MAGNETO-THERMOELASTIC WAVES INDUCED BY A THERMAL SHOCK IN A FINITELY CONDUCTING ELASTIC HALF SPACE

The propagation of magneto-thermoelastic disturbances produced by a thermal shock in a finitely conducting elastic half-space in contact with vacuum is investigated. The boundary of the half-space is subjected to a normal load. Lord-Shulman theory of thermoelasticity [1] is used to account for the interaction between the elastic and thermal fields. Laplace transform on time is used to obtain the short-time approximations of the solutions because of the short duration of 'second sound' effects. It is found that in the half-space the displacement is continuous at the modified dilational and thermal wavefronts, whereas the perturbed magnetic field, stress and the temperature suffer discontinuities at these locations. The perturbed magnetic field, is, however, discontinuous at the Alf'ven-acoustic wavefront in vacuum.


INTRODUCTION
The generation of magneto-thermoelastic waves by a thermal shock in a perfectly conducting half-space in contact with vacuum was investigated by Kaliski and Nowacki [2].Both media were supposed to be permeated by a primary uniform magnetic field.But the influence of coupling between the temperature and strain fields was neglected.The coupling between temperature and strain fields was taken into account by Massalas and Dalamangas [3].Then Roychoudhuri and Chatterjee [4,5] extended the problem [3] in generalized thermoelasticity by using the thermal relaxation time of Lord-Shulman theory 1] and the theory of Green and Lindsay [6], involving two relaxation times.
Later, Sharma and Dayal Chand [7] studied transient generalized magneto-thermoelastic waves in a perfectly conducting elastic half-space due to a normal load acting on the boundary of the half-space using the generalized theory of thermoelasticity developed by 2. PROBLEM FORMULATION We assume that a magneto-thermoelastic wave is produced in an elastic half-space xl 0 due to a normal load and a thermal shock applied on xt -0.The simplified linear equations of electrodynamics of slowly moving bodies having finite conductivity are the following [10] Po Oh VxE C 0t 4:.-.
Vh ----g (1) V .h - whereE denotes the electric field, h is the perturbation of the magnetic field, H0 is the initial constant magnetic field, " denotes the current density vector, ' denotes the displacement vector, I-is the magnetic permeability, X is the electrical conductivity and C is the velocity of light.
The linear form of the displacement equations of motion including electromagnetic effect and the modified form of Fourier's law of heat conduction in the context of LrdoShulman theory [1] of thermoelasticity are, .v+(x+ )(.u-3 +[() go]-o-o (2) pc(0 + o0) + To(A + 0A) KVO (3) where K, are the Lam6 constants, ?(3.+ 2)ar, ctr is the coefficient of linear thermal expansion, 0-T-To, T is the absolute temperature, To is the uniform temperature of the body in its natural state, K , denotes coefficient of heat conduction, C, is the specific heat at constant strain, p is the mass density, Cv is the sp.heat at constant volume, and o is the thermal relaxation time, A is the dilation.
The equations (1), after elimination of E and j give, v '-'--6",,'0) (4) where I]-C The magneto-thermoelastic wave propagated in the medium xl > 0 is assumed to depend on xl and time t.Furthermore it is assumed that the initial magnetic field vector is directed along the x3-axis i.e.
H (0, 0,H3) where//3 is a constant.Under these assumptions, equations (1) lead to (ao ao pC + +VTo +o._= -K x at oxdt a2h3 Oh3 a2ui For clarity, we shall use the notations, u -u, xx in the following. Since, the elastic medium is in contact with the vacuum, equations ( 5)-( 7) have to be supplemented by the electrodynamic equations in vacuum.
In vacuum, the system of equations of electrodynamics reduce to the following

BOUNDARY CONDITIONS
The components of Maxwell's stress tensor in the elastic medium T n and in vacuum a are given by Tn -hfl /,l ; The normal stress in the elastic medium is obtained as (9) o-(Z.+ 21-t)'-y0 The boundary conditions are assumed as o,+Tlt-'tl-OoH(t) on x-x'-0 (10) (11) e-,hon x-x'-o and the thermal boundary condition is assumed as 0(0,t)-0oH(/) on x-x'-O where H(t) is the Heaviside unit function.

SHORT TIME APPROXIMATION
The inversion of the Laplace transform is very difficult because of the dependency of on s.To reduce these difficulties, we use some approximate methods.The thermal relaxation effects are short-lived.Accordingly we concentrate our attention on small time approximations.For large s, ',2 -m + BI'z + Dl'2 s where W, (P2 +/-Fla)'a/v (45) Bi,2 [P, (P,P2 2)/Fxa]/2q(P2 ++-1"1/2) la (46) Dx,2-[+/-P2/Fla(pxP2-2)2/F3a-(P +_(P,P2-2)/rla)2/2(P2+_r'a)]/4v(P2+/-r'a)v2 (47) F-P-4Xo', P-l+e, P2"l+to'+exo' From the short-time solutions it is observed that the solution consists of three waves--the modified elastic wave travelling with velocity V 1, the modified thermal wave travelling with velocity V2 and Alf'ven acoustic wave moving with velocity L The terms containing H x-g represent the contribution of the elastic wave in the vicinity of the wave front VF., the terms with HO: ) represent the contribution of the thermal wave in the vicinity of the wavefront -V2x and the terms with HO:-a') represent the contribution of the All'yen acoustic wave in the vicinity of the wavefront ,'-! ,.
We observe that, in the solid the displacement is continuous at the modified elastic and thermal wavefronts, but the temperature, total stress and perturbed magnetic field suffer discontinuities at the two wave-fronts, whereas in the vacuum the perturbed magnetic field suffers discontinuity at the Alf'ven acoustic wavefront.
The discontinuities are given by v?v [z z-.,,,, is"

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning 2), (3) and (4) then reduce to 0ul H ah OO 02u (+2)0x 4n ax, Y" p'

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation