THERMOELASTIC WAVES IN AN INFINITE SOLID CAUSED BY A LINE HEAT SOURCE

ABSTILCT. The generalized thermoelasticity theory recently developed by Green and Naghdi is employed to investigate thermoelastic interactions caused by a continuous line heat source in a homogeneous isotropic unbounded solid. Hankel-Laplace transform techni’que is used to solve the problem. Explicit expressions, for stress and temperature fields, are obtained for small time approximation. Numerical values are displayed graphicay. Our results show that this theory predicts an infinite speed for heat propagation in general, and includes the second sound phenomena as a special case.

theory of thennoelasticity, which predicts a finite speed for heat propagation.Lord and   Shulman [1], based on a modified Fourier's law, developed a generalized theory of thermoelasticity whose governing system of equations are entirely hyperbolic and hence predict finite speed for heat propagation.Green and Lindsay [2], based on an entropy production inequality proposed by Green and Laws [3], developed a temperature-rate dependent thermoelasticity that includes the temperature-rate among constitutive variables and also predicts a finite speed for heat propagation.The applications of these theories have been examined extensively by many authors (see [4]- [9]).
Recently Green and Naghdi [10] re-examined the basic postulates of thermechanics and postulated three type of constitutive repose functions for the thermal phenomena.The nature of these three types of constitutive equations is [11] such that when the respective theories are linearized, type theory is the same as the classical heat conduction theory (based on Fourier's law); type II theory predicts a finite speed for heat propagation and involves no energy dissipation; type HI theory permits propagation of thermal signals at both infinite and finite speeds [12] and there is structural difference between these field equations and those developed in [1] and [2].
The aim of the present paper is to study the thermoelastic interactions caused by a continuous line heat source in a homogeneous and isotropic infinite solid by employing the above mentioned type HI theory.We use Hankel-Laplace transform to find the explicit expressions for stress and temperature fields in small time intervals.Our results indicate that this theory generally predicts diffusion type of heat propagation and includes the wave type of heat propagation as a special case.The counterparts of this problem in the context of theories developed in [1] and [2] have been studied respectively by Sherief and Anwar [8] and Chandrasekharaiah and Murthy [9].Due to the structural difference between these theories, our results differ from those obtained in [8] and [9].

PRELIMINARIES
We consider a homogeneous isotropic elastic solid occupying the whole space.The governing system of equations in thermoelasticity of type III developed in [12] are ( -)uj,ij + ui,jj-"0,i " Pfi Pli oc-}-")'0.i,io( -k,i -k*,ii, ffij Uk'kij + Ui'j "}" Uj'i)-"ij ' where A and are Lame constants, /--Ef/(l-2u), E is Young's modulus, P is the Poisson ratio, * is the coefficient of volume expansion, u are the components of the displacement vector, p is the mass density, is the temperature deviation above the initial temperature g., ffij are the compomemts of stree tensor, k is the thermal conductivity, k* is a constant, c is the specific heat for processes with invariant strain tensor, Q and /i are, respectively, the heat source and the components of the body force, measured per unit volume.
In above equations, the notation of Cartesian tensor is employed, superposed dots denote the time derivatives and a comma followed by the index denotes the partial derivative with respect to z i.Using the nondimensional variables where is a standard length, a. is a standard speed, p. is a standard mass density, the basic equations (2.1)-(2.3)reduce to the following (dropping primes for convenience): pli a2Uj,ij " Ui,jj-a20,i + ,ii " a4,ii + Pas{ Pa6 + azi,i o'ij Uk,kij + (Ui, -i-Uj,i)- where gt)is the Dirac delta function, H(t)is the Heaviside unit step function, r= (+)V2 and Q. is a constant.The resulting thermoelastic interactions are axisymmetric in nature so that the displacement vector has only the radial component u u{r,t) and the stress tensor has only two components r and % which are normal stresses in the radial' and transverse directions.
In the context of the problem considered, the regularity conditions are taken as and the initial conditions at 0 are u 0 0. (3.3) Transforming equations (2.4)-(2.6)into cylindrical coordinates, with fi 0, we obtain Applying the Laplace transform, defined by (r, p)= f(R) r, t)exp{--pt}dt, Re(p) > O, to equations (3.11)-(3.13)under the homogeneous initil conditions, with Q given by (7), we find that _2 r (bP where A 1/2b4Q.]" and b pa r We may rewrite equation (3.16) in the form where , and are roots of the characteristic equation Applying the Hankel transform defined by (,p) fo (R)   where J0 is the Bessel function of the first kind and of zero order, to equation ( 23), we Using the inverse Hankel transform defined by in equation (3.19), we obtain (r,p) 1 +Ao4p[Ko(AIr) K0(A'r)]/(A-A)' (3.20)   where K is the modified Bessel function of the second kind and of zero order.
Using the equation (3.20) and the following recurrence relations of the modified Bessel functions of the second kind K0(r K,(r), and rK(r)] rKo(r), where K(r) is the modified Bessel function of second kind and of order one, the equations (3.14) and (3.15) become O'r-1 +Aa,p iZ (-1)'''_-bp'Ko(Air) + AiK(Air) /(A-A), Taking the Laplace transform of both sides of equation (3.9) and using the equation ( 26), we find that The system of equations (3.21)-(3.23)gives rise to the solutions of stress and temperature fields in Laplace transform domain.The solutions in (r, t) domain can be obtained by inverting the Laplace transform.For this reason we have resorted to the case of small-time approximation.From equation (3.18), we find that (4.1) For large p, expanding the above equation binomially in ascending powers of 1/p and retaining only necessary terms, we obtain From [13], we find the following inverse Laplace transforms .'-'{,") . 'O,"Ko(,)) -)co,x(/), ( where I'(n) is the Gamma function.Using the shift property of the Laplace transform and the expansion technique for large p, we obtain 't{p"Ko[(b,oP-l-bu)} ez'p{-mt} e{_mL}.'-l{(p-,+ mp "2 + m2p'3)Ko(btoPr)},  Similiarly we obtain (4.12) Substituting from equations (4.9), (4.11) and (4.12) into equation (4.10), we obtain Applying similiar procedure to other terms and making use of the following identities X-'{K,()} .ff "{p"Kt()} /"/'(t-)(-) ' Due to the presence of the exponential integral and the exponent functions in the expressions of equations (4.18)-(4.20), it can be seen that the effect of input ( 7) is felt throughout the medium instantly and hence it shows that according to this theory heat travels at an infinite speed.

