Internat. J. Math. & Math. Sci.VOL. 20 NO. 2 (1997) 323-334

323

THERMOELASTIC WAVES IN AN INFINITE SOLIDCAUSED BY A LINE HEAT SOURCE

RANJIT S. DHALIWAL, SAMIR R. MAJUMDAR and JUN WANGDepartment of Mathematics and Statistics

University of Calk,aryCalary, Alberta, Canada T2N 1N4

(Received October 19, 1994 and in revised form March 15, 1995)

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 temperaturefields, 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.

KEY WORDS AND PHB.ASES. Generalized thermoelasticity, thermoelastic waves.

1992 AMS SUBJECT CLASSIFICATION CODES. 73, 80

1. INTRODUCTION.In recent years many papers have been devoted to the development of the generalized

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].

324 R.S. DHALIWAL, S. R. MAJIAR AND J. UANG

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].2. 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 Plioc-}- ")’0.i,i o( -- k,i - k*,ii,

ffij Uk’kij + Ui’j "}" Uj’i)- "ij’

(2.1)

(2.2)

(2.3)

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 zi. 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)-

(2.4)

(2.6)

with

THERMOELASTIC WAVES DUE TO LINE HEAT SOURCE 325

3. SOLIPI’ION IN LAPLACE TRANSFORM DOMAINIn the present paper we consider an ifinite solid contaizfing a line heat source situated

along the z3-axis that is,

and Q 6(r)H(t) (3.1)

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) andthe 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

(% %, 0-- (0, o, 0) r (R), (3.2)

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

(3.4)

(3.)

(3.6)

(3.7)

Introducing the thermoelastic potential function defined by

equations (3.4) and (3.5) reduce to

(3.9)

326 R. S. DHALIWAL, S. R. MAJUMDAR AND J. WANG

(3.10)

1where V’= + . Using equations (3.9)and (3.10), equations (3.6) and (3.7) reduce

to

Elimination of 0 between equations (3.9) and (3.10) gives

+ b4 O,

(3.13)

where

b galas/(as+l).

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

_2r (bP

(b’+h)fv + hf]’ + -(r) o,

(3.14)

(3.15)

(3.16)

where A 1/2b4Q.]" and b par We may rewrite equation (3.16) in the form

(7’- )(V’ 2]) -1A $(r) (3.17)+ a4)’

where , and are roots of the characteristic equation

THERMOELASTIC WAVES DUE TO LINE HEAT SOURCE 327

btp+b2 1 b-t O.zl+a4 p p2z + l+atp (3.18)

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

find that

A (3.19)(’ + )(’ + %)(’P) 1 + op"

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),

A .[(3.21)

(3.22)

Taking the Laplace transform of both sides of equation (3.9) and using the equation (26),we find that

328 R. S. DHALIWAL, S. R. MAJUMDAR AND J. WANG

The system of equations (3.21)-(3.23) gives rise to the solutions of stress andtemperature fields in Laplace transform domain. The solutions in (r, t) domain can be

obtained by inverting the Laplace transform.4. SMALL-TIME SOLUTION IN (r,t) DOMAIN

It is a formidable task to find the inverse Laplace transform of equations (3.21)-(3.23).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

1 bloP + bw bop1/2, (4.2)

by neglecting terms of O(p’2), where

blo-- (bl/4)1/2= bs/(2+l) 1/’, b,o-- (b3)1/2,bu {blb b3- b)/(bl4)/2.

Similiarly, we have

(]-])C 1+c4p)(4.3)

where

b (2bs blb)/b.Substituting from equations (4.2) and (4.3) into equations (3.21)-(3.23), we obtain

THERMOELASTIC WAVES DUE TO LINE HEAT SOURCE 329

From [13], we find the following inverse Laplace transforms

.’-’{,")

. ’O,"Ko(,)) -)co,x(/),

(4.7)

(4.8)

(4.9)

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)}, (4.;o)

where m btt/bto. Using the convolution theorem of Laplace transform and the formulas

(4.7) and (4.8), we get

.2 ";Tp’2K0( bt0Pr)} .2’ "tTp’2}, .ff "tTKo(btoPr)}[fl’C2)],[H(t-bxor)(t2-.b2tor2)’;/fot(t-s)H(s-btor)(s-bor2)’l/ds

H(t-bxor)[t cosh’x[tl(btor)]

Similiarly we obtain

(4.12)

330 R. S. DHALIWAL, S. R. MAJUMDAR AND J. WANG

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-)(-) ’1’,

(4.14)

(4.15)

(4.16)

(4.17)

we find that for small values of time

where E,{z} f2 e:rp{-s}ds, z > 0, is the exponential integral and

1 [Sb’ 4bgblr 4boblbb,obr 3b,obh 4b,bo bobborrl tL + + + +

+

t[mb 2mbobb bobbob] (4.21)bobbtob,j + +

bobtb + 3btobt 1

12mbtb- 12mbobxbj + 16bob: 8biobobtx(4.22)

Sbo Sbob,b],+ (4.3)

