Frictional contact of an arc crack in an isotropic homogeneous material due to uniaxial loading

~

Pergamon FRICTIONAL ISOTROPIC

Engineering FractureMechanicsVol. 50, No. 1, pp. 121-130, 1995

0013-7944(94)00138-3

CONTACT

Copyright © 1994 Elsevier ScienceLtd Printed in Great Britain. All rights reserved 0013-7944/95 $9.50+ 0.00

OF AN ARC CRACK

HOMOGENEOUS UNIAXIAL

MATERIAL

IN AN

DUE TO

LOADING

RU-MIN CHAOt and MING-YING RAU Department of Naval Architecture and Marine Engineering, National Cheng Kung University, Tainan, Taiwan 70101, R.O.C. Abstract--In this paper we continue the work of Chao and Laws [Q. J/Mech. appl. Math. 45(4), 629~540 (1992)] and study the partial contact problem of an arc crack in an infinite isotropic elastic solid under uniaxial loading at infinity. We extend the analysis to include the effect of friction on the crack surfaces and consider the contact crack surfaces in slip and no-slip contact conditions. Formulation of the problem is based on the integral equation. The solutions are compared with the Muskhelishvili [Some Basic Problems of the Mathematical Theory of Elasticity. P. Noordhoff Ltd., Groningen] open-crack solution and the earlier Chao and Laws frictionless contact solution. We find that the stress intensity factors at the open crack tip do not change very much for the frictional contact case when compared with the frictionless contact solution. We also find that Kit at the closed crack tip changes significantly as the friction coefficient varies. The size of contact zone is also affected by both the loading orientation and the coefficient of friction.

INTRODUCTION THE PROBLEMof an arc-shaped crack in an infinite isotropic homogeneous elastic material subject to uniaxial loading at infinity was solved by Muskhelishvili [2]. This solution has been discussed extensively by many authors, including England [3], Sih et al. and Erdogan [4], Savin [5], Tada et aL [6], Rooke and Cartwright [7] and Sih [8]. All the authors cited above assumed that the crack will open for any orientation of the applied load. The solution was constructed based on this assumption and it was found that for certain orientations of the applied load at infinity, the crack surfaces are in partial contact, thus contradicting the assumption on which the solution was based. Toya [9] was the first researcher to describe the contact phenomenon of the arc crack when he worked with interfacial crack. However, Toya [9] did not suggest a solution to the problem of partial contact and dealt only with those regions in which the crack remained open. Chao and Laws [l] adopted Muskhelishvili's complex potential method to study the contact problem of an arc-shaped crack, and they reduced the problem to the solution of two singular integral equations. With a frictionless contact condition and a requirement for single-valued displacements, the two singular integral equations were solved numerically, and the contact length and stress intensity factors at the crack tips were found. There are a number of papers discussing the frictional contract problem of a straight crack in an infinite isotropic elastic solid (for example, Bowie and Freese [10], Comninou and Dundurs [l l, 12] and Dundurs and Cominou [13]), but none deal with arc-shaped cracks. In this paper, we discuss the arc-shaped crack subject to uniaxial loading at infinity with the crack surfaces are partially in frictional contact. Formulation of the problem is based on singular integral equations, and the crack is considered a continuous distribution of dislocation densities. This method is also used by Comninou and Dundurs [l 1, 12]. The same formulation can be applied to the problem of any number of arc cracks that have different sizes located on the circle of radius a. In our problem, only one arc crack is considered, and the formulation is reduced to the same singular integral equations given by Chao and Laws [1], who derived them in terms of complex variable method. Assuming that the applied load at infinity increases monotonically and the frictional force is small in the contact zone, then the slipping condition and Coulomb's friction law apply to the contact crack surfaces. The slipping condition requires that the relative crack surfaces' displacement in the radial direction must be zero. In this case, the shear stress ahead of the closed crack tip has inverse tTo whom correspondence should be addressed. 121

