New theory for functionally graded (FG) shell based on expansion of the equations of elasticity for functionally graded materials (GFMs) into Legendre polynomials series has been developed. Stress and strain tensors, vectors of displacements, traction and body forces have been expanded into Legendre polynomials series in a thickness coordinate. In the same way functions that describe functionally graded relations has been also expanded. Thereby all equations of elasticity including Hook-s law have been transformed to corresponding equations for Fourier coefficients. Then system of differential equations in term of displacements and boundary conditions for Fourier coefficients has been obtained. Cases of the first and second approximations have been considered in more details. For obtained boundary-value problems solution finite element (FE) has been used of Numerical calculations have been done with Comsol Multiphysics and Matlab.
[1] Chung Y.L., Chi S.H. Mechanical behavior of functionally graded
material plates under transverse load - Part I: Analysis // International
Journal of Solids and Structures, 2006, 43, P. 3657-3674.
[2] Chung Y.L., Chi S.H. Mechanical behavior of functionally graded
material plates under transverse load - Part II: Numerical results //
International Journal of Solids and Structures, 2006, 43 P. 3675-3691.
[3] Gulyaev V.I., Bazhenov, V. A. , Lizunov, P. P. The Nonclassical Theory
of Shells and Its Application to the Solution of Engineering Problems. -
L'vov: Vyshcha Shkola, 1978.- 190 p.
[4] Ebrahimi F, Sepiani H. Transverse shear and rotary inertia effects on the
stability analysis of functionally graded shells under combined static and
periodic axial loadings // Journal of Mechanical Science and
Technology,. 2010, 24, P. 2359-2366.
[5] Pelekh B.L., Suhorolskiy M.A. Contact problems of the theory of elastic
anisotropic shells. - Kiev: Naukova dumka. 1980. - 216 p.
[6] Reddy J.N., Praveen G.N. Nonlinear transient thermoelastic analysis of
functionally graded ceramic-metal plates // International Journal of
Solids and Structures, 1998, 35, P. 4457-4476.
[7] Reddy J. N. Mechanics of laminated composite plates and shells: theory
and analysis. Second ed. - CRC Press LLC, 2004. - 855 p.
[8] Shah AG, Mahmood T, Naeem MN. Vibrations of FGM thin cylindrical
shells with exponential volume fraction law // Applied Mathematics and
Mechanics, 2009, 30, P. 607-615.
[9] Shen H-S. Functionally graded materials : nonlinear analysis of plates
and shells. - CRC Press, Taylor & Francis Group; 2009.
[10] Shiota I, Miyamoto, Y. Functionally Graded Materials 1996. In:
Prosiding of 4th International Symposium on Functionally Graded
Materials. - Tokyo, Japan. : Elsevier: 1997. - 803 p.
[11] Suresh S, Mortensen A. Fundamentals of functionally graded materials.
In: Processing and Thermomechanical Behavior of Graded Metals and
Metal-Ceramic Composites. - London.: IOM Communications Ltd.:
1998. - 165 p.
[12] Vekua I.N. Some General Methods for Constructing Various Versions of
the Theory of Shells. - Moskow: Nayka: 1982. - 288 p.
[13] Xiao J.R., Gilhooley .DF., Batra R.C., McCarthy M.A., Gillespie J.W.
Analysis of thick functionally graded plates by using higher-order shear
and normal deformable plate theory and MLPG method with radial basis
functions // Compos Struct. 2007, P. 80:539-5352.
[14] Zozulya V.V. Contact cylindrical shell with a rigid body though the heatconducting
layer // Doklady Akademii Nauk Ukrainskoy SSR. 1989, 10,
P. 48-51.
[15] Zozulya V.V. The combines problem of thermoelastic contact between
two plates though a heat conducting layer // Journal of Applied
Mathematics and Mechanics. 1989,53(5), P. 622-627.
[16] Zozulya V.V. Contact cylindrical shell with a rigid body though the heatconducting
layer in transitional temperature field // Mechanics of Solids,
1991, 2, P. 160-165.
[17] Zozulya V.V. Nonperfect contact of laminated shells with considering
debonding between laminas in temperature field // Theoretical and
Applied mechanics 2006, 42, P.92-97.
