STUDY OF MAGNETOTHERMODYNAMIC STRESS AND PERTURBATION OF MAGNETIC FIELD VECTOR IN AN ORTHOTROPIC SOLID CYLINDER

BAGADE S.H.¹, VARGHESE V.², KHOBRAGADE N.W.³*
¹Department of Maths, MJP Educational Campus, RTM Nagpur University, Nagpur- 440033, MS, India.
²Department of Maths, MJP Educational Campus, RTM Nagpur University, Nagpur- 440033, MS, India.
³Department of Maths, MJP Educational Campus, RTM Nagpur University, Nagpur- 440033, MS, India.
* Corresponding Author : khobragadenw@gmail.com

Received : 10-12-2012     Accepted : 18-12-2012     Published : 26-12-2012
Volume : 3     Issue : 3       Pages : 134 - 137
J Stat Math 3.3 (2012):134-137

In this paper, we investigate the problem of magnetothermoelastic responses and perturbation of magnetic field vector in a conducting thermoelastic orthotropic cylinder subjected to rapid temperature change due to thermal shocks. The Laplace transform and finite Marchi- Zgrablich integral transform techniques are used to obtain the analytical expression for Magnetothermodynamic stresses and perturbation response of an axial magnetic field vector in the orthotropic cylinders. Graphical representation of magnetothermostress and perturbation of magnetic field vector shows focusing effect which is then discussed.

Orthotropic Cylinder, Magnetothermoelastic, Perturbation of magnetic field vector Focusing effect.

74J25, 74H99, 74D99

Whenever electromagnetic pulses or thermal shocks of any kind or radiant energy in the form of γ-rays are incident on a conducting orthotropic solid cylinder then absorption of these pulses leads to rapid change on the temperature T(r,t). Also the interactions between the deformations and the applied magnetic field give rise to the theory of dynamic coupled magnetothermoelasticity. Increasing attention is being devoted to this theory due to its applications in the field of engineering, magnetic structural elements, magnetic storage elements etc. The theory is also of great importance in the branches of physics which includes plasma physics, Geophysics and the corresponding measurement techniques of magnetoelasticity.
That magnetoelastic effect can be one of the reasons for the change in electromagnetic field and temperature distribution in Seismic Zones of lithosphere in course of their activation was proposed by Novik [7] . The macroscopic theory for analyzing magnetothermoelastic response in type-II superconductor was studied by Zhou [12] . Dhaliwal et al [1] dealt with the generalized elastic theory applied to the problems of magnetothermoelastic waves produced by thermal shocks in an infinite elastic solid with cylindrical cavity. Ezzat [2] found the distribution of thermal stresses and temperature in perfectly conducting half spaces when suddenly heated in absence of magnetic field. In Sherief et al [8] , the Laplace transform technique was used to find the distribution of thermal stresses and temperature in a thermoelastic, electrically conducting half spaces under sudden thermal shock and permeated by magnetic field. The interactions between deformations and magnetic field in conducting orthotropic cylinder is considered by adding electromagnetic force in Kraus [4] . Solodyak et al [9] presented the analytical method for obtaining electromagnetic and temperature field and stresses in ferromagnetic solid subjected to electromagnetic field.
Till date most of the investigations about magnetothermoelasticity are done with respect to infinite isotropic structure and investigations on magnetothermoelasticity in finite orthotropic structures have been very few. In this paper we attempt to determine the response histories of magnetic thermo stresses and perturbation of magnetic field vector in finite orthotropic cylinder (solid) under sudden temperature change to a constant temperature and permeated by uniform primary magnetic field. The Laplace transform and finite Marchi-Zgrablich transform techniques have been used to analyze the above problem.
In the numerical results presented, it is seen that the response of Magnetothermodynamic stresses and perturbation of magnetic field vector near the center of an orthotropic cylinder reaches a peak value. This is called the wave-focusing effect. This focusing effect is reduced by magnetothermoelastic waves colliding with each other at the center of orthotropic cylinder and continuously accumulating at the center.

