Deformation of modified couple stress thermoelastic diffusion in a thick circular plate due to heat sources
Kumar Rajneesh 1, Devi Shaloo 2*
1 Department of Mathematics
Kurukshetra University
Kurukshetra, India2 Department of Mathematics & Statistics
Himachal Pradesh University
Shimla, India
*E-mail: shaloosharma2673@gmail.com
Received:
Received: 22 June 2018; revised: 30 December 2019; accepted: 30 December 2019; published online: 31 December 2019
DOI: 10.12921/cmst.2018.0000034
Abstract:
The aim of present study is to present a mathematical model for predicting the results for displacements, stress components, temperature change and chemical potential with considering independently a particular type of heat source. The general solution for the two-dimensional problem of a thick circular plate with heat sources in modified couple stress thermoelastic diffusion has been obtained in the context of one and two relaxation times. Laplace and Hankel transforms technique is applied to obtain the solutions of the governing equations. Resulting quantities are obtained in the transformed domain. The numerical inversion technique has been used to obtain the solutions in the physical domain. Effects of time on the resulting quantities are shown graphically.
Key words:
heat sources, Laplace and Hankel transforms, modified couple stress theory, thermoelasticity, thick circular plate
References:
[1] R.S. Lakes, Dynamical study of couple stress effects in human compact bone, Journal of Biomechanical Engineering 104, 6–11 (1982).
[2] D.C.C. Lam, F. Yang, A.C.M. Chong, J. Wang, P. Tong, Experiments and theory in strain gradient elasticity, Journal of Mechanics and Physics of Solids 51, 1477–1508 (2003).
[3] F. Yang, A.C.M. Chong, D.C.C. Lam, P. Tong, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures 39, 2731–2743 (2002).
[4] S.K. Park, X.L. Gao, Bernoulli–Euler beam model based on a modified couple stress theory, Journal of Micromechanics and Micro engineering 16, 23–55 (2006).
[5] M. Simsek, J.N. Reddy, Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory, International Journal of Engineering Sciences 64, 37–53 (2013).
[6] M. Shaat, F.F. Mahmoud, X.L. Gao, A.F. Faheem, Sizedependent bending analysis of Kirchhoff nano-plates based on a modified couple-stress theory including surface effects, International Journal of Mechanical Sciences 79, 31–37 (2014).
[7] A. Arani Ghorbanpour, M. Abdollahian, H.M. Jalaei, Vibration of bioliquid-filled microtubules embedded in cytoplasm including surface effects using modified couple stress theory, Journal of Theoretical Biology 367, 29–38 (2015).
[8] H. Darijani, A.H. Shahdadi, A new shear deformation model with modified couple stress theory for microplates, Acta Mechanica 226, 2773–2788 (2015).
[9] W.Y. Gang, L.W. Hui, N. Liu, Nonlinear bending and postbuckling of extensible microscale beams based on modified couple stress theory, Applied Mathematical Modelling 39, 117–127 (2015).
[10] I.S. Podstrigach, Differential equations of the problem of thermodiffusion in isotropic deformed solid bodies, Doklady. Akademii. Nauk Ukrainskoi. SSR 169–172 (1961).
[11] W. Nowacki, Dynamical problems of thermo diffusion in solids I, Bulletin of Polish Academy of Science and Technology 22, 55–64 (1974).
[12] W. Nowacki, Dynamical problems of thermo diffusion in solids II, Bulletin of Polish Academy of Science and Technology 22, 129–135 (1974).
[13] W. Nowacki, Dynamical problems of thermo diffusion in solids III, Bulletin of Polish Academy of Science and Technology 22, 257–266 (1974).
[14] W. Nowacki, Dynamical problems of thermo diffusion in solids, Engineering Fracture Mechanics 8, 261–266 (1976).
[15] H.H. Sherief, H. Saleh, F. Hamza, The theory of generalized thermoelastic diffusion, International Journal of Engineering Sciences 42, 591–608 (2004).
[16] H.H. Sherief, H. Saleh, A half-space problem in the theory of generalized thermoelastic diffusion, International Journal of Solids and Structures 42, 4484–4493 (2005).
[17] R. Kumar, T. Kansal, Propagation of Lamb waves in transversely isotropic thermoelastic diffusion plate, International Journal of Solids and Structures 45, 5890–5913 (2008).
[18] N.M. El-Maghraby, A.A. Abdel-Halim, A generalized thermoelasticity problem for a half space with heat sources under axisymmetric distributions, Australian Journal of Basic and Applied Sciences 4, 3803–3814 (2010).
[19] H.W Lord, Y. Shulman, A generalized dynamical theory of thermoelasticity, Journal of Mechanics and Physics of Solids 15, 299–309 (1967).
[20] J.J. Tripathi, G.D. Kedar, K.C. Deshmukh, Dynamic problem of generalized thermoelasicity for a semi-infinite cylinder with heat sources, Journal of Thermoelasticity 2, 1–8 (2014).
[21] J.J. Tripathi, G.D. Kedar, K.C. Deshmukh, Generalized thermoelastic diffusion problem in a thick circular plate with axisymmetric heat supply, Acta Mechanica 226, 2121–2134 (2015).
[22] G. Honig, U. Hirdes, A method for the numerical inversion of the Laplace transforms, Journal of Computational and Applied Mathematics 10, 113–132 (1984).