[18] Zozulya VV. Laminated shells with debonding between laminas in
temperature field // International Applied Mechanics, 2006, 42(7), P.
842-848.
[19] Zozulya V.V. Mathematical Modeling of Pencil-Thin Nuclear Fuel Rods.
In: Gupta A., ed. Structural Mechanics in Reactor Technology. -
Toronto, Canada. 2007. p. C04-C12.
[20] Zozulya V. V. Contact of a shell and rigid body though the heatconducting
layer temperature field // International Journal of
Mathematics and Computers in Simulation. 2007, 2, P. 138-45.
[21] Zozulya V.V. Contact of the thin-walled structures and rigid body though
the heatconducting layer. In: Krope J, Sohrab, S.H., Benra F.-K., eds.
Theoretical and Experimental Aspects of Heat and Mass Transfer.
Acapulco, Mexico.: WSEAS Press, 2008. p. 145-50.
[22] Zozulya V. V. Heat transfer between shell and rigid body through the thin
heat-conducting layer taking into account mechanical contact. In:
Sunden B., Brebbia C.A. eds. Advanced Computational Methods and
Experiments in Heat Transfer X.- Southampton: WIT Press,. 2008, 61, P.
81-90.
[23] Zozulya V.V., Aguilar M. Thermo-elastic contact and heat transfer
between plates and shells through the heat-conducting layer. In: Sunden
B., Brebbia C.A. eds. Advanced computational methods in heat transfer
VI .- Southampton: WIT Press, 2000, 3. P. 85-94.
[24] Zozulya V.V., Borodenko Yu.N. Thermoplastic contact of rigidly fixed
shell with a rigid body though the heat-conducting layer // Doklady
Akademii Nauk Ukrainskoy SSR. 1991, 7, P.47-53.
[25] Zozulya V.V. Borodenko, Yu.N. Connecting problem on contact of
cylindrical shells with a rigid body in temperature though the heatconducting
layer // Doklady Akademii Nauk Ukrainskoy SSR. 1992, 4,
p. 35-41.
[1] Chung Y.L., Chi S.H. Mechanical behavior of functionally graded
material plates under transverse load - Part I: Analysis // International
Journal of Solids and Structures, 2006, 43, P. 3657-3674.
[2] Chung Y.L., Chi S.H. Mechanical behavior of functionally graded
material plates under transverse load - Part II: Numerical results //
International Journal of Solids and Structures, 2006, 43 P. 3675-3691.
[3] Gulyaev V.I., Bazhenov, V. A. , Lizunov, P. P. The Nonclassical Theory
of Shells and Its Application to the Solution of Engineering Problems. -
L'vov: Vyshcha Shkola, 1978.- 190 p.
[4] Ebrahimi F, Sepiani H. Transverse shear and rotary inertia effects on the
stability analysis of functionally graded shells under combined static and
periodic axial loadings // Journal of Mechanical Science and
Technology,. 2010, 24, P. 2359-2366.
[5] Pelekh B.L., Suhorolskiy M.A. Contact problems of the theory of elastic
anisotropic shells. - Kiev: Naukova dumka. 1980. - 216 p.
[6] Reddy J.N., Praveen G.N. Nonlinear transient thermoelastic analysis of
functionally graded ceramic-metal plates // International Journal of
Solids and Structures, 1998, 35, P. 4457-4476.
[7] Reddy J. N. Mechanics of laminated composite plates and shells: theory
and analysis. Second ed. - CRC Press LLC, 2004. - 855 p.
[8] Shah AG, Mahmood T, Naeem MN. Vibrations of FGM thin cylindrical
shells with exponential volume fraction law // Applied Mathematics and
Mechanics, 2009, 30, P. 607-615.
[9] Shen H-S. Functionally graded materials : nonlinear analysis of plates
and shells. - CRC Press, Taylor & Francis Group; 2009.
[10] Shiota I, Miyamoto, Y. Functionally Graded Materials 1996. In:
Prosiding of 4th International Symposium on Functionally Graded
Materials. - Tokyo, Japan. : Elsevier: 1997. - 803 p.
[11] Suresh S, Mortensen A. Fundamentals of functionally graded materials.
In: Processing and Thermomechanical Behavior of Graded Metals and
Metal-Ceramic Composites. - London.: IOM Communications Ltd.:
1998. - 165 p.
