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Abstract
In the present work, flexural analysis of a thin to moderately thick FGM plate subjected to transverse loads have been studied using finite element method. The formulation is developed based on the First order Shear Deformation Theory (FSDT). The mechanical properties are assumed to vary continuously through the thickness of the plate and obey a power law distribution of the volume fraction of the constituents. FGM’s are typically heterogeneous in nature and a homogenization scheme is generally adopted for the analysis. To ascertain the effect of homogenization schemes on the material properties, the variation of volume fraction through the thickness have been computed using two different material homogenization techniques; namely the Rule of Mixtures and Mori-Tanaka scheme. In calculating the effective material properties through the thickness numerical integration have been used to evaluate the integrals which is easier and faster when compared to symbolic integration. A detailed discussion comparing the results from both the homogenization schemes have been presented. In addition to that a detailed parametric study bringing out the effect of boundary conditions, loading intensities and volume fraction index have been presented. Convergence tests and comparison studies have been carried out to demonstrate the efficiency of the present formulation.
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References
- Zuiker, J.R., "Functionally Graded Materials: Choice of Micro Mechanics Model and Limitations in Property Variation", Composites Engineering, 5 (7), 1995, pp.807-819.
- Reiter, T. and Dvorak, G.J., "Micro Mechanical Modeling of Functionally Graded Materials", IUTAM Symposium on Transformation Problem in Composite and Active Materials, Kluwer Academic, London, 1997, pp.173-184.
- Gasik, M.M., "Micro Mechanical Modeling of Functionally Graded Materials", Computational Materials Science, 13, 1998, pp.42-55.
- Birman, V. and Byrd, L.W., "Modeling and Analysis of Functionally Graded Materials and Structures", Applied Mechanics Reviews, 60, 2007, pp.195216.
- Hashin, Z. and Shtrikman, S., "On Some Variational Principles in Anisotropic and Non-homogeneous Elasticity", Journal of Mechanics and Physics of Solids, 10 (4), 1962, pp.335-342.
- Pan, E., "Exact Solution for Functionally Graded Anisotropic Elastic Composite Laminate", Journal of Composite Materials, 37(21), 2003, pp.19031920.
- Willis, J.R., "Bounds and Self-consistent Estimates for the Overall Properties of Anisotropic Composites", Journal of Mechanics and Physics of Solids, 25 (3), 1977, pp.185-202.
- Reddy, J.N. and Chin, C.D., "Thermo Mechanical Analysis of Functionally Graded Cylinders and plates", Journal of Thermal Stresses, 21, 1998, pp.593-626.
- Reddy, J.N., "Analysis of functionally graded Plates", International Journal of Numerical Methods in Engineering, 47(1-3), 2000, pp.663-684.
- Reddy, J.N. and Cheng, Z-Q., "Three Dimensional Thermomechanical Deformations of Functionally Graded Rectangular Plates", European Journal of Mechanics A/Solids, 20, 2001, pp.841-855.
- Praveen, G.N. and Reddy, J.N., "Nonlinear transient Thermoelastic Analysis of Functionally Graded Ceramic-metal Plates", Journal of Solids and Structures, 35(33), 1998, pp.4457-4476.
- Vel, S.S. and Batra, R.C., "Exact Solution for Thermoelastic Deformations of Functionally Graded Thick Rectangular Plates", AIAA Journal, 40(7), 2002, pp.1421-1433.
- Qian, L.F., Batra, R.C. and Chen, L.M., "Static and Dynamic Deformations of tHick Functionally Graded Elastic Plates by Using Higher-order Shear and Normal Deformable Plate Theory and Meshless Local Petrov-Galerkin Method", Composites Part B: Engg, 35, 2004, pp.685-697.
- Ferreira, A.J.M., Batra, R.C., Rouque, C.M.C., Qian, L.F. and Martins, P.A.L.S., "Static Analysis of Functionally Graded Plates Using Third-order shear deformation Theory and a Meshless Method", Composite Structures, 69, 2005a, pp.449-457.