[23] W.H. Press, S.A. Teukolsky, W.T. Vellerling, B.P. Flannery, Numerical recipes, Cambridge University Press, 1986.
[24] R.S. Daliwal, A. Singh, Dynamical coupled thermoelasticity, Hindustan Publishers, Delhi, 1980.
The aim of present study is to present a mathematical model for predicting the results for displacements, stress components, temperature change and chemical potential with considering independently a particular type of heat source. The general solution for the two-dimensional problem of a thick circular plate with heat sources in modified couple stress thermoelastic diffusion has been obtained in the context of one and two relaxation times. Laplace and Hankel transforms technique is applied to obtain the solutions of the governing equations. Resulting quantities are obtained in the transformed domain. The numerical inversion technique has been used to obtain the solutions in the physical domain. Effects of time on the resulting quantities are shown graphically.
Key words:
heat sources, Laplace and Hankel transforms, modified couple stress theory, thermoelasticity, thick circular plate
References:
[1] R.S. Lakes, Dynamical study of couple stress effects in human compact bone, Journal of Biomechanical Engineering 104, 6–11 (1982).
[2] D.C.C. Lam, F. Yang, A.C.M. Chong, J. Wang, P. Tong, Experiments and theory in strain gradient elasticity, Journal of Mechanics and Physics of Solids 51, 1477–1508 (2003).
[3] F. Yang, A.C.M. Chong, D.C.C. Lam, P. Tong, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures 39, 2731–2743 (2002).
[4] S.K. Park, X.L. Gao, Bernoulli–Euler beam model based on a modified couple stress theory, Journal of Micromechanics and Micro engineering 16, 23–55 (2006).
[5] M. Simsek, J.N. Reddy, Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory, International Journal of Engineering Sciences 64, 37–53 (2013).
[6] M. Shaat, F.F. Mahmoud, X.L. Gao, A.F. Faheem, Sizedependent bending analysis of Kirchhoff nano-plates based on a modified couple-stress theory including surface effects, International Journal of Mechanical Sciences 79, 31–37 (2014).
[7] A. Arani Ghorbanpour, M. Abdollahian, H.M. Jalaei, Vibration of bioliquid-filled microtubules embedded in cytoplasm including surface effects using modified couple stress theory, Journal of Theoretical Biology 367, 29–38 (2015).
[8] H. Darijani, A.H. Shahdadi, A new shear deformation model with modified couple stress theory for microplates, Acta Mechanica 226, 2773–2788 (2015).
[9] W.Y. Gang, L.W. Hui, N. Liu, Nonlinear bending and postbuckling of extensible microscale beams based on modified couple stress theory, Applied Mathematical Modelling 39, 117–127 (2015).
[10] I.S. Podstrigach, Differential equations of the problem of thermodiffusion in isotropic deformed solid bodies, Doklady. Akademii. Nauk Ukrainskoi. SSR 169–172 (1961).
[11] W. Nowacki, Dynamical problems of thermo diffusion in solids I, Bulletin of Polish Academy of Science and Technology 22, 55–64 (1974).
[12] W. Nowacki, Dynamical problems of thermo diffusion in solids II, Bulletin of Polish Academy of Science and Technology 22, 129–135 (1974).
[13] W. Nowacki, Dynamical problems of thermo diffusion in solids III, Bulletin of Polish Academy of Science and Technology 22, 257–266 (1974).
[14] W. Nowacki, Dynamical problems of thermo diffusion in solids, Engineering Fracture Mechanics 8, 261–266 (1976).
[15] H.H. Sherief, H. Saleh, F. Hamza, The theory of generalized thermoelastic diffusion, International Journal of Engineering Sciences 42, 591–608 (2004).
[16] H.H. Sherief, H. Saleh, A half-space problem in the theory of generalized thermoelastic diffusion, International Journal of Solids and Structures 42, 4484–4493 (2005).
[17] R. Kumar, T. Kansal, Propagation of Lamb waves in transversely isotropic thermoelastic diffusion plate, International Journal of Solids and Structures 45, 5890–5913 (2008).
[18] N.M. El-Maghraby, A.A. Abdel-Halim, A generalized thermoelasticity problem for a half space with heat sources under axisymmetric distributions, Australian Journal of Basic and Applied Sciences 4, 3803–3814 (2010).
[19] H.W Lord, Y. Shulman, A generalized dynamical theory of thermoelasticity, Journal of Mechanics and Physics of Solids 15, 299–309 (1967).
[20] J.J. Tripathi, G.D. Kedar, K.C. Deshmukh, Dynamic problem of generalized thermoelasicity for a semi-infinite cylinder with heat sources, Journal of Thermoelasticity 2, 1–8 (2014).
[21] J.J. Tripathi, G.D. Kedar, K.C. Deshmukh, Generalized thermoelastic diffusion problem in a thick circular plate with axisymmetric heat supply, Acta Mechanica 226, 2121–2134 (2015).
[22] G. Honig, U. Hirdes, A method for the numerical inversion of the Laplace transforms, Journal of Computational and Applied Mathematics 10, 113–132 (1984).
[23] W.H. Press, S.A. Teukolsky, W.T. Vellerling, B.P. Flannery, Numerical recipes, Cambridge University Press, 1986.
[24] R.S. Daliwal, A. Singh, Dynamical coupled thermoelasticity, Hindustan Publishers, Delhi, 1980.