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A micromechanical analysis of fiber crack propagation in composite materials due to transverse tensile loads
Engineering ~racrure A4echanic.s Vol. 45, No. 6, pp. 813-829, Printed in Great Britain.
1993
0013-7944/93 $6.00 + 0.00 c, 1993 Pergamon Press Ltd.
A MICROMECHANICAL ANALYSIS OF FIBER CRACK PROPAGATION IN COMPOSITE MATERIALS DUE TO TRANSVERSE TENSILE LOADS YONG Department
of Engineering
LI XU and KENNETH
L. REIFSNIDER
Science and Mechanics, Virginia Polytechnic Blacksburg, VA 24061, U.S.A.
Institute
and State University,
Abstract-The
fiber crack propagation in composites due to transverse tensile loads is studied using a micromechanical model and Linear Elastic Fracture Mechanics. To approach the problem, a three domain cylindrical model is introduced to simulate the fiber cracking. The model problem is then solved by the dislocation and singular integral equation techniques. The stress intensity factors of the fiber crack are calculated for various situations. It is found that fiber anisotropy has hardly any effect on the fiber crack “reverse composites” (composites in which fiber is less stiff than the matrix, such as propagation: Nicalon/SiC ceramic composite) virtually eliminate fiber crack propagation.
INTRODUCTION THE BEHAVIOR of a unidirectionally
reinforced composite subjected to transverse tensile loading remains as a particularly significant problem in material technology. This is because of the very low transverse stiffness and strength characteristics of such materials relative to their stiffness and strength in the direction of reinforcement orientation. Since it is often the transverse properties that limit the performance of the composite system, there is a high potential for improvement of behavior through study of this problem. In the study of the composite transverse strengths, a number of investigators have conducted investigations and presented their results [l-8]. A brief review of the historical development of micromechanical analyses, as applied to the problem of transverse behavior of a unidirectional composite, can be found in [l ,5,8]. In general, finite element models are used in those studies, and matrix elastic-plastic behavior is modeled; analyses are continued until the material failure occurs, and transverse strengths are thus estimated. Among those studies, Adams [6] first introduced crack initiation and propagation in the model, i.e. a crack initiated at the fiber-matrix interface, propagated partially around the fiber and then across the matrix. In [7], a special element was developed which allowed shear failure to occur at a predefined stress followed by friction slip. In [8], a special element was developed to enable the interface to be modeled with both shear and tensile failure criteria with a quadratic interaction formula. In this paper, the fiber crack propagation in composites due to transverse tensile loads is studied using a micromechanical model and Linear Elastic Fracture Mechanics (LEFM). These tiny fiber cracks can also be regarded as microcracking formed in the process of manufacturing due to mismatch of fiber and matrix thermal expansion coefficients. The objective of this study is to investigate the effect of fiber volume fraction, fiber anisotropy, and fiber-matrix mechanical property mismatch on the fiber crack propagation. It should also be mentioned that Erdogan and Gupta’s solution of a general inclusion-crack problem [9] also includes some results on inclusion-crack behavior. To approach the problem, a three-domain cylindrical model is introduced to simulate the fiber cracking. The model problem is then solved by the dislocation and singular integral equation techniques. The stress intensity factors of the fiber crack are calculated for various situations. It is found that fiber anisotropy has hardly any effect on the fiber crack propagation; “reverse composites” (composites in which fiber is less stiff than the matrix, such as Nicalon/SiC ceramic composite) virtually eliminate fiber crack propagation.
MODELING
OF A FIBER
CRACKING
To study the behavior of the fiber crack, a cylindrical model Fig. 1, the model consists of three domains, defined as follows. 813
is introduced.
As is shown
in
814
YONG
LI XU and K. L. REIFSNIDER
~o~ui~ I: 0 G r d R, the fiber (radius R), with a radial crack embedded from -u to a (0 < a c R). The fiber is assumed to be transversely isotropic with Young’s moduli Ef and ED. and Poisson’s ratios v, and yD,, where Ef is Young’s modulus in the x-_V plane (see Fig. I), and E,, is Young’s modulus in the z direction (fiber axial direction). v~-is the Poisson’s ratio in the .Y;I-’plane. and vD, is the general Poisson’s ratio (strain in the X-J plane due to stress in z direction). Accordingly. we have the following relation:
where ljfi3, v,.,~are the general Poisson’s ratios (strain in z direction induced by stress in x or J direction, respectively). Domain II: the surrounding matrix, R < Y G h; isotropic material with Young’s modulus E,,, and Poisson’s ratio v,,, with
where yf is the fiber volume fraction of the composite. Doma& III: the composite. h < Y < 1%;transversely isotropic material with material constants determined as follows:
where EC is Young’s modulus in the X-J plane (see Fig. l), and EC3is Young’s modulus in the 3 direction (fiber axial direction). v,, is the Poisson’s ratio in the X-.-J plane, and v,~, is the general Poisson’s ratio (strain in the .u-;v plane due to stress in z direction). Also, we have the following relation: \‘C 3I --=----, “r 13 1’,y (7) E,.?-
E,
E;
Radial Crack Composite
0 0
b) Coordinate system
a) Model of fiber cracking
Fig. 1. Modeling
of fiber cracking.
815
Micromechanical analysis of fiber crack propagation
where vc13, vcz3 are the general Poisson’s ratios (strain in z direction due to stress in x or y direction, respectively). The composite is subjected to a uniform unidirectional traction co at infinity. For plane problems of the transversely isotropic material, when we consider the stress and displacement distributions in the x-y isotropy plane, it can be shown (see Appendix A) that the compatibility equation is still a biha~onic equation. We make the following transformations. For the fiber (domain I), let
Ef_
Ff -- E,$ii ’ EP
,= l
(8)
Then, for the plane strain case, we have the following constitutive equations:
l (Qe- Vf- , CT,,1. tge=Ey_i
(9)
For the composite (domain III), let
EC
EC
EC_,= 1
--
41
v, + -
Ec3
EC& ’ “‘-‘=
(10)
1 _ &vi,
Ec3
Ec3
and also for the plane strain case, we have f&r =
--!(@rr - v,-,Gd EC-,
cet?=&oee-vc-*~,F). ‘
f
It can also be shown that l+v,_,=1+v,Er-r
Er
13-v,-, 3 -=-. 4-r
l-+VC
W)
&
Therefore, /q-r=
E,- ,
J&-t
4
2(1 +v,_,)=m=pfV
pC-J=2(l
EC
+VC_,)==2(1 +vJSPC,
(13)
where pf and ~1~are shear moduli of the fibers and composite in the x-y plane, res~ctively.
FOR~LATIO~
OF THE MODEL
PROBLEM
To approach the model problem, we first seek the corresponding dislocation solution (see Fig. 2). Then, by superposition with the untracked geometry case (see Fig. 3), we reduce the original problem to a perturbation problem in which the geometry and the singufar nature of the fiber crack remain the same, but the only Ioading is the crack surface self-equilibrating pressure. After that, we integrate the dislocation solution, in other words, we use the dislocation solution as a Green’s function to integrate along [-a, a] to generate a singular integral equation. The stress intensity factors of the fiber crack are related to the solution of the singular integral equation and may be solved by Gaussian quadrature to a desired accuracy.
816
YONG
Ll XU and K. L. REIFSNIDER
Matrix
X(x11
Dislocation
Composite
f.~g. 1. The wrrespondlng
(a) Dislocatiort
d~slo~atron
problem.
solrrtiorz
The geometry of the dislocation case is shown in dislocation case, instead of a crack, an edge dislocation in the fiber at r = t, 0 = 0. The details of the dislocation we only give the hoop stress o,,~,(Y, H) in the fiber as
Fig. 2. It is seen that m the corresponding with Burgers’ vector (0, h, . 0) is embedded solution can be found in Appendix B. Here follows:
with K, iwhere co,, together
with unknowns
a) original problem
3 - l’/ , .-.. -. lfrj, :
(1st
c,,?, h,,, and h,,,. can be determined
c) perturbatcon case
b) untracked geometry
Fig. 3. Superposition
of the crack
by the following
problem.
