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Effective longitudinal shear modulus of a unidirectional fiber composite containing interfacial cracks
EFFECTIVE LONGITUDINAL SHEAR MODULUS A UNIDIRECTIONAL FIBER COMPOSITE CONTAINING INTERFACIAL CRACKS
OF
HOSG TENG Department
of Theoretical and Applied Mechanics. Univerzity at UrbanaChampaIgn. Urbana. IL 61801. U.S.A.
of Illinois
Abstract-The problem of evaluating the elTective longitudinal shear modulus of a unidirectional tiber composite containing fiber-matrix interfacial cracks is considered. The generalized self-consistent scheme is employed in the formulation of the problem. The resulting mixed boundary value problem leads to a system of’dual series equations. which can then be reduced to Fredholm integral equations of the first kind wirh a logarithmically singular kernel. The reduced longitudinal shear modulus is calculated by solving the governing weakly singular integral equations.
INTRODUCTION
The presence of many cracks in a composite material may cause reduction in its stiffness. thus degrading the integrity
of the material.
The problem of analysing stiffness
reduction due to the presence of cracks. like other
crack probtcms for composite materials.
may bc attacked by
two
fundamcntalty
approaches. One is fhe macromcchanics approach that approximates positc materials
as
homogeneous but anisotropic
ditTcrcnr
hctcrogcncous com-
media. The olhcr is the micromcchanics
approach which incvitabty tcads to the trcatmcnt of cracks in dissimilar matcriats. A large number of solutions exist in the tcchnicat titcraturc for dissimilar matcriats containing cracks
of
various locations and orientations.
As it is welt rccognizcd that cracks arc more
tikcly to dcvctop at intcrfaccs belwccn two dilrcrcnt
constituent
materials of a composite
medium. a considcrabtc amount of work has been dcvotcd to the analysis cracks [SW Comninou investigations
(1990)
for rcvicw and further
of interfacial
rcfcronccs]. Howcvcr,
most of the
wcrc concerned with cracks near or at an intcrfacc of IWO semi-inlinitc
rcndcring no direct applications to the probtcm of stilTncss
media
reduction caused by interface
cracks prcscnl in composite materials. Although
a number of studies has been made on the cvatuation of cfTcclivc ctasGc
moduti of cracked homogeneous materials [for instance. Dctamerer (11(II. (1975). (1979)], stilt
Hocnig
the work on cracked fiber composites in the light of micromcchanics analysis
rctativcly
longitudinal
rare in the literature. Young’s modulus
of an
Highly
approxima~c trcatmcnts
aligned short-libcr
tibcr-end cracks and a unidirectional Takao (81~1. (1987) and St&f (19&l),
reinforced composite wcakcned b>
fiber composite with broken libers wcrc given b> respectively, in which the singular
nature of the stress
licld in the vicinity of a crack has not been taken into account. The problem atastic moduti of unidirectional
is
of the reduction in
of calculating
fiber composites containing matrix cracks was considcrcd
by Laws (‘I rrl. ( 1983). To
permit a comparatively
simple analytical
formulation,
in the prcscnt paper we
consider the probtcm of evaluating the reduced longitudinal
shtar
rcctionat fiber composite containing
eeneratizcd self-consistent
interfacial
cracks. The
modulus
of a unidi-
scheme is employed to evaluate the effective shear modulus of the composite weakened by those cracks. This method has been applied by Christensen and Lo (1979) to calculate elastic moduti of fiber-reinforced composites with perfect interface, and later extended to imperfect interface conditions
(Hashin.
1990). The self-consistent
analysis
has also been
applied rather frequently to calculate elastic moduti of cracked homogeneous materials. In the two-dimensional crack configuration of the present problem. cracks are assumed to take place along the entire length of the fiber, which may not happen in general. The
15x2
TESC
How
resulting
mixed boundary value problem leads to a system of dual series equations, which
can then be reduced to Fredholm singular
kernel. Although
Fredholm logarithmic
integral equations of the first kind with a logarithmically
great difficulty
is generally to be expected in numerically
integral equations of the first singularity
numerical solution.
solving
kind with a smooth kernel. the presence of the
in the current problem makes the integral equations amenable to
The reduced longitudinal
shear modulus is then calculated by solving
the governing \veakl> singular integral equations. Problems related to fiber-reinforced
composites under longitudinal
studied by many authors (Adams and Doner. and Dollar.
1967: Budiansky
shearing have been
and Carrier.
1953: Steif
19%). The dual series approach adopted in this paper has been used in solving
various miued boundary value problems such as the separation of an inclusion infinite matris
from an
(Keer t’t trl.. 1973) and the bending of cracked beams (Westmann and Yang,
1967) and plates (Keer and Sve. 1970).
FORMULATION
Consider a unidirectional the fibers containing
OF
fiber-reinforced
interfacial
THE
PROBLEXI
composite as shown in Fig. I with some of
cracks. Both the fibers and the matrix
are taken to be
homo_cencous. isotropic and linearly elastic. with shear modulus of G, and G,,. respcctivcly. It is ~~ssumcd that the interface cracks arc randomly locatcd so that the composite material remains transversely
isotropic.
It is further assumtxi
that fibers contain only sin&
but the crack siyc may v;lry from fiber to tibcr. As illustrated crack at the tibcr m:Itris
intcrl’lcc is dclincd
mc;1surcd by the angle r with the halfcrack To dctcrminc the clrcctivc longitudin;ll
by the angle
/I. and the extent of the crack is
Icngth c = W. whcrc (I is the fihcr radius. shear modulus G,. ;I longitudinal
r2, = T,, is applictl to the composite matcri;~l ;IS illustrated that of
lhc
longitudinal
:tnti-pl;tnc
strain deformation.
cracks.
in Fig. 2. the location of the
in Fig.
