Nonlinear analysis of hyperelastic plates using first-order shear deformation plate theory and a meshless method

Document Type : Research Article

Authors

1 Department of mechanics, Tarbiat Modares University, Tehran, Iran

2 Department of Mechanics, Tarbiat Modares University, Tehran, Iran

Abstract

In this paper, the static analysis of hyperelastic plates under uniform and sinusoidal distributed loading is investigated. Right Cauchy-Green deformation tensor and Lagrange strains are used to derive the nonlinear strain relations. Also, the first-order shear deformation plate theory is considered. For the first time, the governing equations of hyperelastic plates using the neo-Hookean strain energy function are derived. The Lagrange equation is utilized to implement the variational method on potential energy function. The governing nonlinear differential equations are discretized using the meshless collocation method and radial basis functions. The thin plate spline basis function is applied for deriving shape functions of the meshless method. The results are compared to the results of the finite element method. The static analysis is investigated on hyperelastic plates for uniform and sinusoidal loading and various thicknesses. Additionally, the effect of thickness is studied on the deflection of the hyperelastic plates. The results show an acceptable accuracy for static analysis of hyperelastic plates under uniform and sinusoidal loading; also, the stress contour is the same in both methods. Consequently, the meshless collocation method using the thin-plate spline basis function is an adequate method for analyzing FSDT hyperelastic plates due to no integration and imposing boundary conditions directly.

