Theoretical and Applied Fracture Mechanics 42 (2004) 155–170
www.elsevier.com/locate/tafmec
Crack-tip stress fields in functionally graded materials
with linearly varying properties
N. Jain, C.E. Rousseau, A. Shukla
*
Department of Mechanical Engineering and Applied Mechanics, University of Rhode Island, Kingston, RI 02881, USA
Abstract
Crack-tip stress fields for a stationary crack along or inclined to the direction of property gradation in functionally
graded materials (FGMs) are obtained through an asymptotic analysis coupled with WestergaardÕs stress function
approach. The elastic modulus of the FGM is assumed to vary linearly along the gradation direction. The first six terms
for a crack along the direction of property variation and first four terms for a crack inclined to the direction of property
variation in the expansion of the stress field are derived to explicitly bring out the influence of nonhomogeneity on the
structure of the stress field. Using these stress fields, contours of constant maximum shear stress and constant out of
plane displacement are generated and the effect of inclination of property gradation direction on these contours is discussed. The strain energy density criterion is applied to obtain critical conditions for crack initiation and the effect of
property gradation is discussed. It is shown that the materials with varying properties can offer more resistance to crack
propagation and will suppress crack growth in some situations.
2004 Elsevier Ltd. All rights reserved.
1. Introduction
In todayÕs highly demanding technological environment, one of the main challenges in new material design appears to be combining irreconcilable thermo-mechanical and strength properties in the same component. A functionally graded material (FGM) is a composite consisting of two or more phases, which
is fabricated such that its composition varies in some spatial direction. The design is intended to take
advantage of certain desirable features of each of the constituent phases.
*
Corresponding author. Tel.: +1 401 874 2283; fax: +1 401 874 2355.
E-mail address: shuklaa@egr.uri.edu (A. Shukla).
0167-8442/$ - see front matter 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.tafmec.2004.08.005
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N. Jain et al. / Theoretical and Applied Fracture Mechanics 42 (2004) 155–170
The knowledge of crack growth in FGMs is important in order to evaluate their integrity. Several
researchers have established that the near-tip stress field around cracks in FGMs retains the inverse square
root singularity [1,2]. Assuming an exponential spatial variation of the elastic modulus, the asymptotic
stress field for stationary cracks along the gradient in FGMs were developed [3]. Later, stress field equations
for a crack inclined to the direction of the exponentially varying property gradation were derived [4]. However, in several instances, the YoungÕs modulus variation is linear [5], hence, it is necessary to obtain the
stress field associated with the stationary cracks in materials with linearly varying properties.
In the present work, stress field for stationary cracks aligned along or inclined to the direction of
property variation in an FGM with linearly varying elastic modulus is developed through an asymptotic
analysis coupled with WestergaardÕs stress function approach. The first six terms in the expansion of the
stress field for opening and shearing mode loading are obtained when the crack is along the direction of
property gradation. The first four terms for a crack inclined to the direction of property variation are
obtained.
The stress fields developed in the first part of the paper is then used in strain energy density criterion [6,7]
to predict the critical conditions of the crack initiation. Subsequently the effect of nonhomogeneity on the
crack initiation behavior is discussed.
2. Crack along the direction of property gradation
In this section stress field for cracks along the direction of property gradation are discussed.
2.1. Theoretical formulation
The stress function approach of solving plane crack problems in homogeneous materials leads to a biharmonic equation, which can be satisfied, by the real and imaginary parts of a carefully chosen analytical
complex function. Because of the material inhomogeneity inherent to FGMs, the governing differential
equation contains several lower order differential terms that complicate the solution, leading to an almost
universal assumption of exponential gradation in FGMs, in an attempt to simplify the problem. Thus,
stress field expressions for cracks in FGMs with linear spatial variation of elastic modulus have not yet been
derived. The following derivation will fill this gap. An asymptotic analysis is performed to obtain explicit
expressions for the stress field in a series form.
Consider the plane elasticity problem containing a finite crack on y = 0 plane as shown in Fig. 1. Previous studies indicate that the influence of the PoissonÕs ratio on the stress fields is not very significant [1].
Therefore, to make analysis tractable, the PoissonÕs ratio, m, is assumed to be constant and the YoungÕs
modulus is assumed to vary linearly in the x direction as
EðxÞ ¼ E0 ð1 þ dxÞ;
ð1Þ
where E0 is the modulus at the crack-tip (x = 0) and d is the nonhomogeneity parameter. Expressing the inplane stress components in terms of AiryÕs stress function F(x, y), the compatibility equation can be written
as
2
ð1 þ dxÞ r2 ðr2 F Þ 2ð1 þ dxÞd
o
o2 F
ðr2 F Þ þ 2d2 ðr2 F Þ 2d2 ð1 þ mÞ 2 ¼ 0;
ox
oy
ð2Þ
where r2 is the Laplacian operator. It may be observed from Eq. (2) that the first term is the bi-harmonic
equation and the additional lower order differentials are due to the nonhomogeneity. The solution for Eq.
