Stress intensity factors of two bonded half-plane problem with a point heat source

ELSEVIER

Nuclear Engineeringand Design 160 (1996) 97-109

Stress intensity factors of two bonded half-plane problem with a point heat source C . K . C h a o , S.J. C h e n Department of Mechanical Engineering, National Taiwan Institute of Technology, Taipei, Taiwan 106, Taiwan

Received4 April 1995; revised 27 June 1995

Abstract The two-dimensional thermoelastic crack problem of two bonded dissimilar media with a point heat source is considered in this paper. Based on the complex variable theory and the method of analytical continuation, the problem is formulated by two stress functions and a temperature function for each material medium which are enforced to satisfy the interface condition. Furthermore, the si:gular integral equations are derived by taking some density functions along the crack border in a way that the traction-free condition is satisfied on the crack surface. Numerical results for both half-plane and two bonded half-plane problems associated with a curved crack or a line crack under a point heat source are presented and" provided in graphic form.

1. Introduction When the flow of heat is disturbed by the presence of inhomogeneities such as cracks, local intensifications of temperature gradient or stresses will occur at the sharp edges of inhomogeneities and consequently result in material failure through crack propagation. The singular character of the two-dimensional thermal stresses at the tips of a line crack in an infinite medium was first discussed by Sih (1962). Thermal stress singularities at crack tips in a semi-infinite medium has been investigated by Sekine (1977) and Tweed and Lowe (1979). Recently, Chen and Hasebe (1992) considered t'le curve crack problem in an infinite plate by solving singular integral equations. All the solutions presented in the literature as mentioned above are, however, restricted to the he-

mogeneous problem. In this paper, the problem of bonded dissimilar materials containing a crack subjected to a point heat source is considered. It is known that the crack problem associated with at least one separate boundary surface or irregularly shaped cracks is a case that a closed-form solution is impossible to achieve and which will have to rely on some numerical approach. One of the most widely used methods in sol~ing this complicated problem is based on the formulation of singular integral equations directly by using the related Green's function (such as dislocation or concentrated force solution) in conjunction with the technique of superposition. This methed has clear advantages in solving the problem by applying numerical treatment. In the present approach, a temperature dislocation function is introduced to formulate a singular integral equation for the

0029-5493/96/$15.00 © 1996 ElsevierScienceS.A. All rights reserved SSDI 0029-5493(95)01101-3

C.K. Chao, S.J. Chen /Nuclear Engineeringand Design 160 (1996) 97-109 heat concluction problem and edge dislocation functions are applied for the thermoelastic problem. Furthermore, interdependent relations of the related complex potential functions are established through the application of analytical continuation such that the requirement of the interface condition will be satisfied. Some numerical examples of two bonded half-plane and half plane problems associated with a single curved crack (or line crack) under a point heat source are given to illustrate the use of the present approach. The results compared with those of the homogeneous ease given in the literature show that the method proposed here is effective, simple and general.

2. T e m p e r a t u r e

field

For a two-dimensional steady state heat conduction problem, the temperature potentials associated with bi-material media can be represented by two complex functions Ol(z) and 0z(Z) which satisfy the Laplace equation within the domain Sl and $2 respectively as shown in Fig. 1. In order to formulate the boundary conditions, the resultant heat flux Qy and temperature Tj for each material medium may be expressed as

Qj = [

(qxydy - qyjdx) = - k)Im[Oj(z)]

(1)

d

Tj = Re[0fiz)]

(2)

where Re and lm denote the real part and imaginary part of the bracketed expression respectively. The quantities qxj, qyj in Eq. (1) are the components of heat flux in the x- and y-directions respectively, and kj stands for the heat conductivity w i t h j = 1 for S~ a n d j = 2 for $2. If there exists a curved crack (or line crack) L is one of the two dissimilar media, say $1, it is convenient to express the complex potential O~(z) as

O,(z) = O,p(z) + O,c(z)

(3)

where Olp and 0to are the principal part and complement part of the complex function respectively. Since the temperature and resultant heat flux are continuous across the interface L*, it implies [0~(t) + 01(0] + = [02(0 + 02(01 -

(5)

kt - k 2 - -

02(z) = ~

//////

[

x,x.~

Fig. I. A curved crack in two bonded dissimilar materials.

