Introduction of stress functions around a circular interface crack between dissimilar materials

~

Engineering Fracture Mechanics Vol. 53, No. 4, pp. 661-673, 1996

Pergamon

0013--7944(95)t)0043-7

INTRODUCTION CIRCULAR

OF STRESS

INTERFACE

Copyright © 1996 ElsevierScience Ltd. Printed in Great Britain. All rights reserved 0013-7944/96 $15.00+ 0.00

FUNCTIONS

CRACK

BETWEEN

AROUND

A

DISSIMILAR

MATERIALS Y. MURASE and K. NAKAGAWA Department of Civil Engineering, Faculty of Engineering, Gifu University, Gifu, Japan T. HACHIYA Prefectural Office of Aichi, Japan and S. J. DUAN North China Institute of Water Conservancy and Hydroelectric Power, Handan, Hebei, P. R. C.

Abstract--The stress concentration around a circular interface crack between two dissimilar elastic planes is analysed in this paper. The constructive algorithm of the group of general stress functions which satisfy the crack boundary conditions is introduced. From that, six analytic solutions are selected to the problem of uniform tension acting at infinity. Then, the values and characters of the stress distribution and the crack opening displacements are investigated, which is by the following comparisons: the results for three different types of weight functions; the energy release rates (J-integrals); and the numerical results by finite element method.

INTRODUCTION As IS well known, the unbonded part along the interface between two dissimilar elastic bodies is called the interface crack in fracture mechanics. Some analytic solutions to such problems have been obtained by Williams [1], Rice and Sih [2], England [3] and Erdogan [4], but with a rapid oscillatory singularity of the stress at the crack tip. Another type of solution is proposed by Comninou [5] to remove the oscillations from the tip, in which a contact zone is introduced with a closed shape and in frictionless contact, but the contact zone is quite small, which has the same length as the order 10 -4 of magnitude of the grain size of the material. Authors have proposed two methods to analyse the straight line or the circular interface crack problems. One is by the approximate finite Fourier series solutions [7-9]; and another is by the closed stress functions [10, 11]. A fracture process zone is introduced near the crack tip with finite stress concentration and smooth closed crack opening displacement. However, the shape of stress concentration is dependent on the weight function and the process zone length, these have not been discussed yet in the above references. It is also necessary to examine the solutions and to get the new stress functions to express other stress fields. In this paper, the circular interface crack problem is considered and discussed. The formation and the group of the general stress functions which satisfy the interface crack boundary conditions will be shown. Then, six solutions will be superposed to express the crack problem loaded by uniform tension at infinity. At last, the characters of the stress distribution and the crack opening displacements near the crack tip will be investigated, which is by the comparisons of: the results in different weight functions; the fracture energy release rates (J-integrals) to express stress variable and fracture criterion; and the numerical results by finite element methods. F O R M A T I O N OF CRACK OPENING FUNCTION

The discussed mechanical model (Fig. 1) is that two dissimilar elastic materials are bonded in the circular direction r = a, but [0[ < co (from z~ to :~t) is unbonded and represents an interface 661

662

Y. MURASE et al.

f

t

~

f~f ~

*-- ~

t

I

) oot ,oe

"

'r - ~ f

i nt, e r f a c e

I i ne\

o,a

X 0o

f

examples a =10crn plane st;pain) ~ =20"

E1 =2.06X105 ul =0.3

B =10"

E2=2.94XlO

Go=O. IMPa

1'2=0.167

MPe"~ J 4" MPa

Fig. 1. Circular interface crack between dissimilar media.

crack. A crack opening along any curve from z~ to i~ can be formed by the following complex function branch

Ho(z,z,,f,) = "~L

(1)

\ z - z,

z = x + iy

z~ = ae ~''.

As z~ and :?1are the points along the circular interface, and the imaginary part is not continuous at crack tips with a jump (+in), we take a weight integral method [6] to transform the function into a smooth changed one near the tips. The crack tip z~ and ~ in function Ha(z, z~, ~) is taken as the integral variable and integrated under the defined weight functions. At present, the second, third, and fourth power weight functions [p2(t), p3,(t), p3b(t), p,(t)] are taken as examples (Fig. 2) and then the corresponding crack opening functions H2(z)[10, 11], H3(2) and H4(2) are obtained as follows:

H (z ) =

Ho(z ,t,,t2)pk~(t Opk2(t2)dt,dt2

(2)

=-2

2nd power

p2,(t)=c(t( a2- z,)(t ) ( z2), p22(t) = c ~ -- f,

.:"')',<. 133b(t) 134(0

/'

.o-°.