(2) is obtained through an asymptotic analysis coupled with WestergaardÕs stress function approach as explained in the following sections.
N. Jain et al. / Theoretical and Applied Fracture Mechanics 42 (2004) 155–170
157
Fig. 1. Reference coordinate (u = 0 for straight crack).
2.2. Asymptotic analysis
The crack-tip coordinates are scaled to fill the entire field of observation using the transformation
g1 = x/e, and g2 = y/e in the scaled coordinates (g1, g2), Eq. (2) takes the form
2
ð1 þ dg1 eÞ r2 ðr2 F Þ 2ð1 þ dg1 eÞed
o
o2 F
ðr2 F Þ þ 2d2 e2 ðr2 F Þ 2d2 e2 ð1 þ mÞ 2 ¼ 0:
og1
og2
ð3Þ
It is assumed at this stage that the stress function F(eg1, eg2) can be expanded in powers of the parameter e
as follows:
1
1
X
X
eðmþ3=2Þ /m ðg1 ; g2 Þ þ
eðnþ2Þ wn ðg1 ; g2 Þ:
ð4Þ
F ðeg1 ; eg2 Þ ¼
m¼0
n¼0
The first part of the above series /m represents the singular series and the second part containing wn accounts for the finiteness of the domain. The above series, upon substitution into Eq. (3), leads to an infinite
series involving differential equations associated with each power of e as written below.
1 X
o
mþ32Þ 2
mþ52Þ
2
2
2
2
ð
ð
r ðr /m Þ þ e
2dg1 r ðr /m Þ 2d
e
ðr /m Þ
og1
m¼0
7
o
o2 / m
þ eðmþ2Þ d2 g21 r2 ðr2 /m Þ 2d2 g1
ðr2 /m Þ þ 2d2 ðr2 /m Þ 2d2 ð1 þ mÞ
og1
og22
1
X
o
eðnþ2Þ r2 ðr2 wn Þ þ eðnþ3Þ 2dg1 r2 ðr2 wn Þ 2d
ðr2 wn Þ
þ
og
1
n¼0
o
o2 w
þ eðnþ4Þ d2 g21 r2 ðr2 wn Þ 2d2 g1
ðr2 wn Þ þ 2d2 ðr2 wn Þ 2d2 ð1 þ mÞ 2n
¼ 0:
ð5Þ
og1
og2
For Eq. (5) to be valid, the differential equations corresponding to each power of e (e3/2, e2, e5/2, . . .) should
vanish independently. This leads to the following set of differential equations.
m ¼ n ¼ 0;
r2 ðr2 /m Þ ¼ 0; r2 ðr2 wn Þ ¼ 0:
ð6Þ
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N. Jain et al. / Theoretical and Applied Fracture Mechanics 42 (2004) 155–170
o
ðr2 /m1 Þ ¼ 0;
og1
o
r2 ðr2 wn Þ þ 2dg1 r2 ðr2 wn1 Þ 2d
ðr2 wn1 Þ ¼ 0:
og1
m ¼ n ¼ 1; r2 ðr2 /m Þ þ 2dg1 r2 ðr2 /m1 Þ 2d
m; n > 1; r2 ðr2 /m Þ þ 2dg1 r2 ðr2 /m1 Þ 2d
ð7Þ
o
ðr2 /m1 Þ þ d2 g21 r2 ðr2 /m2 Þ
og1
o
o2 /m2
ðr2 /m2 Þ þ 2d2 ðr2 /m2 Þ 2d2 ð1 þ mÞ
¼ 0;
og1
og22
o
r2 ðr2 wn Þ þ 2dg1 r2 ðr2 wn1 Þ 2d
ðr2 wn1 Þ þ d2 g21 r2 ðr2 wn2 Þ
og1
2d2 g1
2d2 g1
ð8Þ
o
o2 wn2
ðr2 wn2 Þ þ 2d2 ðr2 wn2 Þ 2d2 ð1 þ mÞ
¼ 0:
og1
og22
It should be observed that Eq. (6) is identical to the relation obtained for homogeneous materials for which
the solution is available [8]. However, the differential equations (7) and (8), associated with higher powers of
e are no more the classical bi-harmonic equation and contain lower order differentials having the coefficient
d, nonhomogeneity parameter. These are solved in a recursive manner and the solution for opening mode
and shear mode are provided in the following sections.
2.3. Opening mode loading
The solution of Eqs. (7) and (8) consists of a complementary and a particular solution. The complementary solution is obtained from the series expansion of the stress function for homogeneous material and the
particular solution is derived from already known /0. First consider the symmetry of the normal stress
components about the line of the crack and the traction-free crack-face boundary conditions.
