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Boundary element analysis of interface cracks between dissimilar anisotropic materials
BOUNDARY ELEMENT ANALYSIS CRACKS BETWEEN DISSIMILAR MATERIALS
OF INTERFACE ANISOTROPIC
C. L. TAN. Y. L. GAO and F. F. AFAGH Department
of Mechanical
and Aerospace Engineering. Canada KIS 586
Carleton
University.
Ottawa.
Abstract-In this paper. the boundary element method (BEM) is applied to the analysis of interface cracks between dissimilar anisotropic materials in plane elasticity. It is based on the quadratic element formulation and special crack-tip elements which incorporate the proper O(r-’ ‘“:‘) oscillatory traction singularity are employed. A simple expression relating the stress intensity factors to the BEM computed trxtion coetlicients is derived. and this procedure for determining stress intensity factors is validated by several examples. The numerical results obtained are shown to be wry satisfactory even with relatively coarse mesh discretizutions.
I. INTRODUCTION
The
study of cracks that lie along the interface between two ditrercnt
elastic media is
important in providing a better understanding of the integrity of bonded intcrfaccs between dissimilar materials. and for determining thcappropriatc
factors which affect the mechanical
propcrtics of composites and multi-phase solids. In such problems. the near-tip ticlds may bc charactcrizcd by the complex stress intensity factor K = k; + iK,,, and the strcsscs in the vicinity of the intcrfacc crack tip arc oscillatorily
singular. of the order r
’ “‘;‘, whcrc ;’ is
;I bimatcrial property. Also. the tensile and shear dli~t~ arc alwitys COU~IC~.cvcn for single mode loading. As a result. the stress intensity
factors (S.I.F.),
K, and K,,, should not bc
intcrprctccl in the cl:lssic:il scparatc sense as for cracks in homogeneous mutcrials. The bimateri;ll and cxpcrimcntal Kacznski
interface crack problem has bcrn the subject of extensive thcorctical investigations
and Matysiak
in rcccnt years [c.g. Rice (1988).
(1989). Comninou
Jih (1987). Mates VIUI. (1989). Hutchinson Toya
(1990).
previously
Park
and Earmme
(1986)
also received some attention
(1990). Charulambides
Hasebc (‘I tri. (I 987). rt ul. (1989). Sun and
and Evans (1989), Yehia and Sheppard (1988). and Cao and Evans by Williams
(1959).
(1989)]. Erdogan
(1965). Rice and Sih (1965). Perlman and Sih (1967) and Comninou
although (1965).
it has
England
(I 977). among others.
However. these studies based on the linear elastic fracture mechanics approach have concentrated mainly on isotropic bimaterials;
similar investigations
into anisotropic bimaterials
arc relatively limited in number. Reported works in this latter case include those by Willis (197 I). Clements (197 I). Ting (1986). Bassani and Qu (1989a, b), Wu (1990, 1991), Ni and (199 I) itnd Nakagawa Ed td. (1990), all of whom have treated the problem
Ncmat-Nasser analytically.
Numerical
methods have also been employed for the analysis of interface
cracks bctwccn non-isotropic
elastic media. but the finite element method is the technique
almost cxclusivcly used. Among the contributions Yuan (1983).
in this regard are those by Wang and
KUO and Wang (1985). Raju et ul. (1988). Sun and Manoharan
Lin and Hartmann
(1989). and
(1989).
The boundary clcmcnt method (BEM).
alsocommonly
known as the boundary integral
equation (BIE) method, has recently been applied to the study of interface cracks as well. but only in isotropic bimaterials [Yuuki er al.. 1987; Yuuki and Cho. 1989; Lee and Choi, I988 ; Tan and Gao. 1990a. b, I99 I ; Ciao and Tan, I992). In the BE&l approach employed by Yuuki cr (11.(1987) and Yuuki and Cho (1988). an infinite plate made of two dissimilar isotropic solution in the BIE by extrapolation
formulation.
Hetenyi’s solution for a point load in media was used as the fundamental
The stress intensity factors, K, and K,,, were then obtained
techniques based on the computed crack face displacement data. Lee and 3201
L T\X e’l Lli
c
3’0’ _ _
Choi (IYYS), on the other hand. used the conventional quadratic isoparametric correlating
elements. They
the computed nodal displacements
crack-tip to the classical field solution. mesh discretizations.
by conventional
formulation
obtained the stress
of the BEXI
intensity
on the elements adjacent to the interface
In both these approaches. however. very refined
BEM
standards. need to be employed. Tan and Gao
(1990a. b. 1991) and Gao and Tan (1997) also used the conventional BEM but with quarter-point
with
factors by directly
in their studies.
O(r-
’ ‘) traction-singular crack-tip elements. Instead of computing for K, and K,, directly. the modulus. K,,. of the complex stress intensity factor. tvhich may r I also be written as h = A,, e Iv. was obtained. It is worth mentioning here that for the interface crack in isotropic bimaterials.
this quantity K,! is directly related to the strain energy release
rate. In these references. the authors
calculated K,, from expressions
they had derived
relating it to the computed nodal tractions or to the nodal displacements of the quarterpoint crack-tip elements. The phase. $. of the complex stress intensity not determined directly via its relationship elements. These
using a formally
factor. however, was
established procedure. Instead. it was estimated
with the crack face displacements at certain positions positions
were established
from
numerical
several test problems with exact analytical solutions.
experiments
Nevertheless.
on the crack-tip carried out on
good solution
accuracy
for K, and $ was obtained even with relatively coarse mesh designs. lr
‘1is pupcr, the conventional
BEM
with quarter-point
crack-tip elements is applied
.Ilysis of interface cracks bctwecn two dissimilar anisotropic media in two dimcnto thsions. In contrast to the previous treatment by the authors on intcrfacc cracks in isotropic bimntcrinls mcntioncd abovc, the proper O(r ’ ‘*‘;) traction singularity is incorporated in the crack-tip clcmcnts employed. Also. instead of obtaining K,, and I/J. the focus of the study is the direct dctcrmination cxprcssion
of K, and K,,. This is bccausc. as will bc cvidcnt later, any
for A’,, in terms of the computed primary
and the tractions,
variables. namely. the displaccmcnts
more complicated in the anisotropic cast, and it is
bccomcs significantly
also doubtful if the less than formal proccrlurc to cstimatc the phase angle II/ of the complca stress intensity
factor remains applicable. Morcovcr,
the relationship
bctwccn the strain
cncrgy rclcasc rate and K,, is no longer as direct and simple as hcforc. In this study, analytical cxprcssions tractions
which cnablc A’, and K,, to bc calculated directly
on the crack-tip
from the computcd nodal
clcmcnts arc clcrivctl. The veracity of the prcscnt approach is
dcmonstratcd by scvcral cxamplcs involving an interface crack in isotropic ;III~ anisotropic birnaterials.
Before thcsc results
arc prcscnted and discussed, the analytical
basis of the
method cmploycd will first bc shown.
2. UASIC
EQUATIONS
AND
THE
S.I.F.
The development of the equations presented here closely follows that by Bassani and Qu (1959) and Wu (1990). The indicial notation is used, in which the Latin indicts take on the values I. 2 and 3 while the Greek indices only summation
take on the values I and 2. Also,
is implied for rcpcatcd indicts.
For a generally anisotropic
elastic body in Cartesian co-ordinate space s,, the strcss--
strain relation may be written as
dkl
and the Navicr’s equation of equilibrium
Ck,m,lk
=
(1)
Ck,,,,nEn,n
for plane deformation
,.,,,
=
0.
of the body as
(2)
where II,,,. (TV,and I:,,,” arc the displacements. stresses and strains. rcspcctively, and Cklmnare the elastic constants of the material. By introducing the following Fourier transform pair:
BEIM of Interface cracks
3203
(34 f(5)
=
IL
I
\/ln
where i = ,/-
I. and applying
I(<) ee’zr
d<,
(3b)
-XT
them to eqn (2) over the X, co-ordinate.
the following
equation may be obtained :
This equation may also be written in matrix notation as
(5)
where the superscript “T” denotes the transpose of the matrix and
Q = (0,~) = (C,,H) =
s
=
t-y,)
=
(C’,,kZ)
=
c III1
CIII?
c
cI:ll
cIz12
clzlr
C’I III
Cl.I,2
c,,,,
C' Ill1
Cll?2
Cl121
C‘lZlJ
c’,??Z
cl:z,
(‘1122
Cl,?,
L C’
I112
c’ v
=
(I.‘,,)
=
(C,?k?)
=
c:?Ij L c
Equation
(4) or (5) admits particular
1212
?II?
III?
C’lzzr
Cl??,
C’J???
cz22,
c ‘?J??
c’?I?J
.
(6~1)
.
(6b)
I
I
1 .
(6~)
solutions of the form
(7) or. in matrix notation.
(8) provided that a and cz satisfy the cigcnvaluc equation
[<;Q+
= 0.
(9)
= /I<,, cqn (9) bccomcs
+p(S+S’) +p'V]i~(<~)
[Q
=0
(10)
and the ncccssary and sutlicicnt condition for a not having trivial solutions is thus the vanishing of the dctcrminant of the coefficient matrix in eqn (IO). That is
IQ+p(S+S’)+p’Vl which is a sixth-order
polynomial
= 0,
(11)
in p. Since Q, S and V are real and are functions of the
C. L. Ta\
3X4
matenal
constants
(2) is elliptic. be written
rr ul.
CL,““. the roots p, (,V = 1.6). are independent
eqn ( I I) has three pairs of complex
conjugate
of .Z,. And
because eyn
roots. Thus the six roots ma!
as Im(p,,)
Consider
> 0,
now a crack lying along
are made of linear In the vicinitypf
elastic anisotropic
the crack-tip.
pII_?
=p<.
M = 1.3.
the interface
between
(12)
two semi-infinite
I and II, respectively,
materials
solids which
as shown
the stresses ahead of the crack along the interface
in Fig.
I.
are written
LIS
t = t, = (0,: From dimensional cracks
may also
physical
problem.
considerations be written
with
[see Rice (1985)]. reference
These stress intensity
relationship
with
(13)
the stress intensity
to a characteristic
factors
will be written
K(L) = (A-,, and thtir
a>,)‘.
rJz2
factors for interface
dimension,
say L. of the
as
A-,,,)‘-
if-,
(14)
the strcsscs t is
K(l,) =
j’nr-
[O1
(15)
f ‘;
(If))
0;
r
R
I,
et5
whcrc [see Ting (l9M)j
;’ =
I 2X ln
(
)
(17) 1~ CLlII (I 7). W and I) arc the negative is.
that
real and imaginary
parts of ;I complex
matrix
$1,
BE\1
of interface
hI
=
and \I in turn is related to a Hcrmitian of the bimaterial.
crxks
3205
-(IV+iD)
(18)
matrix A which is defined by the elastic constants
More specifically. hl = -i(3)’
‘[.-4’] -‘.
A = (B’+B”)
B’ = i(3)
(19)
-‘.
\=I \I= I
’’ i
L’(
Adj L’(p;)
/J’, ) =
Q'
i
C’(p;,)
C’(/G) = (S’)’
(21)
(2’) (‘3)
+p\.V’.
I. II.
2 =
I or II. rcspectivcly.
where the superscript r = I or Ii refers to material It can be shown that (Ting,
-I. 1
+/‘:.L”.
[S’ + (s’).‘]
+p;
Adj L’(p;,)
(20)
IW6)
so /i in al11 ( 17) is always real. Also, in crln ( 15). lho matrix H is dclincd hy
H[cj = I + II11 [cII’+ ( I - Kc [C],P. whcrc In1 ;IIitl I&z tlcnotc the iiii;lginary
antI real parts. I --
I’= ibl
(25)
rcspcclivcly. I is lhc unit
Iii:Ilrix.
I
P)
*
’ = W+il’>.
It should also be noted that H has the following
(‘7)
properties:
K[l] = I. I+]
and
* I+]
(Xl)
= K[w].
(Xb)
in which the arguments c and 1’ ;lrc tlimcnsionlcss. so that R is also dimcnsionlcss. For
nxIteri;Ils
whcrc the s,-;lxis
C,,?, = C,, ,, = 0 ;lllll c-2 ,,J = c,, compo~icnl
11, ivilh llic coniponcnls
is ;I two-fold
,: = 0. Thcrc
symmetry
axis. the elastic constants
is thus no coupling
0, and 11~.In this c;isc. the matrix
of the displ;rccmcnt I” in cqn (25) can
bc shown to bc
I
0
0
0
1
()
1’: = - [ 0
0
0
1 .