equations:
817
Micromechanical analysis of fiber crack propagation b $ (G - 1)b
-Lb,,*+
2&n
2pL,b
d,, ,
together with unknowns simultaneous equations:
(16)
Cl2 -
2hH
1
4, + 1+
Kr-,
4
1 2)R2 4, - m 2Pf-
(K, + 2)R2
(K,_ t +
Cl2 -
I
by the following
- 2Rd,, =
(K, - 2)R2
1 2)R’ 4, - m 2P,-
1 bo3 = 0; coz+2pC_,b -
c12, d12, c,~, S, and S,,, may be determined
2Rd,, + 2
@_, -
+ 2c02 - f$ = 0
2&
+v
logeR 1
4, - S, 5+lc,_,
t2 --
4
R2
-~ K/_,-
2
1 -lo&R
1
2c +-2bd,,+=O
1 --7Cl2 3.d
(v2)bZd +
_ PC 1
2pc+,b2
I3
_ -c ’ I2 2pI._(b2
I3
I2
%4n
1 2c,2+ (K,+2)b2d 2PllI 2/.4
+s,,=o
-s,,
= 0;
(17)
a,,, and b,, , together with unknowns a,,*, b,2, c,,~, d,,, , c,,~and dn3 (n > 2), can be determined by the following equations: n(1 - n)Rnm2a,, + (n + 1)(2 -n)R”b,,
+ n(n - 1)R”p2a,2 + (n +
+ n(n + 1)R-“-2c,2 n(n - 1)Rnm2a,, + n(n + l)R”b,,
+ (n -
l)(n + 2)R-“d,,2 = -
1 2P,- I
1
nR”-‘%
+ -(I+,2pf_,
1 +-nR”2&
1 -n)R”+‘b,,
---(n 2/&I
m
1
“+‘b,, - $
n-‘a,, + ++‘+“‘)R ,
--
1
m
‘a,,+l(n
,b?.Bn
x(1 + Kf-,)
-Km+
PfLr-,b,.A, x(1 + K,-,)
l)R”f’b,z
2Pln l)R-“+‘d,,=
+K,-
nR”-‘a,,,
_$(n
2Pm?IR-“- IG2 n(n - l)bnm2an2 + (n + l)(n - 2)b”b,, + n(n +
+ n(n - 1)Rpnd,2 =
1
1 -- 2~ ~R-“-‘c,,~ -nR 2Pf- I
/+
- n(n - 1)R”-2a,2 - n(n + l)R”b,, + n(n + 1)R-“-2~,2
--
l)(n - 2)R”b,2
- $
m
(n +
-
b,,C, x(1 + ‘C-t)
1 + K,)R”+‘b,2
-1 _K,)Rm”+‘d,2= m
-
DnbJ n(l + rcf-1)
l)b-‘-2c,2 + (n - l)(n + 2)b-“d,, -n(n+1)b-‘-2c,3-(n-l)(n+2)b-“d,,3=0
n(n - l)b”-2a,2
+ n(n + l)bnb,2 - n(n +
l)b-“-2c,2 - n(n - l)b-“d,,, + n(n +
1)b-‘-2c,3 + n(n - l)b-“d,,
= 0
YONG
818
LI XtJ and K. L. REIFSNIDER
with
where A,, B,,, C+,and D, are functions
of t given by
By superposition with the sofution of the untracked geometry case, which may be found in Appendix C, we can reduce the original crack problem to a perturbation case (see Fig. 3). For the problem under discussion, the boundary conditions for the perturbation case can be written as foifows: T,~~=O,
O<~<.X,
O=O.
CTf,(j(X, 0) = P(X),
-a
U,j(X, 0) = 0.
.^-?r, <
.Y < II,
and
0-n
< x 6 U and
O), and aoof(r. 0) is the fiber hoop stress in the untracked Appendix C). Stnce a crack can be regarded as a pile-up of dislocations, we define
By integrating the dislocation may obtain a singular integral
solution equation
(24)
n<.u<~~,
where p(r> = --a&r,
we
(23)
(25) geometry
case (see
eq. (24).
(14) along --a < x < a, t) = 0. and applying with a Cauchy kernel as follows:
(27) with the Fredholm
kernel
K(X, t) given
K(X, t) = r(12+4_r) 1 c
i
by
2COl+ 6ril, + i [n(n - 1)x”--%,, + (n -t- l)(n + 2)X%,,] ?
where car, d, t ~ a,, and 6,, are functions
of t determined
by eqs (16)-( 18)
,
(28)
Micromechanical analysis of fiber crack propagation
819
For an embedded crack, it follows that a f(t) dt = 0. s --4
(29)
The embedded fiber crack has a square-root singularity, and the functionf(x)
may be written
as
Wf f(x) = [(x + a)(a - x)]‘i2 ’ where F(x) is a bounded function at the crack tips. The stress intensity factors at crack tips x = -a or x = a may be defined according to the conventional definition, and calculated as follows:
k, (-a) = kl (a) = !i?
Jw - aM6m(4 0)
=---_ a+- J-(a) 1++i J;E I
~M~~I~AL
(31)
SOLUTION
The singular equation (27) can be solved using the method introducing the following normalizations: x = up,
where
-a
-l
t=uz,
where
-a
-l
t) = K(P, r),
P(X) = 4(P),
developed
in [lo, 1I]. By
let S(t) = g(r),
f s-
a%
(32)
eqs (27) and (29) become
g(z)
-IT---P
dT
i
+
f
-1
UP, z>sCr)dt =
and
NY1+ Kf-1) 4(P), -lfp
(33)
2&- I
I g(z) dr = 0.
(34)
Sl The function g(z) is singular at 1: = + 1 and, likewise, may be written as
G(z)
(35)
g(r) = (1 _ T2)l:2'
E
e0
d
E,=E_, 1.0
E,/E,=3,
v,~=v,~~=v
In
@_0,9.% 10.8: 8% 0.7: I ,_ 0.6: G 13 0.5:
0.4 0.6 0.8 N~~a~~~~ crack length a/R
1.0
Fig. 4. Effect of V, on fiber crack propagation. EFM 4W-G
_ _._
2
0
0.2 0.4 0.6 0.8 Norma~zed crack length a/R
1.0
Fig. 5. Effect of V, on fiber crack propagation.
820
YONG
Ll XU and K. L. REIFSNIDER
where G(r) is a bounded function at T = i 1. The stress intensity be calculated by the following expression: k,(-u)=k,((/)=
-
factors at the crack tips may then
$;...!\;(‘n)(;(ll,
(36)
/ /
Using the Labatto-Chebyshev integration method [I I]. eqs (33) and (34) can be discretized into a set of N algebraic simultaneous equations as follows:
and t i.,G(r,) = 0 =I,
Table
1. Normalized
stress
intensity
factors vs normalized crack q = E,, . I’, = 1’,,,, E,,‘E,,, =
(37)
lengths
for
various
fiber
volume
fractions.
12
k,(a) Cl, \’
a/R 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Table
F’. = 0
0.6046 0.5898 0.5680 0.5421 0.5144 0.4865 0.4583 0.4284 0.3914
2. Normalized
0
I,‘, = 0.1
P.. = 0.2
I’, = 0.3
E/ = 0.3
1’, = 0.5
I =:0.6
I, = 0.:
0.6475 0.6319 0.6089 0.5816 0.5525 0.5329 0.493 I 0.4614 0.4218
0.6910 0.675 1
0.7353 0.7193 0.6955 0.6671 0.6364 0.6049 0.5724 0.537 I 0.49 16
O.7798 Il.7640 0.7405 Cl.7122 0.68 13 0.6490 0.6152 0.5776 0.5287
0.8’37 0.808 0.7X5R 0.7581 0.7275 0.6949 0.6600 0.6204 0.5679
0.X660 0.x520 ll.X3OX 0.8046 0.7752 0 7432 U.7080 0.667:! 0.61 18
0.9060 0.8938 0.875 I 0.85 16 0.8247 0.7949 0.7613 0.7’1 7 0.6651)
stress
intensity
0.6515 0.6234 0.5933 0.5627 0.5315 0.498 I 0.4559
factors
vs normalized crack lengths E, = E,,. 1’,= “,>, . E,;E,,,=: 3
for
various
fiber
colume
fractions.
@iI, ‘a n:R 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
v,=o L.2834 I.3113 1.3562 1.4163 1.4907 1.5797
1.6874 1.8272 2.0483
r;=o.1 I .2639 1.2912 1.3352 1.3940 I .466? 1.5535 I .6584 1.7943 2.0095
F’,=O.?
r; = 0.3
1,; = 0.4
I.2446 1.2710 1.3134 1.3699 1.4395 1.5222 1.6218 1.7503 I.9541
1.2256 1.2506 I.2905 I .3436 I .4084 I .4850 I .5764 I .6940 1.8812
I .206287 1.229341 1.I?45998 1.314283 I .372605 I .440734 I.521401 1.624792
1.790796
rl=
0.5
I, 185798 1.2063% 1.238694 1.280832 1.331003 I .388736 I .456279 1.542623 I .683559
v, = 0.6 I. 16282 1.18022 I.20735
1.242096 I .28’652 I .328376 1.381059 1.448402 1.561542
)‘,=0.7
1.135664 I.149357
I 170377 I. 196693 1.226564
1.259289 1.296261 1.343804 1.42785
Micromechanical
analysis
of fiber crack
821
propagation
with weights di given by 71
when i = 1 or N
2(N - 1)’
/$ =
(38) wheni=2,3
(NII 1) ’
,...,
N-l,
and zi and p, are roots of the following equations: T,m ,(p,) = 0,
U,_,(r,)=O,
j = 1,2,3,.