122

RU-MIN CHAO and MING-YING RAU

square root singularity, and one of the singular integral equation can be solved explicitly. However, the solution of the remaining equations must be obtained numerically. If the crack surfaces are so rough that they cannot have relative displacements in both radial and tangential directions, the no-slip condition is used to describe the contact behavior, cf. Mak et al. [14]. For the no-slip contact condition, the stresses at the closed crack tip are bounded, and the normal stress at the transition point from no-clip to open is assumed to be zero. Because of the rough surface contact, shear stress is discontinuous at the transition point. The two singular integral equations are solved with the help of a technique due to Erdogan and Gupta [15]. We are thus able to obtain the contact length for any given loading orientation and coefficient of friction. Similar to Chao and Laws [1], the required stress intensity factors at the crack tip can also be found. When compared with Muskhelishivili's solution [2] and Chao and Laws' frictionless contact solution [1], we find that the difference for stress intensity factor at the open crack tip is minimal. However, the mode II stress intensity factor, Kx~, at the closed crack tip goes to zero as the coefficient of friction increases, and it equals zero for the no-slip case. On the basis of the numerical results, the size of contact region increases as the loading orientation and the coefficient of friction increase, and it reaches maximum when the no-slip condition applies. Finally, the frictional slip contact crack problem was analyzed using MSC/NASTRAN finite element software with the help of the gap elements in the contact zone and a crack tip element at the crack tip. The results agree well with the analytic solution. FORMULATION We consider the two-dimensional (2D) problem in which an arc crack of radius a and length 2a$ is present in an infinite isotropic elastic solid. The solid is loaded by uniaxial tension at infinity (see Fig. 1). Based on Chao and Laws [1] solution, we assume that for certain loading orientations, portions of the crack will close. It is, therefore, convenient at the outset to assume also that the

y

T

X B

:Half angle subtended by crack :Ploar angle subtended by contact zone

i

:Loading orientation A, B :Crack tips

T.

T :Loading at infinity Fig. 1. Statementofproblem.

Frictional contact of an arc-shaped crack

123

crack surfaces are loaded by an applied surface traction and that the tractions on the upper and lower surfaces of the crack are equal and opposite. We represent the crack as an array of distributed edge dislocations along the arc. The Muskhelishvili complex potentials for a dislocation with the Burgers vector (bx, by) situated at z = z0 in an infinite media are well-known: ¢(z) = ~ ln(z - z0),

(1)

~0

q~(z) = ~ ln(z - z0) - - - , Z

--

(2)

Z0

where z = x + iy in the complex plane, i = x / ( - 1), and ~ = (p(bx + iby))/(ni(r + 1)). Notice p denotes the shear modulus, x = 3 - 4v for plane strain, k = (3 - v)/(1 + v) for plane stress, and v is Poisson's ratio. Now let the dislocation be at a point on a circle with radius a, that is, Zo = ad °. Then the stresses induced by the dislocation on the circumference, z = ae it, can be easily calculated from the complex potential functions: c

0 -7

a~(7) + iaro(7) = ~abr(O)cot--- f -

c

-

-

0 -7

c

bo(O)+ i~a bo(O) cot ---f-- + a ei(°-r)(b°(O) - ib,(0)), (3)

where br + ibo = e-i°(bx + iby) and c = (2/~)/(n(1 + x)). Denoting by bx(0) and by(0) the densities of the distributed dislocations, we note that dh(0) d----if-'

bx(0)=

dg(0)

by(0)- d-O'

(4), (5)

where

h(O) = Ux(a +, 0) - Ux(a-, O)

(6)

is the relative x-direction displacement of the outer crack face to the inner and

g(O) = Uy(a +, 0) -- Uy(a-, O)

(7)

is the relative y-direction displacement of the crack faces. The tractions on z = ae i~ associated with the unidirectional tension field are ~rrr(7) + i~0(7) =

[1 + cos 2(o) - 7)] + i ~ sin 2(o) - 7),

(8)

where o) is the loading orientation with respect to the x axis. On the basis of (3) and (8), the boundary conditions immediately give the integral equations for the dislocation densities. Normal traction on the crack surface is given by N(7)=-~a f + ~ b ' ( 0 ) c ° t ( ~ - f - ) d 0 - C f

~ a ~br(O) dO +{[l+c°s2(o) -Y)]'

171<+

(9)

and shear traction on the crack surface is given by S(7)=-~a

bo(O)cot

dO +-~cos2(o) - 7 ) ,

171 < ¢ .