and the only nonvanishing
shear stress
I. The problclm
is
displacement is the
component I(‘. It follows that
(1) whcrc f2, and rT2, arc rcspcctivcly the average stress and strain in the composite. Using the avcragc stress thcorcm. it can bc shown that f2, = r,,. Applying the avcragc strain theorem, fJ, can bc cxprcsscd in terms of the libcr and matrix strain avcragcs Z>, and Zyl. respcctivcly. and the clisplaccmcnt jump [II.] across the intcrfnce cracks as
Shear modulus of umdirectionul
1583
fiber composites
with
(3)
whcrc C, is lhc libcr volume fraction. /I is the conipositc spccimcn cross-section dcnotcs the contour outu.;lrd
of all the cracked intcrlilccs.
arca, SC
t12 dcnotcs the .v,-component
normal of the contour and ds is the clcmcnt of arc Icngth of the contour.
the stress
strain relations for the fiber and the matrix matcri;lls
and eliminating
of the Using
the matrix
avcr;igc stress one linds that
and it follows from (I) that
(5)
Equation
(5) constitutes
shear modulus Evaluation
the basic equation for calculating the ctTcctivc longitudinal
of the fiber composite containing intcrfacial
of the average fibcr stress
done by utilizing
cracks under consideration.
and the intcrfacc displacement
the generalized self-consistent
schcmc. The underlying
integral
in (5) is
assumption of the
gcneralizcd self-consistent scheme in the prcscnt context is that the average state of stress and strain in any individual fiber is estimated by embudding ;I composite cylinder, consisting ofa fiber with radius (I and a concentric matrix shell with radius h such that the fiber volume content in the composite cylinder is the same as the gross composite material, in an infinite homogeneous and transversely
isotropic
material having the effective longitudinal
modulus G, yet to be dctcrmincd, and n longitudinal
shear stress
T,,
shear
is applied at infinity.
WC introduce local crack coordinates (s,._T~, .u,) as opposed to the global coordinates (.\I,. .f2. .C>)with s3 = 1 ,, along with the corresponding polar coordinates (r. 0). It is defined
1584
Fig. 3. Generalized self-consistent
with
respect to a single crack
as illustrated
center of the crack arc. It is convenient system. Hence a three-phase
in Fig. 3 with
to consider
boundary
fiber stress and the interface
displacement
over all the cracked
has to bc solved many times according modulus
iterative
procedure
The displacement
function
needs to be solved to obtain
integral
in (5). Since the average intcrfaccs.
this boundary
the average
is taken
self-consistent
model,
shear stress components
where
G denotes
matrix
and the etTective material
The following
the shear modulus.
satisfying
Laplace’s
equation
boundary
condition
?r
(6)
given by
(Ill-
’
T,,: =
The above
provided
an
to solve (5) for G,.
w(r. 0) is harmonic,
r,: = G
over
value problem
v2w = 0 with corresponding
the
sizes. Since the yet unknown
itself of the three-phase
required
passing through
in the local polar coordinate
to cracks with different
G, enters the solution is apparently
the s,-axis
the problem
value problem
all the fibers and the integral cffcctive
scheme model.
G
I >,Ir (7U
expressions
that G is replaced
at infinity
is obtained
are valid
for the fiber,
the
by Gi, G,,, and G, respectively.
with
refcrencc
to the local polar
coordinates r,: = T,, cos /I sin U + r,, sin /I cos 0, The solution
to Laplace’s
equation
(6) is expressed
where d”(r.
U). s = I, 2. are odd and even functions written can be components
stress T&r. 0) = r$‘(r.
0) +$‘(r,
0). Denoting
CIc r c h and r > b by superscripts
the solutions
r -+ -z.
(8)
as nfr. 0) = d”(r.
0) + d”(r,
U).
of 0. respectively. The corresponding t,:r, 0) = r!i’(r. 0) + rji’(r, 0). as in the three
f, m and e respectively,
regions
of 0 c r c a.
we have
(9)
(10)
Shear modulus of unidirectional D”’
w(lb
fiber composites
to cos /I
=.I
n f
sin no+ -rsin G
1585
8
and ,(.IZN =
AL2’Scos n6
f
(12)
n-n
,l.i h
(-12’
-
_
B”‘f n
+ L
cos
ne
f
(13)
(14)
Due to the symmetries. only the region of 0 < 8 < R need be considered. By using (7). the radial shear stress components can be expressed as
(15)
(16)
(17) and
~!f”’= Gf i n.4!,“f - ’ n-
cos nO
I
cosntl
(18)
(19)
D’2’
r:f’= = G,
2 -n-+cosnB+r,sinflcostI.
n- I
Similar expressions may be written for the corresponding tangential components. The solutions are required to satisfy the boundary conditions at r = a
(20) shear stress
(21) (22)
(23) and the boundary conditions at r = h
(24)
1586
The boundary conditions are mixed. and they lead to the following dual series equations
i
nrr”
’ .4b” sin
n1f
’ A;,z’cos no = 0, 0 ,< u 6
n= /
nO = 0.
0 G 0 < x
(36)
and
c “7
z
(78)
I
(2%
whcrc
i =
G,
(30)
G,,,
(31)
(33)
F,(U) =
F?(U) =
SOLUTIOSS
r,, cos p
4
usin 0
(33)
4 r,, sin /j -_. ._____._._- _._ ~_.__ u cos II. I -&-(I -G)V, G,,,
(34)
-‘-.
~
I +G--(1
Of: THE
-G)Pl
DUAL
.---
G,"
SERIES
EQUATIONS
We now proceed to construct the solutions for the dual scrics equations (X)-(27)
and
(38)-(20). For this purpose, let If, (0) and H:(U) dcnotc the ~lnti-sytnmetric and symmetric pnrts of the shear traction along tfx uncrxked portion of the interface. It follows that
(35)
The Fourier coeflicicnts 4,”
and A!,” are given by
Sheur modulus of undirect~onzl
tiber composites
1587
(37)
(35)
and 5
H,(4) d4 = 0.