Consider a long orthotropic solid cylinder of radius a placed in an axial magnetic field H(0,0,HZ). If electromagnetic pulse or γ-rays pulse radiant energy is incident on a solid cylinder, then it undergoes rapid change in temperature T(r,t).
This then leads to interactions between the deformations of the cylinder and the perturbation of the magnetic field vector in orthotropic cylinder.
Assuming that, the magnetic permeability, µ, of the orthotropic solid cylinder equals the magnetic permeability of the medium around it and omitting the displacement electric current, the governing electrodynamic Maxwell’s equation [6] for perfectly conducting body are



where
J is current density, h is perturbation of magnetic field vector (0,0,hZ)
e is perturbation of electric field vector
H is magnetic intensity vector (0,0,HZ)
U is the displacement vector.
Applying the initial magnetic field H(0,0,HZ) to [Eq-1] in (r,Θ,z) i.e. cylindrical polar coordinate system , we get



(2b)





Ur is the radial displacement component.
From [Eq-1] and [Eq-2] and Fung [3] the magneto elastic dynamic equation of orthotropic cylinder becomes



where is the Lorentz force [4] which is given by



From Lekhnitskii [9] the generalized Hookes law for an orthotropic thermoelastic cylinder can be written as



where σr, σΘ are radial and circumferential stress
Ei, Vi,j αi (I, j=r, Θ, z) are the Young’s modulus, Poison’s ratio and coefficient of thermal expansion in radial, circumferential and axial directions respectively.
are the strains.
Considering a generalized plane strain problem, and solving [Eq-5] we get



where engineering constants βij, S1,S2,S3, are as shown



Substituting [Eq-6] into [Eq-3] , the basic displacement equation of magnetothermoelastic motion in an orthotropic cylinder may be expressed as in Wang, et al. [11] .



Where





Omitting the Maxwell tensor on the surface of Orthotropic cylinder, the corresponding boundary conditions are U_r (0,t)=0



The initial conditions are U_r (r,0)=0 (8c)

Let us assume that the general solution to [Eq-8a] may be expressed in the form

U (r,t) = U_s (r,t)+U_d (r,t) (9)

where U_s (r,t) is the static part of solution to [Eq-8] and U_d is the dynamic part of solution to [Eq-8a] . The static part U_ssatisfies the sol-ution with inhomogeneous boundary conditions while the dynamic part U_d satisfies the solution with homogenous boundary conditions.
For the static part U_s (r,t) the governing equations and boundary conditions become



U_s (0,t) = 0 (10b)



[Eq-10a] simplifies to



From [Eq-10d] the solution of equation 10(a) which satisfies the boundary conditions 10(b) is written as


where



The unknown constant C1 in [Eq-11a] can be easily determined from boundary condition (10c). The dynamic part of the solution Ud(r,t) should satisfy the inhomogeneous [Eq-12a] and corresponding homogenous boundary conditions (12b) and (12c) and initial condition (12d).

(12a)

U_d (0,t) = 0 (12b)





where U_s (r,t) is the static solution as on [Eq-11] .
magneto thermoelastic wave speed.
In order to remove the left hand side group of the [Eq-12a] for the homogenous boundary conditions (12b) of the third kind we apply the finite Marchi-Zgrablich integral transform with respect to the radial coordinate r to [Eq-12a] .
Applying finite Marchi-Zgrablich transform we get



where and are Marchi-Zgrablich transform of and respectively where the eigen values are the positive root of the frequency of characteristic equation.



Applying the Laplace transform to [Eq-13] one obtains

(15)

Where S is the Laplace transform parameter &
Using the inverse Laplace transform and its convolution property in [Eq-15] gives

(16)

Applying the finite inverse Marchi-Zgrablich integral transform to [Eq-16] the solution U_d (r,t) is expressed as



where is finite Marchi-Zgrablich integral transform of with respect to the Kernel function and weight function r. Here the Kernel can be defined as



(18)
Where


for i=1,2,3…. (19)

and are Bessel’s function of first and second kind respectively.
By substituting [Eq-17] and [Eq-11a] in [Eq-9] the general solution for the basic governing [Eq-8] becomes U (r,t) = U_s (r,t) + U_d (r,t)

The symmetric magnetothermoelastic problem of an orthotropic solid cylinder placed in uniform axial magnetic field and subjected to thermal shocks due to any source is considered.
Here for the orthotropic solid cylinder the various material constants taken are :
E_z = 150 GP_a, E_r = 7.5 GP_a, E_θ = 20GP_a
v_zr = 0.25, v_zθ = 0.3, v_θr = 0.2

, ,

Magnetic interference wave speed V=1000 m/s .
Further we define some non-dimensional parameters

and

For the sake of study, three values of R i.e. R=0.01, R=0.5, R=0.9 are considered [Fig-1] , [Fig-2] .

