Mode II stress intensity factor for layered material under arbitrary shear crack surface loading

~

Engineering Fracture Mechanics Vol. 55, No. 1, pp. 85-94, 1996

Pergamon

Copyright© 1996ElsevierScienceLtd. Printedin Great Britain.All rightsreserved 0013-7944/96 $15.00+ 0.00

0013-7944(95)00250-2

MODE

II S T R E S S

MATERIAL

INTENSITY

UNDER

FACTOR

ARBITRARY

SURFACE

FOR

SHEAR

LAYERED CRACK

LOADING

SUNG HO KIM Agency for Defense Development,Yusung P.O. Box 35(1-2-4), Taejon, Korea KANG YONG LEEr and MOON BOK PARK Department of Mechanical Engineering, Yonsei University, Seoul 120-749, Korea

Abstract--Amodel is constructed to evaluate the mode II stress intensity factor (SIF) for the layered

material with a center crack under an arbitrary shear crack surface loading. The mixed boundary value problem is formulated by the Fourier integral transform method and a Fredholm integral equation is derived. The integral equation is numerically solved and the effectsof the mode II SIF on the ratio of the shear modulus between each layer, Poisson's ratio and crack length to layer thickness are analyzed. Copyright © 1996 Elsevier Science Ltd

INTRODUCTION THE DETERMINATIONof stress intensity factors (SIFs) for the layered materials that have been widely used for high performance structures is very important in solving fracture problems [1]. Generally, the loading on the crack surface in the layered material can be separated into symmetric and anti-symmetric loading parts. For the symmetric loading cases, Hilton and Sih [2, 3] derived the mode I SIFs for the cracks parallel and normal to the interface in two bonded layers by the integral transform method. Sih and Cben [4] also obtained the mode II SIFs for the same problem [2]. Delale and Erdogan [5] obtained the mode I and III SIFs for two homogeneous elastic half planes bonded through a non-homogeneous layer with a center crack. Kim e t al. [6, 7] treated the cracks for three bonded elastic layers under in-plane, anti-plane shear and uniform tension loadings, respectively. Kassir and Sih [8] presented the analytical solution for a three-dimensional body with an elliptical crack under uniform shear loading on the crack surface. For the anti-symmetric loading cases, Chen [9] obtained the SIF for the anti-symmetric crack surface loading by integral transform method. Recently, Lee et al. [10] considered the three layered material subjected to linearly varying normal crack surface traction. Kim et al. [1 l] did the same problem [10] under an arbitrary anti-symmetric crack surface loading. Also, Lee e t al. [12] studied the multi-layered material under anti-symmetric normal crack surface loading. For both loading cases, Isida [13] obtained the mode I SIF for the crack embedded on the infinite homogeneous plate by Laurent expansion of the complex stress potential. Chen and Chang [14] obtained the mode I and II SIFs for an infinite homogeneous plate by the efficient finite element alternating method. Fan and Keer [15] evaluated the mode I and II SIFs for the infinite strip. In this study, the model is extended to the layered material with a crack under an arbitrary shear crack surface loading. A Fredholm integral equation is derived by the Fourier integral transform method and the mode II SIF is evaluated by solving the integral equation numerically. The effects of the shear modulus ratio between the layers, Poisson's ratio and crack length to layer thickness on the mode II SIF are analyzed.

THEORY Consider a plane strain type layered material with a center crack subjected to an arbitrary shear crack surface loading as illustrated in Fig. 1. It is assumed that the crack surface is parallel "['Author to whom all correspondenceshould be addressed. 85

86

SUNG HO KIM et al.

to the layer interface and each isotropic layer is perfectly bonded. It is also assumed that the outward layers are elastically identical. Using the Fourier integral transformation, stress and displacement components can be formulated for symmetric and anti-symmetric loadings, respectively, in the forms,

1 £~ 0_~. I-sin ix] a~ a"(J) = -#

ay ~

Lcos¢xj-"

(1)

[ sin ~ x l

(lb)

1 ['~2 G

~"~"' = - ~ Jo

t c o s i x j de

l t"" ~ [ - cos Cx1 a.~,,(j) =

uu) =

v
,,+v,,fo.[ nEt

n~

rc Jo ~ 8 y

(1 -

j

(lc)

sin i x J d~

dy

+ vji2G~

][cos,x

(ld)

sin i x _] di/~

(1 - Vj) Oy 3 + ( v i - 2)i 2 Oy

sin ~ x

d

cos ix

f/~ ,

2

Oe)

( j = 1,2).

