International Journal of Fracture (2005) 131:155–172
DOI 10.1007/s10704-004-3636-6
© Springer 2005
Interaction between a radial matrix crack and a three-phase
circular inclusion with imperfect interface in plane elasticity
K. KIM and L.J. SUDAK∗
Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary, Alberta
T2N 1N4, Canada
∗
Author for correspondence. (E-mail: sudakl@enme.ucalgary.ca)
Received 23 February 2004; accepted in revised form 22 September 2004
Abstract. The solution for the elastic three-phase circular inclusion problem plays a fundamental role
in many practical and theoretical applications. In particular, it offers the fundamental solution for the
generalized self-consistent method in the mechanics of composites materials. In this paper, a general
method is presented for evaluating the interaction between a pre-existing radial matrix crack and a
three-phase circular inclusion. The bonding at the inclusion-interphase interface is considered to be
imperfect with the assumption that the interface imperfections are constant. On the remaining boundary, that being the interphase-matrix interface, the bonding is considered to be perfect. Using complex
variable techniques, we derive series representations for the corresponding stress functions inside the
inclusion, in the interphase layer and the surrounding matrix. The governing boundary value problem
is then formulated in such a way that these stress distributions simultaneously satisfy the traction free
condition along the crack face, the imperfect interface conditions and the prescribed asymptotic loading conditions. Stress intensity factor (SIF) calculations are performed at the crack tips for different
material property combinations, imperfect interface conditions and crack positions. The results illustrate convincingly the role of an interphase layer as well as the effects of an imperfect interface on
crack behavior. For instance, when the interphase layer is softer than the inclusion and matrix, the
results show that the radial matrix crack will propagate from the nearby crack tip regardless of the
level of the imperfect (spring-layer) interface parameter. In comparison, when the interphase layer is
stiffer than the inclusion and matrix, the interphase layer will shield the crack from effects of the
imperfect (spring-layer) interface. Hence, these results provide a quantitative description of the interaction problem between a three-phase inclusion with interface imperfections and a radial matrix crack.
Key words: Crack, crack/inclusion interaction, imperfect interface, three-phase inclusion.
1. Introduction
The interaction between cracks in the surrounding matrix and nearby fibers (inclusions) is an important problem when attempting to understand and predict the
strengthening and hardening mechanisms of composite materials. For example, thermal mismatch between the fibers (inclusions) and the surrounding matrix may lead to
high residual stresses in the vicinity of the inclusion-matrix interface. These stresses
can be tensile in nature and lead to crack propagation or interface separation. Therefore, the solution of appropriate elasticity problems, dealing with matrix cracking,
provides valuable insight into the mechanical behavior of composite materials.
During the last several decades, the study of material defects (such as dislocations
and cracks) has received a considerable amount of attention in the literature concerning the mechanical behavior of composite materials. Dundurs and Mura (1964)
156 K. Kim and L.J. Sudak
solved the problem of a circular elastic inclusion near an edge dislocation in terms of
the Airy stress function. They found that under certain conditions the dislocations
might have a stable equilibrium position at some distance away from the interface.
This result suggests a so-called trapping mechanism of the motion of dislocations
near the inclusion. Tamate (1968) discussed the influence of a circular inclusion on
the stress state surrounding a matrix crack subject to tension. He showed that a
compliant inclusion increases the stress intensity factor (SIF) while a stiff inclusion
decreases it; in addition, he also showed that the effect of the inclusion is greater
on the nearby crack tip than on the distant crack tip. Atkinson (1972) and Erdogan
et al. (1974) used the dislocation density method to study the interaction between a
crack and an inclusion. In this case, the derivations of the governing integral equations for the unknown density functions are based on the conditions that mechanical
tractions vanish along the crack face.
In the majority of the aforementioned two-phase (inclusion/matrix) fracture
mechanics models, the basic assumption that has been made is that both displacements and tractions are continuous across the inclusion-matrix interface – the
so-called perfect bonding model. In many practical problems however, various kinds
of interfacial damage, for example, damage arising from imperfect adhesions, microcracks and voids makes the perfect bonding assumption inadequate. In these kinds of
situations, it becomes necessary to model the interface as an imperfectly bonded interface incorporating the effects of interface imperfections (see, Aboudi, 1987; Hashin,
1990; Hashin, 1991; Sudak et al., 1999). Recently, Stagni (1999) investigated the
image glide force on an edge dislocation inside an elliptic inclusion. The matrix-inclusion interface is assumed to be slipping. He showed that, unlike the perfect bonding case where the force field is insensitive to the inclusion shape, for the slipping
interface the shape effects are essential, ranging from a ‘matrix-shielding’ effect to the
appearance of new stable-equilibrium loci. Liu et al. (2001) examined the effects of
an imperfect interface on the SIF calculated at a radial matrix crack in a two-phase
system. Their results show that interface imperfections have a significant effect on the
SIFs. Amenyah et al. (2001) studied the effects of an imperfect interface on the stress
fields inside a circular elastic inclusion containing a pre-existing interior radial crack
subject to thermal loadings. Their results clearly demonstrate that interface imperfections have a pronounced effect on the SIFs at the crack tips and should not be
ignored.
It is well known that the behavior of composite materials (such as load transfer between fiber and matrix) is significantly affected by the presence of an interphase layer (i.e. a non-uniform thin intermediate zone separating fiber from matrix).