(4)

t ~ L"

where the overbar is used to indicate a conjugate complex quantity while the superscripts + and - are used to stand for the boundary values of the temperature potentials as they are approached from S~ and $2 respectively. Substituting Eq. (3) into Eqs. (4) and (5), it is easy to verify that Eqs. (4) and (5) are satisfied by the following complex potentials

Ot(z)=O~p(Z)+ k---t-~O~p(z )

\\\\\\

t e L"

k,[Ot(t) - 0~(t)] + = k2102(t) - 02(0] -

Oiv(z )

zeSt+L*

z e S 2 + L"

(6) (7)

Once the principal part of the complex potential Otp(z) is determined, the temperature potentials associated with the bi-material problem can be obtained with the aid of Eqs. (6) and (7). Referring to Fig. 1, an insulated curved crack is embedded in the upper portion of dissimilar media under the application of a point heat source with a constant rate qo per unit time which is located at arbitrary position z0 within the domain S,. In order to derive the integral equation for temperature field, it is convenient to represent the solution as the sum of the heat flux due to a point

99

C.K. Chao, S.J. Chen / Nuclear Engineering and Design 160 (1996) 97-109

heat source in an unflawed media and a corrective solution as

Oi(z) = O~(z) + O'S(z)

(j = !,2)

(8)

where the superindex f and u denote the flawed and unflawed media respectively. The solution associated with the unflawed bi-material media has been given by (Carslaw, 1959)

iog(z -

Qt(t)=Q~l(t)+Q~(t)=O

o)

(9)

teL

(16)

Substituting Eqs. (11) and (15) into Eq. (16), we have the following singular integral equation k~ - k2

q°

.q-

,,-k, cos,q,

=2~L\-~l

0~'(z) = - 2--~klIog(z -- Zo) 2~k.~2)

implies

:

-I k,+k2 r2 / k,~-~ 72 ) j+co toEL, x,yEL

O~(z)

n(kq~_ k2)iog(z - z0)

(10)

The resultant heat flux Q'~ across the line L in the unflawed media can be obtained by substituting Eqs. (9) and (10) into Eq. (1) as Qr=~

qofL[(c°s'---~+ k, - k2 c~'23d, L\

r,

k,+k,

r2 )

_ s(i n , , + gYk~

Ill)

k rt r2 / J where r~, r2, E~, E2 are defined by z - zo = rt exp(iet)

z - ;~o= r2exp(ie2)

(12)

For the crack problem of dissimilar media, the corresponding temperature functions can be obtained by using the dislocation function as

O{(z ) = -- ~

bo(s )log(z -- t )ds - kl -- k~ k~ + k 2

x fL bo(s)log(z- ,)ds ] Of2(z) =

1 2kl ~

~ri kl-+k2J L bo(s)log(z - t)ds

j l bo(s)ds = 0

(18)

The dislocation function bo(s ) appearing in Eq. (17) with subsidiary condition, Eq.(18), may be solved numerically by applying the appropriate interpolation formulae which will be discussed in detail in the next section. Once we obtain the dislocation function bo(s), the temperature functions 0t(z ), 02(z) can be obtained by substituting Eqs. (9), (13) and (10), (14) respectively into Eq. (8).

3. Thermoe!asfic field (13) (14)

and the related resultant heat flux is given by

For the two-dimensional isotropie thermoelastic problem, resultant forces and displacements can be expressed as follows (Bogdanoff, 1954): - Yj + i X i = ~j(z) -I- z~b)tz) -I- ~ki(z)

09)

2Oj(uj + ivj) = xAbj(z) - zt~j(z) - - ~ j ( Z ) "4" 2Gjfl.igj(g)

Q~l= -~'~;t. b°(s)[ IOglz-tl kl -- k2 i°gl~- z[ld s k, ~--k-;~

(17)

where co is a real constant. In addition, the singlevalued condition of the temperature must be satisfied, i.e.

gj(z) = j" Oj(z)dz

(20)

(21)

(15)

J

where bo(s) is the strength of the temperature dislocation. Since the total heat flux across L must vanish for the condition of an insulated crack, it

where Gj is the shear modulus of elasticity, and Kj = (3 -- vj)/(l + b), #~ = =~ for plane stress and x~ = 3 - 4v,, #~ = (1 + vj)=j for plane strain, with vj being the Poisson's ratio, and ati the thermal ex-

C.K. Chao. S.J. Chen/ Nuclear Engineeringand Design 160 (1996) 97-109

100

pansion coefficients. Similar to the previous approach, the complex potentials pertaining to $1 and $2 are denoted by Ol(z), ~l(z) and ¢2(z), ~-,(z) respectively, and the former pairs can be divided into two parts, i.e.