(If)

where G~ = [Ai(~) + ~ y B ~ ( ~ ) ] c o s h ( i y ) + [Cj(~) + ~yDj(~)] sinh(~y)

The subscript ] = 1 represents the first layer where a crack is located and the subscript j = 2 represents the second layer whose outward edge is stress free. vj and Ej are Poisson's ratio and Young's modulus, respectively. The unknown coefficients Aj(i), Bj(i), Cj(~) and Dr(i ) are to be determined. In the model as shown in Fig. l, consideration of the upper half plane is sufficient due to the symmetricity. The boundary conditions are as follows, tL,(l) = - q ( x ) ,

um = 0 , ( t x l

>a,y=0)

(2c)

arm = a.~2), ( - oo < x < o% y = h~)

(2d)

%.,I = a,,.,.(2), ( - oe < x < oo, y = h.)

(2e)

u,) = u(2), ( - o o

< x < oo, y = h~)

(2f)

vm = v~z), ( - ~

< x < oo, y = h J

(2g)

tr,:,.(z) = O, ( - m < x < oo, y = hE)

(2h)

ayy(21= O, ( - oo < x < o% y = h2),

(2i)

2rid layer I

I

&5-2~ Z ~ q(x) 2a

Ii........ J /~2, ~z

(2b)

O'y~j)= 0, ( - oo < x < 0% y = 0)

/~2, Is2

~I,Vl

(2a)

( - a < x < a, y = O)

--x 1st layer 2rid layer I i

Fig. 1. Geometry and configuration of the model.

,I

Mode II stress intensity factor for layered material

q(x),

where the arbitrary shear crack surface pressure,

87

is defined as

q(x) ,~p,(a) x) zk-~ , x X t~ ,~=p2,(a) x X z' x = + ~,~,P~*-'( -a --

(3)

where P2* and P2, - ~are the given influence coefficients for the symmetric and anti-symmetric loading parts, respectively. By applying the boundary conditions to eq. (1), a pair of dual integral equations is obtained for each loading part in the forms,

M(~

cos

d~ =

sin

( I x I > a)

(4a)

(I x I < a), --

P2k

-

(4b)

I

where F({) -

{{B,({) + C,({)} M(~)

(5a)

M(~) -= ~D,(~)

(Sb)

z~tanh z~ tanh z, - z~ z~tanh z~ - 2(1 - vl - tanh zl(1 - 2vh + z~ 0 0 -

,(~) ~(¢) ~(¢) ~(¢) [ i ~(¢) '(¢)

-

-

4.0

~;,~nt aolhstiml * * * *Fieh~'saolution[17]

3.5

3.0

•~ /z

[•2.5

~Y

3 ~ 2.0

1.5

'

~

~

J

/

/

1.0

o.5

I

~

~

~

~

~

-~ ~

~

~o

Fig. 2. Comparison o f present dimensionless mode 1I SIF with previous solution (~ = O).

(5c)

88

S U N G H O K I M et al.

,4,(¢) = o zj

t a n hz~ 1 t a n hZl -- 1 0 0

T21 731

~=

T41

0 0 21

- 1 -- t a n h zl F

- z~

Ftanh

T44 z2 T64

-

734

-

T24=

- t a n h z, -- 1 -- F t a n h z, F t a n h z2 1

724

Z 1

1 t a n h z2

zJanhz~+l,

(Sd)

-1-zJanhz~,

T31 = 2 t a n h z~(1 - v,) + z,,

734

=

T26=

- z , t a n h z,

726 736 746

(5e)

z2tanh z2 T66 -z,-tanhz,

F[2(1 - v2)tanh z~ + z~]

--

2v,)

7'36 =

- F [ 2 ( 1 - v2) + z , t a n h z d ,

741

T.=

- F[(1 - 2v2) - z , t a n h z d ,

T46 =

T~=

1 + z2tanh z2, T6~ = z: + t a n h z2

(sf)

F =/~l/P2,

(Sg)

=

--

z , t a n h z,(1 -

- F[(1 - 2v2)tanh z, - zt]

zl = ~hl, z2 = ¢h2.

3.0 -v~ =vz =0.3 .......

IJl = v z = 0 . 0

2.5 F=4.0

2.0

'~=~I ~2.