Thus, for m = 0, the solution is same as for homogeneous materials and for higher value of m solution
can be obtained in a recursive manner. The first three terms in the expansion, m = 0, 1, 2, are:
/0 ¼ RefG0 g þ g2 ImfG0 g;
d
/1 ¼ RefG1 g þ g2 ImfG1 g g22 RefG0 g;
2
d
ð5 mÞd2 2
g2 ImfG0 g
/2 ¼ RefG2 g þ g2 ImfG2 g g22 RefG1 g 2
12
d2
d2
þ g1 g22 RefG0 g g22 RefG0 g;
2
4
4
4
4
3
5
7
G0 ¼ A0 12 ; G1 ¼ A1 12 ; and G2 ¼ A2 12 ;
3
15
35
pffiffiffiffiffiffiffi
oG0
;
1 ¼ g1 þ ig2 ; i ¼ 1; G0 ¼
o1
ð9Þ
where G0 ; G1 , and G2 are the first three terms in the series solution of the stress function for nonhomogeneous materials [1]. An are real constants and A0 is proportional to the mode-I stress intensity factor.
For n = 0, 1 and 2 the first three terms of the series solution corresponding to nonhomogeneous materials, are:
N. Jain et al. / Theoretical and Applied Fracture Mechanics 42 (2004) 155–170
159
w0 ¼ g2 ImfH 0 g;
d
w1 ¼ g2 ImfH 1 g g22 RefH 0 g;
2
d
ð5 mÞd2 3
d2
w2 ¼ g2 ImfH 2 g g22 RefH 1 g g2 ImfH 0 g þ g1 g22 RefH 0 g
2
12
2
2
ð2 þ mÞd 2
g2 RefH 0 g;
4
1
1
H 0 ¼ B0 1; H 1 ¼ B1 12 ; and H 2 ¼ B2 13 ;
2
3
ð10Þ
where Bn are real constants. Switching back to the (x, y) coordinates through (3) and (5), a six term expansion for the stresses can be obtained as
rxx ¼
2
1 X
X
d
f2RefP n 4yImfP n g y 2 RefP 0n g
fRefP n g yImfP 0n g þ 2RefQn g yImfQ0n gg 2
n¼0
n¼0
ð5
mÞ
2
þ 2RefQn g 4yImfQn g y 2 RefQ0n gg d f6yImfP 0 g þ 6y 2 RefP 0 g y 3 ImfP 00 g
12
d2
f2xRefP 0 g 4xyImfP 0 g xy 2 RefP 00 g
2
d2
þ 2xRefQ0 g 4xyImfQ0 g xy 2 RefQ00 gg f2RefP 0 g 4yImfP 0 g y 2 RefP 0 gg
4
ð2 þ mÞ 2
d f2RefQ0 g 4yImfQ0 g y 2 RefQ0 gg:
4
2
1 X
X
d 2
0
0
0
0
2
fy RefP n g þ y RefQn gg
ryy ¼
fRefP n g þ yImfP n g þ yImfQn gg 2
n¼0
n¼0
þ 6yImfQ0 g þ 6y 2 RefQ0 g y 3 ImfQ00 gg þ
ð11Þ
ð5 mÞ 2 3
d2
d fy ImfP 00 g þ y 3 ImfQ00 gg þ f2y 2 RefP 0 g þ xy 2 RefP 00 g þ 2y 2 RefQ0 g
12
2
2
d
ð2
þ
mÞ
d2 fy 2 RefQ0 gg:
ð12Þ
þ xy 2 RefQ00 gg fy 2 RefP 0 gg 4
4
2
1 X
X
d
0
0
rxy ¼
fyRefP n g þ yRefQn g þ ImfQn gg þ
f2yRefP n g y 2 ImfP 0n g þ 2yRefQn g
2
n¼0
n¼0
ð5 mÞ 2
d2
0
2
d f3y 2 ImfP 0 g þ y 3 RefP 00 g þ 3y 2 ImfQ0 g þ y 3 RefQ00 gg þ f4yRefP 0 g
y ImfQn gg 12
4
þ 4xyRefP 0 g 2y 2 ImfP 0 g 2xy 2 ImfP 00 g 2yRefP 0 g þ y 2 ImfP 0 gg þ
þ 4xyRefQ0 g 2y 2 ImfQ0 g 2xy 2 ImfQ00 gg þ
d2
f4yRefQ0 g
4
ð2 þ mÞ 2 2
d fy ImfQ0 g 2yRefQ0 gg;
4
ð13Þ
where, Pn = Anzn1/2, Qn = Bnzn.
The complex functions used in the above Eqs. (11)–(13) Pn and Qn in the (x, y) coordinates are counterparts of the functions Gn and Hn, respectively. Note that by setting d to zero, the stresses collapse to their
homogeneous counterparts. It can also be observed that the stress field contains terms having PoissonÕs
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N. Jain et al. / Theoretical and Applied Fracture Mechanics 42 (2004) 155–170
ratio, unlike homogeneous materials. This is due to the presence of the PoissonÕs ratio-dependent coefficient
in the governing Eq. (2) as opposed to the bi-harmonic equation for homogeneous materials.
