Interaction due to Hall Current and Rotation in a Modified Couple Stress Elastic Half-Space due to Ramp-type Loading
Kumar Rajneesh 1*, Devi Shaloo 2
1 Department of Mathematics, Kurukshetra University Kurukshetra
Kurukshetra, India
∗E-mail: rajneesh_kuk@rediffmail.com2 Department of Mathematics & Statistics, Himachal Pradesh University Shimla
Shimla, India
E-mail: shaloosharma2673@gmail.com
Received:
Received: 29 April 2015; revised: 21 December 2015; accepted: 21 December 2015; published online: 29 December 2015
DOI: 10.12921/cmst.2015.21.04.007
Abstract:
The present investigation is to focus on the effect of Hall current and rotation in a modified couple stress theory of elastic half space due to ramp-type loading in a homogeneous, isotropic, thermoelastic diffusive medium. The mathematical formulation is prepared for different theories of thermoelastic diffusion, including the Coriolis and centrifugal forces. The Laplace and Fourier transforms techniques are applied to obtain the solutions of the governing equations. The components of
displacement, stresses, temperature change and mass concentration are obtained in the transformed domain. The numerical inversion technique has been used to obtain the solutions in the physical domain. Effects of Hall current and rotation are shown on the resulting quantities. Some particular cases are also discussed in the present problem.
Key words:
generalized thermoelasticity, Hall current and rotation, Laplace and Fourier transforms, modified couple stress, ramp-type Loading
References:
[1] R.A. Toupin, Elastic materials with couple-stresses. Arch. Rational Mech. and Anal., 11, 385-414, (1962).
[2] R.D. Mindlin, H. F. Tiersten, Effects of couple-stresses in linear elasticity. Arch. Rational Mech. and Anal., 11, 415-448, (1962).
[3] W.T. Koiter, Couple-stresses in the theory of elasticity. Proc. R. Neth. Acad. Sci., 67, 17-44, (1964).
[4] J. Zhao, C. Wanji, B. Ji, A weak continuity condition of FEM for axisymmetric couple stress theory and an 18-DOF triangular
axisymmetric element. Finite Elements in Analysis and Design, 46(8), 632-644, (2010).
[5] J.N. Reddy, Microstructure-dependent couple stress theories of functionally graded beams. J. Mech. Phys. Solids, 59, 2382–2399,
(2011).
[6] A.R. Hadjesfandiari, G. F. Dargush, Boundary element formulation for plane problems in couple stress elasticity. Numerical methods
in Engineering, 89(5), 618-636, (2012).
[7] 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).
[8] M. R. Shankar, S. Chandrasekar, T. N., Farris Interaction between dis-locations in a couple stress medium. ASME J. Appl. Mech., 71,
546-550, (2004).
[9] C. Babaoglu, S. Erbay, Two-dimensional wave packets in an elastic solid with couple stresses. Int. J. Non-Linear Mech., 39, 941-949,
(2004).
[10] S. Diebels, H. Steeb, Stress and couple stress in foams. Comput. Mater. Sci., 28, 714-722, (2003).
[11] M.A. Kulesh, V. P. Matveenko, I. N. Shardakov, Parametric analysis of analytical solutions to one and two dimensional problems in
couple-stress theory of elasticity. ZAMM J. Appl. Math. Mech., 83, 238-248, (2003).
[12] S.K. Park, X. L. Gao, Bernoulli–Euler beam model based on a modified couple stress theory. J. of Micromech. and Micro engg., 16,
2355, (2006).
[13] H.M. Ma, X. L. Gao, J. N. Reddy, A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J.
of the Mech. and Phys. of Solids, 56, 3379- 3391, (2008).
[14] M. Marin, On the minimum principle for dipolar materials with stretch. Nonlinear Analysis: Real World Applications, 10, 1572-1578,
(2009).
[15] G.C. Tsiatas, A new Kirchhoff plate model based on a modified couple stress theory. International Journal of Solids and Structures,
46, 2757–2764, (2009).
[16] H.M. Ma, X. L. Gao, J. N. Reddy, A non-classical Mindlin plate model based on a modified couple stress theory. Acta. Mech. 220,
217–35, (2011).