i=1,2,3
,...,
N-2
..,N - 1
and
z=fl,
(39)
where TN_, (x) and U,,_ Z(x) are Chebyshev polynomials of first and second kinds, respectively. RESULTS
AND DISCUSSION
By solving the simultaneous equations (37) and using eq. (36), the stress intensity factor of the crack can be calculated. When V,= 0, Ef= EB, v,= v~,, the problem degenerates to that of an isotropic fiber embedded in an infinite matrix (Erdogan and Gupta’s problem [9]). It can be seen in Figs 4 and 5, as well as in Tables 1 and 2, that the pressent results match their results exactly. Normalized stress intensity factors (k, (a))/(a,&) of the fiber crack are calculated for various situations, and results are given in Figs 4-7 and Tables 14. The effect of fiber volume fraction on the normalized stress intensity factors is given in Figs 4 and 5 and Tables 1 and 2. The effect of fiber-matrix mechanical property mismatch on the crack propagation is given in Fig. 6 and Table 3. The effect of fiber mechanical anisotropy on the fiber crack behavior is presented in Fig. 7 and Table 4. The results in Fig. 6 and Table 3 can be understood in two ways. One interpretation is that for a given matrix stiffness, the stiffer the fiber, the more likely the fiber crack will propagate; Table 3. Normalized stress intensity factors vs normalized crack lengths for various E,/E, fractions. E, = E,, . v, = v,~, E,I-&
Vf
alR
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
k,(a) GO a 1.oOOo 1.1717 1.2827 1.3622 1.4224 1.4696 1.5077 1.5390
1.5653 1.5876 1.6068 1.6235 1.6382 1.6512 1.6628 1.6732 1.6826 1.6911 1.6988 1.7059 1.7124 1.7184 1.7239 1.7291 1.7338 1.7383 1.7424 1.7463 1.7500 1.7534
Table 4. Normalized stress intensity factors crack lengths for various E,,/E, fractions. v,=v,,,, V,=O.6
vs normalized E,= Em = 10,
k,(a) alR 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
E,,IEf=
1
1.3184 1.3544 1.4126
1.4907 1.5876 1.7048 1.8503 2.0495 2.4082
G E,,/E,=4 1.3204 1.3558 1.4128 1.4893 1.5838 1.6978 1.8386 2.0307 2.3751
E,,/E,=
9
1.3209 I .3562 1.4131
1.4892 1.5833 1.6968
1.8368 2.0276 2.3696
YONG
822
LI XU and K. L. REIFSNIDER
the secondary interpretation is that for a given fiber stiffness, the stiffer the matrix, the less likely the fiber crack propagation will take place. It is seen by comparing results in Figs 4 and 5 that. from the viewpoint of avoiding fiber cracking, “reverse composites” (composites in which the fiber is less stiff than the matrix, for example, in a Nicalon/SiC ceramic composite) will be a better choice. It is also observed that the fiber mechanical anisotropy seems to have little efl‘ect on the fiber cracking (for transverse loads). Acknowledgement-This study is supported by the National Science Foundation (Center for High Performance Polymertc Adhesives and Composites) under grant No. DMR 8809714, and by the Virginia Institute for Material Systems
REFERENCES normal loading of a unidirectional composite. .I. c’onrpes. ~a/er. I, 152 164 (1967). D. F. Adams, Inelastic analysis of a unidirectional composite subjected to transverse normal loading. J. cnmpo,s. bfarer 4, 310 (1973). R. L. Foye, Theoretical post-yielding behavior of composite laminates, part I-inelastic micromechanics. J. conrpo.,. Mater. 7, 178 (1973). D. F. Adams, Elastoplastic crack propagation in a transversely loaded unidirectional composite. J. cotnpos. ilfarer. 8, 38 (1974). G. P. Sendeckyj (Ed.), Micromechanics. Academic Press. New York (1974). D. F. Adams, A micromechanical analysis of crack propagation in an elastoplastic composite material. Fiber .%I. Technol. 7, 237-256 (1974). D. R. Owen and J. F. Lyness, Investigation of bond failure in fiber-reinforced materials by the finite element method. Fiber Sci. Technol. 5, 129-141 (1972). M. R. Wisnom, Factors affecting the transverse tensile strength of unidirectional continuous silicon carbide fibre reinforced 6061 aluminum. J. compos. Mater. 24, 707-726 (1990). F. Erdogan and G. D. Gupta, The inclusion problem with a crack crossing the boundary. Inr. J. Frucrurr 11, 13- 17 (1975). F. Erdogan, G. D. Gupta and T. S. Cook, Numerical solution of singular integral equations, m Mechanics oJ‘Fra(,lure. Volume 1: Methods of Analysis and Solurions qf Crack Problems (Edited by G. C. Sih), pp. 368425. Noordhoff, Leyden (1973). N. I. Iaokimidis and P. S. Theocaris, On the solution of collocation points for the numerical solution of singular integral equations with generalized kernels appearing in elasticity problems. Comput. Sfructwzr 11, 289-295 (1980). I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals. Series, and Products. Academic Press, New York (1980).
[l] D. F. Adams and D. R. Doner, Transverse [2] [3] [4] [S] [6] [7] [8] [9] [lo]
[l I] [12]
APPENDIX
A
The following is a justification of the fact that, even though an elastic body is transversely isotropic, in the isotropy plane x-y, the compatibility equation is still a biharmonic equation, VV@ = 0. Using the same coordinate system as in Fig. 1. the constitutive equations for a transversely isotropic material are
where E is Young’s modulus in the x-1 plane, E3 is the Young’s plane, %I, v23, v32 and Ye, are the general Poisson’s ratios. Accordingly, we have
Using
the identities
For the plane strain
in (A2), we can rewrite
case, we have
(Al
) as follows:
modulus
in the -_ direction.
I*is Poisson’s
ratio in the I
i‘
Micromechanical analysis of fiber crack propagation
a23
Substituting (A4) into (A3), we may obtain cc: = E3
(AS)
Substituting (AS) into (A3a) and (A3b) yields
(fw with
It can be readily shown that
Assume the Airy stress function Cgand (A9) Then
7Z.V =
31 + “h). =---* 2(f + v) Peg E
E
axay
WO)
Substituting (AlO) and (A8) into the compatibility equation as follows:
a5 ,. a4,, -2+-i-I-=o’ ay
a*yyy axay
(All)
=o.
(AU
we obtain ;vv@
Therefore, the compatibility equation is still a biharmonic equation.
APPENDIX B. FORMULATION OF THE DISLOCATION PROBLEM As is shown in Fig. 2, in the corresponding dislocation case, instead of the crack, there is an edge dislocation with Burgers’ vector (0,6,, 0) embedded in the fiber at r = t, B = 0. To formulate the problem, plane polar coordinates are used and Michell’s general (interior, annulus and external) solutions are utilized for all three of the domains, respectively. However, due to the presence of the dislocation in the fiber, the stress function for the fiber is constructed as the sum of Michell’s interior solution and that of an infinite plane containing an edge dislocation. For the problem under consideration, the stresses and displacements should also satisfy the following symmetry conditions: o,,(r, @ = urr(r, -8)
(Bt)
%lJP. @)= Qo&, - 0)
(B2)
r,,,o(r.0) = -r&r, ~~(6 8) = a,(~,
-8)
-8)
uct@,0) = --Il&, -0).
(B3) (B4) (BS)
After taking regufarity and symmetry conditions into consideration, stresses and displacements in the fiber, matrix and composite can be written as follows. (a) Fiber By referring to Michell’s interior stress function and by superposition with the solution of an infinite media embedded edge dislocation, the stresses and displacements in the fiber may be written as follows:
u,,,(r,8)=al!l(r,0)+a!31(r,0) 0~ (rr 0) = u&X(rr 0) + c&I,(r, 0)
824
LI XU and K. L. REIFSNIDER
YONG
ii I,
(l’.
fll
:y
IPO0) rl .
f
7f:;‘lr.l)).
(Bhl
where the first term on the right side in all the above equatmns is the stress or displacement obtamed from Michell’s interior stress function; the seeond term is the stress or displacement of an infinite media with an embedded edge dislocation (see the following):
where co,, d,,, S2, a,,, and h,,,are integral
constants. 2’ sin2 (Jtr cos 0 - t) iBl?i ~r~+r~-WfOSO)‘
----
1
co5 0 - i)(r -
f co5
0) iHl.3)
(P
+
/‘-
7rt
cos
H)-
7(r cos 0 - r)(r - I cos O)-' (V'-kr?-2rt costi):
(614)
/r sin-fi
(BIS!
In order to combine the infinite plane solution at interface Y = R, we now express the displacements in terms of the following Fourier series:
with the Fourier
coefficients
determined
by
with Michell’s interior solution and to satisfy the continuity condition in eqs (B I 5) and (B 16). the stresses in eqs (B 12) and (B 13). with I = R
Micromechanical analysis of fiber crack propagation
825
u$‘(R, 0) dB
(824)
n(l +X1-,)2 * C,(r) =-.--b,. ~* o u$‘(R 1 0)cos nfl d0
(825)
X(i +K,-,)2 D,(t)=-%--
(B24)
s
*
u&R.
@sin no d8.
* s0
With the aid of tables in 1121,integrals in eqs (821 j(B26) can be evaluated in closed form and the results are as follows:
(827)
A (+(n+?)r”” ”
nt”-’ --3 (n3.22)
(B28)
R”
R”+?
B,(t) = - g
(B29)
1 - $+f
B,(t)=
(B30)
B,o=-(n~~~t~‘+‘+~~,
(n-&2)
(B31)
C&f = f
(~32) 1 + ti,-, + !!CZ!