2.4. Shear mode loading
Following the same procedure as the previous section and keeping the inherent nature of shear mode
problem, the first six terms of the expansion for the stress field can be obtained.
The first three terms in the expansion (m = 0, 1, 2) are
/0 ¼ g2 RefJ 0 g;
d
/1 ¼ g2 RefJ 1 g g22 ImfJ 0 g;
2
d
ð5 mÞd2 3
d2
ð2 þ mÞd2 2
g2 RefJ 0 g þ g1 g22 ImfJ 0 g g2 ImfJ 0 g;
/2 ¼ g2 RefJ 2 g g22 ImfJ 1 g þ
2
12
2
4
4
4
4
3
5
7
J 0 ¼ C 0 12 ; J 1 ¼ C 1 12 ; and J 2 ¼ C 2 12 ;
3
15
35
ð14Þ
where Cn are real constants and C0 is proportional to the mode-II stress intensity factor.
The first three terms in the expansion (n = 0, 1, 2) are:
w0 ¼ ImfL0 g g2 RefL0 g;
d
w1 ¼ ImfL1 g g2 RefL1 g g22 ImfL0 g;
2
d
ð5 mÞd2 3
d2
d2
g2 RefL0 g þ g1 g22 ImfL0 g g22 ImfL0 g;
w2 ¼ ImfL2 g g2 RefL2 g g22 ImfL1 g þ
2
12
2
4
1
1
L0 ¼ D0 1; L1 ¼ D1 12 ; and L2 ¼ D2 13 ;
2
3
ð15Þ
where, Dn are real constants. The first six terms of the expansion for the stress field become:
rxx ¼
2
X
f2ImfRn g þ yRefR0n g þ ImfS n g þ yRefS 0n gg
n¼0
1 X
d
n¼0
2
f2ImfRn g þ 4yRefRn g y
2
ImfR0n g
þ 2ImfS n g 4yRefS n g y
2
ImfS 0n gg
ð5 mÞ 2
d f6yRefR0 g 6y 2 ImfR0 g y 3 RefR00 g þ 6yRefS 0 g 6y 2 ImfS 0 g y 3 RefS 00 gg
12
d2
þ f2xImfR0 g þ 4xyRefR0 g xy 2 ImfR00 g þ 2xImfS 0 g þ 4xyRefS 0 g xy 2 ImfS 00 gg
2
d2
ð2 þ mÞ 2
d f2RefS 0 g 4yImfS 0 g y 2 ImfS 0 gg: ð16Þ
f2RefR0 g 4yImfR0 g y 2 ImfR0 gg 4
4
2
1 X
X
d 2
0
0
0
0
2
fy ImfRn g y ImfS n gg
ryy ¼
fyRefRn g þ ImfS n g yRefS n gg 2
n¼0
n¼0
þ
ð5 mÞ 2 3
d2
d fy RefR00 g þ y 3 RefS 00 gg þ f2y 2 ImfR0 g þ xy 2 ImfR00 g þ 2y 2 ImfS 0 g
12
2
2
d
ð2
þ
mÞ
d2 fy 2 ImfS 0 gg:
þ xy 2 ImfS 00 gg fy 2 ImfR0 gg 4
4
þ
ð17Þ
N. Jain et al. / Theoretical and Applied Fracture Mechanics 42 (2004) 155–170
161
2
1 X
X
d
0
0
2
0
2
f2yImfRn g þ y RefRn g þ 2yImfS n g þ y RefS n gg
rxy ¼
fRefRn g yImfRn g ImfS n gg þ
2
n¼0
n¼0
þ
ð5 mÞ 2
d2
d f3y 2 RefR0 g y 3 ImfR00 g þ 3y 2 RefS 0 g y 2 ImfS 00 gg þ f4yImfR0 g þ 4xyImfR0 g
12
2
þ 2y 2 RefR0 g þ 2xy 2 RefR00 gg þ
2yImfS 0 g y 2 RefS 0 gg d2
f4yImfS 0 g þ 4xyImfS 0 g þ 2y 2 RefS 0 g þ 2xy 2 RefS 00 g
4
ð2 þ mÞ 2
d f2y 2 RefR0 g þ 2yImfR0 gg;
4
ð18Þ
where, Rn = Cn zn1/2, Sn = Dn zn. The complex functions used in the above Eqs. (16)–(18) Rn and Sn, are
counterparts of the functions Jn and Ln, respectively in the (x, y) coordinates. It can be noticed that by setting d to zero, the stresses collapse to their homogeneous counterparts.
3. Crack inclined to the direction of property gradation
In this section stress fields for a crack inclined to the linear property gradation in FGMs under quasistatic loading, a geometry which results in an inherently mixed-mode stress field. In homogeneous materials
the solution for a mixed mode problem can be readily obtained by just superimposing the opening and
shear mode components of the stress field. In the present case, due to the presence of the nonhomogeneity
terms, the solution cannot be obtained by simply superimposing the opening and shear mode components
of the stress field.
