# American Institute of Mathematical Sciences

June  2012, 1(1): 155-169. doi: 10.3934/eect.2012.1.155

## Modeling of a nonlinear plate

 1 Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China, China

Received  October 2011 Revised  January 2012 Published  March 2012

We consider modeling of a nonlinear thin plate under the following assumptions: (a) the materials are nonlinear; (b) the deflections are small (linear strain displacement relations). When the middle surface is planar, we consider the bending of a plate to establish the strain energy, the equilibrium equations, and the motion equations. For a shell with a curved middle surface in $\mathbb{R}^3$, we derive a nonlinear model where a deformation in three-dimensions is concerned.
Citation: Shun Li, Peng-Fei Yao. Modeling of a nonlinear plate. Evolution Equations & Control Theory, 2012, 1 (1) : 155-169. doi: 10.3934/eect.2012.1.155
##### References:
 [1] M. Amabili, Non-linear vibrations of doubly curved shallowshells,, International Journal of Non-Linear Mechanics, 40 (2005), 683.  doi: 10.1016/j.ijnonlinmec.2004.08.007.  Google Scholar [2] M. Amabili and M. P. Paioussis, Review of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels, with and without fluid-structure interaction,, Appl. Mech. Rev., 56 (2003), 349.  doi: 10.1115/1.1565084.  Google Scholar [3] S. A. Ambartsumian, M. V. Belubekyan and M. M. Minasyan, On the problem of vibrations of nonlinear elastic electroconductive plates in transverse and longitudinal magnetic fields,, International Journal of Nonlinear Mechanics, 19 (1983), 141.  doi: 10.1016/0020-7462(84)90003-9.  Google Scholar [4] G. Y. Bagdasaryan, "Vibrations and Stability of Magnetoelastic Systems,", (Russian), (1999).   Google Scholar [5] S. G. Chai, Stabilization of thermoelastic plates with variable coefficients and dynamical boundary control,, Indian J. Pure Appl. Math., 36 (2005), 227.   Google Scholar [6] _____, Boundary feedback stabilization of Naghdi's model,, Acta Math. Sin. (Engl. Ser.), 21 (2005), 169.   Google Scholar [7] _____, Uniqueness in the Cauchy problem for the Koiter shell,, J. Math. Anal. Appl., 369 (2010), 43.  doi: 10.1016/j.jmaa.2010.02.030.  Google Scholar [8] S. G. Chai and B.-Z. Guo, Analyticity of a thermoelastic plate with variable coefficients,, J. Math. Anal. Appl., 354 (2009), 330.  doi: 10.1016/j.jmaa.2008.12.060.  Google Scholar [9] _____, Feedthrough operator for linear elasticity system with boundary control and observation,, SIAM J. Control Optim., 48 (2010), 3708.  doi: 10.1137/080729335.  Google Scholar [10] _____, Well-posedness and regularity of Naghdi's shell equation under boundary control and observation,, J. Differential Equations, 249 (2010), 3174.   Google Scholar [11] S. G. Chai, Y. X. Guo and P.-F. Yao, Boundary feedback stabilization of shallow shells,, SIAM J. Control Optim., 42 (2003), 239.  doi: 10.1137/S0363012901397156.  Google Scholar [12] S. G. Chai and K. Liu, Observability inequalities for the transmission of shallow shells,, Systems Control Letters, 55 (2006), 726.  doi: 10.1016/j.sysconle.2006.02.004.  Google Scholar [13] S. G. Chai and K. Liu, Boundary feedback stabilization of the transmission problem of Naghdi's model,, J. Math. Anal. Appl., 319 (2006), 199.  doi: 10.1016/j.jmaa.2005.08.032.  Google Scholar [14] C. Y. Chia, Nonlinear analysis of doubly curved symmetrically laminated shallowshells with rectangular platform,, Ing.-Arch., 58 (1988), 252.   Google Scholar [15] I. Chueshov and I. Lasiecka, "Von Kármán Evolution Equations. Well-Posedness and Long-Time Dynamics,", Springer Monographs in Mathematics, (2010).   Google Scholar [16] P.-G. Ciarlet and V. Lods, On the ellipticity of linear membrane shell equations,, J. Math. Pures Appl. (9), 75 (1996), 107.   