A PROBLEM OF THERMAL SHOCK WITH THERMAL RELAXATION

The problem of a semi-infinite medium subjected to thermal shock on its plane boundary is solved using the generalized theory of thermoelastlclty. The expressions for temperature, strain and stress are presented. The results are exhibited graphically and compared with previous results.

time is considered and is welt-known as the modified coupled heat conduction equation.
Here the problem of an isotropic, homogeneous half-space subjected Lo thermal shock on its plane boundary is solved using the Laplace transform technique.
The equations concerning the generalized theory of thermoelasticlty are used to solve the said problem.
The boundary condition for temperature s in the form of exponential heating, a more realistic situation.
After effect[ng the Laplace inversion, the expressions for temperature, strain and stress are obtained.
As a special case, the results due to Danilovskaya [9] for step-type boundary condftlons and that of Sternburg et al [I0] for ramp type boundary condition can be obta|ned.
Further by setting relaxation constant to zero, the results due to Dalmaruya et al [I l] are obtalned.

FORMULATION OF THE PROBLEM.
Consider an Isotropic homogeneous half space, subjected to a thermal disturbance on its boundary.The governing equatlons of the generalized theory of thermoelastlclty for the one dimensional case, are where k, O, C , T are thermal conductivity, density, specific heat, coefficient of e o linear thermal expansion and the relaxation time, respectively.I and are the well known Lame's elastic constants.
Here T, Oxx and u are temperature, stress and displacement, respectively.
The inltal and boundary conditions are (31 + 2)aT o u'(0,t) (I + 2) [I -exp (-t/to)]" (2.5)In the above, To, and t o are constants.The step type boundary condition is obtained when t o 0, i.e.T(0,t) H(t), and the ramp type boundary condition is obtained by expanding the exponential type and neglecting the higher order terms.
Here H(t) is Heavlside unit step function.
The regularity boundary conditions are T(x,t), u(x,t), (x,t) for distance, time, temperature, stress, and displacement respectlvely in equations (2.1) (2.3), we get e" e' E=U' where 'dot' and 'dash' denote differentiation with respect to y and z respectively and Here B and e are the relaxation constant and thermoelastlc coupling constant, respectively.
In the above, po =p ' + I. (a21_ a22) We consider a special case in which the relaxation constant (B) is expressed in terms of the coupling constant e, i.e.
For this value of B, we get R. RAMAMURTHY AND A.V.M. SHARMA The expression Eor stress can be obtained f-ore (2.8c), (3.9) and (3.10).
Taking 'o 0, recovers the results due to Dan[tovskay.i [9]and seLLing 0, recovers the results of Datmaruya Ill].

RESULTS ND DISCUSSION.
The results for temperature, strain and stress distributions ate evaluated numerically and exhibited graphically in figures to 3, for a particular value of the relaxation constant (8) given by --I/(l+).The transport of thermal energy In the medium, i.e. either a diffusion process or a wave Like process depends on the magnitude of the relaxation constant.
It was observed that at low temperatures the magnitude of the relaxation constant becomes signif[cant and the energy equal ion predicts a wave-type phenomenon.
The magnitudes of the coupling and relaxatlon constants were calculated over a range of Intermediale and high temperatures by Lord The values of the coupling paramter are smaller than unity for moat of the materials.
ller for the computation, the relaxation constant (8) was taken as 0.98 and 0.76 (the corresponding values of e are 0.I and 0.31 respectively) and the values 0.25, 0.5 and 2 for the '.The tlme-dependence of the non-dimensional temperature (0), o strain (U'), and stress (Y.)are depleted as a function of non-dlmenslonal time y at the non-dlmenslonal distance z 2 for ' 0.5.o As the relaxation constant increases the correspoading components of temperature, strain and stress decrease.The gap between these corresponding parameters increases, due to the effect of relaxation time unlike that of coupled theory.
It may be entloned that a similar phenomena was observed by Dalmaruya Ill] In case of coupled theory.Moreover due to presence of Heavfslde unit step function, the two discontinuities can occur in temperature and stress at the wavefronts y z and yz and the corresponding accouatlc velocity (v I) and thermal velocity (v 2) at the wavefronts are and I/ respectively.Last but not least, the results obtained here which including the effect of the relaxation time are more general. ACKNOWLEDGEMENT.
The second author wishes to thank the Councll of Scientific and Industrlal Research, New Delhl, for the flnanclal assistance given for his work.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

(3. 3 )
Here O, U are Laplace Lransforms of 0 and U respectively and p Is the transform parameter

First
Round of Reviews March 1, 2009