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.

[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.

[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.

[4]

G. Y. Bagdasaryan, "Vibrations and Stability of Magnetoelastic Systems,", (Russian), (1999).

[5]

S. G. Chai, Stabilization of thermoelastic plates with variable coefficients and dynamical boundary control,, Indian J. Pure Appl. Math., 36 (2005), 227.

[6]

_____, Boundary feedback stabilization of Naghdi's model,, Acta Math. Sin. (Engl. Ser.), 21 (2005), 169.

[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.

[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.

[9]

_____, Feedthrough operator for linear elasticity system with boundary control and observation,, SIAM J. Control Optim., 48 (2010), 3708. doi: 10.1137/080729335.

[10]

_____, Well-posedness and regularity of Naghdi's shell equation under boundary control and observation,, J. Differential Equations, 249 (2010), 3174.

[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.

[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.

[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.

[14]

C. Y. Chia, Nonlinear analysis of doubly curved symmetrically laminated shallowshells with rectangular platform,, Ing.-Arch., 58 (1988), 252.

[15]

I. Chueshov and I. Lasiecka, "Von Kármán Evolution Equations. Well-Posedness and Long-Time Dynamics,", Springer Monographs in Mathematics, (2010).

[16]

P.-G. Ciarlet and V. Lods, On the ellipticity of linear membrane shell equations,, J. Math. Pures Appl. (9), 75 (1996), 107.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations,, in, 448 (1975), 25.

[24]

_____, Linear and quasilinear equations of evolution of hyperbolic type,, Hyperbolicity, (1976), 125.

[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.

[26]

W. T. Koiter, "A Consistent First Approximation in the General Theory of Thin Elastic Shells,", in, (1960), 12.

[27]

A. A. Ilyushin, "Plasticity. Part One. Elasticity-Plastic Deformations,", (Russian), (1948).

[28]

J. E. Lagnese, "Boundary Stabilization of Thin Plates,", SIAM Studies in Applied Mathematics, 10 (1989).

[29]

J. E. Lagnese and J.-L. Lions, "Modelling Analysis and Control of Thin Plates,", Recherches en Mathématiques Appliquées, 6 (1988).

[30]

I. Lasiecka, "Mathematical Control Theory of Coupled PDEs,", CBMS-NSF Regional Conference Series in Applied Mathematics, 75 (2002).

[31]

_____, Uniform stabilizability of a full von Kármán system with nonlinear boundary feedback,, SIAM J. Control, 36 (1998), 1376. doi: 10.1137/S0363012996301907.

[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.

[33]

_____, Finite-dimensionality of attractors associated with von Kármán plate equations and boundary damping,, J. Differential Equations, 117 (1995), 357.

[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.

[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.

[36]

_____, Sharp trace estimate of solutions to Kirchhoff and Euler-Bernoulli equations,, in, (1993), 141.

[37]

_____, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems,", Encyclopedia of Mathematics and its Applications, 74 (2000).

[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.

[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.

[40]

I. Lasiecka, R. Triggiani and W. Valente, Uniform stabilization of spherical shells by boundary dissipation,, Adv. Differential Equations, 1 (1996), 635.

[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.

[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.

[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.

[44]

L. Librescu, "Elastostatics and Kinetics of Anisotropic and Heterogeneous Shell-type Structures,", Noordhoff, (1975).

[45]

M. Mooney, A theory of large elastic deformation,, J. Appl. Phys., 11 (1940), 583. doi: 10.1063/1.1712836.

[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.

[47]

R. W. Ogden, "Nonlinear Elastic Deformations,", Ellis Horwood Series: Mathematics and its Applications, (1984).

[48]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983).

[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.

[50]

_____, Large solutions and smoothing properties for nonlinear thermoelastic systems,, Journal of Differential Equations, 127 (1996), 454.

[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.

[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.

[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.

[54]

M. E. Taylor, "Partial Differential Equations I. Basic Theory,", Second edition, 115 (2011).

[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.

[56]

H. Wu, The Bochner technique in differential geometry,, Mathematical Reports, 3 (1988), 289.

[57]

H. Wu, C. L. Shen and Y. L. Yu, "An Introduction to Riemannian Geometry,", (Chinese), (1989).

[58]

P.-F. Yao, On shallow shell equations,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 697. doi: 10.3934/dcdss.2009.2.697.

[59]

_____, "Modeling and Control in Vibrational and Structual Dynamics. A Differential Geometric Approach,", Chapman & HALL/CRC Applied Mathematics and Nonlinear Science Series, (2011).

[60]

_____, Observability inequalities for the Euler-Bernoulli plate with variable coefficients,, in, 268 (2000), 383.

[61]

_____, Global smooth solutions for the quasilinear wave equation with boundary dissipation,, J. Differential Equations, 241 (2007), 62.

[62]

_____, Observability inequalities for shallow shells,, SIAM J. Contr. and Optim., 38 (2000), 1729. doi: 10.1137/S0363012999338692.

[63]

_____, The ellipticity of the elliptic membrane,, Acta Anal. Funct. Appl., 3 (2001), 322.

[64]

_____, The rigid displacement lemma for elliptic membrane,, Higher Mathematics Reports, 40 (2001), 1.

[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.

[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.

[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.