After rearrangement, the results of the foregoing manipulations can be summarized as

[z-~,. (z ) +

¢,(z) = ¢,. (z) +

2Gi G2fll k2 - kl 4 Gi + xl G-----k, 2 + k~ gn, (z)

~,(z) = ~,~(z) + ,ARz) ~,(z) = ~=p(z) + ~,~(z)

~(z)] z¢S, + L* (29)

(22)

Substituting Eq. (22) into Eqs. (19) and (20) to enforce traction continuity and displacement continuity across the interface L*. there follows

¢,,(z)=

¢%-~

xt G 2 - K 2 G I

G, + K2G, :,p(z)-- z4~',c(z)

2G, G2# . . . .

2G, G2fl. . . .

g~ptz~ - ~

g:zj

{~,p(t) + ¢~(t) + [ t ~ . ( t ) + ¢qp(t)l + [ t ~ ( t ) zeSl + L* + ~blc(t)]+ = {~2(t) + tq~2(t) + #~(t)} -

(23)

~b2(z)

and

(1 +Xl)G2

= G2 +~--x2G-~~,.(z) + ~

{tOtG2[t~lp(t) + q~lc(t)] -- G2[tqb'~p(t) + @lp(t)] -

-

2-~G~_~2 g2(z)

G2[t~'=~(t)Jr ~bt¢(t)] 4- 2G= G2fl=

@..

~b'" ( 1 4 - K , ) G , . . . 2G, G2 ~tz; = ~ ~p,ptz) + G, + ~G~ X LSlglp (z) -- ~2g2(z)] __

ZES 2 + L* (25)

Kt G2 -- K2G,

zC~;~(z) + ¢,,~(z)

A Gi4-xiG2kl~-~2glp(z)

z~SI+L*

(32)

~l(z) = ~2(z) = ~,p(z)

2G~ G2 "+ G2 + s:2G~

#,(z) = ~2(z) = @,p(z)

(33) (34)

Moreover, if one lets G2 = 0 for the half-plane problem, the above results reduce to

~292(Z)]

z~S2+L*

(31)

Now, we have completed a general solution to the thermoelastic problem once the complex functions ~b,~(z) and @,p(z) are given. If one lets G~ = G2 in the foregoing equations for the homogeneous problem, the above results become

and

--

g,p(z)

zES2+L*

[z~',,(z) + ¢,,. (z)] - zO'~(z)

~'-;- ,qO---~, ,A,(z)

X L~lglp(Z)

(30)

2GiG2fll k2- kl

(24)

By applying the continuation theorem, the following relationships are established as (Chao, 1995)

__

(I + l q ) G 2

:zj = ~

x [g=~(t) + g~c(t)]} + = {~¢2G~O2(t) - Gl x [t~b~(t) + @2(0] + 2G~ G2fl2g2(t)} -

2GIG2Pl

(26)

~,(z) = ~,.(z)

--

[ z ~ . ( z ) + ~,.(z)]

(35)

and

~,~(z) = G 2 - aL [z~i.(z) + ¢,,.(z)l

¢,,(z) = ¢,,,,(z) - 4~,.(z)

G~ + x~ 02

z¢S2 4- L* •

. .

(27)

(l + t':I)G2 . . . .

z¢;(z) + ~o2~z~= ~

tzq,~,tz) + ¢% (z)] z ~ S 2 + L*

(28)

+ z ~ p ( z ) +-~p(z) + ~p(z)]

(36)

Consider a curved crack L to be situated in the upper half plane as indicated in Fig. 1, the complex functions ~,p(Z) and ¢lp(Z) can be expressed as

C.K. Chao, S.J. Chen /Nuclear Engineering and Design 160 (1996) 97-109 (k'v (z) = ~

iG~

f

~

Jc [bt(s) + ib2(s)] log(z - t) ds (37)

and 0,p(Z) = ~

- iGI

fL

[b,(s) - ib2(s)] log(z - t) ds

.... i__G.L _ ~ [bt(s) + ib2(s)]t ds n(l + K,) Jr. z -- t

(38)

where bl(s) and b2(s) indicate the components of the displacement discontinuities across the dislocation line. Now, the singular integral equations for the stress field can be obtained by substituting Eqs. (37) and (38) into Eqs. (29) and (30) and using Eq. (19), there follows

fL Kt(G t, to, to)[bl(s) + ib2(s)] (is

St :(zt.a,.k~ .u0

dm

J I"

))))))

\,.o

_.