--

(3)

e2

.>,:-'-\

, ~ , ,

:l ~ , . % /

(d

-7-

)]

\\ "'\.

133a(t)

'X \\ .P 2(t).

(.d + I~

Fig. 2. Weight functions (four types).

Introduction of stress functions 3rd p o w e r

4th p o w e r

c(t - z,)(t Zz)(t - z3), _,~a~ _~a'

(~

p31(t)=

(a 2 p,2(,)=c

T

) J7

-

p4,(t)=c(t- z,)Z(t- z2)2 z,) ( 7

(4)

) 1~

_V/a 2 -

2 (5)

- 52

pkj(t).

w h e r e c is an u n k n o w n coefficent with different values for

Hz(z) =

663

fl; f -:_,:' ~1log (z-ZL)pz,p,:dzLdzu z - zu f z)

1 I - {3(z, - z) - (z2 - z)}(z2 - z ) q o g ( z 2 - z) + {3(z2

=Uc0

a2

,z z,~,z z,,log~z z,+{+ z)-(z, az)) z~

Z

log zt

-- Z

+ ~ o o l LV- {3(~ - z) - ( ~ - z)}(f~ - z)21og(~ - z) + {3(z-~ - z) - ( E

Z

z)}

{(o,) ( a2))( ~)~ (~ a') -{3(~- -~)-(~ a~)l(~ a;)2,og(~ a:)1 _

X (z2-z)21og(~-z)+

3 ~

z

--

~

z

~-

log

z

+~b~ z,,{+~Zl Z,~z, (~' a2)(~ ~)}] __

a 2

(6)

in which Co = (z, - z2) 3,

H3(z)=

e0 = (5, - 52) 3.

~i:fi" (z-~,~,,dzL~z~ ~log

= ~

l[

z-zu

/

{(z, -- z) 2 - 2(z2 + z3 - 2z)(z, - z) + 6(z2 -

z)(z3 -- z)}(z,

- {(z2 - z) 2 - 2(z, + z3 - 2z)(z2 - z) + 6(z, - z)(z3 -

a2 {(Z,-

z

Z3

a2 -

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z)}(z2 -

- z)21og(z~ - z) z)21og(z2 - z)

a2 z)(Z3-

a2 --))

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666

MURASE et

Y.

Table 1. Q(z)[F.(z) = Q(z)gl(z), Power Type

al. F,.(z) = Q(z)g~(z)] Q(z)

1

1

1

2

(zo) a+z

4

(z + a ) i ( z -

1

4

a)

(z + z)(,)(a -- a ) a

"

" n

Z

n

In eqs (6)-(8), zv = ae~.zt = a e - ~ . It is necessary that the weight function value is a real number at any point from z, to z2 or f, to 22 along the circular r = a and its integration is unit in the region (z~, z2) or (~,,~2). Other weight functions pk,(t,zl,z2) and pk2(t,f~,~2) can also be defined as long as the above conditions are satisfied.

(9)

The above crack opening functions are the base to construction of the stress functions.

S T R E S S F U N C T I O N T O SATISFY T H E B O U N D A R Y C O N D I T I O N S Boundary conditions The lowest limit of the boundary conditions are supposed as following. In the real crack r = a and 101 < co, ar = 0,v,, = 0. Both the stresses and displacements must be continuous across the circular interface 101___ o~ + / L In the process zone co < 101 < co +/~ the components ar and r,~ are continuous and the crack opening displacements u and v are discontinuous and smoothed.

Table 2. Mode of function g,(z), g,_(z)

Mode

gl(z)

g,.(z)

1

.l;(z)

./.;(z)

2 3

f,(z) if,-(z)

f,(z) ill(z)

4

i[k(z)

i]j(z)

Introduction of stress functions

667

Table 3. Analytic solution for uniform tension problem [F~(z) = Q(z) gffz), F,_(z) = Q(z)gffz)] Mode of function

Q(z)

No. 1 2

4

6

Power

Type

Function

Mode

g~(z)

g2(z)

Combined coefficient

1 1

I 1

1 I

3 4

i~(z) ifi(z)

ifi(z) ifi(z)

0.2822 0.2598

i~(z)

~(z)

0.3719

ifi(z)

V~(z)

0.1100

fi(z)

~(z)

o.3719

fi(z)

fi(z)

0.1 I00

1

2

1

+ z

3

i

a -

4

2

Element functions

T h e e l e m e n t f u n c t i o n s ~ ( z ) (k = 1 ~ 4) to satisfy the b o u n d a r y c o n d i t i o n s a b o u t the c i r c u l a r i n t e r f a c e a r e d e f i n e d b y the c r a c k o p e n i n g f u n c t i o n H ( z ) a n d the bielastic c o n s t a n t ~ as