‘,‘f’=g&c(t)=!
n
_(n
2i
3 tZ m=~R’--_ D&j=;
I
-
+ I -ti,-,y-
-...(n - l)R”-’
xi_,+5
log,R (n +2)1”+’ +(n+l)R”+’
@~2)
1 +
q_,- 1 - 1--logzR
(B33)
(B34)
(B35)
-l-(-l)nill (n>2). +q_,,u (n - ---f 1n(l
(n -
1-li_,fPt)R”-’
(n c2)t”+’
(n + I)R’+’
..-
-
2(& -1)
’
(B34)
’
(b,i Mc2tri.v The stresses and displacements in the matrix can be written as - 2’12+ 2d,,r
CT,,? (r, Q) = !!E + 2c,, + r2
C
r3
-S[~,n(nl)r’-‘cb,,(n
q),Jr,
0) = - “,? 0’ f -C ’ .*_ 3 + (‘:;A+ T +-y[u,n(n-f)r”-?
>
cos 0 +I)(n
-2)r”+c,gt(n
+l)r-‘“+2)+dnZ(n
- I)(n +2)r-“]cosnB
(837)
-I)@--2)~“]costzU
(B38)
6d ,? r ) cos 0
+b,zfnfI)fn+2)r”fc,,zn(n+l)r-‘^*”+d~,(n
+ b,zn(n + i)tn-
~,~n(rt +
l)rm~(“C2)-d,2n(n
- l)rm”]sinnO
(B39)
I I - bozr + (tini,, - Ik,,,r + c,::I? - + (IX,,, - 2)d,?rZ cos U
u,?(r, 0) = -!-
b”,
I
f I[ - nu,? r’s-. ’ -(n -K,,,f I)h,,Zr”+‘+nc’,,zr-‘l-’ 1
+(n +K,,-
l)dnSr-“+‘]cosnO
(B40)
I +~[-~na,zr”-‘+(n+ti,+l)b,~r”+‘fn~~2r~”-i+(n-i-ti,,)d~zr-“+’]sinn# where bOz. co2.cs2,dlZ. &. c+, h,, ml and dn2 are all integral constants.
(B41)
826
YONG
Lt XU and K. L. REIFSNIDER
(c) Composite
The stresses and the displacements
whereb,,,
ct3,
in the external
S.,, c,,~ and d,, are all integral
composite
can be written
as follows:
constants.
(d) Boundary conditions
After assuming the basic form of the solutions, the unknown constants in these expressions applying the following boundary and continuity conditions: n,,,(R, 0) = a,,,(R.
may be determined
by
tj)
T,,ilfR.H)=5,(12(R,H) l~~~(R,~)=~,~~R.~) zt<,j(R, 0) = u,>,(R. 0) rr,,,(b, 8) = “r,J(h, 0) r,02ih. 0 ) = Tr,ii(h. H ) u,L(h. Ul = tc,,(h, 0) n,,,(h. 0) = u,,,th. 0).
After equations:
substituting
eqs (B7)-fBl6)
and (837)~(B46)
into (B47). we obtain the following h 02 %, ---R’
1B471
three sets of simultaneous
29 hl z
(848)
(B49)
Microm~hani~I n(1 -t~)R”-~u,,
+ (n + I)(2 -n)R”b,,
+n(n
- l)R”-2a,z c(n + I)(n - 2)R”bnz +n(n
n(n - I)R”-*a,,
+n(n
827
analysis of fiber crack propagation
+ 1)R-“-2c,2i-(n
- l)(n +2)R-“d,,=
-
+ l)R”b,,, - n(n - I)R^-*Q,, - n(n + i)R”b, e-.tbj An
+n(pt+l)R1”-2e,,+n(n-1)R-“d,,=
--!--~R”-la,,+.--l-(~r_,7% 2P,- I
I-n)R*“b.,+InR”-‘~~~+~(~
m
%4n
I
.-!.- nR”- $I,, + --$, 2i+ I 2Pf-,
+ 1 f rc/_,)R”+ lb,,, - -!- nR”- ‘a,, -&I m 2&
+x,_
+ I)b-“-‘c,,+(n
))R-“+‘dnz=
2&l
b.C - ---:= n(i + Xf-,)
+ 1 + q,,)R”+‘bn2
_L,R-‘-l,_._‘,, 2Pm i- l)(n -2)b”b,,,+n(n
n(I + q-,)
-IC,,,+ l)R”+‘bj,*
_.~nR-+.$-(n 2fh!
n(n - l)b”-‘un2+(n
ttf-tbhi x(1 + K,-,)
_ 1 -K,)R-“+‘dx?=
-
2Pm
D,b, n(1 -I-Kf-,f
- I)(n +2)b-“d, - n(n + 1)b -“-*c,, - (n - I)(n + 2)b -“dnj = 0
n(n-I)b”-
a,2fn(n+I)b”b,,-n(n+I)b-“-2c,,-n(n-l)b-”d,z+n(n+I)b-“-2c,,fn(n-I)b-“d,,=O
2
I -~b”-‘a,+-t(n+l+K,)b”+~~~2+-i-nb-”-i~~~+~(~-l-K,)b-n+~d~~ 2Pm 2&i 2& 2&n I
--
2/L,
nb -*- ‘c,, -I;i.(n
-$f--)B-n+id~~=O.
NO)
with & PW=2(1 iv,)’
K,=3-4V,,
4-t
x,._,=
pe-‘=2(1+v,_,f’
&_,=
E,-, 2t1 + v,-,) ’
3 -v,-*
K(_, = -
1 + v, - g
3 -4v,+,
and A,, B,, C, and D, are functions of t given by eqs (B27)-(B36). After determining all these constants, stresses and displacements may be obtained by eqs (B6)-(816) and (B37)-(B46).
APPENDIX C. UNCRACKED GEOMETRY SOLUTION The geometry and loads of the untracked case are the same as those of the cracked case, as is shown in Fig. 1, except that there is no crack in the fiber. To formulate the untracked problem, we assume proper stress functions for the fiber, the matrix and the external composite, and then match the boundary and continuity conditions. The unknown constants in these stress and displacement expressions can be determined. The stresses and displacements in all three domains can thus be obtained. After taking regularity and symmetry conditions into consideratjon, stresses and disp~a~ments in the fiber, matrix and composite can be written as follows. (a) Fiber o,,,(r, @) = 2c,, - Za,, cos 28
Kfi
(r,,,,,(r, @) = 2c,, + (2azI + 12bz,r2jcos 28
(C2)
f&r,
(C3)
8) = (2az, + 6b,, r’)sin 28 +(3-ri,_,)bZ,r3]cos28j
u,,(r,@)=-!---
{(K,_,- l)e,,r -[2a,,r
u&r, 6) = --!-
[2az, r + (3 + ‘c, _,)b,, r3fsin 28
2P,-,
2&-r
(C4) 0)
828
YONG
LI XU and K. L. REIFSNlDER
with
where L;,,,
uzI and h,, are Integral constants.
fh) Matrix The stresses
T,,l”,
and displacements
in the matrix
(
3c,:
cr.01=
2u22+3h,,r’---~
r4
can be assumed
(I,? r2 1
in the form
sm2lJ
k’ -1
uom(r,t))-j~ 2~:,r+r,:Zi+(3+ri,,)h2~rl-ij-2--I. m
I
il
sin Zfl 1
I
with “*i=zii;:, where ho?.
-)’ ‘“8
K,,,=i-441.
/,,.
coo2, az2.c2?and t& are all integral constants.
(~‘1 Composite The stresses and the displacements
I,,~,tr. 0)
= -
in the external
composite
can be written
as follows.
i
with r; , p, ,E---~~ , k‘s ,-3 -4y 3 I + y ,I where hOi. cz7 and dz, are constants, (d) Boundary
and 0,) is the traction
applied
:.
at infinity.
conditior1.v
Having assumed the basic form of the solutions, the unknown applying the following boundary and continuity conditions: a,.;(R.(JJ
constants
in these expressions
may bc determined
by
=rr,,,,,!R.iJ)
:,!ji ( R. tJ ) =
i, ,/,I, t R. 0 1
rr,,(R.IJ)-t/
,,,,(R.(J)
I,!,,t R, II ) =
it,,,,, i R. 0 I
m,#,,(h,~l)==r7,,, fh, iii i ,,,,,, I/J.0 1= T,,.,(h. f/ 1 itr,,,(h. 0) = u,,(k 0) rr,,,,,(h. (J) = u,i, (h. (J 1
(Cl61
~icromechan~~l
analysis of fiber crack propagation
829
By so doing, we may obtain the following two sets of s~muitaneous equations: (Cl-i-I)
(C17-2)
fC17-3)
(C17-4)
(018-i)
(Cl&2) a,, R --+-2jl_, Pf- I
(q_ 8- 3)R 3b,, _ t R%, Pm
c,, (I + KJdv (Km- 3)R3bx -““-_-~:=o P”,R3 2&R 2Pm
(Cl&3)
(Cl&4)
(Cl%5)
(Cl&6)
(Cl 8-7)
(Cl&S)
After solving the simultaneous equations, stresses and displacements in all three domains can be determined by eqs (CI)-(C 15). (Receiwd
15 May 1992)
1993
0013-7944/93 $6.00 + 0.00 c, 1993 Pergamon Press Ltd.