'~~~/

1.5

1.0

0.5 F~0.25

1

0.0

I

I

I

I

2

4

6

8

• 1

O/I i

(a) ,e=O

0.9 v i ~ v:z : 0 . 3 .......

v l = I~l = 0 . 0

0.8

0.7

$

0.6

0.5

0.4

r-0.25

0.3

O.~o

~

~,

A a/kl (b) B-I

Fig. 3a and b. See caption on p. 89.

10

Mode II stress intensity factor for layered material

89

1.0 vz = v2

~0.3

F --A.O 0.8

a/k~=

2

.

~

~

~0.6

0.4 F=4.0

o.2 o

~'

~

&

10

a/kl (c) B=2

0.6 vz=v2 =0.3 .

.

.

.

.

.

.

Vl = 1~2 ~ 0 . 0

F~4.0

05

$ '~ 0.4

0.3

0.2

I 2

I 4

I 6

I 8

10

a/h~

(~ $=3 Fig. 3. Dimensionless mode II S I F vs

a/h~ for

various F. (a) fl = 0, (b) fl = 1, (c) fl = 2, (d) fl = 3.

By following the method of Copson [16],

~ar dp2k(t)Jo(~t)1

M(~) = .]0 [_q~2k-,(t)J,(~t)]

dt,

(6)

where J0 and Jt are the Bessel functions of the first kind of order 0 and 1, respectively. ~b2k(t) and ~b2k_~(t) are to be determined. By eq. (6), eq. (4a) is satisfied automatically and eq. (4b) is reduced to the form of the Fredholm integral equation of the second kind as follows, za (P2k(Z) [ (P2k(a)l L(P2k _ ,(,)j+ I ~K(')[(P:k-,(z)] d~= [-f2k-fzk(a) ,(a)_]l

(7a)

where K('r,a) =

~[F(~/a) - llFJ°(°tz)J°(~ca)l d~

-LJ,(~,oJ,(~o-)j

(7b)

90

S U N G H O K I M et al. f2,(a)

--

( 2 k - 1)!! a 2 k + (2~c!!)

(2k -

~, ,(~)-

(7c)

h'2

1)!! ax,_ ,,2

(7d)

In eq. (7), the following were defined, t = at

s = aa

,/u(t) =- ,~v/~p_~,~,(O,

~ =- ~ a

x = aY

,¢u-,(t) = ,~,/-~p~,_ ,~,_ ,(0.

(8)

To solve the Fredholm integral equation, the Gaussian-Laguerre integration technique is utilized. At t., = a., (m = 1, 2 . . . . . N a ) , ~P2,(T,,,) and ~P2,-~(t.,) can be determined numerically as follows,

£ [6me+ K('c.,,ze)]F cP2*(Z")]W('c,.)= I f2k(re) 1 (e =1,-- No)

.,=,

L~

,(~o,)J

-f~,-,e~)J

3.0 ltl ~ IJ2 = 0 . 3

25

.......

1~1 = V2 = 0 . D

20 F

~

~

~

"

- ~

~

a/ill -~.

a/kl ffi2.0

a/kt ~ffi1.0

1.0

0.5

/'=0.25

O00

I

I

1

2

I

,3

a/l,, (a) affi0

09 ut •v2 •0.3

.......

v~ ffi a,2 = 0 . 0

a/A1 ffi4.0

<

0.8

0.7

¢~I.0

rffi4.0

r=l.

3 0.6 ~.. 0.5

0.4

0.30

I

1

I

2 a/t~

I

3

(b) a = l

Fig. 4a a n d b. See caption p. 91.

(9a) '

Mode II stress intensity factor for layered material

91

1.0

. . . . . . . Vl= Is2~ 0.0 0.9

0.8

Ffl.O

.

0/

3 0.6

=1.0

05

0.4

0.25 I I

0.3

I 2

I 3

(c) #=2 0.7 vx = v 2 = 0 . 3 . . . . . . .

Vl

= V

2

zO.O

06

a/k1ffi4.0 0.5 .r'-~4.0

$

Fffi 1.0

~k~=3.0 a/k~-2.0

0.4

~

U

a/kl =1.0

0.3

0.2

I I

I 2

I 3

(c) ~'=3

F i g . 4. Dimensionless mode I1 S I F vs

a/h2 for

various F . (a) fl = 0, (b) # = 1, (c) # = 2, (d) # = 3.

where N,

K(zm,re) = ~ , = , Z ~ , [ F ( ~ , / a )

-

Lj.(~#.,)j.(~#..)AW(~,).