HEAT PROPAGATION WITH A FINITE SPEED
As we have pointed out earlier, this theory generally predicts a diffusion type of heat propagation and includes the finite wave propagation of heat as a special case.In this section we consider this special case when k* ) k, that is, a 0, b = 0 and equation (4.1) becomes i Ci, 1, where c.
Making use of identities (4.8) and (4.15), we find the solution for stress and temperature fields as In this special case, solutions (5.6)-(5.8)indicate that thermal signals propagate at a finite speed since the Heaviside unit step function appears in all terms.

NUMERICAL RESULTS AND CONCLUSIONS
The numerical values of stress and temperature fields at time 0.5 have been calculated and displayed in figure 1-3 along the r-axis.To obtain these numerical values, we have taken that b 5.2, b3 3.35, b 2.25, and b] 2.54.In the general case, we assumed that 4 3.1 and b 2.8.
From Fig. 1 and 2, we note that in the general case the magnitudes of the radial stress and the tangential stress decrease fromat r 0 to zero as r tends to infinity without Fig. 1.

germml cse
Or/A es.r for 0.
genera! spechs!case Fig. 2. <r/A vs. r for 0.5 general case special case 0.4 0.6 0.8 Fig. 3. 30/A vs. r for 0.5 any jumps.However, in the undamped case, the corresponding magnitudes suffer two infinite jumps at r tic 0.237113 and r tic 0.576053.From Fig. 3, we see that in the genera/ case the temperature decreases from +(R) at r 0 to zero as r tends to infinity while the temperature in the undamped case suffers two infinit jumps at r 0.237113 and r = 0.576053 and vanishes for r > 0.576053.
It is also apparent that, in general, this theory predicts a diffusion type of thermal propagation.The values of stress and temperature fields damp out gradually as r increses.
In the special case (undamped case) when k* ) k, this theory predicts finite speed for heat propagation.In this special case, stress and temperature fields vanish identically for r > t/c 0.576053.
We also note that in the special case, both the stress and the temperatme fields have finite values at r 0, which is quite unusual for a continuous line heat source input given by (3.1).However, if we set k 0 in the heat conduction equation (2.2), we obtain a hyperbolic heat equation which contains ( In this case we get an impulsive heat source ( when we take Q as a continuous heat source given by (3.1).According to solutions (5.6)-(5.8), the precise values for stress and temperature fields at r 0 are <rrlA =-l/t., IA =-l/t, O/A lit.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
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: 'ION IN LAPLACE TRANSFORM DOMAIN In the present paper we consider an ifinite solid contaizfing a line heat source situated along the z3-axis that is,
Using the convolution theorem of Laplace transform and the formulas