Keywords

Main Subjects


[1] R.W.Ogden, Nonlinear Elastic Deformations, Dover Publications, New York, 1997.
[2] T. Shearer, A new strain energy function for the hyperelastic modelling of ligaments and tendons based on fascicle microstructure, Journal of Biomechanics, 48(2) (2015) 290-297.
[3] K. Upadhyay, G. Subhash, D. Spearot, Visco-hyperelastic constitutive modeling of strain rate sensitive soft materials, Journal of the Mechanics and Physics of Solids,  (2019) 103777-103777.
[4] L. Liu, Y. Li, Mechanics of Materials A visco-hyperelastic softening model for predicting the strain rate e ff ects of 3D-printed soft wavy interfacial layer, Mechanics of Materials, 137(May) (2019) 103128-103128.
[5] S. Fahimi, M. Baghani, M.-r. Zakerzadeh, A. Eskandari, Developing a visco-hyperelastic material model for 3D finite deformation of elastomers, Finite Elements in Analysis & Design, 140(July) (2017) 1-10.
[6] R.M. Chen, Some nonlinear dispersive waves arising in compressible hyperelastic plates, International Journal of Engineering Science, 44(18-19) (2006) 1188-1204.
[7] P.B. Gonçalves, R.M. Soares, D. Pamplona, Nonlinear vibrations of a radially stretched circular hyperelastic membrane, Journal of Sound and Vibration, 327(1-2) (2009) 231-248.
[8] I. Dayyani, M.I. Friswell, S. Ziaei-Rad, E.I. Saavedra Flores, Equivalent models of composite corrugated cores with elastomeric coatings for morphing structures, Composite Structures, 104 (2013) 281-292.
[9] S. Faghihi, A. Karimi, M. Jamadi, R. Imani, R. Salarian, Graphene oxide/poly(acrylic acid)/gelatin nanocomposite hydrogel: Experimental and numerical validation of hyperelastic model, Materials Science and Engineering: C, 38 (2014) 299-305.
[10] R. Gupta, D. Harursampath, Dielectric elastomers: Asymptotically-correct three-dimensional displacement field, International Journal of Engineering Science, 87 (2015) 1-12.
[11] I.B. Badriev, G.Z. Garipova, M.V. Makarov, V.N. Paimushin, R.F. Khabibullin, Solving Physically Nonlinear Equilibrium Problems for Sandwich Plates with a Transversally Soft Core, 36(4) (2015) 474-481.
[12] P. Balasubramanian, G. Ferrari, M. Amabili, Z.J.G.N. del Prado, Experimental and theoretical study on large amplitude vibrations of clamped rubber plates, International Journal of Non-Linear Mechanics, 94 (2017) 36-45.
[13] A.I. Yusuf, N.M. Amin, Determination of Rayleigh Damping Coefficient for Natural Damping Rubber Plate Using Finite Element Modal Analysis,  (2015).
[14] M. Amabili, P. Balasubramanian, I.D.B.G. Ferrari, R. Garziera, K. Riabova, Experimental and numerical study on vibrations and static deflection of a thin hyperelastic plate, Journal of Sound and Vibration, (September) (2016).
[15] I. Breslavsky, M. Amabili, M. Legrand, Physically and Geometrically Nonlinear Vibrations of Thin Rectangular Plates, 3(2) (2012) 1-2.
[16] I.D. Breslavsky, M. Amabili, M. Legrand, Nonlinear vibrations of thin hyperelastic plates, Journal of Sound and Vibration, 333(19) (2014) 4668-4681.
[17] P. Balasubramanian, G. Ferrari, M. Amabili, Identification of the viscoelastic response and nonlinear damping of a rubber plate in nonlinear vibration regime, Mechanical Systems and Signal Processing, 111 (2018) 376-398.
[18] F. Alijani, M. Amabili, Non-linear vibrations of shells: A literature review from 2003 to 2013, International Journal of Non-Linear Mechanics, 58(i) (2014) 233-257.
[19] J. Dervaux, P. Ciarletta, M. Ben Amar, Morphogenesis of thin hyperelastic plates: A constitutive theory of biological growth in the Föppl–von Kármán limit, Journal of the Mechanics and Physics of Solids, 57(3) (2009) 458-471.
[20] P.H. Wen, Y.C. Hon, Geometrically nonlinear analysis of Reissner-Mindlin plate by meshless computation, CMES - Computer Modeling in Engineering and Sciences, 21(3) (2007) 177-191.
[21] K.M. Liew, L.X. Peng, S. Kitipornchai, Nonlinear analysis of corrugated plates using a FSDT and a meshfree method, Computer Methods in Applied Mechanics and Engineering, 196(21-24) (2007) 2358-2376.
[22] M. Naffa, H.J. Al-Gahtani, RBF-based meshless method for large deflection of thin plates, Engineering Analysis with Boundary Elements, 31(4) (2007) 311-317.
[23] J. Singh, K.K. Shukla, Nonlinear flexural analysis of laminated composite plates using RBF based meshless method, Composite Structures, 94(5) (2012) 1714-1720.
[24] J. Singh, K.K. Shukla, Nonlinear flexural analysis of functionally graded plates under different loadings using RBF based meshless method, Engineering Analysis with Boundary Elements, 36(12) (2012) 1819-1827.
[25] F. Liu, J. Zhao, Upper bound limit analysis using radial point interpolation meshless method and nonlinear programming, International Journal of Mechanical Sciences, 70 (2013) 26-38.
[26] Z.X. Lei, L.W. Zhang, K.M. Liew, Meshless modeling of geometrically nonlinear behavior of CNT-reinforced functionally graded composite laminated plates, Applied Mathematics and Computation, 295 (2017) 24-46.
[27] V.N.V. Do, C.H. Lee, Quasi-3D higher-order shear deformation theory for thermal buckling analysis of FGM plates based on a meshless method, Aerospace Science and Technology, 82-83(September) (2018) 450-465.
[28] E. Barbieri, L. Ventura, D. Grignoli, E. Bilotti, A meshless method for the nonlinear von Kármán plate with multiple folds of complex shape: A bridge between cracks and folds, Computational Mechanics, 64(3) (2019) 769-787.
[29] H. Nourmohammadi, B. Behjat, Geometrically nonlinear analysis of functionally graded piezoelectric plate using mesh-free RPIM, Engineering Analysis with Boundary Elements, 99(June 2018) (2019) 131-141.
[30] K.W. Cassel, Variational Methods with Applications in Science and Engineering, 2004.
[31] J. Ghaboussi, D.A. Pecknold, X.S. Wu, Nonlinear Computational Solid Mechanics, CRC Press, 2017.
[32] G.R. Liu, Meshfree methods, CRC Press, 2006.