generalized thermoelastic longitudinal waves in unbounded medium

28

Generalized Thermoelastic Longitudinal Waves in Unbounded Medium

By HARINDER SINGH 1) and A. SINGH 2)

Summary - In this paper the generalized thermoelastic longitudinal waves and the temperature field set up due to coupling of the displacement and the temperature fields, with heat wave travelling with certain finite velocity, in an unbounded medium are studied. The thermoelastic displacement potential and the temperature field at any point are obtained in terms of the surface integrals involving the potential, the temperature and their normal derivatives.

X i - -

I1 -=

V -

0 t =-

T o -

T -

C e - -

2 , # -

~ o -

N o t a t i o n

the cartesian coordinate system, i = 1, 2, 3.

(ui) the displacement vector.

(0/~ xi) the del operator. O/~t the derivative with respect to time. the temperature corresponding to the natural state of zero stress and strain.

Absolute temperature.

the specific heat.

LamEs constants.

the density.

c~ - coefficient of linear thermal expansion.

K - thermal conductivity coefficient.

~ k k ~ g U

% - the relaxation time.

1. I n t r o d u c t i o n

A large number of general thcorem~ h~,~o_ already been obtained in the coupled theory of thermoelasticity based on the thermodynam~-c~ oe irreversible processes. This paper considers the propagat ion of longitudinal waves in an unbounctea i~otrooic homogeneous elastic medium arising due to the coupling of displacement and the

temperature fields. In the classical theory of thermoelasticity it is assumed that the heat wave travels with an infinite velocity, which is physically impossible. In the present

1) Lecture Mathematics Department, Khalsa College, Amritsar. India. 2) Reader, Mathematics Department, Punjabi University, Patiala. India.

Generalized Thermoelastic Longitudinal Waves in Unbounded Medium 29

paper we take the heat conduction equation in the generalized form by including the terms corresponding to relaxation. The heat wave satisfying this equation has a finite, though large, velocity. It has been assumed that there are no heat sources inside the medium and no body forces are acting onit. This paper generalizes the results obtained by NOWACKI [4] 3) and therefore is a further generalization of Kirchhoff's theorem.

The thermoelastic displacement potential and the temperature at any point inside the medium has been obtained in terms of the surface integrals of the potential, the temperature and their normal derivatives. The approximate formulae based on the preturbation theory have also been obtained.

A special case for the semi-infinite medium with a plane as the boundary, with the displacement potential and temperature prescribed on the boundary, has also been discussed in a separate section.

2. Thermoelastic equations

Consider an isotropic homogeneous elastic medium occupying a certain region R § Assuming that there is no heat source inside the medium the generalized heat conduction equation is [6]

K V zT=o~ o G ( o t T + z oO~t T ) + ( 3 ; t + 2 # ) e T o ( O ,Gk+zo32ekk). (1)

Here the term (3 2 +2 #) c~T o 3 t Ekk represents the effect of coupling the displacement field with the temperature field. The terms containing % correspond to relaxation.

The heat wave satisfying this generalized equation has a velocity x/K/(~o o c e %) which although large, is finite.

The thermoelastic equation in the vector form and in the absence of body forces is

(~, +/,t) V(V u) + #V 2 u = (3 ;t + 2 y ) czV T + Go 0~ u. (2)

For convenience we introduce the dimensionless quantities

' K-Q~ CeFI~ + 2YT/z x i _ _ x i L ~o -J

t ,__GoCe~+2P t K fro

T - T o T ~ -

To U' 0~0 CeF/~ -[- 2/~T/2 Go

= J

In the following discussion we neglect the primes.

i = 1 ,2 ,3 .

u .

3) Numbers in brackets refer to References, page 37.

30 Harinder Singh and A. Singh (Pageoph,

Writing (1) and (2) in the non-dimensional form we get

v ~ T = (0, + ~ 02) T + ~(0, + �9 02) ~+k

where

(Z +/~) V(Vu) + # V 2 u = (2 + 2 # ) (V T + 0 2 u)

Oo Ce Z + 21t z = z o - - - - Relaxation constant

K o o

(3 2 + 2 ~)2 ~2 To e = Thermoelastic coupling constant

~o o ce(2 + 2 #)

Representing the displacement u in the form

u = V ~ + V x ~

(3)

(4)

the system of equations (3) and (4) can be reduced to

( v 2 - 0, - + 02) r = ~(0, + + 02) ~kk

(V ~ -- 0, ~) ~ = r

(V 2 1 2 ) - F a ~ , = o

where c 2 =/l/(c5 +2 #).