[17] M. Asghari, Geometrically nonlinear micro-plate formulation based on the modified couple stress theory. Int. J. Engg. Sci., 51,
292–309, (2012).
[18] M. Marin and G. Stan, Weak solutions in elasticity of dipolar bodies with stretch. CARPATHIAN J. of Mathematics, 29(1), 33-40,
(2013).
[19] 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. Int. J. of Engg. Sci., 64, 37–53, (2013).
[20] M. Marin, R. P. Agarwal and S. R. Mahmoud, Nonsimple material probems addressed by the Lagrange’s identity. Boundary Value
Problems, Article No. 135, (2013), DOI: 10.1186/1687-2770-2013-135.
[21] M. Mohammad-Abadi, A. R. Daneshmehr, Size dependent buckling analysis of micro beams based on modified couple stress theory
with high order theories and general boundary conditions. Int.J. of Engg. Sci., 74, 1–14, (2014).
[22] M. Shaat, F. F. Mahmoud, X.-L. Gao, A. F. Faheem, Size-dependent bending analysis of Kirchhoff nano-plates based on a modified
couple-stress theory including surface effects. Int. J. of Mech. Sci., (2013), http://dx.doi.org/10.1016/j.ijmecsci.2013.11.022i.
[23] A. Arani Ghorbanpour, M. Abdollahian and H. M. Jalaei, Vibration of bioliquid-filled microtubules embedded in cytoplasm including
surface effects using modified couple stress theory. J. of Theoret. Biology, 367, 29-38, (2015).
[24] Yong-Gang Wang, Wen-Hui Lin, Ning Liu, Nonlinear bending and post-buckling of extensible microscale beams based on modified
couple stress theory. App. Math. Model.39 117–127, (2015).
[25] Ia. S. Podstrigach, Differential equations of the problem of thermodiffusion in isotropic deformed solid bodies. Dop. Akad. Nauk Ukr.
SSR, 169-172, (1961).
[26] W. Nowacki, Dynamical problems of thermo diffusion in solids I. Bull Acad. Pol. Sci. Ser. Sci, Tech., 22, 5564, (1974a).
[27] W. Nowacki, Dynamical problems of thermo diffusion in solids II. Bull Acad. Pol. Sci. Ser. Sci, Tech., 22, 129-135, (1974b).
[28] W. Nowacki, Dynamical problems of thermo diffusion in solids III. Bull Acad. Pol. Sci. Ser. Sci, Tech, 22, 257-266, (1974c).
[29] W. Nowacki, Dynamical problems of thermo diffusion in solids. Engg. Frac. Mech., 8, 261-266, (1976).
[30] H.H. Sherief, H. Saleh, F. Hamza, The theory of generalized thermoelastic diffusion. Int. J. Engg. Sci., 42, 591-608, (2004).
[31] H.H. Sherief, H. Saleh, A half-space problem in the theory of generalized thermoelastic diffusion. Int. J. of solid and structures,42,
4484-4493, (2005).
[32] R. Kumar, T. Kansal, Propagation of Lamb waves in transversely isotropic thermoelastic diffusion plate. Int. J. Solid Struc.,45,
5890-5913, (2008).
[33] L. Knopoff, The interaction between elastic wave motion and a magnetic field in electrical conductors. J. Geophys. Res. 60, 441–456,
(1955).
[34] P. Chadwick, Ninth Int. Congr. Appl. Mech. 7, 143, (1957).
[35] S. Kaliski, J. Petykiewicz, Equation of motion coupled with the field of temperature in a magnetic field involving mechanical and
electrical relaxation for anisotropic bodies. Proc. Vibr. Probl. 4, 1, (1959).
[36] M. Zakaria, Effects of Hall Current and Rotation on Magneto Micropolar Generalized Thermoelasticity due to Ramp-Type Heating.
International Journal of Electromagnetics and Applications, 2, 24-32, (2012).
[37] M. Zakaria, Effect of Hall current on generalized magneto-thermoelasticity micropolar solid subjected to ramp-type heating. Int.
Appl. Mech., 50, 130-144, (2014).
