Abstract
Dissimilar orthotropic stiffness coefficient variations are a characteristic feature of unidirectionally reinforced fiber composites with a variable fiber volume fraction, but have not been commonly considered in the literature. The objective of this work is to account for them and obtain a three-dimensional elasticity solution with specific reference to simply supported rectangular plates. The analysis involves the solution of variable coefficient governing equations using the power series approach. For a graphite–epoxy plate with a sandwich-like configuration, results useful as a benchmark for future comparisons are tabulated for a specific power law variation of the volume fraction. It is shown that the thickness-wise variations of displacements and stresses are significantly nonlinear, and such variations are not captured correctly by the classical plate theory. Further, on the basis of the elasticity solution, the relative benefit of using a sandwich-like configuration versus a homogeneous plate is shown to depend on the span-to-thickness ratio and to decrease significantly as the plate becomes thick.
Similar content being viewed by others
References
Sankar, B.V.: An elasticity solution for functionally graded beams. Compos. Sci. Technol. 61(5), 689–696 (2001)
Venkataraman, S., Sankar, B.V.: Elasticity solution for stresses in a sandwich beam with functionally graded core. AIAA J. 41(12), 2501–2505 (2003)
Kashtalyan, M.: Three-dimensional elasticity solution for bending of functionally graded rectangular plates. Eur. J. Mech. A/Solids 23(5), 853–864 (2004)
Kashtalyan, M., Menshykova, M.: Three-dimensional elastic deformation of a functionally graded coating/substrate system. Int. J. Solids Struct. 44(16), 5272–5288 (2007)
Zenkour, A.M.: Benchmark trigonometric and 3-D elasticity solutions for an exponentially graded thick rectangular plate. Arch. Appl. Mech. 77(4), 197–214 (2007)
Huang, Z.Y., Lu, C.F., Chen, W.Q.: Benchmark solutions for functionally graded thick plates resting on Winkler–Pasternak elastic foundations. Compos. Struct. 85(2), 95–104 (2008)
Xu, Y., Zhou, D.: Three-dimensional elasticity solution of functionally graded rectangular plates with variable thickness. Compos. Struct. 91(1), 56–65 (2009)
Zhong, Z., Shang, E.: Closed-form solutions of three-dimensional functionally graded plates. Mech. Adv. Mater. Struct. 15(5), 355–363 (2008)
Woodward, B., Kashtalyan, M.: Three-dimensional elasticity solution for bending of transversely isotropic functionally graded plates. Eur. J. Mech. A/Solids 30(5), 705–718 (2011)
Pan, E.: Exact solution for functionally graded anisotropic elastic composite laminates. J. Compos. Mater. 37(21), 1903–1920 (2003)
Liu, W., Zhong, Z.: Three-dimensional analysis of simply supported functionally graded plate with arbitrary distributed elastic modulus. Tsinghua Sci. Technol. 14(Suppl. 2), 58–63 (2009)
Zhong, Z., Yu, T.: Two-dimensional analysis of functionally graded beams. AIAA J. 44(12), 3160–3164 (2006)
Zhong, Z., Yu, T.: Analytical solution of a cantilever functionally graded beam. Compos. Sci. Technol. 67(3–4), 481–488 (2007)
Ding, H.J., Huang, D.J., Chen, W.Q.: Elasticity solutions for plane anisotropic functionally graded beams. Int. J. Solids Struct. 44(1), 176–196 (2007)
Yang, B., Ding, H.J., Chen, W.Q.: Elasticity solutions for functionally graded plates in cylindrical bending. Appl. Math. Mech. Engl. Ed. 29(8), 999–1004 (2008)
Nie, G.J., Zhong, Z., Chen, S.: Analytical solution for a functionally graded beam with arbitrary graded material properties. Compos. Part B 44(1), 274–282 (2013)
Daouadji, T.H., Henni, A.H., Tounsi, A., Abbes, A.B.E.: Elasticity solution of a cantilever functionally graded beam. Appl. Compos. Mater. 20(1), 1–15 (2013)
Benguediab, S., Tounsi, A., Abdelaziz, H.H., Meziane, M.A.A.: Elasticity solution for a cantilever beam with exponentially varying properties. J. Appl. Mech. Tech. Phys. 58(2), 354–361 (2017)
Bhaskar, K., Varadan, T.K.: Theory of Isotropic/Orthotropic Elasticity. CRC Press, Boca Raton (2009)
Jha, D.K., Kant, T., Singh, R.K.: A critical review of recent research on functionally graded plates. Compos. Struct. 96, 833–849 (2013)
Swaminathan, K., Naveenkumar, D.T., Zenkour, A.M., Carrera, E.: Stress, vibration and buckling analyses of FGM plates—a state-of-the-art review. Compos. Struct. 120, 10–31 (2015)
Reddy, K.S.K., Kant, T.: Three-dimensional elasticity solution for free vibrations of exponentially graded plates. J. Eng. Mech. 140(7), 04014047 1-9 (2014)
Carrera, E., Brischetto, S., Robaldo, A.: Variable kinematic model for the analysis of functionally graded material plates. AIAA J. 46(1), 194–203 (2008)
Giunta, G., Belouettar, S., Carrera, E.: Analysis of FGM beams by means of classical and advanced theories. Mech. Adv. Mater. Struct. 17(8), 622–635 (2010)
Pagano, N.J.: Influence of shear coupling in cylindrical bending of anisotropic laminates. J. Compos. Mater. 4(3), 330–343 (1970)
Miracle, D.B., Donaldson, S.L. (eds.): ASM Handbook, Vol 21: Composites. ASM International, Novelty (2001)
Jones, R.M.: Mechanics of Composite Materials. Taylor and Francis, New York (1999)
Bhaskar, K., Varadan, T.K.: Plates: Theories and Applications. Wiley, Chichester (2014)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Bhaskar, K., Ravindran, A. Elasticity solution for orthotropic FGM plates with dissimilar stiffness coefficient variations. Acta Mech 230, 979–992 (2019). https://doi.org/10.1007/s00707-018-2341-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-018-2341-x