s moduli vary continuously in the thickness direction according to the volume fraction of constituents, which are mathematically modeled as an exponential function. The governing equations of equilibrium for the FG plates are derived on the basis of a HOSNT assuming varying material properties. Numerical solutions are obtained by the use of the Navier solution method. Several examples of isotropic, orthotropic, and FG plates are presented. The accuracy of the numerical solutions has been compared with the solutions obtained by other models and the exact three-dimensional (3D) elasticity solutions.
Get full access to this article
View all available purchase options and get full access to this article.
References
Carrera, E., Brischetto, S., and Robaldo, A. (2008). “Variable kinematic model for the analysis of functionally graded material plates.” AIAA J., 46(1), 194–203.
Cheng, Z. Q., and Batra, R. C. (2000a). “Deflection relationships between the homogeneous Kirchhoff plate theory and different functionally graded plate theories.” Arch. Mech., 52(1), 143–158.
Cheng, Z. Q., and Batra, R. C. (2000b). “Exact correspondence between eigenvalues of membranes and functionally graded simply supported polygonal plates.” J. Sound Vibrat., 229(4), 879–895.
Della Croce, L., and Venini, P. (2004). “Finite elements for functionally graded Reissner–Mindlin plates.” Comput. Methods Appl. Mech. Eng., 193, 705–725.
Dym, C. L., and Shames, I. H. (1973). Solid mechanics: A variational approach, McGraw Hill, New York.
GhannadPour, S. A. M., and Alinia, M. M. (2006). “Large deflection behavior of functionally graded plates under pressure loads.” Compos. Struct., 75(1–4), 67–71.
Hildebrand, F. B., Reissner, E., and Thomas, G. B. (1949). Notes on the foundations of the theory of small displacements of orthotropic shells, National Advisory Committee for Aeronautics, Washington, DC.
Jha, D. K., Kant, T., and Singh, R. K. (2012). “A critical review of recent research on functionally graded plates.” Compos. Struct., 96, 833–849.
Kant, T. (1982). “Numerical analysis of thick plates.” Comput. Methods Appl. Mech. Eng., 44(4), 1–18.
Kant, T., and Manjunatha, B. S. (1988). “An unsymmetric FRC laminate finite element model with 12 degrees of freedom per node.” Eng. Computat., 5(3), 300–308.
Kant, T., and Manjunatha, B. S. (1994). “On accurate estimation of transverse stress in multilayer laminates.” Comp. Struct., 50(3), 351–365.
Kant, T., and Swaminathan, K. (2002). “Analytical solutions for the static analysis of laminated composite and sandwich plates based on a higher-order refined theory.” Compos. Struct., 56(4), 329–344.
Kashtalyan, M. (2004). “Three-dimensional elasticity solution for bending of functionally graded rectangular plates.” Eur. J. Mech. A., 23(5), 853–864.
Koizumi, M. (1993). “The concept of FGM.” Ceram. Trans., 34, 3–10.
Matsunaga, H. (2008). “Free vibration and stability of functionally graded plates according to a 2-D higher-order deformation theory.” Compos. Struct., 82, 499–512.
Mian, A. M., and Spencer, A. J. M. (1998). “Exact solutions for functionally graded and laminated elastic materials.” J. Mech. Phys. Solids, 46(12), 2283–2295.
Mindlin, R. D. (1951). “Influence of rotary inertia and shear in flexural motions of isotropic elastic plates.” J. Appl. Mech., 18, 1031–1036.
Nguyen, T. K., Sab, K., and Bonnet, G. (2008). “First-order shear deformation plate models for functionally graded materials.” Compos. Struct., 83(1), 25–36.
Pagano, N. J. (1969). “Exact solutions for composite laminates in cylindrical bending.” J. Compos. Mater., 3, 398–411.
Pagano, N. J. (1970). “Exact solutions for rectangular bidirectional composite and sandwich plates.” J. Compos. Mater., 4, 20–34.