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.

[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.

[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.

[4]

G. Y. Bagdasaryan, "Vibrations and Stability of Magnetoelastic Systems,", (Russian), (1999).

[5]

S. G. Chai, Stabilization of thermoelastic plates with variable coefficients and dynamical boundary control,, Indian J. Pure Appl. Math., 36 (2005), 227.

[6]

_____, Boundary feedback stabilization of Naghdi's model,, Acta Math. Sin. (Engl. Ser.), 21 (2005), 169.

[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.

[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.

[9]

_____, Feedthrough operator for linear elasticity system with boundary control and observation,, SIAM J. Control Optim., 48 (2010), 3708. doi: 10.1137/080729335.

[10]

_____, Well-posedness and regularity of Naghdi's shell equation under boundary control and observation,, J. Differential Equations, 249 (2010), 3174.

[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.

[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.

[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.

[14]

C. Y. Chia, Nonlinear analysis of doubly curved symmetrically laminated shallowshells with rectangular platform,, Ing.-Arch., 58 (1988), 252.

[15]

I. Chueshov and I. Lasiecka, "Von Kármán Evolution Equations. Well-Posedness and Long-Time Dynamics,", Springer Monographs in Mathematics, (2010).

[16]

P.-G. Ciarlet and V. Lods, On the ellipticity of linear membrane shell equations,, J. Math. Pures Appl. (9), 75 (1996), 107.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations,, in, 448 (1975), 25.

[24]

_____, Linear and quasilinear equations of evolution of hyperbolic type,, Hyperbolicity, (1976), 125.

[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.

[26]

W. T. Koiter, "A Consistent First Approximation in the General Theory of Thin Elastic Shells,", in, (1960), 12.

[27]

A. A. Ilyushin, "Plasticity. Part One. Elasticity-Plastic Deformations,", (Russian), (1948).

[28]

J. E. Lagnese, "Boundary Stabilization of Thin Plates,", SIAM Studies in Applied Mathematics, 10 (1989).

[29]

J. E. Lagnese and J.-L. Lions, "Modelling Analysis and Control of Thin Plates,", Recherches en Mathématiques Appliquées, 6 (1988).

[30]

I. Lasiecka, "Mathematical Control Theory of Coupled PDEs,", CBMS-NSF Regional Conference Series in Applied Mathematics, 75 (2002).

[31]

_____, Uniform stabilizability of a full von Kármán system with nonlinear boundary feedback,, SIAM J. Control, 36 (1998), 1376. doi: 10.1137/S0363012996301907.

[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.

[33]

_____, Finite-dimensionality of attractors associated with von Kármán plate equations and boundary damping,, J. Differential Equations, 117 (1995), 357.

[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.

[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.

[36]

_____, Sharp trace estimate of solutions to Kirchhoff and Euler-Bernoulli equations,, in, (1993), 141.

[37]

_____, "Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems,", Encyclopedia of Mathematics and its Applications, 74 (2000).

[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.

[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.

[40]

I. Lasiecka, R. Triggiani and W. Valente, Uniform stabilization of spherical shells by boundary dissipation,, Adv. Differential Equations, 1 (1996), 635.

[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.

[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.

[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.

[44]

L. Librescu, "Elastostatics and Kinetics of Anisotropic and Heterogeneous Shell-type Structures,", Noordhoff, (1975).

[45]

M. Mooney, A theory of large elastic deformation,, J. Appl. Phys., 11 (1940), 583. doi: 10.1063/1.1712836.

[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.

[47]

R. W. Ogden, "Nonlinear Elastic Deformations,", Ellis Horwood Series: Mathematics and its Applications, (1984).

[48]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983).

[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.

[50]

_____, Large solutions and smoothing properties for nonlinear thermoelastic systems,, Journal of Differential Equations, 127 (1996), 454.

[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.

[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.

[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.

[54]

M. E. Taylor, "Partial Differential Equations I. Basic Theory,", Second edition, 115 (2011).

[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.

[56]

H. Wu, The Bochner technique in differential geometry,, Mathematical Reports, 3 (1988), 289.

[57]

H. Wu, C. L. Shen and Y. L. Yu, "An Introduction to Riemannian Geometry,", (Chinese), (1989).

[58]

P.-F. Yao, On shallow shell equations,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 697. doi: 10.3934/dcdss.2009.2.697.

[59]

_____, "Modeling and Control in Vibrational and Structual Dynamics. A Differential Geometric Approach,", Chapman & HALL/CRC Applied Mathematics and Nonlinear Science Series, (2011).

[60]

_____, Observability inequalities for the Euler-Bernoulli plate with variable coefficients,, in, 268 (2000), 383.

[61]

_____, Global smooth solutions for the quasilinear wave equation with boundary dissipation,, J. Differential Equations, 241 (2007), 62.

[62]

_____, Observability inequalities for shallow shells,, SIAM J. Contr. and Optim., 38 (2000), 1729. doi: 10.1137/S0363012999338692.

[63]

_____, The ellipticity of the elliptic membrane,, Acta Anal. Funct. Appl., 3 (2001), 322.

[64]

_____, The rigid displacement lemma for elliptic membrane,, Higher Mathematics Reports, 40 (2001), 1.

[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.

[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.

[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.

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