Generalized Thermoelastic Functionally Graded Spherically Isotropic Solid Containing a Spherical Cavity Under Thermal Shock

M. K. Ghosh; M. Kanoria

Volume 29 Issue 10

Turn off MathJax

Article Contents

Abstract

References

Article Navigation > Applied Mathematics and Mechanics > 2008 > 29(10): 1147-1160

M. K. Ghosh, M. Kanoria. Generalized Thermoelastic Functionally Graded Spherically Isotropic Solid Containing a Spherical Cavity Under Thermal Shock[J]. Applied Mathematics and Mechanics, 2008, 29(10): 1147-1160.

Citation:

PDF( 574 KB)

Generalized Thermoelastic Functionally Graded Spherically Isotropic Solid Containing a Spherical Cavity Under Thermal Shock

M. K. Ghosh¹,
M. Kanoria²

1.
Department of Mathematics, Serampore College, Serampore, Hooghly-712 201, India;

Received Date: 2008-02-13
Rev Recd Date: 2008-06-30
Publish Date: 2008-10-15

Abstract

Abstract

The determination of thermoelastic displacement,stresses and temperature in a functionally graded spherically isotropic infinite elastic medium having a spherical cavity in the context of the linear theory of generalized thermoelasticity with two relaxation time parameters(Green and Lind-say theory)are concerned with.The surface of the cavity is stress free and is subjected to a time dependent thermal shock.The basic equations were written in the form of a vecto-rmatrix differential equation in the Laplace transform domain which was then solved by eigenvalue approach.The numerical inversion of the transforms was carried out using Bellman method.The displacement,stresses and temperature were computed and presented graphically.It is found that the variation of thermo-physical properties of a material strongly influences the response to loading.A comparative study with the corresponding homogeneous material has also been made.
- generalized thermoelasticity,
- functionally graded material(FGM),
- Green-Lindsay theory,
- vector-matrix differential equation,
- Bellman method

FullText(HTML)

References(36)