Pandya, B. N., and Kant, T. (1988). “Higher order shear deformable theories for flexural of sandwich plates: Finite element evaluations.” Int. J. Solids Struct., 24(12), 1267–1286.
Plevako, V. P. (1971). “The theory of elasticity of inhomogeneous media.” J. Appl. Math. Mech., 35(5), 806–813.
Reddy, J. N. (1997). Mechanics of laminated composite plates: Theory and analysis, CRC, New York.
Reddy, J. N. (2000). “Analysis of functionally graded plates.” Int. J. Numer. Methods Eng., 47(1–3), 663–684.
Reddy, J. N. (2002). Energy principles and variational methods in applied mechanics, 2nd Ed., Wiley, New York.
Reddy, J. N., and Cheng, Z. Q. (2001). “Three-dimensional thermo mechanical deformations of functionally graded rectangular plates.” Eur. J. Mech. Solids, 20(5), 841–855.
Reissner, E. (1945). “The effects of transverse shear deformation on bending of elastic plates.” J. Appl. Mech., 12(2), A69–A77.
Rogers, T. G., Watson, P., and Spencer, A. J. M. (1995). “Exact three-dimensional elasticity solutions for bending of moderately thick inhomogeneous and laminated strips under normal pressure.” Int. J. Solids Struct., 32(12), 1659–1673.
Shimpi, R. P., and Patel, H. G. (2006). “A two variable refined plate theory for orthotropic plate analysis.” Int. J. Solids Struct., 43(22-23), 6783–6799.
Sladek, J., Sladek, V., Zhang, Ch., Krivacek, J., and Wen, P. H. (2006). “Analysis of orthotropic thick plates by meshless local Petrov–Galerkin (MLPG) method.” Int. J. Numer. Methods Eng., 67(13), 1830–1850.
Srinivas, S., Joga Rao, C. V., and Rao, A. K. (1970). “An exact analysis for vibration of simply supported homogeneous and laminated thick rectangular plates.” J. Sound Vibrat., 12(2), 187–199.
Srinivas, S., and Rao, A. K. (1970). “Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates.” Int. J. Solids Struct., 6(11), 1463–1481.
Suresh, S., and Mortensen, A. (1998). Fundamentals of functionally graded materials, IOM Communications, London.
Tanigawa, Y. (1995). “Some basic thermoelastic problems for nonhomogeneous structural materials.” Appl. Math. Mech., 48(6), 287–300.
Timoshenko, S., and Goodier, J. (1951). Theory of elasticity, McGraw Hill, New York.
Timoshenko, S., and Woinowsky-Krieger, S. (1959). Theory of plates and shells, 2nd Ed., McGraw Hill, New York.
Tsukamoto, H. (2003). “Analytical method of inelastic thermal stresses in a functionally graded material plate by a combination of a micro- and macromechanical approaches.” Composites Part B, 34(6), 561–568.
Vel, S. S., and Batra, R. C. (2002). “Exact solution for thermoelastic deformations of functionally graded thick rectangular plates.” AIAA J., 40(7), 1421–1433.
Vel, S. S., and Batra, R. C. (2004). “Three-dimensional exact solution for the vibration of functionally graded rectangular plate.” J. Sound Vibrat., 272, 703–730.
Wen, P. H., Sladek, J., and Sladek, V. (2011). “Three-dimensional analysis of functionally graded plates.” Int. J. Numer. Methods Eng., 87(10), 923–942.
Zhang, C., and Zhong, Z. (2007). “Three-dimensional analysis of a simply supported functionally graded plate based on Haar wavelet method.” Acta Mech. Solida Sin., 28(3), 217–223.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 139 • Issue 12 • December 2013
Pages: 1663 - 1680
Copyright
© 2013 American Society of Civil Engineers.
History
Received: Sep 5, 2011
Accepted: Feb 5, 2013
Published online: Feb 7, 2013
Published in print: Dec 1, 2013
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.