A MICROMECHANICAL ANALYSIS OF FIBER CRACK PROPAGATION IN COMPOSITE MATERIALS DUE TO TRANSVERSE TENSILE LOADS YONG Department
of Engineering
LI XU and KENNETH
L. REIFSNIDER
Science and Mechanics, Virginia Polytechnic Blacksburg, VA 24061, U.S.A.
Institute
and State University,
Abstract-The
fiber crack propagation in composites due to transverse tensile loads is studied using a micromechanical model and Linear Elastic Fracture Mechanics. To approach the problem, a three domain cylindrical model is introduced to simulate the fiber cracking. The model problem is then solved by the dislocation and singular integral equation techniques. The stress intensity factors of the fiber crack are calculated for various situations. It is found that fiber anisotropy has hardly any effect on the fiber crack “reverse composites” (composites in which fiber is less stiff than the matrix, such as propagation: Nicalon/SiC ceramic composite) virtually eliminate fiber crack propagation.
INTRODUCTION THE BEHAVIOR of a unidirectionally
reinforced composite subjected to transverse tensile loading remains as a particularly significant problem in material technology. This is because of the very low transverse stiffness and strength characteristics of such materials relative to their stiffness and strength in the direction of reinforcement orientation. Since it is often the transverse properties that limit the performance of the composite system, there is a high potential for improvement of behavior through study of this problem. In the study of the composite transverse strengths, a number of investigators have conducted investigations and presented their results [l-8]. A brief review of the historical development of micromechanical analyses, as applied to the problem of transverse behavior of a unidirectional composite, can be found in [l ,5,8]. In general, finite element models are used in those studies, and matrix elastic-plastic behavior is modeled; analyses are continued until the material failure occurs, and transverse strengths are thus estimated. Among those studies, Adams [6] first introduced crack initiation and propagation in the model, i.e. a crack initiated at the fiber-matrix interface, propagated partially around the fiber and then across the matrix. In [7], a special element was developed which allowed shear failure to occur at a predefined stress followed by friction slip. In [8], a special element was developed to enable the interface to be modeled with both shear and tensile failure criteria with a quadratic interaction formula. In this paper, the fiber crack propagation in composites due to transverse tensile loads is studied using a micromechanical model and Linear Elastic Fracture Mechanics (LEFM). These tiny fiber cracks can also be regarded as microcracking formed in the process of manufacturing due to mismatch of fiber and matrix thermal expansion coefficients. The objective of this study is to investigate the effect of fiber volume fraction, fiber anisotropy, and fiber-matrix mechanical property mismatch on the fiber crack propagation. It should also be mentioned that Erdogan and Gupta’s solution of a general inclusion-crack problem [9] also includes some results on inclusion-crack behavior. To approach the problem, a three-domain cylindrical model is introduced to simulate the fiber cracking. The model problem is then solved by the dislocation and singular integral equation techniques. The stress intensity factors of the fiber crack are calculated for various situations. It is found that fiber anisotropy has hardly any effect on the fiber crack propagation; “reverse composites” (composites in which fiber is less stiff than the matrix, such as Nicalon/SiC ceramic composite) virtually eliminate fiber crack propagation.
MODELING
OF A FIBER
CRACKING
To study the behavior of the fiber crack, a cylindrical model Fig. 1, the model consists of three domains, defined as follows. 813
is introduced.
As is shown
in
814
YONG
LI XU and K. L. REIFSNIDER
~o~ui~ I: 0 G r d R, the fiber (radius R), with a radial crack embedded from -u to a (0 < a c R). The fiber is assumed to be transversely isotropic with Young’s moduli Ef and ED. and Poisson’s ratios v, and yD,, where Ef is Young’s modulus in the x-_V plane (see Fig. I), and E,, is Young’s modulus in the z direction (fiber axial direction). v~-is the Poisson’s ratio in the .Y;I-’plane. and vD, is the general Poisson’s ratio (strain in the X-J plane due to stress in z direction). Accordingly. we have the following relation:
where ljfi3, v,.,~are the general Poisson’s ratios (strain in z direction induced by stress in x or J direction, respectively). Domain II: the surrounding matrix, R < Y G h; isotropic material with Young’s modulus E,,, and Poisson’s ratio v,,, with
where yf is the fiber volume fraction of the composite. Doma& III: the composite. h < Y < 1%;transversely isotropic material with material constants determined as follows:
where EC is Young’s modulus in the X-J plane (see Fig. l), and EC3is Young’s modulus in the 3 direction (fiber axial direction). v,, is the Poisson’s ratio in the X-.-J plane, and v,~, is the general Poisson’s ratio (strain in the .u-;v plane due to stress in z direction). Also, we have the following relation: \‘C 3I --=----, “r 13 1’,y (7) E,.?-
E,
E;
Radial Crack Composite
0 0
b) Coordinate system
a) Model of fiber cracking
Fig. 1. Modeling
of fiber cracking.
815
Micromechanical analysis of fiber crack propagation
where vc13, vcz3 are the general Poisson’s ratios (strain in z direction due to stress in x or y direction, respectively). The composite is subjected to a uniform unidirectional traction co at infinity. For plane problems of the transversely isotropic material, when we consider the stress and displacement distributions in the x-y isotropy plane, it can be shown (see Appendix A) that the compatibility equation is still a biha~onic equation. We make the following transformations. For the fiber (domain I), let
Ef_
Ff -- E,$ii ’ EP
,= l
(8)
Then, for the plane strain case, we have the following constitutive equations:
l (Qe- Vf- , CT,,1. tge=Ey_i
(9)
For the composite (domain III), let
EC
EC
EC_,= 1
--
41
v, + -
Ec3
EC& ’ “‘-‘=
(10)
1 _ &vi,
Ec3
Ec3
and also for the plane strain case, we have f&r =
--!(@rr - v,-,Gd EC-,
cet?=&oee-vc-*~,F). ‘
f
It can also be shown that l+v,_,=1+v,Er-r
Er
13-v,-, 3 -=-. 4-r
l-+VC
W)
&
Therefore, /q-r=
E,- ,
J&-t
4
2(1 +v,_,)=m=pfV
pC-J=2(l
EC
+VC_,)==2(1 +vJSPC,
(13)
where pf and ~1~are shear moduli of the fibers and composite in the x-y plane, res~ctively.
FOR~LATIO~
OF THE MODEL
PROBLEM
To approach the model problem, we first seek the corresponding dislocation solution (see Fig. 2). Then, by superposition with the untracked geometry case (see Fig. 3), we reduce the original problem to a perturbation problem in which the geometry and the singufar nature of the fiber crack remain the same, but the only Ioading is the crack surface self-equilibrating pressure. After that, we integrate the dislocation solution, in other words, we use the dislocation solution as a Green’s function to integrate along [-a, a] to generate a singular integral equation. The stress intensity factors of the fiber crack are related to the solution of the singular integral equation and may be solved by Gaussian quadrature to a desired accuracy.
816
YONG
Ll XU and K. L. REIFSNIDER
Matrix
X(x11
Dislocation
Composite
f.~g. 1. The wrrespondlng
(a) Dislocatiort
d~slo~atron
problem.
solrrtiorz
The geometry of the dislocation case is shown in dislocation case, instead of a crack, an edge dislocation in the fiber at r = t, 0 = 0. The details of the dislocation we only give the hoop stress o,,~,(Y, H) in the fiber as
Fig. 2. It is seen that m the corresponding with Burgers’ vector (0, h, . 0) is embedded solution can be found in Appendix B. Here follows:
with K, iwhere co,, together
with unknowns
a) original problem
3 - l’/ , .-.. -. lfrj, :
(1st
c,,?, h,,, and h,,,. can be determined
c) perturbatcon case
b) untracked geometry
Fig. 3. Superposition
of the crack
by the following
problem.