[12] Vekua I.N. Some General Methods for Constructing Various Versions of
the Theory of Shells. - Moskow: Nayka: 1982. - 288 p.
[13] Xiao J.R., Gilhooley .DF., Batra R.C., McCarthy M.A., Gillespie J.W.
Analysis of thick functionally graded plates by using higher-order shear
and normal deformable plate theory and MLPG method with radial basis
functions // Compos Struct. 2007, P. 80:539-5352.
[14] Zozulya V.V. Contact cylindrical shell with a rigid body though the heatconducting
layer // Doklady Akademii Nauk Ukrainskoy SSR. 1989, 10,
P. 48-51.
[15] Zozulya V.V. The combines problem of thermoelastic contact between
two plates though a heat conducting layer // Journal of Applied
Mathematics and Mechanics. 1989,53(5), P. 622-627.
[16] Zozulya V.V. Contact cylindrical shell with a rigid body though the heatconducting
layer in transitional temperature field // Mechanics of Solids,
1991, 2, P. 160-165.
[17] Zozulya V.V. Nonperfect contact of laminated shells with considering
debonding between laminas in temperature field // Theoretical and
Applied mechanics 2006, 42, P.92-97.
[18] Zozulya VV. Laminated shells with debonding between laminas in
temperature field // International Applied Mechanics, 2006, 42(7), P.
842-848.
[19] Zozulya V.V. Mathematical Modeling of Pencil-Thin Nuclear Fuel Rods.
In: Gupta A., ed. Structural Mechanics in Reactor Technology. -
Toronto, Canada. 2007. p. C04-C12.
[20] Zozulya V. V. Contact of a shell and rigid body though the heatconducting
layer temperature field // International Journal of
Mathematics and Computers in Simulation. 2007, 2, P. 138-45.
[21] Zozulya V.V. Contact of the thin-walled structures and rigid body though
the heatconducting layer. In: Krope J, Sohrab, S.H., Benra F.-K., eds.
Theoretical and Experimental Aspects of Heat and Mass Transfer.
Acapulco, Mexico.: WSEAS Press, 2008. p. 145-50.
[22] Zozulya V. V. Heat transfer between shell and rigid body through the thin
heat-conducting layer taking into account mechanical contact. In:
Sunden B., Brebbia C.A. eds. Advanced Computational Methods and
Experiments in Heat Transfer X.- Southampton: WIT Press,. 2008, 61, P.
81-90.
[23] Zozulya V.V., Aguilar M. Thermo-elastic contact and heat transfer
between plates and shells through the heat-conducting layer. In: Sunden
B., Brebbia C.A. eds. Advanced computational methods in heat transfer
VI .- Southampton: WIT Press, 2000, 3. P. 85-94.
[24] Zozulya V.V., Borodenko Yu.N. Thermoplastic contact of rigidly fixed
shell with a rigid body though the heat-conducting layer // Doklady
Akademii Nauk Ukrainskoy SSR. 1991, 7, P.47-53.
[25] Zozulya V.V. Borodenko, Yu.N. Connecting problem on contact of
cylindrical shells with a rigid body in temperature though the heatconducting
layer // Doklady Akademii Nauk Ukrainskoy SSR. 1992, 4,
p. 35-41.
@article{"International Journal of Chemical, Materials and Biomolecular Sciences:50379", author = "V. V. Zozulya", title = "A High Order Theory for Functionally Graded Shell", abstract = "New theory for functionally graded (FG) shell based on expansion of the equations of elasticity for functionally graded materials (GFMs) into Legendre polynomials series has been developed. Stress and strain tensors, vectors of displacements, traction and body forces have been expanded into Legendre polynomials series in a thickness coordinate. In the same way functions that describe functionally graded relations has been also expanded. Thereby all equations of elasticity including Hook-s law have been transformed to corresponding equations for Fourier coefficients. Then system of differential equations in term of displacements and boundary conditions for Fourier coefficients has been obtained. Cases of the first and second approximations have been considered in more details. For obtained boundary-value problems solution finite element (FE) has been used of Numerical calculations have been done with Comsol Multiphysics and Matlab.
", keywords = "Shell, FEM, FGM, legendre polynomial.", volume = "5", number = "11", pages = "924-6", }