Google Scholar [17] G. Friesecke, R. James and S. Müller, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence,, Arch. Ration. Mech. Anal., 180 (2006), 183.  doi: 10.1007/s00205-005-0400-7.  Google Scholar [18] G. Friesecke, R. James, M. G. Mora, and S. Müller, Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence,, C. R. Math. Acad. Sci. Paris, 336 (2003), 697.  doi: 10.1016/S1631-073X(03)00028-1.  Google Scholar [19] Y. Guo, S. G. Chai and P. F. Yao, Stabilization of elastic plates with variable coefficients and dynamical boundary control,, Quart. of Appl. Math., 60 (2002), 383.   Google Scholar [20] Y. X. Guo and P. F. Yao, Stabilization of Euler-Bernoulli plate equation with variable coefficients by nonlinear boundary feedback,, J. Math. Anal. Appl., 317 (2006), 50.  doi: 10.1016/j.jmaa.2005.12.006.  Google Scholar [21] D. Hasanyan, N. Hovakimyan, A. J. Sasane and V. Stepanyan, Analysis of nonlinear thermoelastic plate equations,, Proceedings of the 43rd IEEE Conference on Decision and Control, 2 (2004), 1514.   Google Scholar [22] T. von Kármán, The engineer grapples with non-linear problems,, Bull. Amer. Math. Soc., 46 (1940), 615.  doi: 10.1090/S0002-9904-1940-07266-0.  Google Scholar [23] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations,, in, 448 (1975), 25.   Google Scholar [24] _____, Linear and quasilinear equations of evolution of hyperbolic type,, Hyperbolicity, (1976), 125.   Google Scholar [25] R. Kirby and Z. Yosibash, Solution of von Kármán dynamic non-linear plate equations using a pseudo-spectral method,, Computer Methods in Applied Mechanics and Engineering, 193 (2004), 575.  doi: 10.1016/j.cma.2003.10.013.  Google Scholar [26] W. T. Koiter, "A Consistent First Approximation in the General Theory of Thin Elastic Shells,", in, (1960), 12.   Google Scholar [27] A. A. Ilyushin, "Plasticity. Part One. Elasticity-Plastic Deformations,", (Russian), (1948).   Google Scholar [28] J. E. Lagnese, "Boundary Stabilization of Thin Plates,", SIAM Studies in Applied Mathematics, 10 (1989).   Google Scholar [29] J. E. Lagnese and J.-L. Lions, "Modelling Analysis and Control of Thin Plates,", Recherches en Mathématiques Appliquées, 6 (1988).   Google Scholar [30] I. Lasiecka, "Mathematical Control Theory of Coupled PDEs,", CBMS-NSF Regional Conference Series in Applied Mathematics, 75 (2002).   Google Scholar [31] _____, Uniform stabilizability of a full von Kármán system with nonlinear boundary feedback,, SIAM J. Control, 36 (1998), 1376.  doi: 10.1137/S0363012996301907.  Google Scholar [32] _____, Uniform decay rates for the full von Kármán system of dynamic thermoelasticity with free boundary conditions and partial boundary dissipation,, Comm. Partial Differential Equations, 24 (1999), 1801.   Google Scholar [33] _____, Finite-dimensionality of attractors associated with von Kármán plate equations and boundary damping,, J. Differential Equations, 117 (1995), 357.   Google Scholar [34] I. Lasiecka, Sara Maad and Amol Sasane, Existence and exponential decay of solutions to a quasilinear thermoelastic plate system,, NoDEA Nonlinear Differential Equations and Applications, 15 (2008), 689.   Google Scholar [35] I. Lasiecka and R. Triggiani, Exact controllability of the Euler-Bernoulli equation with boundary controls for displacement and moment,, J. Math. Anal. Appl., 146 (1990), 1.  doi: 10.1016/0022-247X(90)90330-I.  Google Scholar [36] _____, Sharp trace estimate of solutions to Kirchhoff and Euler-Bernoulli equations,, in, (1993), 141.   Google Scholar [37] _____, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems,", Encyclopedia of Mathematics and its Applications, 74 (2000).   Google Scholar [38] _____, Uniform stabilization of a shallow shell model with nonlinear boundary feedbacks,, J. Math. Anal. Appl., 269 (2002), 642.  