S ~:(r<,r'..k°. ~)

Fig. 2. Divisionand nodal distributionfor the curved crack. + L K2(t' ~' to, to,)[b,(s) - ib2(s)] ds

k2 - k~

×~

+ .I,. K3(t, t, lo, to)bo(s) ds = c~ + ic2

[(t o -

t) log(to - i') - to]

(39)

2k,.p~.. [(to- 7) (kl

where c~ and c, are real constants and the kernels in Eq. (39) are 2~G, KI(t , i', to, i0) = ~ x

x, G2 - K2GI loglt0 -- t[ + G2 + K2GI

.. iGt

. log(?o--t)

G2 -- G~

K2(t'i"t°'i'°)

%

× log(to -

i ' ) - to]t

(42)

In addition, the single-valued condition of the displacement must be satisfied, i.e.

iGt

Flog(to- t-) L

with

(t -- ?')(to-- i'o)1

~;:- ;o~ /

(40)

iGi t -- to G2 - Gi it(1 -I-tel)t--t0 + G I + ~ 2

iG,

[ to-io

t-i,

X ~ L - t_/o+i._,,,] (41)

K3(t, t, to, i'o)

k2)P,

n| l -F -I~ ltT,1)

+ ~, ¥-~-,o2,~(f¥ ~,) ×

+

iGI G2fll

{

1c(K2G, + G,) _[(t° - i') ~2G~ + G, x log(to - t) - to] + K,----G2+-~,"

ao(S) =

bo(~)d~

In order to solve the dislocation functions bo(s), bl(s) and b2(s) appearing in the above singular integral equations, the interpolation formulae along the crack which is approximated by N line se~,A~nents(Fig. 2) are used to perform the numerical calculation. Since the temperature dislocation bo and displacement dislocations b~(s), b2(s) have the square-root singularity at the vicinity of the crack tip, the interpolation formulae in local coordinates st and sx for each crack tip element arc thus defined as

C.K. Chao, S.J. Chen / Nuclear Engineering and Design 160 (1996) 97-109

102

F(2"W o,(~,)=b,,OL~ / - l 1+b,,, a,(s~)

= a ....

F( L \

~ /

-

( i = 0 , 1,2)

1]+b .....

zo

Meanwhile, the interpolation formulae for the intermediate segment in local coordinates sj(2 <~j<~ N - l) are taken as •

2 d j - sj

b,(sj)=h,.i_,--~-j+b,.j-~j

sj

Y

))))))))

r1

r°

x

( i = 0 , 1,2)

(46) where dy(l <~j<~N) are the half length for each line segment and bia(0 ~
Km -- iKim = ( 2~ )3n( 2dN )l/2 exp( -- io~n)(bt. N + ib2.1v ) (48) where the angles ~^ and c¢B are defined in Fig. 1.

Fig. 3. An insulated line crack in two bonded dissimilar materials. 4. Results and discussion

The accuracy of the present study can be demonstrated by considering the following special cases. First, an insulated line crack of length 2l with/? = 90 ° (Fig. 3) in the homogeneous medium under a remote point heat source located at :co/ 1= 1000 and yo/h = 1 is considered. Table 1 displays the comparisons between the calculated and exact values of the thermal stress intensity factors for ~iffe~'ent numbers of line segment with equally divided space. It can be seen that the proposed m,::thod provides very reliable results even for a coarse mesh division, i.e. N = 10. Next, we consider an insulated circular-arc crack with the half angle 0 = 30 ° and the radius a subject to a remote point heat source (xo/a = 1000, yo/h = 1) as displayed in Fig. 4. Fig. 5 shows that the calculated values of the stress intensity factors yield very good accuracy compared with the exact results given by Chao and Chen (1993) for the number of line segment N = 30.

Table I Comparisions between the calculated and exact values of the thermal stress intensity factors for different discretizationsN

N

K~a/(K.)cx.c,

Km/(K.)cx,~,

Kua/(Knhxac,

Kue/(g, hx~,

I0 20 30

0.0000 0.0000 0.0000

0.0000 0.0000 0.0000

0.9910 0.9982 0.9991

-0.9910 -0.9982 -0.9991

C.K. Chao. S.J. Chen / Nuclear Engineering and Design 160 (1996) 97-109 Y

I S, :(Gt,ta,.s,,~a) \\\\\\\\ ////////

i

Fig. 4. An insulated circular-arc crack in two bonded dissimilar materials.

