Fundamental Solution for the Plane Problem in Magnetothermoelastic Diffusion Media
Kumar Rajneesh *, Chawla Vijay **
Department of Mathematics, Kurukshetra University
Kurukshetra-136119, Haryana (India)
E-mails: *rajneesh_kuk@rediffmail.com, **vijay.chawla@ymail.com
Received:
Received: 22 May 2012; revised: 3 May 2013; accepted: 17 May 2013; published online: 23 September 2013
DOI: 10.12921/cmst.2013.19.4.195-207
OAI: oai:lib.psnc.pl:456
Abstract:
The aim of the present paper is to study the fundamental solution in orthotropic magneto- thermoelastic diffusion media. With this objective, firstly the two-dimensional general solution in orthotropic magnetothermoelastic diffusion media is derived. On the basis of thegeneral solution, the fundamental solution for a steady point heat source in an infinite and a semi-infinite orthotropic magnetothermoelastic diffusion material is constructed by four newly introduced harmonic functions. The components of displacement, stress, temperature distribution and mass concentration are expressed in terms of elementary functions. From the present investigation, some special cases of interest are also deduced and compared with the previously obtained results. The resulting quantities are computed numerically for infinite and semi-infinite magnetothermoelastic material and presented graphically to depict the magnetic effect.
Key words:
fundamental solution, infinite, magnetothermoelastic diffusion, orthotropic, semi-infinite
References:
[1] H.J.Ding, B.Chen, J. Liang, General Solutions for Coupled
Equations in Piezoelectric Media, International Journal of
Solids and Structures 33, 2283-2298 (1996).
[2] M.L.Dunn, H.A.Wienecke, Half Space Green’s Functions
for Transversely Isotropic Piezoelectric Solids, Journal of
Applied Mechanics 66, 675-679 (1999).
[3] E. Pan, F. Tanon, Three Dimensional Green’s Functions
in Anisotropic Piezoelectric Solids, International Journal of
Solids and Structures 37, 943-958 (2000).
[4] H.J.Ding, F.L.Guo, and P.F.Hou, A general Solution for
Piezothermoelasticity of Transversely Isotropic Piezoelec-
tric Materials and its Applications, International Journal of
Engineering Science 38, 1415-1440 (2000).
[5] B. Sharma, Thermal Stresses in Transversely Isotropic Semi-
Infinite Elastic Solids, ASME Journal of Applied Mechanics
23, 86-88 (1958).
[6] W.Q. Chen, H.J. Ding, D.S. Ling, Thermoelastic field of
Transversely Isotropic Elastic Medium Containing a Penny-
Shaped Crack: Exact Fundamental Solution, International
Journalof Solids and Structures 41, 69-83 (2004).
[7] P.F. Hou, A.Y.T. Leung, C.P. Chen, Green’s Functions for
Semi-Infinite Transversely Isotropic Thermoelastic Materi-
als, ZAMM Z. Angew. Math. Mech. 1, 33-41 (2008).
[8] P.F. Hou, L. Wang, T. Yi, 2D Green’s Functions for Semi-
Infinite Orthotropic Thermoelastic Plane, Applied Mathe-
matical Modeling 33, 1674-1682 (2009).
[9] P.F. Hou, H. Sha, C.P. Chen, 2D General Solution and Fun-
damental Solution for Orthotropic Thermoelastic Materials,
Engineering.Analysis with Boundary Elements 45, 392-408
(2008).
[10] S. Kaloski, W. Nowacki, Wave Propagation of Thermo-
Magneto-Microelasticity, Bulletin of the Polish Academy of
Sciences Technical Sciences 18 155-158 (1970).
[11] M.I.A.Othman, Y. Song, Reflection of Magneto-
Thermoelastic Waves with Two-Relaxation Times and Tem-
perature Dependent Elastic Moduli, Applied Mathematical
Modeling 32,483-500 (2008).
