Laminated
composites are widely used in many engineering industries such as aircraft,
spacecraft, boat hulls, racing car bodies, and storage tanks. We analyze the 3D
deformations of a multilayered, linear elastic, anisotropic rectangular plate
subjected to arbitrary boundary conditions on one edge and simply supported on
other edge. The rectangular laminate consists of anisotropic and homogeneous
laminae of arbitrary thicknesses. This study presents the elastic analysis of
laminated composite plates subjected to sinusoidal mechanical loading under
arbitrary boundary conditions. Least square finite element solutions for
displacements and stresses are investigated using a mathematical model, called
a state-space model, which allows us to simultaneously solve for these field
variables in the composite structure’s domain and ensure that continuity
conditions are satisfied at layer interfaces. The governing equations are
derived from this model using a numerical technique called the least-squares finite element method (LSFEM). These LSFEMs seek to minimize the squares of the governing equations
and the associated side conditions residuals over the computational domain. The
model is comprised of layerwise variables such as displacements, out-of-plane
stresses, and in-plane strains, treated as independent variables. Numerical results are
presented to demonstrate the response of the laminated composite plates under
various arbitrary boundary conditions using LSFEM and compared with the 3D
elasticity solution available in the literature.
References
[1]
Vel, S.S. and Batra, R. (1999) Analytical Solution for Rectangular Thick Laminated Plates Subjected to Arbitrary Boundary Conditions. AIAA Journal, 37, 1464-1473.
https://doi.org/10.2514/2.624
[2]
Reddy, J.N. (1984) A Simple Higher-Order Theory for Laminated Composite Plates. Journal of Applied Mechanics, 51, 745-752. https://doi.org/10.1115/1.3167719
[3]
Reddy, J.N. (2003) Mechanics of Laminated Composite Plates and Shells: Theory and Analysis. CRC Press, Boca Raton. https://doi.org/10.1201/b12409
[4]
Reddy, J.N. (2019) Introduction to the Finite Element Method. McGraw-Hill Education, New York.
[5]
Pagano, N. and Hatfield, S.J. (1972) Elastic Behavior of Multilayered Bidirectional Composites. AIAA Journal, 10, 931-933. https://doi.org/10.2514/3.50249
[6]
Pagano, N.J. (1970) Exact Solutions for Rectangular Bidirectional Composites and Sandwich Plates. Journal of Composite Materials, 4, 20-34.
https://doi.org/10.1177/002199837000400102
[7]
Moleiro, F., Mota Soares, C.M., Mota Soares, C.A. and Reddy, J.N. (2011) A Layerwise Mixed Least-Squares Finite Element Model for Static Analysis of Multilayered Composite Plates. Computers & Structures, 89, 1730-1742.
https://doi.org/10.1016/j.compstruc.2010.10.008
[8]
Tauchert, T. and Howard, H. (2012) Least-Squares Residual Solution for Displacement Control of a Composite Piezothermoelastic Cylinder. Journal of Thermal Stresses, 35, 61-74. https://doi.org/10.1080/01495739.2012.637742
[9]
Burns, D. and Batra, R.C. (2021) Static Deformations of Fiber-Reinforced Composite Laminates by the Least-Squares Method. In: Challamel, N., Kaplunov, J. and Takewaki, I., Eds., Modern Trends in Structural and Solid Mechanics 1: Statics and Stability, Publisher, city, 1-16. https://doi.org/10.1002/9781119831891.ch1
[10]
Srinivas, S. and Rao, A. (1970) Bending, Vibration and Buckling of Simply Supported Thick Orthotropic Rectangular Plates and Laminates. International Journal of Solids and Structures, 6, 1463-1481.
https://doi.org/10.1016/0020-7683(70)90076-4
[11]
Vel, S.S. and Batra, R. (2000) The Generalized Plane Strain Deformations of Thick Anisotropic Composite Laminated Plates. International Journal of Solids and Structures, 37, 715-733. https://doi.org/10.1016/S0020-7683(99)00040-2
[12]
Reissner, E. (1984) On a Certain Mixed Variational Theorem and a Proposed Application. International Journal for Numerical Methods in Engineering, 20, 1366-1368. https://doi.org/10.1002/nme.1620200714
[13]
Reissner, E. (1986) On a Mixed Variational Theorem and on Shear Deformable Plate Theory. International Journal for Numerical Methods in Engineering, 23, 193-198.