Substituting integation
(39)
(37) and (38) into (27) and (29). respectively. and changing the order of
and summation.
we arrive at the following
Frcdholm
integral equations of the
(43)
(44)
(45)
(46)
The kernels of the integral equations (40) and (-!I) contain ;! logarithmic
singularity
To see this. Ict (43) and (44) be rewritten as
KI;,(U,(b) = (I +
j.),,$, 1, sin rr0 sin nrb+Zi(
I -G’)
_$, j-yc
1; sin no sin f@
r (47)
(48)
How
I %Y
Since L;- < I. the second series in (17) and
summed
TESG
(48) are bounded
while the first ones can be
exactly to yield I
i - sin nU sin ncj = jlog “=I ‘7
$ I cos nocos I?4
t,-,= I?
Hence (49) and (50) shou
= - jlog
sin~JJ1-jlog/sin~l
(49)
?sinf~~-~log~2sin~/
(50)
that the kernel functions
become logarithmically
0 = (f,. Note that the functions /;((I) (s = I. 2) on the right-hand-sides well-lxhavcd.
and consequently the solutions
possess loparithmicall~
singular
kernels,
singular
at
of (40) and (41 ) art’
to the integral equations (40) and (4 I ). ivhich
have ;I squaw-root
singularity
at (I = x (Tuck.
19SO). Using (JO) 2nd (50). we \vrite (40) and (-II) as follows:
whcrc A:;
= ( I t i)(/(r,,
C;,,,)sin
/hi:,:'. and
(53)
+
+ 4i.(l
-G)
1+i.
g, (0) = ___(I +L)[I
!/ 2(0 1 =
cz “_,
4 ..-__._+G-(I -G)Vr] 4
(l+i)[l+e-(l-G,V,-]
v; I+&(I-G)Iq
! cos fze cos fIa n
sino
cos 0.
The coctlicicnt :I:,” is dctcrrnincd by the auxiliary equation (41). Hcncc the solution of the dual scrims equations (26).-(27) and (X)-(29) the solution
of the Frcdholm
(54)
(5%
(56)
is rcduwd to
intogral equations (51) and (52).
C~4LCULATIOS OF TfIE Et-FEC-IIVEMODULUS By applying the avtxagc stress theorem. the avcragc fiber stress in (5) can be written as
Shear modulus of unidirectional
fiber composites
I -I T23 = l-,X? ds A r I s,,+s,
I589
(57)
where Ar is the total cross area of all fibers, S, is the contour of all the interfaces without cracks and T3 is the shear traction at the interfaces. Let us assume that for simplicity all fibers have equal radius a. It follows that Ar = iVfrca’, where Nr is the number of fibers. Following the procedure of the generalized self-consistent scheme. the integral over the untracked interfaces can be easily calculated. The result is
Tj.\‘:ds=(I-n,)
4). (L+I)(I+G)+(i.-I)(l-G)c;r”
(58)
where II, = N,/N( is the interface crack number fraction. NCbeing the number of fibers that contain interface cracks. In terms of the local polar coordinates we have from (57) and (58)
(59)
Notice that as before we do not write the explicit dependence of H.,(O) or A,(o) (s = 1.2) on 2. For convenience we will continue to do so when there is no danger of confusion. As discussed earlier, we postulate that the interface cracks are randomly located so that the angle /I varies from 0 to 2~. In addition, we introduce a distribution functionf(z) such that the fraction of cracks whose half crack lengths have values between c and c+dc where c = XI is given by f(r) dr. Thus, the summation in (60) can be replaced by an integration as follows : -I
$=(I
-I?,)
,--
(,.+ I)(1 +C)+(i.-
I
2:
4;.
I)(1 -CR
+2”’
I, ‘(r)dr
where r,a ,< c < a?a gives the range of the interface crack length. Hcncc WC‘have
(62)
How
1590
TEX
In the local polar coordinates. by using the fact that Vr = iV&/A the integral of displacement jump at the cracked interfaces I- L>in (3) can be written in the form
(63)
[by]sin (8 + /?) db,
where [bc] = @(II. 0) - LV’(U,0). It can be shown that the following expression is true :
[,,.I
=
._..__._-!___ :?(I
I +G-(I
-G)Z;
G,,
sin (0+/j)
Inserting (64) into (63). using (37) and (38). and after some manipulation the following
we obtain
result :
As discussed bcforc the summation ;~li~rcmcntioncd distribution
in (65) can be replaced by an integration
function j(r).
After carrying out the integration
through the with rcspsct
to /I, wc obtain
fi,((b)AI,(O,
Thus the etrective longitudinal
4) sin 0 dU dg-2
shear modulus is calculated via the solution of the equation
(67)
NUMERICAL It is well known
very ditiicult
SOLUTIONS
that in general a Fredhom
to solve numerically.
Indeed
prcfcrcnce for reducing dual series equations
ASD
RESULTS
integral equation of the first kind can bc
this fact might
partly
account
to an integral equation
for the past
of the second kind
(Westman and Yang. 1967 ; Keer and Sve. 1970; Keer ef trl., 1973). An effective technique for numerically solving singular integral equations of the first kind with a dominant Cauchy type singularity, which arc frequently encountered in fracture mechanics analysis, has been proposed by Erdogan and Gupta (1972). but no general numerical treatment of the type of the integral equation under present consideration appears to exist in the literature. Nonetheless. the logarithmic singularity present in the current problem makes the integral equations (51) and (52) suitable for a numerical solution that will bc dcscribcd hcrc.
Shear modulus of unidirectional
First note that the solutions To incorporate s = I,‘.
such singularity
contain
the square-root
in the solutions.
1591
fiber cornparks
singularity
we introduce
at the end point 4 = z.
bounded
functions
h,(4),
such that
To solve the integral
equations
[X n] into N subintervals and auxiliary
equation
(51) and (52) in combination with (42) we divide the interval (68) into (51). (52) [&,_ ,, $,I. j = 1.2,. . . . N and by substituting
(42). we have
(69)
(70)
(71)
Then
WC approximate
/r,(tfi) by a step function.