doi: 10.1016/S0022-247X(02)00041-0.  Google Scholar [39] _____, Linear hyperbolic and Petrowski type PDEs with continuous boundary control $\to$ boundary observation open loop map: Implication on nonlinear boundary stabilization with optimal decay rates,, in, 10 (2009), 187.   Google Scholar [40] I. Lasiecka, R. Triggiani and W. Valente, Uniform stabilization of spherical shells by boundary dissipation,, Adv. Differential Equations, 1 (1996), 635.   Google Scholar [41] I. Lasiecka and W. Valente, Uniform boundary stabilization of a nonlinear shallow and thin elastic spherical cap,, J. Math. Anal. Appl., 202 (1996), 951.  doi: 10.1006/jmaa.1996.0356.  Google Scholar [42] M. Lewicka, M. G. Mora and M. R. Pakzad, Shell theories arising as low energy $\Gamma$-limit of 3d nonlinear elasticity,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 9 (2010), 253.   Google Scholar [43] _____, The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells,, Arch. Rational Mech. Anal. (3), 200 (2011), 1023.  doi: 10.1007/s00205-010-0387-6.  Google Scholar [44] L. Librescu, "Elastostatics and Kinetics of Anisotropic and Heterogeneous Shell-type Structures,", Noordhoff, (1975).   Google Scholar [45] M. Mooney, A theory of large elastic deformation,, J. Appl. Phys., 11 (1940), 583.  doi: 10.1063/1.1712836.  Google Scholar [46] R. W. Ogden, Large deformation isotropic elasticity - on the correlation of theory and experiment for incompressible rubberlike solids,, Proc. R. Soc. Lond. A., 326 (1972), 565.  doi: 10.1098/rspa.1972.0026.  Google Scholar [47] R. W. Ogden, "Nonlinear Elastic Deformations,", Ellis Horwood Series: Mathematics and its Applications, (1984).   Google Scholar [48] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983).   Google Scholar [49] J. E. Muñoz Rivera and R. Racke, Smoothing properties, decay, and global existence of solutions to nonlinear coupled systems of thermoelastic type,, SIAM Journal on Mathematical Analysis, 26 (1995), 1547.  doi: 10.1137/S0036142993255058.  Google Scholar [50] _____, Large solutions and smoothing properties for nonlinear thermoelastic systems,, Journal of Differential Equations, 127 (1996), 454.   Google Scholar [51] R. S. Rivlin, A note on the torsion of an incompressible highly-elastic cylinder,, Proc. Cambridge Philos. Soc., 45 (1949), 485.  doi: 10.1017/S0305004100025135.  Google Scholar [52] A. P. S. Selvadurai, Deflections of a rubber membrane,, Journal of the Mechanics and Physics of Solids, 54 (2006), 1093.  doi: 10.1016/j.jmps.2006.01.001.  Google Scholar [53] J. Shivakumar and M. C. Ray, Geometrically nonlinear analysis of antisymmetric angle-ply smart composite plates integrated with a layer of piezoelectric fiber reinforced composite,, Smart Mater. Struct., 16 (2007), 754.  doi: 10.1088/0964-1726/16/3/024.  Google Scholar [54] M. E. Taylor, "Partial Differential Equations I. Basic Theory,", Second edition, 115 (2011).   Google Scholar [55] R. Triggiani, Regularity theory, exact controllability and optimal quadratic cost problem for spherical shells with physical boundary controls,, Special Issue of Control and Cybernetics, 25 (1996), 553.   Google Scholar [56] H. Wu, The Bochner technique in differential geometry,, Mathematical Reports, 3 (1988), 289.   Google Scholar [57] H. Wu, C. L. Shen and Y. L. Yu, "An Introduction to Riemannian Geometry,", (Chinese), (1989).   Google Scholar [58] P.-F. Yao, On shallow shell equations,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 697.  doi: 10.3934/dcdss.2009.2.697.  Google Scholar [59] _____, "Modeling and Control in Vibrational and Structual Dynamics. A Differential Geometric Approach,", Chapman & HALL/CRC Applied Mathematics and Nonlinear Science Series, (2011).   