https://doi.org/10.1002/nme.1620230203
[14]
Carrera, E. (1998) Evaluation of Layerwise Mixed Theories for Laminated Plates Analysis. AIAA Journal, 36, 830-839. https://doi.org/10.2514/2.444
[15]
Carrera, E., Brischetto, S. and Nali, P. (2011) Plates and Shells for Smart Structures: Classical and Advanced Theories for Modeling and Analysis. John Wiley & Sons, New York. https://doi.org/10.1002/9781119950004
[16]
Carrera, E. (2000) A Priori vs. a Posteriori Evaluation of Transverse Stresses in Multilayered Orthotropic Plates. Composite Structures, 48, 245-260.
https://doi.org/10.1016/S0263-8223(99)00112-9
[17]
Carrera, E. (2002) Theories and Finite Elements for Multilayered, Anisotropic, Composite Plates and Shells. Archives of Computational Methods in Engineering, 9, 87-140. https://doi.org/10.1007/BF02736649
[18]
Lage, R.G., Soares, C.M., Soares, C.M. and Reddy, J. (2004) Layerwise Partial Mixed Finite Element Analysis of Magneto-Electro-Elastic Plates. Computers & Structures, 82, 1293-1301. https://doi.org/10.1016/j.compstruc.2004.03.026
[19]
Lage, R.G., Soares, C.M., Soares, C.M. and Reddy, J. (2004) Modelling of Piezolaminated Plates Using Layerwise Mixed Finite Elements. Computers & Structures, 82, 1849-1863. https://doi.org/10.1016/j.compstruc.2004.03.068
[20]
Bathe, K. (1996) Finite Element Procedures. Prentice Hall, Englewood Cliffs.
[21]
Bathe, K.J. (2001) The Inf-Sup Condition and Its Evaluation for Mixed Finite Element Methods. Computers & Structures, 79, 243-252.
https://doi.org/10.1016/S0045-7949(00)00123-1
[22]
Pontaza, J. and Reddy, J. (2004) Mixed Plate Bending Elements Based on Least- Squares Formulation. International Journal for Numerical Methods in Engineering, 60, 891-922. https://doi.org/10.1002/nme.991
[23]
Pontaza, J.P. (2005) Least-Squares Variational Principles and the Finite Element Method: Theory, Formulations, and Models for Solid and Fluid Mechanics. Finite Elements in Analysis and Design, 41, 703-728.
https://doi.org/10.1016/j.finel.2004.09.002
[24]
Moleiro, F., Mota Soares, C.M., Mota Soares, C.A. and Reddy, J.N. (2010) Layerwise Mixed Least-Squares Finite Element Models for Static and Free Vibration Analysis of Multilayered Composite Plates. Composite Structures, 92, 2328-2338.
https://doi.org/10.1016/j.compstruct.2009.07.005
[25]
Moleiro, F., Soares, C.M., Soares, C.M. and Reddy, J. (2008) Mixed Least-Squares Finite Element Model for the Static Analysis of Laminated Composite Plates. Computers & Structures, 86, 826-838. https://doi.org/10.1016/j.compstruc.2007.04.012
[26]
Vel, S.S. and Batra, R. (2002) Exact Solution for Thermoelastic Deformations of Functionally Graded Thick Rectangular Plates. AIAA Journal, 40, 1421-1433.
https://doi.org/10.2514/2.1805
[27]
Demasi, L. (2012) Partially Zig-Zag Advanced Higher Order Shear Deformation Theories Based on the Generalized Unified Formulation. Composite Structures, 94, 363-375. https://doi.org/10.1016/j.compstruct.2011.07.022
[28]
Demasi, L. (2007) Three-Dimensional Closed Form Solutions and Exact Thin Plate Theories for Isotropic Plates. Composite Structures, 80, 183-195.
https://doi.org/10.1016/j.compstruct.2006.04.073
[29]
Moleiro, F., Mota Soares, C.M. and Carrera, E. (2019) Three-Dimensional Exact Hygro-Thermo-Elastic Solutions for Multilayered Plates: Composite Laminates, Fibre Metal Laminates and Sandwich Plates. Composite Structures, 216, 260-278.
https://doi.org/10.1016/j.compstruct.2019.02.071
[30]
Moleiro, F., Carrera, E., Li, G., Cinefra, M. and Reddy, J.N. (2019) Hygro-Thermo- Mechanical Modelling of Multilayered Plates: Hybrid Composite Laminates, Fibre Metal Laminates and Sandwich Plates. Composites Part B: Engineering, 177, Article ID: 107388. https://doi.org/10.1016/j.compositesb.2019.107388
[31]
Mathew, C. and Ejiofor, O. (2023) Mechanics and Computational Homogenization of Effective Material Properties of Functionally Graded (Composite) Material Plate FGM. International Journal of Scientific and Research Publications, 13, 128-150.
https://doi.org/10.29322/IJSRP.13.09.2023.p14120