[&, _ , , (6,] by :I constant
11:” (s = I, 2). which
i.e. replace
it within
the jth
suhintcrval
gives
(73
(73)
(74)
If we now evaluate
the integrals
carry out the integration
in (72) and (73) at the mid-point
in (74) we arrive
i=lT
at the following
* -, . . . ,
of each subinterval
and
systems of linear equations
N
whcrc
(78)
(79)
1592
HVSG TESG
(80)
gl” = S$(@,)
(8’) s=
1.2.
A modified version of the so-called Chebyshev mesh is used. giving
b-f
4, = n-(n-I)COS
j=O.l
2.v *
’ . . IV. , _.
(52)
It has the feature of allocating more mesh points near the crack tip 0 = 2, which is obviously desirable in view of the expected stress singularity
there.
The functions K,(f),. 4) in (78) become unbounded at (b = 0,. thus conventional rature formulae
will not yield sufficient accuracy. To circumvent
the kernels (53) and (54) in the following
such dificulty.
quad-
we write
manner
s=I
.-
whcrc
(84)
’
4i.(l-G’)
+
R($(U,fj)
~‘IYW)
= --log
C
I +i
-“!
--
na, 1+e-(I-G)v;
’ sin nO sin nfb n
= -log(U+c~~-logW-~~-log~2P-u-$Jl
sin (O+ 4)/Z (2n_u_#))(o+(b),2
/ -log/
4i.( 1 -G) +
1 +i.
(85)
(86)
/
“;;:,;;2 V;! % 1 1+G(l--G)V; “_,
! cos nU cos n4. n
(87)
It follows that
(88)
s =
1.2;
j=17
._....,i
v.
The first integrals in (88) can bc intcgratcd cxnctly and since K’,“(U.cb) is well behaved, through a change of variable II = W/c/~-r. the second integrals may be calculated by any numerical method, for instance. Simpson’s rule is found to be quite etfcctive. For the purpose of illustration. in the subsequent numerical calculations we consider a simple situation when the interface crack length is uniformly distributed between 0 and 2nc1with the distribution function given by
IS93
Shear modulus of unidirectional fiber composites
\
V,Eo.SS
0.5 -
OL 0
L 0.6
I 0.4
I 0.2
I 0.8
ne Fig. 4. Effective longitudinal shear modulus vs fraction of fibers containing interface cracks. v, = 0.55.
f(l)=;.
(89)
O
Calculations of the effective shear modulus were carried out for various interface crack number ~r~ictions, ratios of ~ber-rn~~trix moduli and fiber volume contents. The bisection method was found to bc quite eKcctivc in solving tqn (671 for G,. The results arc presented in Figs 4--R. Figure
4 shows the variation of G, with intcrfacc crack number fraction n, for various
ratios GJG,,. The results indicate th;tt the ri\tc of reduction in longitudinal incrcascs with the incrcnsc of GJG,,. crack number fraction
for various
Figures
valurs of C’(. Note that for a given 11,. the ilbsoiutc
number of fibers that contain intcrfxc liuction.
shear stilrncss
5-6 show the variation of G, with intcrfxc
Figures 7-8 show the variation
cracks increases with the incrcasc of fiber volume of G, ivith tibcr volume fraction for various tt,.
The results presented in Figs 4-8 indicate that in the cast when all fibers contain interface cracks, i.e. II, = I, the composite loses ail the longitudinal by the fibers. In int~rprcting
stiITncss reinforc~Ri~nt
provided
these results. one should notice that in general. the number of
interfacial cracks in the composite may well depend on the Ievet of appiied load as well as fiber volume content. and the crack fcngth is likely to obey more complicated distribution
0.7 ”
0
G,/Gr,, -10.0
0.2
0.6
0.4
0.8
1
nc Fig. 5. Elective longitudinal shear modulus vs fraction of fibers containing interface cracks. G,;G, = 10.0.
HONC TEW
1594
3.6
1.6
0.4
0.6
0.8
nc Fig. 6. EtTcctive longitudmal
shear modulus
vs fraction
of libers wntnining
interface
G, G,,, = X.0.
0’
I
0
I
0.2
1
I
0.4
0.6
0.8
“I
,,,, G,/G,
#’
=20.0
perfect bonding
01 0
0.2
0.6
0.4
”f
0.8
cracks.
Shear modulus of unidirectional
functions
(e.g.. Gaussian distribution)
fiber composites
than the simple uniform
distribution
1595
function
used
here for numerical examples.
REFERENCES .AdJms. D. F. and Doncr. R. (IYh7). Longitudinal shear loaiing of a unidirecttonal composite. /. Conrp. .Il~rur. I. 4 17. Budlansk!. B. and Carrier. G. F. t I9S-t).High stresses in stlfT-liber composites. 1. .-I&. .\Itv/t. 51. 733.-735. Chrlstcnxn. R. i%l. and Lo. K. H (lY7Y). Solutions for effective shear propertics of three-phase sphere and cylinder mod&. J. .\/dr. Ph,rr. .Sdi~(s 27. 3 15%??O. Comninou. %1. (IWO). An o\criieu of intcrf.ice cracks. &~qnq fiucr. .\lc*ch. 37. 197-20X. Drlameter. W. R.. Hcrrm:lnn. G. and Barnctt. D. Xl ( IY75). Weakening ofan cI;Islic solid b) a rcct,mgular array of cracks. J Appl. .llcdr. 42. 74 X0. Erdogan. F. 2nd Cupta. G. D. (lY72). On numcrlcal solution of singular Integral equatwns. Q. .4ppl. .I/crrh. 30. 5?! 547.
H.wnlg. A ( lY7Y) Elastic m~xluli of non-randomI> cracked body. In/. J. Sdrtlr S/r~rc/ttrc~IS. I?7 154. Kxr. L. hl.. Dundurx. J. and Ki;lttlkomol. K. ( IUi?). Separation of ;I smooth Inclusion from ;L matrix. ffrr. J. &uc/rru.kr I I. 1221 I:??. liccr. L. hl and SW. C. ( 1070). On the bending of cracked plates. hrr. J. Sditbv Sfrucfrrr~.v 6, I545 1550. L.Iu~. N . Wcwk. G J. Lund tIcja7i. M. (lW3). Stllfness chanscs in unidirccttonal cumpusircs caused hy crack ‘(yFtcm’i. ~/C~C~/I. .\/lr/c,r. 2, II!? 137.