(5)

(6)

(7)

3. Integral representation of the solution of the equation for thermoelastic displacement potential

We denote the internal region occupied by the medium as R + and the surface bounding it as A. By R- we mean the complement of R + with respect to the whole space. We shall study the propagation of longitudinal waves, as represented by the equation (6), in the region R +. So we shall try to find an expression for the function ~/' at any point x = (xi) of R + in terms of surface integrals of the functions T, q~ and their normal derivatives.

The functions ~b and T are assumed to be regular in R +, having first and second order derivatives in the closure of R + and the satisfying homogeneous initial conditions. Thus we consider the equations (5)and (6)with no singularities in the region R + u A.

Eliminating Tf rom the equations (5) and (6) we get

[(V 2 - 82) (V 2 - 8, - + 02) - e(3t + ~ 02) V ~3 �9 = 0. (8)

Applying the Laplace transform

oo

~(x, p) = f ~(x, t) exp (-- p t) dt 0

Vol. 101, 1972/IX) Generalized Thermoelastic Longitudinal Waves in Unbounded Medium 31

to both sides of(8) we get N~2~1 N22 ~(x, p) = 0 (9)

where u--2& = V 2_ 2i2. i= 1, 2; and 2~ and 222 are roots of the equation

;~ - ; [ 1 + ~ + (1 + ~ + ~ ~) p] ;? + p3 + ~ p4 = 0 .

Consider the solution G (x, y, t) of the equation

[( V2 - 82) ( V2 - Qt - z 8 2 ) - e(St + z 8 2) V 2] G(x, y, t) (10)

= - ~(x - y) ~(t)

in an infinite region. The right hand side of the equation (10) corresponds to an instantaneous point impulse at the point y. This impulse can be taken as equivalent to an instantaneous heat source concentrated at that point. The initial conditions satis- fied by G (x, y, t) are assumed to be homogeneous. Applying the Laplace transform to (10) we get

D ~& D ~ 8(x , y, p) = - ,~(x - y ) . (11)

Solution of this equation is [1, 2] e - 2 a ~ _ e -220

G(x, y, p) - 4 ~ ~(2 2 - 2~) (12)

where ~0 2 = (X 1 - - y l ) 2 @ ( X 2 - - y 2 ) 2 -]-(X 3 - - y 3 ) 2 .

The function G describes an outgoing spherical wave concentrated at point y. We obtain ~b in R § in terms of surface integrals from the identity

f (cD~ DL ~ ~D~ m ~ c)d~ - - ) . 2

. (13)

= ~ [ 8 v ~ ~ - ~ v 4 c - (x~ + x~) ( c v ~ ~ - ~ v ~ c ) ] d~

where dv=dx~ dx2 dx3. Using the formula for iterated Laplacian

f (G V 4 ~ - ~ V 4 G) dv

v

8n - 8n 8n (V2 ~) - V2 8n dA A

and Green's theorem

( 6 v ~ - ~ v ~G) d~ J \ 8 n - 8 n j d A v A


in (13) we get

f (C []#, :7#~ ~ - ~ []#, ~ 6) dv

v

= D2 a a n - - a~ (ff]2 a) + (7 ~n (v2 ~) - v2 ~ ~n dA

A

where [] 2 ~ g 2 _ 2 2 _ _ ,~ 2 ; and dA is the area element of the surface. Using (9) and (11) in (14) we get

a~ U~ (v~ m) - v ~ ~ an_] . ( i s )

A

Now the temperature T and the potential ~b are related by equation (6). Taking its Laplace transform we get

(V 2 _ p2) ~ = T. (16)

Substituting for V 2 ~ from (16) in (15) we get

-- ina a~ T aa a 8Ul~ ub ~ (F-l# a)]dA (17)

fory~R + and fory~R- ~(y, p)-O; where [] ,~--V2-2~-2~ +p2 Taking the inverse Laplace transform of (17) we get

t

an ~n (18) 0 A

+ G(x, y, t - t') 3,1)(x, q~(x, t') aG(x' y' t - tt)I dA(x ) l " "

fo ryeR + andfory~R- ~(y, t )=0, whereG--L-~([~# Z)--L-a([--]#L(G). Formula (18), obtained with in the frame work of coupled thermoelasticity, taking

into consideration the relaxation effect also is similar to Kirchhoff's theorem in the classical elastokinetics. When the processes giving rise to the deformation are time harmonic formula (18) is analogue of Helmholtz's theorem of elastokinetics [3].

