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An analytical solution for the static bending of smart laminated composite and functionally graded plates with and without porosity

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Abstract

In this paper, an analytical solution for smart laminated composite and functionally graded material plates with and without through-thickness porosity is presented. The kinematics of the deformation in smart structures is modelled through the newly proposed five non-polynomial higher-order shear deformation theories. The relative performance and accuracy of the different theories are assessed for the static bending response of the smart structures under a combined electromechanical loading. The proposed theories assume a nonlinear variation of the transverse shear strain through the thickness of the plate with transverse shear stress-free top and bottom surfaces. The governing differential equations of the plate derived through Hamilton’s principle are solved by Navier’s solution technique with simply supported boundary conditions. To demonstrate the accuracy and applicability of the proposed higher-order shear deformation theories, a wide range of numerical examples for static bending under mechanical and electrostatics loads are considered. The accuracy of the present theories is compared against the three-dimensional elasticity solution, and thereafter, several benchmark solutions for the functionally graded plates with and without porosity are reported.

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References

  1. Curie, J., Curie, P.: Développement par compression de l’électricité polaire dans les cristaux hémièdres à faces inclinées. Bull. Minér. 3, 90–93 (1880)

    MATH  Google Scholar 

  2. Gandhi, M.V., Thompson, B.D.: Smart Materials and Structures. Springer, Berlin (1992)

    Google Scholar 

  3. Pagano, N.J.: Exact solutions for rectangular bidirectional composites and sandwich plates. J. Compos. Mater. 4, 20–34 (1970)

    Article  Google Scholar 

  4. Pagano, N.J., Hatfield, S.J.: Elastic behaviour of multilayered bidirectional composites. AIAA J. 10, 931–933 (1972)

    Article  Google Scholar 

  5. Ray, M.C., Bhattacharya, R., Samanta, B.: Exact solutions for static analysis of intelligent structures. AIAA J. 31(9), 1684–1691 (1993)

    Article  MATH  Google Scholar 

  6. Heyliger, P.: Static behaviour of laminated elastic/piezoelectric plates. AIAA J. 32, 2481–2484 (1994)

    Article  MATH  Google Scholar 

  7. Heyliger, P.: Exact solutions for simply supported laminated piezoelectric plates. ASME J. Appl. Mech. 64, 299–306 (1997)

    Article  MATH  Google Scholar 

  8. Mallik, N., Ray, M.C.: Exact solutions for the analysis of piezoelectric fiber reinforced composites as distributed actuators for smart composite plates. Int. J. Mech. Mater. Des. 1, 347–364 (2004)

    Article  Google Scholar 

  9. Dube, G.P., Kapuria, S., Dumir, P.C.: Exact piezothermoelastic solution of simply supported orthotropic circular cylindrical panel in cylindrical bending. Arch. Appl. Mech. 66, 537–554 (1996)

    Article  MATH  Google Scholar 

  10. Dube, G.P., Kapuria, S., Dumir, P.C.: Exact piezothermoelastic solution of simply supported orthotropic flat panel in cylindrical bending. Int. J. Mech. Sci. 38, 1161–1177 (1996)

    Article  MATH  Google Scholar 

  11. Kapuria, S., Dumir, P.C., Sengupta, S.: Exact piezothermoelastic axisymmetric solution of a finite transversely isotropic cylindrical shell. Comput. Struct. 61, 1085–1099 (1996)

    Article  MATH  Google Scholar 

  12. Reddy, J.N.: Mechanics of Laminated Composite Plates and Shells: Theory and Analysis. CRC Press, Boca Raton (2003)

    Book  Google Scholar 

  13. Reissner, E.: The effect of transverse shear deformation on the bending of elastic plates. ASME J. Appl. Mech. 12, 69–77 (1945)

    Article  MathSciNet  MATH  Google Scholar 

  14. Mindlin, R.D.: Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. ASME J. Appl. Mech. 18, 31–38 (1951)

    Article  MATH  Google Scholar 

  15. Kugler, S., Fotiu, P.A., Murin, J.: The numerical analysis of FGM shells with enhanced finite elements. Eng. Struct. 49, 920–935 (2013)

    Article  Google Scholar 

  16. Kugler, S., Fotiu, P.A., Murin, J.: Enhanced functionally graded material shell finite elements. J. Appl. Math. Mech. 94, 72–84 (2014)