OF
HOSG TENG Department
of Theoretical and Applied Mechanics. Univerzity at UrbanaChampaIgn. Urbana. IL 61801. U.S.A.
of Illinois
Abstract-The problem of evaluating the elTective longitudinal shear modulus of a unidirectional tiber composite containing fiber-matrix interfacial cracks is considered. The generalized self-consistent scheme is employed in the formulation of the problem. The resulting mixed boundary value problem leads to a system of’dual series equations. which can then be reduced to Fredholm integral equations of the first kind wirh a logarithmically singular kernel. The reduced longitudinal shear modulus is calculated by solving the governing weakly singular integral equations.
INTRODUCTION
The presence of many cracks in a composite material may cause reduction in its stiffness. thus degrading the integrity
of the material.
The problem of analysing stiffness
reduction due to the presence of cracks. like other
crack probtcms for composite materials.
may bc attacked by
two
fundamcntalty
approaches. One is fhe macromcchanics approach that approximates positc materials
as
homogeneous but anisotropic
ditTcrcnr
hctcrogcncous com-
media. The olhcr is the micromcchanics
approach which incvitabty tcads to the trcatmcnt of cracks in dissimilar matcriats. A large number of solutions exist in the tcchnicat titcraturc for dissimilar matcriats containing cracks
of
various locations and orientations.
As it is welt rccognizcd that cracks arc more
tikcly to dcvctop at intcrfaccs belwccn two dilrcrcnt
constituent
materials of a composite
medium. a considcrabtc amount of work has been dcvotcd to the analysis cracks [SW Comninou investigations
(1990)
for rcvicw and further
of interfacial
rcfcronccs]. Howcvcr,
most of the
wcrc concerned with cracks near or at an intcrfacc of IWO semi-inlinitc
rcndcring no direct applications to the probtcm of stilTncss
media
reduction caused by interface
cracks prcscnl in composite materials. Although
a number of studies has been made on the cvatuation of cfTcclivc ctasGc
moduti of cracked homogeneous materials [for instance. Dctamerer (11(II. (1975). (1979)], stilt
Hocnig
the work on cracked fiber composites in the light of micromcchanics analysis
rctativcly
longitudinal
rare in the literature. Young’s modulus
of an
Highly
approxima~c trcatmcnts
aligned short-libcr
tibcr-end cracks and a unidirectional Takao (81~1. (1987) and St&f (19&l),
reinforced composite wcakcned b>
fiber composite with broken libers wcrc given b> respectively, in which the singular
nature of the stress
licld in the vicinity of a crack has not been taken into account. The problem atastic moduti of unidirectional
is
of the reduction in
of calculating
fiber composites containing matrix cracks was considcrcd
by Laws (‘I rrl. ( 1983). To
permit a comparatively
simple analytical
formulation,
in the prcscnt paper we
consider the probtcm of evaluating the reduced longitudinal
shtar
rcctionat fiber composite containing
eeneratizcd self-consistent
interfacial
cracks. The
modulus
of a unidi-
scheme is employed to evaluate the effective shear modulus of the composite weakened by those cracks. This method has been applied by Christensen and Lo (1979) to calculate elastic moduti of fiber-reinforced composites with perfect interface, and later extended to imperfect interface conditions
(Hashin.
1990). The self-consistent
analysis
has also been
applied rather frequently to calculate elastic moduti of cracked homogeneous materials. In the two-dimensional crack configuration of the present problem. cracks are assumed to take place along the entire length of the fiber, which may not happen in general. The
15x2
TESC
How
resulting
mixed boundary value problem leads to a system of dual series equations, which
can then be reduced to Fredholm singular
kernel. Although
Fredholm logarithmic
integral equations of the first kind with a logarithmically
great difficulty
is generally to be expected in numerically
integral equations of the first singularity
numerical solution.
solving
kind with a smooth kernel. the presence of the
in the current problem makes the integral equations amenable to
The reduced longitudinal
shear modulus is then calculated by solving
the governing \veakl> singular integral equations. Problems related to fiber-reinforced
composites under longitudinal
studied by many authors (Adams and Doner. and Dollar.
1967: Budiansky
shearing have been
and Carrier.
1953: Steif
19%). The dual series approach adopted in this paper has been used in solving
various miued boundary value problems such as the separation of an inclusion infinite matris
from an
(Keer t’t trl.. 1973) and the bending of cracked beams (Westmann and Yang,
1967) and plates (Keer and Sve. 1970).
FORMULATION
Consider a unidirectional the fibers containing
OF
fiber-reinforced
interfacial
THE
PROBLEXI
composite as shown in Fig. I with some of
cracks. Both the fibers and the matrix
are taken to be
homo_cencous. isotropic and linearly elastic. with shear modulus of G, and G,,. respcctivcly. It is ~~ssumcd that the interface cracks arc randomly locatcd so that the composite material remains transversely
isotropic.
It is further assumtxi
that fibers contain only sin&
but the crack siyc may v;lry from fiber to tibcr. As illustrated crack at the tibcr m:Itris
intcrl’lcc is dclincd
mc;1surcd by the angle r with the halfcrack To dctcrminc the clrcctivc longitudin;ll
by the angle
/I. and the extent of the crack is
Icngth c = W. whcrc (I is the fihcr radius. shear modulus G,. ;I longitudinal
r2, = T,, is applictl to the composite matcri;~l ;IS illustrated that of
lhc
longitudinal
:tnti-pl;tnc
strain deformation.
cracks.
in Fig. 2. the location of the
in Fig.
and the only nonvanishing
shear stress
I. The problclm
is
displacement is the
component I(‘. It follows that
(1) whcrc f2, and rT2, arc rcspcctivcly the average stress and strain in the composite. Using the avcragc stress thcorcm. it can bc shown that f2, = r,,. Applying the avcragc strain theorem, fJ, can bc cxprcsscd in terms of the libcr and matrix strain avcragcs Z>, and Zyl. respcctivcly. and the clisplaccmcnt jump [II.] across the intcrfnce cracks as
Shear modulus of umdirectionul
1583
fiber composites
with
(3)
whcrc C, is lhc libcr volume fraction. /I is the conipositc spccimcn cross-section dcnotcs the contour outu.;lrd
of all the cracked intcrlilccs.