equations:
817
Micromechanical analysis of fiber crack propagation b $ (G - 1)b
-Lb,,*+
2&n
2pL,b
d,, ,
together with unknowns simultaneous equations:
(16)
Cl2 -
2hH
1
4, + 1+
Kr-,
4
1 2)R2 4, - m 2Pf-
(K, + 2)R2
(K,_ t +
Cl2 -
I
by the following
- 2Rd,, =
(K, - 2)R2
1 2)R’ 4, - m 2P,-
1 bo3 = 0; coz+2pC_,b -
c12, d12, c,~, S, and S,,, may be determined
2Rd,, + 2
@_, -
+ 2c02 - f$ = 0
2&
+v
logeR 1
4, - S, 5+lc,_,
t2 --
4
R2
-~ K/_,-
2
1 -lo&R
1
2c +-2bd,,+=O
1 --7Cl2 3.d
(v2)bZd +
_ PC 1
2pc+,b2
I3
_ -c ’ I2 2pI._(b2
I3
I2
%4n
1 2c,2+ (K,+2)b2d 2PllI 2/.4
+s,,=o
-s,,
= 0;
(17)
a,,, and b,, , together with unknowns a,,*, b,2, c,,~, d,,, , c,,~and dn3 (n > 2), can be determined by the following equations: n(1 - n)Rnm2a,, + (n + 1)(2 -n)R”b,,
+ n(n - 1)R”p2a,2 + (n +
+ n(n + 1)R-“-2c,2 n(n - 1)Rnm2a,, + n(n + l)R”b,,
+ (n -
l)(n + 2)R-“d,,2 = -
1 2P,- I
1
nR”-‘%
+ -(I+,2pf_,
1 +-nR”2&
1 -n)R”+‘b,,
---(n 2/&I
m
1
“+‘b,, - $
n-‘a,, + ++‘+“‘)R ,
--
1
m
‘a,,+l(n
,b?.Bn
x(1 + Kf-,)
-Km+
PfLr-,b,.A, x(1 + K,-,)
l)R”f’b,z
2Pln l)R-“+‘d,,=
+K,-
nR”-‘a,,,
_$(n
2Pm?IR-“- IG2 n(n - l)bnm2an2 + (n + l)(n - 2)b”b,, + n(n +
+ n(n - 1)Rpnd,2 =
1
1 -- 2~ ~R-“-‘c,,~ -nR 2Pf- I
/+
- n(n - 1)R”-2a,2 - n(n + l)R”b,, + n(n + 1)R-“-2~,2
--
l)(n - 2)R”b,2
- $
m
(n +
-
b,,C, x(1 + ‘C-t)
1 + K,)R”+‘b,2
-1 _K,)Rm”+‘d,2= m
-
DnbJ n(l + rcf-1)
l)b-‘-2c,2 + (n - l)(n + 2)b-“d,, -n(n+1)b-‘-2c,3-(n-l)(n+2)b-“d,,3=0
n(n - l)b”-2a,2
+ n(n + l)bnb,2 - n(n +
l)b-“-2c,2 - n(n - l)b-“d,,, + n(n +
1)b-‘-2c,3 + n(n - l)b-“d,,
= 0
YONG
818
LI XtJ and K. L. REIFSNIDER
with
where A,, B,,, C+,and D, are functions
of t given by
By superposition with the sofution of the untracked geometry case, which may be found in Appendix C, we can reduce the original crack problem to a perturbation case (see Fig. 3). For the problem under discussion, the boundary conditions for the perturbation case can be written as foifows: T,~~=O,
O<~<.X,
O=O.
CTf,(j(X, 0) = P(X),
-a
U,j(X, 0) = 0.
.^-?r, <
.Y < II,
and
0-n
< x 6 U and
O), and aoof(r. 0) is the fiber hoop stress in the untracked Appendix C). Stnce a crack can be regarded as a pile-up of dislocations, we define
By integrating the dislocation may obtain a singular integral
solution equation
(24)
n<.u<~~,
where p(r> = --a&r,
we
(23)
(25) geometry
case (see
eq. (24).
(14) along --a < x < a, t) = 0. and applying with a Cauchy kernel as follows:
(27) with the Fredholm
kernel
K(X, t) given
K(X, t) = r(12+4_r) 1 c
i
by
2COl+ 6ril, + i [n(n - 1)x”--%,, + (n -t- l)(n + 2)X%,,] ?
where car, d, t ~ a,, and 6,, are functions
of t determined
by eqs (16)-( 18)
,
(28)
Micromechanical analysis of fiber crack propagation
819
For an embedded crack, it follows that a f(t) dt = 0. s --4
(29)
The embedded fiber crack has a square-root singularity, and the functionf(x)
may be written
as
Wf f(x) = [(x + a)(a - x)]‘i2 ’ where F(x) is a bounded function at the crack tips. The stress intensity factors at crack tips x = -a or x = a may be defined according to the conventional definition, and calculated as follows:
k, (-a) = kl (a) = !i?
Jw - aM6m(4 0)
=---_ a+- J-(a) 1++i J;E I
~M~~I~AL
(31)
SOLUTION
The singular equation (27) can be solved using the method introducing the following normalizations: x = up,
where
-a
-l
t=uz,
where
-a
-l
t) = K(P, r),
P(X) = 4(P),
developed
in [lo, 1I]. By
let S(t) = g(r),
f s-
a%
(32)
eqs (27) and (29) become
g(z)
-IT---P
dT
i
+
f
-1
UP, z>sCr)dt =
and
NY1+ Kf-1) 4(P), -lfp
(33)
2&- I
I g(z) dr = 0.
(34)
Sl The function g(z) is singular at 1: = + 1 and, likewise, may be written as
G(z)
(35)
g(r) = (1 _ T2)l:2'
E
e0
d
E,=E_, 1.0
E,/E,=3,
v,~=v,~~=v
In
@_0,9.% 10.8: 8% 0.7: I ,_ 0.6: G 13 0.5:
0.4 0.6 0.8 N~~a~~~~ crack length a/R
1.0
Fig. 4. Effect of V, on fiber crack propagation. EFM 4W-G
_ _._
2
0
0.2 0.4 0.6 0.8 Norma~zed crack length a/R
1.0
Fig. 5. Effect of V, on fiber crack propagation.
820
YONG
Ll XU and K. L. REIFSNIDER
where G(r) is a bounded function at T = i 1. The stress intensity be calculated by the following expression: k,(-u)=k,((/)=
-
factors at the crack tips may then
$;...!\;(‘n)(;(ll,
(36)
/ /
Using the Labatto-Chebyshev integration method [I I]. eqs (33) and (34) can be discretized into a set of N algebraic simultaneous equations as follows:
and t i.,G(r,) = 0 =I,
Table
1. Normalized
stress
intensity
factors vs normalized crack q = E,, . I’, = 1’,,,, E,,‘E,,, =
(37)
lengths
for
various
fiber
volume
fractions.
12
k,(a) Cl, \’
a/R 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Table
F’. = 0
0.6046 0.5898 0.5680 0.5421 0.5144 0.4865 0.4583 0.4284 0.3914
2. Normalized
0
I,‘, = 0.1
P.. = 0.2
I’, = 0.3
E/ = 0.3
1’, = 0.5
I =:0.6
I, = 0.:
0.6475 0.6319 0.6089 0.5816 0.5525 0.5329 0.493 I 0.4614 0.4218
0.6910 0.675 1
0.7353 0.7193 0.6955 0.6671 0.6364 0.6049 0.5724 0.537 I 0.49 16
O.7798 Il.7640 0.7405 Cl.7122 0.68 13 0.6490 0.6152 0.5776 0.5287
0.8’37 0.808 0.7X5R 0.7581 0.7275 0.6949 0.6600 0.6204 0.5679
0.X660 0.x520 ll.X3OX 0.8046 0.7752 0 7432 U.7080 0.667:! 0.61 18
0.9060 0.8938 0.875 I 0.85 16 0.8247 0.7949 0.7613 0.7’1 7 0.6651)
stress
intensity
0.6515 0.6234 0.5933 0.5627 0.5315 0.498 I 0.4559
factors
vs normalized crack lengths E, = E,,. 1’,= “,>, . E,;E,,,=: 3
for
various
fiber
colume
fractions.
@iI, ‘a n:R 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
v,=o L.2834 I.3113 1.3562 1.4163 1.4907 1.5797
1.6874 1.8272 2.0483
r;=o.1 I .2639 1.2912 1.3352 1.3940 I .466? 1.5535 I .6584 1.7943 2.0095
F’,=O.?
r; = 0.3
1,; = 0.4
I.2446 1.2710 1.3134 1.3699 1.4395 1.5222 1.6218 1.7503 I.9541
1.2256 1.2506 I.2905 I .3436 I .4084 I .4850 I .5764 I .6940 1.8812
I .206287 1.229341 1.I?45998 1.314283 I .372605 I .440734 I.521401 1.624792
1.790796
rl=
0.5
I, 185798 1.2063% 1.238694 1.280832 1.331003 I .388736 I .456279 1.542623 I .683559
v, = 0.6 I. 16282 1.18022 I.20735
1.242096 I .28’652 I .328376 1.381059 1.448402 1.561542
)‘,=0.7
1.135664 I.149357
I 170377 I. 196693 1.226564
1.259289 1.296261 1.343804 1.42785
Micromechanical
analysis
of fiber crack
821
propagation
with weights di given by 71
when i = 1 or N
2(N - 1)’
/$ =
(38) wheni=2,3
(NII 1) ’
,...,
N-l,
and zi and p, are roots of the following equations: T,m ,(p,) = 0,
U,_,(r,)=O,
j = 1,2,3,.
