Effects of Hall Current and Rotation in Modified Couple Stress Generalized Thermoelastic Half Space due to Ramp-Type Heating

Journal of Solid Mechanics

مقاله 6، دوره 9، شماره 3، آذر 2017، صفحه 527-542   

نوع مقاله: Research Paper

نویسندگان

R Kumar^{* 1}؛ Sh Devi²؛ V Sharma²

¹Department of Mathematics, Kurukshetra University, Kurukshetra, India

²Department of Mathematics & Statistics, Himachal Pradesh University, Shimla, India

چکیده

The objective is to study the deformation in a homogeneous isotropic modified couple stress thermoelastic rotating medium in the presence of Hall current and magnetic field due to a ramp-type thermal source. The generalized theories of thermoelasticity developed by Lord Shulman (L-S, 1967) and Green Lindsay (G-L, 1972) are used to investigate the problem. Laplace and Fourier transform technique is applied to obtain the solutions of the governing equations. The displacements, stress components, temperature change and mass concentration are obtained in the transformed domain. Numerical inversion technique has been used to obtain the solutions in the physical domain. Effects of Hall current and rotation are shown in a resulting quantities. Some special cases of interest are also deduced. 

کلیدواژه‌ها

Modified couple stress؛ Generalized thermoelasticity؛ Laplace and Fourier transforms؛ Ramp-Type heating؛ Hall current and magnetic effect

مراجع

[1] Voigt W., 1887, Theoretische Studien über die Elasticitätsverhältnisse der Krystalle ,Göttingen, Dieterichsche Verlags-buchhandlung , German.
[2] Cosserat E., Cosserat F., 1909, Theory of Deformable Bodies, Hermann et Fils, Paris.
[3] Mindlin R. D., Tiersten H. F., 1962, Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis 11: 415-448.
[4] Toupin R. A., 1962, Elastic materials with couple-stresse, Archive for Rational Mechanics and Analysis 11: 385-414.
[5] Mindlin R. D., 1964, Micro-structure in linear elasticity, Archive for Rational Mechanics and Analysis 16: 51-78.
[6] Koiter W. T., 1964, Couple-stresses in the theory of elasticity, Proceedings of the National Academy of Sciences 67: 17-44.
[7] Lakes R. S., 1982, Dynamical study of couple stress effects in human compact bone, Journal of Biomechanical Engineering 104: 6-11.
[8] Lam D. C. C., Yang F., Chong A. C. M., Wang J., Tong P., 2003, Experiments and theory in strain gradient elasticity, Journal of the Mechanics and Physics of Solids 51: 1477-508.
[9] Yang F., Chong A. C. M., Lam D. C. C., Tong P., 2002, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures 39: 2731-2743.
[10] Park S. K., Gao X. L., 2006, Bernoulli–Euler beam model based on a modified couple stress theory, Journal of Micromechanics and Microengineering 16: 2355.
[11] Ma H. M., Gao X. L., Reddy J. N., 2008, A microstructure-dependent Timoshenko beam model based on a modified couple stress theory, Journal of the Mechanics and Physics of Solids 56: 3379-3391.
[12] Ke L. L., Wang Y. S., 2011, Size effect on dynamic stability of functionally graded micro beams based on a modified couple stress theory, Composite Structures 93: 342-350.
[13] Chen W., Li L., Xu M., 2011, A modified couple stress model for bending analysis of composite laminated beams with first order shear deformation, Composite Structures 93: 2723-2732.
[14] Asghari M., 2012, Geometrically nonlinear micro-plate formulation based on the modified couple stress theory, International Journal of Engineering Science 51: 292-309.
[15] Simsek M., Reddy J. N., 2013, Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory, International Journal of Engineering Science 64: 37-53.
[16] Mohammad-Abadi M., Daneshmehr A. R., 2014, Size dependent buckling analysis of micro beams based on modified couple stress theory with high order theories and general boundary conditions, International Journal of Engineering Science 74: 1-14.
[17] Shaat M., Mahmoud F. F., Gao X. L., Faheem A. F., 2013, Size-dependent bending analysis of Kirchhoff nano-plates based on a modified couple-stress theory including surface effects, International Journal of Mechanical Sciences 79: 31-37.
[18] Nowacki W., 1974a, Dynamical problems of thermo diffusion in solids I, Bulletin of the Polish Academy of Sciences - Technical Sciences 22: 55-64.
[19] Nowacki W., 1974b, Dynamical problems of thermo diffusion in solids II, Bulletin of the Polish Academy of Sciences - Technical Sciences 22: 129-135.
[20] Nowacki W., 1974c, Dynamical problems of thermo diffusion in solids III, Bulletin of the Polish Academy of Sciences - Technical Sciences 22: 257-266.
[21] Nowacki W., 1976, Dynamical problems of thermo diffusion in solids, Engineering Fracture Mechanics 8: 261-266.
[22] PodstrigachIa S., 1961, Differential equations of the problem of thermodiffusion in isotropic deformed solid bodies, Dopovidi Akademii Nauk Ukrainskoi SSR 1961: 169-172.
[23] Sherief H. H., Saleh H., Hamza F., 2004, The theory of generalized thermoelastic diffusion, International Journal of Engineering Science 42: 591-608.
[24] Sherief H. H., Saleh H., 2005, A half-space problem in the theory of generalized thermoelastic diffusion, International Journal of Solids and Structures 42: 4484-4493.
[25] Kumar R., Kansal T., 2008, Propagation of Lamb waves in transversely isotropic thermoelastic diffusion plate, International Journal of Solids and Structures 45: 5890-5913.
[26] Knopoff L., 1955, The interaction between elastic wave motion and a magnetic field in electrical conductors, Journal of Geophysical Research 60: 441-456.
[27] Chadwick P., 1957, Ninth, International Congress of Theoretical and Applied Mechanics 7: 143.
[28] Kaliski S., Petykiewicz J., 1959, Equation of motion coupled with the field of temperature in a magnetic field involving mechanical and electrical relaxation for anisotropic bodies, Proceeding Vibration Problems.
[29] Zakaria M., 2012, Effects of Hall current and rotation on magneto-micropolar generalized thermoelasticity due to ramp-type heating, International Journal of Applied Electromagnetics 2: 24-32.
[30] Zakaria M., 2014 Effect of Hall current on generalized magneto-thermoelasticity micropolar solid subjected to ramp-type heating, International Applied Mechanics 50: 130-144.
[31] Honig G., Hirdes U., 1984, A method for the numerical inversion of the Laplace transform, Journal of Computational and Applied Mathematics 10: 113-132.
[32] Press W. H., Teukolsky S. A., Vellerling W. T., Flannery B. P., 1986, Numerical Recipes , Cambridge University Press.