an elastoplastic analysis of the interface … · an elastoplastic analysis of the interface ... an...

J. Mcch. Phy\. Solids Vol. 39, No. 3, pp. 31 l-344, 1991 Prmkd in Great Britain.

0022-5096iYI $3.00+0.00 I( 1991 Per&mm Press plc

AN ELASTOPLASTIC ANALYSIS OF THE INTERFACE CRACK WITH CONTACT ZONES

N. ARAVAS and S. M. SHARMA

Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA IYlO4, U.S.A.

AESTRACT

AN ELASTOPLASTIC solution for the interface crack with contact zones is presented in this paper. The elastic asymptotic solution is discussed first and approximate expressions for the Comninou stress intensity factors of a Griffith interface crack are obtained. It is shown that the elastic asymptotic solutions based on the assumption of a closed crack tip predict material interpenetration : material elements near the crack tip are “turned inside out” and cross the bimaterial interface. The problem of a crack along the interface of an elastoplastic material and a rigid substrate is analyzed in detail. Any contact between the crack surface and the substrate is assumed to be frictionless. An elastoplastic asymptotic solution for a closed crack tip under plane strain conditions is presented. It is shown that the asymptotic solution is separable in r and 0, where (r, 0) are polar coordinates at the crack tip. A boundary layer approach is used to study the near tip elastoplastic fields under small-scale yielding conditions. The finite element method is used for the solution of the small-scale yieldmg problem and the validity of the developed plastic asymptotic solution is verified numerically. Large-scale yielding solutions for a Griffith interface crack with contact zones are also presented.

1. INTRODUCTION

THE EARLY elasticity solutions of the interface crack problem were developed for an open crack, i.e. for a crack with traction free surfaces (ERDOGAN, 1963; SALGANIK,

1963 ; ENGLAND, 1965 ; ERDOGAN, 1965 ; RICE and SIH, 1965). These solutions involve the oscillatory singularities found by WILLIAMS (1959) and predict wrinkling of the crack faces and overlapping of the materials, which contradicts the assumption ot traction-free crack surfaces (ENGLAND, 1965; MALYSHEV and SALGANIK, 1965). The extent of material interpenetration depends strongly on the type of far field loading. For an interface crack in a tension field the interpenetration region is limited to a very small fraction of the crack length, whereas for the case of remote shear the open crack solutions predict material overlapping over half the crack length (WILLIS, 1972;

COMNINOU, 1978). Similar situations in which elastic solutions for open cracks predict overlapping of the crack surfaces in homogeneous linear elastic materials have been reported by BOWIE and FREESE (1976) and THEOCARIS (1986, 1987).

An alternative approach to the interface crack problem was presented by COMNINOU

311

312 N. AKA~AS and S. M. SHAKMA

( 1977a. 1978) and COMNINOU and S~‘HMUESER ( 1979) who considered the problem 01 a crack along the interface of two semi-infinite homogeneous isotropic linear elastic media and examined the possibility of frictionless contact along the crack face. The Comninou solutions predict the existence of two contact zones, one at each tip. the size of which depends very strongly on the type of remote loading. For the case of remote tension, the size of the contact zones is a very small fraction of the crack length, whereas in the presence of far field shear one of the contact zones extends over a substantial part of the crack. The stress singularity is of the square root type and the asymptotic solution is mode-II-like in the sense that the shear stress is the only singular stress component ahead of the crack. An analytical solution to the interface crack problem with contact zones was presented recently by GAUTESEN and DUNDURS (1987, 1988). A uniqueness proof for the solution of such contact problems has been given by SHIELD (1982).

In this paper, we consider the problem of a crack lying along the interface of an elastoplastic medium and a rigid substrate. The possibility of frictionless contact of the crack surface with substrate is considered. A discussion of the elastic solution is presented in Section 2. It is shown that, in cases where the size of the contact zone is small compared to the crack length, the traditional solution based on the assumption of traction-free crack faces is still valid over distances that are larger than the contact zone size but still small compared to the crack length. It is also shown that asymptotic solutions based on the assumption of a closed crack tip also predict material interpenetration of a different kind--~material elements near the crack tip are “turned inside out” and cross the bimaterial interface. This puts some limitations on the applicability of the Comninou asymptotic solution when the predicted size of the contact zone is small compared to the crack length. An elastoplastic asymptotic solution for a closed crack tip under plane strain conditions is presented in Section 3. It is shown that the asymptotic solution is separable in r and 0, where (r, H) are polar coordinates at the crack tip. A boundary layer approach is used to study the near tip elastoplastic fields under small-scale yielding conditions. The finite element method is used for the solution of the small-scale yielding problem and the validity of the developed plastic asymptotic solution is verified numerically. Large-scale yielding solutions for a Griffith crack lying along the interface of an elastoplastic medium and a rigid substrate are also obtained. When remote tension is applied. the crack opens completely when the size of the plastic zone becomes of the order of the contact zones predicted by the elastic solution. In the case of remote shear, there are two contact zones, one at each tip; the larger contact zone is of the order of the crack length. whereas the smaller one is always a small fraction of the crack length.

7 _. THE ELASTIC SOLUTION

The elastic solution of an interface crack with contact zones is discussed in this section. A finite element solution is developed first and it is used to motivate a discussion of the relevant stress intensity factors and other aspects of the asymptotic elastic solution.

An elastoplastic solution is presented in Section 3.

Interface crack with contact zones 313

2.1. A,finite element solution

We consider the plane strain problem of a crack lying along the interface of an elastic medium and a rigid substrate (Fig. 1). The elastic medium is homogeneous linear isotropic with shear modulus G and Poisson’s ratio v. Remote tension a:,. is applied in the direction normal to the crack and a uniform tension a’, = VCJ: /( I -v) is also applied parallel to the interface at infinity so that the extensional strain E.,, is continuous across the interface. The remote shear is assumed to vanish, i.e. o,?, = 0.

A solution to this problem was presented by COMNINOU (1977a) who showed that the crack opens everywhere except for small regions near the crack tips where the crack faces come in contact which was assumed to be frictionless. The asymptotic Comninou solution is summarized in Appendix A, and the traditional asymptotic solution based on the assumption of traction free crack faces is given in Appendix B.

An analytical solution to the interface crack problem with contact zones was presented recently by GAUTESEN and DUNDURS (1987, 1988). For the case of a rigid substrate and remote tension their results for the Comninou stress intensity factor K,, at the right crack tip and the size c of the contact zones reduce to

K,, = -O;c,.J7W[(l+4&‘)‘~2+O(k’)], (2.1. I)

c z 8 exp [ (

-f tanP’(2c)+; , a )I

where

1 E= ‘,

(2.1.2)

(2.1.3)

2a is the crack length, K = 3 -4~ for plane strain, and K = (3 - v)/( 1 + v) for plane stress. The corresponding complex stress intensity factor for the open crack is given by (RICE and SIH, 1965 ; RICE, 1988)

K = ~~~Jxa( I + 2ie)(2a) -I’:. (2.1.4)

FIG. 1. Crack along the interface of a deformable medium and a rigid substrate.