[12] P.F. Hou, T. Yi, L.Wang, 2D General Solution and
Fundamental Solution for Orthotropic Electro-Magneto-
Thermoelastic Materials, Journal of Thermal Stresses 31,
807-822 (2008).
[13] W. Nowacki, Dynamical Problem of Thermodiffusion
in Solid – I, Bulletin ofpolish Academy of Sciences Series,
Science and Technology 22, 55-64 (1974).
[14] W. Nowacki, Dynamical Problem of Thermodiffusion
in Solid – II, Bulletin of Polish Academy of Sciences Series,
Science and Technology 22, 129-135 (1974).
[15] W. Nowacki, Dynamical Problem of Thermodiffusion
in Solid – III, Bulletin of Polish Academy of Sciences Se-
ries, Science and Technology 22, 275-276 (1974).
[16] W. Nowacki, Dynamical Problems of Thermodiffusion in
Solids, Proc. Vib. Prob. 15, 105-128 (1974).
[17] H.H. Sherief, H. Saleh, A Half Space Problem in the Theory
of Generalized Thermoelastic Diffusion, International Jour-
nal of Solids and Structures 42, 4484-4493 (2005).
[18] R. Kumar, T. Kansal, Propagation of Lamb waves in trans-
versely isotropic Thermoelastic diffusive plate, Int. J. Solid
Struct. 45 (2008), pp. 5890-5913.
[19] R. Kumar, V. Chawla , A Study of Plane Wave Propagation
in Anisotropic Three-Phase-Lag and Two-Phase-Lag Model,
International Communication in Heat and Mass Transfer 38,
1262-1268 (2011).
[20] R. Kumar, V. Chawla, A Study of Fundamental Solution
in Orthotropic Thermodiffusive Elastic Media, International
Communication in Heat and Mass Transfer 38, 456-462
(2011).
[21] R. Kumar, V. Chawla, Green’s Functions in Orthotropic
ThermoelasticDiffusion Media, Engineering Analysis with
Boundary Elements 36, 1272-1277 (2012).
[22] M.A. Ezzat, State approach to generalized magneto-
thermoelastic with two-relaxation times in a medium of per-
fect conductivity, International Journal of Engineering Sci-
ence 35, 741-752 (1997).
[23] A.C. Eringen, Foundations and Solids, Microcontinuum
Field Theories, Springer-Verlag, New York 1999.
The aim of the present paper is to study the fundamental solution in orthotropic magneto- thermoelastic diffusion media. With this objective, firstly the two-dimensional general solution in orthotropic magnetothermoelastic diffusion media is derived. On the basis of thegeneral solution, the fundamental solution for a steady point heat source in an infinite and a semi-infinite orthotropic magnetothermoelastic diffusion material is constructed by four newly introduced harmonic functions. The components of displacement, stress, temperature distribution and mass concentration are expressed in terms of elementary functions. From the present investigation, some special cases of interest are also deduced and compared with the previously obtained results. The resulting quantities are computed numerically for infinite and semi-infinite magnetothermoelastic material and presented graphically to depict the magnetic effect.
Key words:
fundamental solution, infinite, magnetothermoelastic diffusion, orthotropic, semi-infinite
References:
[1] H.J.Ding, B.Chen, J. Liang, General Solutions for Coupled
Equations in Piezoelectric Media, International Journal of
Solids and Structures 33, 2283-2298 (1996).
[2] M.L.Dunn, H.A.Wienecke, Half Space Green’s Functions
for Transversely Isotropic Piezoelectric Solids, Journal of
Applied Mechanics 66, 675-679 (1999).
[3] E. Pan, F. Tanon, Three Dimensional Green’s Functions
in Anisotropic Piezoelectric Solids, International Journal of
Solids and Structures 37, 943-958 (2000).
[4] H.J.Ding, F.L.Guo, and P.F.Hou, A general Solution for
Piezothermoelasticity of Transversely Isotropic Piezoelec-
tric Materials and its Applications, International Journal of
Engineering Science 38, 1415-1440 (2000).