Google Scholar [60] _____, Observability inequalities for the Euler-Bernoulli plate with variable coefficients,, in, 268 (2000), 383.   Google Scholar [61] _____, Global smooth solutions for the quasilinear wave equation with boundary dissipation,, J. Differential Equations, 241 (2007), 62.   Google Scholar [62] _____, Observability inequalities for shallow shells,, SIAM J. Contr. and Optim., 38 (2000), 1729.  doi: 10.1137/S0363012999338692.  Google Scholar [63] _____, The ellipticity of the elliptic membrane,, Acta Anal. Funct. Appl., 3 (2001), 322.   Google Scholar [64] _____, The rigid displacement lemma for elliptic membrane,, Higher Mathematics Reports, 40 (2001), 1.   Google Scholar [65] Z. Yosibash, R. M. Kirby and D. Gottlieb, Collocation methods for the solution of von-Kármán dynamic non-linear plate systems,, J. Comput. Phys., 200 (2004), 432.  doi: 10.1016/j.jcp.2004.03.018.  Google Scholar [66] Y. X. Zhang and K. S. Kim, Linear and geometrically nonlinear analysis of plates and shells by a new refined non-conforming triangular plate/shell element,, Computational Mechanics, 36 (2005), 331.  doi: 10.1007/s00466-004-0625-6.  Google Scholar [67] Z.-F. Zhang and P.-F. Yao, Global smooth solutions of the quasi-linear wave equation with internal velocity feedback,, SIAM J. Control Optim., 47 (2008), 2044.  doi: 10.1137/070679454.  Google Scholar

show all references

##### References:
 [1] M. Amabili, Non-linear vibrations of doubly curved shallowshells,, International Journal of Non-Linear Mechanics, 40 (2005), 683.  doi: 10.1016/j.ijnonlinmec.2004.08.007.  Google Scholar [2] M. Amabili and M. P. Paioussis, Review of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels, with and without fluid-structure interaction,, Appl. Mech. Rev., 56 (2003), 349.  doi: 10.1115/1.1565084.  Google Scholar [3] S. A. Ambartsumian, M. V. Belubekyan and M. M. Minasyan, On the problem of vibrations of nonlinear elastic electroconductive plates in transverse and longitudinal magnetic fields,, International Journal of Nonlinear Mechanics, 19 (1983), 141.  doi: 10.1016/0020-7462(84)90003-9.  Google Scholar [4] G. Y. Bagdasaryan, "Vibrations and Stability of Magnetoelastic Systems,", (Russian), (1999).   Google Scholar [5] S. G. Chai, Stabilization of thermoelastic plates with variable coefficients and dynamical boundary control,, Indian J. Pure Appl. Math., 36 (2005), 227.   Google Scholar [6] _____, Boundary feedback stabilization of Naghdi's model,, Acta Math. Sin. (Engl. Ser.), 21 (2005), 169.   Google Scholar [7] _____, Uniqueness in the Cauchy problem for the Koiter shell,, J. Math. Anal. Appl., 369 (2010), 43.  doi: 10.1016/j.jmaa.2010.02.030.  Google Scholar [8] S. G. Chai and B.-Z. Guo, Analyticity of a thermoelastic plate with variable coefficients,, J. Math. Anal. Appl., 354 (2009), 330.  doi: 10.1016/j.jmaa.2008.12.060.  Google Scholar [9] _____, Feedthrough operator for linear elasticity system with boundary control and observation,, SIAM J. Control Optim., 48 (2010), 3708.  doi: 10.1137/080729335.  Google Scholar [10] _____, Well-posedness and regularity of Naghdi's shell equation under boundary control and observation,, J. Differential Equations, 249 (2010), 3174.   Google Scholar [11] S. G. Chai, Y. X. Guo and P.-F. Yao, Boundary feedback stabilization of shallow shells,, SIAM J. Control Optim., 42 (2003), 239.  doi: 10.1137/S0363012901397156.  Google Scholar [12] S. G. Chai and K. Liu, Observability inequalities for the transmission of shallow shells,, Systems Control Letters, 55 (2006), 726.  doi: 10.1016/j.sysconle.2006.02.004.  Google Scholar [13] S. G. Chai and K. Liu, Boundary feedback stabilization of the transmission problem of Naghdi's model,, J. Math. Anal. Appl., 319 (2006), 199.  doi: 10.1016/j.jmaa.2005.08.032.  