3 I-I N ;\I< \\,\\ and S M Stj>\i<1\1,\

The finite element model is constructed using X-node isoparametric plane strain elenicnts with 2 x 2 Gauss integration. Intcrfrice elements ;II-c introduced along the crack fxe. thus allowing for possible contact with the rigid substr;ltc. Because of symmetry. only one half of the elastic medium needs to be modclcd. The crack length is 2~1 and the height and half width of the elastic body arc‘ taken to bc IOOrr. A scnli-

circular arrangement of wedge-shaped (collapsed) elements ia used around the cracA tip. The radial length of each of the wedge-shaped elements is IO “‘II and all crack tip nodes are tied. Semi-circular strips of elements with 4 strips spanning each decade of radial length surround the wedge shaped elements in the region v c 0. Itr. where I’

is the radial distance from the crack tip. There ;Ire I3 quaI si/ed elements within each strip spanning the angular region from 0 to TI. A total of 034 elements is LIS& in the

calculations which arc carried out for ;I value of Poisson’s ratio I’ = 0.2. The ABAQUS general purpose finite t’lcment program (HIHBITT. 19X4) is LIX~ for

the computations. The sire of the contact Lone is found to lx 4.39 x IO “O which agrees well with the analytical result 4. II x IO “O of (2.1.2). b’igure 2 shows the calculated normalized stress component n,, o,‘, together with CoMiuvor:‘s (1977~1) asymptotic solution and the traditional asymptotic solution based on the assumption

of :I traction-free crack fax along the radial lint (0 = 4. I ) for txdial distances ~mallct

than the contact zone si/e. The open circles in Fig. 2 and in ~111 subscqut‘nt ligurcs in

the paper indicate the results of finite &mcnt solutions. The numcric:~l and ~‘omnino~~

-12.0 -11.0 -10.0 -9.00 -8.00 -7.00

1 og “ a


asymptotic solutions agree very well for values of r that are much smaller than the contact zone size. It is interesting to note that for 8 # 0 singular compressive stresses

develop ahead of the crack even though remote tension is applied. Figure 3 shows the radial variation of q,,,/acV on B = 4.1” for larger Y values together with the Comninou asymptotic solution and the traditional asymptotic solution. The size c of the contact zone behind the crack is also indicated in Fig. 3. It should be noted that the scale along the stress axis is different in Figs 2 and 3. The normal stress G,.,. changes sign and approaches the traditional asymptotic solution for values of Y larger than the contact zone size. All stress components behave in a similar way, i.e. they agree with

Comninou’s asymptotic solution over distances smaller than the contact zone size, whereas for larger r they agree well with the traditional asymptotic solution.

It can be easily verified that the same holds true when both the upper and lower materials are deformable. This clearly shows that the Comninou and the traditional asymptotic solutions do not necessarily exclude each other-it is simply that their regions of validity are different. Figure 4 shows a schematic representation of the several regions of dominance where it is shown that the Comninou asymptotic solution is the relevant solution sufficiently close to the tip, whereas the traditional asymptotic solution is valid for larger r, a suggestion that has been also made by ATKINSON (1982) and RICE (1988) who introduced the concept of ‘small-scale contact’.

It should be emphasized that the above conclusions hold true provided that the

0.100 r

0.050 -

$$ x 10-3 o YY

-0.050 -

0

0

0

0

0

-8.00 -6.00 -4.00 -2.00

FIG. 3. Radial variation of g,., along 0 = 4.1 The open circles indicate the results of the finite clement solution. Note that the scale along the stress axis is different in Figs 2 and 3.

conhct mne is much smaller than the crack Icngth (small-scale contact). which iz

indeed the USC when t-emote tension is applied. Whetxxcr there is substantial remolc

shear, however. the six of the contact Lone becomes of the order of the crack length

and the Comninou solution becomes the only relevant claatic aymptotic solution.

’ ’ -.-. Till’ l~l~l~Ttr/lt .ctl‘l’.v.\ illlm\il~~~ Jrlc~fo,..\. 11111/c/~ .\1111/11 .\c~rrllJ c’r~/lfLlc’r

The magnitude of the t\vo asymptotic solutions discussed in the ptxxious section i\

given bc the corrcspondinp stress intcnsit>, fxlot-s, namely Contninou‘h h’,, and the c complex stress intensity factor K of the trditional solution (Rtc‘t,. 19Xx). Ilndcr the

assumption of frictionless contact. RKY’S ( 196X) J-integral is path independent and it caii l-c used to rclatc the aforemontiuncd stress intunsit\j factors.

The value of ./ (Lvhich is also the cI;tsIic energy rcloase ralc) in rcgion I 01‘ I’ig. 4

is giLen by (e.g. MAI.?SHI.L and SAI.C;AUIK. l965)

li , I c

(, =li,+l I c;\ . i= I.?. >; =

1Tl Ill

c; / G,

(2.2.7) li 1 I .

L

G 7 c; ,

and subscripts 1 and 2 refer to the corresponding tnatcrials as labeled in Fig. 4.

Evaluation of J in region II of Fig. 4 leads to (COMNINOI.. 1977a)

J= C’, i(‘Y

- K,. I6 cash’ (TIC)

(2.2.1)

Interface crack with contact zones

Path independence of the J-integral then implies that

317

K, = i-M. (2.2.4)

The above relationship has been stated in the literature in several forms by COMNINOU (1977a) who calculated a ‘comparison stress intensity factor,’ by ATKINSON (1982) who used matched asymptotic expansions to relate the two stress intensity factors, and by PARK and EARMME (1986) and ZYWICZ (1988) who also made use of the J- integral.

2.3. The GrlBth interface crack with contact zones

We consider a crack of length 2a along the interface between two isotropic linear elastic media toaded at infinity as shown in Fig. 5. For definiteness, we assume that the material combination is such that E > 0, A solution to that problem has been given by COMNINOU and SCIIMUESER (1979) who showed that there exist two contact zones, one at each tip, the size of which strongly depends on the direction of the remote loading. The Comninou stress intensity factors should be such that compressive normal stresses develop along the contact zone, namely

Ki, > 0 and K;, < 0, (23.1)

where K!, and KI, are the stress intensity factors at the left and right crack tips, respectively. It should be noted that the relationships (2.3.1) hold independently of the direction of the remote loading. A through discussion of the solution to this problem has been given by COMNINOU and DUNDURS (1980).