3.1. Theoretical formulation
Consider the plane elasticity problem containing a finite crack on the y = 0 plane as shown in Fig. 1. The
YoungÕs modulus is assumed to vary linearly in the x1 direction as given in Eqs. (19), (21) and the PoissonÕs
ratio, m, is assumed to be constant. The property gradient direction is at an angle u to the y = 0 plane.
Eðx1 Þ ¼ E0 ð1 þ dx1 Þ;
ð19Þ
where E0 is the YoungÕs modulus at the crack-tip (x = x1 = 0) and d is the nonhomogeneity parameter having dimension (Length)1. Eq. (19) can be written in terms of (x, y) coordinates by using simple coordinate
transformation as
Eðx; yÞ ¼ E0 ð1 þ ax þ byÞ;
d
a ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
1 þ tan2 u
d tan u
b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :
1 þ tan2 u
ð20Þ
ð21Þ
Defining the in plane stress components (rij, i, j 2 {x, y}) in terms of AiryÕs stress function F(x, y) and using
HookeÕs law, the compatibility equation takes the following form:
o
o
ð1 þ ax þ byÞ2 r2 ðr2 F Þ 2 a ð1 þ ax þ byÞ ðr2 F Þ 2 b ð1 þ ax þ byÞ ðr2 F Þ
ox
oy
2
2
o2 F
2
2o F
2
2
2o F
þ 2 ða þ b Þ ðr F Þ 2 ð1 þ mÞ a
þb
2ab
¼ 0:
oy 2
ox2
ox oy
ð22Þ
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N. Jain et al. / Theoretical and Applied Fracture Mechanics 42 (2004) 155–170
3.2. Asymptotic analysis
As described in the previous section a new set of coordinates (g1,g2) is introduced, so Eq. (22) can now be
expressed in terms of scaled coordinates as
o
o
2
ð1 þ aeg1 þ beg2 Þ r2 ðr2 F Þ 2að1 þ aeg1 þ beg2 Þe
ðr2 F Þ 2bð1 þ aeg1 þ beg2 Þe
ðr2 F Þ
og1
og2
o2 F
o2 F
o2 F
þ 2ða2 þ b2 Þe2 ðr2 F Þ 2ð1 þ mÞe2 a2 2 þ b2 2 2ab
¼ 0:
og1 og2
og2
og1
Assuming an expansion identical to the one given by Eq. (4), Eq. (23) becomes:
1 X
o
o
mþ32Þ 2
mþ52Þ
2
2
2
2
2
ð
ð
r ðr /m Þ þ e
2ðag1 þ bg2 Þr ðr /m Þ 2a
e
ðr /m Þ 2b
ðr /m Þ
og1
og2
m¼0
7
o
o
2
þ eðmþ2Þ ðag1 þ bg2 Þ r2 ðr2 /m Þ 2aðag1 þ bg2 Þ
ðr2 /m Þ 2bðag1 þ bg2 Þ
ðr2 /m Þ
og1
og2
2
o2 /m
o2 / m
2 o /m
þ 2ða2 þ b2 Þðr2 /m Þ 2ð1 þ mÞ a2
þ
b
2ab
og22
og21
og1 g2
1
X
o
o
2
2
2
2
2
ðnþ2Þ 2
ðnþ3Þ
e
r ðr wn Þ þ e
2ðag1 þ bg2 Þr ðr wn Þ 2a
ðr wn Þ 2b
ðr wn Þ
þ
og1
og2
n¼0
o
o
þ eðnþ4Þ ðag1 þ bg2 Þ2 r2 ðr2 wn Þ 2aðag1 þ bg2 Þ
ðr2 wn Þ 2bðag1 þ bg2 Þ
ðr2 wn Þ
og1
og2
o2 w
o2 w
o2 wn
þ 2ða2 þ b2 Þðr2 wn Þ 2ð1 þ mÞ a2 2n þ b2 2n 2ab
¼ 0:
og1 g2
og2
og1
ð23Þ
ð24Þ
For Eq. (24) to be valid, the differential equations corresponding to each power of e (e3/2, e2, e5/2 . . .) should
vanish independently. This leads to the following set of differential equations:
m ¼ n ¼ 0;
r2 ðr2 /m Þ ¼ 0;
r2 ðr2 wm Þ ¼ 0;
o
o
ðr2 /m1 Þ 2b
ðr2 /m1 Þ ¼ 0;
og1
og2
o
o
r2 ðr2 wn Þ þ 2ðag1 þ bg2 Þr2 ðr2 wn1 Þ 2a
ðr2 wn1 Þ 2b
ðr2 wn1 Þ ¼ 0:
og1
og2
ð25Þ
m ¼ n ¼ 1; r2 ðr2 /m Þ þ 2ðag1 þ bg2 Þr2 ðr2 /m1 Þ 2a
ð26Þ
Since the crack is inclined to the direction of property gradation, the stress field near the crack-tip is mixed
mode. However for m = 0, due to the continuous variation of the elastic properties the opening and shear
mode components of the stress field can be superposed in the same way as that for a mixed mode crack in
homogenous material. Hence for m = 0, the first term in the expansion can be written as
/0 ¼ RefG0 g þ g2 ImfG0 g g2 RefJ 0 g:
ð27Þ
For higher values of m, the solution can be obtained in a recursive manner using Eq. (26):
a
b
/1 ¼ RefG1 g þ g2 ImfG1 g g22 RefG0 g þ g22 ImfG0 g g2 RefJ 1 g
2
2
a 2
b 2
g2 ImfJ 0 g g2 RefJ 0 g;
2
2
where G0 ; J 0 ; G1 , and J 1 are previously defined.