Consequently, the study of three-phase (fiber/interphase/matrix) elastic inclusions is
of great practical and theoretical importance (see, Christensen and Lo, 1979; Benveniste et al., 1989; Luo and Weng, 1989; Christensen, 1990). In view of this, the
problem of a three-phase inclusion interacting with material defects (such as dislocations and cracks) is receiving an increasing amount of attention in the literature
(see, Huang et al., 1994; Huang et al., 1996). In fact, recent works have shown that
radial matrix cracking is the most common and dangerous mode of failure (see, Lu
et al., 1991; Müller and Schmauder 1993; Chandra et al., 1997). Figure 1 clearly
depicts extensive radial matrix cracking occurring in a MoSi2 /SiC composite system
(see Lu et al., 1991 for details). Luo and Chen (1991) obtained solutions for the
Interaction between a radial matrix crack and a three-phase circular inclusion 157
Figure 1. Electron micrograph showing radial matrix cracking in a MoSi2 /SiC system (from Lu et al.
1991).
stress fields due to an edge dislocation embedded within the interphase layer of a
three-phase inclusion. It is shown that, in contrast with the two-phase model adopted
by Dundurs and Mura (1994), the three-phase model allows the dislocation to have
a stable equilibrium position under much less stringent combinations of the material
constants. Xiao and Chen (2001) obtained solutions for the elastic stress fields due
to an exterior edge dislocation located near a three-phase circular inclusion. They
showed that when the interphase layer is thick the elastic properties of the inclusion
have no significant effect on the force on the dislocation. Alternatively, if the interphase layer is thin, both the shear modulus and Poisson’s ratio of the inclusion and
interphase layer can influence the equilibrium position and the stability of the dislocation. The related problem of a screw dislocation interacting with a three-phase
circular inclusion is solved by Xiao and Chen (2000), Xiao and Chen (2001) studied
the interaction between a radial matrix crack and a three-phase circular inclusion.
They used the solution of an edge dislocation near the coated inclusion (see Xiao
and Chen, 2001) as the Green’s function and reformulated the problem into a set of
singular integral equations. Cheeseman and Santare (2001) investigated the problem
of a radial and circumferential matrix crack interacting with a circular inclusion surrounded by an interphase layer. The crack is modeled as a distribution of dislocations yielding a singular integral equation. They showed that a compliant interphase
layer increases the mode-I SIF for radial cracks while a stiff interphase layer shields
the crack from the inclusion relative to the no-interphase case. However, like the
two-phase problem, the majority of the three-phase fracture mechanics models only
consider the classical perfect bonding conditions.
Recently, Wang and Shen (2002), obtained an elastic solution derived in a decoupled manner for the interaction problem between an edge dislocation and a threephase circular inclusion with circumferentially homogeneous sliding interface. They
used the dislocation density method to model the crack in the interphase layer and
they showed how the sliding interface influences the SIF. Sudak (2003) obtained a
series respresentation for the elastic stress fields due to the presence an exterior screw
dislocation interacting with a three-phase circular inclusion. The results demonstrate
that the relative thickness of the interphase layer, the influence of the imperfect
158 K. Kim and L.J. Sudak
y
Matrix, S2
µ2, n2
Imperfect Interphase
2
R1
R0
0
a
Inclusion, S0
µ0, n0
Perfect Interphase
Interphase Layer, S1
µ1, n1
b
x
d
Radial crack
1
Figure 2. A three-phase circular inclusion with a radial matrix crack.
bonding condition and the material property combinations were manifested by their
effects on the equilibrium position and subsequent stability of the dislocation.
In most of the problems dealing with the interaction between a two-phase or
three-phase inclusion and a crack under either perfect or imperfect bonding assumptions, the existing most popular method is the dislocation-density method, which
assumes that the crack can be modeled as a distribution of dislocations with
unknown density. Aside from an earlier study involving a slipping interface (see
Wang and Shen, 2002), to our knowledge, the three-phase inclusion/crack interaction
remains to be investigated in the case of imperfect bonding. This can be attributed
to the fact that the extension of the dislocation density method to study three-phase
inclusion/crack interactions with imperfect bonding requirements meets two major
difficulties. First, the fundamental solution for the interaction between an isolated
dislocation and a three-phase inclusion with a generally imperfect interface is not yet
available and secondly, numerical solutions of the resulting singular integral equations
for imperfect interfaces is extremely challenging.
The objective of the present paper is to develop a simple series method to study
the interaction between a radial matrix crack and a three-phase circular inclusion
under the assumption of imperfect bonding at the inclusion-interphase interface. The
results clearly demonstrate that the series method is simple and effective in describing the role of the interphase layer and the effects of the interface imperfections on
the radial matrix crack for a variety of different material property combinations and
crack positions.
2. Problem formulation
Consider a domain in R 2 , infinite in extent, containing a single circular elastic
inclusion, which is bonded to an elastic matrix through a single coaxial circular interphase layer. The linearly elastic material occupying the inclusion, the interphase layer
and the matrix are assumed to be homogeneous and isotropic with associated shear
modulus µ0 , µ1 and µ2 , respectively. The inclusion, with center at the origin of the
coordinate system and radius R0 , occupies a region denoted by S0 . The interphase
layer, with radius R1 , occupies a region denoted by S1 and the matrix with a preexisting radial crack occupies a region that is denoted by S2 . The inclusion-interphase
Interaction between a radial matrix crack and a three-phase circular inclusion 159
interface and the interphase-matrix interface is denoted by the curve k (k = 0, 1),
respectively (see Figure 2). Furthermore, unless otherwise stated, the subscripts 0, 1,
and 2 will be used to denote quantities in S0 , S1 and S2 , respectively.