4. I. Half-plane problem Based on the derivations of the previous section, the solutions associated with the half-plane problem can be obtained from the solutions given for the bi-material problem if one lets k2 = 6;2 = 0. Both a line crack and circular-arc crack are considered to illustrate the thermal stress intensity factors in the half-plane medium under an arbitrarily located point heat source. The geometric configurations are indicated in Figs. 3 and 4. In the following study, the dimensionless stress intensity factor is used, which is defined as KI corII1 K;t .... ~= [2fl,Gim3/2qo~/n/(l +/~,)kl ] where m = 1 for a line crack and m = a for a circular-arc crack problem. The effect of the location of a point heat source on the thermal stress intensity factors K, and gll for a line crack can be seen from Figs. 6 and 7. It is shown that both KI and Kn change sign as a point heat source moves from the left side to the right side of the crack surface. Note that the stress intensity factors attain their maximum values near the crack surface and experience a jump when crossing the crack surface. This phenomenon can be further explained by observing the variations of the factors with the angle ~b as shown in Figs. 8 and 9. The influence of the distance between a line crack and the bounding surface on the stress intensity fac-

!03

tors can be found in Fig. 10. As expected, the magnitude of the local stresses increases with decreasing distance between the crack and the half-plane surface. Fig. I l illustrates the opening mode and in-plane shear mode ~tress intensity factors against the location of a point heat source for an insulated circular-arc crack. Again, the stress intensity factors change sign when a point heat source moves across the crack surface.

4.2. Two bonded '+alf-plane problem Referring to Figs. 3 and 4, a line crack and a circular-arc crack are considered separately to be situated in the upper portion of dissimilar media. In what follows, k,/k2 and B~/~2 are fixed at i. The dimensionless stress intensity factors against the location of a point heat source for G2/G, = 0.5 and G2/Gt = 2.0 are shown in Figs. 12 and 13 respectively. It is interesting to see that the factors change sign for the case G2/GI = 2.0 compared with the case G2/Gt = 0.5 while the geometric parameters are kept unchanged. The effects o f the material properties and geometric parameters on the thermal stress intensity factors for a circulararc crack problem are exhibited in Figs. 14 and 15. In general, the local stresses near the crack tips increase with decreasing G2/GI. Note that the mode-I stress intensity factor begins to increase with the crack angle and attains its maximum value around 0 = 74° and then decreases as the crack angle further increases as indicated in Fig. 14. This is because the presence of an insulated crack surface may be able to partially shield the heat flux as the crack angle 0 extends beyond 74° and consequently results in decreasing the thermal energy intensification around the crack tip.

5. Conclusions The thermal stress intensity factors for two bonded half-plane problem and half-plane problem under the application of a point heat source are obtained by solving singular integral equations with logarithmic singular kernels which are established from the equilibrium conditions o f resultant force and heat flux across the crack

1.00

..... o.9o

(e=ac 0 Ku (N=30)

j-

K~

. . . . .

" ~ 0.80

+

i ~ . 0.70 0.60

~

0.~

0.30 0.20 0.10

o.oo 0.0

' ~ * I t l l * l ' ' h l ' l l ' l ' ' l l l l l l 10.0

20.0

80.0

40.0

50.0

80.0

70.0

80.0

90.0

100.0

hell (z~Ote (0) Fig. 5. Comparisons between the calculated and exact values of the thermal stress intensity factors for the circular-arc crack. 0.05

i

Ku

0.04

0.02

0.02

0.01

"~

0.00

-0.01

-0.02

'i --0,04 --8.0

IIll'|lll'lJllllll~llllilllllllDllltllllllllllllllllli --5.0

-4.0

--8.0

-2.0

-1.0

0.0

1.0

2.0

8.0

4.0

6.0

Fig. 6. The mode l stress intensity factors vs. the horizontal distance between a line crack and a point heat source in the halt-plane medium: 0,o/I = h/l = 1.2, # = - 6 0 ° ) .

C.K. Chao, S.J. Chen / Nuclear Engineering and Design 160 0996) 97-109

~Km

0.50

0.40

0.30

0.20

0.10

• ,,'/! ~

-0.00

\

-0.10

7 #/

\

-0.20

-0.30

llllllllllltllllllllllllllllllll[lll

-0.40 -6.0

-5.0

-4.0

-3.0

-2.0

llllllllllll - 1.0

0.0

~.0

2.0

3.0

llll 4-.0

l 5.0

xo/z Fig. 7. ]'he mode !1 stress intensity factors vs. the horizontal distance between a line crack and a point heat source in the half-plane medium: (yo/I = h//= 1.2, p = - 60°).