References

[1]	Biot M A. Thermoelasticity and irreversible thermodynamics[J].J Appl Phys,1956,27(3):240-253. doi: 10.1063/1.1722351
[2]	Chadwick P.Thermoelasticity,the Dynamical Theory[M].In:Hill R,Sneddon I N,Eds.Progress in Solid Mechanics.Vol 1.Amsterdam: North Holland,1960.
[3]	Lord H，Shulman Y.A generalized dynamical theory of thermoelasticity[J].Mech Phys Solid,1967,15(5):299-309. doi: 10.1016/0022-5096(67)90024-5
[4]	Green A E，Lindsay K A. Thermoelasticity[J].J Elast,1972,2(1):1-7. doi: 10.1007/BF00045689
[5]	Tzou D Y. Experimental support for the lagging behavior in heat propagation[J].J Thermophys Heat Transf,1995,9(4):686-693. doi: 10.2514/3.725
[6]	Mitra K, Kumar S，Vedaverg A. Experimental evidence of hyperbolic heat conduction in processed meat[J].J Heat Transfer,ASME,1995,117(3):568-573. doi: 10.1115/1.2822615
[7]	Chandrasekharaiah D S. Thermoelasticity with second sound, a review[J].Appl Mech Rev,1986,39(3):355-375. doi: 10.1115/1.3143705
[8]	Bahar L, Hetnarski R. State space approach to thermoelasticity[J].J Thermal Stresses,1978,1(1):135-145. doi: 10.1080/01495737808926936
[9]	Ezzat M. Fundamental solution in thermoelasticity with two relaxation times for cylindrical regions[J].Internat J Engrg Sci,1995,33(14):2011-2020. doi: 10.1016/0020-7225(95)00050-8
[10]	Hetnarski R B，Ignaczak J.Generalized thermoelasticity response of semi-space to a short laser pulse[J].J Thermal Stresses,1994,17(3):377-396. doi: 10.1080/01495739408946267
[11]	Bagri A, Eslami M R.Generalized coupled thermoelasticity of disks based on the Lord-Shulman model[J].J Thermal Stresses,2004,27(8):691-704. doi: 10.1080/01495730490440127
[12]	Kar A, Kanoria M.Thermo-elastic interaction with energy dissipation in an unbounded body with a spherical hole[J].International Journal of Solids and Structures,2007,44(9):2961-2971. doi: 10.1016/j.ijsolstr.2006.08.030
[13]	Das N C，Lahiri A. Thermoelastic interactions due to prescribed pressure inside a spherical cavity in an unbounded medium[J].Ind J Pure Appl Math,2000,31(1):19-32.
[14]	Kar A, Kanoria M. Thermoelastic interaction with energy dissipation in a transversely isotropic thin circular disc[J].European Journal of Mechanics, A/Solids,2007，26（6）：969-981. doi: 10.1016/j.euromechsol.2007.03.001
[15]	Ghosh M K，Kanoria M. Generalized thermoelastic problem of a spherically isotropic infinite elastic medium containing a spherical cavity[J].J Thermal Stresses,2008,31(8):665-679. doi: 10.1080/01495730802193872
[16]	Aboudi J, Pindera M J，Arnold S M. Thermo-inelastic response of functionally graded composites[J].International Journal of Solids and Structures,1995, 32(12):1675-1710. doi: 10.1016/0020-7683(94)00201-7
[17]	Wetherhold R C，Seelman S, WANG Jian-zhong. The use of functionally graded materials to eliminate or control thermal deformation[J].Composites Science and Technology,1996,56(9):1099-1104. doi: 10.1016/0266-3538(96)00075-9
[18]	Sugano Y. An expression for transient thermal stress in a nonhomogeneous plate with temperature variation through thickness[J].Ingenieur Archiv,1987,57(2):147-156. doi: 10.1007/BF00541388
[19]	Qian L F， Batra R C. Transient thermoelastic deformations of a thick functionally graded plate[J].J Thermal Stresses,2004,27(8):705-740. doi: 10.1080/01495730490440145