arca, SC
t12 dcnotcs the .v,-component
normal of the contour and ds is the clcmcnt of arc Icngth of the contour.
the stress
strain relations for the fiber and the matrix matcri;lls
and eliminating
of the Using
the matrix
avcr;igc stress one linds that
and it follows from (I) that
(5)
Equation
(5) constitutes
shear modulus Evaluation
the basic equation for calculating the ctTcctivc longitudinal
of the fiber composite containing intcrfacial
of the average fibcr stress
done by utilizing
cracks under consideration.
and the intcrfacc displacement
the generalized self-consistent
schcmc. The underlying
integral
in (5) is
assumption of the
gcneralizcd self-consistent scheme in the prcscnt context is that the average state of stress and strain in any individual fiber is estimated by embudding ;I composite cylinder, consisting ofa fiber with radius (I and a concentric matrix shell with radius h such that the fiber volume content in the composite cylinder is the same as the gross composite material, in an infinite homogeneous and transversely
isotropic
material having the effective longitudinal
modulus G, yet to be dctcrmincd, and n longitudinal
shear stress
T,,
shear
is applied at infinity.
WC introduce local crack coordinates (s,._T~, .u,) as opposed to the global coordinates (.\I,. .f2. .C>)with s3 = 1 ,, along with the corresponding polar coordinates (r. 0). It is defined
1584
Fig. 3. Generalized self-consistent
with
respect to a single crack
as illustrated
center of the crack arc. It is convenient system. Hence a three-phase
in Fig. 3 with
to consider
boundary
fiber stress and the interface
displacement
over all the cracked
has to bc solved many times according modulus
iterative
procedure
The displacement
function
needs to be solved to obtain
integral
in (5). Since the average intcrfaccs.
this boundary
the average
is taken
self-consistent
model,
shear stress components
where
G denotes
matrix
and the etTective material
The following
the shear modulus.
satisfying
Laplace’s
equation
boundary
condition
?r
(6)
given by
(Ill-
’
T,,: =
The above
provided
an
to solve (5) for G,.
w(r. 0) is harmonic,
r,: = G
over
value problem
v2w = 0 with corresponding
the
sizes. Since the yet unknown
itself of the three-phase
required
passing through
in the local polar coordinate
to cracks with different
G, enters the solution is apparently
the s,-axis
the problem
value problem
all the fibers and the integral cffcctive
scheme model.
G
I >,Ir (7U
expressions
that G is replaced
at infinity
is obtained
are valid
for the fiber,
the
by Gi, G,,, and G, respectively.
with
refcrencc
to the local polar
coordinates r,: = T,, cos /I sin U + r,, sin /I cos 0, The solution
to Laplace’s
equation
(6) is expressed
where d”(r.
U). s = I, 2. are odd and even functions written can be components
stress T&r. 0) = r$‘(r.
0) +$‘(r,
0). Denoting
CIc r c h and r > b by superscripts
the solutions
r -+ -z.
(8)
as nfr. 0) = d”(r.
0) + d”(r,
U).
of 0. respectively. The corresponding t,:r, 0) = r!i’(r. 0) + rji’(r, 0). as in the three
f, m and e respectively,
regions
of 0 c r c a.
we have
(9)
(10)
Shear modulus of unidirectional D”’
w(lb
fiber composites
to cos /I
=.I
n f
sin no+ -rsin G
1585
8
and ,(.IZN =
AL2’Scos n6
f
(12)
n-n
,l.i h
(-12’
-
_
B”‘f n
+ L
cos
ne
f
(13)
(14)
Due to the symmetries. only the region of 0 < 8 < R need be considered. By using (7). the radial shear stress components can be expressed as
(15)
(16)
(17) and
~!f”’= Gf i n.4!,“f - ’ n-
cos nO
I
cosntl
(18)
(19)
D’2’
r:f’= = G,
2 -n-+cosnB+r,sinflcostI.
n- I
Similar expressions may be written for the corresponding tangential components. The solutions are required to satisfy the boundary conditions at r = a
(20) shear stress
(21) (22)
(23) and the boundary conditions at r = h
(24)
1586
The boundary conditions are mixed. and they lead to the following dual series equations
i
nrr”
’ .4b” sin
n1f
’ A;,z’cos no = 0, 0 ,< u 6
n= /
nO = 0.
0 G 0 < x
(36)
and
c “7
z
(78)
I
(2%
whcrc
i =
G,
(30)
G,,,
(31)
(33)
F,(U) =
F?(U) =
SOLUTIOSS
r,, cos p
4
usin 0
(33)
4 r,, sin /j -_. ._____._._- _._ ~_.__ u cos II. I -&-(I -G)V, G,,,
(34)
-‘-.
~
I +G--(1
Of: THE
-G)Pl
DUAL
.---
G,"
SERIES
EQUATIONS
We now proceed to construct the solutions for the dual scrics equations (X)-(27)
and
(38)-(20). For this purpose, let If, (0) and H:(U) dcnotc the ~lnti-sytnmetric and symmetric pnrts of the shear traction along tfx uncrxked portion of the interface. It follows that
(35)
The Fourier coeflicicnts 4,”
and A!,” are given by
Sheur modulus of undirect~onzl
tiber composites
1587
(37)
(35)
and 5
H,(4) d4 = 0.
Substituting integation
(39)
(37) and (38) into (27) and (29). respectively. and changing the order of
and summation.
we arrive at the following
Frcdholm
integral equations of the
(43)
(44)
(45)
(46)
The kernels of the integral equations (40) and (-!I) contain ;! logarithmic
singularity
To see this. Ict (43) and (44) be rewritten as
KI;,(U,(b) = (I +
j.),,$, 1, sin rr0 sin nrb+Zi(
I -G’)
_$, j-yc
1; sin no sin f@
r (47)
(48)
How
I %Y
Since L;- < I. the second series in (17) and
summed
TESG
(48) are bounded
while the first ones can be
exactly to yield I
i - sin nU sin ncj = jlog “=I ‘7
$ I cos nocos I?4
t,-,= I?