(10)

To get the above two integral equations, the condition where the total Burgers vector equals zero has been applied. That is,

f~e'°(b,(O)+ibo(O))dO=O.~

(11)

As one can see, eqs (9) and (10) are identical to the singular integral equations given by Chao and Laws [1], who derived the above integral equations by means of complex variable method. Notice the integral is taken over the entire crack contour and therefore, with slight modification, the same formulation can be applied to the problem of arbitrary number and size of cracks on the circle of radius a subject to uniaxial tension at infinity. This problem will be addressed in a future study. EFM

~o/I--1

124

RU-MIN CHAO and MING-YING RAU

In the contact region, ~b - q < 7 < $, where q is the polar angle subtended by the contact region, the traction N(7) and S(7) are to be determined, and the contact condition Ur(a+, 7) -- Ur(a-, 7) = 0

(12)

has to be enforced. The contact condition of eq. (12) was proved by Chao [16] to be identical to eq. (17) of Chao and Laws [1]. If the uni-directional loading at infinity increases monotonically and the friction is weak, then we assume that slippage takes place along the entire contact region and follows Coulomb's friction law. That is, IS(7)l --gqN(~,)l.

(13)

Here, we have assumed that the outer crack surface slips over the inner crack surface in the 0-direction and that f is the coefficient of friction. According to Comninou and Dundurs [17], f = f k =f~ in the static problem, where fk and f~ represent the coefficients of kinetic and static friction, respectively. If the contact surfaces are so rough that the slippage of the crack surfaces is prohibited in the contact zone, then the following no-slip condition is added: uo(a +, 7) - uo(a-, 7) = 0.

(14)

The boundary conditions in the open zone are N(7 ) = 0,

S(7) = 0,

(15), (16)

and Ur(a+, 7) -- Ur(a , 7) /> 0.

(17)

Besides the above boundary conditions, the normal traction in the contact zone must be negative, i.e. a.(~) ~<0. NUMERICAL ANALYSIS AND RESULTS By rewriting the normalized angular coordinate ~ = tan - - 7/2 tan ~/2'

then the two singular integral equations can be written in terms of a scalar variable with range [ - 1, 1]:

.2fl

l f_ l -B,(¢)d ~'~ ~ =-~-

~B,(~) 2d~ +--n 2 q f ' _ t l +B2(q~2)2d~ + ½ [ l + c o s 2 ( o 9 - 2 t a n - ' ( q ~ ) ) l - N ( ( ) , -! 1+172~

(18) l f_ ,~--L-~ B2(~) d ~ = ~t/2f~ ,1 ~B2(~) +t/2~ 2dg +½sin2(o9 - 2 t a n - ' ( t / ~ ) ) - g ( ~ ) ,

(19)

where q = tan~b/2, b,:(O) =a/C(Bl(~)), bo(O) = a/c(B2(~)), and )Q(~)= (N(O))/T and ,~(~) = (S(O))/T are the normalized traction on the crack surface. Similarly, the slip or no-slip boundary conditions can be normalized in the same way, but they are skipped here for simplicity. We consider the orientation, o9, of the applied load at infinity to lie between 0 ° and 90 ° so that crack closure is limited to the lower crack tip B. The material considered here is isotropic, and according to Comninou and Dundurs[17], Dundurs and Comninou[18], and Chao and Laws [1], the shear stress ahead of the open crack tip A and the closed crack tip B has inverse-square root singularity for the frictional slip contact case. Therefore, eq. (19) can actually be inverted to get the explicit solution with an unknown constant. The solution is later substituted into eq. (18), and the final form of eq. (18) is