(9b)

brae is Kronecker's delta, r,, and ~ are the abscissas for Gaussian and Laguerre integrations, respectively, and r~ is collocation point. W(a~) and W(%) are the weight factors for Laguerre and Gaussian, respectively, with N~ and N~ being the number of integration points. After examining the convergency of the solution, N/and Na are selected as 30 and 110, respectively. The SIFs for both sides of the crack tips are calculated in the forms,

K,,R = lim x / 2 n a ( i - 1)a..., (if.0)

(10a)

KHL = ~lim- w / - 2na(i + 1)a..,,<,,(i.0).

(10b)

92

H O K I M et al.

SUNG

KII R and K,,L represent the mode II SIFs for the right and left crack tips, respectively. After some integral manipulation for e~,(£,0), the mode II SIFs are obtained in the forms, where

K,,R= x ~ - a [ ~°(1)p° + k=~ {~2~(1)P2k--4~2k ,(l)p2k '}1

(lla)

~,1~= ~ I o o . ~ o +

.~b,

~,~ : ~ . ~ . ~ +

o~ ,~..~k_l~1 .

NUMERICAL RESULTS AND DISCUSSION For the infinite and homogeneous material, K(z,a) in eq. (7) approaches zero because F(~/a) = 1. Thus, ~2~.(a) and ~2k-,(a) become f2~ ~(a). In this case, the SIFs are as follows,

=

2.5

20r

(p,o -

, gp3 3 + ' ):

~p, + 5 p ~ -

--~-___t--_-_--_U_"

alh~ =2.5, alh2=2.0

a/h~ =5.0, N ~ =2.0

05 - - v ~

=0.2. ~ =0.4

. . . . . . . vl =0.4, v2=0.2 0.0

I

I

I

I

2 F

3

(0) ,8=0 0.9

08

a/h~ = 2.5. a/~ = Z.0

a/h~ = 5.0. a/~ ffi2.0

07

0.6

0.5

0.4 a]k~ =2.0, a//~ = 0.1

a/k1 = 5.0, a/k2 ~O.l

0.3 - - u l

0.2

L

J

I

2

=0.2. ~ •0.4

....... ux =0.4, ~ =0.2 I

3

F (b) 8 = ]

Fig. 5a and b. See caption on p. 93.

(12a)

93

M o d e II stress intensity factor for layered material

0.9

0.8

0.7

......

3 0.6 ~.

:k:[i-iii~Oil"[[[:i~

-~-

..;;; .::'-. . . . . . . . . .

0.5

0.4

--ul .......

0.3 0

i

h

I

2

=,i

•0.2. ~ =0.4 •0.4, ~ =0.2

3

F

(4 B=2 0.60

--ul ~0.2,~ =0.4 . . . . . . . ul =0.4. ~ =0.2

0.55

050 J

[~ 0.45

~0.40 0.35

0.30

0.25

'

0

'

1

2

,

3

1" (d)/~=3

Fig. 5. Dimensionless m o d e II S I F vs F for various Poisson's ratios. (a) fl = 0, (b) fl = 1, (c) fl = 2, (d)

fl=3. k=,

(

~ 2"~')I~" "" {P2k + P 2 k - l }

1 1 3 )x/~ Po+ ~ p , + ~ p 2 + g P 3 + " " ,

(12b)

(2k - 1)!! - (2k - 1) × (2k - 3)...3 x 1

(13a)

(2k)!! = (2k) × ( 2 k - 2)...4 × 2.

(13b)

=

where

In the case of the cracked homogeneous strip material, the dimensionless mode II SIF for the right crack tip, ~f}/(pax//-~) is evaluated with fl = 0 as a function of a/h2 as shown in Fig. 2 and is in good agreement with Ficher's [17] within 1.25%. When two layers are elastically dissimilar, the numerical parameters for the dimensionless mode II SIF are a/ht, a/h2, F, v~ and v2. For fixed a/h2, the dimensionless mode II SIF for the right crack tip is evaluated as a function of a/h~ for various F as shown in Fig. 3. In the case when the

94

SUNG HO KIM et al.

first layer is stiffer than the second (F > 1.0), the dimensionless mode II SIF increases as a/h~ increases. On the other hand, it decreases as a/h~ increases when the first layer is softer than the second (F < 1.0). Also, the corresponding dimensionless mode II SIF decreases as the order of each term of polynomial loading (fl) increases. For fixed a/h~, the dimensionless mode II SIF is evaluated as a function of a/h2 for various F as shown in Fig. 4. It increases as a/h2 increases for all values of F. F = 1.0 represents the case of homogeneous material. If the first layer is thinner, the effect of the shear modulus ratio is more pronounced. The effect of Poisson's ratio on the dimensionless mode II SIF is shown in Fig. 5. If the first layer is thinner, the effect of Poisson's ratio is more significant as discussed by Keer and Guo [18]. CONCLUSIONS