(29)
Also. c2, and A’,,, disrlppear for plant problems. so only K, and K,, need be considered. From eqns (29). (3) ilnd (I 5). the stress intensity factors in terms of the stresses become
3206
C. L. T\\
PI d
where
R*[($“]=
“]P
Re[(i)“]I+Im[(i)
=cos[;:In(i)][(!)
:I-sin[;.ln(;)][iII
,‘lj].
(31)
It is evident from eqns (30) and (31) that when r 4 0. the stresses exhibit an oscillatory singularity.
as was mentioned earlier. In the case when ;’ = 0. that is. when the elastic hod)
is homogeneous. R’ becomes ;L unity matrix and the stress intensity
factors are then of the
classical separate form for the two distinct modes of crack deformation.
3. t3Ehl DETERMINATION
The analytical basis of the BEM partial differential equation written fundamental
OF TIIE
in elastostatics
S.I f-.
is the transformation
equations valid over the cntiro elastic solution for just
solutions
its boundary
with
the Rctti
I-. In the direct formulation. Raylcigh
rcciprocill
appropriate limiting operations. result in the bOliIldil~y displaccmcnts II, and tractions
of the governing
domain Q into an integral
work
use of the unit load thcorcm will.
through
integral equation (BIE) relating the
I, at r. 7‘11~ 131E for two dimensions
nuy
hc written
in
indicial notation ;IS :
C,,,(/‘)~f,(i’)f
whcrc
sI
Ir,(V)?‘,,,(I’.C))ctr(l))
il
/,(C))U,,,(I’.Q)lll‘(C)).
U,,,( i’. Q) :~ntl ?‘,,,(I’. Q) ;Irc the ftlntl;lnlcnti~l
placcnicnts and tractions. inlinitc
=
body. Detitils
rcspcctivcly.
solutions
in the s,Jircction
of their derivation
2.p = 1.3.
(3’)
which rcprcscnt the dis-
at I’ in ;I plant ho1ni~gc1lcous
for the anisotropic USC have been given by Cruse
(198X) and their explicit forms have also been prcscntcd by Tan and Gao (1991). Ah.
in
eqn (32). the value of c’,,,( I’) dcpcnds on the local gcomctry of I- iit the point I’. To solve the BIE
numerically.
the boundary r of the solution
domain is dividctl into
11series. or “mesh” of lint clcmcnts. Over ci1ch of those clcmcnts, the boundary gcomctry. displacements
and tractions
may be written.
respective nodal KI~UCS and the quiidriitic three nodes, two at the ends anti N’(i)
as in the present work,
shape functions
OIW at the mid-point :
nnd
in terms of their
N’ (i). The ckmcnts each ~:IVC the associated shape functions
arc
N’(i)
= i<(i- I).
N?(i)
= I -<‘.
iv’(<) = Li(l +i).
(33)
Substitution of the isoparametric rcprcscntations of the gcomotry and functions into cqn (31) will result in ;I set of linear algebraic equations for the unknoivn tractions and displaccmcnts iit the nodcs on the boundary of the solution
domain. Thcsc equations may
then be solved using standard matrix solution tcchniqucs. If the elastic domain is made up of scvcral pica-rvisc dilTcrcnt materials, it may bo divided into sovcral sub-rcgions. each with corresponding mutcrial properties. A BIE is written for each sub-region and the appropriate continuity and equilibrium conditions are applied at the common interface boundarics. before the linear algebraic equations arc solved. This multi-region approach may illso bc used to model general crack problems in
of tnkrlrtce crdckr
REM
homogeneous bodies (Blctndford more sub-regions with the tirtiticial
et ul., 1981 I whereby ;L domain is divided into two or interface boundaries containing the crack plane.
It has been well established in BEM the use of the traction-sin&r for the stress intensity
3107
linear elastic xntlysis
quarter-point
of fracture problems thut
crack-tip elements will yield accurate results
factors [see. e.g. Martinez
and Dominguez
(195-t). Tan
and &IO
( 1990:~ b. IYY I. IYY?)]. This is because the near-tip fields ;tre more accurately represented over these elements. It hus also been uidely shown that by simply shifting nodes of the quadratic isoparametric
elements adjxent
points (set Fig. 2). using the shape functions vuriation
the mid-point
to the crack tip to the quarter-
given in eqn (33). the following
form of the
for the displacements ;tnd tractions over the elements is obtained :
(3-l) vvhcrc the superscripts denote the nodes shown in Fig. 2. :tnd I is the length of the cracktip clcmcnt. In xidition.
if the shape functions associated with the nodal tractions for the
element ahead of the crxk-tip, O(r
;~s shoun in Fig. 2. a lrc multiplied
by ,/1/r,
the tractions ;tre
’ ‘). That is.
= ?,“N’(<)
J
I r +?,“N’(();
+?,"N'(;).
It c;~n hc further vcrilictl II121 the coniputctl notl;~l v;~Iucs ol’“tr;lctions”.
(35)
1;, on this lraclion
singuhtr clcmcnt ;irc rclatctl lo the physical values /; ;~s follows :
?,I’ = lim !I”
, 10
For cracks in hon~ogcncous mutcri;tls. blxofllcs
(a)
(bl
J r
I’
the bimutcrial
property ;’ = 0, ccln (31) then
320x
L.
C
T.\-. VI ul (37)
SO
K(L) = ii = J’rcrt
= \/ 2nli”‘.
(35)
Or
in which the classical stress intensity factors are directly obtained from the computed nodal value of the traction coefficient ?!‘I at the crack tip. In the case of interface cracks between dissimilar
isotropic
materials,
P, , = Pz2 = 0 and PII = -P:,
= I in eqn (31). It then
becomes
R’[(~)“]=cos[~ln(L)1[~
which clearly has the oscillatory Gross intensity
filCtOr.
is not oscillatory cxprcssod
ilS
(WU.
:I-sin[;ln(1)1[:
form
ilS
r + 0. Howcvcr,
the modulus
(40)
A-,, of the complex
in form. I%csidcs. the strain cncrgy rclcasc rate. C;. which Illily gcncrnlly bc I900)
:
to Ki. This
led the ;luthors
of h’, and A’,, directly in their previous studies (Tan and (;a~, illltl
(30)
dctincd as
c;ln bc shown to bc directly proportional (30). (30)
A].
K,,
can
hc
to dclcrminc A’,,instcxl
1990~1.h. IWI).
I:rom cqns
shown to be rcl;ltcd to the computcil nodal traction cocliicicnts
AS : ii,, = J2nl[(r7,“)?+(r7?“)‘]’
(43)
?.
For an intcrlhcc crack bctwccn dissimilar anisotropic miltcriills. I’,,, # 0. (r, // = I, 2). It can then be eilsily vcrilicd using cqns (31) and (36) that A’,,will not huvc the sitmc simple form of crln (42) in terms of the computed traction cocliicicnts.
A new niodilicd
function is therefore introduced into eqn (35). which incorporates the oscillatory the traction singularity singular quarter-point
at the intcrfxc
crack-tip. The variation
shape
nature 01
of ths tractions over the
crack-tip element is now taken to bc:
ivhere pr is the computed traction coeflicicnt at the cth node of the crack-tip element with this new modified shape function. It can be easily verified that the relationship between ?’ and the physical tractions I: is as follows :
BEM of mterhce cracks
For two different characteristic
3x9
dimensions L, and LI. the corresponding
K(L,)
L, Ii’ .)I*
= R’
L [(
K(L:).
(46)
Thus. from eqns (28). (30). (45) and (46). using the crack-tip the characteristic
stress inten-
and K(L2) may be related according to Wu (1990) :
sity factors K(L,)
element length 1. as one of
dimensions.
K(L)
=
R’
” K>I ;
K(I)
1;.&&I”. = Lr>I = R*[(~J’]-~-R*[(;)~“].J~.R[(;~~].;,~)
R’
(47)
[(
whcrc i’” = [I-,‘I ?z’)]” are the computed
traction cocflicionts at the crack-tip
node. Equa-
tion (47) may bc written explicitly as
(4X)
As can bc seen from this equation, singularity
arc now removed.
the numerical diflicultics associated with the oscillatory
Furthermore,
it enables the stress intensity
obtained directly and simply from the BEM computed
factors to bc
traction cocfiicicnts at the crack-tip
IlOClC.
4. NUMERICAL The results for four test problems, hcrc to dcmonstratc
EXAMPLES
all with exact analytical
solutions, arc prcscntcd
the veracity of the BEM approach described above for obtaining stress
intensity factors of bimatcrial
interface cracks. Of these four problems, isotropic bimaterials
were trcatcd in two of them. This was. in part, to verify the soundness of the technique when the material propcrtics reduce to the case of isotropy, and also because of the paucity of exact closed-form
solutions for anisotropic
these test problems,
one other problem
application
bimaterial
was trcatcd
intcrfacc cracks. In addition
in this study to illustrate
of the method. such as in the micromechanics
analysis of multi-phase
to
a typical materials.
It involves an elliptical inclusion with a debond crack in an anisotropic matrix. In the numerical trcatmcnt of the problems. all the quadratic quarter-point crack-tip clcmcnts have the same length I for a given problem case. This length I was typically taken to bc IO”’ of the modellcd crack length N. The clcmcnts adjacent to these crack-tip clcmcnts were gradually increased in size awily from the crack-tip illong the bimaterial interface. AS will be seen below. a relatively small number of boundary elements was used for each of thcsc problems considered.
All the computations
computer using single precision arithmetic.
were carried out on the Honeywell
DPS8/70
3210
C. L. T \\
,‘I
,I/.
a, -
I?gurc 3 shows the lirst test prohlm trc;ltctl. ILIIW~~. that 01‘ ;III inlinitc isotropic hillliltcri~ll f-kilt2 with an intcrliicc crack of Icngtli 2~. sutjcclcd to direct strcsscs at intinily. In the HEM ~noclcl. ;I linitc bimatcrial plate was considcrctl illstc;d. but its height and width wcrc t:tkcn to bc 20 times the six of the crack. Thus the cllkcts 01’the linitc bountl~~rics GIII bccxpcctcd to bc not signilicant. The problm was ;~~ulyscd unclcr conditions ul’pl;~nc stress. Also. with rdcrcncc to I?g. 3, th~iIppIid strcssrr, was taken to bc [I*~ - (I:‘,:‘I~,)~,I(a,),whcrc: E, am1 Y, arc: the Young’s ~mxlt~lt~s ilntl Poisson’s ratio. rcspcctivcly. 01’ material 2. This was lo cnsurc continuity conditions Ibr llic strain I: , , along the bimatcrial intcrlilcc. The bound;lry dcmcnt mesh omploycd Ibr this problem is show11 in I:ig. -I whcrc only hall 01’
Box A
Box A
r - -- - - -: -
1
I -I
I
I
I
BESI of interhcv Table
I. Norrnalizcd
stress intensir
bctors
I‘or an intcrbce
cracks crack on an infinite bimaterial
rcmolc strcs.sc*j-Problem R, b,, ..