Frictional contact of an arc-shaped crack

1 f' Bt(~)d~+

rc~/(1 .q_ )/2)

-1

i T ~ T ~ "~-

de - - ~ -

125

-i l

+~2)/2d~

2)/D + ½[1 + cos 2(o) - 2 tan-t()/O)] - ~r(~) _ const, = ,/(1 + )/~)

(20)

where )/'

f '

-

,.... ,~,j ( l - ,~')

const = ~/(I + )/2)j_1 sin 2(0 - 2 tan- ()/z)) ~-~ ~

dR

and D is a constant to be determined. B~(~) has inverse-square root singularity at the open crack tip and is bounded at the closed crack tip. We then apply the numerical technique of Erdogan and Gupta [15] to the solution of the singular integral eq. (20). The discretized form of eq. (20) provides m equations with m unknown of B I (¢) and n unknown of contact pressure. The contact condition of eq. (12) gives exactly another n equations. Given the size of the contact zone, that is, given n contact points, the two uniqueness conditions provide the remaining conditions to solve for the m + n unknowns, the constant D, and the loading orientation. As described in Chao and Laws [1], this procedure is similar to that of Comninou [19] and is not ideal, but it suffices to give an accurate result. For the no-slip case, because of the rough contact, the closed crack surfaces stick together from the beginning to the end of the loading process. Since no slip has previously propagated into the tip of the crack, shear stress is also bounded at the closed crack tip. In such a case, both B I (~) and B2(¢) are bounded at the contact crack tip and are unbounded at the open crack tip. The no-slip conditions of (12) and (14) imply that B l (~) and B 2(~) are also zero within the no-slip zone. Notice that eq. (19) cannot be inverted as in the previous case because of the unknown function S(¢) and its nonlinear behavior. Therefore, eqs (18) and (19) have to be solved simultaneously by numerical technique. At first, we try to solve the two singular integral equations with all the no-slip conditions and the single-valued displacement conditions and find that the number of equations is one less than the number of unknowns. According to Comninou and Dundurs [17], the normal traction at the transition point from no-slip to open is zero. With this extra condition, the problem is solved and the solution gives a discontinuous shear stress at the transition point of no-slip to open. We believe that the discontinuous shear stress at the transition point is mainly due to the rough contact surface and the fact that shear stress at the transition point is finite on the contact part, but is zero on the open part. The same frictional slip contact problem was tested by MSC/NASTRAN with the help of a gap element and a crack tip element. When applying the gap element, the axial stiffness (KA) is chosen to match the frictionless contact solution of Chao and Laws [1] (Fig. 2). Then the value of axial stiffness for the gap element is set to 1021 for the subsequent frictional contact case. For a given arc crack with half crack angle ~b, when the loading orientation co falls into the close region described in Toya [9] and Chao and Laws [1], contact will be established at the lower tip B because we always choose o9 to lie between 0 ° and 90 °. On further increase of the orientation, the contact length will gradually increase as shown in Figs 3 and 4. Notice the fractional contact length A is defined as tan q/2 A-- - 2 tan ~b/2" As one can see from the figures, the friction force has the tendency to increase the size of the contact region. The size of the contact region increases with the friction coefficient and the loading orientation. We next turn our attention to the stress intensity factors at the crack tips and define the normalized mode I, II stress intensity factors as

K,, K. KI, 1(.2 = Tx/(rr a sin 4~)'

respectively. Then Figs 5 and 6 show Kt at the open crack tip A for various contact conditions and values of crack size, q~. Figure 7 shows/(2 at A. It can be seen from the previous two graphs that the effect of friction has little influence on KI (A) and/(2 (A). At the closed crack tip B, Kt

126

RU-MIN CHAO and MING-YING RAU

<

0.3

= 60 °

,~

,,:::

45 °

=

.i,a

Ill

////'

I::I Q.I

, .'~'

0.2

, =

a0 °

<,,,a 0

0

0.1

°J,-4

,/

/

//{Y' ,.7 ' 0.0

I 0

I

'"

I

'

I

20

10

"~'/"'1"''

I

///I ! '

I

50

60

70

30

40

Load

orientation,

'

I

'

80

90

(0

Fig. 2. Comparison between Chao and Laws[l] frictionlcss contact solution (show by lines) and NASTRAN solution (show by symbols).