- Kashtalyan, M., "Three-dimensional Elasticity Solution for Bending of Functionally Graded Rectangular Plates", European Journal of Mechanics A/Solids, 23, 2004, pp.853-864.
- Elishakoff, I. and Gentilini, C., "Three-dimensional Flexure of Rectangular Plates Made of Functionally Graded Materials", ASME, Journal of Applied Mechanics, 72, 2005, pp.788-791.
- Zenkour, A.M., "Generalized Shear Deformation Theory for Bending Analysis of Functionally Graded Plates", Applied Mathematical Modeling, 30, 2006, pp.67-84.
- Matsunaga, H., "Stress Analysis of Functionally Graded Plates Subjected to Thermal and Mechanical Loadings", Composite Structures, 87, 2009, pp.344357.
- Sh. Hosseini-Hashemi, Rokni Damavandi Taher, H., Akhavan, H. and Omidi, M., "Free Vibration of Functionally Graded Rectangular Plates Using Firstorder Shear Deformation Plate Theory", Applied Mathematical Modelling, 34, 2010, pp.1276-1291.
- Talha, M. and Singh, B.N., "Static Response and Free Vibration Analysis of FGM Plates Using Higher Order Shear Deformation Theory", Applied Mathematical Modeling, 34 (12), 2010, pp.3991-4011.
- Singha, M.K., Prakash, T. and Ganapathi, M., "Finite Element Analysis of Functionally Graded Plates Under Transverse Load", Finite Elements in Analysis and Design, 47, 2011, pp.453-460.
- Gupta, A. and Talha, M., "Recent Developments in Modeling and Analysis of Functionally Graded Materials and Structures", Progress in Aerospace Sciences, 79, 2015, pp.1-14.
- Klusemann, B. and Svendsen, B., "Homogenization Methods for Multi-phase Elastic Composites: Comparison and Benchmarks", Tech. Mech, 30 (4), 2010, pp.374-386.
- Mori, T. and Tanaka, K., "Average Stress in Matrix and Average Elastic Energy of Materials with Misfitting Inclusions", Acta Metallurgica, 21, 1973, pp.571-574.
- Benveniste, Y., "A New Approach to the Application of Mori-Tanaka’s Theory in Composite Materials", Mechanics of Materials, 6, 1987, pp.147-157.
- Hui-Shen Shen and Zhen-Xin Wang., "Assessment of Voigt and Mori-Tanaka Models for Vibration Analysis of Functionally Graded Plates", Composite Structures, 94, 2012, pp.2197-2208.
- Memar Ardestani, M., Soltani, B. and Sh. Shams., "Analysis of Functionally Graded Stiffened Plates Based on FSDT Utilizing Reproducing Kernel Particle Method", Composite Structures, 112, 2014, pp.231-240.
- Reddy, J.N., "Microstructure-dependent Couple Stress Theories of Functionally Graded Beams", J
- Mech Phys Solids, 59, 2011, pp.2382-2399.
- Neves, A.M.A., Ferreira, A.J.M., Carrera, E., Cinefra, M., Roque, C.M.C., Jorge, R.M.N. and Soares,
- C.M.M., "Static, Free Vibration and Buckling Analysis of Functionally Graded Plates Using a Quasi-3D
- Higher-order Shear Deformation Theory and a Meshless Technique", Composites Part B: Engineering, 44 (1), 2013, pp.657-674.
- Carrera, E., Brischetto, S., Cinefra, M.and Soave, M., "Effects of Thickness Stretching in Functionally
- Graded Plates and Shells", Composites Part B: Engineering, 42 (2), 2011, pp.123-133.
- Arbind, A. and Reddy, J.N., "Nonlinear Analysis of Functionally Graded Microstructure Dependent
- Beams", Composite Structures, 98, 2013, pp.272-281.