Hence (49) and (50) shou
= - jlog
sin~JJ1-jlog/sin~l
(49)
?sinf~~-~log~2sin~/
(50)
that the kernel functions
become logarithmically
0 = (f,. Note that the functions /;((I) (s = I. 2) on the right-hand-sides well-lxhavcd.
and consequently the solutions
possess loparithmicall~
singular
kernels,
singular
at
of (40) and (41 ) art’
to the integral equations (40) and (4 I ). ivhich
have ;I squaw-root
singularity
at (I = x (Tuck.
19SO). Using (JO) 2nd (50). we \vrite (40) and (-II) as follows:
whcrc A:;
= ( I t i)(/(r,,
C;,,,)sin
/hi:,:'. and
(53)
+
+ 4i.(l
-G)
1+i.
g, (0) = ___(I +L)[I
!/ 2(0 1 =
cz “_,
4 ..-__._+G-(I -G)Vr] 4
(l+i)[l+e-(l-G,V,-]
v; I+&(I-G)Iq
! cos fze cos fIa n
sino
cos 0.
The coctlicicnt :I:,” is dctcrrnincd by the auxiliary equation (41). Hcncc the solution of the dual scrims equations (26).-(27) and (X)-(29) the solution
of the Frcdholm
(54)
(5%
(56)
is rcduwd to
intogral equations (51) and (52).
C~4LCULATIOS OF TfIE Et-FEC-IIVEMODULUS By applying the avtxagc stress theorem. the avcragc fiber stress in (5) can be written as
Shear modulus of unidirectional
fiber composites
I -I T23 = l-,X? ds A r I s,,+s,
I589
(57)
where Ar is the total cross area of all fibers, S, is the contour of all the interfaces without cracks and T3 is the shear traction at the interfaces. Let us assume that for simplicity all fibers have equal radius a. It follows that Ar = iVfrca’, where Nr is the number of fibers. Following the procedure of the generalized self-consistent scheme. the integral over the untracked interfaces can be easily calculated. The result is
Tj.\‘:ds=(I-n,)
4). (L+I)(I+G)+(i.-I)(l-G)c;r”
(58)
where II, = N,/N( is the interface crack number fraction. NCbeing the number of fibers that contain interface cracks. In terms of the local polar coordinates we have from (57) and (58)
(59)
Notice that as before we do not write the explicit dependence of H.,(O) or A,(o) (s = 1.2) on 2. For convenience we will continue to do so when there is no danger of confusion. As discussed earlier, we postulate that the interface cracks are randomly located so that the angle /I varies from 0 to 2~. In addition, we introduce a distribution functionf(z) such that the fraction of cracks whose half crack lengths have values between c and c+dc where c = XI is given by f(r) dr. Thus, the summation in (60) can be replaced by an integration as follows : -I
$=(I
-I?,)
,--
(,.+ I)(1 +C)+(i.-
I
2:
4;.
I)(1 -CR
+2”’
I, ‘(r)dr
where r,a ,< c < a?a gives the range of the interface crack length. Hcncc WC‘have
(62)
How
1590
TEX
In the local polar coordinates. by using the fact that Vr = iV&/A the integral of displacement jump at the cracked interfaces I- L>in (3) can be written in the form
(63)
[by]sin (8 + /?) db,
where [bc] = @(II. 0) - LV’(U,0). It can be shown that the following expression is true :
[,,.I
=
._..__._-!___ :?(I
I +G-(I
-G)Z;
G,,
sin (0+/j)
Inserting (64) into (63). using (37) and (38). and after some manipulation the following
we obtain
result :
As discussed bcforc the summation ;~li~rcmcntioncd distribution
in (65) can be replaced by an integration
function j(r).
After carrying out the integration
through the with rcspsct
to /I, wc obtain
fi,((b)AI,(O,
Thus the etrective longitudinal
4) sin 0 dU dg-2
shear modulus is calculated via the solution of the equation
(67)
NUMERICAL It is well known
very ditiicult
SOLUTIONS
that in general a Fredhom
to solve numerically.
Indeed
prcfcrcnce for reducing dual series equations
ASD
RESULTS
integral equation of the first kind can bc
this fact might
partly
account
to an integral equation
for the past
of the second kind
(Westman and Yang. 1967 ; Keer and Sve. 1970; Keer ef trl., 1973). An effective technique for numerically solving singular integral equations of the first kind with a dominant Cauchy type singularity, which arc frequently encountered in fracture mechanics analysis, has been proposed by Erdogan and Gupta (1972). but no general numerical treatment of the type of the integral equation under present consideration appears to exist in the literature. Nonetheless. the logarithmic singularity present in the current problem makes the integral equations (51) and (52) suitable for a numerical solution that will bc dcscribcd hcrc.
Shear modulus of unidirectional
First note that the solutions To incorporate s = I,‘.
such singularity
contain
the square-root
in the solutions.
1591
fiber cornparks
singularity
we introduce
at the end point 4 = z.
bounded
functions
h,(4),
such that
To solve the integral
equations
[X n] into N subintervals and auxiliary
equation
(51) and (52) in combination with (42) we divide the interval (68) into (51). (52) [&,_ ,, $,I. j = 1.2,. . . . N and by substituting
(42). we have
(69)
(70)
(71)
Then
WC approximate
/r,(tfi) by a step function.
[&, _ , , (6,] by :I constant
11:” (s = I, 2). which
i.e. replace
it within
the jth
suhintcrval
gives
(73
(73)
(74)
If we now evaluate
the integrals
carry out the integration
in (72) and (73) at the mid-point
in (74) we arrive
i=lT
at the following
* -, . . . ,
of each subinterval
and
systems of linear equations
N
whcrc
(78)
(79)
1592
HVSG TESG
(80)
gl” = S$(@,)
(8’) s=
1.2.