0.35

Frictionless 0.30 --

solution[

1]

Frictional

contact,

f=0.4

..........

Frictional

contact,

f=0.fl

.................................

<1

t~

Frictional contact, f = l . 0

0.25 --

; NASTRAN

Z~

................... ; N A S T R A N

A

~k

~< .//'*

No-slip

contact

......................

//"

q~ ...

0.20 -O

/,/

.#J

oC~

/

0.15 -i/

O

i.•"

..'"

/

.l"

/"

// ,,

0.10 --

.....

.-

..

f..

/

/i

/.. ./"

/

.~ f//

j/

/-

/

j

// ~," ..y" i

Cr.

0.05 - /.~.:...>

0.00

' 30

I

'

40

I

'

I

60

Load

orientation,

60

=

Fig. 3. Contact length near tip B, for various contact conditions; for ~ = 45 °.

' 70

Frictional contact of an arc-shaped crack

127

0.35

Frlctlonless contact[I] 0.$0 --

.<]

0.25 --

./

Frictional contact, f=0.4

........

Frictional co,,tact, NAS~aN solution

.......................... /x A

f=0.8

Frictional contact, f=l.0 NASTRAN solution

//

/

•

No-shp

.,...J

contact

0

. / //

0 .,=.~ ,,,.J

// /.

,."

/

/' .I

...

/

.-"

,.

.....

//

jK/ /..

/

/ /#,. /

/

.I /." ./.' ,././

/ // ,./t ./.../ / • /. // • / / / /'/ t'" .// 1 "~ // ./ ./" // ./ / ..." / /." ~ // 1

0.15 --

., ./

//" ,.'~ ..........

/

.................../"

0.20 --

/.

//

.............. ~ ,

./

/

1 1 ~

i~.

0.10 --

/ / y

c~

/.

i/ i"//, • /"y." /'/'...." // ..//./ / /

/.:,/// /-// / ,..-..-~,/

0.05 --

.~:.'..~

0.00 10

I

I

I

I

I

20

30

40

50

50

Load

70

orientation,

Fig. 4. C o n t a c t length near tip B, for various contact conditions; for 4~ = 60 °. 1.0

0.8

t~ 0.6 -

~d

Muskhelishvili

QJ ~4

....................... Frictionless

solution contact,

f=0.O

°¢=q

0.4 o

.............

Frictional

..................

No-slip

contact,

f=0.8

contact

z

0.2-

0,0

; 0

I 10

'

I 20

Load Fig. 5. M o d e

'

I 30

'

I

'

I

40

orientation,

50

'

I 80

70

co

I stress intensity factor at the crack tip A for various contact conditions; ~b -- 45 °.

128

RU-MIN

CHAO

and

MING-YING

RAU

1.0

.......:::::::::::!!i!!!:::.::):::!::!:!!_;:(:i:/:: f " "" ..........

0.8-

.:f'5:"" 7

0.6~=~

N ~=~

4":"

Muskhelishvili solution

0.4-

.....................Frictionless contact, O

............

Frictional contact,

............

No-slip contact

f=0.0 f=0,8

Z 0.2

0.0

'

I I0

0

'

I 20

'

Load

[ 30

'

] 40

'

I 50

'

I 80

' 70

orientation,

Fig. 6. Mode I stress intensity factor at the crack tip A for various contact conditions; ~b = 60 °.