Y

3°

~

.1

S1:(Cl'l~':' )r Fig. 8. Local coordinate system at tip A of a line crack in the half-plane medium.

~

0.10 I 0.08

h/z= 1.5 - u/l=7

0.05

h~ -o.oo

lit

Ill

I t t t i l t l l

50

100

Iltl~il

150

ettltlt

¢

200

till

250

Fig. 9. The mode I stress intensity factors vs. the angle 0 with

t Ill

300 r/I

850

= 1.0.

0.200 0.175

'~

, . - . p=-20: p=-40: p=-40: ..... p=-60 o = ~ ~-= p = - 6 0

0.150 0.125

iI

0.100

~ \

(Lip {tip A B (tip (Lip (Lip

0.075 O.OGO 0.025

.

0.000 "1 LO

-:= z ~2.0

3.0

4.0

h/l

5.0

6.0

-

- • ~1

7.0

Fig. 10. The mode l stress intensity factors vs. the distance from the center of the line crack to the bounding surface of the half-plane medium: (xo/I = O, yo/I = O. I ).

o.3o

~" ~ A'm I~'~ *'*'*'*-* A'nD

o.~

I'~

•

-0.30

-0"40-6 0

50

4.0

-3.0

-2.0

-1.0

0.0

t.O

2.0

3.0

4.0

5.0

XO~G[. Fig. I I.inThe i and medium: mode II (Yo/a stress =intensity factors vs. the horizontal distance between the circular-arc crack and a point heat source the mode half-plane h/a = 1,5, 0 = 60*).

0.04I

G~/G,=0.5 ......

KL~

0.02

0~0!

'~ 0.00

~0.01

Fig. 12. The medium: (Yell mode ~ ho/i I ~stress 1.0, intensity ~ = 30°). Factors vs. the horizontal distance between a line crack and a point heat source in the half-plane

C~/C, =2.'0 t,

KI, i

0.005

0.000

-0.003

-0.005

-0.008

-o.o10 -0.0

in,llrllllltrll,l,lnn,llt~ll -5.0 -4.0 -3.0 -2.0

-1.0 0.0 Xo/l

r¢,llltllllltll,llllllll 1.0 2.0 8.0

4.0

6.0

Fig. 13. The mode I stress intensity factors vs. the horizontM distance between a line crack and a point heat sourc.~ in the half-plane medium: (yo/I = ho/I = 1.0, # = 30°).

o.15 , /

..... . . . . .

G~GI=2.0 a~,'a,=~.o

/-

cjc,=o.5

".,,,

O.lO

0.05

0.00 o

to

20

30

40

eo

0o

To

.o

oo

loo

H(z[I Angle (0) Fig. i4. ]'he mode I stress intensity factors at tip A vs. the half angle 0 of the circular-~,:: crack in the two bonded materials:

(Xo/a = 2 . 0 , yo]a = 0.5, h/a = 1.5).

C.K. Chao, S.J. Chen / Nuclecr Engim,~.ring and Design 160 (1996) 97-109

0.S0

. . . . .

c

109

Gz/~G,=Z.O = G~/G,=I.O G~/'G,=O.5

0.15

0.10

0.05

0.oo ~ 0

,

, 10

, 20

,

,,,,

80

,ill

,,i,

,,,,

l,,,

50 80 ?0 HaZf Angle (0)

40

,,,,

O0

lllllI

O0

100

Fig. 15. The mode I stress intensity factors at tip B vs. the half angle 0 of the circular-arc crack in the two bonded materials: (xo/a = 2.0, y./a = 0.5, h/a = 1.5).

surface. Both the temperature and displacement complex potentials are formulated such that the continuity conditions across the interface are satisfied. C o m p a r i s o n s o f the results with existing analytical solutions s h o w that the m e t h o d proposed here is effective, simple and general.

References J.L. Bogdanoff, J. Appl. Mech. 21 (1954) 88.

H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids. Clarendon Press, London, 1959. C.K. Chao and M.H. Shen, int. J. Solids Strucl. 30 0993) 3041. C.K. Chao and M.H. Shen, Int. J. Solids Struct., 32 (1995) 3537. Y.Z. Chen and N. Hasebe, J. Therm. S~ress. 15 (1992) 519. H. Sekine, J. Appl. Mech. 44 (1977) 637. G.C. Sih, J. Appl. Mech. 29 (1962) 587. J. Tweed and S. Lowe, Int. J. Eng. Sci. 17 (1979) 357.