Probabilistic Evaluation on the Free Vibration of Functionally Graded Material Plates Using 3D Solution and Meta-Model Methods
Authors
Abstract
This paper presents a probabilistic assessment on the free vibration analysis of functionally graded material plates, including layers with magneto-electro-elastic properties, using the 3D solution and surrogate models. The plate is located on an elastic foundation and the intra-layer slipping effect is also considered in the analysis by employing the generalized intra-layer spring model. Due to the high computational cost of the 3D solution in calculating the free vibration frequency of the plate, surrogate models are used. The meta models including kriging method, radial fundamental function method and polynomial response surface method are used to construct the surrogate model. For surrogate models training, the results of the three-dimensional solving method are used. The elastic foundation hardness, the intra-layer slipping effect, the material properties index, and the layer densities are considered as the variables with uncertainty. The three-dimensional solution method is validated through a comparison with other available reference. The results obtained through the surrogate models have been compared to those of the 3D solution formulation, showing a good agreement. The effects of some parameters including the elastic foundation hardness, the intra-layer slipping effect, the density of each layer, and the material properties index on the fundamental frequency of functionally graded material plates are investigated. By using three-dimensional solution method and Kriging Surrogate Model, it is shown that the shear and transverse components of elastic foundation hardness and the density of each layer have the greatest effect on the fundamental frequency of the functionally graded material plates.
Keywords
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2. Nan, C. W., “Magnetoelectric Effect in Composites of Piezoelectric and Piezomagnetic Phases”, Physics Review B, Vol. 50, pp. 6082–6088, 1994.
3. Benviste, Y., “Magnetoelectric Effect in fibrous Composites with Piezoelectric and Piezomagnetic Phases”, Physics Review B, Vol. 51, pp. 16424–16427, 1995.
4. Pan, E., “Exact Solution for Simply Supported and Multilayered Magneto- Electro- Elastic Plates”, Journal of Applied Mechanics, Vol. 68, pp. 608–618, 2001.
5. Pan, E., and Heyliger, P. R., “Exact Solutions for Magnetoelectroelastic Laminates in Cylindrical Bending”, International Journal of Solids and Structures, Vol. 40, pp. 6859–6876, 2003.
6. Pan, E., and Heyliger, P. R., “Free Vibration of Simply- Supported and Multilayered Magneto- Electro- Elastic Plates”, Journal of Sound and Vibration, Vol. 252, pp. 429–442, 2002.
7. Pan, E., and Han, F., “Exact Solution for Functionally Graded and Layered-Magneto- Electro- Elastic Plates”, International Journal of Engineering Science, Vol. 43, No. 3, pp. 321- 339, 2005.
8. Cheraghi, N., and Lazgy Nazargah, M., “An Exact Bending Solution for Functionally Graded Magneto-Electro-Elastic Plates Resting on Elastic Foundations with Considering Interfacial Imperfections”, Modares Mechanical Engineering; Vol. 15, No. 12, pp. 346–356, 2016 (in persian).
9. Jiangyi, C., Hualing, C., and Ernian, P., “Free Vibration of Functionally Graded, Magneto- Electro- Elastic and Multilayered Plates”, Acta Mechanica Solida Sinica, Vol. 19, No. 2, pp. 160– 166, 2006.
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11. Lezgy- Nazargah, M., “A Three- Dimensional Peano Series Solution for the Vibration of Functionally Graded Piezoelectric Laminates in Cylindrical Bending”, Scientia Iranica , Vol. 23, No. 3, pp. 788– 801, 2016.
12. Lezgy- Nazargah, M., “Fully Coupled Thermo- Mechanical Analysis of Bi- Directional FGM Beams Using NURBS Isogeometric Finite Element Approach”, Aerospace Science and Technology, Vol. 45, pp. 154–164, 2015.
13. Vafakhah, Z., and Navayi Neya, B., “An Exact Three Dimensional Solution for Bending of Thick Rectangular FGM Plate”, Composites Part B: Engineering Vol. 156, No. 1, pp. 72– 87, 2019.
14. Hong, C.C., “GDQ Computation for Thermal Vibration of Thick FGM Plates by Using Fully Homogeneous Equation and TSDT”, Thin- Walled Structures, Vol. 135, pp. 78– 88, 2019.