The stress intensity factor (SIF) for a center crack under an arbitrary shear crack surface loading in three bonded elastic layers is analyzed by the Fourier integral transform method. The following conclusions are obtained from the numerical analysis: 1. The SIF solutions for the homogeneous material coincide with the previous results. 2. The dimensionless mode II SIF decreases as the order of each term of polynomial loading increases. 3. For a fixed half crack length to the second layer thickness ratio, the dimensionless mode II SIF increases with the increase of half crack length to the first layer thickness ratio when the first layer is stiffer than the second. 4. For a fixed half crack length to the first layer thickness ratio, the dimensionless mode II SIF increases with the increase of half crack length to second layer thickness ratio for all values of shear modulus ratio and if the first layer is thinner, the effect of the shear modulus ratio is more pronounced. 5. If the first layer is thinner, the effect of Poisson's ratio is more significant. REFERENCES [1] F. Erdogan, Stress intensity factors. J. appl. Mech. 50, 992-1002, (1983). [2] P. D. Hilton and G. C. Sih, A sandwiched layer of dissimilar material weakened by crack like imperfections. Proc. Fifth South Eastern Conf. on the Theoretical and Applied Mechanics (Edited by G. L. Rogers, S. C. Kranc and E. G. Henneke), Vol. 5, pp. 123-149 (1970). [3] P. D. Hilton and G. C. Sih, A laminate composite with a crack normal to the interface. Int. J. Solids Structures 7, 913-930, (1971). [4] G. C. Sih and E. P. Chen, Cracks in composite materials. Mechanics of Fracture VI, Martinus Nijhoff, Hague (1981). [5] F. Delale and F. Erdogan, On the mechanical modeling of the interfacial region in bonded half-planes. J. appl. Mech. 55, 317-324, (1988). [6] S. H. Kim, J. H. Oh and J. W. Ong, Stress intensity factors for center cracked laminated composites under uni-axial tension (in Korean). Trans. Korean Society of Mech. Engrs 15, 1611-1619, (1991). [7] S. H. Kim, J. H. Oh and J. W. Ong, Stress intensity factors for center cracked laminated composites under shear loading (in Korean). Trans. Korean Society of Mech. Engrs 16, 838-848, (1992). [8] M. K. Kassir and G. C. Sih, Three-dimensional stress distribution around an elliptical crack under arbitrary loadings. J. appl. Mech. 33, 601~611 (1966). [9] Y. Z. Chen, Crack problem in plane elasticity under antisymmetric loading. Int. J. Fracture 41, R29-R34, (1989). [10] K. Y. Lee, S. H. Kim and M. B. Park, Stress intensity factors for layered material under anti-symmetric loading (in Korean). Trans. Korean Society of Mech. Engrs 18, 1382-1387, (1994). [11] S. H. Kim, K. Y. Lee and M. B. Park, Mode i stress intensity factors for layered materials under anti-symmetric loadings. Engng Fracture Mech., in press (1994). [12] K. Y. Lee, S. H. Kim and M. B. Park, Stress intensity factors for multi-layered material under anti-symmetric polynomial loading (in Korean). Trans. Korean Society of Mech. Engrs 18, 1999-2010, (1994). [13] M. lsida, Elastic analysis of cracks and stress intensity factors (in Japanese). Fracture Mechanics and Strength o f Materials 2, 128, (1976). [14] W. H. Chen and C. S. Chang, Analysis of two dimensional fracture problems with multiple cracks under mixed boundary conditions. Engng Fracture Mech. 34, 921-934, (1994). [15] H. Fan and L. M. Keer, Two-dimensional contact on an anisotropic elastic half-space. J. appl. Mech. 61,250-255, (1983) [16] E. T. Copson, On certain dual integral equations. Proc. of Glasgow Mathematical Association 5, 19-24, (1964). [17] W. B. Ficber, Compendium of Stress Intensity Factors, pp. 20-27. Hillingdon Press, Uxbridge (1976). [18] L. M. Keer and Q. Guo, Stress analysis for symmetrically loaded bonded layers. Int. J. Fracture 43, 69-81, (1990). (Received 2 February 1995)