For plane deformations, the elastic stresses and the associated displacements can
be given in terms of two complex potentials ϕ(z) and ψ(z) as Muskhelishvili (1963):
2µ(ur + iuθ ) = e−iθ κϕ(z) − zϕ (z) − ψ(z) ,
σrr + σθθ = 2 ϕ (z) + ϕ (z) ,
(1)
σrr − iσrθ = ϕ (z) + ϕ (z) − e2iθ z̄ϕ (z) + ψ (z) ,
where z = x + iy = reiθ is the complex coordinate, κ = 3 − 4ν for plane strain and κ =
(3 − ν)/(1 + ν) for plane stress. Here µ and ν are the shear modulus and Poisson’s
ratio, respectively. In addition, the Cartesian components of resultant force acting on
the left of an arbitrary arc AB in the elastic body is given by Muskhelishvili (1963):
B
Fx + iFy = − ϕ(z) + zϕ (z) + ψ(z) ,
(2)
A
where [f (∗)]BA = f (A) − f (B), is independent of the path.
Let us further assume that the inclusion is imperfectly bonded along the circular
curve 0 , whereas the interphase layer is perfectly bonded along the circular curve 1 .
In view of this, the condition along 0 is given by Sudak and Mioduchowski (2002):
σrr − iσrθ = 0,
σrr = mur − mu0r ,
σrθ = nuθ − nu0θ , z ∈ 0 ,
(3)
and the condition along 1 is given by:
σrr − iσrθ = 0,
ur = 0, uθ = 0, z ∈ 1 ,
(4)
where m and n are non-negative and constant interface parameters. Physically,
these interface parameters characterize the strength, stiffness and overall degree
of adhesion along the material interface and they are described by a simple,
straight-forward constitutive relationship (see, Bigoni et al., 1998). The expressions
∗ = (∗)1 − (∗)0 and ∗ = (∗)2 − (∗)1 denote the jump across k (k = 0, 1), respectively and u0 is the displacement induced by the uniform stress-free eigenstrains
0
(εx0 , εy0 , εxy
) prescribed within the inclusion which might be the result of thermal mismatch between the matrix and the inclusion. In addition, note that when m = n = 0,
Equation (3) represents the traction-free boundary condition, and if m = n = ∞, condition (3) corresponds to a perfectly bonded interface. In fact, the interface model
given above can be viewed as a thin compliant elastic interphase layer (see Hashin,
1990) or a continuous distribution of linear springs (see Achenbach and Zhu, 1989;
Achenbach and Zhu, 1991). The only restriction is that the interface parameters, m
and n, are positive in the latter case and m > n in the former case. One of the advantages of the imperfect interface model used in (3) is that it permits representation of
intermediate states of bonding between the inclusion and the interphase layer from
perfect bonding to complete debonding.
160 K. Kim and L.J. Sudak
It has been shown in Sudak et al. (1999), the displacements induced by the uniform eigenstrains can be written in the following form:
m+n
m−n
(ε2 − iε3 )z2 +
R03 (ε2 + iε3 ), z ∈ 0 , (5)
(mu0r − inu0θ ) = mR0 ε1 +
2R0
2z2
where
ε1 =
εx0 + εy0
2
,
ε2 =
εx0 − εy0
2
,
0
ε3 = εxy
.
As will be seen below, adequate use of complex variable representation leads to a
concise formulation of the current boundary value problem. Thus, the boundary
value problem, given by Equations (4) and (3), take the following form, respectively
2
2 R1
R1
µ2
µ2
2
(1 + κ2 )zϕ2 (z) = 1 + κ1
+ zψ1
,
zϕ1 (z) + 1 −
z ϕ1
µ1
µ1
z
z
2
R1
µ2
(1 + κ2 ) zψ2 (z) + R12 ϕ2 (z) = κ2 − κ1
zϕ1
µ1
z
µ2 + κ2 +
zψ1 (z) + R12 ϕ1 (z) , z ∈ 1 ,
µ1
2
2
R
R02
z
z2
0
ϕ1 (z) + ϕ1
(6)
− zϕ1 (z) − 2 ψ1 (z) = ϕ0 (z) + ϕ0
− zϕ0 (z) − 2 ψ0 (z),
z
z
R0
R0
R02
R0
z
(m − n)(1 + κ1 ) ϕ1 (z) + (m + n)(1 + κ1 ) ϕ1
z
R0
z
R02
µ1 R0
z2 = 4µ1 ϕ0 (z) + ϕ0
− zϕ0 (z) − 2 ψ0 (z) + (m − n) 1 + κ0
ϕ0 (z)
z
µ0 z
R0
2
R
R02
µ1
R
0
0
+ (m − n) 1 −
+ ψ0
R0 ϕ0
µ0
z
z
z
µ1 z R02
+ (m + n) 1 + κ0
ϕ
µ0 R0 0 z
µ1
z
2µ1 (m + n)(ε2 − iε3 ) 2
+ (m + n) 1 −
R0 ϕ0 (z) + ψ0 (z) + 4mµ1 R0 ε1 +
z
µ0
R0
R0
2µ1 (m − n)(ε2 + iε3 )R03
+
, z ∈ 0 .
z2
Note that all functions appearing on both sides of Equation (6) have been written
with the understanding that they may be represented as power series expansions in
the variable z. Consequently, condition (6) requires that the power series expansion
on both sides of (6), have the same coefficients.