[20]	Lutz M P，Zimmerman R W. Thermal stresses and effective thermal expansion coefficient of a functionally graded sphere[J].J Thermal Stresses,1996,19(1): 39-54. doi: 10.1080/01495739608946159
[21]	Ye G R, Chen W Q，Cai J B.A uniformly heated functionally graded cylindrical shell with transverse isotropy[J].Mechanics Research Communication,2001,28(5):535-542. doi: 10.1016/S0093-6413(01)00206-3
[22]	Chen W Q, Wang X，Ding H J. Free vibration of a fluid-filled hollow sphere of a functionally graded material with spherical isotropy[J].Journal of the Acoustical Society of America,1999,106(5):2588-2594. doi: 10.1121/1.428090
[23]	Ding H J, Wang H M，Chen W Q.Analytical thermo-elastodynamic solutions for a nonhomogeneous transversely isotropic hollow sphere[J].Archive of Applied Mechanics,2002,72(8):545-553. doi: 10.1007/s00419-002-0225-x
[24]	Chen W Q, Ding H J，Wang X. The exact elasto-electric field of a rotating piezoceramic spherical shell with a functionally graded property[J].International Journal of Solids and Structures,2001,38(38/39):7015-7027. doi: 10.1016/S0020-7683(00)00394-2
[25]	Wang B L，Mai Y W. Transient one dimensional heat conduction problems solved by finite element[J].International Journal of Mechanical Sciences,2005,47(2):303-317. doi: 10.1016/j.ijmecsci.2004.11.001
[26]	Shao Z S, Wang T J，Ang K K. Transient thermo-mechanical analysis of functionally graded hollow circular cylinders[J].J Thermal Stresses,2007,30(1):81-104. doi: 10.1080/01495730600897211
[27]	Mallik S H，Kanoria M. Generalized thermo-elastic functionally graded solid with a periodically varying heat source[J].International Journal of Solids and Structures,2007,44(22/23):7633-7645. doi: 10.1016/j.ijsolstr.2007.05.001
[28]	Bagri A, Eslami M R. A unified generalized thermoelasticity formulation; application to thick functionally graded cylinders[J].J Thermal Stresses,2007,30(9/10):911-930. doi: 10.1080/01495730701496079
[29]	Chen W Q. Stress distribution in a rotating elastic functionally graded material hollow sphere with spherical isotropy[J].Journal of Strain Analysis for Engineering Design,2000,35(1):13-20. doi: 10.1243/0309324001513973
[30]	Obata Y, Noda N.Steady thermal stresses in a hollow circular cylinder and a hollow sphere of a functionally graded material[J].J Thermal Stresses,1994,17(5):471-487. doi: 10.1080/01495739408946273
[31]	Ootao Y，Tanigawa Y.Transient thermoelastic problem of a functionally graded cylindrical panel due to nonuniform heat supply[J].J Thermal Stresses,2007,30(5):441-457. doi: 10.1080/01495730601146394
[32]	Das N C, Lahiri A, Sen P K.Eigenvalue approach to three dimensional generalized thermoelasticity[J].Bulletin Calcutta Math Soc,2006,98(4):305-318.
[33]	Nowacki W.Dynamic Problems of Thermoelasticity[M].Warszawa: Polish Scientific Publishers, 1975.
[34]	Wang H M, Ding H J，Chen Y M. Thermoelastic dynamic solution of a multilayered spherically isotropic hollow sphere for spherically symmetric problems[J].Acta Mechanica,2005,173(1/4):131-145.
[35]	Bellman R, Kolaba R E， Lockette J A.Numerical Inversion of the Laplace Transform[M].New York:American Elsevier Pub Co, 1966.
[36]	Dhaliwal R S，Sing A.Dynamic Coupled Thermoelasticity[M].Delhi:Hindustan Publ, 1980.