FIG. 5. Crack along the interface of two deformable media. The contour used for the evaluation of the J-integral is also shown.

31s N. ARAVA\ and S M. SHAKW

As discussed in the previous section, when remote tension is applied. the si;7c of the

contact mnes is a very small fraction of the crack length and the conditions of small-

scale contact are met. When remote shear in the direction shown in Fig. 5 is added.

Ihe size of the left contact zone increases. whereas that of the right one decreases.

Therefore, at least at one of the tips of the Griflith interfaze crack. the small-scale

contact conditions are always satisfied.

We consider next the J-integral. which must vanish when evaluated around the

closed contour shown in Fig. 5. It can be casilv shown thal Ihis leads to

(IV\,)’ = (/I;,):. (2.i.7)

which. in view of (2.3. I ). can be written ;15

/cl,- Iv;,. (2.i3)

:I simple relationship that appears to have escaped notice in the past. lising the abo\,c

result it can be easily shown that if the remote .shcwr traction component is reversed.

K\, and K;, do not change, whereas the contact zones Ilip.

The above result can bc now used to derive approximate expressions for the

Conininou stress intensity factors (I/ ho//7 ('I'L/i'/i /i/n as follou~ Assuming that the

direction of the rcmotc loading is :IS shown in Fig. 5. the small-scale contact conditions

arc met at the right crack lip and the complex stress intensity factor of the traditional

solution al that tip is given by (RICK and SIH. 1965 : RICY. 19x8)

K - (c~‘, + irr{, )( I + ?i;:)(Zrl) ‘, I TICI. (33.4)

l‘he right-hand side of (2.3.4) is only an ~ippl-o~imation to the exact ~alut2 of ti. for

it is derived on the assumption of Iraclion-free crack faces. 1.~‘. without taking intcl

account the disturbance caused by Ihe presence of the contact /ones. llsing (2.7.3).

(3.3. I ). (1.3.3) and (2.3.4) one can write

where

/.-= (rr,‘,?+o,,,‘)’ !. (2.3.0)

In the case of remole tension ((r,‘, = 0) (3.3.5) rcduccs to the analytical result 01’

GAI.TFX:Y and DUVXXS ( 19X7) (their equation (4.31)). When combined uzmion and

shear are applied at infinity (with (T,‘, > 0), a comparison with the rt‘sults oI’GA~I I isi k and DUTXDURS (1988) shows Uiat Ihe maximum difTercnce between the predictions of‘

(2.3.5) and the values plotted in Figs 4 and 5 in their paper is i”cl.

In this scclion WC consider the C’omninou asymptotic elastic displacement ticld neat

the tip of 21 crack lying along the interface of ;I homogeneous linear elastic medium

and ;I rigid substrate, and show that this asymptotic solution also predicts nmm-ial

interpcnctl-ation in a region near the crack tip. Referring to Fig. 6. o~rrlapping ofrhc


FIG 6. Schematic representation of the near crack tip region.

Y A

CLOSED CRACK TIP

materials is avoided when

UV(X>Y) +.Y > 0 (Y > 0) (2.4.1)

at all points of the deformable medium. The y-component of the displacement field is given by (see Appendix A)

where

1 ii,(t), K) = ~--~~

(

0 30

Ii+1 cos ~ -cos -

2 2 > .

(2.4.2)

(2.4.3)

Taking into account that K,, must be negative for the asymptotic solution to be valid, we can write (2.4.1) as

Lj,.(o; K) < - g sin t?J27rr (0 < 0 < E), r -+O. (2.4.4) II

Since Y can be arbitrarily small, a necessary and sufficient condition for no material interpenetration as Y + 0 is that

z?,(O; ti) < 0 for 0 < 8 < 71. (2.4.5)

The function C,,(H; K), however, is positive in the region 0 < 0 < 7-r and, therefore, the above solution predicts material interpenetration near the crack tip. In fact, using (2.4.4) we obtain the following estimate for the extent of the material interpenetration zone :

0 38 z

“‘2 -cos I )I (0 < fl < 7-c). (2.4.6)

The maximum value of vi is

max r, = (2.4.7)

320 N. AKAVI\S and S. M. SHAllblh

and occurs for (I = 71. Equation (2.4.6) provides an approximate tneasut-e of the c.xtent

of the interpretation region, for it is based only on the first term of the asymptotic

solution near the crack tip. Consideration of higher order Icrms will modify the

predicted shape of that region but it will not affect our conclusion of material inter-

penetration. A plot of the estimated region of material points that cross the .\--axis is

shown in Fig. 7.

The interpenetration in this case is a consequence of the singular comprcssivc strain5

that develop in the near crack tip region. The material in the intcrpenctration region

is “turned inside out” and the Jacobian of the transformation that carries material

points from (.v,j‘) to (-\-f/t ,..r+r/, ) is ncgativc in that region. The linearixd com-

patibility equations are still satisfied and thcdisplacement is single valued at all points,

but this does not prevent the Jacobian from being negative. It should bc noted that

linear elasticity solutions arc based on the assumption of infinitesimal displacement

gradients which implies that the aforetncntioned Jacobian is close to unity. The abo\e

“anomaly”. thcrefot-c. is a conscquencc of the USC of linear kinematics in ;I rcgion

where large strains develop.

The Comninou asymptotic solution is physically inadmissible for values ol‘r of the

order of F:“” or stnallct-. This asymptotic solution is only valid for \-atues of I’ grcatcr

than I.y. provided of course Iha( I’, “‘,” is much smaller than the silt: of the contact

yone. II is interesting to note. however. that the six of the contact ~oneb depends on

the direction of the remotc ioading and not on its magnitude ((;At’TI:St% atid

DLNIIIIRS. 19x7. 19Xx). whereas the six of the intcrpenctr~ttion region incrcascs \vith

increasing A’,,. Therefore. \vhcn the applied load (i.e. K,,) is large eno~~gh. so that

T”’ bccotncs of the order of the conlact mtlc. the ~‘otnninott asqtnptotic solution

becomes physically inadmissible for all r.

As an example, we consider the Griflith interface crack on ;I rigid substrate in ;I

Icnsion field (Fig. I. ni,!, = 0). In this cast.