I
E, E:
I
0
5 ‘0 100
0.075 O.Io-1 0.1 I4
the physical
problem
and 68 nodes
were
3211
nlI
&,;a,,
ELKI
BE&l
?b Error
I.O(N) I .IWNJ I.013 I.015
I.010
I.0
I .024
1.8
-0.098
l.O?I I.017
I.8 0.2
-0.136 -0.I-a
was modelled
by virtue
used to represent
isotropic plate undrr
(i)
Exact 0
of symmetry.
the two
nLI
BEM
?C Error
0 -0.096 -0.141 -0.155
0 -2.0 3.7 J.7
A total of 34 boundary
sub-regions
with
elements
the respective
material
properties. Four different values of E,‘&: were treated. namely. I. 5. 20 and 100; the Poisson’s ratios V, and r1 were taken to be the same however. as 0.3. These ratios thus correspond to the following ~~lucs for the bimateriul constant 7 in eqn (16) : 0.0.075.0. IO4 and 0. I 14. respectively. The esact solution
for this problem
(48). the stress intensity coetlicients teristic Table
factors
at the crack-tip
dimension
node of the traction-singular
L was taken
I lists the normali&
from
to bc
cqwl
stress intensity
quarter-point
and their noll-dilllcnsiorlalizcrl
K,ia,.Jno
for K,
GillgC
Y0llllg’s
01’ tllc
l?gurc 0 shows matrix which is
a11
III:I~C
and K,,/(T,Jx~I.
moduli
riltich
results arc
according plotted
to cqn (42) in the DEM
in Fig. 5 against
the EJE2
trcatcd.
isotropic. elastic circular inclusion, radius U. cmbsddcd in an inlinitc 01’
;Inothcr isotropic elastic material. A circular
an angle 211 exists as shown.
subjcotcd
to remote
bctwccn
tension d. In the BEM
biaxial
the two elastic
media,
nanicly.
arc dcbond
crack,
and the matrix
is
analysis, the intinitc body was modclled
as a square with side lengths 20 times the diamctcr of the inclusion. E,/E
as obtained
listed arc the cor-
‘I’hc errors of the HEM results wcrc again Icss than 5”/u
spanning
of 0 were trcatcd.
traction
The charac-
less than 5% for K,,. The strain energy
and
vi~lucs arc shown
with the closed form solution.
for tllC
element.
ratios considered. Also
rclcasc rate. G, for the dilYcrcnt casts was also calculated ratio
Using eqn
the BEM computed
of Rice and Sih (1965). As can be seen, the BEM
exact solutions
very good indeed. with less than _ “%, error ~tutly.
from
to the half-crack length (I in the calculations. factors,
analysis for the dilTcrcnt E:/E,
the REM
responding
has been given by Rice and Sih (1965).
K, and A;, were obtained
0 = 30 , 60 . 90 allcl IX
Four diflizrent
values
‘. For each of these angles. the same
ratios as in the previous c?camplc,namely. I, 5. 20 and 100, were again considered.
The Poisson’s
ratios for both materials were also taken as 0.3 but plane strain conditions
were
7’hC
ilSSlllllCd.
corresponding
O.OY4 and 0.091. rcspcctivcly.
Fig. 5. Variation
of y. the bimatcrial constant, were thus 0. 0.061, 7 shows a typical BEM mesh used to model half of
Villll~S
Figure
of tllc normxli/cd strain cncrgy release rate. G/([(I -vi)/E,](n,nu): Young’s modulus ratio. E,iE,-Problem (i).
with the
& a
Matntal II
Fig. 6. An Inlcrfxc
tllc tOtill
physicill probltm, Of 3.5
crxk
bc~wccn ;I circular subjccl la bixkil
atlvilntilgc
boundary ClCIllCIltS
Tahlc 2 prcscnts the fjEM
e a
e
isotropic inclusion ;md ;m inlinile lcnsion-Problem (ii).
hcing tiikcn of symmetry.
It h:is
isotropic m;llrix
sub-regions with ;I
two
70 nodes.
illld
computed villucs of K,/a,/nu
itnd K,,/rr,~~tr,
and the cor-
responding enact results by f’crlmiln ilnd Sih (1967). The d~oractcristic dimension L for the stress intensity
li\ctor computations
magnitudes ol’thccrror This
in the fjEM
wits titkcn
11s u.
AS
can by seen from the table, the
solutions wcrc alI within 5”/;1for iIll thccascsconsidcrcd.
was also true li)r the computed strain cncrgy rd~:lsC rate. G, the variations
with L’,/E2 Ibr tha tlifrcront
vnlucs
of II arc shown
i’rohl~vtr (iii) : Atr irrlcf$rc.e crtrck hrnwti -f‘hc third
dissitrdtrr
mcrleritil.~ c~pcnetl by itrlcrrrtrl prcs.wrc~
test problem considercd was that of’ LI crack. length
internal pressure u,, ilt its
~‘ICCS,
20
and subjcctcd to
which lit illong the straight interface between
Boa A \ I
Fig. 7. BEM
c_____
n!
ol‘which
in Fig. 8.
I
mesh fur Problem (ii).
two
dissimilar
BEM Table
1. Normshxd
stress intensity
factors
inclusion
of mtcrface
for a circular
in an infinite
E,
0
E,
;
EWC1
BEM
I
0 0.06 I 0.0x.8 O.OY~
0.6-w
0.625 0.859 0.Y3-1 0.955
0
0.64-I
0.061 0.0X-l
O.YSh I.080
100
V.OYI
I 5 20 IO0
0 0.061 0.084 O.OYZ
30
5 10 IO0
60
I 5 21)
00
I
I’0
5 20 I cm
.v,-.v: ;Ixt’S.
as shnw
lymg
along
the interface
body-Problem
of an isotropic
elastic
(ii)
;;
K,, o,\ “;b Error
ncl
Enact
BEM
-2.3 -2.8 -0.1 0.2
-0.172 -0.121 - O.OXY - 0.077
-0.166 -0.126 -0.091 -0.077
-3.5 I.4 2.2 0
I.106
0.631 O.Y5l I.069 1.lO.S
-2.0 -3.5 - I.0 -0.1
- 0.372 -0.459 -0.460 -0.457
-0.364 -0.448 - 0.456 -0.452
-2.2 -2.4 -0.Y - I.1
0.47 I 0.7Y7 0.Y I P 0.957
0.463 0.7iY O.Yl2 0.Y57
-1.7 -2.3 -0.7 0
-0.171 -0.6SO -0.73x -0.754
-0.457 -0.664 -0.725 -0.739
- 3.0 -2.4 - I.2 -2.0 -3.‘) -0.1 - I.4 -2.6
0.8X-l 0.938 0.953
0
0.265
0.261
- I.5
-0.460
-0.432
0.06 I 0.0x-l O.OY2
O.JY7 0.610 0.653
O.JY7 0.613 0.657
0 0.5 0.6
-0.70x -0.Hl3 -0.851
-0.707 -0.802 - 0.82’)
semi-inlinitc bodies, ils shown in Fig. with the orthogonal
arc crack
isotropic
K, rJ,\
3213
cracks
9. In
matcri;ll ilxcs. ST-sr. in the lipurc. To
9.0 Error
this problem, m:iterii\l I was treated ils anisotropic rotated by an ilngle 0 with respect to the global reduce the number of vilri;lblc paramctcrs in the
prohlcm. m;~tcrial 2 wils trciltcd iIS isotropic. It ~h~ttld. howcvcr, be emphasized that this dots not tlctract from the vitlidity of the HEM tcchniquc cmploycd. For the purpose of illllslri~tii~n.
the ClilStiC propcrtics of sin&
crystal ~llLmlinil N20r
while those of fully stabilizotl polycrystal zirconia. I’Sl’ 230,. cnginccring conStiilltS for the Ibrmcr, ci\lcttliltd Simmons ;I1111 Willlg ( I97 I ). iIIX ;IS li)llOWS :
. , , = 345 Gl’u.
I:*
v::
= 0. I3 I,
= 516 (21% a ,
= 345 GPa.
vf, = 0.196.
j/:2,: = 0,
‘I::,) = -59 GPa.
= I73 Gl’il,
whcrc the aStcriSkS dcnotc that these Villl~cS art’ with refucncc to the orthogonal UC’S directions;
E& is the Young’s 09
modulus in the x:-direction;
material
CT2 is the shear modulus
II
08
0
01
l2o’l
t
-1
9
IO
20
loo
El ‘5 Fig.
8. Variation
I,
the clnstic complianccs given in
ET,
v:, = 0.362,
t/:2., = 59 Gl’u, CT,
L’:>
from
wcrc used for m:ltCriiil
wcrc used for mutcrial 2. The
of the
normalized young’s
strain modulus
energy release rate. ratio ,5,/E,-Problem
G/(1( I -vi)/E,](n:n~)) (ii).
with
the
C. L
c’t ul
alumina anJ
Fig. 0. An intcrlilcc crack bctwecn nnisolrqw
in the .\-:-_vr
T\\
FSP zir~oni;l--Probl~ins
As ;I matter
to by Lekhnitskii ol’ interest.
(1Y62)
as the codlicimts
the isotropic
values
dA~:OI
h:ivc hccn given in the liter:itiirc
zirconia.
which
is isotropic.
In tlic niinicrkil ‘II
“qlld
to 2Ot/. I:iciirc c
it has ;I lotal
ol’J0
lhc ;in;llysis.
Also.
wcrc wnsidcrcd.
a linito
IO shows
clcnicnts
anil
li~ur rlil~crcnt nanicly.
Ibr
iis 2%
the corresponding
analysis.
0 = 0
J
ratio Lvhich is defined as the compressive strain in
plane ; I$ is the Poisson’s
the .\-f-direction due to a unit cxtcnsional strain in the x~dircction arc rel*crred
(iii) and (I\
the Young’s G&i
values
arc I92 Gki
clcnicnt
I’I;inc
ol’0.
niotluIi~s
body was trcatctf
tlic hounrl;iry
02 noilcs.
influcncc
strain
the angle
mudi
,
L___.
i((
&xX
kind. ratio
As for tfic f:sf’
and 0.3. iI3pcctively. with
width
cniploycd
cvnrlitions of oricnkition
,
t/5,.r
tirst
and Poisson’s
Il’;inti
height
Ibr the prohhi
wcrc again
assi~nNxl
01’ Lhc mitcrial
, 30 . 00 aid 90 . I’hc c?cact dosctl-lorin
r---
ol’thc
anti ().?J, rcspcctivdy.
vducs
hiiiintcrial
; and the quantities
of mutual
soliition
; in
;IXCS. Ibr the
BEM of mrcrface cracks
3215
Tahlc 3. Nornulizcd stress Intensity lkton and strain energy release rates for a pressurized interfucc crack_betut~n FSP ZrO, and anisotropic AIIO, half-planes. K=fJ ,,, .1u: G,, = :tJ,,+d,,+d,, -d:,,l,,_,,RZ
30
Enact
BEM
Ya Error
0.992 0.9YY -0.107 0.107
I .OOO 0.99’ -0.107 0.101
0.8 -0.7 0.0 j.6
0.478 0.419
O.~Yj O.-w
I.4 0.8
0.97Y 0.902 0.193
0.994
k R
0.186
I.5 I.3 -3.6
Gk, KH R GE G,,
-0.193 0.47s 0.479
-0.186 0.491 0.4x9
-3.62.7 2.1
I ,000
I .OOj
0.5 0.3 -5.1 -5.7 0.8 0.6
E;: R
0.0443
hy K:,
O.Oo7X
60
K,’
K
A’? b E;,;/A’ Kl: ,rc (i ‘!G,, G”‘G 0 YO
EyK E;,:I’lc A’r;:K
illtcrlsity
I .OOh O.YYX 0.10.5
O.YY? 0. I I I -0.1 II
Thus,
0.7 0.6 -5.J
-0. I05 0.50’ o.w
O.J’)h 0.495
factors of intcrfacc cracks bctwccn anisotropic
hccn given by Wu (IWO). rcspcct lo
0.n!? - 0.033 0.51h 0.525
O.YYY
f; ‘, G,, (;“iC; II
stress
I .W?
I.000 0.035 -0.035 0.52’ 0.52’
K,‘ik
0.0~5-l
1.005
the accuracy of the HEM
-5.-l I.1 I .o
dissimilar
solutions
is shown in TilhlC 3, whers lhr supcrscripls
lhcsc rcsulls. This
half plants has
can be
with
nsscsscd
“A” ilIIcl
in
“11”
A’,. K,, and (I’ dcnotc the two crack tips shown in Fig. 9. The characterizing dimension usccl in the stress intensity
factor computations
here
was
plalc. Also. lhc paramclcr G,, iiscil for non-dimcnsioiiing
where (7,,,are the cocllicicnts
of the lj
CV.
L
the width of the bimatcrial
lhs strain cncrgy lI2lCilSC
rilk
G is
matrix defncd in eqn (37) and are material parameters.
For the single crystal propcrtics of AlzOl considered, when 0 = 0 , they are as follows
‘7,, =
1.375x
IO :,
d,? =
4.041 x IO- ‘,
(72,=
4.041 x IO ‘,
(722=
1.328x
The accuracy of the BEM errors
IO ?.
(50)
results for A’, and G can be seen to be very good indeed. the
were all less than 3%.
Those
for A’,( were slightly
higher however, but then. the
of magnitude
smaller than K, for this problem.