.~(z) = c o s h { ( l + i~t)H} - cosh{(1 - i~)H} = i 2 s i n h ( H ) s i n ( ~ H ) J~(z) = sinh{(1 + i ~ ) n } + s i n h { ( l - i s ) H } = 2 s i n h ( H ) c o s ( ~ n ) f i ( z ) = icosh{(1 + i~)H} + icosh{(1 - i~)H} = i2cosh(H)cos(ctH) f 4 ( z ) = isinh{(1 + i~)H} + i s i n h { ( l - i s ) H } = 2 c o s h ( H ) s i n ( u H )

_._~y

process

/

o r

c e line

[MPa]

0.4 process z o n e

g

fflO

-0.4

Fig. 3. (a) Stress a,. (b) Stress ar (crack detail).

(10)

668

Y. MURASE et al.

1 a = - log ~z

G22 + G-~t Ki 1

(11)

'

in which Ej, G~, vt are the elastic constants for the inside, while E2, G2, v2 for the outside; and K = 3 - 4v to the plane strain while K = (3 -v)/(1 + v) to the plane stress. The above functions satisfy the lowest limit of boundary conditions, so we called them as element functions for stress functions of circular interface crack. General type o f the stress functions

The stress function for the plane problem can be expressed by analytic functions ~(z) and ~(z), and the stress components adopt their real parts. VW2W(z,z) = 0 W=z~ + •

(12)

ax = O2W/3Y 2 = 2~" - (~" + z q ~") a.,, = 02W/OX 2 = 27j' - (~" + z ~ " ) z.,,, = - O z w / o x o Y = - i(~" + z~")

(13)

2G(u - iv) --- Kql(z) - z ~ ' ( z ) -- ~'(z).

(14)

For the inside, =

+

• ,(z) =

+

05)

and for the outside, =

+ D

ck2(z)

(16)

• 2(z) = D2~e~(z) + D22~2(z)

in which, D . ~ D22 are arbitrary constants. It is shown that from the stress continuity conditions along the interface, D , = _+ D21,

DiE = _ D22

(17)

and from the displacement continuity conditions, K2 + 1 D~-----! =

D,2

Ki + 1)

---G-~2 + G----~

+

tn) tanh ( ~~-

(18)

.

( K 2 - 1 K , - 1 ) G 2G,

Basic solutions of the stress function c~j, c~j, Fj

For determining the stress functions, the thj, rk/(j = 1,2) can be arbitrary combined by the type Q(z) as shown in Table 1 and the mode of functions g~(z), g2 (z) as shown in Table 2. ckl(z)=zF,(z)

~bl,(z)= -a2Fl~z)

dp2(z) = zF2(z)

~p2(z) =

Fl(z) = Q(z)gt(z)

a2

a~F~(z) - 2 z F2(z) F2(z) = Q(z)g2(z).

1

(19) (20)

It is difficult to get rk2(z) from ~(z), but ~b2(z) is not necessary for determining the stress and displacement components. No matter how the modes and types are combined, the stress function can be found which satisfies the lowest limit of boundary conditions along the interface crack, but the high power Q(z)

Introduction of stress functions

669

__~y

process

-50tur

[

z°nerack zone x

l

O

)rocess zone

X

- process S czonem

~

7 Fig. 4. (a) Displacement U,. (b) Displacement U, (crack detail). is divergent as z--* oo. The general solutions presented here can express the states of expanding (or shrinking), torsion and uniform tension. EXAMPLES AND REMARKS Numerical examples in the case o f uniform tension at infinity

In this case, six analytic solutions are selected and combined (Table 3), in which the coefficients are determined by the least square method for stress tr, inclination value along the circular interface. Here, the fourth powers weight function is used, while the case in the second powers weight function has been shown in ref. [11]. A plane strain model (Fig. 1) is considered, the inside is steel while the outside concrete, E, =2.06 x 105 MPa, v~ = 0 . 3 for r < a and E 2 = E U 7 , v2=0.167 for r > a , a = 10cm, ~ = 2 0 °, fl = 10° and a0 = 0.1 MPa. The calculation results are shown in Fig. 3 (stress tr,) and Fig. 4 (displacement U,). For easy understanding, two types of figures with different scales are drawn. (a) ar = 0 is completely tenable in the crack opening part; while in the process zone there are no oscillatory singularities with finite stress distribution. (b) The displacement U, is formed with a smooth crack opening shape.