,.llh\ ~ (’ (3.4.S)


requires that

r Qv, 4(ti+ 1) -c e (lf4EZjirSexp

[ ( -& , tan-’ (2~)+ i )I , (2.4.9)

where (2.4.7), (2.1 .I) and (2.1.2) have been used. For plane strain conditions, the above relation becomes

CJ:, << 10m2G for v = 0.2, (2.4.10)

o.;! << 10 ‘G for v = 0.3, (2.4.11)

and

a:,. << lO_‘G for v = 0.4. (2.4.12)

For values of the applied tension that violate (2.4.9), the region where finite strains develop near the crack tip is of the order of the predicted contact zone and it is not clear whether the local finite strains and any inelastic deformations will open the crack tip or simply modify the size of the predicted contact zone. In this connection, we mention that KNOWLES (1981) considered the problem of a mode-11 traction-free crack in a homogeneous isotropic compressible elastic material and showed that, within the framework of finite elastostatics, the solution may or may not predict material interpenetration depending on the properties of the elastic material under consideration. KNOWLES and STERNBERC; (1983) also considered the problem of a traction-free interface crack between two dissimilar incompressible Neo-Hookean sheets and showed that, within the nonlinear theory of elasostatic plane stress, the near tip stress field is free of oscillatory singularities and that the crack opens smoothly near its ends.

Returning to the Comninou solution for the Griffith crack,we mention that when substantial remote shear is applied, similar problems may arise at the tip with the small contact zone. At the other crack tip, however, where the contact zone is of the order of the crack length, the Comninou solution is the only relevant linear elastic asymptotic solution.

It should be noted that similar conclusions can be drawn if one considers a crack along the interface of two dissimilar linear elastic media. Assuming that the crack lips are in frictionless contact, we can easily show that the solution predicts interpenetration of the type discussed in this section in the near crack tip region of both materials.

At the risk of becoming redundant, it is thought important to mention once more that the aforementioned problems associated with the asymptotic displacement field are related to the finite strains that develop in the near tip region, and that all asymptotic solutions based on linear kinematics are not valid in the near tip region where strains become 0( 1) or larger. Furthermore, most of the linear elastic asymptotic solutions with a mode-II component predict near tip material overlapping of the type discussed in this section; such solutions include the mode-IT crack in a homogeneous material, the traditional solution of a traction-free interface crack, and the Comninou solution for the interface crack, where it can be easily shown that the aforementioned Jacobian becomes negative as r + 0. This observation, however, does

‘9, N A~.4\.\4 and S M. SH-\IW~

noi invalidate these asymptotic solutions it simply iiicilns that they iii-e not relevant

very close to the tip where finite strains develop: at larger distances from the tip. where strains arc infinitesimal. the asymptotic solutions that are based on the linearized theory arc still valid. In fact. the linear asymptotic solutions provide the boundal-! conditions in a bounda~-): layer formulation of the ‘small SXIC nonlinear crock problem’ which examines the nonlinear ell‘ccts in the near crack tip region (RI<,]:. 1967: KVOWLFS, 1981). It is important to rcalia. however. that the C‘omninou solution C;III

be iis~d in ;I boundary layer formulation only if the near tip linite strain region i\ much smaller than the predicted contact /one size.

The :~bovc discussion suggests that for furlher progress to\hards ;I more complete understanding ofintcrfiicial fracture it is nuxssar~ to consider nonlinear elTcct5 in the

nwr tip region. such ;IS nonlinear constitiiti\c theorics (SHIH ad ASIN). I%-#. 1%‘)) and nonlinear kinematics ( KXOWI.I:S and STIXNIWKG. 1083). togcthcr ~\ith ;I niorc

detailed description ofthe structure and properties ofthe bimaterial interface (BASSAZI

and Qu. 1990). The efl’ccts of plastic deformation in the near tip region arc discussed in detail in

the following s&ion.

-7. AN ELASI-OPI.ASTK SOLIITION

We consider 21 plane strain crack along the interface between a homogeneou\ isotropic elastoplastic material and ;I rigid substrate. The constitutive behavior of the deformable medium is described by the ./,-deformation theory for a Rambcrg Osgood uniaxial stres+strain behavior. nameI>

(3.1)

\vhcre E is the infinitesimal strain tensor. 6 is the C’auchy stress tensor and CJ’ its

deviatoric part. ci,, is the Kronecker delta. E is Young’s modulus, v is Poisson’s ratio. CYC is a material constant, II is the hardening cxponenl (I < II < II_ ). CT,, is the yield

stress. I:,, = CT,,. ‘E. and (T, is the Mises cquivalcnt stress defined 21s

Under plane strain conditions the only non-zero stress. strain and displacement

components are (T,, . (T,,~,. 0.;. (T,,~. i:,, . I:,,,,. i:,,). and II, . II,,. where I is the coordinate vxis normal to the plane of deformation. The problem is formulated in terms of ;I stress

function r/;(l-. 0) and the in-plane stress components are defined by


(3.5)

The plane strain condition 6,: = 0 implies that

(3.6)

which implicitly defines u;.. The equilibrium equations are satisfied automatically. The partial differential equation governing the stress function is obtained by eliminating the strains from the compatibility equation

(3.7)

The resulting partial differential equation is

where

and

%? = 3 (VA’ + aS,’ + o:&, + &), (3.9)

(3.10)

3.1. An asymptotic solution under plane strain conditions

The crack face is assumed to be in frictionless contact with the rigid substrate near the crack tip. Referring to Fig. 6, the boundary conditions for the asymptotic problem are

u,.(r, 0) = 0 ~+j(r,O) = 0 as r -+ 0, (3.1.1)

o(,,(r, 7-c) = 0 z+,(r, x) = 0 as r + 0. (3.1.2)

We attempt an asymptotic expansion of the solution of (3.8) in the form

U(r,H) = r’O,(0)+rfOi,(B)+..., as r-+0, (3.1.3)

where s < t -c ....

J ii (ii / 8

I/, = 1X(,

t 1 %X,,cT,, I ,,,, ,.I ii,, 1, Ii, (ii ; 77). (3. I .h)


2n+l d20, d,, = __

n+l Q+m,

n(2n+ 1) rl 500 = @+I)’ 1

and

n doi, cJrH = - - -~

n+l do’

(3.1.16)

(3.1.17)

(3.1.18)

The leading order boundary conditions resulting from (3.1.1) and (3.1.2) are

2n+l - d’& 6nfl do, --Yu,+~~~~=O, (n+l)* dQ (n+ I)-

p+d$‘=O on O=O (3.1.19)

and

do, d%, _= d8 0, ~3 = 0 on 0 = rt. (3.1.20)

The above homogeneous differential equation and boundary conditions define 0, to within a multiplicative constant the magnitude of which is determined from the condition $“’ = 1 and its sign is chosen such that BYTE < 0. It should be also mentioned that if the solution is not separable in r and 0, so that s = (2n+ l)/(n+ I) is not an eigenvalue of the problem, then the only solution of (3.1. IO), (3.1.19) and (3.1.20) will be the trivial solution 0, = 0.