[5] B. Sharma, Thermal Stresses in Transversely Isotropic Semi-
Infinite Elastic Solids, ASME Journal of Applied Mechanics
23, 86-88 (1958).
[6] W.Q. Chen, H.J. Ding, D.S. Ling, Thermoelastic field of
Transversely Isotropic Elastic Medium Containing a Penny-
Shaped Crack: Exact Fundamental Solution, International
Journalof Solids and Structures 41, 69-83 (2004).
[7] P.F. Hou, A.Y.T. Leung, C.P. Chen, Green’s Functions for
Semi-Infinite Transversely Isotropic Thermoelastic Materi-
als, ZAMM Z. Angew. Math. Mech. 1, 33-41 (2008).
[8] P.F. Hou, L. Wang, T. Yi, 2D Green’s Functions for Semi-
Infinite Orthotropic Thermoelastic Plane, Applied Mathe-
matical Modeling 33, 1674-1682 (2009).
[9] P.F. Hou, H. Sha, C.P. Chen, 2D General Solution and Fun-
damental Solution for Orthotropic Thermoelastic Materials,
Engineering.Analysis with Boundary Elements 45, 392-408
(2008).
[10] S. Kaloski, W. Nowacki, Wave Propagation of Thermo-
Magneto-Microelasticity, Bulletin of the Polish Academy of
Sciences Technical Sciences 18 155-158 (1970).
[11] M.I.A.Othman, Y. Song, Reflection of Magneto-
Thermoelastic Waves with Two-Relaxation Times and Tem-
perature Dependent Elastic Moduli, Applied Mathematical
Modeling 32,483-500 (2008).
[12] P.F. Hou, T. Yi, L.Wang, 2D General Solution and
Fundamental Solution for Orthotropic Electro-Magneto-
Thermoelastic Materials, Journal of Thermal Stresses 31,
807-822 (2008).
[13] W. Nowacki, Dynamical Problem of Thermodiffusion
in Solid – I, Bulletin ofpolish Academy of Sciences Series,
Science and Technology 22, 55-64 (1974).
[14] W. Nowacki, Dynamical Problem of Thermodiffusion
in Solid – II, Bulletin of Polish Academy of Sciences Series,
Science and Technology 22, 129-135 (1974).
[15] W. Nowacki, Dynamical Problem of Thermodiffusion
in Solid – III, Bulletin of Polish Academy of Sciences Se-
ries, Science and Technology 22, 275-276 (1974).
[16] W. Nowacki, Dynamical Problems of Thermodiffusion in
Solids, Proc. Vib. Prob. 15, 105-128 (1974).
[17] H.H. Sherief, H. Saleh, A Half Space Problem in the Theory
of Generalized Thermoelastic Diffusion, International Jour-
nal of Solids and Structures 42, 4484-4493 (2005).
[18] R. Kumar, T. Kansal, Propagation of Lamb waves in trans-
versely isotropic Thermoelastic diffusive plate, Int. J. Solid
Struct. 45 (2008), pp. 5890-5913.
[19] R. Kumar, V. Chawla , A Study of Plane Wave Propagation
in Anisotropic Three-Phase-Lag and Two-Phase-Lag Model,
International Communication in Heat and Mass Transfer 38,
1262-1268 (2011).
[20] R. Kumar, V. Chawla, A Study of Fundamental Solution
in Orthotropic Thermodiffusive Elastic Media, International
Communication in Heat and Mass Transfer 38, 456-462
(2011).
[21] R. Kumar, V. Chawla, Green’s Functions in Orthotropic
ThermoelasticDiffusion Media, Engineering Analysis with
Boundary Elements 36, 1272-1277 (2012).
[22] M.A. Ezzat, State approach to generalized magneto-
thermoelastic with two-relaxation times in a medium of per-
fect conductivity, International Journal of Engineering Sci-
ence 35, 741-752 (1997).
[23] A.C. Eringen, Foundations and Solids, Microcontinuum
Field Theories, Springer-Verlag, New York 1999.