It is also cvidcnt from the results that. for a given BEM
mesh employed. the accuracy of
numerical
valuss
of K,, arc about an order
the solutions obtainccl was 1101dcpcndcnt on the orientation hc lflC cast.
of the material axes. as should
The fourth test problem analysed was similar to that in Problem (iii) in all respects except for the loading condition on the crack faces. Instead of internal pressure acting on the crack faces. they were subjected to a uniform shear stress r,, as shown in Fig. 9. The boundary element mesh used was also the same bs in the previous example,
as shown in I:ig. IO. Table J lists the results lix
the norni:ili~ctl
slrcss intensity
I1ictors.
Jm, ;~nrt the norm~~liz~tl strain cncrgy rclcasc rate (;/c;,,. (~7,~ +J, 2+t7:, +;7::)(rim/)/i). The cocllicicnls c7,,,have the s;inic nunicrical
k,/r,,Jntr G,, =
and
previously
K,,/T,,
uscd. In this prohlcm,
the niagniludcs
thosr of A’,. It is again ofsignificancc
whcrc values
of K,, arc ;ibout an ortlcr grcatcr than
to note that the HEM
results wcrc very good indeed.
thus validating the tcchniquc cmploycd. Prol~l~vtr
(I*) : Att rllipticctl
ittclir.siort
The tinal problem trcakd inlinilr
anisotropic
matrix
and
The gcomctry of the inclusion respectively, anisotropic
dtt itrl~~r/kc
conlainilig
itt (III trtti.voIroyic
elastic inclusion
trtrrlri.v
tx~bcddccl in an
an intcrfacc dcbond crack as sliown in Fig. I I.
is defmcd by the semi-major
mutrix
is subject Lo rcmotc uniform
20,
hcrc while the matrix
and semi-minor
;IXCS, (I and h.
radial tension 6. The elastic inclusion
the U’;IS
was taken to have the S;IIW elastic propcrtics
crystal A120, given carlicr. Five dilrcrcnl
sin&
i7wc~k
and fhc extent of the dcbond crack is mcasurcd by the :~nglc 31. Also,
taken 10 bc FSI’ of
wit/r
W;IS that of itn’cllipticill
ellipse aspccl
rillios.
hlu.
au-c
illl~llySCd,
namely, h/r/ = I /3, I /2. I. _ 7 and 3. For cilch of thcsc gcomclrics, thrco siycs of the intcrfacc crack wcrc trc;iIcd;
they correspond to II = 30 . 00 and 90 . III the HEM analysis, the infinite matrix wils modcllcd iis iI linitc circular cylinder with
radius IOn. Also. the material LWS directions wcrc taken to coincide with those of the global Cart&n ;IXCS. For the purp0sc of comparison, rcpc:lt computer runs wc’rc matIc with the matrix
assu~~~cd
to
bc
isotropic
A1201 for cilch of the abovemcntioncd
gcomckc C;ISCS.
Figure I2 shows a typical BEM mesh cmploycd for the problem whcrc ;I maximum of 57 clcmcnts and IO4 noda wcrc uscd. The
results
for the stress
intensity
factors arc shown
in Tahlc
5: they have been
normalized with rcspcct lo 17~,‘7r1(1 and the characlarizing dimension I!, uscrl in lhcir computations wils N. It is worth noting from this table that the stress intensity Ihctor K, and K,, it1 the two crack-tips “A” and “13” can bc significantly difrcrcnt from one another in magnitude because of the m;~teriill anisotropy. They arc also in deviation from the corresponding results
obtained \vhen the matrix
was assumed to bc isotropic.
Figures
13(a).
3’Tli
BEM of interface cracks
Fig.
I I. An elliptical
FSP ZrO,
inclusion in an anisotropic Problem (v).
(h) ;mJ (c) show the variations
of the norm&al
aspect ratio. h/cr. of the elliptical inclusion, coctlicicnts dz,, in the normalizing
AI,O,
matrix with an interfxc
strain
cncrgy rclcasc rate with
for the three &bond
W;IS
the
crack sizes andyscd. The
paramctcr wcrc the same ils bcforc ;Is given in cqn (SO).
Of intcrcst to note is that the strain cncrgy rclcasc rills obtain4 USC
crnck-
closer to that iIt crack-tip “1~” in the anisotropic
for thL; isotropic
matrix
matrix cast, for ;I given crack
si/c.
5. CONCLUSIONS
The multi-region
boundary element method (BEM)
for plant anisotropy
employed in the analysis of intcrfacc cracks between dissimilar
Fig. 12. BEM mesh for Problem (v).
motcrinls.
hits bcon
It was based
0
A
2 b IO
(b) 0 3-
0 2-
< 0
I .-
0
\
Crock
tw P
I 2
I 0
I
I
3
b/o
( 0 3-
0 2 ‘-
00 . (3
0
I-
0
L
I 2
I I
1
3
b/a
Fig. 13. Variation
ol’ ths ~wrn~;tlir.cJ strain cncrgy rclcasc raw. C,iG,,. with the sllipsc qxct h/u. (a) Or II = 30 . (b) li)r 0 = 60 . and (c) I’m U = 00
on the quadratic isoparamctric clcmcnt formulation elements which incorporate the proper oscillatory O(r With
rxrio.
and spccinl quarter-point crack-tip ’ ’ +I;‘) traction singularity were used.
these elements, ;l simple expression relating the stress intensity
rxtors
to the BEM
computed nodal traction cocflicicnts at the interface crack tip could bc obtained. This provided ;I very quick and cfiicicnt means or obtaining the stress intensity factors for interface cracks between dissimilar anisotropic bodies. The tcchniquc was validated by four test problems in this paper. two of which have a crack between dissimilar
isotropic materials
while in the other two, material anisotropy was considered. Very good solution accuracy for the stress intensity factors and the energy release rates were obtained even with modest mesh discretizations. The results for n fifth problem, namely that of an elliptic elastic
BEM
3219
of interface cracks
Table 5. Normalized stress intensitv factors for an interface crack between an elhptical FSP ZrO, zirconia inclusion and an inlinite alumina Ala matrix subjected lo uniform radial tension-Problem (v). R = o,‘na (NOTE: For the isotropic matrix. K,, is +ve for crack-up A: -ve for cracktip E.)
0
bll
k:
R
30
I3
0.315
I ’ I2 3
0.412 O.W,Y D.405 0.429
60
I3 If? I 1 3
0.526 0.543 0.420 0.439 O..sK?
YO
I 3 Ii? I ?. 3
0. I35 0. I74 0.273 0.442 0.5lc-l
Anisotropic k”II/‘R
matrix
Isotropic
matrix
K;, K
K,
0.443 0.594
-0.047 -0.092 -0.1 I6 -0.266 -0.241
0.369 0.429 0.494 0.461 0.536
0.039 0.059 0.155 0.279 0.314
0.009 0.059 0.“5 O.%h 0.263
0.5’0 0.530 0.503 0.53 I 0.610
-0.145 -0.239 -0.236 -0.331 -0.412
0.556 0.580 0.469 0.495 0.610
0.094 0.159 0.29 I 0.334 0.363
0.“9 ().I& 0.2x0 0.351 0.34 I
0.273 o.zn3 0.369 0.5J7 0.6X9
-0.249 -0.283 -0.330 -0.3X8 -0.3x1
0.200 0.22 I 0.320 0.498 0.641
0.28 I 0.305 0.338 0.406 0.392
0. I42 0.02’ 0.260 0.139 0.3Y6
K: f 0.358 0.411 0.4X2
K,, (+I
inclusion in an anisotropic matrix and containing an intcrfacc dcbond crack. have also bcon prcscntcd. It providcrl iln illustration
of the pritctic;tl uscfulncss of the boundary
nicthotl in the fracture mechanics analysis of multi-phwc
clcmcnt
materials cvcn in anisotropy.
Ihswni. J. I.. amI Qu. J. ( IWh). I:inilc cr;lck WI him;ttcri;d ;~nd hicrystal iu~erfaccs, J. ,VC*C/I./‘/r~.v. So/& 37. 435 453. Iklssani. J. L. antI Qu. J. ( I’JXOh). On elasticity solulion.\ for cracks on him;ltcri;tl and bicrystal inlcrfaces. ~V/lrrc*r. S1.i. /:‘qyq I\ 107, I77 I X-1. Bl;u~dford. Ci. I:.. Iugr;dfe:r. A. R. ;IIICILiggctt. J. A. (I98 I). Two dimsnsional stress intensity Elclor computations using the boundary clrmsn~ mcrhod. /II/. J. NUUIU. Al~rlr. G!y,tq 17. 3X7 404. C’ao. II. C’. ;ml Ev:~ns. A. G. (I9X’J). Au expcrimcn1:tl study of the fracture rcsislancc of bimaterial interfaces. .\/C*C/r. ,\/urc*r. 7. ‘95 30-l. C’h;lr;d;lmhiJcs. I’. G.. Lund. J., I‘:v:tus. A. G. and McMccking. R. M. (IOXY). A ICSI specimen for dctcrmining rhc fr:iclurc rcsistancc of binia~sri;~l intcrfaccs. J. .-Ip/~l. i(/zc/r. 56. 77 X2. C~CI~IIIS. D. L. (I971 ). A crack hetwccn dissimilar anisolropic media. I~tr. J. /Gryrtq .‘Gi. 9. 257-265. Comniuou. M. ( 1977). The intsrfacc crack. 1. ,-1/j/)1.,\/cc~/I. 44. 63 I -636. Comninou. M. ( 1990). An wcrvicw ol’ in~rrl:;cc crwks. .Grqqy Frrrcr. ,%/&I. 37, I97 -20X. Cruse. T. A. ( I WX). /hwdw,v /z’l~t~tc*t~r A~rtrl~sis irr C’o~rrprtrtrrio~ttrl Fritrc~uw illcc/uarit:v. K luwcr Academic Publishers. Dordrccht. The Nsthsrlands. England. A. I I. (1965). A crack hctwcen dissimilar media. J. Appl. ,UCV/I. 32. 400 402. Erdogan. F. (1965). Stress dislrihution in hondcd dissimilar materials with cracks. J. App/. +/et/r. 32.403 410. Go. Y. I.. and Tan. c’. L. (1902).Dctcrmin;l~ion of ch;traclcri&lg paramslcrs for bimntcrial imcrbcc cracks using Ihs boundary clcmcnt method. /31q,1qFrrrcr. ;\/cT/I. 41. 779 7X4. Ilassbc. N.. Dkumur:l. Xl. and Nak;lmura. T. (19X7). Stress analysis ol’dcbonding and a crack around ;L circular rigid inclusion. f~fr. J. FrtrcI. 32. I60 1x3. Ilutchinson. J. W. and Evans. A. G. (19x9). On the mixed mods fraclurc rcsisrancc of intsrfaccs. A&r ~C/c*/rtr//. 37. 909 0 16. Kacznski. A. and Macysiak. S. J. (19X9). A syswn of inhzrfacc cracks in periodically laycrcd cl;lsGc composite. !.‘~:,rcl,qFrtrcr. h/w/t. 32. 745 756. Kuo. A. Y. and Wang. S. S. (19X5). A dynamic finite clcmcnl analysis of intcrfacial cracks in composites. In /k/otnirtctrirm trd /khotrc/i~tg I)/’ .\/orrc*riu/.r, ASTM STP X76. pp. 5 .-34. Lee. K. Y. and Choi. II. J. ( IOXX). Boundary clcmcnt ;tnalysis of swcss intcnsily factors for bimatcrial inlcrfacc cracks. /Grqqq Frcwr. ,\/dr. 29. 46 I 472. Lskhnitskii. S. G. ( 1963). T’/rc~Jr,~of Eltrrricir~ rrl’rrn Arri.~~~/ropicE/a.vrir &U/Y. t loldcn-Day. San Francisco. Lin. K. Y. and Ilartmann (19X9). Numerical analysis of stress sinpularitics al ;I bonded anisotropic wcdgc. Engng Frtrcr. .Mdf. 32. $1 I 21-I. hlarrinez. J. and Domingucz. J. (I0X-l). On the UK’ of quar(cr-point boundary clcmcnls for stress intensity factor compu~;l~ions. /#ft. J. ,Vrrnwr. .\/crh. Eqqrfq 20. I94 I - 1950. hl;llos. P. P. L.. McMccking. R. hl.. Charalamhidcs. P. G. and Drory. M. D. (19X9). A method for calculating stress intcnqitics in him;ltcrial fracture. IN. J. Frtrc/. JO. 235 254. Nak;lg;lw;l, K.. Anma. T. and DU;III. S. J. (1990). A m;tthcmatical approach ol’ the intcrfucc crack between dissimihr ;~niso~ropic composik materials. Etrqy/ Frtrcf. .\/~*c/I.36. 4.19 449.