By using the Laplace transform and finite Marchi-Zgrablich transforms, the solutions for displacement, stress components and magnetic field vector perturbations were found out for conducting orthotropic solid cylinder placed in axial magnetic field. Large amplitude of stresses and magnetic field vector perturbations near the center indicate focusing effect. By using this knowledge we can design various magnetoelastic elements for specific engineering requirements. The knowledge can also be used for proper selection of materials for in designing and manufacturing field; and to control the magnetothermoelastic stresses and magnetic field vector perturbations.

The authors are thankful to University Grant Commission, New Delhi for providing the partial financial assistance under major research project scheme.

[1] Dhaliwal R.S. and Saxena H.S. (1991) Journal of Thermal Stresses, 14, 353-369.  
» CrossRef   » Google Scholar   » PubMed   » DOAJ   » CAS   » Scopus  

[2] Ezzat M.A. (1965) Journal of Thermal Stresses, 20, 617-633.  
» CrossRef   » Google Scholar   » PubMed   » DOAJ   » CAS   » Scopus  

[3] Fung Y.C. (1965) Foundation of Solid Mechanics, Prentice Hall, Inc Englewood Clifts, USA.  
» CrossRef   » Google Scholar   » PubMed   » DOAJ   » CAS   » Scopus  

[4] Kraus J.D. (1984) Electro Magnetics, McGraw Hill, Inc, USA.  
» CrossRef   » Google Scholar   » PubMed   » DOAJ   » CAS   » Scopus  

[5] Lekhnitskii S.G. (1981) Theory of Elasticity on Anisotropic Body, Mir Publishers Moscow.  
» CrossRef   » Google Scholar   » PubMed   » DOAJ   » CAS   » Scopus  

[6] Marchi E. and Zgrablich G. (1966) Bulletin of the Calcutta Mathematical Society, 58, 73-76.  
» CrossRef   » Google Scholar   » PubMed   » DOAJ   » CAS   » Scopus  

[7] Novik O.B. (1996) Transaction of the Russian Academy of Sciences, Earth Science Section 341A , 5-11.  
» CrossRef   » Google Scholar   » PubMed   » DOAJ   » CAS   » Scopus  

[8] Sherief H.H., Ezzat M.A. (1996) International Journal of Solids and Structures, 33, 4449-4459.  
» CrossRef   » Google Scholar   » PubMed   » DOAJ   » CAS   » Scopus  

[9] Solodyak M., Gachkevich A. (1996) Mathematical Methods in Electromagnetic Theory, Conference Proceedings, 219-222.  
» CrossRef   » Google Scholar   » PubMed   » DOAJ   » CAS   » Scopus  

[10] Sneddon I.N. (1974) The use of Integral Transforms, McGraw Hill Book, Co.  
» CrossRef   » Google Scholar   » PubMed   » DOAJ   » CAS   » Scopus  

[11] Wang X., Dai H.L. (2004) International Journal of Engineering Sciences., 42, 539-556.  
» CrossRef   » Google Scholar   » PubMed   » DOAJ   » CAS   » Scopus  

[12] Zhou S.A. (1994) International Journal of Applied Electromagnetic in Material, 53, 211-222.  
» CrossRef   » Google Scholar   » PubMed   » DOAJ   » CAS   » Scopus  

	Fig. 1- Perturbation response histories of magnetic field vector in an orthopedic cylinder subjected to thermal shock and permeated by a uniform axial magnetic field, where α=0.01m, R=r/α, τ=tV/α
	Fig. 2- Response histories circumferential stress in an orthopedic cylinder subjected to thermal shock and permeated by a uniform axial magnetic field, where α=0.01m, R=r/α, τ=tV/α