Google Scholar [14] C. Y. Chia, Nonlinear analysis of doubly curved symmetrically laminated shallowshells with rectangular platform,, Ing.-Arch., 58 (1988), 252.   Google Scholar [15] I. Chueshov and I. Lasiecka, "Von Kármán Evolution Equations. Well-Posedness and Long-Time Dynamics,", Springer Monographs in Mathematics, (2010).   Google Scholar [16] P.-G. Ciarlet and V. Lods, On the ellipticity of linear membrane shell equations,, J. Math. Pures Appl. (9), 75 (1996), 107.   Google Scholar [17] G. Friesecke, R. James and S. Müller, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence,, Arch. Ration. Mech. Anal., 180 (2006), 183.  doi: 10.1007/s00205-005-0400-7.  Google Scholar [18] G. Friesecke, R. James, M. G. Mora, and S. Müller, Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence,, C. R. Math. Acad. Sci. Paris, 336 (2003), 697.  doi: 10.1016/S1631-073X(03)00028-1.  Google Scholar [19] Y. Guo, S. G. Chai and P. F. Yao, Stabilization of elastic plates with variable coefficients and dynamical boundary control,, Quart. of Appl. Math., 60 (2002), 383.   Google Scholar [20] Y. X. Guo and P. F. Yao, Stabilization of Euler-Bernoulli plate equation with variable coefficients by nonlinear boundary feedback,, J. Math. Anal. Appl., 317 (2006), 50.  doi: 10.1016/j.jmaa.2005.12.006.  Google Scholar [21] D. Hasanyan, N. Hovakimyan, A. J. Sasane and V. Stepanyan, Analysis of nonlinear thermoelastic plate equations,, Proceedings of the 43rd IEEE Conference on Decision and Control, 2 (2004), 1514.   Google Scholar [22] T. von Kármán, The engineer grapples with non-linear problems,, Bull. Amer. Math. Soc., 46 (1940), 615.  doi: 10.1090/S0002-9904-1940-07266-0.  Google Scholar [23] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations,, in, 448 (1975), 25.   Google Scholar [24] _____, Linear and quasilinear equations of evolution of hyperbolic type,, Hyperbolicity, (1976), 125.   Google Scholar [25] R. Kirby and Z. Yosibash, Solution of von Kármán dynamic non-linear plate equations using a pseudo-spectral method,, Computer Methods in Applied Mechanics and Engineering, 193 (2004), 575.  doi: 10.1016/j.cma.2003.10.013.  Google Scholar [26] W. T. Koiter, "A Consistent First Approximation in the General Theory of Thin Elastic Shells,", in, (1960), 12.   Google Scholar [27] A. A. Ilyushin, "Plasticity. Part One. Elasticity-Plastic Deformations,", (Russian), (1948).   Google Scholar [28] J. E. Lagnese, "Boundary Stabilization of Thin Plates,", SIAM Studies in Applied Mathematics, 10 (1989).   Google Scholar [29] J. E. Lagnese and J.-L. Lions, "Modelling Analysis and Control of Thin Plates,", Recherches en Mathématiques Appliquées, 6 (1988).   Google Scholar [30] I. Lasiecka, "Mathematical Control Theory of Coupled PDEs,", CBMS-NSF Regional Conference Series in Applied Mathematics, 75 (2002).   Google Scholar [31] _____, Uniform stabilizability of a full von Kármán system with nonlinear boundary feedback,, SIAM J. Control, 36 (1998), 1376.  doi: 10.1137/S0363012996301907.  Google Scholar [32] _____, Uniform decay rates for the full von Kármán system of dynamic thermoelasticity with free boundary conditions and partial boundary dissipation,, Comm. Partial Differential Equations, 24 (1999), 1801.   Google Scholar [33] _____, Finite-dimensionality of attractors associated with von Kármán plate equations and boundary damping,, J. Differential Equations, 117 (1995), 357.   Google Scholar [34] I. Lasiecka, Sara Maad and Amol Sasane, Existence and exponential decay of solutions to a quasilinear thermoelastic plate system,, NoDEA Nonlinear Differential Equations and Applications, 15 (2008), 689.   Google Scholar [35] I. Lasiecka and R. Triggiani, Exact controllability of the Euler-Bernoulli equation with boundary controls for displacement and moment,, J. Math. Anal. Appl., 146 (1990), 1.  doi: 10.1016/0022-247X(90)90330-I.  