×

A mathematical approach to solving an inverse thermoelastic problem in a thin elliptic plate. (English) Zbl 1425.35234

Summary: This article investigates the inverse thermoelasticity of an elliptical plate for determining the temperature distribution and its associated thermal stresses by mean of integral transform techniques. Furthermore, by considering a circle as a special kind of ellipse, it is seen that the temperature distribution and history in a circular solution can be drawn as a special case of the present mathematical solution. The numerical results obtained using these computational tools are accurate enough for practical purposes.

MSC:

35R30 Inverse problems for PDEs
35Q74 PDEs in connection with mechanics of deformable solids
74K20 Plates
74F05 Thermal effects in solid mechanics
44A10 Laplace transform
PDFBibTeX XMLCite
Full Text: Link

References:

[1] G. B. Jeffery, Plain stress and plane strain in bipolar co-ordinates, Philosophical Transactions Royal Society, 221 (1920) 265 - 293.
[2] S. Ghosh, On the solution of the equations of elastic equilibrium suitable for elliptic boundaries, Transactions of the American Mathematical Society 32(1) (1930) 47 - 60. · JFM 56.0688.05
[3] N. W. McLachlan, Theory and Application of Mathieu function, Clarendon press, Oxford, 1947. · Zbl 0029.02901
[4] E. T. Kirkpatric, W. F. Stokey, Transient heat conduction in elliptical plate and cylinders, Trans. ASME J. Heat Transfer 81 (1959) 54 - 60.
[5] R. K. Gupta, A finite transform involving Mathieu functions and its application, Proc. Net. Inst. Sc., India, Part A 30(6)(1964) 779 - 795. · Zbl 0161.10001
[6] N. K. Choubey, Heat conduction in a hollow elliptic cylinder with radiation, National Heat and Mass Transfer Conference, 4th, Roorkee, India, November 21-23, 1977, Proceedings. (A79-13576 03-34) Meerut, India, Sari
[7] Y. Sugano, Y. Kondoh, H. Yano, An analytical for a plane thermal stress problem expressed in elliptical coordinates, JSME, 56 (1990) 78 - 83.
[8] F. Erdo˘gdu, M. O. Balaban, K. V. Chau, Modeling of Heat conduction in elliptical cross section: I. Development and testing of the model, J. of Food Engineering 38 (1998a) 223 - 229.
[9] F. Erdo˘gdu, M. O. Balaban, K. V. Chau, Modeling of Heat conduction in elliptical cross section: II. Adaptation to thermal processing of shrimp, J. of Food Engineering 38 (1998b) 223 - 229.
[10] A. R. El Dhaba, A. F. Ghaleb, M. S. Abou-Dina, A problem of plane, uncoupled linear thermoelasticity for an infinite, elliptical cylinder by a boundary integral method, Journal of Thermal Stresses 26(2) (2003) 93 - 121.
[11] K. Sato, Heat conduction in an infinite elliptical cylinder during heating or cooling, Proceedings of the 55t hJapan National congress on Theoretical and Applied Mechanics 55 (2006) 157 - 158 .
[12] E. Marchi, G. Zgrablich, Heat conduction in hollow cylinders with radiation 14(II) (1964) 159 - 164. · Zbl 0168.36305
[13] P. Bhad, V. Varghese, L. Khalsa, Transient thermoelastic problem in a confocal elliptical disc with internal heat sources, Advances in Mathematical Sciences and Applications, Tokyo, Japan 25(1) (2016) 43 - 61. · Zbl 1487.74024
[14] P. Bhad, V. Varghese, L. Khalsa, Heat source problem of thermoelasticity in an elliptic plate with thermal bending moments, Journal of Thermal Stresses 40(1) (2017) 96 - 107.
[15] P. Bhad, V. Varghese, L. Khalsa, Thermoelastic-induced vibrations on an elliptical disk with internal heat sources, Journal of Thermal Stresses 40(4) (2017) 502 - 516.
[16] P. Bhad, V. Varghese, L. Khalsa, A modified approach for the thermoelastic large deflection in the elliptical plate, Archive of Applied Mechanics 87(4) (2017) 767 - 781.
[17] T. Dhakate, V. Varghese, L. Khalsa, Integral transform approach for solving dynamic thermal vibrations in the elliptical disk, Journal of Thermal Stresses 40(9) (2017) 1093 - 1110.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.