A modified version of the so-called Chebyshev mesh is used. giving
b-f
4, = n-(n-I)COS
j=O.l
2.v *
’ . . IV. , _.
(52)
It has the feature of allocating more mesh points near the crack tip 0 = 2, which is obviously desirable in view of the expected stress singularity
there.
The functions K,(f),. 4) in (78) become unbounded at (b = 0,. thus conventional rature formulae
will not yield sufficient accuracy. To circumvent
the kernels (53) and (54) in the following
such dificulty.
quad-
we write
manner
s=I
.-
whcrc
(84)
’
4i.(l-G’)
+
R($(U,fj)
~‘IYW)
= --log
C
I +i
-“!
--
na, 1+e-(I-G)v;
’ sin nO sin nfb n
= -log(U+c~~-logW-~~-log~2P-u-$Jl
sin (O+ 4)/Z (2n_u_#))(o+(b),2
/ -log/
4i.( 1 -G) +
1 +i.
(85)
(86)
/
“;;:,;;2 V;! % 1 1+G(l--G)V; “_,
! cos nU cos n4. n
(87)
It follows that
(88)
s =
1.2;
j=17
._....,i
v.
The first integrals in (88) can bc intcgratcd cxnctly and since K’,“(U.cb) is well behaved, through a change of variable II = W/c/~-r. the second integrals may be calculated by any numerical method, for instance. Simpson’s rule is found to be quite etfcctive. For the purpose of illustration. in the subsequent numerical calculations we consider a simple situation when the interface crack length is uniformly distributed between 0 and 2nc1with the distribution function given by
IS93
Shear modulus of unidirectional fiber composites
\
V,Eo.SS
0.5 -
OL 0
L 0.6
I 0.4
I 0.2
I 0.8
ne Fig. 4. Effective longitudinal shear modulus vs fraction of fibers containing interface cracks. v, = 0.55.
f(l)=;.
(89)
O
Calculations of the effective shear modulus were carried out for various interface crack number ~r~ictions, ratios of ~ber-rn~~trix moduli and fiber volume contents. The bisection method was found to bc quite eKcctivc in solving tqn (671 for G,. The results arc presented in Figs 4--R. Figure
4 shows the variation of G, with intcrfacc crack number fraction n, for various
ratios GJG,,. The results indicate th;tt the ri\tc of reduction in longitudinal incrcascs with the incrcnsc of GJG,,. crack number fraction
for various
Figures
valurs of C’(. Note that for a given 11,. the ilbsoiutc
number of fibers that contain intcrfxc liuction.
shear stilrncss
5-6 show the variation of G, with intcrfxc
Figures 7-8 show the variation
cracks increases with the incrcasc of fiber volume of G, ivith tibcr volume fraction for various tt,.
The results presented in Figs 4-8 indicate that in the cast when all fibers contain interface cracks, i.e. II, = I, the composite loses ail the longitudinal by the fibers. In int~rprcting
stiITncss reinforc~Ri~nt
provided
these results. one should notice that in general. the number of
interfacial cracks in the composite may well depend on the Ievet of appiied load as well as fiber volume content. and the crack fcngth is likely to obey more complicated distribution
0.7 ”
0
G,/Gr,, -10.0
0.2
0.6
0.4
0.8
1
nc Fig. 5. Elective longitudinal shear modulus vs fraction of fibers containing interface cracks. G,;G, = 10.0.
HONC TEW
1594
3.6
1.6
0.4
0.6
0.8
nc Fig. 6. EtTcctive longitudmal
shear modulus
vs fraction
of libers wntnining
interface
G, G,,, = X.0.
0’
I
0
I
0.2
1
I
0.4
0.6
0.8
“I
,,,, G,/G,
#’
=20.0
perfect bonding
01 0
0.2
0.6
0.4
”f
0.8
cracks.
Shear modulus of unidirectional
functions
(e.g.. Gaussian distribution)
fiber composites
than the simple uniform
distribution
1595
function
used
here for numerical examples.
REFERENCES .AdJms. D. F. and Doncr. R. (IYh7). Longitudinal shear loaiing of a unidirecttonal composite. /. Conrp. .Il~rur. I. 4 17. Budlansk!. B. and Carrier. G. F. t I9S-t).High stresses in stlfT-liber composites. 1. .-I&. .\Itv/t. 51. 733.-735. Chrlstcnxn. R. i%l. and Lo. K. H (lY7Y). Solutions for effective shear propertics of three-phase sphere and cylinder mod&. J. .\/dr. Ph,rr. .Sdi~(s 27. 3 15%??O. Comninou. %1. (IWO). An o\criieu of intcrf.ice cracks. &~qnq fiucr. .\lc*ch. 37. 197-20X. Drlameter. W. R.. Hcrrm:lnn. G. and Barnctt. D. Xl ( IY75). Weakening ofan cI;Islic solid b) a rcct,mgular array of cracks. J Appl. .llcdr. 42. 74 X0. Erdogan. F. 2nd Cupta. G. D. (lY72). On numcrlcal solution of singular Integral equatwns. Q. .4ppl. .I/crrh. 30. 5?! 547.
H.wnlg. A ( lY7Y) Elastic m~xluli of non-randomI> cracked body. In/. J. Sdrtlr S/r~rc/ttrc~IS. I?7 154. Kxr. L. hl.. Dundurx. J. and Ki;lttlkomol. K. ( IUi?). Separation of ;I smooth Inclusion from ;L matrix. ffrr. J. &uc/rru.kr I I. 1221 I:??. liccr. L. hl and SW. C. ( 1070). On the bending of cracked plates. hrr. J. Sditbv Sfrucfrrr~.v 6, I545 1550. L.Iu~. N . Wcwk. G J. Lund tIcja7i. M. (lW3). Stllfness chanscs in unidirccttonal cumpusircs caused hy crack ‘(yFtcm’i. ~/C~C~/I. .\/lr/c,r. 2, II!? 137.