4. Solution of heat equation

We shall determine the temperature field T(x, t) associated with the potential (x, t). Consider the solution of the equation

[ ( v 2 - ~ ) ( v 2 - a, - �9 a, ~) - ~(a, + �9 a~) v 2] H ( x , y, t) = - (V 2 - a~) ~(x - y ) ~(t) (19)

Vol.101, 1972/IX) Generalized Thermoelastlc Longitudinal Waves in Unbounded Medium 33

having a singularity in the infinite elastic space. Here it is further assumed that an instantaneous heat source has been applied at the point y and that H(x, y, t) satisfies homogeneous initial conditions.

Applying Laplace transform to (19) we get

D2~ V]~. 2 H(x, y, p) = - D~ 6(x - y) (20)

where D 1 -= V 2 _pZ. The solution of (20) is given by

(Z~ - p2) e-~,~o _ ()o~ _ p2) e-a2.o H(x, y, p) = D~ G(x, y, p) = 4 x 0(2~ - Z 2) (21)

The identity analogous to (14) in this case is

f ( []LD DL )d

_a,~ ~ a ([_12/~) +/? 3 dA. = D 2Ha - an A

Using (9), (16) and (20) in (22) we get

f(Y'P) = L an an + [N~ an - On ( "~ I~) dA. (23) A

Taking the inverse transform of (23) we get t

0 A

'9 c) (x, c) + f t ( x , y , t - an

all(x, y, t - t') T(x, t') an

#ft(x, anY't-t ')} dA(x) l dr'

(24)

fo ry~R + and f o r y e R - T(y, t ) - 0 , whe re /4 - L-1 ([i]~ H) Equation (24) may be regarded as a generalization of the Green's theorem in the

theory of heat conduction to the problems in the coupled thermoelasticity.

5. Approximations for the potential q~ and T

The determination of ~b and Tfrom the results (18) and (24) involves the evaluation of the inverse transforms of G /~ and their derivatives which may be quite difficult in the general case. To avoid this difficulty and to have an approximate idea of the functions q~ and T, we expand the functions G and H as infinite series in the powers of small parameters e and r and their products. We express G and Has

G = Goo + (z Go~ + ~ Glo) + (z 2 Go2 +'c ~ Gll + 82 G2o) + " " (25)

H = Hoo + (~ Hal + ~//lo) + (~2 Ha2 + ~ ~ Hll + ez//2o) + ... . (26)

3 P A G E O P H 101 (1972/IX)


Also we have G - [ v = - (~ + ~) ( r + , r G. (27)

Substituting (25) and (27) in (18) and neglecting the terms involving ~2, ~2 T /~ and their higher powers we get

i[f{ ~ ( y , t ) = G ~ 1 7 6 On + N G ~ 1 7 6 - 0 A

t

8T T - - + 8 Glo On - On

0 A

8~ cb O__ ( N G o o ) } d A l d t , On On

+ (N Glo - at' G~176 0n8~ �9 an0 (N Glo - 0t' Goo)} dA 1 dr' (28)

t

+ ~ G~ On T On 0 A

+ ( N Gol-02~ oO)~nn - q~nn(N G~ Goo ) dA dt'

where N=- V 2 -8r ' . When z = 0, (28) becomes (6.2) of [4] and when ~ = a = 0, it gives the stress potential

in the theory of thermal stresses. To the same order of approximation we get from (24) and (26)

t

T(y, t) = 14~176 8~ - T On 0 A

fEf{ + ~ H~o On 0 A

Oq~ + N Hoo On - - - # &z ( N H~176 ) dA dt'

8Hlo H 8r - - -- T On . + (N Hlo - Ot, oo) ~ n-

+ �9 Hol ---0n T 0n + (N Ho~ - 0~ Hoo) U~e~ 0 A

O (N Hot--O 2 Hoo)}dA]dt ' .