Google Scholar [36] _____, Sharp trace estimate of solutions to Kirchhoff and Euler-Bernoulli equations,, in, (1993), 141.   Google Scholar [37] _____, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems,", Encyclopedia of Mathematics and its Applications, 74 (2000).   Google Scholar [38] _____, Uniform stabilization of a shallow shell model with nonlinear boundary feedbacks,, J. Math. Anal. Appl., 269 (2002), 642.  doi: 10.1016/S0022-247X(02)00041-0.  Google Scholar [39] _____, Linear hyperbolic and Petrowski type PDEs with continuous boundary control $\to$ boundary observation open loop map: Implication on nonlinear boundary stabilization with optimal decay rates,, in, 10 (2009), 187.   Google Scholar [40] I. Lasiecka, R. Triggiani and W. Valente, Uniform stabilization of spherical shells by boundary dissipation,, Adv. Differential Equations, 1 (1996), 635.   Google Scholar [41] I. Lasiecka and W. Valente, Uniform boundary stabilization of a nonlinear shallow and thin elastic spherical cap,, J. Math. Anal. Appl., 202 (1996), 951.  doi: 10.1006/jmaa.1996.0356.  Google Scholar [42] M. Lewicka, M. G. Mora and M. R. Pakzad, Shell theories arising as low energy $\Gamma$-limit of 3d nonlinear elasticity,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 9 (2010), 253.   Google Scholar [43] _____, The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells,, Arch. Rational Mech. Anal. (3), 200 (2011), 1023.  doi: 10.1007/s00205-010-0387-6.  Google Scholar [44] L. Librescu, "Elastostatics and Kinetics of Anisotropic and Heterogeneous Shell-type Structures,", Noordhoff, (1975).   Google Scholar [45] M. Mooney, A theory of large elastic deformation,, J. Appl. Phys., 11 (1940), 583.  doi: 10.1063/1.1712836.  Google Scholar [46] R. W. Ogden, Large deformation isotropic elasticity - on the correlation of theory and experiment for incompressible rubberlike solids,, Proc. R. Soc. Lond. A., 326 (1972), 565.  doi: 10.1098/rspa.1972.0026.  Google Scholar [47] R. W. Ogden, "Nonlinear Elastic Deformations,", Ellis Horwood Series: Mathematics and its Applications, (1984).   Google Scholar [48] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983).   Google Scholar [49] J. E. Muñoz Rivera and R. Racke, Smoothing properties, decay, and global existence of solutions to nonlinear coupled systems of thermoelastic type,, SIAM Journal on Mathematical Analysis, 26 (1995), 1547.  doi: 10.1137/S0036142993255058.  Google Scholar [50] _____, Large solutions and smoothing properties for nonlinear thermoelastic systems,, Journal of Differential Equations, 127 (1996), 454.   Google Scholar [51] R. S. Rivlin, A note on the torsion of an incompressible highly-elastic cylinder,, Proc. Cambridge Philos. Soc., 45 (1949), 485.  doi: 10.1017/S0305004100025135.  Google Scholar [52] A. P. S. Selvadurai, Deflections of a rubber membrane,, Journal of the Mechanics and Physics of Solids, 54 (2006), 1093.  doi: 10.1016/j.jmps.2006.01.001.  Google Scholar [53] J. Shivakumar and M. C. Ray, Geometrically nonlinear analysis of antisymmetric angle-ply smart composite plates integrated with a layer of piezoelectric fiber reinforced composite,, Smart Mater. Struct., 16 (2007), 754.  doi: 10.1088/0964-1726/16/3/024.  Google Scholar [54] M. E. Taylor, "Partial Differential Equations I. Basic Theory,", Second edition, 115 (2011).   Google Scholar [55] R. Triggiani, Regularity theory, exact controllability and optimal quadratic cost problem for spherical shells with physical boundary controls,, Special Issue of Control and Cybernetics, 25 (1996), 553.   Google Scholar [56] H. Wu, The Bochner technique in differential geometry,, Mathematical Reports, 3 (1988), 289.   Google Scholar [57] H. Wu, C. L. Shen and Y. L. 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