Finite Crack in a Thermoelastic Transversely Isotropic Medium Under Green-Naghdi Theory

Subject Areas : Applied Mechanics

S.K Panja ¹ , S.C Mandal ²

1 - Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India
2 - Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India

Received: 2021-09-20 Accepted : 2021-12-13 Published : 2022-03-30

Keywords: Wave propagation, G-N theory, Thermoelastic, Finite crack, Transversely isotropic medium,

Abstract :

In this paper, we have studied a model of finite linear Mode-I crack in a thermoelastic transversely isotropic medium under Green Naghdi theory. The crack is subjected to a prescribed temperature and a known tensile stress. The plane boundary surface is considered as isothermal and all the field variables are sufficiently smooth. The heat conduction equation is written under two temperature theory (2TT) for Green Naghdi model which contains absolute temperature as well as conductive temperature. The analytical expressions of displacement components, stress components and temperature variables are obtained by normal mode analysis and matrix inversion method. Comparisons have been made within Green Naghdi (G-N) theory of type I, type II and type III for displacement, stress and absolute temperature variables against the crack width for a transversely isotropic material (Cobalt) by virtues of graphs. Also, Comparison have been made among displacement, thermal stress and absolute temperature for different depths.

References:

1.  Dhaliwal R.S., 1980, External crack due to thermal effects in an infinite elastic solid with a cylindrical inclusion, Thermal Stresses in Severe Environments 1980: 665-692.


2.  Ueda S., 2003, Thermally induced fracture of a piezoelectric laminate with a crack normal to interfaces, Journal of Thermal Stresses 26(4): 311-331.


3.  Hasanyan D., Librescu L., Qin Z., Young R.D., 2005, Thermoelastic cracked plates carrying nonstationary electrical current, Journal of Thermal Stresses 28(6-7): 729-745.


4.  Elfalaky A., Abdel-Halim A. A., 2006, A mode-I crack problem for an infinite space in thermoelasticity, Journal of Applied Sciences 6(3): 598-606.


5. Othman M.I., Atwa S.Y., 2013, 2-D problem of a Mode-I crack for a generalized thermoelasticity under Green-Naghdi theory, Meccanica 48(6): 1543-1551.


6.  Nowacki W., 2013,Thermoelasticity, Elsevier.


7.  Nowinski J.L., 1978, Theory of Thermoelasticity with Applications, Springer.


8.  Biot M.A., 1956, Thermoelasticity and irreversible thermodynamics, Journal of Applied Physics 27(3): 240-253.


9.  Lord H.W., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids 15(5): 299-309.


10.  Dhaliwal R.S., Sherief H.H., 1980, Generalized thermoelasticity for anisotropic media, Quarterly of Applied Mathematics 38(1): 1-8.


11.  Green A.E., Laws N., 1972, On the entropy production inequality, Archive for Rational Mechanics and Analysis 45(1): 47-53.


12.  Green A.E., Naghdi P.M., 1992, On undamped heat waves in an elastic solid, Journal of Thermal Stresses 15(2): 253-264.


13.  Green A.E., Naghdi P.M., 1993, Thermoelasticity without energy dissipation, Journal of Elasticity 31(3): 189-208.


14.  Othman M.I., Atwa S.Y., Farouk R.M., 2009, The effect of diffusion on two dimensional problem of generalized thermoelasticity with Green-Naghdi theory, International Communications in Heat and Mass Transfer 36(8): 857-864.


15. Sarkar N., Atwa S.Y., 2019, Two-temperature problem of a fiber-reinforced thermoelastic medium with a Mode-I crack under Green-Naghdi theory, Microsystem Technologies 25(4): 1357-167.


16. Lata P., Kaur I., 2019, A study of transversely isotropic thermoelastic beam with Green-Naghdi Type-II and Type-III theories of thermoelasticity, Applications & Applied Mathematics 14(1): 270-283.


17.  Kaur I., Lata P., 2020, Axisymmetric deformation in transversely isotropic magneto-thermoelastic solid with Green-naghdi III due to inclined load, International Journal of Mechanical and Materials Engineering 15(1): 1-9.


18.  Sur A., Kanoria M., 2019, Elasto-thermodiffusive response in a two-dimensional transversely isotropic medium, Mechanics of Advanced Composite Structures 6(2): 95-104.


19. Singh A.K., Guha S., 2020, Reflection of plane waves from the surface of a piezothermoelastic fiber-reinforced composite half-space, Mechanics of Advanced Materials and Structures 2020: 1-13.


20. Guha S., Singh A.K., 2020, Effects of initial stresses on reflection phenomenon of plane waves at the free surface of a rotating piezothermoelastic fiber-reinforced composite half-space, International Journal of Mechanical Sciences 181:


21.  Kumar R., Sharma N., Lata P., 2016, Thermomechanical interactions in transversely isotropic magnetothermoelastic medium with vacuum and with and without energy dissipation with combined effects of rotation, vacuum and two temperatures, Applied Mathematical Modelling 40(13-14): 6560-6575.


22.  Kumar R., Sharma N., Lata P., 2016, Thermomechanical interactions due to hall current in transversely isotropic thermoelastic with and without energy dissipation with two temperatures and rotation, Journal of Solid Mechanics8(4): 840-858.


23.  Sharma N., Kumar R., Lata P., 2015, Disturbance due to inclined load in transversely isotropic thermoelastic medium with two temperatures and without energy dissipation, Material Physics and Mechanics 22(2): 107-117.


24.  Lata P., Kaur I., 2019, Transversely isotropic thick plate with two temperature and GN type-III in frequency domain,Coupled Systems Mechanics 8(1): 55-70.


25.  Youssef H.M., 2011, Theory of two-temperature thermoelasticity without energy dissipation, Journal of Thermal Stresses 34(2): 138-146.


26. Atwa S.Y., 2014, Generalized magneto-thermoelasticity with two temperature and initial stress under Green-Naghdi theory, Applied Mathematical Modelling 38(21-22): 5217-5230.


27. Dhaliwal R.S., Singh A., 1980, Dynamic Coupled Thermoelasticity, Hindustan Publishing Corporation.