(29)

To determine ~b and Twe have to determine the functions Goo, Go1, Glo, Hoo, Hoa and//1 o- 22 and 2 z are functions of e, ~ and p, where a and z are small. Therefore expanding 22 and 222 in Taylor's series involving powers of e, z and their products

Vol. lOl, 1972/IX) Generalized Thermoelastic Longitudinal Waves in Unbounded Medium 35

and retaining only the terms containing e and ~, we get p2

2,2 ~ p2 + _ _ g p - - 1

~2 P z2,~ p . . . . . e + z p 2 p - - 1

P 2 1 ~ P + - - e

2(p - 1)

2 2 ~ p l / 2 1 2 ( p - l ) + �89 p z

1 1 1) (p p + l _ P 1 (2~ - 22) - * g p ( p _ 1 ()2 ~ + p - i ~ "

Substituting the values of 2, and 22 in e -z'e and e -02"~ we get

e-~1-~ ~ 1 - � 8 9

e-ZZ~ ~ l + �89 e O _ _ _ �89 z 4 p 3/z e -e , /p . p - 1

Using (30) and (31) in (12) we get e-eP _ e-opl/2

aoo 4 rc 4 P(P - 1)

G l ~ 1)2 L \ p - - 1 ~+ e - O P - - - - - 1 14p112 e-~~

G o l - 4 ~ 4 ( p _ l ) k p _ i �89 ~/2 e - o . ~'~ .

The quantity Go o = L-1 {Goo } is given by the formula [4, 5]

1 1 Goo = ~ (e t-~ - 1) H ( t - Q) - 4 7c ~ [U(4 , t) - erfc (4/2xfi)]

and G~o =L- ~{G~o} and Go1 =L-~{Go~} are given by

Glo = - _ _ _ 47c 4

1 Go1 - ~ 4

{ [ ( t - 9 ) z - ( t - Q) (1 - 4 / 2 ) + 13 e t - e - 1} H(t - 4) ] /

1 [ ( t 2 _ t + l + 4 2 / 2 ) U(4, t ) _ { ( 3 t _ 2 ) 4 V ( 4 , t)[ + 4 ~ 4 ]

- erfc (4/2x/0 - .o x/t /rc e -a~/'t]

(30)

(31)

(32)

(33)

- - - - [ ( t - - O) e t - ~ H ( t - Q) - t U(O, t) + o~ V(O , t) + 1 4 ( r c t ) - , / 2 e-•2/4 t]

(34)

Vol. 101, 1972/IX) Generalized Thermoelastic Longitudinal Waves in Unbounded Medium 3 5

and retaining only the terms containing e and ~, we get pZ

22 ~ p2 + _ _ p - 1

~ ~ p - P - - - e + z p 2 p - 1

P 2 ( p - 1)

22~p~/2 1 - 2 ( p _ _ l ) + �89 p ~

(2~ - 2~)-* ~ p ( p - 1) ~ (p - 1) 2" + - p - 1 ~

Substituting the values of 21 and 22 in e -~e and e -a~~ we get

I e-;~Q ~, 1 - - � 8 9

e-X~,o~ l + � 8 9 ~ z f f p 3 / 2 e-Q,/p.

Using (30) and (31) in (12) we get

e-aP _ e-Qpa/2 Goo

4 ;z 0 P ( P - 1)

G,o = - 4 rc 0 P(P - 1) 2 k\p - i + e-~P - - 1 - �89 0 p~/2 e-e~,*/~

lre-o ( ) 1 6o* - 4 ~z o (p _ l ) k p 7 -1 p ~ i - l o p */z e-~V~/~ .

The quantity Goo = L-* {Goo } is given by the formula [4, 5]

1 Goo-- ;47001 (e,_~_ 1 ) H ( t - O) - ~ o [ U ( o , t ) - e r f c ( o / 2 ~ / ~ t ) ]

and G~o =L-*{G,o} and Go, =L-*{6o,} are given by

1 G,o - { [ ( t - 0) 2 - ( t - 0) (1 - 0/2) + 1] e t - ~ 1} H ( t - O)

4rc 0 1

+ 4 r e 0 [ ( t 2 - t + l +0e/2) e (o , t ) - � 8 9

- erfc (ff/2x/i) - 0 x / t [ ~ e -~

1 Go, -- 4To

(30)

(31)

(32)

(33)

- - - [ ( t - e ) e ' -~ H ( t - O) - t C~(O, t) + 0 V ( e , t) + �89 ~(,~ t) - ' / ~ e - ~ / ~ ' 3

(34)


where H (x) is the Heaviside step function and

U(O, t) = 2- e-~ erfc( Q ' - - x/i + e~erfc Q + x/t \2 x/t

V(~,t)=�89176 ~ xfi) - e~ erfc ( ~ t + x f i ) l .