15. Jha, D.K., Tarun Kant, and Singh, R.K., “A Critical Review of Recent Research on Functionally Graded Plates”, Composite Structures, Vol. 96, pp. 833–849, 2013.
16. Yang, J., Liew, K.M., and Kitipornchai, S., “Second- Order Statistics of The Elastic Buckling of Functionally Graded Rectangular Plates”, Composites Science and Technology, Vol. 65, pp. 1165–1175, 2005.
17. Damásio, F.R., Silva, T.A.N., Carvalho, A., and Loja, M.A.R., “on The Characterization of Parametric Uncertainty on FGM Plates”, 10th International Conference on Composite Science and Technology, Lisboa, Portugal, 2015.
18. Carvalho, A., Silva, T., Ramos Loja, M.A., and Damgsio, F.R., “Assessing the Influence of Material and Geometrical Uncertainty on The Mechanical Behavior of Functionally Graded Material Plates”. Mechanics of Advanced Materials and Structures, Vol. 24, No. 5, pp. 417- 426, 2017.
19. García- Macías, E., Castro- Triguero, R.I., Friswell, M., Adhikari, S., and Saez, A., “Metamodel- Based Approach for Stochastic Free Vibration Analysis of Functionally Graded Carbon Nanotube Reinforced Plates”, Composite Structures, Vol. 152, pp. 183– 198, 2016.
20. Hosseini, S.M., and Shahabian, F., “Reliability of Stress Field In Al–Al2O3 Functionally Graded Thick Hollow Cylinder Subjected to Sudden Unloading, Considering Uncertain Mechanical Properties”. Materials & Design, Vol. 31, No. 8, pp. 3748– 3760, 2010.
21. Noh, Y.J., Kang, Y.J., Youn, S.J., Cho, J.R., and Lim, O.K., “Reliability- Based Design Optimization of Volume Fraction Distribution in Functionally Graded Composites”, Computational Materials Science, Vol. 69, pp. 435– 442, 2013.
22. Shaker, A., Wael, A., Tawfik, M., and Sadek, E., “Stochastic Finite Element Analysis of the Free Vibration of Functionally Graded Material Plates”, Computational Mechanics,Vol. 41, No. 5, pp. 707– 714, 2008.
23. Koehler, J. R., Owen, A. B., “9 Computer experiments”, Handbook of Statistics Elsevier, Vol. 13, pp. 261–308, 1996.
24. Mitchell, T.J., and Morris, M.D., “Bayesian Design and Analysis of Computer Experiments: Two Examples”, Statistica Sinica, Vol. 2, pp. 359–79, 1992.
25. Fang, H., and Horstemeyer, M., “Global Response Approximation with Radial Basis Functions”, Engineering Optimization, Vol. 38, No. 4, pp. 407–424, 2006.
26. Box, GEP., and Wilson KB, “The Exploration and Exploitation of Response Surfaces: Some General Considerations and Examples”, Biometrics, Vol. 10, pp. 16–60, 1954.
27. Das, P.K., and Zheng Y., “Improved Response Surface Method and ikts Application to Stiffened Plate Reliability Analysis”, Engineering Structures, Vol. 22, No. 5, pp. 544–51, 2000.
28. Roussouly, N., Petitjean, F., and Salaun, M., “A New Adaptive Response Surface Method for Reliability Analysis”, Probabilistic Engineering Mechanics, Vol. 32, pp. 103–15, 2013.
29. Khuri, A.I., and Mukhopadhyay, S., “Response Surface Methodology”, WIREs Computional Statistics, Vol. 2, pp.128–49, 2010.
30. Gaxiola-Camacho, J. R., Haldar, A., Azizsoltani, H., Valenzuela-Beltran, F., and Reyes-Salazar, A., “Performance-Based Seismic Design of Steel Buildings Using Rigidities of Connections”, ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering, Vol. 4, No. 1, pp. 1–14, 2018.
31. Metropolis, N., and Ulam, S., “The Monte Carlo Method”, Journal of the American Statistical Association, Vol. 44, pp. 335–41, 1949.
32. Nowak, A.S., and Collins, K.R., Reliability of Structures, McGraw- Hill, New York, 2000.
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