[38] G. Honig, U. Hirdes, A method for the numerical inversion of the Laplace transforms. J. Comput. Appl. Math. 10(1), 113-132, (1984).
[39] W.H. Press, S.A. Teukolsky, W.T. Vellerling, B.P. Flannery, Numerical recipes (Cambridge: University Press), (1986).
The present investigation is to focus on the effect of Hall current and rotation in a modified couple stress theory of elastic half space due to ramp-type loading in a homogeneous, isotropic, thermoelastic diffusive medium. The mathematical formulation is prepared for different theories of thermoelastic diffusion, including the Coriolis and centrifugal forces. The Laplace and Fourier transforms techniques are applied to obtain the solutions of the governing equations. The components of
displacement, stresses, temperature change and mass concentration are obtained in the transformed domain. The numerical inversion technique has been used to obtain the solutions in the physical domain. Effects of Hall current and rotation are shown on the resulting quantities. Some particular cases are also discussed in the present problem.
Key words:
generalized thermoelasticity, Hall current and rotation, Laplace and Fourier transforms, modified couple stress, ramp-type Loading
References:
[1] R.A. Toupin, Elastic materials with couple-stresses. Arch. Rational Mech. and Anal., 11, 385-414, (1962).
[2] R.D. Mindlin, H. F. Tiersten, Effects of couple-stresses in linear elasticity. Arch. Rational Mech. and Anal., 11, 415-448, (1962).
[3] W.T. Koiter, Couple-stresses in the theory of elasticity. Proc. R. Neth. Acad. Sci., 67, 17-44, (1964).
[4] J. Zhao, C. Wanji, B. Ji, A weak continuity condition of FEM for axisymmetric couple stress theory and an 18-DOF triangular
axisymmetric element. Finite Elements in Analysis and Design, 46(8), 632-644, (2010).
[5] J.N. Reddy, Microstructure-dependent couple stress theories of functionally graded beams. J. Mech. Phys. Solids, 59, 2382–2399,
(2011).
[6] A.R. Hadjesfandiari, G. F. Dargush, Boundary element formulation for plane problems in couple stress elasticity. Numerical methods
in Engineering, 89(5), 618-636, (2012).
[7] 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).
[8] M. R. Shankar, S. Chandrasekar, T. N., Farris Interaction between dis-locations in a couple stress medium. ASME J. Appl. Mech., 71,
546-550, (2004).
[9] C. Babaoglu, S. Erbay, Two-dimensional wave packets in an elastic solid with couple stresses. Int. J. Non-Linear Mech., 39, 941-949,
(2004).
[10] S. Diebels, H. Steeb, Stress and couple stress in foams. Comput. Mater. Sci., 28, 714-722, (2003).
[11] M.A. Kulesh, V. P. Matveenko, I. N. Shardakov, Parametric analysis of analytical solutions to one and two dimensional problems in
couple-stress theory of elasticity. ZAMM J. Appl. Math. Mech., 83, 238-248, (2003).
[12] S.K. Park, X. L. Gao, Bernoulli–Euler beam model based on a modified couple stress theory. J. of Micromech. and Micro engg., 16,
2355, (2006).
[13] H.M. Ma, X. L. Gao, J. N. Reddy, A microstructure-dependent Timoshenko beam model based on a modified couple stress theory. J.
of the Mech. and Phys. of Solids, 56, 3379- 3391, (2008).
[14] M. Marin, On the minimum principle for dipolar materials with stretch. Nonlinear Analysis: Real World Applications, 10, 1572-1578,
(2009).
[15] G.C. Tsiatas, A new Kirchhoff plate model based on a modified couple stress theory. International Journal of Solids and Structures,
46, 2757–2764, (2009).
[16] H.M. Ma, X. L. Gao, J. N. Reddy, A non-classical Mindlin plate model based on a modified couple stress theory. Acta. Mech. 220,
217–35, (2011).
[17] M. Asghari, Geometrically nonlinear micro-plate formulation based on the modified couple stress theory. Int. J. Engg. Sci., 51,
292–309, (2012).
[18] M. Marin and G. Stan, Weak solutions in elasticity of dipolar bodies with stretch. CARPATHIAN J. of Mathematics, 29(1), 33-40,
(2013).
