Axisymmetric Problem of Thick Circular Plate with Heat Sources in Modified Couple Stress Theory

Document Type: Research Paper

Authors

1 Department of Mathematics, Kurukshetra University , Kurukshetra, Haryana, India

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

Abstract

The main aim is to study the two dimensional axisymmetric problem of thick circular plate in modified couple stress theory with heat and mass diffusive sources. The thermoelastic theories with mass diffusion developed by Sherief et al. [1] and kumar and Kansal [2] have been used to investigate the problem. Laplace and Hankel transforms technique is applied to obtain the solutions of the governing equations. The displacements, stress components, temperature change and chemical potential are obtained in the transformed domain. Numerical inversion technique has been used to obtain the solutions in the physical domain. Effects of couple stress on the resulting quantities are shown graphically. Some particular cases of interest are also deduced.

Keywords


Sherief H. H., Saleh H., Hamza F., 2004, The theory of generalized thermoelastic diffusion, International Journal of Engineering Science 42: 591-608.
[2] Kumar R., Kansal T., 2008, Propagation of Lamb waves in transversely isotropic thermoelastic diffusion plate, International Journal of Solids and Structures 45: 5890-5913.
[3] Voigt W., 1887, Theoretische Studien über die Elasticitätsverhältnisse der krystalle, Göttingen Dieterichsche Verlags- uchhandlung.
[4] Cosserat E., Cosserat F., 1909, Theory of Deformable Bodies, Hermann et Fils, Paris.
[5] Mindlin R. D., Tiersten H. F., 1962, Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis 11: 415-448.
[6] Toupin R. A., 1962, Elastic materials with couple-stresses, Archive for Rational Mechanics and Analysis 11: 385-414.
[7] Koiter W. T., 1964, Couple-stresses in the theory of elasticity, Proceedings of the Royal Netherlands Academy of Arts and Science 67: 17-44.
[8] Lakes R. S., 1982, Dynamical study of couple stress effects in human compact bone, Journal of Biomechanical Engineering 104: 6-11.
[9] 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: 1477-1508.
[10] 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.
[11] 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.
[12] 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.
[13] Shaat M., Mahmoud F. F., Gao X. L., Faheem A. F., 2014, 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.
[14] Ghorbanpour A. Arani, Abdollahian M., Jalaei H. M., 2015, Vibration of bioliquid-filled microtubules embedded in cytoplasm including surface effects using modified couple stress theory, Journal of Theoretical Biology 367: 29-38.
[15] Darijani H., Shahdadi A. H., 2015, A new shear deformation model with modified couple stress theory for microplates, Acta Mechanica 226: 2773-2788.
[16] Wang Y. G., Lin W. H., Ning L., 2015, Nonlinear bending and post-buckling of extensible microscale beams based on modified couple stress theory, Applied Mathematical Modelling 39: 117-127.
[17] Podstrigach I. S., 1961, Differential equations of the problem of thermodiffusion in isotropic deformed solid bodies, Dop Akad Nauk Ukr SSR 169-172.
[18] Nowacki W., 1974a, Dynamical problems of thermo diffusion in solids I.  Bulletin de l'Academie Polonaise des Sciences, Serie des Sciences Techniques 22: 55-64.
[19] Nowacki W., 1974b, Dynamical problems of thermo diffusion in solids II. Bulletin de l'Academie Polonaise des Sciences, Serie des Sciences Techniques 22: 129-135.
[20] Nowacki W., 1974c, Dynamical problems of thermo diffusion in solids III. Bulletin de l'Academie Polonaise des Sciences, Serie des Sciences Techniques 22: 257-266.
[21] Nowacki W., 1976, Dynamical problems of thermo diffusion in solids, Engineering Fracture Mechanics 8: 261-266.
[22] 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.
[23] El-Maghraby N. M., Abdel-Halim A. A., 2010, A generalized thermoelsticity problem for a half space with heat sources under axisymmetric distributions, Australian Journal of Basic and Applied Sciences 4: 3803-3814.
[24] Lord H. W., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids 15: 299-309.
[25] Tripathi J. J., Kedar G. D., Deshmukh K. C., 2014, Dynamic problem of generalizedthermoelasicity for a semi-infinite cylinder with heat sources, Journal of Thermoelasticity 2: 1-8.
[26] Tripathi J. J., Kedar G. D., Deshmukh K. C., 2015, Generalized thermoelastic diffusion problem in a thick circular plate with axisymmetric heat supply, Acta Mechanica 226: 2121-2134.
[27] Honig G., Hirdes U., 1984, A method for the numerical inversion of the Laplace transform, Journal of Computational and Applied Mathematics 10: 113-132.
[28] Press W. H., Teukolsky S. A., Vellerling W. T., Flannery B. P., 1986, Numerical Recipes , Cambridge University Press.