Key Engineering Materials Vols. J83-I87 (2000) pp. 73-78© 2000 Trans Tech Publications. Switzerland

Kinking out of a Mixed Mode Interface Crack

T. Ikeda\ Y. Komohara^, A. Nakamura^ and N. Miyazaki^^ Department of Material Process Engineering, Kyushu University.

6-10-1 Hakozaki, Higashl-ku, Fukuoka 812-8581, Japan^Nippon Mektron, Ltd., 1-12-15 Shiba-Dalmon, Minato-ku, Tokyo 105-0012, Japan

^Yodogawa Steel Works Ltd., Yodoko Build., 4-1-1 Minami Honmachi,Cyuo-ku, Osaka 541-0054, Japan

Keywords: Crack Kinking, Fracture Toughness, Interface Crack, Mixed Mode, Residual Stress

A B S T R A C T

In this study we focus on the mixed mode fracture toughness and the kink angle of an interfacecrack. We measure residual stress and perform mixed mode fracture tests for three types of interfacecrack. Each mixed mode fracture toughness including residual stress is successfully described bystress intensity factors and for each interface crack. The kink angle of each interface crack alsocan be expected by the stress intensity factors using the modified maximum hoop stress criterion.

1 . S T R E S S I N T E N S I T Y F A C T O R S A N D C R A C K K I N K I N G O F A N I N T E R F A C E C R A C K

1.1 Stress Intensity Factors of an Interface Crack

The asymptotic solution of the stress distribution around an interface crack as shown in Fig. 1was proposed by Erdogan [1]. The stress along the A:-axis is shown as

^,(K2+1) + PJ(K|+I) ' ^,(K2 + 1) + P2(K,+1)

e = (l/27r)ln(l- p/l+p'

(2)

(3)

K ; = 3 - 4 v , { P l a n e s t r a i n )K, = (3-v,j/[l +v,j (Plane stress) (4)

7 4 Fracture and Strength of Solids

where a and P are Dundurs's parameters [2], £ is the bimaterial constant, and Pi, |i2. Vi and Vj areshear moduli and Poisson's ratios for respective materials. Cyy and are stresses, Ki and are thestress intensity factors (SIF) for respective mode and i is the complex number (/2= -I). / is an arbitarycharacteristic number. Argument of the stress in Eq. (1) is shown as

arg (o^ +i<T .) -y +eln (5)

7 is argument of the complex stress intensity factor Kj+iKu which is defined as Fig. 2 and

7 = sign (K„) cos- K { - n<y <7 i ) . (6)

sign(/i:;;) = l(A:„>0), (7)

where K, is Kf + K], . y is easily transformed for another value of 1 =1 [3].

Y = 7 + E l n W / / i ) ( 8 )

It is obvious by Eq. (5) that on A:-axis corresponds to KnlKi at rs= 4. In other words, KnlKjcharacterizes the ratio of shear stress and normal stress, o^ylOyy, at r= l| . It is difficult to decide thesuitable length of If.. However, it is obvious that we should use a fixed value for 1^, because a set of K/and Kfi cannot express unique stress field around a crack tip for different lengths of /*. Rice [4]recommended to select the length of 4 to be Ipm.

1.2 Crack Kinking Angle

He et al.[5,6] proposed the maximum energy release rate criterion for an interface crack betweendissimilar materials. They analytically obtained the kinking angle based on this model for an interfacecrack between dissimilar materials whose Dundurs parameter P in Eq. (2) is zero. The combination ofmaterials whose P is zero is limited to the combination of similar materials. Geubelle and Knauss<8)

Fig. 1 Coordinates system around aninterface crack tip.

Fig. 2 Definition of y on thefi e l d .

Key Engineering Materials Vols. 183-187 7 5

applied the maximum energy release rate criterion to an interface crack between dissimilar materialswhose P is not zero using the finite element method. They slightly extended the crack with differentkinking angle, and obtained the energy release for each angle. The crack kinking angle expected bythis theory depends on the crack extension length.

On the other hand, Yuki and Xu [8] proposed the maximum hoop stress criterion. The distributionof hoop stress Oee around an interface crack as shown in Fig. 1

O a a i " " i B c o s \ | / - C s i n \ } / ( j = \ , 2 ) ( 9 )2v27urcosh (en]

\ | / = e l n ( r / / ^ ) ( 1 0 )where Oeey is hoop stress in the area of material j, the functions B and C are given as

2 c o s cos 0 + 2esin 0 c o sU w ,

- c o s fe+Y (11)

C(0,e.Y] = W, 2sin f + 7 + cos 0 + 2esin 0 s m + -^sinW , §0 + 7 (12)

w ^ e - « s - 0 > ,

They simplified Eq. (9) by assumming 4^=0 because of the small value of e.