[19] 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. Int. J. of Engg. Sci., 64, 37–53, (2013).
[20] M. Marin, R. P. Agarwal and S. R. Mahmoud, Nonsimple material probems addressed by the Lagrange’s identity. Boundary Value
Problems, Article No. 135, (2013), DOI: 10.1186/1687-2770-2013-135.
[21] M. Mohammad-Abadi, A. R. Daneshmehr, Size dependent buckling analysis of micro beams based on modified couple stress theory
with high order theories and general boundary conditions. Int.J. of Engg. Sci., 74, 1–14, (2014).
[22] M. Shaat, F. F. Mahmoud, X.-L. Gao, A. F. Faheem, Size-dependent bending analysis of Kirchhoff nano-plates based on a modified
couple-stress theory including surface effects. Int. J. of Mech. Sci., (2013), http://dx.doi.org/10.1016/j.ijmecsci.2013.11.022i.
[23] A. Arani Ghorbanpour, M. Abdollahian and H. M. Jalaei, Vibration of bioliquid-filled microtubules embedded in cytoplasm including
surface effects using modified couple stress theory. J. of Theoret. Biology, 367, 29-38, (2015).
[24] Yong-Gang Wang, Wen-Hui Lin, Ning Liu, Nonlinear bending and post-buckling of extensible microscale beams based on modified
couple stress theory. App. Math. Model.39 117–127, (2015).
[25] Ia. S. Podstrigach, Differential equations of the problem of thermodiffusion in isotropic deformed solid bodies. Dop. Akad. Nauk Ukr.
SSR, 169-172, (1961).
[26] W. Nowacki, Dynamical problems of thermo diffusion in solids I. Bull Acad. Pol. Sci. Ser. Sci, Tech., 22, 5564, (1974a).
[27] W. Nowacki, Dynamical problems of thermo diffusion in solids II. Bull Acad. Pol. Sci. Ser. Sci, Tech., 22, 129-135, (1974b).
[28] W. Nowacki, Dynamical problems of thermo diffusion in solids III. Bull Acad. Pol. Sci. Ser. Sci, Tech, 22, 257-266, (1974c).
[29] W. Nowacki, Dynamical problems of thermo diffusion in solids. Engg. Frac. Mech., 8, 261-266, (1976).
[30] H.H. Sherief, H. Saleh, F. Hamza, The theory of generalized thermoelastic diffusion. Int. J. Engg. Sci., 42, 591-608, (2004).
[31] H.H. Sherief, H. Saleh, A half-space problem in the theory of generalized thermoelastic diffusion. Int. J. of solid and structures,42,
4484-4493, (2005).
[32] R. Kumar, T. Kansal, Propagation of Lamb waves in transversely isotropic thermoelastic diffusion plate. Int. J. Solid Struc.,45,
5890-5913, (2008).
[33] L. Knopoff, The interaction between elastic wave motion and a magnetic field in electrical conductors. J. Geophys. Res. 60, 441–456,
(1955).
[34] P. Chadwick, Ninth Int. Congr. Appl. Mech. 7, 143, (1957).
[35] S. Kaliski, J. Petykiewicz, Equation of motion coupled with the field of temperature in a magnetic field involving mechanical and
electrical relaxation for anisotropic bodies. Proc. Vibr. Probl. 4, 1, (1959).
[36] M. Zakaria, Effects of Hall Current and Rotation on Magneto Micropolar Generalized Thermoelasticity due to Ramp-Type Heating.
International Journal of Electromagnetics and Applications, 2, 24-32, (2012).
[37] M. Zakaria, Effect of Hall current on generalized magneto-thermoelasticity micropolar solid subjected to ramp-type heating. Int.
Appl. Mech., 50, 130-144, (2014).
[38] G. Honig, U. Hirdes, A method for the numerical inversion of the Laplace transforms. J. Comput. Appl. Math. 10(1), 113-132, (1984).
[39] W.H. Press, S.A. Teukolsky, W.T. Vellerling, B.P. Flannery, Numerical recipes (Cambridge: University Press), (1986).