[MPa] 0.3

[xlO-5cm]

2nd -- ...... 4 th

l"x ~-~~'-',

~,~

0.2 or

Ur

/ :1!7 /'/

0.1

~',~\

2nd 4 th . . . . 3rd (a) . . . . . 3rd (b)

~,S t

'

20

,

'

I

30 () [deg]

Fig. 5. ~, for different weight functions.

20 ' 30 (~[deg] Fig. 6. U, for different weightfunctions.

670

Y. MURASE et al. node 2035 element 1553

O"o

t Ph~

t ",

t

t

t

t

~ I

I

\,

t

t

I'

/

crack

,~

__jo E

~

u C

crack DPoce88

60cm

detail

zone e l e m e n t

$0 ~

Go process

zone(detai

t)

Fig. 7. FEM model.

Comparison of the calculation results for different weight functions Figures 5 and 6 are, respectively, used to demonstrate the stress and displacement distribution along the process zone for different weight functions. We can see there are no very large differences in the extreme values.

The examination for FEM results and the J-integrals For checking the correction of the proposed analytic solution, the numerical results by the finite element method and the energy release rate as the crack extension (J-integral) are presented and compared. Figure 7 is the structure and F E M mesh. Within the process zone, different Young's modulus (strain softening) to the other part is assumed. In detail, one case is assumed E = 2058 MPa and another case is changed from E = 2.06 x 105 MPa to 21 MPa (from the effective crack tip to the physical crack tip) in the linear case. The comparisons of the FEM and the present analytic results are shown in Figs 8 and 9. xlO-Scm]

5 Ur 4

///.

-

.~

3

,L I,%

2

/,"

1 0

I II proce88

crack

II I proce88

zone

zone analytic ....

FEM(E=

zone 8olution constant)

-----

FEM

(E=

Fig. 8. Displacement U, (reduce inside from outside).

varl

able)

Introduction of stress functions

[M Pa] 0,6 CTr

671

/

0.5

/

\

0.¢

f-~

0.3--

I

'

/

i J

0.2

$' , '

O.q 0

/

/

/

.-\

-L

/

/

//

I

zone c r a c k zone process ~ a n a l y t i c solution . . . . F E M ( E = constant) - - - - - F E M ( E = v a r i a b l e ) Fig. 9. Stress

~, (comparison

of FEM).

[xlO-5cm]

[MPa] r

,

r

,

~

,

i

I

'

r

i

0.3 or

'

I

.

B=IO° (constant)-

0.2 Ur

0.1

co~<______~=

25° ..

/ / / / / / 13=10~ (constant)

1'0

2'0

3'o

4'o

1'0

O[deg]

Fig. 10. a, varying with different co.

20

3'0

4'0 8 [deg]

Fig. I I. U, varying with different co

T w o types o f energy release rates are defined: one is the case o f the physical c r a c k extension with c o n s t a n t process zone length; a n d a n o t h e r is the case o f the process zone extension with c o n s t a n t physical c r a c k length. A t present, the following two cases are calculated: fl = 10 ° with 2to = 3 0 ° , 3 4 ° , . . . 50°; a n d 209 = 4 0 ° with fl = 5 °, 7 ° , . . . 15 ° . U, a n d trr d i s t r i b u t i o n s are, respectively, s h o w n in Figs 10-13. T h e J - i n t e g r a l s (the energy release rates) are c o m p u t e d f r o m the c h a n g e r a t i o o f the stress a n d the displacement: to = 15 °, fl = l0 ° with Ato = 1°, 2 °, . . . l0 ° a n d to = 20 ° , fl = 5 ° with Aft = 1°, 2 °, . . . 10 °. These are s h o w n in Figs 14 a n d 15. (a) C o m p a r i s o n o f the results b y F E M a n d the present solution. W e can see t h a t the d i s p l a c e m e n t U, d i s t r i b u t i o n has a similar shape, b u t the stress a, is very different. This m a y be because o f the process zone a s s u m p t i o n in F E M a n d the f o u r t h p o w e r s weight function used in the present case.

[MPa]

[xlO-Scm] '

OAf '

'

'

'

Ur 2

10

20

30

Fig. 12. tr, varying with different ft. EFM

53/4---L

I

4 ~

40 8 [deg]

ol

'

i

i

~=20"(constant)

~ 1 5 "

4.0 e [deg] Fig. 13. /Jr varying with different B.

1'o

2b

3'o

672

Y. MURASE et al. I x 1 0 -4" N / c i

m] i

[xl o-* N/c m] i

i

i

i

~

i

i

I

;,=2~"(~on~ant; 13 = 5" A

6=I"

(constant) ~10"

J ~=15"

(constant)

.......

2nd 4 th

B =1o"

(constant)

....

3 r d (a)

A~=I" i

i

~10" i

i

-- . . . . ~

i

i

I

I

d

I

I

2nd .......