(13)

^ slK]^Kl2V2nrcosh (en)(Tm. = J—'—- 5(0,e.Y) (14)

J ®

-eln(474)Master curve (ro=

Shifted curve (ro= 4")

Fig. 3 Shift of the estimated kink angle with thevalue of Tp.

Load Angle

®exp(")

Fig. 4 Round shape mixed mode interfacecrack specimen and kink angle.

(r= 12mm for case 1, 13mm for case 2 and 3).

7 6 Fracture and Strength of Solids

CTeej takes maximum value if /30 = 0. However, the value of 4 in Eq. (10) cannot be ignored evenif e is small, because the absolute value of ln(r/4) increases with decreasing r. For example, if e= -0.033 and r/lk= 0.001,0054^ and sin4^ are equal to 0.974 and 0.228 respectively.

We modified maximum hoop stress criterion proposed by Yuuki and Xu [8]. An interface crackkinks out in the direction of maximum hoop stress at some fixed distance, tq, from a crack tip. Thedirection of maximum hoop stress can be obtained easily if ro= Ik because T takes zero at r= Ik-

According to this theory, the expected crack kinking angle ©omax for "0= 4 can be illustrated as 'mastercurve' in Fig. 3. The angle ©omax for another tq, ro= 4', can be obtained by shifting the master curve byAy= -e ln(4'/4) according to Eq. (8) as Fig. 3.

2. MIXED MODE FRACTURE TESTS OF INTERFACE CRACKS

Fracture tests were performed using round shapespecimens with mixed mode interface crack asshown in Fig. 4. Two combinations of materials, Aluminum - Epoxy resin A (Combination I) andAluminum - Epoxy resin B (Combination 2) were used for these specimens. Material properties ofAluminum, Epoxy resin A and Epoxy resin B are given in Table 1. A semicircular aluminum platewas set in a circular shape cast, then epoxy resin was pored into the cast. The cast was kept at 120Cfor 16 hours to cure the resin. The splicing surface of an aluminum plate was roughened by a piece ofemery paper beforehand and release agent was applied on a part of the surface for making an interfacecrack .

Residual stress along a jointed interface is sometimes very great. Wc evaluated the residualstress by measuring the released strain, AEt, of the end notched joint specimen at the delamination. Thereleased strain, AEt, was measured by a strain gauge attached to the surface of a specimen as shown inFig. 5. The relative expansion ratio, Apexp,. between material 1 and 2 is expected by the beam theory.The SIF's of a round shape specimen caused by residual stress were calculated from APexp. by thevirtual crack extension method (VCEM) with the finite element method (FEM) which developed inthe previous study [9]. They are indicated in Table 2 for Combination 1 and Combination 2.

Mixed mode fracture tests were performed using the round shape specimens as shown in Fig. 4and a universal testing machine (Shimazu Autograph). The load angles in Fig. 4 were selected as 30deg., 60 deg., 90 deg., 120deg. and 150 deg. The rate of displacement was Imm/min for all cases. Thefracture load was determined as maximum load because the fracture modes of all specimens werebrittle. The actual SIF's at the fracture can be obtained from the fracture load and the relative expansionratio, Apcxp, using the VCEM [9]. The crack kinking angle was measured as the angle of the tangent

, " . ~ V »„ -Ma'tertara" »„*■ \

(mm)Fig. 5 End notched joint specimen for measuring residual stress.

Key Engineering Materials Vols. 183-187 7 7

Ta b l e I M a t e r i a l c o n s t a n t s .

M a t e r i a l Yo u n g ' s P o i s s o n ' sModulus (GPa) Ratio

A l u m i n u m

Resin A

R e s i n B

73.1

3.35

3.84

0 . 3 2

0.43

0 . 3 7

Table 2 Measured released strain and stress intensityfactors caused by residual stress (1= 10 ^m).