C. i. TAs
I”0 ___ Nl. L. and Nemat-Sasser. .\lcch. Phi.% .%/i&5 39.
S. (1991).
et ul.
Interface cracks in anisotropic
dissimilar materials:
an analyttc soiutmn. /
I I )_ I 14.
Puk.
J. H. and Earmme. Y. Y. (19X6). Application ofconservation integrals to interfacial crack problems. .\/r( h .\lurcr. 5, 261-276. Pcrlmun. A. B. and Sih. G. C. (1967). Elastostatic problems of curvllineur cracks m bonded dlssimllar m;LtcTIals. Inr. J. Et+/ny Sc.1.5. YJ_~X67. Rdju. I. S.. Crew J. H. and Arnmpour. M. A. (1988). Convergence of strain energy release r;lte components for edge-delaminated composite laminates. Enyny Frucr. .Mech. 30. 383-396. Rlcc. J. R. ( 19x8). Elastic fracture mechanics concepts for interracial cracks. J. Appl. .Mdr. 55. 98 103. Rw. J. R. and Sih. G. C. ( 1965). Plane problems of cracks in dissimilar media. J. Appl. Mech. 3.2.4l&t23. Simmons. G. and Wang. H. (1971). Single Cr~srcrl Elusric Comrun~s und Calcultrred Aq,qrcqute PuI~wI;~,,T. ._I H~~lhrk. MIT Press, Cambridge. MA. Sun. C. T. and Jih C. J. (IYX7). On strain energy release rates for interfacial cracks in bimaterial media. Enqny Frwr. ,rlcc/r. 2X. 13-10. Sun. C. T. and Slanoharan. M. G. (1989). Strain energy release rates of an interfacial crack between two orthotropic solids. J. Conrp. .Utrtcr. 23. 460-178. Tan. C. L. and Gao. Y. L. ( IY9Oa). Treatment of bimaterial interface crack problems using the boundary element method. Ettqny Frtrcr. Md. M. 9 19-937. Tan. C. L. and Gao. Y. L. (I990b). Stress intensity factors for cracks at spherical inclusions by the boundary integral equation method. J. S~ruin Anal. 25. 197-206. Tan. C. L. and Gao. Y. L. (I991 ). Axisymmetric boundary integral equation analysis of interface cracks between disstmtlnr materials.. Contpctfcrfionnl IVech. 7, 3X1-396. Tan. C. L. and Gao. Y. L. (1902). Boundary element analysisof planeanisotropic bodies with stressconcentrations and cracks. Irrr. J. Conrp. S/rwr. 20. 17-28. Ting. T. C. T. (19%). Explicit solution and invariance of the singularities at an interface crack in anisotropic composites. III/. 1. Sditls S/r~rcfrrrc.c22. 965-9X3. Toya. hl. ( 1900). Fracture mechanics of intcrfaccs. frrr. J. JSME33 Series I. 413 424. Wang. S. S. and Suan. F. G. (19X3).A hyhritl linitc clcmcnt approach to composite Iamina~c elasticity problems with singularities. J. .4ppl. A/N/I. SO, 835 X44. Williams. M. L. (IWO).The rtrcsscs around a fault or crack in dissimilar media. NtrN. Sc~i.wro/. Si~c. ,,(,tr. 49. IO9 20-t. Willis. J. R. (197 I). Fracture mechanics of intcrfacial cracks. 1. ,M~I. Ph~s. S~~litl.,19, 353 -36X. Wu. K, <‘. ( IOOO). Stress intcnsily I’tctor :III~ cncryy rclcasc rate for intcrfacial cracks hctwccn dissimilar anisotropic nwlcrlals. 1. :lpp/. Mdr. 57. XX2 XX6. Wu. K;. C’. ( 1991 ). Explicit cmck-tip Ii&Is of :III cntcnding intcrfacc crack in an anisotroplc him;ltcrial. Iptr. J. .S~didVSlrrrt~lwcT 27. 4.55 Jhh. Ychia. N. A. It. and Shcppartl. M. S. ( 19x8). Automatic crack growth tracking of himatcrial intcrfacc cracks. Ifr/. J. Frm~r 37, I’5 135. Yuuhi. R. and Cho. S. Il. (19,X0). EHicicnt boundary clement analysis of strrss intensity factors for interface cracks Frwr. Md. 30. I 79 I XX. in d~ssmular m;ttcrials. fh/try
Y uuhi. R.. Cho. S. II.. M;~ISI~GO. T. and Kisu. I I. (19X7). Uscfulncss
of I Ictcnyi’s solution for boundary clcmcnt anal> sis of crack prohlcms in dissimilar materials. In Ho/l* o/Frw~w~* ,&!rc#rtrrtic.virr ~%lotkw Tc*c/wrok+~~(Edit4 by G. C. Sih. II. Nisitani and T. Ishihara). pp. X23 X33. Elscvirr. Amsterdam.
OF INTERFACE ANISOTROPIC
C. L. TAN. Y. L. GAO and F. F. AFAGH Department
of Mechanical
and Aerospace Engineering. Canada KIS 586
Carleton
University.
Ottawa.
Abstract-In this paper. the boundary element method (BEM) is applied to the analysis of interface cracks between dissimilar anisotropic materials in plane elasticity. It is based on the quadratic element formulation and special crack-tip elements which incorporate the proper O(r-’ ‘“:‘) oscillatory traction singularity are employed. A simple expression relating the stress intensity factors to the BEM computed trxtion coetlicients is derived. and this procedure for determining stress intensity factors is validated by several examples. The numerical results obtained are shown to be wry satisfactory even with relatively coarse mesh discretizutions.
I. INTRODUCTION
The
study of cracks that lie along the interface between two ditrercnt
elastic media is
important in providing a better understanding of the integrity of bonded intcrfaccs between dissimilar materials. and for determining thcappropriatc
factors which affect the mechanical
propcrtics of composites and multi-phase solids. In such problems. the near-tip ticlds may bc charactcrizcd by the complex stress intensity factor K = k; + iK,,, and the strcsscs in the vicinity of the intcrfacc crack tip arc oscillatorily
singular. of the order r
’ “‘;‘, whcrc ;’ is
;I bimatcrial property. Also. the tensile and shear dli~t~ arc alwitys COU~IC~.cvcn for single mode loading. As a result. the stress intensity
factors (S.I.F.),
K, and K,,, should not bc
intcrprctccl in the cl:lssic:il scparatc sense as for cracks in homogeneous mutcrials. The bimateri;ll and cxpcrimcntal Kacznski
interface crack problem has bcrn the subject of extensive thcorctical investigations
and Matysiak
in rcccnt years [c.g. Rice (1988).
(1989). Comninou
Jih (1987). Mates VIUI. (1989). Hutchinson Toya
(1990).
previously
Park
and Earmme
(1986)
also received some attention
(1990). Charulambides
Hasebc (‘I tri. (I 987). rt ul. (1989). Sun and
and Evans (1989), Yehia and Sheppard (1988). and Cao and Evans by Williams
(1959).
(1989)]. Erdogan
(1965). Rice and Sih (1965). Perlman and Sih (1967) and Comninou
although (1965).
it has
England
(I 977). among others.
However. these studies based on the linear elastic fracture mechanics approach have concentrated mainly on isotropic bimaterials;
similar investigations
into anisotropic bimaterials
arc relatively limited in number. Reported works in this latter case include those by Willis (197 I). Clements (197 I). Ting (1986). Bassani and Qu (1989a, b), Wu (1990, 1991), Ni and (199 I) itnd Nakagawa Ed td. (1990), all of whom have treated the problem
Ncmat-Nasser analytically.
Numerical
methods have also been employed for the analysis of interface
cracks bctwccn non-isotropic
elastic media. but the finite element method is the technique
almost cxclusivcly used. Among the contributions Yuan (1983).
in this regard are those by Wang and
KUO and Wang (1985). Raju et ul. (1988). Sun and Manoharan
Lin and Hartmann
(1989). and
(1989).
The boundary clcmcnt method (BEM).
alsocommonly
known as the boundary integral
equation (BIE) method, has recently been applied to the study of interface cracks as well. but only in isotropic bimaterials [Yuuki er al.. 1987; Yuuki and Cho. 1989; Lee and Choi, I988 ; Tan and Gao. 1990a. b, I99 I ; Ciao and Tan, I992). In the BE&l approach employed by Yuuki cr (11.(1987) and Yuuki and Cho (1988). an infinite plate made of two dissimilar isotropic solution in the BIE by extrapolation
formulation.
Hetenyi’s solution for a point load in media was used as the fundamental
The stress intensity factors, K, and K,,, were then obtained
techniques based on the computed crack face displacement data. Lee and 3201
L T\X e’l Lli
c
3’0’ _ _
Choi (IYYS), on the other hand. used the conventional quadratic isoparametric correlating
elements. They
the computed nodal displacements
crack-tip to the classical field solution. mesh discretizations.
by conventional
formulation
obtained the stress
of the BEXI
intensity
on the elements adjacent to the interface
In both these approaches. however. very refined
BEM
standards. need to be employed. Tan and Gao
(1990a. b. 1991) and Gao and Tan (1997) also used the conventional BEM but with quarter-point
with
factors by directly
in their studies.
O(r-
’ ‘) traction-singular crack-tip elements. Instead of computing for K, and K,, directly. the modulus. K,,. of the complex stress intensity factor. tvhich may r I also be written as h = A,, e Iv. was obtained. It is worth mentioning here that for the interface crack in isotropic bimaterials.
this quantity K,! is directly related to the strain energy release
rate. In these references. the authors
calculated K,, from expressions
they had derived
relating it to the computed nodal tractions or to the nodal displacements of the quarterpoint crack-tip elements. The phase. $. of the complex stress intensity not determined directly via its relationship elements. These
using a formally
factor. however, was
established procedure. Instead. it was estimated
with the crack face displacements at certain positions positions
were established
from
numerical
several test problems with exact analytical solutions.
experiments
Nevertheless.
on the crack-tip carried out on
good solution
accuracy
for K, and $ was obtained even with relatively coarse mesh designs. lr
‘1is pupcr, the conventional
BEM
with quarter-point
crack-tip elements is applied
.Ilysis of interface cracks bctwecn two dissimilar anisotropic media in two dimcnto thsions. In contrast to the previous treatment by the authors on intcrfacc cracks in isotropic bimntcrinls mcntioncd abovc, the proper O(r ’ ‘*‘;) traction singularity is incorporated in the crack-tip clcmcnts employed. Also. instead of obtaining K,, and I/J. the focus of the study is the direct dctcrmination cxprcssion
of K, and K,,. This is bccausc. as will bc cvidcnt later, any
for A’,, in terms of the computed primary
and the tractions,
variables. namely. the displaccmcnts
more complicated in the anisotropic cast, and it is
bccomcs significantly
also doubtful if the less than formal proccrlurc to cstimatc the phase angle II/ of the complca stress intensity
factor remains applicable. Morcovcr,
the relationship
bctwccn the strain
cncrgy rclcasc rate and K,, is no longer as direct and simple as hcforc. In this study, analytical cxprcssions tractions
which cnablc A’, and K,, to bc calculated directly
on the crack-tip
from the computcd nodal
clcmcnts arc clcrivctl. The veracity of the prcscnt approach is
dcmonstratcd by scvcral cxamplcs involving an interface crack in isotropic ;III~ anisotropic birnaterials.
Before thcsc results
arc prcscnted and discussed, the analytical
basis of the
method cmploycd will first bc shown.
2. UASIC
EQUATIONS
AND
THE
S.I.F.
The development of the equations presented here closely follows that by Bassani and Qu (1959) and Wu (1990). The indicial notation is used, in which the Latin indicts take on the values I. 2 and 3 while the Greek indices only summation
take on the values I and 2. Also,
is implied for rcpcatcd indicts.