The uniform remote loading at infinity is described by:
ϕ2 (z) = Az + O(1),
ψ2 (z) = Bz + O(1), |z| → ∞,
(7)
Interaction between a radial matrix crack and a three-phase circular inclusion 161
where A is a given real number and B is a given complex number which are deter∞
mined by the remote principal stresses (σx∞ , σy∞ , σxy
) as follows:
σx∞ + σy∞ = 4A,
∞
σx∞ − iσxy
= 2A − B.
(8)
∞
In the case of a uniaxial load normal to the crack, σx∞ = 0, σy∞ = σ ∞ and σxy
= 0,
then A and B are given by:
A=
σ∞
,
4
B = B = 2A =
σ∞
.
2
(9)
A convenient method used to analyze problems with circular boundaries is the
series method. However, in the present problem, the domain S2 contains a crack of
length 2 so that ϕ2 (z) and ψ2 (z) are not analytic outside the circle 1 . As a result,
ϕ2 (z) and ψ2 (z) cannot be expanded in a standard Laurent series in S2 . To overcome
this difficulty, a method based on analytic continuation (see Muskhelishvili, 1963) is
employed to express ϕ2 (z) and ψ2 (z) in terms of two new functions which are analytic outside the circle 1 and can subsequently be expanded into standard Laurent
series. To this end, denote by D the domain outside the circle 1 minus the matrix
crack 2. Clearly, ϕ2 (z) and ψ2 (z) are analytic in D but not in S2 . In view of (2) and
the traction free condition along the crack-face 2, we can express this condition in
the upper and lower half planes as follows:
ϕ2 (z)+ + zϕ2 (z)+ + ψ2 (z)+ = 0,
z ∈ 2+ ,
ϕ2 (z)− + zϕ2 (z)− + ψ2 (z)− = 0,
z ∈ 2− .
(10)
According to the symmetry principle of analytical continuation (i.e. ϕ(z̄)+ = ϕ(z)−
across the real axis) we can rewrite (10) in a convenient form as follows:
−
ϕ2 (z)+ + zϕ2 (z) + ψ2 (z) = 0, z ∈ 2,
+
ϕ2 (z)− + zϕ2 (z) + ψ2 (z) = 0, z ∈ 2.
(11)
Subtracting the above two equations yields:
+
−
ϕ2 (z)+ − zϕ2 (z) + ψ2 (z) = ϕ2 (z)− − zϕ2 (z) + ψ2 (z) ,
z ∈ 2.
Then, if we define a new analytic function X(z) in S2 as follows:
X(z) = ϕ2 (z) − zϕ2 (z) + ψ2 (z) ,
(12)
(13)
Equation (12) will imply that X(z) is continuous across the crack 2. Hence, X(z) is
analytic in D which is the domain outside the circle 1 minus the matrix crack 2,
thus, it can be expanded into a Laurent series in D as follows:
X(z) = −Bz +
∞
k=1
ak z−k ,
z ∈ D,
(14)
162 K. Kim and L.J. Sudak
where ak (k = 1, 2, . . . ) are the undetermined complex coefficients. Furthermore, the
remaining crack face boundary condition (11) can be written as follows:
+ −
(15)
ϕ2 (z) + zϕ2 (z) + ψ2 (z) + ϕ2 (z) + zϕ2 (z) + ψ2 (z) = 0, z ∈ 2.
Using Plemlj’s formula Muskhelishvili (1963), Equation (15) can be defined in terms
of another new analytic function Y (z) in S2 as follows:
√
(z − a)(z − b) Y (z) =
ϕ2 (z) + zϕ2 (z) + ψ2 (z) ,
(16)
z
√
√
where (z − a)(z − b)+ = − (z − a)(z − b)− = 0, z ∈ 2 = [a, b] .
From Equations (15) and (16), it is clear that Y (z) is continuous across 2 and analytic in D and can be expanded into a Laurent series in D as follows:
∞
Y (z) = 2A + B z +
bk z−k , z ∈ D,
(17)
k=1
where bk (k = 1, 2, . . . ) are the undetermined complex coefficients. Consequently, from
Equations (13) and (16), the complex potentials ϕ2 (z) and ψ2 (z) defined in the matrix
containing the crack can be written in terms of X(z) and Y (z) as follows:
zY (z)
X(z)
,
ϕ2 (z) = √
+
2
2 (z − a)(z − b)
(18)
ψ2 (z) = ϕ2 (z) − zϕ2 (z) − X(z).
(19)
and
z
in Equation (18) is a multi-valued function across
The term F (z) = √
2 (z − a)(z − b)
the crack face but analytic in the domain 1 ∪ S1 ∪ 0 ∪ S0 . It can be expanded in a
Taylor series as follows:
∞
z
z
ck z k ,
F (z) = √
= (−a + z)−1/2 (−b + z)−1/2
2 (z − a)(z − b) 2
k=1
(20)
where the undetermined coefficients c1 , c2 , c3 . . . are determined in terms of the crack
tip positions a and b.
