### Stress Waves in a Generalized Thermo Elastic Polygonal Plate of Inner and Outer Cross Sections

Document Type : Research Paper

Author

Department of Mathematics, Karunya University, Coimbatore-641 114, Tamil Nadu, India

Abstract

The stress wave propagation in a generalized thermoelastic polygonal plate of inner and outer cross sections is studied using the Fourier expansion collocation method. The wave equation of motion based on two-dimensional theory of elasticity is applied under the plane strain assumption of generalized thermoelastic plate of polygonal shape, composed of homogeneous isotropic material.  The frequency equations are obtained by satisfying the irregular boundary conditions along the inner and outer surface of the polygonal plate. The computed non-dimensional wave number and wave velocity of triangular, square, pentagonal and hexagonal plates are given by dispersion curves for longitudinal and flexural antisymmetric modes of vibrations. The roots of the frequency equation are obtained by using the secant method, applicable for complex roots.

Keywords

[1] Nagaya K., 1981, Simplified method for solving problems of plates of doubly connected arbitrary shape, Part I: Derivation of the frequency equation, Journal of Sound and Vibration 74(4): 543-551.
[2] Nagaya K., 1981, Simplified method for solving problems of plates of doubly connected arbitrary shape, Part II: Applications and experiments, Journal of Sound and Vibration 74(4): 553-564.
[3] Nagaya K., 1981, Dispersion of elastic waves in bar with polygonal cross-section, Journal of Acoustical Society of America 70(3): 763-770.
[4] Nagaya K., 1983, Vibration of a thick walled pipe or ring of arbitrary shape in its Plane, Journal of Applied Mechanics 50: 757-764.
[5] Nagaya K., 1983, Vibration of a thick polygonal ring in its plane, Journal of Acoustical Society of America 74(5): 1441-1447.
[6] Lord H.W., Shulman Y., 1967, A generalized dynamical theory of thermo elasticity, Journal of Mechanics of Physics of Solids 5: 299-309.
[7] Dhaliwal R.S., Sherief H.H., 1980, Generalized thermo elasticity for anisotropic media, Quartely Applied Mathematics 8(1): 1-8.
[8] Green A.E., Laws N., 1972, On the entropy production inequality, Archive of Rational Mechanical Analysis 45: 47-53.
[9] Green A.E., Lindsay K.A.,1972,Thermo elasticity, Journal of Elasticity 2: 1-7.
[10] Suhubi E.S., 1964, Longitudinal vibrations of a circular cylindrical coupled with a thermal field, Journal of Mechanics of Physics of Solids 12: 69-75.
[11] Erbay E.S., Suhubi E.S., 1986, Longitudinal wave propagation of thermoelastic cylinder, Journal of Thermal Stresses 9: 279-295.
[12] Sharma J.N., Sharma P.K., 2002, Free vibration analysis of homogeneous transversely isotropic thermoelastic cylindrical panel, Journal of Thermal Stresses 25: 169-182.
[13] Sharma J.N., Kumar R., 2004, Asymptotic of wave motion in transversely isotropic plates, Journal of Sound and Vibration 274: 747-759.
[14] Ashida F., Tauchert T.R., 2001, A general plane-stress solution in cylindrical coordinates for a piezoelectric plate, International Journal of Solids and Structures 30: 4969-4985.
[15] Ashida F.,2003,Thermally-induced wave propagation in piezoelectric plate, Acta Mechanica 161: 1-16.
[16] Tso Y.K., Hansen C.H., 1995, Wave propagation through cylinder/plate junctions, Journal of Sound and Vibration 186(3): 447-461.
[17] Heyliger P.R., Ramirez G., 2000, Free vibration of Laminated circular piezoelectric plates and disc, Journal of Sound and Vibration 229(4): 935-956.
[18] Gaikward M.K., Deshmukh K.C., 2005, Thermal deflection of an inverse thermoelastic problem in a thin isotropic circular plate, Journal of Applied Mathematical Modelling 29: 797-804.
[19] Varma K.L., 2002, On the propagation of waves in layered anisotropic media in generalized thermo elasticity, International Journal of Engineering Sciences 40: 2077-2096.
[20] Ponnusamy P., 2007, Wave propagation in a generalized thermoelastic solid cylinder of arbitrary cross- section, International Journal of Solids and Structures 44: 5336-5348.
[21] Ponnusamy P., 2013, Wave propagation in a piezoelectric solid bar of circular cross-section immersed in fluid, International Journal of Pressure Vessels and Piping 105: 12-18.
[22] Jiangong Y., Bin W., Cunfu H., 2010, Circumferential thermoelastic waves in orthotropic cylindrical curved plates without energy dissipation, Ultrosonics 53(3):416-423.
[23] Jiangong Y., Tonglong X., 2010, Generalized thermoelastici waves in spherical curved plates without energy dissipation, Acta Mechanica 212: 39-50.
[24] Jiangong Y., Xiaoming Zh., Tonglong X., 2010, Generalized thermoelastici waves in functionally graded plates without energy dissipation, Composite Structures 93(1): 32-39.
[25] Kumar R. , Chawla V., Abbas I.A., 2012, Effect of viscosity on wave propagation in anisotropic thermoelastic medium with three-phase-lag model, Theoretical and Applied Mechanics 39(4): 313-341.
[26] Kumar R., Abbas I. A., 2014, Response of thermal source in initially stressed generalized thermoelastic half-space with voids, Journal of Computational and Theoretical Nanoscience 11: 1-8.
[27] Kumar R., Abbas I. A., Marin M., 2015, Analytical numerical solution of thermoelastic interactions in a semi-infinite medium with one relaxation time, Journal of Computational and Theoretical Nanoscience 12: 1-5.
[28] Ponnusamy P., Selvamani R., 2012, Dispersion analysis of generalized magneto-thermoelastic waves in a transversely isotropic cylindrical panel, Journal of Thermal Stresses 35: 1119-1142.
[29] Ponnusamy P., Selvamani R., 2013, Wave propagation in magneto thermo elastic cylindrical panel, European Journal of Mechanics-A solids 39: 76-85.
[30] Selvamani R., Ponnusamy P., 2013, Wave propagation in a generalized thermo elastic plate immersed in fluid, Structural Engineering and Mechanics 46(6): 827-842.
[31] Selvamani R., Ponnusamy P., 2014, Dynamic response of a solid bar of cardioidal cross-sections immersed in an inviscid fluid, Applied Mathematics and Information Sciences 8(6): 2909-2919.
[32] Mirsky I., 1964, Wave propagation in a transversely isotropic circular cylinders, Part I: Theory, Part II: Numerical results, Journal of Acoustical Society of America 37(6): 1016-1026.
[33] Antia H.M., 2002, Numerical Methods for Scientists and Engineers, Hindustan Book Agency, New Delhi.