THERMOELASTIC WAVES DUE TO LINE HEAT SOURCE 331

t 5bob,bo] bot/,fir4 1(4.24)

1[+ (4bbl12blob,4bobblo-bobblob)] + ,[(16b+lbobb-16bob,bo48bob+bobbob Sbobbob] +

011

+ 8b,b,o[( to tt to to tt’bbto-12bobbtobt)

b 1 11 1

12bbobbo) bobit/(bo), (4.26)1 [8b + 18bo 12b boblbor]% t bbsb-

+ t[8b 8bb’b 4bb] bb’b]’ (4.27)

,., ,[bob_3, bob:o] + + 5, Ibo,./b,o/b + e[(bte)-’ bob,o +

1 r (7b2b2+lObbsb 2b2b 2

[(12btobt+4bobobobb) + bobzbob]+t’[Tb,+4bobtblob,2mbob,bl-2m’b:] ++

0 (bob + mb/bt bobo 3btobxt/b bobtobtx3 21 11+ bob + bJb 3bobob.)

1 1 bobo) 1b+ bob bo/b bot,o

(4.28)

(4.29)

(4.30)

(4.32)

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 feltthroughout the medium instantly and hence it shows that according to this theory heattravels at an infinite speed.5. HEAT PROPAGATION WITH A FINITE SPEED

As we have pointed out earlier, this theory generally predicts a diffusion type of heat

332 R.S. DHALIWAL, S. R. MAJLDAR AND J. WANG

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, 2, (5.1)

where

c lib, "t" (b] 4b,)1//2) 1/’(.2)

Using equation (5.1) in equations (3.21)-(3.23), we obtain

c. aa/A ,Z,(-1)i"( b])Ko(ciPr (5.s)

where c. (b 4b3) 1/.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.

6. NUMERICAL RESULTS AND CONCLUSIONSThe 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 from - at r 0 to zero as r tends to infinity without

THERMOELASTIC WAVES DUE TO LINE HEAT SOURCE 333

Fig.1.

germml cse

Or/A es. r for 0.

genera!spechs! case

Fig.2. <r/A vs. r for 0.5

general casespecial 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 twoinfinite 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 infinitywhile the temperature in the undamped case suffers two infinit jumps at r 0.237113 andr = 0.576053 and vanishes for r > 0.576053.

It is also apparent that, in general, this theory predicts a diffusion type of thermalpropagation. 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 heatpropagation. 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 havefinite 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 obtaina hyperbolic heat equation which contains (

In this case we get an impulsive heat source ( when we take Q as a continuous heatsource given by (3.1). According to solutions (5.6)-(5.8), the precise values for stress andtemperature fields at r 0 are

<rrlA =-l/t., IA =-l/t, O/A lit.

334 R.S. DHALIWAL, S. R. MAJUIAR AND J. WANG

REFERENCE

1. H. Lord and Y. Shulman, A Generalized Dyn.amical Theory Of The.rm0clasticity, J.Mech. Phys. Solids, 15, 299-309, 1967.

2., A.E. Green and K. A. Lindsay, Thermoela.sticity, J. Elasticity, 2, 1-7, 1972.

3. A.E. Green and N. Laws, On The Entrovy Production Inc_uality, Arch. Rat. Mech.Anal., 45, 47-53, 1972.

4. H.H. Sherief and R. S. Dhaliwal, Ge.neraliz .One-dimensional Thermal-shockProblem For Small Times, J. Thermal Stresses, 4, 407-420, 1981.

5. J. Wang and R. S. Dhaliwal, Fund.amen.tal Solutions Of The GeneralizedThermoelastic Equations, J. Thermal Stresses, 16, 135-161, 1993.

6. R.S. Dhaliwal and J. G. Rokne, One-dimensional Thermal Shock Problem WithTwo Relaxation Times. J. Thermal Stresses, 12, 259-279, 1989.

7. J. Ignaczak, A .S.t.r0n Discontinuity Wave In Thermoelasticity With.. Reltxati0nTimes, J. Thema] Stresses, 8, 25-40, 1985.

8. H.H. Sherief and M. N. Anwar, Problem In Generalized Thermoelasticity, J.Thermal Stresses, 9, 165-181, 1986.

9. D. S. Chandrasekharaiah and H. N. Murthy, Temverature-rate-dpendentThermoelastic Interac.tio..ns Due To A Line Heat Source, Acta Mechanica. 89,1-12, 1991.

10. A. E. Green and P. M. Naghdi, A Re-examinat.ion Of .The Basic P..stultes OfThermomechanics, Proc. Roy. Soc. London Set. A, 432, 171-194, 1991.

11. A. E. Green and P. M. Naghdi, Thermoelasticity Without Ener,T Dissivation,J. Elasticity, 31, 189-208, 1993.

12. A.E. Green and P. M. Naghdi, On Undamved Heat Wves In An Elastic Solid, J.Thermal Stresses, 15, 253-264, 1992.

13. A. Erdelyi, T.ables Of. Integral Transforms, Vol.1, 2, McGraw-Hill, 1954.

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