The complex potential ϕ1 (z) and ψ1 (z) are analytic within the interphase layer and
can be expanded into a standard Laurent series in domain S1 as follows:
ϕ1 =
∞
k
dk z ,
ψ1 =
k=−∞
∞
ek z k ,
(21)
k=−∞
also the complex potential ϕ0 (z) and ψ0 (z) are analytic within the inclusion and can
be expanded into a Taylor series in domain S0 as follows:
ϕ0 =
∞
k=0
fk zk ,
ψ0 =
∞
k=0
gk zk
(22)
Interaction between a radial matrix crack and a three-phase circular inclusion 163
where dk , ek , fk , and gk are the undetermined complex coefficients. Further, using the
expansion of F (z), we can expand ϕ2 (z) and ψ2 (z) at the interface as infinite Laurent
series in positive and negative powers of z. Finally, having expressed all six complex
potentials in series form, the problem is reduced to determining the unknown coefficients ak , bk , dk , ek , fk , and gk such that the four interface conditions (6) are satisfied
on 0 and 1 , respectively.
3. Numerical procedure and results
The undetermined coefficients ak , bk , dk , ek , fk , and gk are determined by substituting
truncated series representations of the complex potentials into the interface conditions. This yields four-semi infinite series from which a set of linear algebraic equations can be obtained by comparing coefficients for the powers of z and then solved
numerically to determine the coefficients ak , bk , dk , ek , fk , and gk . In this paper, a set
of 27 equations is obtained by comparing coefficients of powers of z from z4 to z−2 .
It can be shown that the present three-phase imperfect interface model reduces to the
two-phase imperfect interface model given by Liu et al. (2000). In addition, it can be
shown numerically that the present three-phase imperfect interface model reduces to
the classical three-phase perfect bonding condition when m = n = ∞. Consequently,
the SIFs calculated from the present three-phase perfect bonding model are consistent with those given by Xiao and Chen (2001). This confirms the validity of the
present series method.
In this paper, we examine the effects of two imperfect interface conditions: (i)
the case when m = n – the so-called spring-layer imperfect interface, and (ii) the case
when m = ∞, n = f inite– the so-called sliding imperfect interface, in particular, the
example when the sliding imperfect interface takes the form m = ∞, n = 0 will be
discussed.
3.1. Stress intensity factor
The SIF is characterized by the elastic stress distribution near the crack tip. In the
present paper, we assume the SIF is a property used to determine the direction of
crack propagation.
To determine the expressions for the SIF, it is noted that the stresses in the surrounding matrix can be written in Cartesian form as follows:
σxx = Re ϕ2 (z) + ϕ2 (z) − zϕ2 (z) − ψ2 (z) ,
σxy = Im ϕ2 (z) + ϕ2 (z) − zϕ2 (z) − ψ2 (z) ,
σyy = Re ϕ2 (z) + ϕ2 (z) + zϕ2 (z) + ψ2 (z) .
(23)
Substituting (18) and (19) into (23) and considering the leading order terms only,
it follows that for a mode-I crack the stresses in the neighborhood of the crack tip
z = a are given by:
164 K. Kim and L.J. Sudak
∞
1 1−
kbk+1 a −(k+1) + O(r10 ),
2aσ ∞
k=1
∞
∞ θ1 1
θ1
1 2aσ
5
σyy = − √ √
sin − sin 5
1−
kbk+1 a −(k+1) + O r10 , (24)
∞
2 8
2
2aσ
r1 2 8
k=1
∞
∞
2aσ
θ1 1
θ1
1 1
σxy = − √ √
kbk+1 a −(k+1) + O r10 ,
cos − cos 5
1−
∞
2 8
2
2aσ
r1 2 8
2aσ ∞
σxx = − √ √
r1 2
θ1 1
θ1
3
sin + sin5
8
2 8
2
k=1
where z − a = r1 eiθ1 (0 ≤ θ1 ≤ 2π).
Similarly, the stresses around crack tip z = b are given by:
∞
θ2 1
θ2
1 2bσ ∞
3
−(k+1)
σxx = √ √
+ O r20 ,
cos + cos 5
1−
kbk+1 b
∞
2 8
2
2bσ
r1 2 8
k=1
∞
∞ 0
θ2 1
θ2
1
2bσ
5
−(k+1)
σyy = √ √
+
O
r2 , (25)
cos − cos 5
1−
kb
b
k+1
2 8
2
2bσ ∞
r1 2 8
k=1
∞
θ2 1
θ2
1 2bσ ∞
1
σxy = − √ √
sin − sin 5
1−
kbk+1 b−(k+1) + O r20 ,
∞
2 8
2
2bσ
r1 2 8
k=1
where z − b = r2 eiθ2 (−π ≤ θ2 ≤ π).
Remark 1. The square bracketed expressions appearing in (24) and (25) represents
the influence of the three-phase inclusion and the imperfect interface condition on the
SIF.