The differential equation (3.1.10) is solved numerically for different values of the hardening exponent using a Galerkin-finite-element technique. Two-node isoparametric elements with four Gauss integration points are used in the computations and the resulting set of nonlinear algebraic equations is solved using Newton’s method. Four degrees of freedom per node are used and the interpolation of 0, within each element is defined by a seventh-order polynomial in terms of the nodal values of o,, d~,ldf). d’a,/dB’, and d’U,/dB’. A total of 180 one-degree elements are used in the computations. Similar numerical techniques have been also used recently by SYMINGTON et ul. (1989) for the determination of asymptotic crack tip fields.

The angular variation of the in-plane stress components and the Mises equivalent stress associated with the dominant singularity are shown in Fig. 8 for n = 3 and M = IO. The stress distribution of the hardening solution for large n closely approxi- mates that given by the perfect plasticity solution as can be seen from a comparison of the n = 50 solution of Fig. 9 with plots of Fig. 10 which is based on the slip line field solution of PARKS and ZYWICZ (1989) (with a,,,(8 = 0) > 0). The components of the stress deviator instead of total stress are plotted in Figs 9 and 10, since the asymptotic slip line solution of Fig. 10 defines the stress field to within an arbitrary constant pressure which can only be determined from the solution of a complete boundary value problem.

The dependence of Z, on the hardening exponent is shown in Fig. 1 I. The solution of (3.1.10) shows that the condition otiO(t) = 7~) < 0 requires that

a,,,(0 = 0) > 0 for all values of the hardening exponent in the region 1 < n < az. Put

1 .oa

0.500

0.

-0.500

-1.00

-1.50

I“-/______._...- ,f,

“\ n=3

::::..1”.--:-I;__‘I

0. 60.0 120. 180.

0 (degrees)

1 .a0

osuo

0.

-0SCKl

-1 .oo

-1.50 5

-2.00

-2.50 60.0 120. I x0.

0 (degrees)


-0.800 I I I I I I, I I0 I I I, 1 I I

0. 60.0 120. 180.

6 (degrees)

FIG. 9. Angular variation of deviatoric stress components at the crack tip for n = 50

327

in other words, a necessary condition for the interface crack to remain closed near its tip, where plastic yielding occurs and the singular term on the right-hand side of (3.1.5) is the dominant term, is that G,(~(@ = 0) > 0. On the other hand, the asymptotic elastic solution requires the opposite, i.e. K,, < 0 or ~~~~(0 = 0) < 0 (see Appendix A) ; some interesting implications of this condition are discussed in the section that follows.

We conclude this section with a discussion of the higher order terms in the expansion (3.1.3). Formally, these terms are determined by substituting (3.1.3) into the governing equation (3.8) and the boundary conditions (3.1.1) and (3.1.2), collecting terms of like powers of r, and solving the resulting hierarchy of problems for the unknown function 0,. 0;. etc. Elasticity enters the solution to second order or higher. where the linear elastic term (1 -v’)V”U is balanced by higher order plastic nonlinear terms in (3.8). A detailed discussion of the higher order terms in the expansion (3.1.3) is presented in a separate pubtication (SHARMA and ARAVAS, 1990). Here, we simply present a heuristic argument for the determination of the second term in (3.1.3). Because of the stress singularity at the crack tip, the local plastic strains are much larger than the elastic ones and the total strains are equal to the plastic strains to leading order which, in turn, implies that the volumetric strain vanishes to leading order as r + 0. Because of incompressibility as r -+ 0, the asymptotic traction boundary condition (T,(~(T, n) = 0 is not sufficient for the complete determination of the leading order stress field which can be determined only to within a constant pressure p. The additional constant pressure can be treated as a higher order term in the expansion

32x

for stress. This observation together with the linite element solutions presented in the Ihllowinp section suggest that for II 3 3 WC‘ can \vritc the expansions for C ’ and CJ ~15

(.(,.. 0) = ,.I’(,’ ” I”. “fi, ((1) + J,“.J + . . . . ;,\ ,. --f () (3.121)

and

It can be readily shown that. for values of the hardening exponent II 2 2. the two-

term expansions (3.1.2 1) and (3.121) areconsistent to second order with the governing equation (3.8). the plane strain condition (3.6). and the boundary conditions (3. I I )


0.600

L

0.400

0.200

0.

n

FIG. I I. The dependence of I,, on the hardening exponent n

and (3. I .2). For values of n in the region 1 < n < 2 the second term in the expansion (3.1.22) is also singular in r (SHARMA and ARAVAS, 1990).

For small values of the hardening exponent and for distances sufficiently close to the tip, the asymptotic solution for the stress field can be accurately described by the singular term on the right-hand side of (3.1.22) alone. For higher values of n, how-

ever, the constant pressure term plays an important role even for small values of I’, since as IZ increases the strength of the stress singularity decreases, and in the limit- ing case of perfect plasticity (n = a) both terms on the right-hand side of (3.1.22) are of the same order. The effects of the constant pressure term are addressed in detail in Section 3.2.

3.2. Small-scale yieldining.finite element ana1,vsi.s

In order to verify the asymptotic solutions developed in the previous section, we carry out finite element calculations under small-scale yielding conditions. We consider the near tip region of a crack lying along the interface of an elastoplastic medium and a rigid substrate and use a boundary layer approach to study the near tip elastoplastic fields. The Comninou asymptotic elastic displacement field (with K,, < 0) is applied on a semicircular boundary remote from the tip. The boundary conditions ahead of the crack are

U, = u,) = 0 on 0 = 0. (3.2.1)

330 !‘d :\KA\.\S anti !, M. %I\KN\

Interl’ace elements :II-c introduced along the crack face. thus allowing for possible contact with the rigid substrate. The coetlicicnt of friction between the elastoplastic medium and the substrate is assumed to bc /era.

We mention again that the Comninou solution and lhc elvstopla~tic solution\ developed in this section arc the relevant asymptotic solutions \vhen the G/c of the contact zone is of the order of the characteristic length of the gcomctr! unciu

consideration (say. the shorter of crack Icngth. untracked ligament hi/e. etc.).

IJsing dimensional analysis one can easily show that the solution of the problem

must bc of the tbrm

Lvhcre F is ;I dimcnsionlcss tensor ~alucd function. Nine-node Lagrangian isoparametric elements with 3 x 3 Gauss integration arc

cased in the computations. To avoid numerical dilficulties. which arise Vrom the nearI\.

incompressible deformation of the material. the so-called B-bar metl~od is used (HUGHES. 19x0. 1987). i.e. the dcviatoric part of the strain tensor is calculated at the 3 x 3 G~LISS points whereas the volumetric part is evaluated at the 2 x 2 Gauss point” and then interpolated~extr~lp~~l~lted to the 3 x 3 Gauss points. These clcments ha\,c been successfully cased in the past to obtain accurate solutions for ductile fracture problems (SHIH and NEEI~ILMAN. 19X4; SHIH and ASAN). 19X8. 1989). The mcA

design is similar to that described in Section 2. I and the radial length of the crack tip elements is IO “‘times the outermost radius where the asymptotic elastic displaccmcnt field is imposed.