- Cowper, G.R., "The Shear Coefficient in Timoshenko’s Beam Theory", Journal of Applied Mechanics, 1966, pp.335-340.
- Reddy, J.N., "Mechanics of Laminated Composite Plates and Shells: Theory and Analysis", Second
- Edition, CRC Press, Boca Raton, FL, 2004.
- Reddy, J.N., "Theory of Elastic Plates and Shells", Second Edition, CRC Press, 2007.
- Reddy, J.N., "An Introduction to the Finite Element Method", Third Edition, McGraw-Hill, New York,
References
Zuiker, J.R., "Functionally Graded Materials: Choice of Micro Mechanics Model and Limitations in Property Variation", Composites Engineering, 5 (7), 1995, pp.807-819.
Reiter, T. and Dvorak, G.J., "Micro Mechanical Modeling of Functionally Graded Materials", IUTAM Symposium on Transformation Problem in Composite and Active Materials, Kluwer Academic, London, 1997, pp.173-184.
Gasik, M.M., "Micro Mechanical Modeling of Functionally Graded Materials", Computational Materials Science, 13, 1998, pp.42-55.
Birman, V. and Byrd, L.W., "Modeling and Analysis of Functionally Graded Materials and Structures", Applied Mechanics Reviews, 60, 2007, pp.195216.
Hashin, Z. and Shtrikman, S., "On Some Variational Principles in Anisotropic and Non-homogeneous Elasticity", Journal of Mechanics and Physics of Solids, 10 (4), 1962, pp.335-342.
Pan, E., "Exact Solution for Functionally Graded Anisotropic Elastic Composite Laminate", Journal of Composite Materials, 37(21), 2003, pp.19031920.
Willis, J.R., "Bounds and Self-consistent Estimates for the Overall Properties of Anisotropic Composites", Journal of Mechanics and Physics of Solids, 25 (3), 1977, pp.185-202.
Reddy, J.N. and Chin, C.D., "Thermo Mechanical Analysis of Functionally Graded Cylinders and plates", Journal of Thermal Stresses, 21, 1998, pp.593-626.
Reddy, J.N., "Analysis of functionally graded Plates", International Journal of Numerical Methods in Engineering, 47(1-3), 2000, pp.663-684.
Reddy, J.N. and Cheng, Z-Q., "Three Dimensional Thermomechanical Deformations of Functionally Graded Rectangular Plates", European Journal of Mechanics A/Solids, 20, 2001, pp.841-855.
Praveen, G.N. and Reddy, J.N., "Nonlinear transient Thermoelastic Analysis of Functionally Graded Ceramic-metal Plates", Journal of Solids and Structures, 35(33), 1998, pp.4457-4476.
Vel, S.S. and Batra, R.C., "Exact Solution for Thermoelastic Deformations of Functionally Graded Thick Rectangular Plates", AIAA Journal, 40(7), 2002, pp.1421-1433.
Qian, L.F., Batra, R.C. and Chen, L.M., "Static and Dynamic Deformations of tHick Functionally Graded Elastic Plates by Using Higher-order Shear and Normal Deformable Plate Theory and Meshless Local Petrov-Galerkin Method", Composites Part B: Engg, 35, 2004, pp.685-697.
Ferreira, A.J.M., Batra, R.C., Rouque, C.M.C., Qian, L.F. and Martins, P.A.L.S., "Static Analysis of Functionally Graded Plates Using Third-order shear deformation Theory and a Meshless Method", Composite Structures, 69, 2005a, pp.449-457.
Kashtalyan, M., "Three-dimensional Elasticity Solution for Bending of Functionally Graded Rectangular Plates", European Journal of Mechanics A/Solids, 23, 2004, pp.853-864.
Elishakoff, I. and Gentilini, C., "Three-dimensional Flexure of Rectangular Plates Made of Functionally Graded Materials", ASME, Journal of Applied Mechanics, 72, 2005, pp.788-791.