Relative Articles

[1]	ZHAO Yingzhi, TANG Huaiping, LAI Zedong, ZHANG Jiajie. Free Vibration Analysis of Porous 2D Functionally Graded Material Microbeams on Winkler's Foundation[J]. Applied Mathematics and Mechanics, 2023, 44(11): 1354-1365. doi: 10.21656/1000-0887.440050
[2]	ZHAO Yi, TIAN Xiaogeng. Thermomechanical Responses of YSZ Under Ultrashort Thermal Shock Based on the L-S Generalized Thermoelastic Theory[J]. Applied Mathematics and Mechanics, 2023, 44(7): 784-796. doi: 10.21656/1000-0887.430134
[3]	WANG Pingyuan, LI Cheng, YAO Linquan. Bending and Buckling of Functionally Graded Nanoplates Based on the Nonlocal Strain Gradient Theory[J]. Applied Mathematics and Mechanics, 2021, 42(1): 15-26. doi: 10.21656/1000-0887.410188
[4]	WANG Xue, ZHAO Weidong. Geometrically Nonlinear Analysis of Functionally Graded Beams Under Thermomechanical Loading[J]. Applied Mathematics and Mechanics, 2019, 40(5): 508-517. doi: 10.21656/1000-0887.390201
[5]	WANG Ying-ze, WANG Qian, LIU Dong, GU Li-ping. Generalized Solution to the Thermoelastic Problem of Finite-Thickness Hollow Cylinders Subjected to Thermal Shock[J]. Applied Mathematics and Mechanics, 2016, 37(6): 644-654. doi: 10.3879/j.issn.1000-0887.2016.06.010
[6]	YANG Yun-fang, YANG Bo, CHEN Wei-qiu, DING Hao-jiang. Elasticity Solutions for Cylindrical Bending of Functionally Graded Plates[J]. Applied Mathematics and Mechanics, 2015, 36(12): 1257-1264. doi: 10.3879/j.issn.1000-0887.2015.12.004
[7]	WANG Ying-ze, WANG Qian, LIU Dong, SONG Xin-nan. Generalized Thermoelastic Solutions to the Problems of Thermal Shock on Elastic Half Space[J]. Applied Mathematics and Mechanics, 2014, 35(6): 640-651. doi: 10.3879/j.issn.1000-0887.2014.06.006
[8]	CHEN Guang-hua, CHEN Guang-ming, DAI Zhi-hua. Modified Domain Decomposition Method for Hamilton-Jacobi-Bellman Equations[J]. Applied Mathematics and Mechanics, 2010, 31(12): 1496-1502. doi: 10.3879/j.issn.1000-0887.2010.12.010
[9]	LI Shi-rong, SU Hou-de, CHENG Chang-jun. Free Vibration of Functionally Graded Material Beams With Surface-Bonded Piezoelectric Layers in Thermal Environment[J]. Applied Mathematics and Mechanics, 2009, 30(8): 907-918. doi: 10.3879/j.issn.1000-0887.2009.08.003
[10]	PENG Xu-long, LI Xian-fang. Thermoelastic Analysis of a Functionally Graded Annulus With an Arbitrary Gradient[J]. Applied Mathematics and Mechanics, 2009, 30(10): 1135-1142. doi: 10.3879/j.issn.1000-0887.2009.10.001
[11]	Payel Das, M. Kanoria. Magneto-Thermo-Elastic Waves in an Infinite Perfectly Conducting Elastic Solid With Energy Dissipation[J]. Applied Mathematics and Mechanics, 2009, 30(2): 213-220.
[12]	Praveen Ailawalia, Naib Singh Narah. Effect of Rotation in Generalized Thermoelastic Solid Under the Influence of Gravity With an Overlying Infinite Thermoelastic Fluid[J]. Applied Mathematics and Mechanics, 2009, 30(12): 1415-1426. doi: 10.3879/j.issn.1000-0887.2009.12.003
[13]	Abdul Ghafar Shah, Tahir Mahmood, Muhammad Nawaz Naeem. Vibrations of FGM Thin Cylindrical Shells With Exponential Volume Fraction Law[J]. Applied Mathematics and Mechanics, 2009, 30(5): 567-574. doi: 10.3879/j.issn.1000-0887.2009.05.007
[14]	YANG Bo, DING Hao-jiang, CHEN Wei-qiu. Elasticity Solutions for Functionally Graded Plates in Cylindrical Bending[J]. Applied Mathematics and Mechanics, 2008, 29(8): 905-910.
[15]	Sunita Deswal, Suman Choudhary. Two-Dimensional Interactions Due to Moving Load in Generalized Thermoelastic Solid With Diffusion[J]. Applied Mathematics and Mechanics, 2008, 29(2): 188-202.
[16]	ZHAO Feng-qun, WANG Zhong-min, LIU Hong-zhao. Thermal Post-buckling Analyses of Functionally Graded Material Rod[J]. Applied Mathematics and Mechanics, 2007, 28(1): 53-60.
[17]	Rajneesh Kumar, Manjeet Singh. Propagation of Plane Waves in Thermoelastic Cubic Crystal Material With Two Relaxation Times[J]. Applied Mathematics and Mechanics, 2007, 28(5): 561-574.
[18]	WANG Lin-xiang, Roderick V. N. Melnik. Differential-Algebraic Approach to Coupled Problems of Dynamic Thermoelasticity[J]. Applied Mathematics and Mechanics, 2006, 27(9): 1036-1046.
[19]	Hamdy M. Youssef. Dependence of the Modulus of Elasticity and the Thermal Conductivity on the Reference Temperature in Generalized Thermoelasticity for an Infinite Material With a Spherical Cavity[J]. Applied Mathematics and Mechanics, 2005, 26(4): 431-436.
[20]	Pi Dao-hua, Jin Li-hua. Structural Function Theory of Generalized Variational Principles for Linear Elastic Materials with Voids[J]. Applied Mathematics and Mechanics, 1991, 12(6): 531-540.