3.2. Spring-layer imperfect interface
Under uniaxial loading normal to the crack face (i.e. A = σ ∞ /4, B = B = 2A = σ ∞ /2),
let us consider the case when the homogeneously imperfect inclusion-interphase interface is represented by the condition where the normal and tangential spring-factor
type interface parameters are equal (i.e. m = n). Physically, this means that the same
degree of imperfection is assumed in both the normal and tangential directions at
the interface 0 – the so-called spring-layer imperfect interface. For convenience, we
introduce a non-dimensional parameter, M = mR0 /µ1 which characterizes the effectiveness of the bonding at the interface in transferring load between the inclusion
and the interphase. A very small value of M (say M = 0.01) corresponds to a debonded inclusion and values of M between 0.1 and 100 are assumed to correspond
to an imperfect bonding condition. A large value of M (say M > 100) corresponds
to the case of a perfect bond between the interphase and inclusion. Poisson’s ratio
is assumed constant and given by ν0 , ν1 , ν2 = 1/3, respectively. The crack length 2 is
fixed as = R1 .
3.2.1. Influence of inclusion stiffness on crack behavior
Figures 3 and 4 illustrate the changes of the normalized mode-I SIF, KI /KI (noinclusion), at crack tip “a” and crack tip “b” for various values of the interface
Interaction between a radial matrix crack and a three-phase circular inclusion 165
1.8
µ2 / µ1 = 1.02
µ0 / µ2 = 4
R1 / R0 = 1.5
KI /KI(no inclusion)
1.6
M=0.01
M=0.1
M=1
M=10
M=100
M=0.01
M=0.1
M=1
M=10
M=100
crack tip
"a"
crack tip
"b"
1.4
1.2
1
0.8
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
d/R1
Figure 3. Effect of the crack tip interphase-matrix distance and the imperfect spring-layer interface
parameter on the normalized mode-I SIF when the inclusion is stiff.
1.8
µ2 / µ1 = 0.98
µ0 / µ2 = 0.25
R1 / R0 = 1.5
KI /KI(no inclusion)
1.6
M=0.01
M=0.1
M=1
M=10
M=100
M=0.01
M=0.1
M=1
M=10
M=100
crack tip
"a"
crack tip
"b"
1.4
1.2
1
0.8
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
d/R1
Figure 4. Effect of the crack tip interphase-matrix distance and the imperfect spring-layer interface
parameter on the normalized mode-I SIF when the inclusion is soft.
parameter M when the inclusion is stiff and soft, respectively. Note the parameter
√ KI
is the actual calculated mode-I SIF and the parameter KI (no-inclusion) = σ ∞ π is
the mode-I SIF for the same crack in a homogeneous matrix material without inclusion. It is clear from these figures that the inclusion has a significant effect on the
SIF at crack tip “a” but not on crack tip “b”. On the other hand, the influence
of the stiff inclusion decreases with increasing crack tip interphase-matrix distance
– d/R1 . In fact, when the inclusion is stiffer than the surrounding interphase layer
and matrix, Figure 2 shows that as the interface parameter M decreases the SIF at
crack tip “a” increases whereas for crack tip “b” there is virtually no corresponding
effect on the SIF. It is clear that the presence of a stiff inclusion with an imperfect
(spring-layer) interface can either increase the SIF at crack tip “a” (for example when
M = 0.01, 0.1 or 1) or reduce it when M = 10 or 100. This suggests that for small
values of the imperfect interface parameter the stiff inclusion has a higher tendency
to promote crack extension relative to the no inclusion case. On the other hand, for
large values of the imperfect interface parameter the tendency for crack propagation,
166 K. Kim and L.J. Sudak
KI /KI(no inclusion)
1.1
µ2 / µ0 = 1.02
µ1 / µ2 = 4
R1 / R0 = 1.5
1.05
M=0.01
M=0.1
M=1
M=10
M=100
M=0.01
M=0.1
M=1
M=10
M=100
crack tip
"a"
crack tip
"b"
1
0.95
0.9
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
d/R1
Figure 5. Effect of the crack tip interphase-matrix distance and the imperfect spring-layer interface
parameter on the normalized mode-I SIF for a stiff interphase layer.
1.8
µ2 / µ0 = 0.98
µ1 / µ2 = 0.25
R1 / R0 = 1.5
KI /KI(no inclusion)
1.6
M=0.01
M=0.1
M=1
M=10
M=100
M=0.01
M=0.1
M=1
M=10
M=100
crack tip
"a"
crack tip
"b"
1.4
1.2
1
0.8
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
d/R1
Figure 6. Effect of the crack tip interphase-matrix distance and the imperfect spring-layer interface
parameter on the normalized mode-I SIF for a soft interphase layer.
as a result of the stiff inclusion, is unlikely – the interphase layer shields the crack
from the effects of the imperfect (spring-layer) interface.
For the case when the inclusion is softer than the surrounding interphase layer
and matrix (Figure 4), the SIF at crack tip “a” is always larger than the SIF at
crack tip “b”. In fact, the SIF at crack tip “a” increases significantly as the crack
approaches the interphase layer regardless of the level of the imperfect interface
parameter. This suggests that a soft inclusion has a higher tendency to promote crack
extension, originating from the nearby crack tip, relative to the no inclusion case –
this is similar to the two-phase problem encountered in Liu et al. (2001).