The ABAQUS general purpose tinite element program is used for the computations (HI~~ITT. 1984). It should bc mentioned that the O-node Lagrangian element is not included in the ABAQUS “element library”. This code, however, provides ;I general interface so that the user may introduce his own clement by using a “user subroutine”. The constitutive calculations are part of the element definition and the deformation plasticity model discussed in the beginning of Section 3 was also implemented in ABAQUS using the same “user subroutine”. The values of the material properties used in the computations arc E/n,, = 300, 1’ = 0.9. and x = 0.1.

The finite clement solutions show that. for all II > I. the crack face remains in

contact with the substrate except for ;I small region right at the tip where the crack opens. The siLe of the open region is always a small fraction of the p:nstic /one si;/e. Spccifc values for the sixe of the open region are reported Inter in this section. The reason for which the crack opens at its tip is explained in the following. The stress intensity factor K,, of the Comninou asymptotic solution tnust be negutivc so that compressive stresses develop on 0 = TZ whcrc the crack face is in contact with the rigid substrate. This. in turn. implies that (T,,,(U = 0) < 0 as I --t /- in our boundary layct formulation of the small-scale yielding problem. The finite element solutions show that. for all II > I. the shear stress component o,,,(t) = 0) remains negative for all I’. Therefore, according to the discussion of the last paragraph in Section 3. I. the crack must open near its tip. This conclusion is reached if one considers the singular term only on the right-hand side of (3.121). For larger I’ values. however. the constant


pressure term on the right-hand side of (3.1.22) becomes important, and a necessary condition for the crack tip to remain closed is

(3.2.3)

which does not necessarily require that o,,,(B = 0) > 0. In fact, the finite element solution shows that for larger values of r the constant compressive hydrostatic stress p makes o,,,(B = 7~) negative within the plastic zone and keeps the crack tip closed, even though a,.,,(0 = 0) < 0. A detailed description of the finite element results is given in the following.

Plastic zones determined from finite element solutions are shown in Fig. 12 for n = 3 and n = 10. The plastic zone is defined as the region in which crC > (T(,.

Figure 13 shows the computed radial variation of the Mises equivalent stress C-J, along the radial line 0 = 86.3’~’ for II = 3. In that figure, the size of the plastic zone on 0 = 0 is indicated by yp and the size of the open portion of the crack by d. The asymptotic elastic and plastic solutions are also plotted in Fig. 13. The finite element solution agrees well with the asymptotic elastic solution in the elastic region and follows closely the asymptotic plastic solution within the plastic zone for values of r in the region d < I’ < r,/lOO, which appears to be the region of dominance of the plastic asymptotic solution. The angular variation of the Mises equivalent stress for values of r in the region d < r < u,/lOO is shown in Fig. 14, where the normalized calculated oer namely o,/ (a,[J/(cq,crOZn~)] I’(“+ ‘), is plotted together with a”, of the asymptotic solution. The finite element solution agrees well with the analytical asymptotic solution presented in Section 3.1. Figure 15 focuses on the plastic zone and shows that the calculated uC deviates from the asymptotic solution for values of r less than d where the crack opens. A closer examination of the solution shows that, for r < cl,

the angular variation of all stress components depends on r, which implies that the solution is not separable in r and 8 in the open portion of the crack.

-0.2 0 0.2 0.4 0.6

FIG. 12. Plastic zones for n = 3 and n = IO

3.00

1.00

log 2 1 .oo fin

0.

0.900

0.800

ge

0.700

0.600

n=3

Interfxe crack with contact zones 333

6.00

3.00

0.

-6.00 -4.00 -2.00 0.

FIG. 15. Radial variation of normalized Mises equivalent stress C, along II = 86.3 for 1~ = 3. The open circles indicate the results of the finite element solution.

The radial variation of the hydrostatic stress component of the calculated stress field is plotted in Fig. 16. The asymptotic elastic and plastic solutions are also included in Fig. 16. As discussed earlier in this section, the constant hydrostatic stress p on the right-hand side of (3.1.22) plays an important role and should not be neglected. Curve I in Fig. 16 represents the singular term of the asymptotic solution, whereas curve II is the sum of the singular and the constant terms on the right-hand side of (3.1.22). The magnitude of the constant p is determined by matching the numerical and the analytical solution at one point inside the plastic zone ; the calculated p value is indeed independent of the particular point chosen for the matching as long as the radial distance of that point from the crack tip is larger than d and smaller than r,/lOO, which is in the region of dominance of the plastic asymptotic solution. Figure 17 focuses on the plastic zone and shows that the calculated hydrostatic stress agrees well with curve II of the asymptotic solution for values of y in the region d < r < r,/lOO and deviates from it for values of r less than d where the crack opens. The angular variation of the hydrostatic stress component for values of r in the region d < r < r,/lOO is shown in Fig. 18, where the normalized calculated ohk/3, namely (grx/3 -~)/-(o,[J/(a~,~o,,Z,,r)] WI+ “1, is plotted t ogether with gkh/3 of the asymptotic solution. Again, the finite element solution agrees well with the analytical asymptotic solution presented in Section 3.1.

Similar plots for n = 10 are shown in Figs 19-24. Figures 19 and 20 show the

12.0 n=3

1.00

0.600

0.400

0.200


n=3

335

0.

0. 60.0 120. 180.

e (degrees)

FIG. 18. Angular variation of normalized hydrostatic stress ci,,/3 at the crack tip for n = 3. The open circles indicate the results of the tinitc element solution.

1.50

1.00

0.500

log 2 CO

0.

-0.500

-1.00

-14.0 -10.0 -6.00 -2.00 2.00

log (KI&o)2

FIG. 19. Radial variation of normalized Mises equivalent stress C, along H = 86.3 for n = 10. The open circles indicate the results of the finite element solution.

326 N. AKAVAS and S. M. SHAKMA

log = 300

2.00

1.00

0.

-1.00

-2.00

-14.0 -10.0 -6.00 -2.00 2.00

FIG. 20. Radial variation ofnormalked hydrostatic stress oii, 3 along 0 = X6.3 I’orfI : IO. The open urcle\ Indicate the results of the finite element solution. Curve I represents the singular term in the asymptotic solution, whereas curve 11 is the sum of the singular and constant terms on the right-hand side of (3. I .27).

radial variation of the normalized crC and CT ,,i !3, whereas Figs 21 and 22 show the corresponding angular variations in the region r/c Y < r,/lOO. Figures 23 and 24 focus on the plastic zone and show that the finite element solution agrees well with the plastic asymptotic solution in the region cl < I’ < I’,,, ~100. For values of r less than d the solution appears to be non-separable in Y and 0.