i=1,2,3
,...,
N-2
..,N - 1
and
z=fl,
(39)
where TN_, (x) and U,,_ Z(x) are Chebyshev polynomials of first and second kinds, respectively. RESULTS
AND DISCUSSION
By solving the simultaneous equations (37) and using eq. (36), the stress intensity factor of the crack can be calculated. When V,= 0, Ef= EB, v,= v~,, the problem degenerates to that of an isotropic fiber embedded in an infinite matrix (Erdogan and Gupta’s problem [9]). It can be seen in Figs 4 and 5, as well as in Tables 1 and 2, that the pressent results match their results exactly. Normalized stress intensity factors (k, (a))/(a,&) of the fiber crack are calculated for various situations, and results are given in Figs 4-7 and Tables 14. The effect of fiber volume fraction on the normalized stress intensity factors is given in Figs 4 and 5 and Tables 1 and 2. The effect of fiber-matrix mechanical property mismatch on the crack propagation is given in Fig. 6 and Table 3. The effect of fiber mechanical anisotropy on the fiber crack behavior is presented in Fig. 7 and Table 4. The results in Fig. 6 and Table 3 can be understood in two ways. One interpretation is that for a given matrix stiffness, the stiffer the fiber, the more likely the fiber crack will propagate; Table 3. Normalized stress intensity factors vs normalized crack lengths for various E,/E, fractions. E, = E,, . v, = v,~, E,I-&
Vf
alR
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
k,(a) GO a 1.oOOo 1.1717 1.2827 1.3622 1.4224 1.4696 1.5077 1.5390
1.5653 1.5876 1.6068 1.6235 1.6382 1.6512 1.6628 1.6732 1.6826 1.6911 1.6988 1.7059 1.7124 1.7184 1.7239 1.7291 1.7338 1.7383 1.7424 1.7463 1.7500 1.7534
Table 4. Normalized stress intensity factors crack lengths for various E,,/E, fractions. v,=v,,,, V,=O.6
vs normalized E,= Em = 10,
k,(a) alR 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
E,,IEf=
1
1.3184 1.3544 1.4126
1.4907 1.5876 1.7048 1.8503 2.0495 2.4082
G E,,/E,=4 1.3204 1.3558 1.4128 1.4893 1.5838 1.6978 1.8386 2.0307 2.3751
E,,/E,=
9
1.3209 I .3562 1.4131
1.4892 1.5833 1.6968
1.8368 2.0276 2.3696
YONG
822
LI XU and K. L. REIFSNIDER
the secondary interpretation is that for a given fiber stiffness, the stiffer the matrix, the less likely the fiber crack propagation will take place. It is seen by comparing results in Figs 4 and 5 that. from the viewpoint of avoiding fiber cracking, “reverse composites” (composites in which the fiber is less stiff than the matrix, for example, in a Nicalon/SiC ceramic composite) will be a better choice. It is also observed that the fiber mechanical anisotropy seems to have little efl‘ect on the fiber cracking (for transverse loads). Acknowledgement-This study is supported by the National Science Foundation (Center for High Performance Polymertc Adhesives and Composites) under grant No. DMR 8809714, and by the Virginia Institute for Material Systems
REFERENCES normal loading of a unidirectional composite. .I. c’onrpes. ~a/er. I, 152 164 (1967). D. F. Adams, Inelastic analysis of a unidirectional composite subjected to transverse normal loading. J. cnmpo,s. bfarer 4, 310 (1973). R. L. Foye, Theoretical post-yielding behavior of composite laminates, part I-inelastic micromechanics. J. conrpo.,. Mater. 7, 178 (1973). D. F. Adams, Elastoplastic crack propagation in a transversely loaded unidirectional composite. J. cotnpos. ilfarer. 8, 38 (1974). G. P. Sendeckyj (Ed.), Micromechanics. Academic Press. New York (1974). D. F. Adams, A micromechanical analysis of crack propagation in an elastoplastic composite material. Fiber .%I. Technol. 7, 237-256 (1974). D. R. Owen and J. F. Lyness, Investigation of bond failure in fiber-reinforced materials by the finite element method. Fiber Sci. Technol. 5, 129-141 (1972). M. R. Wisnom, Factors affecting the transverse tensile strength of unidirectional continuous silicon carbide fibre reinforced 6061 aluminum. J. compos. Mater. 24, 707-726 (1990). F. Erdogan and G. D. Gupta, The inclusion problem with a crack crossing the boundary. Inr. J. Frucrurr 11, 13- 17 (1975). F. Erdogan, G. D. Gupta and T. S. Cook, Numerical solution of singular integral equations, m Mechanics oJ‘Fra(,lure. Volume 1: Methods of Analysis and Solurions qf Crack Problems (Edited by G. C. Sih), pp. 368425. Noordhoff, Leyden (1973). N. I. Iaokimidis and P. S. Theocaris, On the solution of collocation points for the numerical solution of singular integral equations with generalized kernels appearing in elasticity problems. Comput. Sfructwzr 11, 289-295 (1980). I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals. Series, and Products. Academic Press, New York (1980).
[l] D. F. Adams and D. R. Doner, Transverse [2] [3] [4] [S] [6] [7] [8] [9] [lo]
[l I] [12]
APPENDIX
A
The following is a justification of the fact that, even though an elastic body is transversely isotropic, in the isotropy plane x-y, the compatibility equation is still a biharmonic equation, VV@ = 0. Using the same coordinate system as in Fig. 1. the constitutive equations for a transversely isotropic material are
where E is Young’s modulus in the x-1 plane, E3 is the Young’s plane, %I, v23, v32 and Ye, are the general Poisson’s ratios. Accordingly, we have
Using
the identities
For the plane strain
in (A2), we can rewrite
case, we have
(Al
) as follows:
modulus
in the -_ direction.
I*is Poisson’s
ratio in the I
i‘
Micromechanical analysis of fiber crack propagation
a23
Substituting (A4) into (A3), we may obtain cc: = E3
(AS)
Substituting (AS) into (A3a) and (A3b) yields
(fw with
It can be readily shown that
Assume the Airy stress function Cgand (A9) Then
7Z.V =
31 + “h). =---* 2(f + v) Peg E
E
axay
WO)
Substituting (AlO) and (A8) into the compatibility equation as follows:
a5 ,. a4,, -2+-i-I-=o’ ay
a*yyy axay
(All)
=o.
(AU
we obtain ;vv@
Therefore, the compatibility equation is still a biharmonic equation.
APPENDIX B. FORMULATION OF THE DISLOCATION PROBLEM As is shown in Fig. 2, in the corresponding dislocation case, instead of the crack, there is an edge dislocation with Burgers’ vector (0,6,, 0) embedded in the fiber at r = t, B = 0. To formulate the problem, plane polar coordinates are used and Michell’s general (interior, annulus and external) solutions are utilized for all three of the domains, respectively. However, due to the presence of the dislocation in the fiber, the stress function for the fiber is constructed as the sum of Michell’s interior solution and that of an infinite plane containing an edge dislocation. For the problem under consideration, the stresses and displacements should also satisfy the following symmetry conditions: o,,(r, @ = urr(r, -8)
(Bt)
%lJP. @)= Qo&, - 0)
(B2)
r,,,o(r.0) = -r&r, ~~(6 8) = a,(~,
-8)
-8)
uct@,0) = --Il&, -0).
(B3) (B4) (BS)
After taking regufarity and symmetry conditions into consideration, stresses and displacements in the fiber, matrix and composite can be written as follows. (a) Fiber By referring to Michell’s interior stress function and by superposition with the solution of an infinite media embedded edge dislocation, the stresses and displacements in the fiber may be written as follows:
u,,,(r,8)=al!l(r,0)+a!31(r,0) 0~ (rr 0) = u&X(rr 0) + c&I,(r, 0)
824
LI XU and K. L. REIFSNIDER
YONG
ii I,
(l’.
fll
:y
IPO0) rl .
f
7f:;‘lr.l)).
(Bhl
where the first term on the right side in all the above equatmns is the stress or displacement obtamed from Michell’s interior stress function; the seeond term is the stress or displacement of an infinite media with an embedded edge dislocation (see the following):
where co,, d,,, S2, a,,, and h,,,are integral
constants. 2’ sin2 (Jtr cos 0 - t) iBl?i ~r~+r~-WfOSO)‘
----
1
co5 0 - i)(r -
f co5
0) iHl.3)
(P
+
/‘-
7rt
cos
H)-
7(r cos 0 - r)(r - I cos O)-' (V'-kr?-2rt costi):
(614)
/r sin-fi
(BIS!
In order to combine the infinite plane solution at interface Y = R, we now express the displacements in terms of the following Fourier series:
with the Fourier
coefficients
determined
by
with Michell’s interior solution and to satisfy the continuity condition in eqs (B I 5) and (B 16). the stresses in eqs (B 12) and (B 13). with I = R
Micromechanical analysis of fiber crack propagation
825
u$‘(R, 0) dB
(824)
n(l +X1-,)2 * C,(r) =-.--b,. ~* o u$‘(R 1 0)cos nfl d0
(825)
X(i +K,-,)2 D,(t)=-%--
(B24)
s
*
u&R.
@sin no d8.
* s0
With the aid of tables in 1121,integrals in eqs (821 j(B26) can be evaluated in closed form and the results are as follows:
(827)
A (+(n+?)r”” ”
nt”-’ --3 (n3.22)
(B28)
R”
R”+?
B,(t) = - g
(B29)
1 - $+f
B,(t)=
(B30)
B,o=-(n~~~t~‘+‘+~~,
(n-&2)
(B31)
C&f = f
(~32) 1 + ti,-, + !!CZ!