Substitution of (30) and (31) in expression (21) for H(x, y, p) gives

1 1~ o o - - e - q "/ P

4uQ 1 [ { p - i

/ ] 1 o - 4 ~ r - 1) 2 p e-eP + ~ 2 x/P

1 3/2 e-O,/p. / / o~ = - ~ p

- - - 1 } e-Q'/P 1

The inverse Laplace transforms of these functions are found to be

1 Hoo - e -~2/4t (35)

8(~ t) 3/2

Hlo = ~ (t -- ~ + 1) e t-~ H(t - ~) - (t + 1) U(r t) + 2 ~/~tt e Q2/4t (36)

1 V o~ 6~~ (2~)~] Hol - e -o~/4t . (37)

While considering the stresses and strains in the theory of thermal stresses one disregards the coupling effect. Therefore by setting r = ~ = 0 in (28) and (29) we get stress potential and the temperature distribution in the theory of thermal stresses.

The first term in Goo, Glo, Go1 and Hlo represents an outgoing spherical elastic

wave centered at the point y and propagating with velocity x/ (2 +2 #)/~o. The other terms in each of these functions represent a diffusion wave. The functions Hoo and Hol are diffusive in nature.

6. Case of ha~" space

We consider the half space x3 >0. In this case the bounding surface causes the reflection waves the effect of which must be added to the wave developed at the source point y. This has been done here by using the method of images. The appropriate solution of (10), when �9 and Tare prescribed on the boundary is

1 _ e- 2~ e- 2r] Gl(x' Y' P) - 4 ~r(~ - 2a a) k ~ r (38)

where r 2 -- (x 1 - y 0 2 + (x2-y2) 2 + (x3 +Ya) 2

Vo]. 101, 1972/IX) Generalized Thermoelastic Longitudinal Waves in Unbounded Medium 37

Yl, Yz, -Y3 being the coordinates of the image of the source at point y = (Yi) with respect to the surface x3 = 0. It is then found on the boundary

g i (x , y, p) = 0", rq 2~ (7, = 0

, ,Ee .... e . . 1 On 4/r(~, 2 - 22) 0y 3 r 1

(D 2 G 1 ) - t a [ ( p Z ) c2 )e -~m _ ( p 2 _ 2 ~ ) e - Z m ]

an 4 ~z(,1, 2 - ),22) ay 3 r a

where r, = (x, - y l ) 2 + (x2 _y2 + 2.

Changing G to G, in (18) and using the above results we get t

fir{ ' } J @(y, t) =-- T(x, t') art G~(x, y, t - t') + @(x, t') a G,(x, y, t - t') dA dt' on

0 A

(39) here dA = dxl dx2 and the integration over A is taken for - o9 < x~, x2 < ~ .

Similarly in the case of transformed heat equation (20) we take the solution to be

, [ {e? e?}] Iq~(x 'Y 'P)-4~(221 2~ ) (22 -P2)~e - - ' ;~ -e - - ' * [~ (22 - P2)

- ( 0 r )

Then the temperature is given by t

fir{ a ~iI(x,y,t_lt)}dAldl. a H , ( ~ , y, t - t ') + ~,(x, c ) U~ g ( y , t) = - g(x, t') an o A

(40) here dA =dx, dx2 and the integration over A is taken for - oo <Xa, x2 < oo.

In (39) and (40) the inverse Laplace transform of normal derivatives of G~ and H, are involved. These can found out by applying the results (32) to (36) as the Greens functions G 1 and H 1 are similar in structure as G and H a s introduced by (12) and (21).

REFERENCES

[1] W. NOWACKI, Some dynamic problems in thermoelasticity, Arch. Mech. Stos., 11 (1959), 259-283. [2] W. NOWACKI, Certain dynamic problems in thermoelastieity, Bull. Acad. Polon. Sci. Ser. Sci.

Tech. 13 (1956), 657-666. [3] W. NOWACKt, Quelques thdor~mes de la thermodlasticitd, Rev. Roumaine Sci-Tech. Set. Mech.

Appl. 11 (1966), 1173-1183. [4] W. NOWACKI, Thermoelastic waves in an unbounded medium, Polish Academy of Sciences Wars-

z a w a .

[5] R. B. HETNARSKr, Solution of the eoupled problem of thermoelasticity in the form of series o f functions, Arch. Mech. Stos. 16 (1964), 913-941.

[6] H. W. LORD and Y. SHULMAN, A generalized dynamical theory of thermoelasticity, J. Mech. Phy. Solids 15 (1967), 299-309.

(Received 4th February 1972)

generalized thermoelastic longitudinal waves in unbounded medium

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