For a generally anisotropic
elastic body in Cartesian co-ordinate space s,, the strcss--
strain relation may be written as
dkl
and the Navicr’s equation of equilibrium
Ck,m,lk
=
(1)
Ck,,,,nEn,n
for plane deformation
,.,,,
=
0.
of the body as
(2)
where II,,,. (TV,and I:,,,” arc the displacements. stresses and strains. rcspcctively, and Cklmnare the elastic constants of the material. By introducing the following Fourier transform pair:
BEIM of Interface cracks
3203
(34 f(5)
=
IL
I
\/ln
where i = ,/-
I. and applying
I(<) ee’zr
d<,
(3b)
-XT
them to eqn (2) over the X, co-ordinate.
the following
equation may be obtained :
This equation may also be written in matrix notation as
(5)
where the superscript “T” denotes the transpose of the matrix and
Q = (0,~) = (C,,H) =
s
=
t-y,)
=
(C’,,kZ)
=
c III1
CIII?
c
cI:ll
cIz12
clzlr
C’I III
Cl.I,2
c,,,,
C' Ill1
Cll?2
Cl121
C‘lZlJ
c’,??Z
cl:z,
(‘1122
Cl,?,
L C’
I112
c’ v
=
(I.‘,,)
=
(C,?k?)
=
c:?Ij L c
Equation
(4) or (5) admits particular
1212
?II?
III?
C’lzzr
Cl??,
C’J???
cz22,
c ‘?J??
c’?I?J
.
(6~1)
.
(6b)
I
I
1 .
(6~)
solutions of the form
(7) or. in matrix notation.
(8) provided that a and cz satisfy the cigcnvaluc equation
[<;Q+
= 0.
(9)
= /I<,, cqn (9) bccomcs
+p(S+S’) +p'V]i~(<~)
[Q
=0
(10)
and the ncccssary and sutlicicnt condition for a not having trivial solutions is thus the vanishing of the dctcrminant of the coefficient matrix in eqn (IO). That is
IQ+p(S+S’)+p’Vl which is a sixth-order
polynomial
= 0,
(11)
in p. Since Q, S and V are real and are functions of the
C. L. Ta\
3X4
matenal
constants
(2) is elliptic. be written
rr ul.
CL,““. the roots p, (,V = 1.6). are independent
eqn ( I I) has three pairs of complex
conjugate
of .Z,. And
because eyn
roots. Thus the six roots ma!
as Im(p,,)
Consider
> 0,
now a crack lying along
are made of linear In the vicinitypf
elastic anisotropic
the crack-tip.
pII_?
=p<.
M = 1.3.
the interface
between
(12)
two semi-infinite
I and II, respectively,
materials
solids which
as shown
the stresses ahead of the crack along the interface
in Fig.
I.
are written
LIS
t = t, = (0,: From dimensional cracks
may also
physical
problem.
considerations be written
with
[see Rice (1985)]. reference
These stress intensity
relationship
with
(13)
the stress intensity
to a characteristic
factors
will be written
K(L) = (A-,, and thtir
a>,)‘.
rJz2
factors for interface
dimension,
say L. of the
as
A-,,,)‘-
if-,
(14)
the strcsscs t is
K(l,) =
j’nr-
[O1
(15)
f ‘;
(If))
0;
r
R
I,
et5
whcrc [see Ting (l9M)j
;’ =
I 2X ln
(
)
(17) 1~ CLlII (I 7). W and I) arc the negative is.
that
real and imaginary
parts of ;I complex
matrix
$1,
BE\1
of interface
hI
=
and \I in turn is related to a Hcrmitian of the bimaterial.
crxks
3205
-(IV+iD)
(18)
matrix A which is defined by the elastic constants
More specifically. hl = -i(3)’
‘[.-4’] -‘.
A = (B’+B”)
B’ = i(3)
(19)
-‘.
\=I \I= I
’’ i
L’(
Adj L’(p;)
/J’, ) =
Q'
i
C’(p;,)
C’(/G) = (S’)’
(21)
(2’) (‘3)
+p\.V’.
I. II.
2 =
I or II. rcspectivcly.
where the superscript r = I or Ii refers to material It can be shown that (Ting,
-I. 1
+/‘:.L”.
[S’ + (s’).‘]
+p;
Adj L’(p;,)
(20)
IW6)
so /i in al11 ( 17) is always real. Also, in crln ( 15). lho matrix H is dclincd hy
H[cj = I + II11 [cII’+ ( I - Kc [C],P. whcrc In1 ;IIitl I&z tlcnotc the iiii;lginary
antI real parts. I --
I’= ibl
(25)
rcspcclivcly. I is lhc unit
Iii:Ilrix.
I
P)
*
’ = W+il’>.
It should also be noted that H has the following
(‘7)
properties:
K[l] = I. I+]
and
* I+]
(Xl)
= K[w].
(Xb)
in which the arguments c and 1’ ;lrc tlimcnsionlcss. so that R is also dimcnsionlcss. For
nxIteri;Ils
whcrc the s,-;lxis
C,,?, = C,, ,, = 0 ;lllll c-2 ,,J = c,, compo~icnl
11, ivilh llic coniponcnls
is ;I two-fold
,: = 0. Thcrc
symmetry
axis. the elastic constants
is thus no coupling
0, and 11~.In this c;isc. the matrix
of the displ;rccmcnt I” in cqn (25) can
bc shown to bc
I
0
0
0
1
()
1’: = - [ 0
0
0
1 .
(29)
Also. c2, and A’,,, disrlppear for plant problems. so only K, and K,, need be considered. From eqns (29). (3) ilnd (I 5). the stress intensity factors in terms of the stresses become
3206
C. L. T\\
PI d
where
R*[($“]=
“]P
Re[(i)“]I+Im[(i)
=cos[;:In(i)][(!)
:I-sin[;.ln(;)][iII
,‘lj].
(31)
It is evident from eqns (30) and (31) that when r 4 0. the stresses exhibit an oscillatory singularity.
as was mentioned earlier. In the case when ;’ = 0. that is. when the elastic hod)
is homogeneous. R’ becomes ;L unity matrix and the stress intensity
factors are then of the
classical separate form for the two distinct modes of crack deformation.
3. t3Ehl DETERMINATION
The analytical basis of the BEM partial differential equation written fundamental
OF TIIE
in elastostatics
S.I f-.
is the transformation
equations valid over the cntiro elastic solution for just
solutions
its boundary
with
the Rctti
I-. In the direct formulation. Raylcigh
rcciprocill
appropriate limiting operations. result in the bOliIldil~y displaccmcnts II, and tractions
of the governing
domain Q into an integral
work
use of the unit load thcorcm will.
through
integral equation (BIE) relating the
I, at r. 7‘11~ 131E for two dimensions
nuy
hc written
in
indicial notation ;IS :
C,,,(/‘)~f,(i’)f
whcrc
sI
Ir,(V)?‘,,,(I’.C))ctr(l))
il
/,(C))U,,,(I’.Q)lll‘(C)).
U,,,( i’. Q) :~ntl ?‘,,,(I’. Q) ;Irc the ftlntl;lnlcnti~l
placcnicnts and tractions. inlinitc
=
body. Detitils
rcspcctivcly.
solutions
in the s,Jircction
of their derivation
2.p = 1.3.
(3’)
which rcprcscnt the dis-
at I’ in ;I plant ho1ni~gc1lcous
for the anisotropic USC have been given by Cruse
(198X) and their explicit forms have also been prcscntcd by Tan and Gao (1991). Ah.
in
eqn (32). the value of c’,,,( I’) dcpcnds on the local gcomctry of I- iit the point I’. To solve the BIE
numerically.
the boundary r of the solution
domain is dividctl into
11series. or “mesh” of lint clcmcnts. Over ci1ch of those clcmcnts, the boundary gcomctry. displacements
and tractions
may be written.
respective nodal KI~UCS and the quiidriitic three nodes, two at the ends anti N’(i)
as in the present work,
shape functions
OIW at the mid-point :
nnd
in terms of their
N’ (i). The ckmcnts each ~:IVC the associated shape functions
arc
N’(i)
= i<(i- I).
N?(i)
= I -<‘.
iv’(<) = Li(l +i).
(33)
Substitution of the isoparametric rcprcscntations of the gcomotry and functions into cqn (31) will result in ;I set of linear algebraic equations for the unknoivn tractions and displaccmcnts iit the nodcs on the boundary of the solution
domain. Thcsc equations may
then be solved using standard matrix solution tcchniqucs. If the elastic domain is made up of scvcral pica-rvisc dilTcrcnt materials, it may bo divided into sovcral sub-rcgions. each with corresponding mutcrial properties. A BIE is written for each sub-region and the appropriate continuity and equilibrium conditions are applied at the common interface boundarics. before the linear algebraic equations arc solved. This multi-region approach may illso bc used to model general crack problems in
of tnkrlrtce crdckr
REM
homogeneous bodies (Blctndford more sub-regions with the tirtiticial
et ul., 1981 I whereby ;L domain is divided into two or interface boundaries containing the crack plane.
It has been well established in BEM the use of the traction-sin&r for the stress intensity
3107
linear elastic xntlysis
quarter-point
of fracture problems thut
crack-tip elements will yield accurate results
factors [see. e.g. Martinez
and Dominguez
(195-t). Tan
and &IO
( 1990:~ b. IYY I. IYY?)]. This is because the near-tip fields ;tre more accurately represented over these elements. It hus also been uidely shown that by simply shifting nodes of the quadratic isoparametric
elements adjxent
points (set Fig. 2). using the shape functions vuriation
the mid-point
to the crack tip to the quarter-
given in eqn (33). the following
form of the
for the displacements ;tnd tractions over the elements is obtained :
(3-l) vvhcrc the superscripts denote the nodes shown in Fig. 2. :tnd I is the length of the cracktip clcmcnt. In xidition.
if the shape functions associated with the nodal tractions for the
element ahead of the crxk-tip, O(r
;~s shoun in Fig. 2. a lrc multiplied
by ,/1/r,
the tractions ;tre
’ ‘). That is.
= ?,“N’(<)
J
I r +?,“N’(();
+?,"N'(;).
It c;~n hc further vcrilictl II121 the coniputctl notl;~l v;~Iucs ol’“tr;lctions”.
(35)
1;, on this lraclion
singuhtr clcmcnt ;irc rclatctl lo the physical values /; ;~s follows :
?,I’ = lim !I”
, 10
For cracks in hon~ogcncous mutcri;tls. blxofllcs
(a)
(bl
J r
I’
the bimutcrial
property ;’ = 0, ccln (31) then
320x
L.
C
T.\-. VI ul (37)
SO
K(L) = ii = J’rcrt
= \/ 2nli”‘.
(35)
Or
in which the classical stress intensity factors are directly obtained from the computed nodal value of the traction coefficient ?!‘I at the crack tip. In the case of interface cracks between dissimilar
isotropic
materials,
P, , = Pz2 = 0 and PII = -P:,
= I in eqn (31). It then
becomes
R’[(~)“]=cos[~ln(L)1[~
which clearly has the oscillatory Gross intensity
filCtOr.
is not oscillatory cxprcssod
ilS
(WU.
:I-sin[;ln(1)1[:
form
ilS
r + 0. Howcvcr,
the modulus
(40)
A-,, of the complex
in form. I%csidcs. the strain cncrgy rclcasc rate. C;. which Illily gcncrnlly bc I900)
:
to Ki. This
led the ;luthors
of h’, and A’,, directly in their previous studies (Tan and (;a~, illltl
(30)
dctincd as
c;ln bc shown to bc directly proportional (30). (30)
A].
K,,
can
hc
to dclcrminc A’,,instcxl
1990~1.h. IWI).
I:rom cqns
shown to be rcl;ltcd to the computcil nodal traction cocliicicnts
AS : ii,, = J2nl[(r7,“)?+(r7?“)‘]’
(43)
?.
For an intcrlhcc crack bctwccn dissimilar anisotropic miltcriills. I’,,, # 0. (r, // = I, 2). It can then be eilsily vcrilicd using cqns (31) and (36) that A’,,will not huvc the sitmc simple form of crln (42) in terms of the computed traction cocliicicnts.
A new niodilicd
function is therefore introduced into eqn (35). which incorporates the oscillatory the traction singularity singular quarter-point
at the intcrfxc
crack-tip. The variation
shape
nature 01
of ths tractions over the
crack-tip element is now taken to bc:
ivhere pr is the computed traction coeflicicnt at the cth node of the crack-tip element with this new modified shape function. It can be easily verified that the relationship between ?’ and the physical tractions I: is as follows :
BEM of mterhce cracks
For two different characteristic
3x9
dimensions L, and LI. the corresponding
K(L,)
L, Ii’ .)I*
= R’
L [(
K(L:).