4th

....

3 r d (a)

.....

3 r d (b)

3 r d (b) i

i

I

I

lO A co[deg]

Fig. 14. J-integral value due to Aco.

o

I

I

I

~

I

I

I

L

J

1o

A 13[deg] Fig. 15. J-integral value due to A/L

(b) When the process zone length is constant, the stress concentration value and crack opening displacement near the crack tip tend to be large as the physical crack length increases. (c) When the physical crack length is constant, the stress concentration value at the crack tip tends to infinity as the process zone tends to zero. (d) The J-integral values in all of the different weight functions present in the linear variation are very similar to each other. We may conclude that there are no very large differences from the energy conception, although the stress distribution shape near the crack tip has some changes.

CONCLUSIONS

The general stress function The practical general analytic solution for the circular interface crack problem is presented in this paper. The stress oscillation at the crack tip is successfully vanished by introduction of a process zone with finite stress concentration. (a) The function ~ (z) formation is for vanishing the stress oscillatory singularity at the crack tip, and the weight function p/(t) introduction is for getting the function with finite gradient at t = to and t = o9 + fl, and with a smoothed shape in whole. (b) The function J~(Hj) is the basic element of the stress functions, which can create crack opening along the circular case and satisfy the interface crack boundary conditions. Where k = 1 ,-~ 4 is subject to the crack problems of tension, torsion, etc. (c) The basic solution of stress function formed by Fj(z) multiplied by z or a/z can create the symmetric or antisymmetric stress field. Q(z) is expressed as (z/a + a/z) or i(z/a - a/z) to the nth power, which becomes the real or pure imaginary value of cos (nO) or sin (nO) along the circular interface. (d) The arbitrary appointed stress distribution within the process zone can be realized by superposition of some group of the stress functions, which depends on the weight functions. (e) From the numerical analysis, we find that the solution for concentration force Px or Py acting at the plane of the circular area [9] can not be obtained by the present general solutions. Other types of solutions are needed for this problem.

Conclusion due to the numerical results (a) From the numerical results by three types of defined weight functions, we can find that there are no differences in characteristics. We can also conclude this may be correct for high powers weight functions. (b) As a checking method to the present solutions, we gave~out the FEM numerical results and computed the J-integral values. Although there is a problem that the modeling condition is different from the FEM and the present, it shows the solutions approximation by their certain extent identities. This is also proved by the J-integrals. It is difficult to determine which weight function is more suitable for simulation of the fracture process in fact, but the stress distribution may not influence the results from the energy conception.

Introduction of stress functions

673

(c) Our main objective in this paper is to obtain the stress function on mathematics, where the whole field is considered as elastic. We show that it is possible to express the fracture process zone approximately by the analytic solutions. However, how a large process zone should be taken in steel, concrete, rock or other materials is necessary to investigate in the future. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

M. L. Williams, The stresses around a fault or crack in dissimilar media. Bull. Seism. Soc. Am. 49, 19~204 (1959). J. R. Rice and G. C. Sih, Plane problems of cracks in dissimilar media. J. appl. Mech. 32, 418-423 (1965). A. H. England, A crack between dissimilar media. J. appl. Mech. 32, 400-402 (1965). F. Erdogan, Stress distribution in bonded dissimilar materials with cracks. J. appl. Mech. 32, 403-410 (1965). M. Comninou, The interface crack. J. appl. Mech. 44, 631 636 (1977). S. J. Duan and K. Nakagawa, Stress functions with finite stress concentration at the crack tips for a central cracked panel. Engng Fracture Mech. 29, 517-526. S. J. Duan, H. Yazaki and K. Nakagawa, A crack at the interface of an elastic half plane and rigid body. Engng Fracture Mech. 32, 573-580 (1989). S. J. Duan, H. Yazaki, K. Fujii and K. Nakagawa, A mathematical approach of the interface crack with a fracture process zone. Research Report, Faculty of Engineering, Gifu University, No. 41, pp. 1 10 (1991). K. Fujii, Y. Kato, S. J. Duan and K. Nakagawa, Stress analysis around a partially bonded rigid cylinder in an elastic medium with process zones. Engng Fracture Mech. 45, 31-38 (1993) Y. Murase and K. Nakagawa, Stress analysis around a circular interface crack between dissimilar media. JSCE, No. 483, 1 26~ 41-49 (1994-1) Y. Murase, S. J. Duan and K. Nakagawa, Stress analysis around a circular interface crack between dissimilar media loaded by uniform tension at infinity. Engng Fracture Mech. 48, 325-337 (1994)

(Received 14 July 1994)