3.2.2. Influence of interphase stiffness on crack behavior
Figures 5 and 6 illustrate the influence of the interphase layer stiffness on crack
behavior for a range of imperfect interface conditions and crack tip interphase-matrix
distances. In the case of the interphase layer being stiffer than the matrix and inclusion, Figure 4 shows that the SIF at crack tip “b” is virtually unaffected whereas
Interaction between a radial matrix crack and a three-phase circular inclusion 167
KI /KI(no inclusion)
1.3
1.2
µ1/µ2=0.01
µ1/µ2=0.05
µ1/µ2=0.1
µ1/µ2=1
µ1/µ2=10
µ1/µ2=20
M=∞
µ2 / µ0 = 0.98
d / R1 = 0.5
1.03
1.07
1.1
1
0.9
1.01
1.05
1.09
1.11
1.13
1.15
1.17
1.19
R 1/R 0
Figure 7. Effect of the interphase layer thickness on the normalized mode-I SIF at the nearby crack
tip for various shear moduli ratios and a perfect bonding condition.
at crack tip “a” the SIF is significantly reduced with decreasing d/R1 (i.e. as the
crack approaches the interface 1 ). In fact, the SIF’s at crack tip “a” are always
smaller than the SIF’s at crack tip “b” regardless of the level of the imperfect interface parameter. This suggests that the stiff interphase layer tends to shield the crack
from the effects of the imperfect (spring-layer) interface condition.
For the case of the interphase layer being softer than inclusion and matrix (Figure 6), the SIF’s at crack tip “a” are always larger than the SIF’s at crack tip “b” for
all values of the interface parameter. In fact, the SIF at crack tip “a” increases with
decreasing d/R1 . This suggests that a compliant interphase layer, with an imperfect
(spring-layer) interface, has a higher tendency to promote crack growth towards the
interphase layer.
3.2.3. Influence of interphase thickness
Figure 7 shows the normalized mode-I SIF at crack tip “a”, under the condition of
perfect bonding along both interfaces as a function of the relative thickness of the
interphase layer (R1 /Ro ) and for various shear moduli ratios. The normalized modeI SIF is considered as KI /KI (no-interphase). From this figure, the results illustrate
that a soft interphase layer produces an increase in the SIF as compared to the nointerphase case. In fact, as the thickness of the soft interphase layer increases, the
SIF increases substantially which will promote crack extension from the nearby crack
tip. On the other hand, for the stiff interphase layer, the SIF at the nearby crack tip
decreases with increasing thickness of the interphase layer. In fact, the rigid interphase layer shields the crack from the effects of the inclusion; therefore, the thickness
of a stiff interphase layer has little influence on the nearby crack tip. This is consistent with the results obtained in Cheeseman and Santare (2001). In comparison with
the imperfect (spring-layer) interface condition (Figure 8), the interphase layer thickness does show some influence on the nearby crack tip when the inteprhase layer is
soft but not as dramatic as the perfect bonding model. When the interphase layer is
stiff, the SIF is reduced as compared to the no-interphase case. This leads to a shielding effect of the nearby crack tip from the effects of the imperfect interface. Hence,
the results clearly illustrate that the thickness of the interphase layer regardless of
168 K. Kim and L.J. Sudak
KI /KI(no inclusion)
1.3
µ1/µ2=0.01
µ1/µ2=0.05
µ1/µ2=0.1
µ1/µ2=1
µ1/µ2=10
µ1/µ2=20
1.2
M = 10
µ2 / µ0 = 0.98
d / R1 = 0.5
1.1
1
0.9
1.01
1.03
1.05
1.07
1.09
1.11
1.13
1.15
1.17
1.19
R 1/R 0
Figure 8. Effect of the interphase layer thickness and an imperfect spring-layer interface on the normalized Mode-I SIF at the nearby crack tip for various shear moduli ratios.
whether it is soft or stiff will have little influence on crack extension when an imperfect (spring-layer) interface is considered.
3.3. Sliding imperfect interface
In this case, the homogeneous imperfect inclusion-interphase interface is represented
by the condition where the normal and tangential spring-factor type interface parameters are characterized by the condition m = ∞, n = f inite – the so-called sliding imperfect interface. Physically, this is important for modeling certain features
of material behavior (such as grain-boundary sliding and damage occurring in the
circumferential direction at constituent interfaces). As an example of this more general condition, we consider the case when m = ∞, n = 0. This simple condition
allows one to gain insight into the effects of interface separation (or debonding
between constituent phases) on the mechanical behavior of composites (see Benveniste (1984)).
3.3.1. Influence of inclusion stiffness on crack behavior
Figure 9 shows the changes of normalized mode-I SIF at crack tip “a” and crack
tip “b” as a function of the crack tip interphase-matrix distance for a soft and stiff
inclusion, respectively. In the case of a soft inclusion, the SIF at crack tip “a” is
always larger than the SIF at crack tip “b”. In fact, as the crack approaches the
interphase layer, the SIF at crack tip “a” increases whereas the SIF at crack tip
“b” remains unchanged. This suggests that for a crack in close proximity to a soft
inclusion with a sliding imperfect interface of the form m = ∞, n = 0, there is a tendency for crack extension to initiate from crack tip “a”. In the case of a stiff inclusion, the SIF at crack tip “a” is smaller than at crack tip “b” – which shows no
change in SIF. In fact, the SIF at crack tip “a” decreases as the crack tip interphasematrix distance decreases. Clearly, the results show that a stiff inclusion tends to
shield the crack from the effects of a sliding imperfect interface of the form m = ∞,
n = 0.