A schematic representation of the several regions of dominance is shown in Fig. 2s.

We conclude this section with a discussion of the limit case of an elastic -perfectly- plastic material (n = x). A slip line solution to this problem has been presented by PARKS and ZYWICZ ( 1989) (see also Z~wrcz, 1988). As mentioned in Section 3. I. the asymptotic slip line solution determines the stress field to within an arbitrary constant pressure which can only be determined from the solution of a complete boundary value problem. This can be also seen in (3. I .22), where the two terms on the right- hand side become of the same order in r as II + CC and the additional pressure cannot be determined from the asymptotic solution alone. The finite element small-scale yielding solution shows that o,(H = 0) < 0 and rr,,,,(H = n) < 0 for all I’ and that the crack tip remains closed everywhere.


0.800 -

ge

0.600 -

1.00

0

h n = 10

0.500 1 ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ 0. 60.0 120. 180.

8 (degrees)

FIG. 21, Angular Gariation of normalized Mises equivalent stress d, at the crack tip for n = IO. The open circles indicate the results of the finite element solution.

3.3. Large-scale yielding solutions

In this section we consider a center-cracked-panel with the crack lying along the interface of an elastoplastic medium and a rigid substrate. The constitutive behavior of the deformable medium is described in terms of the Jz.-deformation theory for a Ramberg-Osgood uniaxial stress-strain relation presented in Section 3. The material properties used in the calculations are E/o0 = 300, v = 0.2, cx = 0.1 and n = 3. Any contact between the crack surface and the rigid substrate is assumed to be frictionless.

The solutions are obtained using the finite element method. The dimensions of the specimen and the mesh design are the same as those described in Section 2.1. All crack tip nodes are tied and the radial length of the crack tip wedge-shaped elements is 10 ‘“a, 2a being the crack length. The nine-node Lagrangian elements described in the previous section are used in the computations. Interface elements are introduced along the crack face, thus allowing for possible contact with the rigid substrate.

Referring to Fig. 1, we consider first the case in which the applied remote load is tensile, namely a;* > 0, a_FV = 0, and o.:~ = ~a.;,./(1 -v). For values of the applied tension up to a:,. = 3.33 X. 10 4~0, the size of the contact zones is equal to that predicted by the elastic solution, i.e. c = 4.39 x lo- ‘a; at that load level the extent of the plastic zone along 8 = n is ri = 1.39 x lo- 7u and the size of the open portion of the crack very close to the tip is d = IO- “0 (see Fig. 25). For values of the applied

&k/3

1.80

1.60

1.40

1.20

1.00

0. 60.0 120. i 80.

ok - ‘To

3.00

2.00

1 .oo

0.

-5.0 -3.00 -1.m I s-i0

Ths


ukk -

3ao

4.00

3.cKJ

2.00

1.00

0.

n = 10

-5.00 -3.00 -1.00 1.00

FIG. 24. Radial variation of normalized hydrostatic stress rkn/3 along 0 = 86.3 for n = IO. The open circles indicate the results of the finite element solution. Curve I represents the singular term in the asymptotic solution. whereas curve II is the sum of the singular and constant terms on the right-hand side of (3.1.22).

Comninou osympt.

/,/////~/,/~//“/~‘~‘~‘~~““““” ““““’

kdhr,

FIG. 25. Schematic representation of the near crack tip region (not to scale)

3-K) N. AR,LAS a~lci S M. SH:\Rhlh

tension larger than 3.33 x IO ‘n,, the si/c of the contact lone c decreases whet-cas the

siLc of the near-tip open portion tl increases. Finally. at o,‘, = 7. I1 x IO ‘(Tag the crack

opens completely and the calculaGons arc terminated : at that load level. the si/e of

the plastic mnc along 0 = 7r is I$ = 4.39 x IO ‘(1.

Sunitnari;ling. wc‘ nicntion that. when the far lield loading is predominantly tcnsilc. the size of the contact /one is always a small fraction of the crack length and that it dect-cases 21s Ihe size of the crack tip plastic tone increases. The above analysis also

suggests that in most practical cast’s Ihc interface crack in ;I tension held can he treated

as an open crack. We consider ncnl the cast of remote shear. in which UT;, .> O.and ci,‘, = (50 = 0.

The applied shear traction is gradually increased from /c’ro to ;I ~alur: of CT,, =

(T,, , 3 whcrc the cnlculalinns arc Icrniinatcd. It is found that the si/e of Ihc conLicl /one al the /cl/i crack tip remains constant al all load levels considcrcd and equal to Ihal

predicted by the clnstic solution. namely c = 0.6847r1. It is intercsCng to note that the

left crack tip opens locally; the size of the open pot-bon increascs \vith increasing load

and it t-caches it value of tl’ = 3.98 x IO ‘(I \vhen 0’ ,i = (T,,, b 1. At the Icft crack tip. the region of dominance of the asymptotic solution developed in Section 3. I appear5

to be tl’ < I’ < IO ‘(1. where I‘ is the radial distance from the left crack tip. The si/e of Ihc r(g/lt contact zone increases as the applied shear increases and it reaches ;I wlw

of C’ = IO :I/ when o:, = o,,., 1. Based on the sniall-scale \,ielding solutions prc-

senIcd in Section 3.2 one might argue that the right crack tip is also open locally : the siLe of the right open part. however. will be ;I small fraction of the right contact /one (tl’ << IO ‘(I) and cannot be captured by the mesh used in the linitc element calculations. It appears that for tnosl practical cases the right crack tip can bc treated

as an open tip; the siLe of Ihe left contact rone, however, is of the order of the crack length and the efl-ects of the left contact I-one on the near-tip stress and deformation

fields cannot bc ignored.

4. C’t.osr~lrt-

A detailed analysis of the interface crack problem with frictionless contact LOIICS has been presented in this paper. The substrate was assumed to bc rigid and the results of the analysis can bc used for the study of crack-like defects along the interface of a

ductile metal and ;I brittle substrate. such as it ceramic. A deformation plasticity theory is used in the analysis. The developed solutions can

bc also used as flow theory solutions provided that no unloading occurs and that stressing is proportional within the plastic zone in the corresponding flow theory solutions. A series of finite element calculations using the flow Ihcory of plasticity is now underway so that comparisons can be made.

The coefficient of friction between the crack surface and the substrate was assumed to vanish. Frictional effects are certainly important and will be addressed in a future publication. Here, we tnention that, when frictional forces develop along the crack surface. the J-integral becomes path dependent and the strength of the crack tip singularity decreases as the coefficient of friction increases (COMNINOU, l977b). The near tip frictional forces provide an additional energy dissipation mechanism and


appear to be responsible for the observed increase of the macroscopic critical ‘elastic’ energy release rate for crack growth (EVANS and HUTCHINSON, 1989 ; CAO and EVANS,

1989 ; LIECHTI and CHAI, 1989a, b).