    MathSciNet  Google Scholar 

  17. Ambartsumian, S.A.: On the theory of bending plates. Izv Otd Tech Nauk AN SSSR 5(5), 69–77 (1958)

    MathSciNet  Google Scholar 

  18. Levinson, M.: An accurate, simple theory of the statics and dynamics of elastic plates. Mech. Res. Commun. 7, 343–350 (1980)

    Article  MATH  Google Scholar 

  19. Murthy, M.V.V.: An improved transverse shear deformation theory for laminated anisotropic plates. NASA Technical Paper 1903 (1981)

  20. Reddy, J.N.: A simple higher-order theory for laminated composite plates. ASME J. Appl. Mech. 51, 745–752 (1984)

    Article  MATH  Google Scholar 

  21. Lo, K.H., Christensen, R.M., Wu, E.M.: A high-order theory of plate deformation—part 2: laminated plates. J. Appl. Mech. 44, 669–676 (1977)

    Article  MATH  Google Scholar 

  22. Pandya, B.N., Kant, T.: A consistent refined theory for flexure of a symmetric laminate. Mech. Res. Commun. 14, 107–113 (1987)

    Article  MATH  Google Scholar 

  23. Touratier, M.: An efficient standard plate theory. Int. J. Eng. Sci. 29, 901–916 (1991)

    Article  MATH  Google Scholar 

  24. Stein, M.: Nonlinear theory for plates and shells including the effects of transverse shearing. AIAA J. 24, 1537–1544 (1986)

    Article  MATH  Google Scholar 

  25. Singh, D.B., Singh, B.N.: New higher order shear deformation theories for free vibration and buckling analysis of laminated and braided composite plates. Int. J. Mech. Sci. 131, 265–277 (2017)

    Article  Google Scholar 

  26. Grover, N., Singh, B.N., Maiti, D.K.: New non-polynomial shear-deformation theories for structural behavior of laminated-composite and sandwich plates. AIAA J. 51, 1861–1871 (2013)

    Article  Google Scholar 

  27. Mantari, J.L., Oktem, A.S., Soares, C.G.: A new trigonometric shear deformation theory for isotropic, laminated composite and sandwich plates. Int. J. Solids Struct. 49, 43–53 (2012)

    Article  Google Scholar 

  28. Mantari, J.L., Soares, C.G.: Bending analysis of thick exponentially graded plates using a new trigonometric higher order shear deformation theory. Compos. Struct. 94, 1991–2000 (2012)

    Article  Google Scholar 

  29. Thai, C.H., Ferreira, A.J.M., Bordas, S.P.A., Rabczuk, T., Nguyen-Xuan, H.: Isogeometric analysis of laminated composite and sandwich plates using a new inverse trigonometric shear deformation theory. Eur. J. Mech. A/Solids 43, 89–108 (2014)

    Article  MATH  Google Scholar 

  30. Suganyadevi, S., Singh, B.N.: Assessment of composite and sandwich laminates using a new shear deformation theory. AIAA J. 54, 789–792 (2016)

    Article  Google Scholar 

  31. Kumar, R., Lal, A., Singh, B.N., Singh, J.: New transverse shear deformation theory for bending analysis of FGM plate under patch load. Compos. Struct. 208, 91–100 (2019)

    Article  Google Scholar 

  32. Grover, N., Maiti, D.K., Singh, B.N.: A new inverse hyperbolic shear deformation theory for static and buckling analysis of laminated composite and sandwich plates. Compos. Struct. 95, 667–675 (2013)

    Article  Google Scholar 

  33. Akavci, S.S., Tanrikulu, A.H.: Buckling and free vibration analyses of laminated composite plates by using two new hyperbolic shear-deformation theories. Mech. Compos. Mater. 44, 145 (2008)

    Article  Google Scholar 

  34. Mahi, A., Tounsi, A.: A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates. Appl. Math. Model. 39, 2489–2508 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Joshan, Y.S., Grover, N., Singh, B.N.: A new non-polynomial four variable shear deformation theory in axiomatic formulation for hygro-thermo-mechanical analysis of laminated composite plates. Compos. Struct. 182, 685–693 (2017)

    Article  Google Scholar 

  36. Shi, P., Dong, C., Sun, F., Liu, W., Hu, Q.: A new higher order shear deformation theory for static, vibration and buckling responses of laminated plates with the isogeometric analysis. Compos. Struct. 204, 342–358 (2018)