ð28Þ
N. Jain et al. / Theoretical and Applied Fracture Mechanics 42 (2004) 155–170
163
Similarly, for n = 0 and 1, the solutions can be written as below in terms of the functions H 0 , L0 , L0 , H 1 ,
L1 , and L1 defined in previous section.
w0 ¼ g2 ImfH 0 g þ ImfL0 g g2 RefL0 g:
ð29Þ
a
b
w1 ¼ g2 ImfH 1 g þ ImfL1 g g2 RefL1 g g22 ImfH 0 g þ g22 RefH 0 g
2
2
a 2
b 2
ð30Þ
g2 ImfL0 g g2 RefL0 g:
2
2
Substitution of Eqs. (27)–(30) into Eq. (4) leads to a four-term expansion for stresses in the (x, y) coordinate
system.
rxx ¼
1
X
½RefP n g yImfP 0n gþ2ImfRn g þ yRefR0n g þ 2RefQn g þ yImfQ00 g þ ImfS n g þ yRefS 0n g
n¼0
2
3
y2
y2
0
0
RefP
ImfR
RefP
g
þ
2yImfP
g
g
ImfR
g
2yRefR
g
þ
g
0
0
0
0
0
0 7
6
2
2
7
þ a6
4
5
2
2
y
y
RefQ0 g þ 2yImfQ0 g RefQ00 g ImfS 0 g 2yRefS 0 g þ ImfS 00 g
2
2
2
3
2
2
y
y
0
0
ImfP
ImfP
RefR
g
þ
2yRefP
g
g
RefR
g
þ
2yImfR
g
þ
g
0
0
0
0
0
0
6
7
2
2
7:
þ b6
4
5
y2
y2
0
0
þ ImfQ0 g þ 2yRefQ0 g ImfQ0 g RefS 0 g þ 2yImfS 0 g þ RefS 0 g
2
2
ryy ¼
1
X
n¼0
RefP n g þ yImfP 0n g yRefR0n g þ yImfQ0n g þ ImfS n g yRefS 0n g
ð31Þ
y 2
y2
y2
y2
RefP 00 g ImfR00 g RefQ00 g ImfS 00 g
2
2
2
2
2
2
2
2
y
y
y
y
0
0
0
0
ImfP 0 g RefR0 g þ ImfQ0 g RefS 0 g :
þb
2
2
2
2
þa
rxy ¼
ð32Þ
1
X
yRefP 0n g þ RefRn g yImfR0n g þ ImfQn g þ yRefQ0n g þ yImfS 0n g
n¼0
2
3
y2
y2
0
0
6 yRefP 0 g 2 ImfP 0 g þ yImfR0 g þ 2 RefR0 g 7
7
þ a6
4
5
2
y2
y
yRefQ0 g ImfQ00 g þ yImfS 0 g þ RefS 00 g
2
2
2
3
2
y
y2
0
0
yImfP
RefP
ImfR
g
g
þ
yRefR
g
g
0
0
0
0 7
6
2
2
7:
þ b6
4
5
2
2
y
y
0
0
yImfQ0 g RefQ0 g þ yRefS 0 g ImfS 0 g
2
2
ð33Þ
The complex functions used in the above Eqs. (31)–(33) Pn, Qn, Rn, and Sn in the (x, y) coordinates, which
are counterparts of the functions Gn, Hn, Jn, and Ln, respectively, are as described in previous section.
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4. Discussion on solutions
The above-developed analytic expressions were used to study the effect of nonhomogeneity and property
gradation angle, u, on the structure of crack-tip stress fields and out of plane displacement fields. The contours of constant maximum stress and constant out of plane displacement near the crack-tip were generated
for mixed mode loading condition which are related to the two most widely used experimental techniques
for fracture studies namely photoelasticity and coherent gradient sensing, respectively.