Interaction between a radial matrix crack and a three-phase circular inclusion 169
1.2
µ2 / µ1 = 0.98
R1 / R0 =1.5
KI /KI(no inclusion)
1.1
µ0/µ2=0.25
µ0/µ2=4
µ0/µ2=0.25
µ0/µ2=4
crack tip
"a"
crack tip
"b"
1
0.9
0.8
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
d/R1
Figure 9. Normalized mode-I SIF at the crack tips vs. crack tip interphase-matrix distance for a soft
and stiff inclusion for the sliding interface case m = ∞, n = 0.
1.2
KI /KI(no inclusion)
µ1/µ2=0.25
µ1/µ2=4
µ1/µ2=0.25
µ1/µ2=4
crack tip
"a"
µ2 / µ0 = 0.98
R1 / R0 =1.5
crack tip
"b"
1.1
1
0.9
0.8
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
d/R1
Figure 10. Normalized mode-I SIF at the crack tips vs. crack tip interphase-matrix distance for a soft
and stiff interphase layer for the sliding interface case m = ∞, n = 0.
3.3.2. Influence of interphase layer stiffness on crack behavior
Figure 10 illustrates the changes in the normalized mode-I SIF at crack tip “a” and
crack tip “b”, respectively as a function of the crack tip interphase-matrix distance
when the interphase layer is soft and stiff, respectively. The current results suggest
that for a crack in close proximity to a soft interphase layer with a sliding imperfect
interface of the form m = ∞, n = 0, there is a tendency for crack extension to initiate
from crack tip “a”. When the interphase layer is stiff, the crack is shielded from the
effects of the sliding imperfect interface of the form m = ∞, n = 0.
3.3.3. Influence of interphase thickness
Figure 11 illustrates the normalized mode-I SIF at crack tip “a” for the sliding
imperfect interface given by m = ∞, n = 0 as a function of the relative thickness of
the interphase layer (R1 /Ro) and for various shear moduli ratios. The results illustrate that a soft interphase layer produces an increase in the SIF as compared to
170 K. Kim and L.J. Sudak
KI /KI(no inclusion)
1.3
µ1/µ2=0.01
µ1/µ2=0.05
µ1/µ2=0.1
µ1/µ2=1
µ1/µ2=10
µ1/µ2=20
1.2
µ2 / µ0 = 0.98
d / R1 = 0.5
1.1
1
0.9
1.01
1.03
1.05
1.07
1.09
1.11
1.13
1.15
1.17
1.19
R 1/R 0
Figure 11. Effect of the interphase layer thickness and the sliding interface case m = ∞, n = 0 on the
normalized mode-I SIF at the nearby crack tip for various shear moduli ratios.
the no-interphase case. In fact, as the thickness of the soft interphase layer increases,
the SIF increases substantially. Clearly, the sliding imperfect interface has a significant effect on the tendency for crack extension from the nearby crack tip. On the
other hand, for a stiff interphase layer, the SIF at the nearby crack tip decreases with
increasing thickness of the interphase layer. In fact, the interphase layer shields the
crack from the effects of the sliding imperfect interface. Hence, the thickness of a stiff
interphase layer has little influence on the nearby crack tip.
4. Conclusion
The model of an imperfect interface is applied to the analysis of the interaction
problem between a radial matrix crack and a three-phase circular inclusion in plane
elasticity. Unlike previous works which used the complicated and often cumbersome
dislocation density method to analyze the interaction problem between inclusion
and crack, the current paper develops a simple semi-analytic solution to determine
the effects of an imperfect interface on the corresponding stress functions inside
the inclusion, inside the inteprhase layer and in the surrounding matrix. Numerical results clearly illustrate that the imperfect interface condition, at the inclusioninterphase boundary, has a significant effect on the SIF especially at the nearby crack
tip. In the case when the interphase layer is softer than the inclusion and matrix, the
results show that the radial matrix crack will always originate from the nearby crack
tip and propagate towards the interphase layer regardless of the level of the imperfect
(spring-factor) interface parameter. In comparison, when the interphase layer is stiffer
than the inclusion and matrix, the interphase layer will shield the crack from effects
of the imperfect (spring-factor) interface. In addition, the thickness of the interphase
layer has a somewhat limited influence on the crack as compared to the perfect bonding case.
In the case when the imperfect interface is defined by the sliding interface, in particular, when the sliding interface is given by m = ∞, n = 0, the results illustrate that
a crack in close proximity to a compliant interphase layer will have a higher tendency for crack initiation as compared to a stiff interphase layer which shields the
Interaction between a radial matrix crack and a three-phase circular inclusion 171
crack from the effects of the sliding imperfect interface. Moreover, the thickness of
the interphase layer significantly contributes to the behavior of the crack, in particular, when the interphase layer is soft the SIF increases substantially which increases
the tendency for crack extension from the nearby crack tip.
The current results indicate convincingly that the influence of the interphase layer
as well as the effect of the imperfect bonding condition has a pronounced effect on
the SIF at the crack tip and should not be ignored.
Acknowledgement
This work has been supported by the Natural Sciences and Engineering Research
Council of Canada through grant NSERC No. 249516.
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