Zenkour, A.M., "Generalized Shear Deformation Theory for Bending Analysis of Functionally Graded Plates", Applied Mathematical Modeling, 30, 2006, pp.67-84.
Matsunaga, H., "Stress Analysis of Functionally Graded Plates Subjected to Thermal and Mechanical Loadings", Composite Structures, 87, 2009, pp.344357.
Sh. Hosseini-Hashemi, Rokni Damavandi Taher, H., Akhavan, H. and Omidi, M., "Free Vibration of Functionally Graded Rectangular Plates Using Firstorder Shear Deformation Plate Theory", Applied Mathematical Modelling, 34, 2010, pp.1276-1291.
Talha, M. and Singh, B.N., "Static Response and Free Vibration Analysis of FGM Plates Using Higher Order Shear Deformation Theory", Applied Mathematical Modeling, 34 (12), 2010, pp.3991-4011.
Singha, M.K., Prakash, T. and Ganapathi, M., "Finite Element Analysis of Functionally Graded Plates Under Transverse Load", Finite Elements in Analysis and Design, 47, 2011, pp.453-460.
Gupta, A. and Talha, M., "Recent Developments in Modeling and Analysis of Functionally Graded Materials and Structures", Progress in Aerospace Sciences, 79, 2015, pp.1-14.
Klusemann, B. and Svendsen, B., "Homogenization Methods for Multi-phase Elastic Composites: Comparison and Benchmarks", Tech. Mech, 30 (4), 2010, pp.374-386.
Mori, T. and Tanaka, K., "Average Stress in Matrix and Average Elastic Energy of Materials with Misfitting Inclusions", Acta Metallurgica, 21, 1973, pp.571-574.
Benveniste, Y., "A New Approach to the Application of Mori-Tanaka’s Theory in Composite Materials", Mechanics of Materials, 6, 1987, pp.147-157.
Hui-Shen Shen and Zhen-Xin Wang., "Assessment of Voigt and Mori-Tanaka Models for Vibration Analysis of Functionally Graded Plates", Composite Structures, 94, 2012, pp.2197-2208.
Memar Ardestani, M., Soltani, B. and Sh. Shams., "Analysis of Functionally Graded Stiffened Plates Based on FSDT Utilizing Reproducing Kernel Particle Method", Composite Structures, 112, 2014, pp.231-240.
Reddy, J.N., "Microstructure-dependent Couple Stress Theories of Functionally Graded Beams", J
Mech Phys Solids, 59, 2011, pp.2382-2399.
Neves, A.M.A., Ferreira, A.J.M., Carrera, E., Cinefra, M., Roque, C.M.C., Jorge, R.M.N. and Soares,
C.M.M., "Static, Free Vibration and Buckling Analysis of Functionally Graded Plates Using a Quasi-3D
Higher-order Shear Deformation Theory and a Meshless Technique", Composites Part B: Engineering, 44 (1), 2013, pp.657-674.
Carrera, E., Brischetto, S., Cinefra, M.and Soave, M., "Effects of Thickness Stretching in Functionally
Graded Plates and Shells", Composites Part B: Engineering, 42 (2), 2011, pp.123-133.
Arbind, A. and Reddy, J.N., "Nonlinear Analysis of Functionally Graded Microstructure Dependent
Beams", Composite Structures, 98, 2013, pp.272-281.
Cowper, G.R., "The Shear Coefficient in Timoshenko’s Beam Theory", Journal of Applied Mechanics, 1966, pp.335-340.
Reddy, J.N., "Mechanics of Laminated Composite Plates and Shells: Theory and Analysis", Second
Edition, CRC Press, Boca Raton, FL, 2004.
Reddy, J.N., "Theory of Elastic Plates and Shells", Second Edition, CRC Press, 2007.
Reddy, J.N., "An Introduction to the Finite Element Method", Third Edition, McGraw-Hill, New York,