    Article  Google Scholar 

  37. Karama, M., Afaq, K.S., Mistou, S.: Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity. Int. J. Solids Struct. 40, 1525–1546 (2003)

    Article  MATH  Google Scholar 

  38. Aydogdu, M.: A new shear deformation theory for laminated composite plates. Compos. Struct. 89, 94–101 (2009)

    Article  Google Scholar 

  39. Mantari, J.L., Oktem, A.S., Soares, C.G.: Static and dynamic analysis of laminated composite and sandwich plates and shells by using a new higher-order shear deformation theory. Compos. Struct. 94, 37–49 (2011)

    Article  Google Scholar 

  40. Mantari, J.L., Oktem, A.S., Soares, C.G.: A new higher order shear deformation theory for sandwich and composite laminated plates. Compos. B Eng. 43, 1489–1499 (2012)

    Article  Google Scholar 

  41. Lekhnitskii, S.G.: Strength calculation of composite beams. Vestnikinzheni Tekhnikov, 9 (1935)

  42. Nosier, A., Kapania, R.K., Reddy, J.N.: Free vibration analysis of laminated plates using a layerwise theory. AIAA J. 31, 2335–2346 (1993)

    Article  MATH  Google Scholar 

  43. Chou, P.C., Carleone, J.: Transverse shear in laminated plate theories. AIAA J. 11, 1333–1336 (1973)

    Article  Google Scholar 

  44. Murakami, H.: Laminated composite plate theory with improved in-plane responses (1986)

  45. Karama, M., Harb, B.A., Mistou, S., Caperaa, S.: Bending, buckling and free vibration of laminated composite with a transverse shear stress continuity model. Compos. B Eng. 29, 223–234 (1998)

    Article  Google Scholar 

  46. Thai, C.H., Ferreira, A.J.M., Wahab, M.A., Nguyen-Xuan, H.: A generalized layerwise higher-order shear deformation theory for laminated composite and sandwich plates based on isogeometric analysis. Acta Mech. 227, 1225–1250 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  47. Shimpi, R.P., Ghugal, Y.M.: A new layerwise trigonometric shear deformation theory for two-layered cross-ply beams. Compos. Sci. Technol. 61(9), 1271–1283 (2001)

    Article  Google Scholar 

  48. Carrera, E.: Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking. Arch. Comput. Methods Eng. 10, 215–296 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  49. Demasi, L.: ∞ 3 Hierarchy plate theories for thick and thin composite plates: the generalized unified formulation. Compos. Struct. 84, 256–270 (2008)

    Article  Google Scholar 

  50. Mallik, N., Ray, M.C.: Effective coefficients of piezoelectric fiber-reinforced composites. AIAA J. 41, 704–710 (2003)

    Article  Google Scholar 

  51. Ray, M.C., Mallik, N.: Finite element analysis of smart structures containing piezoelectric fiber-reinforced composite actuator. AIAA J. 42, 1398–1405 (2004)

    Article  Google Scholar 

  52. Ray, M.C., Sachade, H.M.: Finite element analysis of smart functionally graded plates. Int. J. Solids Struct. 43, 5468–5484 (2006)

    Article  MATH  Google Scholar 

  53. Tzou, H.S., Tseng, C.I.: Distributed piezoelectric sensor/actuator design for dynamic measurement/control of distributed parameter systems: a piezoelectric finite element approach. J. Sound Vib. 138, 17–34 (1990)

    Article  Google Scholar 

  54. Mitchell, J.A., Reddy, J.N.: A refined hybrid plate theory for composite laminates with piezoelectric laminae. Int. J. Solids Struct. 32, 2345–2367 (1995)

    Article  MATH  Google Scholar 

  55. Shiyekar, S.M., Kant, T.: Higher order shear deformation effects on analysis of laminates with piezoelectric fibre reinforced composite actuators. Compos. Struct. 93, 3252–3261 (2011)

    Article  Google Scholar 

  56. Joshan, Y.S., Santapuri, S., Grover, N.: Analysis of laminated piezoelectric composite plates using an inverse hyperbolic coupled plate theory. Appl. Math. Model. 82, 359–378 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  57. Kapuria, S.: A coupled zigzag third-order theory for piezoelectric hybrid cross-ply plates. ASME J. Appl. Mech. 71, 604–614 (2004)