‘,‘f’=g&c(t)=!
n
_(n
2i
3 tZ m=~R’--_ D&j=;
I
-
+ I -ti,-,y-
-...(n - l)R”-’
xi_,+5
log,R (n +2)1”+’ +(n+l)R”+’
@~2)
1 +
q_,- 1 - 1--logzR
(B33)
(B34)
(B35)
-l-(-l)nill (n>2). +q_,,u (n - ---f 1n(l
(n -
1-li_,fPt)R”-’
(n c2)t”+’
(n + I)R’+’
..-
-
2(& -1)
’
(B34)
’
(b,i Mc2tri.v The stresses and displacements in the matrix can be written as - 2’12+ 2d,,r
CT,,? (r, Q) = !!E + 2c,, + r2
C
r3
-S[~,n(nl)r’-‘cb,,(n
q),Jr,
0) = - “,? 0’ f -C ’ .*_ 3 + (‘:;A+ T +-y[u,n(n-f)r”-?
>
cos 0 +I)(n
-2)r”+c,gt(n
+l)r-‘“+2)+dnZ(n
- I)(n +2)r-“]cosnB
(837)
-I)@--2)~“]costzU
(B38)
6d ,? r ) cos 0
+b,zfnfI)fn+2)r”fc,,zn(n+l)r-‘^*”+d~,(n
+ b,zn(n + i)tn-
~,~n(rt +
l)rm~(“C2)-d,2n(n
- l)rm”]sinnO
(B39)
I I - bozr + (tini,, - Ik,,,r + c,::I? - + (IX,,, - 2)d,?rZ cos U
u,?(r, 0) = -!-
b”,
I
f I[ - nu,? r’s-. ’ -(n -K,,,f I)h,,Zr”+‘+nc’,,zr-‘l-’ 1
+(n +K,,-
l)dnSr-“+‘]cosnO
(B40)
I +~[-~na,zr”-‘+(n+ti,+l)b,~r”+‘fn~~2r~”-i+(n-i-ti,,)d~zr-“+’]sinn# where bOz. co2.cs2,dlZ. &. c+, h,, ml and dn2 are all integral constants.
(B41)
826
YONG
Lt XU and K. L. REIFSNIDER
(c) Composite
The stresses and the displacements
whereb,,,
ct3,
in the external
S.,, c,,~ and d,, are all integral
composite
can be written
as follows:
constants.
(d) Boundary conditions
After assuming the basic form of the solutions, the unknown constants in these expressions applying the following boundary and continuity conditions: n,,,(R, 0) = a,,,(R.
may be determined
by
tj)
T,,ilfR.H)=5,(12(R,H) l~~~(R,~)=~,~~R.~) zt<,j(R, 0) = u,>,(R. 0) rr,,,(b, 8) = “r,J(h, 0) r,02ih. 0 ) = Tr,ii(h. H ) u,L(h. Ul = tc,,(h, 0) n,,,(h. 0) = u,,,th. 0).
After equations:
substituting
eqs (B7)-fBl6)
and (837)~(B46)
into (B47). we obtain the following h 02 %, ---R’
1B471
three sets of simultaneous
29 hl z
(848)
(B49)
Microm~hani~I n(1 -t~)R”-~u,,
+ (n + I)(2 -n)R”b,,
+n(n
- l)R”-2a,z c(n + I)(n - 2)R”bnz +n(n
n(n - I)R”-*a,,
+n(n
827
analysis of fiber crack propagation
+ 1)R-“-2c,2i-(n
- l)(n +2)R-“d,,=
-
+ l)R”b,,, - n(n - I)R^-*Q,, - n(n + i)R”b, e-.tbj An
+n(pt+l)R1”-2e,,+n(n-1)R-“d,,=
--!--~R”-la,,+.--l-(~r_,7% 2P,- I
I-n)R*“b.,+InR”-‘~~~+~(~
m
%4n
I
.-!.- nR”- $I,, + --$, 2i+ I 2Pf-,
+ 1 f rc/_,)R”+ lb,,, - -!- nR”- ‘a,, -&I m 2&
+x,_
+ I)b-“-‘c,,+(n
))R-“+‘dnz=
2&l
b.C - ---:= n(i + Xf-,)
+ 1 + q,,)R”+‘bn2
_L,R-‘-l,_._‘,, 2Pm i- l)(n -2)b”b,,,+n(n
n(I + q-,)
-IC,,,+ l)R”+‘bj,*
_.~nR-+.$-(n 2fh!
n(n - l)b”-‘un2+(n
ttf-tbhi x(1 + K,-,)
_ 1 -K,)R-“+‘dx?=
-
2Pm
D,b, n(1 -I-Kf-,f
- I)(n +2)b-“d, - n(n + 1)b -“-*c,, - (n - I)(n + 2)b -“dnj = 0
n(n-I)b”-
a,2fn(n+I)b”b,,-n(n+I)b-“-2c,,-n(n-l)b-”d,z+n(n+I)b-“-2c,,fn(n-I)b-“d,,=O
2
I -~b”-‘a,+-t(n+l+K,)b”+~~~2+-i-nb-”-i~~~+~(~-l-K,)b-n+~d~~ 2Pm 2&i 2& 2&n I
--
2/L,
nb -*- ‘c,, -I;i.(n
-$f--)B-n+id~~=O.
NO)
with & PW=2(1 iv,)’
K,=3-4V,,
4-t
x,._,=
pe-‘=2(1+v,_,f’
&_,=
E,-, 2t1 + v,-,) ’
3 -v,-*
K(_, = -
1 + v, - g
3 -4v,+,
and A,, B,, C, and D, are functions of t given by eqs (B27)-(B36). After determining all these constants, stresses and displacements may be obtained by eqs (B6)-(816) and (B37)-(B46).
APPENDIX C. UNCRACKED GEOMETRY SOLUTION The geometry and loads of the untracked case are the same as those of the cracked case, as is shown in Fig. 1, except that there is no crack in the fiber. To formulate the untracked problem, we assume proper stress functions for the fiber, the matrix and the external composite, and then match the boundary and continuity conditions. The unknown constants in these stress and displacement expressions can be determined. The stresses and displacements in all three domains can thus be obtained. After taking regularity and symmetry conditions into consideratjon, stresses and disp~a~ments in the fiber, matrix and composite can be written as follows. (a) Fiber o,,,(r, @) = 2c,, - Za,, cos 28
Kfi
(r,,,,,(r, @) = 2c,, + (2azI + 12bz,r2jcos 28
(C2)
f&r,
(C3)
8) = (2az, + 6b,, r’)sin 28 +(3-ri,_,)bZ,r3]cos28j
u,,(r,@)=-!---
{(K,_,- l)e,,r -[2a,,r
u&r, 6) = --!-
[2az, r + (3 + ‘c, _,)b,, r3fsin 28
2P,-,
2&-r
(C4) 0)
828
YONG
LI XU and K. L. REIFSNlDER
with
where L;,,,
uzI and h,, are Integral constants.
fh) Matrix The stresses
T,,l”,
and displacements
in the matrix
(
3c,:
cr.01=
2u22+3h,,r’---~
r4
can be assumed
(I,? r2 1
in the form
sm2lJ
k’ -1
uom(r,t))-j~ 2~:,r+r,:Zi+(3+ri,,)h2~rl-ij-2--I. m
I
il
sin Zfl 1
I
with “*i=zii;:, where ho?.
-)’ ‘“8
K,,,=i-441.
/,,.
coo2, az2.c2?and t& are all integral constants.
(~‘1 Composite The stresses and the displacements
I,,~,tr. 0)
= -
in the external
composite
can be written
as follows.
i
with r; , p, ,E---~~ , k‘s ,-3 -4y 3 I + y ,I where hOi. cz7 and dz, are constants, (d) Boundary
and 0,) is the traction
applied
:.
at infinity.
conditior1.v
Having assumed the basic form of the solutions, the unknown applying the following boundary and continuity conditions: a,.;(R.(JJ
constants
in these expressions
may bc determined
by
=rr,,,,,!R.iJ)
:,!ji ( R. tJ ) =
i, ,/,I, t R. 0 1
rr,,(R.IJ)-t/
,,,,(R.(J)
I,!,,t R, II ) =
it,,,,, i R. 0 I
m,#,,(h,~l)==r7,,, fh, iii i ,,,,,, I/J.0 1= T,,.,(h. f/ 1 itr,,,(h. 0) = u,,(k 0) rr,,,,,(h. (J) = u,i, (h. (J 1
(Cl61
~icromechan~~l
analysis of fiber crack propagation
829
By so doing, we may obtain the following two sets of s~muitaneous equations: (Cl-i-I)
(C17-2)
fC17-3)
(C17-4)
(018-i)
(Cl&2) a,, R --+-2jl_, Pf- I
(q_ 8- 3)R 3b,, _ t R%, Pm
c,, (I + KJdv (Km- 3)R3bx -““-_-~:=o P”,R3 2&R 2Pm
(Cl&3)
(Cl&4)
(Cl%5)
(Cl&6)
(Cl 8-7)
(Cl&S)
After solving the simultaneous equations, stresses and displacements in all three domains can be determined by eqs (CI)-(C 15). (Receiwd
15 May 1992)