(46)
Thus. from eqns (28). (30). (45) and (46). using the crack-tip the characteristic
stress inten-
and K(L2) may be related according to Wu (1990) :
sity factors K(L,)
element length 1. as one of
dimensions.
K(L)
=
R’
” K>I ;
K(I)
1;.&&I”. = Lr>I = R*[(~J’]-~-R*[(;)~“].J~.R[(;~~].;,~)
R’
(47)
[(
whcrc i’” = [I-,‘I ?z’)]” are the computed
traction cocflicionts at the crack-tip
node. Equa-
tion (47) may bc written explicitly as
(4X)
As can bc seen from this equation, singularity
arc now removed.
the numerical diflicultics associated with the oscillatory
Furthermore,
it enables the stress intensity
obtained directly and simply from the BEM computed
factors to bc
traction cocfiicicnts at the crack-tip
IlOClC.
4. NUMERICAL The results for four test problems, hcrc to dcmonstratc
EXAMPLES
all with exact analytical
solutions, arc prcscntcd
the veracity of the BEM approach described above for obtaining stress
intensity factors of bimatcrial
interface cracks. Of these four problems, isotropic bimaterials
were trcatcd in two of them. This was. in part, to verify the soundness of the technique when the material propcrtics reduce to the case of isotropy, and also because of the paucity of exact closed-form
solutions for anisotropic
these test problems,
one other problem
application
bimaterial
was trcatcd
intcrfacc cracks. In addition
in this study to illustrate
of the method. such as in the micromechanics
analysis of multi-phase
to
a typical materials.
It involves an elliptical inclusion with a debond crack in an anisotropic matrix. In the numerical trcatmcnt of the problems. all the quadratic quarter-point crack-tip clcmcnts have the same length I for a given problem case. This length I was typically taken to bc IO”’ of the modellcd crack length N. The clcmcnts adjacent to these crack-tip clcmcnts were gradually increased in size awily from the crack-tip illong the bimaterial interface. AS will be seen below. a relatively small number of boundary elements was used for each of thcsc problems considered.
All the computations
computer using single precision arithmetic.
were carried out on the Honeywell
DPS8/70
3210
C. L. T \\
,‘I
,I/.
a, -
I?gurc 3 shows the lirst test prohlm trc;ltctl. ILIIW~~. that 01‘ ;III inlinitc isotropic hillliltcri~ll f-kilt2 with an intcrliicc crack of Icngtli 2~. sutjcclcd to direct strcsscs at intinily. In the HEM ~noclcl. ;I linitc bimatcrial plate was considcrctl illstc;d. but its height and width wcrc t:tkcn to bc 20 times the six of the crack. Thus the cllkcts 01’the linitc bountl~~rics GIII bccxpcctcd to bc not signilicant. The problm was ;~~ulyscd unclcr conditions ul’pl;~nc stress. Also. with rdcrcncc to I?g. 3, th~iIppIid strcssrr, was taken to bc [I*~ - (I:‘,:‘I~,)~,I(a,),whcrc: E, am1 Y, arc: the Young’s ~mxlt~lt~s ilntl Poisson’s ratio. rcspcctivcly. 01’ material 2. This was lo cnsurc continuity conditions Ibr llic strain I: , , along the bimatcrial intcrlilcc. The bound;lry dcmcnt mesh omploycd Ibr this problem is show11 in I:ig. -I whcrc only hall 01’
Box A
Box A
r - -- - - -: -
1
I -I
I
I
I
BESI of interhcv Table
I. Norrnalizcd
stress intensir
bctors
I‘or an intcrbce
cracks crack on an infinite bimaterial
rcmolc strcs.sc*j-Problem R, b,, ..
I
E, E:
I
0
5 ‘0 100
0.075 O.Io-1 0.1 I4
the physical
problem
and 68 nodes
were
3211
nlI
&,;a,,
ELKI
BE&l
?b Error
I.O(N) I .IWNJ I.013 I.015
I.010
I.0
I .024
1.8
-0.098
l.O?I I.017
I.8 0.2
-0.136 -0.I-a
was modelled
by virtue
used to represent
isotropic plate undrr
(i)
Exact 0
of symmetry.
the two
nLI
BEM
?C Error
0 -0.096 -0.141 -0.155
0 -2.0 3.7 J.7
A total of 34 boundary
sub-regions
with
elements
the respective
material
properties. Four different values of E,‘&: were treated. namely. I. 5. 20 and 100; the Poisson’s ratios V, and r1 were taken to be the same however. as 0.3. These ratios thus correspond to the following ~~lucs for the bimateriul constant 7 in eqn (16) : 0.0.075.0. IO4 and 0. I 14. respectively. The esact solution
for this problem
(48). the stress intensity coetlicients teristic Table
factors
at the crack-tip
dimension
node of the traction-singular
L was taken
I lists the normali&
from
to bc
cqwl
stress intensity
quarter-point
and their noll-dilllcnsiorlalizcrl
K,ia,.Jno
for K,
GillgC
Y0llllg’s
01’ tllc
l?gurc 0 shows matrix which is
a11
III:I~C
and K,,/(T,Jx~I.
moduli
riltich
results arc
according plotted
to cqn (42) in the DEM
in Fig. 5 against
the EJE2
trcatcd.
isotropic. elastic circular inclusion, radius U. cmbsddcd in an inlinitc 01’
;Inothcr isotropic elastic material. A circular
an angle 211 exists as shown.
subjcotcd
to remote
bctwccn
tension d. In the BEM
biaxial
the two elastic
media,
nanicly.
arc dcbond
crack,
and the matrix
is
analysis, the intinitc body was modclled
as a square with side lengths 20 times the diamctcr of the inclusion. E,/E
as obtained
listed arc the cor-
‘I’hc errors of the HEM results wcrc again Icss than 5”/u
spanning
of 0 were trcatcd.
traction
The charac-
less than 5% for K,,. The strain energy
and
vi~lucs arc shown
with the closed form solution.
for tllC
element.
ratios considered. Also
rclcasc rate. G, for the dilYcrcnt casts was also calculated ratio
Using eqn
the BEM computed
of Rice and Sih (1965). As can be seen, the BEM
exact solutions
very good indeed. with less than _ “%, error ~tutly.
from
to the half-crack length (I in the calculations. factors,
analysis for the dilTcrcnt E:/E,
the REM
responding
has been given by Rice and Sih (1965).
K, and A;, were obtained
0 = 30 , 60 . 90 allcl IX
Four diflizrent
values
‘. For each of these angles. the same
ratios as in the previous c?camplc,namely. I, 5. 20 and 100, were again considered.
The Poisson’s
ratios for both materials were also taken as 0.3 but plane strain conditions
were
7’hC
ilSSlllllCd.
corresponding
O.OY4 and 0.091. rcspcctivcly.
Fig. 5. Variation
of y. the bimatcrial constant, were thus 0. 0.061, 7 shows a typical BEM mesh used to model half of
Villll~S
Figure
of tllc normxli/cd strain cncrgy release rate. G/([(I -vi)/E,](n,nu): Young’s modulus ratio. E,iE,-Problem (i).
with the
& a
Matntal II
Fig. 6. An Inlcrfxc
tllc tOtill
physicill probltm, Of 3.5
crxk
bc~wccn ;I circular subjccl la bixkil
atlvilntilgc
boundary ClCIllCIltS
Tahlc 2 prcscnts the fjEM
e a
e
isotropic inclusion ;md ;m inlinile lcnsion-Problem (ii).
hcing tiikcn of symmetry.
It h:is
isotropic m;llrix
sub-regions with ;I
two
70 nodes.
illld
computed villucs of K,/a,/nu
itnd K,,/rr,~~tr,
and the cor-
responding enact results by f’crlmiln ilnd Sih (1967). The d~oractcristic dimension L for the stress intensity
li\ctor computations
magnitudes ol’thccrror This
in the fjEM
wits titkcn
11s u.
AS
can by seen from the table, the
solutions wcrc alI within 5”/;1for iIll thccascsconsidcrcd.
was also true li)r the computed strain cncrgy rd~:lsC rate. G, the variations
with L’,/E2 Ibr tha tlifrcront
vnlucs
of II arc shown
i’rohl~vtr (iii) : Atr irrlcf$rc.e crtrck hrnwti -f‘hc third
dissitrdtrr
mcrleritil.~ c~pcnetl by itrlcrrrtrl prcs.wrc~
test problem considercd was that of’ LI crack. length
internal pressure u,, ilt its
~‘ICCS,
20
and subjcctcd to
which lit illong the straight interface between
Boa A \ I
Fig. 7. BEM
c_____
n!
ol‘which
in Fig. 8.
I
mesh fur Problem (ii).
two
dissimilar
BEM Table
1. Normshxd
stress intensity
factors
inclusion
of mtcrface
for a circular
in an infinite
E,
0
E,
;
EWC1
BEM
I
0 0.06 I 0.0x.8 O.OY~
0.6-w
0.625 0.859 0.Y3-1 0.955
0
0.64-I
0.061 0.0X-l
O.YSh I.080
100
V.OYI
I 5 20 IO0
0 0.061 0.084 O.OYZ
30
5 10 IO0
60
I 5 21)
00
I
I’0
5 20 I cm
.v,-.v: ;Ixt’S.
as shnw
lymg
along
the interface
body-Problem
of an isotropic
elastic
(ii)
;;
K,, o,\ “;b Error
ncl
Enact
BEM
-2.3 -2.8 -0.1 0.2
-0.172 -0.121 - O.OXY - 0.077
-0.166 -0.126 -0.091 -0.077
-3.5 I.4 2.2 0
I.106
0.631 O.Y5l I.069 1.lO.S
-2.0 -3.5 - I.0 -0.1
- 0.372 -0.459 -0.460 -0.457
-0.364 -0.448 - 0.456 -0.452
-2.2 -2.4 -0.Y - I.1
0.47 I 0.7Y7 0.Y I P 0.957
0.463 0.7iY O.Yl2 0.Y57
-1.7 -2.3 -0.7 0
-0.171 -0.6SO -0.73x -0.754
-0.457 -0.664 -0.725 -0.739
- 3.0 -2.4 - I.2 -2.0 -3.‘) -0.1 - I.4 -2.6
0.8X-l 0.938 0.953
0
0.265
0.261
- I.5
-0.460
-0.432
0.06 I 0.0x-l O.OY2
O.JY7 0.610 0.653
O.JY7 0.613 0.657
0 0.5 0.6
-0.70x -0.Hl3 -0.851
-0.707 -0.802 - 0.82’)
semi-inlinitc bodies, ils shown in Fig. with the orthogonal
arc crack
isotropic
K, rJ,\
3213
cracks
9. In
matcri;ll ilxcs. ST-sr. in the lipurc. To
9.0 Error
this problem, m:iterii\l I was treated ils anisotropic rotated by an ilngle 0 with respect to the global reduce the number of vilri;lblc paramctcrs in the
prohlcm. m;~tcrial 2 wils trciltcd iIS isotropic. It ~h~ttld. howcvcr, be emphasized that this dots not tlctract from the vitlidity of the HEM tcchniquc cmploycd. For the purpose of illllslri~tii~n.
the ClilStiC propcrtics of sin&
crystal ~llLmlinil N20r
while those of fully stabilizotl polycrystal zirconia. I’Sl’ 230,. cnginccring conStiilltS for the Ibrmcr, ci\lcttliltd Simmons ;I1111 Willlg ( I97 I ). iIIX ;IS li)llOWS :
. , , = 345 Gl’u.
I:*
v::
= 0. I3 I,
= 516 (21% a ,
= 345 GPa.
vf, = 0.196.
j/:2,: = 0,
‘I::,) = -59 GPa.
= I73 Gl’il,
whcrc the aStcriSkS dcnotc that these Villl~cS art’ with refucncc to the orthogonal UC’S directions;
E& is the Young’s 09
modulus in the x:-direction;
material
CT2 is the shear modulus
II
08
0
01
l2o’l
t
-1
9
IO
20
loo
El ‘5 Fig.
8. Variation
I,
the clnstic complianccs given in
ET,
v:, = 0.362,
t/:2., = 59 Gl’u, CT,
L’:>
from
wcrc used for m:ltCriiil
wcrc used for mutcrial 2. The
of the
normalized young’s
strain modulus
energy release rate. ratio ,5,/E,-Problem
G/(1( I -vi)/E,](n:n~)) (ii).
with
the
C. L
c’t ul
alumina anJ
Fig. 0. An intcrlilcc crack bctwecn nnisolrqw
in the .\-:-_vr
T\\
FSP zir~oni;l--Probl~ins
As ;I matter
to by Lekhnitskii ol’ interest.