0.8

0.0-

"~~..,

o.4-

t~ "%,:...... 0.2N ,,..~ o~

0

Z

O.O-

Muskhelishvili solution Frictional contact, f=0.0 -0.2 -

Frictional contact, f=0.8 No-sllp solution

-0.4

I0]

20

3I0

Load

410

610

I

60

70

orientation,

Fig. 7. Mode II stress intensity factor at the crack tip A for various contact conditions; for ~ = 60 c

Frictional contact of an arc-shaped crack

129

0.2

Muskhelishvili .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

solution

Frictionless

contact,

f=0.0

0.0-

................ Cq

Frictional

contact,

...........................F r i c t i o a l

f=0.4

contact,

....! • . " -:'."

f=0.8

,::....,,"

/. /,

Frictional

-0.2

contact,

f=l.O

,

,

,

.

,,.., ." ,.

i/.."ff'/, /~// ,/ }~/ / / //.,',

,,.?>,'

~q -0.4

ZJ.,

Zl."7"' 0

Z -0.6

-O.S

'

I

'

I

10

Fig. 8. M o d e

'

I

'

I

20

30

Load

orientation,

'

I

40

'

50

I

'

60

70

II stress i n t e n s i t y f a c t o r at the c r a c k tip B for v a r i o u s c o n t a c t c o n d i t i o n s ; f o r 4b = 45%

0.2 MuskheIishv/I/ Frictionless

0.0-

solution

contact,

f=0.0 ........

Frictional

contact,

f--0.4

.........................

; NASTP,A N

Frictional

contact,

f~0.8

..............

; NkSTRAN

Frictional

contact,

f=l.0

...................

i NkSTRJ~

©

..."// /I

y$.. Ii

-

..i ..,"

[/3

i/

,, ,." ., .," ,,' /

q~ N -0.4

A O

.,." , , -0.2

:~

/ "~/

-

/ .,'

e: // O

. .--

..,/

/," /"

//

i/"

t, /

/

Z -0.6

-0.S

-

'

I 10

'

I 20

'

I 30

'

I

'

40

I S0

'

I 60

' ?0

Load o r i e n t a t i o n , Fig. 9. M o d e II stress i n t e n s i t y f a c t o r at the c r a c k tip B for v a r i o u s c o n t a c t c o n d i t i o n s ; f o r 4b = 60 ° a n d showing NASTR.AN solution.

130

RU-MIN CHAO and MING-YING RAU

is always zero. The m o d e II stress intensity factor for various friction coefficients and crack sizes can be seen in Figs 8 and 9. F o r the no-slip contact case, K2 is always zero at the closed crack tip B because o f the zero value o f B 2. We conclude that the friction force acts as a resistant force against the slipping o f the contact surfaces, resulting in a decreasing value o f K2. The M S C / N A S T R A N solutions are also plotted in Figs 3, 4 and 9, and they show consistency with respect to the analytic solution.

CONCLUSION The frictional c o n t a c t o f an arc-shaped crack in an infinite isotropic h o m o g e n e o u s elastic material subject to uni-directional loading at infinity is considered in this analysis. W e find that the m o d e I and II stress intensity factors at the open crack tip are less affected by friction than in the classical Muskhelishvili solution [2]. However, the m o d e I and II stress intensity factors at the closed crack tip depend not only on the loading orientation but also on the contact conditions near the crack tip. Generally speaking, when two arc crack surfaces are b r o u g h t into c o n t a c t by the applied load, K~ is always zero and K2 is determined by the contact conditions o f the two contact surfaces. W h e n the friction is small and slippage exists in the whole contact region, K2 decreases to zero as the friction coefficient increases. I f the contact surface is very rough, then the no-slip condition is applied to the c o n t a c t region. Since no slip has previously p r o p a g a t e d into the tip, Kn stress intensity factor is zero at the closed crack tip. Based on the numerical results, we have f o u n d that the size o f the c o n t a c t region increases as the loading orientation o~ and the friction coefficient increases. There are cases when the contact region m a y exist in the middle o f the crack while the two crack tips remain open. In other cases, part o f the contact region m a y satisfy slipping condition while the remaining contact region satisfies the no-slip condition. These cases are n o t included in this analysis and are left for future study. Acknowledgement--This work was supported by the National Science Council under grant NSC-81-0403-558.

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