    Article  MATH  Google Scholar 

  58. Kapuria, S., Kulkarni, S.D.: Static electromechanical response of smart composite/sandwich plates using an efficient finite element with physical and electric nodes. Int. J. Mech. Sci. 51, 1–20 (2009)

    Article  MATH  Google Scholar 

  59. Topdar, P., Chakraborti, A., Sheikh, A.H.: An efficient hybrid plate model for analysis and control of smart sandwich laminates. Comput. Methods Appl. Mech. Eng. 193, 4591–4610 (2004)

    Article  MATH  Google Scholar 

  60. Alaimo, A., Orlando, C., Valvano, S.: Analytical frequency response solution for composite plates embedding viscoelastic layers. Aerosp. Sci. Technol. 92, 429–445 (2019)

    Article  MATH  Google Scholar 

  61. Phung-Van, P., Thai, C.H., Ferreira, A.J.M., Rabczuk, T.: Isogeometric nonlinear transient analysis of porous FGM plates subjected to hygro-thermo-mechanical loads. Thin-Walled Struct. 148, 106497 (2020)

    Article  Google Scholar 

  62. Ray, M.C., Sachade, H.M.: Exact solutions for the functionally graded plates integrated with a layer of piezoelectric fibre-reinforced composite. ASME J. Appl. Mech. 73, 622–632 (2006)

    Article  MATH  Google Scholar 

  63. Thai, H.T., Kim, S.E.: A simple higher-order shear deformation theory for bending and free vibration analysis of functionally graded plates. Compos. Struct. 96, 165–173 (2013)

    Article  Google Scholar 

  64. Merdaci, S., Belghoul, H.: High-order shear theory for static analysis of functionally graded plates with porosities. C. R. Méc. 347, 207–217 (2019)

    Article  Google Scholar 

  65. Wang, Q., Quek, S.T.: Flexural vibration analysis of sandwich beam coupled with piezoelectric actuator. Smart Mater. Struct. 9, 103 (2000)

    Article  Google Scholar 

  66. Wang, Q.: On buckling of column structures with a pair of piezoelectric layers. Eng. Struct. 24, 199–205 (2002)

    Article  Google Scholar 

  67. Srinivas, S., Rao, A.K.: Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. Int. J. Solids Struct. 6, 1463–1481 (1970)

    Article  MATH  Google Scholar 

  68. Neves, A.M.A., Ferreira, A.J.M., Carrera, E., Cinefra, M., Roque, C.M.C., Jorge, R.M.N., Soares, C.M.: Static, free vibration and buckling analysis of isotropic and sandwich functionally graded plates using a quasi-3D higher-order shear deformation theory and a meshless technique. Compos. B Eng. 44, 657–674 (2013)

    Article  Google Scholar 

  69. Carrera, E., Brischetto, S., Robaldo, A.: Variable kinematic model for the analysis of functionally graded material plates. AIAA J. 46, 194–203 (2008)

    Article  Google Scholar 

  70. Wu, C.P., Chiu, K.H., Wang, Y.M.: RMVT-based meshless collocation and element-free Galerkin methods for the quasi-3D analysis of multilayered composite and FGM plates. Compos. Struct. 93, 923–943 (2011)

    Article  Google Scholar 

  71. Zenkour, A.M.: Quasi-3D refined theory for functionally graded porous plates: displacements and stresses. Phys. Mesomech. 23, 39–53 (2020)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, Birla Institute of Technology and Science Pilani, K. K. Birla Goa Campus, Mormugao, 403726, India

    Firojkhan Pathan

  2. Department of Mechanical Engineering, Indian Institute of Technology Indore, Indore, 453552, India

    Sandeep Singh

  3. Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, 600036, India

    Sundararajan Natarajan

  4. Department of Mechanical Engineering, Birla Institute of Technology and Science Pilani, Pilani Campus, Pilani, 333031, India

    Gaurav Watts

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  1. Firojkhan Pathan
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  2. Sandeep Singh
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  4. Gaurav Watts
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Correspondence to Sandeep Singh.

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Pathan, F., Singh, S., Natarajan, S. et al. An analytical solution for the static bending of smart laminated composite and functionally graded plates with and without porosity. Arch Appl Mech 92, 903–931 (2022). https://doi.org/10.1007/s00419-021-02080-3

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