(1Y62)
as the codlicimts
the isotropic
values
dA~:OI
h:ivc hccn given in the liter:itiirc
zirconia.
which
is isotropic.
In tlic niinicrkil ‘II
“qlld
to 2Ot/. I:iciirc c
it has ;I lotal
ol’J0
lhc ;in;llysis.
Also.
wcrc wnsidcrcd.
a linito
IO shows
clcnicnts
anil
li~ur rlil~crcnt nanicly.
Ibr
iis 2%
the corresponding
analysis.
0 = 0
J
ratio Lvhich is defined as the compressive strain in
plane ; I$ is the Poisson’s
the .\-f-direction due to a unit cxtcnsional strain in the x~dircction arc rel*crred
(iii) and (I\
the Young’s G&i
values
arc I92 Gki
clcnicnt
I’I;inc
ol’0.
niotluIi~s
body was trcatctf
tlic hounrl;iry
02 noilcs.
influcncc
strain
the angle
mudi
,
L___.
i((
&xX
kind. ratio
As for tfic f:sf’
and 0.3. iI3pcctively. with
width
cniploycd
cvnrlitions of oricnkition
,
t/5,.r
tirst
and Poisson’s
Il’;inti
height
Ibr the prohhi
wcrc again
assi~nNxl
01’ Lhc mitcrial
, 30 . 00 aid 90 . I’hc c?cact dosctl-lorin
r---
ol’thc
anti ().?J, rcspcctivdy.
vducs
hiiiintcrial
; and the quantities
of mutual
soliition
; in
;IXCS. Ibr the
BEM of mrcrface cracks
3215
Tahlc 3. Nornulizcd stress Intensity lkton and strain energy release rates for a pressurized interfucc crack_betut~n FSP ZrO, and anisotropic AIIO, half-planes. K=fJ ,,, .1u: G,, = :tJ,,+d,,+d,, -d:,,l,,_,,RZ
30
Enact
BEM
Ya Error
0.992 0.9YY -0.107 0.107
I .OOO 0.99’ -0.107 0.101
0.8 -0.7 0.0 j.6
0.478 0.419
O.~Yj O.-w
I.4 0.8
0.97Y 0.902 0.193
0.994
k R
0.186
I.5 I.3 -3.6
Gk, KH R GE G,,
-0.193 0.47s 0.479
-0.186 0.491 0.4x9
-3.62.7 2.1
I ,000
I .OOj
0.5 0.3 -5.1 -5.7 0.8 0.6
E;: R
0.0443
hy K:,
O.Oo7X
60
K,’
K
A’? b E;,;/A’ Kl: ,rc (i ‘!G,, G”‘G 0 YO
EyK E;,:I’lc A’r;:K
illtcrlsity
I .OOh O.YYX 0.10.5
O.YY? 0. I I I -0.1 II
Thus,
0.7 0.6 -5.J
-0. I05 0.50’ o.w
O.J’)h 0.495
factors of intcrfacc cracks bctwccn anisotropic
hccn given by Wu (IWO). rcspcct lo
0.n!? - 0.033 0.51h 0.525
O.YYY
f; ‘, G,, (;“iC; II
stress
I .W?
I.000 0.035 -0.035 0.52’ 0.52’
K,‘ik
0.0~5-l
1.005
the accuracy of the HEM
-5.-l I.1 I .o
dissimilar
solutions
is shown in TilhlC 3, whers lhr supcrscripls
lhcsc rcsulls. This
half plants has
can be
with
nsscsscd
“A” ilIIcl
in
“11”
A’,. K,, and (I’ dcnotc the two crack tips shown in Fig. 9. The characterizing dimension usccl in the stress intensity
factor computations
here
was
plalc. Also. lhc paramclcr G,, iiscil for non-dimcnsioiiing
where (7,,,are the cocllicicnts
of the lj
CV.
L
the width of the bimatcrial
lhs strain cncrgy lI2lCilSC
rilk
G is
matrix defncd in eqn (37) and are material parameters.
For the single crystal propcrtics of AlzOl considered, when 0 = 0 , they are as follows
‘7,, =
1.375x
IO :,
d,? =
4.041 x IO- ‘,
(72,=
4.041 x IO ‘,
(722=
1.328x
The accuracy of the BEM errors
IO ?.
(50)
results for A’, and G can be seen to be very good indeed. the
were all less than 3%.
Those
for A’,( were slightly
higher however, but then. the
of magnitude
smaller than K, for this problem.
It is also cvidcnt from the results that. for a given BEM
mesh employed. the accuracy of
numerical
valuss
of K,, arc about an order
the solutions obtainccl was 1101dcpcndcnt on the orientation hc lflC cast.
of the material axes. as should
The fourth test problem analysed was similar to that in Problem (iii) in all respects except for the loading condition on the crack faces. Instead of internal pressure acting on the crack faces. they were subjected to a uniform shear stress r,, as shown in Fig. 9. The boundary element mesh used was also the same bs in the previous example,
as shown in I:ig. IO. Table J lists the results lix
the norni:ili~ctl
slrcss intensity
I1ictors.
Jm, ;~nrt the norm~~liz~tl strain cncrgy rclcasc rate (;/c;,,. (~7,~ +J, 2+t7:, +;7::)(rim/)/i). The cocllicicnls c7,,,have the s;inic nunicrical
k,/r,,Jntr G,, =
and
previously
K,,/T,,
uscd. In this prohlcm,
the niagniludcs
thosr of A’,. It is again ofsignificancc
whcrc values
of K,, arc ;ibout an ortlcr grcatcr than
to note that the HEM
results wcrc very good indeed.
thus validating the tcchniquc cmploycd. Prol~l~vtr
(I*) : Att rllipticctl
ittclir.siort
The tinal problem trcakd inlinilr
anisotropic
matrix
and
The gcomctry of the inclusion respectively, anisotropic
dtt itrl~~r/kc
conlainilig
itt (III trtti.voIroyic
elastic inclusion
trtrrlri.v
tx~bcddccl in an
an intcrfacc dcbond crack as sliown in Fig. I I.
is defmcd by the semi-major
mutrix
is subject Lo rcmotc uniform
20,
hcrc while the matrix
and semi-minor
;IXCS, (I and h.
radial tension 6. The elastic inclusion
the U’;IS
was taken to have the S;IIW elastic propcrtics
crystal A120, given carlicr. Five dilrcrcnl
sin&
i7wc~k
and fhc extent of the dcbond crack is mcasurcd by the :~nglc 31. Also,
taken 10 bc FSI’ of
wit/r
W;IS that of itn’cllipticill
ellipse aspccl
rillios.
hlu.
au-c
illl~llySCd,
namely, h/r/ = I /3, I /2. I. _ 7 and 3. For cilch of thcsc gcomclrics, thrco siycs of the intcrfacc crack wcrc trc;iIcd;
they correspond to II = 30 . 00 and 90 . III the HEM analysis, the infinite matrix wils modcllcd iis iI linitc circular cylinder with
radius IOn. Also. the material LWS directions wcrc taken to coincide with those of the global Cart&n ;IXCS. For the purp0sc of comparison, rcpc:lt computer runs wc’rc matIc with the matrix
assu~~~cd
to
bc
isotropic
A1201 for cilch of the abovemcntioncd
gcomckc C;ISCS.
Figure I2 shows a typical BEM mesh cmploycd for the problem whcrc ;I maximum of 57 clcmcnts and IO4 noda wcrc uscd. The
results
for the stress
intensity
factors arc shown
in Tahlc
5: they have been
normalized with rcspcct lo 17~,‘7r1(1 and the characlarizing dimension I!, uscrl in lhcir computations wils N. It is worth noting from this table that the stress intensity Ihctor K, and K,, it1 the two crack-tips “A” and “13” can bc significantly difrcrcnt from one another in magnitude because of the m;~teriill anisotropy. They arc also in deviation from the corresponding results
obtained \vhen the matrix
was assumed to bc isotropic.
Figures
13(a).
3’Tli
BEM of interface cracks
Fig.
I I. An elliptical
FSP ZrO,
inclusion in an anisotropic Problem (v).
(h) ;mJ (c) show the variations
of the norm&al
aspect ratio. h/cr. of the elliptical inclusion, coctlicicnts dz,, in the normalizing
AI,O,
matrix with an interfxc
strain
cncrgy rclcasc rate with
for the three &bond
W;IS
the
crack sizes andyscd. The
paramctcr wcrc the same ils bcforc ;Is given in cqn (SO).
Of intcrcst to note is that the strain cncrgy rclcasc rills obtain4 USC
crnck-
closer to that iIt crack-tip “1~” in the anisotropic
for thL; isotropic
matrix
matrix cast, for ;I given crack
si/c.
5. CONCLUSIONS
The multi-region
boundary element method (BEM)
for plant anisotropy
employed in the analysis of intcrfacc cracks between dissimilar
Fig. 12. BEM mesh for Problem (v).
motcrinls.
hits bcon
It was based
0
A
2 b IO
(b) 0 3-
0 2-
< 0
I .-
0
\
Crock
tw P
I 2
I 0
I
I
3
b/o
( 0 3-
0 2 ‘-
00 . (3
0
I-
0
L
I 2
I I
1
3
b/a
Fig. 13. Variation
ol’ ths ~wrn~;tlir.cJ strain cncrgy rclcasc raw. C,iG,,. with the sllipsc qxct h/u. (a) Or II = 30 . (b) li)r 0 = 60 . and (c) I’m U = 00
on the quadratic isoparamctric clcmcnt formulation elements which incorporate the proper oscillatory O(r With
rxrio.
and spccinl quarter-point crack-tip ’ ’ +I;‘) traction singularity were used.
these elements, ;l simple expression relating the stress intensity
rxtors
to the BEM
computed nodal traction cocflicicnts at the interface crack tip could bc obtained. This provided ;I very quick and cfiicicnt means or obtaining the stress intensity factors for interface cracks between dissimilar anisotropic bodies. The tcchniquc was validated by four test problems in this paper. two of which have a crack between dissimilar
isotropic materials
while in the other two, material anisotropy was considered. Very good solution accuracy for the stress intensity factors and the energy release rates were obtained even with modest mesh discretizations. The results for n fifth problem, namely that of an elliptic elastic
BEM
3219
of interface cracks
Table 5. Normalized stress intensitv factors for an interface crack between an elhptical FSP ZrO, zirconia inclusion and an inlinite alumina Ala matrix subjected lo uniform radial tension-Problem (v). R = o,‘na (NOTE: For the isotropic matrix. K,, is +ve for crack-up A: -ve for cracktip E.)
0
bll
k:
R
30
I3
0.315
I ’ I2 3
0.412 O.W,Y D.405 0.429
60
I3 If? I 1 3
0.526 0.543 0.420 0.439 O..sK?
YO
I 3 Ii? I ?. 3
0. I35 0. I74 0.273 0.442 0.5lc-l
Anisotropic k”II/‘R
matrix
Isotropic
matrix
K;, K
K,
0.443 0.594
-0.047 -0.092 -0.1 I6 -0.266 -0.241
0.369 0.429 0.494 0.461 0.536
0.039 0.059 0.155 0.279 0.314
0.009 0.059 0.“5 O.%h 0.263
0.5’0 0.530 0.503 0.53 I 0.610
-0.145 -0.239 -0.236 -0.331 -0.412
0.556 0.580 0.469 0.495 0.610
0.094 0.159 0.29 I 0.334 0.363
0.“9 ().I& 0.2x0 0.351 0.34 I
0.273 o.zn3 0.369 0.5J7 0.6X9
-0.249 -0.283 -0.330 -0.3X8 -0.3x1
0.200 0.22 I 0.320 0.498 0.641
0.28 I 0.305 0.338 0.406 0.392
0. I42 0.02’ 0.260 0.139 0.3Y6
K: f 0.358 0.411 0.4X2
K,, (+I
inclusion in an anisotropic matrix and containing an intcrfacc dcbond crack. have also bcon prcscntcd. It providcrl iln illustration
of the pritctic;tl uscfulncss of the boundary
nicthotl in the fracture mechanics analysis of multi-phwc
clcmcnt
materials cvcn in anisotropy.
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