Document Type: Research Paper

**Authors**

^{1}
Department of Mathematics, Faculty of Science, Zagazig University--- Department of Mathematics, Faculty of Science and Arts, Al-mithnab, Qassim University

^{2}
Department of Mathematics, Faculty of Science and Arts, Al-mithnab, Qassim University

**Abstract**

A model of the equations of two dimensional problems in a half space, whose surface in free of micropolar thermoelastic medium possesses cubic symmetry as a result of a Mode-I Crack is studied. There acts an initial magnetic field parallel to the plane boundary of the half- space. The crack is subjected to prescribed temperature and stress distribution. The formulation in the context of the Lord-Şhulman theory LS includes one relaxation time and Green-Lindsay theory GL with two relaxation times, as well as the classical dynamical coupled theory CD. The normal mode analysis is used to obtain the exact expressions for the displacement, microrotation, stresses and temperature distribution. The variations of the considered variables with the horizontal distance are illustrated graphically. Comparisons are made with the results in the presence of magnetic field. A comparison is also made between the three theories for different depths.

**Keywords**

[1] Eringen A. C., Suhubi E. S., 1964, Nonlinear theory of simplemicropolar solids, International Journal of Engineering Science 2:1-18.

[2] Eringen A. C.,1966,Linear theory of micropolar elasticity, Journal of Applied Mathematics and Mechanics 15: 909-924.

[3] Biot M. A., 1956, Thermoclasticity and irreversible thermodynamics, Journal of Applied Physics 27:240-253.

[4] Lord H. W., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids 15:299-306.

[5] Othman M. I. A., 2002, Lord-shulman theory under the dependence of the modulus of elasticity on the reference temperature in two-dimensional generalized thermo- elasticity, Journal of Thermal Stresses 25:1027-1045.

[6] Green A. E., Lindsay K.A., 1972, Thermoelasticity, Journal of Elasticity 2:1-7.

[7] Green A. E., Laws N., 1972, On the entropy production inequality, Archive for Rational Mechanics and Analysis45:47-53.

[8] Suhubi E. S., 1975, Thermoelastic Solids in Continuum Physics, Part 2, Chapter2, Academic Press, New York.

[9] Othman M. I. A., 2004, Relaxation effects on thermal shock problems in an elastic half-space of generalized magneto-thermoelastic waves, Mechanics and Mechanical Engineering 7:165-178.

[10] Iesan D., 1973, The plane micropolar strain of orthotropic elastic solids, Archives of Mechanics 25:547-561.

[11] Iesan D., 1974, Torsion of anisotropic elastic cylinders, Journal of Applied Mathematics and Mechanics 54:773-779.

[12] Iesan D., 1974, Bending of Orthotropic Micropolar Elastic Beams by Terminal Couples, An State University Lasi 20:411- 418.

[13] Nakamura S., Benedict R., Lakes R., 1984, Finite element method for orthotropic micropolar elasticity, International Journal of Engineering Science 22:319-330.

[14] Kumar R., Choudhary S., 2002, Influence and green's function for orthotropic micropolar containua, Archives of Mechanics 54:185-198.

[15] Kumar R., Choudhary S., 2002, Dynamical behavior of orthotropic micropolar elastic medium, Journal of vibration and control 5:1053- 1069.

[16] Kumar R., Choudhary S., 2002, Mechanical sources in orthotropic micropolar continua, Proceedings of the Indian Academy of Sciences 111(2):133-141.

[17] Kumar R., Choudhary S., 2003, Response of orthotropic micropolar elastic medium due to various sources, Meccanica 38:349- 368.

[18] Kumar R., Choudhary S., 2004, Response of orthotropic micropolar elastic medium due to time harmonic sources, Sadhana 29:83- 92.

[19] Singh B., Kumar R., 1998, Reflection of plane wave from a flat boundary of micropolar generalized thermoelastic half-space, International Journal of Engineering Science 36:865-890.

[20] Singh B., 2000, Reflection of plane sound wave from a micropolar generalized thermoelastic solid half-space, Journal of sound and vibration 235:685-696.

[21] Othman M.I. A., Lotfy KH., 2009, Two-dimensional Problem of generalized magneto-thermoelasticity under the Effect of temperature dependent properties for different theories, Multidiscipline Modeling in Materials and Structures 5:235-242.

[22] Othman M.I. A., Lotfy KH., Farouk R.M., 2009, Transient disturbance in a half-space under generalized magneto-thermoelasticity due to moving internal heat source , Acta Physica Polonica A 116:186-192.

[23] Othman M.I. A, Lotfy KH., 2010, On the plane waves in generalized thermo-microstretch elastic half-space, International Communication in Heat and Mass Transfer 37:192-200.

[24] Othman M.I. A, Lotfy KH., 2009, Effect of magnetic field and inclined load in micropolar thermoelastic medium possessing cubic symmetry, International Journal of Industrial Mathematics 1(2): 87-104.

[25] Othman M.I. A, Lotfy KH., 2010, Generalized thermo-microstretch elastic medium with temperature dependent properties for different theories, Engineering Analysis with Boundary Elements 34:229-237.

[26] Lotfy Kh., 2014, Two temperature generalized magneto-thermoelastic interactions in an elastic medium under three theories, Applied Mathematics and Computation 227:871-888.

[26] Lotfy Kh., 2014, Two temperature generalized magneto-thermoelastic interactions in an elastic medium under three theories, Applied Mathematics and Computation 227:871-888.

[27] Dhaliwal R., 1980, External Crack due to Thermal Effects in an Infinite Elastic Solid with a Cylindrical Inclusion, Thermal Stresses in Server Environments, Doi: 10.1007/978-1-4613-3156-8_41.

[28] Hasanyan D., Librescu L., Qin Z., Young R., 2005, Thermoelastic cracked plates carrying nonstationary electrical current, Journal of Thermal Stresses 28:729-745.

[29] Ueda S., 2003, Thermally induced fracture of a piezoelectric Laminate with a crack normal to interfaces, Journal of Thermal Stresses 26:311-323.

[30] Elfalaky A., Abdel-Halim A. A., 2006, A mode-i crack problem for an infinite space in thermoelasticity, Journal of Applied Sciences 6:598-606.

Volume 5, Issue 3

Summer 2013

Pages 253-269