C a s e Ae(^strain) (%)

K j K fi(MPa^) (MPa/m)

Comb. 1

C o m b . 2

9 6 1

3 8 1

- 0 . 1 0 5

-0.0421

0 . 2 1 4

0 . 11 7

- 0 . 4 5 0

- 0 . 1 7 6

0 . 7

0 . 6

^ 0 . 4

2 0.2~1 03" -0.2D

I3 -0.8

•1

• 1 . 2

• E x p e r i m e n t a l

(J,=IOpm»

•

' k

- S h i f t e d C u r v e

(r(,al.38XiO-'pm)

\ Master Curve (0^\(rj=f,=iOpm)

. 1 1 1 1 1

■1 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4

Yfor/j=10|im (rad.j

Fig. 6 Angle of crack kinking for Case 1.

1 . 2

1

0 . 8

0 . 6

0 . 4

0 .2

0

- 0 . 2

isOdeg.J- | J

20deg.g 90deg.P - . . 6 0 d e g .

■ S O d e g .

' r ' I ' l l

- 0 . 6 - 0 . 4 - 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4

K„ (MPa ym, = 1.4X 10" Vm)

Fig. 8 Mixed mode fracture toughness forCasel (/ = 1.4X10^^m).

1.5

1 . 0

" Or a

— 0 , 5

• oc<0

0 . 0

s -0.5

- 1 . 0

- 1 . 5

Experimental(/,=10fim)

G = 0.67rad.Master Curve Cdg

Shifted Curve (0,(rj=l.lX10"'nm)

- 1 . 0 - 0 . 5 0 , 0 0 . 5 1 . 0 1 . 5

Yfor/ =IOfim [rad.]

0 . 5

0 . 4

- 0 . 3X

0 . 2

n 0 . 1Q m

s

0

• 0 . 1

1 5 ( d e g . " i

120deg.

I SOdeg.

I 6 0 d e g .

. \30deg.

- 0 . 1 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5

fr„{MPaVm, l.lx lO'Vm)

Fig. 7 Angle of crack kinking for Case 2. Fig- 9 Mixed mode fracture toughness forCase 2 (/^= l.lXlO Vm).

7 8 Fracture and Strength of Solids

of an extended crack at the initial crack tip in Fig. 4. The crack kinking angles measured on the bothsides of a specimen were averaged.

The measured crack kinking angles with the master curves and the shifted curves of expectedcrack kinking angles for the combination I and 2 are shown in Fig. 6 and Fig. 7 respectively. Themaster curves do not correspond well with the experimental data in both cases. However, each shiftedcurve fits the experimental data very well. The values of Tq at which the shifted curves fit the experimentaldata best are 1.4 X lO ® |J.m and 1.1 X 10- pm respectively. These values of tq for both combinationsseem to be too small as real distances which characterize the crack kinking. We consider that the valueof tq is a kind of fitting parameter. The mixed mode fracture toughness for both combinations areshown in Fig. 8 and Fig. 9 respectively. The values of are taken as the same as the values of ro for theshifted curves in Fig. 6 and Fig. 7 respectively. In these figures, AT/p 0 corresponds to the crackkinking angle being zero (the crack extended along the interface). The curves of the mixed modefracture toughness are described by elliptical curves as

K , I K „ ) \ [ K „ I K „ c ] ^ = \ ( 1 5 )where Kjc and Kuc are constants for a joint system.

3 . C O N C L U S I O N

(1) The crack kinking angle of an interface crack between dissimilar materials is well described bythe modified maximum hoop stress criterion.

(2) The crack kinking angle fits well with the experimental data if the evaluation distance tq is takenas an appropriate value.

(3) The mixed mode fracture toughness of an interface crack can be described by a elliptical curveif the value of is taken as the value of tq which describes the crack kinking angle well.

R E F E R E N C E S

[1] Erdogan, F., ASME J. Appl. Mech., 32 (1965) p. 403.[2] Dundurs, J., ASME J. Appl. Mech., 36 (1969) p. 650.[3] Wang, J. S. and Suo, Z., Acta Metall. Mater., 38-7 (1990) p. 1279.[4] J. R. Rice, Trans. ASME Series E, J. Appl. Mech., 55 (1988) p. 98.[5] He, M. Y. and Hutchinson, J. W., ASME J. Appl. Mech., 56 (1989) p. 270.[6] He, M. Y.,Bartlett, A. etal., J. Am. Seram. Soc., 74-4 (1991) p. 767.[7] Geubelle, P. H. and Knauss, W. G., ASME J. Appl. Mech., 61 (1994) p. 560.[8] Yuuki, R. and Xu, J. Q., Eng. Frac. Mech., 41 (1992) p. 635.[9] Ikeda, T, Komohara, Y. and Miyazaki, N., Advances in Electronic Packaging, ASME EEP19-2(1998) p. 1137.