4.1. Contours of constant maximum shear stress
Constant maximum shear stress, smax, at each point around the crack-tip can be determined by substituting the stress field equation, obtained in the previous sections, in the relation:
ffi
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r r 2
yy
xx
2
smax ¼
þ rxy :
ð34Þ
2
In an experimental investigation, the constants A0, A1, B0, B1, C0, C1, D0 and D1 of the various terms in the
expansion of the stresses are determined by fitting Eq. (34) to the experimental data. The mode-I and modeII stress intensity factors, KI and KII, and the nonsingular stress, r0x, are related to the constants A0, C0 and
B0 as shown below:
KI
A0 ¼ pffiffiffiffiffiffi ;
2p
K II
C 0 ¼ pffiffiffiffiffiffi ;
2p
rox ¼ 2B0 :
ð35Þ
Contours of smax were obtained for a fixed value of KI, KII and r0x. The remaining constants in the stress
equations, A1, B1, C1, D1 were assigned a value of zero. However the nonhomogeneity specific part of the
higher order term, r1/2 with A0 and C0 as the coefficients are retained. Fig. 2 shows the contours of constant
maximum shear stress around the crack-tip corresponding to KI = KII = 1.0 MPa m1/2, r0x = 0 and d = 5
for three different property gradation angles and for homogeneous material (d = 0). This value of d corresponds to 1.5 times increase of YoungÕs modulus over a distance of 0.1 m along the gradient. The crack is
assumed to be in the negative x-axis and the crack-tip is located at (0, 0). The contour numbers indicate
maximum shear stress, smax, values in MPa. It can be observed from Fig. 2(b) that the introduction of nonhomogeneity has not altered the structure of the contours very near the crack-tip compared to homogeneous material shown in Fig. 2(a). However away from the crack-tip, the size and tilt of contours changes
with the introduction of nonhomogeneity. The nonhomogeneity increases the size of the contours (see contours numbered 2) and the contours tilt towards the crack face. For the same nonhomogeneity factor, as the
property gradation angle (in the anticlockwise direction from positive x-axis as shown in Fig. 1) increases,
the size of contours (see contours of 2 in Fig. 2(b)–(d)) decreases and the contours start tilting away from
the crack face. However, the size of these contours is still larger than those of homogeneous material.
4.2. Contours of constant out of plane displacement
Assuming uniform strain in the out of plane direction for plane stress conditions, the contours of constant
out of plane displacement were drawn by substituting stress field equations in the relation given below:
w¼
mtðrxx þ ryy Þ
;
E0 ð1 þ ax þ byÞ
ð36Þ
where m is PoissonÕs ratio of the material, which is assumed to be constant, t is the thickness of the material
in the z-direction and E0 is YoungÕs modulus at the crack-tip. Here, values of m = 0.3, t = 0.01 m, and
E0 = 1 GPa are used. The effect of inclination of property gradation on contours of out of plane displace-
N. Jain et al. / Theoretical and Applied Fracture Mechanics 42 (2004) 155–170
165
Fig. 2. Contours of constant maximum shear stress near the crack-tip for mixed mode loading in an FGM (KI = KII = 1.0 MPa m1/2
and r0x = 0, thickness = 0.012 m).
ment (in lm) for KI = KII = 1 MPa m1/2, r0x = 0 and d = 5 is shown in Fig. 3. It can be observed that the
size of contours increases and contours tilt towards the compliant side (the region of negative x- and y-axis)
with the introduction of nonhomogeneity as shown in Fig. 3(a)–(c). For u = 45 and 90, the contours tilt
further towards the compliant side and the size of the contour decreases as shown in Fig. 3(c) and (d).
5. Strain energy density criterion applied to FGMs
In materials with varying properties, cracks seldom grow self-similarly, but rather change directions as
they propagate. Thus, the state of stress ahead of the crack is mixed and involves both symmetric and skewsymmetric modes. Classical theories such as the critical energy release rate and the critical stress intensity
factor are not able to predict this phenomenon, as they require self-similar crack growth. Their use with
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Fig. 3. Contours of constant out of plane displacement around the crack-tip for mixed mode loading in an FGM
(KI = KII = 1 MPa m1/2 and r0x = 0, thickness = 0.012 m).
graded materials can potentially lead to negative results. The strain energy density criterion [6,7,9] is more
versatile in that it can predict crack growth in any direction. The strain energy density, dW/dV, criterion is
based on the local density of the strain energy field surrounding a crack. It assumes that fracture initiates
when dW/dV reaches a critical value of (dW/dV)c.
For plain strain conditions the elastic strain energy density of the FGM can be given as
dW
1þm
¼
½ð1 mÞðr2xx þ r2yy Þ 2mðrxx ryy Þ þ 2r2xy :
dV
2E0 ð1 þ kxÞ
ð37Þ
The intensity of the energy density at a critical distance rc from crack-tip can be written as
S ¼ rc ðdW =dV Þ;
ð38Þ
where S is strain energy density factor. The region inside the rc is expected to have a nonlinear and plastic
response and is called the core region.
N. Jain et al. / Theoretical and Applied Fracture Mechanics 42 (2004) 155–170
167
The strain energy density fracture criterion states that:
• Crack initiation takes place in a direction of the minimum strain energy density, Smin, i.e.
oS
¼ 0;
oh
at h ¼ hc :
ð39Þ
• Crack extension occurs when the strain energy density factor reaches a critical value, Sc.