ACKNOWLEDGEMENTS

This work was carried out while both authors were supported by the NSF MRL program at the University of Pennsylvania under Grant No. DMR-8519059. The authors are grateful to Hibbitt, Karlsson and Sorensen, Inc. for provision of the ABAQUS general purpose finite element program.

REFERENCES

ATKINSON, C. BOWIE, 0. L. and FREESE, C. E. BASSANI, J. L. and Qu, J.

CAO, H. C. and EVANS, A. G. COMNINOU, M. COMNINOU, M. COMNINOU, M. COMNINOU, M. and DUNDURS, J. COMNINOU, M. and SCHMUESER, D. ENGLAND, A. H. ERDOGAN, F. ERDOGAN, F. EVANS, A. G. and

HUTCHINSON, J. W. GA-EN, A. K. and DUNDURS, J. GAUTESEN, A. K. and DUNDURS, J. HIBBITT, H. D. HUGHES, T. J. R. HUGHES, T. J. R.

HUTCHINSON, J. W. 1968 KNOWLES, J. K. 1981 KNOWLES, J. K. and 1983

STERNBERG, E. LIECHTI, K. M. and CHAI, Y.-S 1989a Biaxial loading experiments for determining inter-

facial toughness. Engineering Mechanics Research Laboratory Report No. EMRL 89- 3, University of Texas at Austin.

1989b Asymmetric shielding in interfacial fracture within plane shear. Engineering Mechanics Research Laboratory Report No. EMRL 89-4, Uni- versity of Texas at Austin.

LIECHTI, K. M. and CHAI, Y.-S.

1982 ht. J. Fracture 19, 131. 1976 Engng Fracture Mech. 8, 373. 1990 Interfacial discontinuities and average bimaterial

properties, to appear in Scripta Metall. Pro- ceedings of the Acta/Scripta Metallurgica Conference on Bonding, Structure and Mech- anical Properties of Metal/Ceramic Interfaces, University of California, Santa Barbara, January 1989.

1989 Mech. Mater. 7,295. 1977a J. appl. Mech. 44,63 1. 1977b J. appl. Mech. 44,780. 1978 J. appl. Mech. 45,287. 1980 Res. Mech. 1,249. 1979 J. appl. Mech. 46,345. 1965 J. appl. Mech. 32,400. 1963 J. appl. Mech. 30,232. 1965 J. appl. Mech. 32,403. 1989 Acta Metall. 37, 909.

1987 J. appl. Mech. 54,93. 1988 J. appl. Mech. 55, 580. 1984 Nucl. Engng Des. 77,271. 1980 Int. J. Numer. Meth. Engng 15, 1413. 1987 The Finite Element Method. Prentice-Hall, Engle-

wood Cliffs, NJ. J. Mech. Phys. Solids 16, 13. Engng Fracture Mech. 15,469. J. Elasticity 13, 257.

MALYSHEV. H. M. and 1965 SALGANIK, R. L.

PAKK. J. H. and EAKMMI:, Y. Y. 19X6 PAKI<S. D. M. xnd ZY~IC,/. E. 1989

RI(.I~. J. R. 1967

RI(,I. .I. R. 196X Ricx J. R. IOXX Rrc.1.. J. R. and ROS~:NGKI:~, G. F. 196X Rrc~. J. R. and SIH. G. C’. I OhS SAL(;ANIK, R. L. I%3 SHAKMA. S. M. and AKAVAS. N. I990

SHIELD. R. T. 1981 SHIH. C. F. and ASAKO, R. J. I988 SHIH. C. F. and ASAKO, R. J. 19x9 SHIH, C‘. I:. and NI:EIU:MA~. A. I984 SYMINGTOS. M.. OKTK M. and I989

Slllll. c. F.

TIII.O(‘ARIS, P. s. I OX6 Till O(‘AI<IS. P. S. 1987 WII I.IAMS. M. 1.. IUS? WII_I.IA~. M. L. 19sc) WIL.I.IS. J. R. 1977 I\~Wlc?. E. IOXX

We wnsidcr the t~~l~~-ciirncnsion~~i problem 01‘ ;I crack lyin, cr along the intcrlBcc 01‘ ;I homogeneous Iincar elastic medium and ii viqitl substrate. The crack LICC is in frictionless cont;lct with the substrate neal- the tips. The mlturc of’ the elastic fields near the closed tips is rcadil) established by using what has bccomc known in clasticit) ;IS the WII IJAMS (1052) tccl~n~quc. Rcfcrrinp to Fig. 6. the boundary conditions to bc satisticd xrc

I,, (I’. 0) = 0. /lo (I’. 0) = 0 (Al)

2nd

I/,,(/.. T[) = 0. fT5,,,(l’. n) = 0. C/42)

The dominant terms in the asymptotic solution 1.w ihc dcli~rmahlc mcdiw~~ (0 6 11 s n) ;II-c li)und to lx (COMNINOIJ. 1977)


and

(A4)

(A3

(A61

(A7)

where

u is the stress tensor, u is the displacement vector, G is the shear modulus. v is Poisson’s ratio. K = 3-4~ for plane strain and ti = (3 -v)/(l +v) for plane stress. For the above solution to be valid the stress intensity factor K,, should be negative so that compressive stresses develop along the contact zone on 0 = 7~.

APPENDIX B

The asymptotic solutions for an open crack along the interface between a homogeneous isotropic linear elastic material and a ri~qidsubstrate is easily derived from the complex potentials given by RICE (1988). The boundary conditions to be satisfied are (Fig. 6)

II, (r. 0) = 0, U,,(T. 0) = 0 (Bl)

and

(T,,,I(Y, rr) = 0. 0,,,(Y. x) = 0. (B2)

The dominant terms in the asymptotic solution for the deformable medium (0 < 0 < 7~) are found to be

-J I +41:’ sin I) e’“’ ‘) ,’ sm (4-i,+ ‘:j]. (B3)

+Jl +4c’sin 0 eilH ‘) sin &[i+ 2 ( ;“j], (84)

+Jl +4~‘sin 0 e’“‘-“’ cos (91i+ ;j] (B5)

WhCK

I i: = 3n In I<. p = tan ( (21:), (HX)

K is the complex stress intensity [actor as defined by RICE (1988), and (I, is the pha~c an& of

w For the case of a (;rifith crack (Rl<‘l< and %I, 1965 ; Rice. 19%)

K= (CT,‘, +iox’,)(I+2ir:)(h) “\ 7ctl (H9)

and

an elastoplastic analysis of the interface … · an elastoplastic analysis of the interface ... an...

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