A Three-Dimensional Finite Element Analysis of A Two-Axially Pre-Loaded Plate Exposed To A Dynamic Force

Abstract

In this paper, the forced vibration analysis by a harmonically time-dependent force of an elastic plate covered rigidly by a rigid half-plane is given. The plate layer is subjected to bi-axial normal initial force, into lateral sides separately. Here, the preloading state is exactly static and homogeneous. To eliminate the disadvantage of such a nonlinear model, the problem formulation is modeled in terms of the fundamental consideration of the theory of linearized wave in elastic solids under a pre-loaded state (TLWESPS) in a plane-stress case. For this purpose, considering Hamilton’s principles, the system of the partial equations of motion and the boundary-contact conditions are found. Based on the virtual work and the fundamental theorem of the calculus of variation, the three-dimensional finite element method (3D-FEM) is used to understand the dynamic behavior of the plate. A numerical validation process is established based on error norm functions. Next, influences of certain problem parameters such as Young’s modulus, aspect ratio, thickness ratio, pre-loaded parameter, etc. on the frequency mode of the pre-stressed system are given. The numerical investigations show that higher values of Poisson's ratio promote the resonant mode of the plate while increasing the influence of the preloaded parameter on the dynamic response of the plate.

Keywords

• Forced Vibration
• Frequency Response
• Initial Stress
• Finite Element Method
• Resonance Value

References

1. [1] Barber, J. R., “Elasticity”, Dordrecht: Kluwer Academic Publishers, (2002).
2. [2] Lurie, A. I., “Theory of Elasticity”, Springer Science & Business Media, (2010).
3. [3] Southwell, R. V. V., “On the general theory of elastic stability”, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 213(497-508): 187-244, (1914).
4. [4] Biezeno, C. B., and Hencky, H. “On the general theory of elastic stability”, In: Proceedings Koninklijke Nederlandse Akademie van Wetenschappen, 31: 569-592, (1928).
5. [5] Biot, M. A. “XLIII. Non-linear theory of elasticity and the linearized case for a body under initial stress”, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 27(183): 468-489, (1939).
6. [6] Neuber, H., “Die Grundgleichungen der elastischen Stabilität in allgemeinen Koordinaten und ihre Integration”, ZAMM‐Journal of Applied Mathematics and Mechanics, 23(6): 321-330, (1943).
7. [7] Trefftz, E., “Zur theorie der stabilität des elastischen gleichgewichts”, ZAMM‐Journal of Applied Mathematics and Mechanics, 13(2): 160-165, (1933).
8. [8] Green, A. E., Rivlin, R. S., and Shield, R. T., “General theory of small elastic deformations superposed on finite elastic deformations”, Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences, 211(1104): 128-154, (1952).