Unlike to the stress intensity factor, KIC, or energy release rate, GIC, which are only a measure of the
crack-tip stress field, the strain energy density factor, S, is direction sensitive. The difference between K
and S is analogous to the difference between a scalar and vector quantity. It can be said here that the critical
value Scr, provides a complete knowledge of arbitrary crack extension, which involves both the direction of
crack initiation and the fracture toughness.
Explicit expression for strain energy density and strain energy density factor, S, are not provided here
but these can be easily derived by substituting the expression developed for stresses in previous sections into
Eqs. (37) and (38). The fracture angle, hc, at which S possesses a minimum value can be obtained from Eq.
(39). In the context of graded materials strain energy density factor, S, will be a function of radius vector, r.
Since r specifically (outside the core region) may not be zero, geometric singularities will no longer requires
special consideration.
In accordance with the strain energy theory [9], the failure will be determined from critical material value
of the strain energy density and is dependent upon the specific radius to be chosen. Specific choice of critical
radius, rc, requires experimental data for a given material, and hence, knowledge of failure loads for one
geometry may easily be used to determine the failure loads for other geometries.
In the following discussion some representative strain energy density distribution in the vicinity of the
crack-tip are presented. The effect of nonhomogeneity on the critical strain energy density is also discussed.
Fig. 4 shows the angular variation of strain energy density factor under pure mode-I loading. The value
of KI was assumed to be 1 MPa m1/2 and KII = 0. These variations are plotted for five different values of
nonhomogeneity parameter, d = 0, 2, 4, 6, 8. It should be noted here that the minimum strain energy density
occurs when at hc = 0 for the homogeneous case, as shown in [9]. Further the critical angle, hc, remains
equal to zero for all the values of d.
Fig. 4. Angular distribution of strain energy density factor under mode-I loading (g = 0.3, t = 0.01 m, E0 = 1 GPa).
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N. Jain et al. / Theoretical and Applied Fracture Mechanics 42 (2004) 155–170
Fig. 5. Min. strain energy density factor versus nonhomogeneity parameter under mode-I loading.
Fig. 5 shows the effect of nonhomogeneity factor on minimum strain energy density factor under pure
mode-I loading. The variation is plotted for different values of radius r. It can be seen here that for any
given value of r, as the value of nonhomogeneity factor increases, the value of minimum strain energy density corresponding to hc increases. As the nonhomogeneity factor increases the material ahead of crack gets
stiffer, reducing the value of strain energy density at h = hc.
Fig. 6 shows the angular variation of strain energy density factor under mixed mode loading. The value
of KI was assumed to be 1 MPa m1/2 and KII = 1 MPa m1/2. It can be seen here that the critical angle, hc, lies
between 53 and 34 for different values of nonhomogeneity parameter ranging from 0 to 8. As with the
Fig. 6. Angular distribution of strain energy density factor under mixed mode loading.
N. Jain et al. / Theoretical and Applied Fracture Mechanics 42 (2004) 155–170
169
previous case the minimum strain energy density factor decreases with increasing nonhomogeneity
parameter.
6. Conclusions
Crack-tip stress fields for a stationary crack along or inclined to the direction of linearly varying property gradation were obtained through an asymptotic analysis coupled with WestergaardÕs stress function
approach. These stress fields are useful in extracting fracture parameters during the analysis of full field
data around crack-tips in experimental techniques such as photoelasticity and CGS. Using these stress
fields, synthetic contours of constant maximum shear stress, and constant out of plane displacement were
developed. Following conclusions can be made from these contours:
• The contours very near to crack-tip are not affected either by the nonhomogeneity or by the inclination
of property gradation, u. This suggests that the singular term of the stress field is dominant near the
crack-tip. However, the nonhomogeneity and inclination of gradation direction have significant influence on the contours away from the crack-tip.
• The size of the contours of constant maximum shear stress increases and they tilt towards the crack face
due to nonhomogeneity. However for the same nonhomogeneity factor, the size of the contours
decreases and the contours tilt away from the crack face as the angle u (gradation direction) increases.
The size of these contours in the positive y-axis region increases and size of those in the negative y-axis
region decreases for a compressive r0x. The trend is reverse for tensile r0x.
• The contours of constant out-of-plane displacement shifts towards the compliant region when nonhomogeneity is introduced. However, the size of these contours in the negative y-axis region decreases as the
gradation angle increases.
The strain energy density criterion was applied to crack problem in graded materials. The effect of property gradation on critical conditions for onset of crack growth is discussed. It is shown that graded material
can offer more resistance to crack growth and suppress crack growth in some situations.
Acknowledgments
The partial financial support of NSF, under grant number CMS 0244330 is acknowledged. This research
was also supported by NIH grant number P20 RR016457 from the BRIN Program of the National Center
for Research Resources.
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