A Problem of Axisymmetric Vibration of Nonlocal Microstretch Thermoelastic Circular Plate with Thermomechanical Sources

Subject Areas : Engineering

R Kumar ¹ , R Rani ² , A Miglani ³

1 - Department of Mathematices, Kurukshetra University, Kurukshetra, Haryana, India
2 - Department of Mathematices, Choudhary Devilal University, Sirsa, Haryana, India
3 - Department of Mathematices, Choudhary Devilal University, Sirsa, Haryana, India

Received: 2018-09-01 Accepted : 2018-11-14 Published : 2019-03-30

Keywords: Circular plate, Thermoelasticity, Nonlocal microstretch, Laplace and Hankeltransforms, Eigenvalue approach,

Abstract :

In the present manuscript, we investigated a two dimensional axisymmetric problem of nonlocal microstretch thermoelastic circular plate subjected to thermomechanical sources. An eigenvalue approach is proposed to analyze the problem. Laplace and Hankel transforms are used to obtain the transformed solutions for the displacements, microrotation, microstretch, temperature distribution and stresses. The results are obtained in the physical domain by applying the numerical inversion technique of transforms. The results of the physical quantities have been obtained numerically and illustrated graphically. The results show the effect of nonlocal in the cases of Lord Shulman (LS), Green Lindsay (GL) and coupled thermoelasticity (CT) on all the physical quantities.

References:

[1] Kroner E., 1967, Elasticity theory of materials with longe range cohesive forces, International Journal of Solids and Structures 3: 731-742.
[2] Eringen A. C., 1972, Nonlocal polar elastic continua, International Journal of Engineering Science 10: 1-16.
[3] Eringen A. C., 1972, Linear theory of nonlocal elasticity and dispersion of plane waves, International Journal of Engineering Science 10: 425-435.
[4] Eringen A. C., 1967, Nonlocal micropolar field theory, In: Eringen A. C., Continuum Physics, Academic Press New York 4: 205-267.
[5] Edelen D. G. B., Green A. E., Laws N., 1971, Nonlocal continuum mechanics, Rational Mechanics and Analysis 43: 36-44.
[6] Eringen A. C., Edelen D. G. B., 1972, On nonlocal elasticity, International Journal of Engineering Science 10: 233-248.
[7] Eringen A. C., Kim B. S., 1974, Stress concentration at the tip of a crack, Mechanics Research Communication 1: 233-237.
[8] Eringen A. C., Speziale C. G., Kim B. S., 1977, Crack-tip problem in nonlocal elasticity, Journal of Mechanics and Mathematics Solids 25: 339-355.
[9] Eringen A. C., 1978, Line crack subjected to a shear, International Journal of Fracture Mechanics 14: 367-379.
[10] Eringen A. C.,1979, Line crack subjected to a anti-plane shear, Engineering Fracture Mechanics 12: 211-219.
[11] Reid E., Gooding R. J., 1992, Inclusion problem in a two dimensional nonlocal elastic solid, Physical Review B 46: 6045-6049.
[12] Gao J., 1999, An asymmetric theory of nonlocal elasticity, International Journal of Solids and Structures 36: 2959-2971.
[13] Sharma P., Ganti S., 2003, The size dependent elastic state of inclusions in nonlocal elastic solids, Philosophical Magazine Letters 83: 745-754.
[14] Paola M. D., Faillla G., Zingales M., 2010, The mechanically based approach to 3D nonlocal linear elasticity theory: Longe-range central interactions, International Journal of Solids and Structures 47: 2347-2358.
[15] Salehipour H., Shahidi A. R., Nahvi H., 2015, Modified nonlocal elasticity theory for functionally graded materials, International Journal of Engineering Science 90: 44-57.
[16] Peng X. L., Li X. F., Tang G. J., Shen Z. B., 2015, Effect of scale parameter on the deflection of a nonlocal beam and application to energy release rate of a crack, Journal of Applied Mathematics and Mechanics 95: 1428-1438.
[17] Sumelka W., Zaera R., Fernandez-Saez J., 2015, A theoretical analysis of the free axial vibration of nonlocal rods with fractional continuum mechanics, Meccanica 50: 2309-2323.
[18] Vasiliev V.V., Lurie S. A., 2016, On correct nonlocal generalized theories of elasticity, Physical Mesomechanics 19: 269-281.
[19] Chen H., Liu Y., 2016, A nonlocal 3D lattice particle framework for elastic solids, International Journal of Solids and Structures 81: 411-420.
[20] Singh D., Kaur G., Tomar S. K., 2017, Waves in nonlocal elastic solid with voids, Journal of Elasticity 128: 85-114.
[21] Eringen A. C., 1974, Theory of nonlocal thermoelasticity, International Journal of Engineering Science 12: 1063-1077.
[22] Balta F., Suhubi E. S., 1977, Theory of nonlocal generalized thermoelasticity, International Journal of Engineering Science 15: 579-588.
[23] Altan S., 1990, Some theorems in nonlocal thermoelasticity, Journal of Thermal Stesses 13: 207-221.
[24] Wang J., Dhaliwal R. S., 1993, Uniqueness in generalized nonlocal thermoelasticity, Journal of Thermal Stresses 16: 71-77.
[25] Zenkour A. M., Abouelregal A. E., 2014, Nonlocal thermoelastic vibrations for variable thermal conductivity nanobeams due to harmonically varying heat, Journal of Vibroengineering 16: 3665-3678.
[26] Zenkour A. M., Abouelregal A. E., 2015, Nonlocal thermoelastic nanobeams subjected to a sinusoidal pulse heating and temperature dependent physical properties, Microsystem Technologies 21: 1767-1776.
[27] Yu Y., Tian X. G., Xiong Q. L., 2016, Nonlocal thermoelasticity based on nonlocal heat conduction and nonlocal elasticity, European Journal of Mechanics –A/ Solids 60: 238-253.
[28] Eringen A. C., 1984, Plane waves in nonlocal micropolar elasticity, International Journal of Engineering Science 22: 1113-1121.
[29] Eringen A. C., 1999, Microcontinuum Field Theories I: Foundations and Solids, Springer-Verlag, New York.
[30] Lord H. W., Shulman Y., 1967, A generalized dynamical theory of thermoelasticity, Journal of Mechanics and Physics of Solids 15: 299-309.
[31] Eringen A. C., 2002, Nonlocal Continuum Field Theories, Springer Verlag, New York.
[32] Tripathi J. J., Kedar K. D., Deshmukh K. C., 2016, Generalized thermoelastic problem of a thick circular plate subjected to axisymmetric heat supply, Journal of Solid Mechanics 8: 578-589.
[33] Kiris A., Inan E., 2008, On the identification of microstretch elastic moduli of materials by using vibration data of plates, International Journal of Engineering Sciences 46: 585-597.
[34] Tomar S. K., Khurana A., 2008, Elastic waves in an electro-microelastic solid, International Journal of Solids and Structures 45: 276-302.
[35] Dhaliwal R. S., Singh A., 1980, Dynamical Coupled Thermoelasticity, Hindustan Publication Corporation, New Delhi.
[36] Green A. E., Lindsay K. A., 1972, Thermoelasticity, Journal of Elasticity 2: 1-7.