9. [9] Guz, A. N., “Three-dimensional theory of elastic stability under finite subcritical deformations”, Soviet Applied Mechanics, 8(12): 1308-1323, (1972).
10. [10] Zubov, L. M., “Theory of small deformations of prestressed thin shells: PMM”, Journal of Applied Mathematics and Mechanics, 40(1): 73-82, (1976).
11. [11] Tiersten, H. F., “Perturbation theory for linear electroelastic equations for small fields superposed on a bias”, The Journal of the Acoustical Society of America, 64(3): 832-837, (1978).
12. [12] Ogden, R. W., “Nonlinear Elastic Deformations”, Ellis Horwood/Halsted Press, New York, (1984).
13. [13] Akbarov, S., and Guz, A. N., “Mechanics of Curved Composites (Vol. 78)”, Springer Science & Business Media (2000).
14. [14] Reddy, J. N., “Mechanics of Laminated Composite Plates and Shells: Theory and Analysis”, CRC press, (2003).
15. [15] Guz, A. N., “Fundamentals of the 3D Theory of Stability of Deformable Bodies”, Springer, New York, (1999). [Translated from Russian by M. Kashtalian.]
16. [16] Akbarov, S. D., “Dynamics of Pre-strained Bi-material Elastic Systems: Linearized Three-dimensional Approach”, Springer, (2015).
17. [17] Kepceler, T., “Torsional wave dispersion relations in a pre-stressed bi-material compounded cylinder with an imperfect interface”, Applied Mathematical Modelling, 34(12): 4058-4073, (2010).
18. [18] Zamanov, A., and Agasiyev, E., “Dispersion of lamb waves in a three-layer plate made from compressible materials with finite initial deformations”, Mechanics of Composite Materials, 46(6): 583-592, (2011).
19. [19] Eröz, M., “The stress field problem for a pre-stressed plate-strip with finite length under the action of arbitrary time-harmonic forces”, Applied Mathematical Modelling, 36(11): 5283-5292, (2012).
20. [20] Akbarov, S. D., Hazar, E., and Eröz, M., “Forced vibration of the pre-stressed and imperfectly bonded bi-layered plate strip resting on a rigid foundation”, Computers, Materials and Continua, 36(1): 23-48, (2013).
21. [21] Hu, W. T., Xia, T. D., and Chen, W. Y., “Influence of lateral initial pressure on axisymmetric wave propagation in hollow cylinder based on first power hypo-elastic model”, Journal of Central South University, 21(2): 753-760, (2014).
22. [22] Ilhan, N., and Koc, N., “Influence of polled direction on the stress distribution in piezoelectric materials”, Structural Engineering and Mechanics: An International Journal, 54(5): 955-971, (2015).
23. [23] Kurt, I., Akbarov, S. D., and Sezer, S., “Effect of uniaxial initial stresses, piezoelectricity and third order elastic constants on the near-surface waves in a stratified half-plane”, Journal of Thermal Engineering, 3(4): 1346-1357, (2016).
24. [24] Yeşil, U. B., “Forced and natural vibrations of an orthotropic pre-stressed rectangular plate with neighboring two cylindrical cavities”, Comput. Mater. Continua, 53(1): 1-22, (2017).
25. [25] Daşdemir, A., “A mathematical model for forced vibration of pre-stressed piezoelectric plate-strip resting on rigid foundation”, MATEMATIKA: Malaysian Journal of Industrial and Applied Mathematics, 34(2): 419-431, (2018).
26. [26] Daşdemir, A., “Dynamic response of a bi-axially pre-stressed bi-layered plate resting on a rigid foundation under a harmonic force”, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 234(3): 784-795, (2020).
27. [27] Selim, M. M., “Dispersion relation for transverse waves in pre-stressed irregular single-walled carbon nanotubes”, Physica Scripta, 95(11): 115218, (2020).
28. [28] Babych, S. Y., and Glukhov, Y. P., “On one dynamic problem for a multilayer half-space with initial stresses”, International Applied Mechanics, 57(1): 43-52, (2021).
29. [29] Mandi, A., Kundu, S., and Pal, P. C.,"Surface wave scattering analysis in an initially stressed stratified media." Engineering Computations, 38(8): 3153-3173, (2021).
30. [30] Bagirov, E. T., “On the influence of the inhomogeneous residual stresses on the dispersion of axisymmetric longitudinal waves in the hollow cylinder”, Mechanics, 42(8): 25-35, (2022).
31. [31] Kumawat, S., Praharaj, S., and Vishwakarma, S. K., “Dispersion of torsional surface waves in a threefold concentric compounded cylinder with imperfect interface”, Waves in Random and Complex Media, 1-26, (2022).
32. [32] Daşdemir, A., “Effect of interaction between polarization direction and inclined force on the dynamic stability of a pre-stressed piezoelectric plate”, Mechanics Research Communications, 131: 104150, (2023).
33. [33] Veliyev, Q. J., and Ipek, C., “The Influence of the Material Properties of an Inhomogeneous Pre-Stressed Hollow Cylinder Containing an Inviscid Fluid on the Dispersion of Quasi-Scholte Waves”, International Applied Mechanics, 59(5): 619-629, (2023).
34. [34] Dehghanian, Z., Fallah, F., and Farrahi, G. H., “Wave propagation analysis in pre-stressed incompressible hyperelastic multi-layered plates using a plate theory”, European Journal of Mechanics-A/Solids 103: 105141, (2024).
35. [35] Daşdemir, A., and Eröz, M., “Forced vibration of a bi-axially pre-stressed plate subjected to a harmonic point force and resting on a rigid foundation”, Transactions of the Canadian Society for Mechanical Engineering, 43(3): 333-343, (2019).
36. [36] Reddy, J. N., “Energy Principles and Variational Methods in Applied Mechanics”, New York, John Wiley & Sons, (2017).
37. [37] Chandrupatla, T., and Belegundu, A., “Introduction to Finite Elements in Engineering”, Third Edition, Prentice Hall, Upper Saddle River, NJ, (2002).