Upload
mohammed-abu-sufian
View
213
Download
0
Embed Size (px)
Citation preview
7/28/2019 anlas00
1/13
International Journal of Fracture 104: 131143, 2000.
2000 Kluwer Academic Publishers. Printed in the Netherlands.
Numerical Calculation of Stress Intensity Factors in Functionally
Graded Materials
G. ANLAS, M. H. SANTARE and J. LAMBROSDepartment of Mechanical Engineering, University of Delaware, Newark, DE 19716, U.S.A.; Permanentaddress: Bogazici University, Bebek, Istanbul, Turkey.
Received 3 August 1999; accepted 7 January 2000
Abstract. The finite element method is studied for its use in cracked and uncracked plates made of functionally
graded materials. The material property variation is discretized by assigning different homogeneous elastic prop-
erties to each element. Finite Element results are compared to existing analytical results and the effect of mesh
size is discussed. Stress intensity factors are calculated for an edge-cracked plate using both the strain energy
release rate and the J-contour integral. The contour dependence of J in an inhomogeneous material is discussed.
An alternative, contour independent integral J is calculated and it is shown numerically that J, the strain energyrelease rate G, and the limit of J as approaches the crack tip (where is the contour of integration) are
all approximately equal. A simple method, using a relatively coarse mesh, is introduced to calculate the stress
intensity factors directly from classical J-integrals by obtaining lim0 J.
Key words: J-integrals, stress intensity factors, uncracked FGM plate.
1. Introduction
There is a class of materials which have continuously varying mechanical properties. Although
this type of material exists commonly in nature, there has been recent interest in manufacturing
them for specific engineering applications. Such materials are called functionally graded ma-
terials because their properties have a spatial gradient. Functionally graded materials (FGMs)
are very attractive for demanding applications such as thermal and wear protective coatings,
yet their behavior needs to be better understood to fully exploit their characteristics.
Analytical work on functionally graded materials goes back as early as the late 1960s when
soil was modeled as a nonhomogeneous material by Gibson (1967). More recently, Delale and
Erdogan (1983) analytically studied the crack problem in an infinite plane where the elastic
properties varied exponentially in the direction of the crack. They showed that the asymptotic
cracktip stress field possesses the same square root singularity seen in homogeneous materials.
In 1987, Eischen studied the crack-tip-singular behavior of the stress field in a nonhomo-
geneous infinite plane by using an eigenfunction expansion technique. He verified that the
leading term of the asymptotic expansion for stresses was square-root singular. This result was
further confirmed by Jin and Noda (1994) for materials with piecewise differentiable propertyvariations. In 1994, Konda and Erdogan studied the behavior of an infinite cracked plane with
exponential properties gradients in both the in-plane directions. Using a similar technique,
Erdogan and Wu (1997) studied various far-field loadings of an infinite FGM strip. They used
an exponentially varying Youngs Modulus, E, but kept the Poissons ratio, , constant. This
approximation was based on the earlier work of Delale and Erdogan (1983) who showed that
the effect of a variation of is negligible. The work by Erdogan and Wu (1997) is one of the
7/28/2019 anlas00
2/13
132 G. Anlas, N. H. Santare and J. Lambros
few fracture solutions available for a finite width FGM. For this reason, their results are used
as the basis of comparison in the current study. For a comprehensive review of fracture of
functionally graded materials, see Erdogan (1985).
In 1996, Jin and Batra studied the crack tip stress field, strain energy release rate and stress
intensity factors in a ceramic-metal FGM. In two separate studies, Gu and Asaro (1997a, b)
analyzed fracture and crack deflection in functionally gradient materials.
In contrast to the above-described growing body of analytical studies, there are relatively
few experimental and numerical investigations of fracture of functionally graded materials. In
a recent publication, Gu et al. (1999) discussed a methodology for numerical determination
of stress intensity factors in FGMs. They studied the effect of material nonhomogeneity in
numerical computations of the J-integral. They concluded that the traditional version of the
J-integral can provide accurate results provided that it is evaluated very close to the crack
tip, using very small elements ( 105b, where b is the crack length). Li et al. (2000),show an experimental-numerical hybrid method for evaluating the strain energy release rate in
laboratory-scale FGMs. This method doesnt require a highly refined mesh but does require
experimental data. Experimental studies have in general been hampered by difficulties asso-
ciated with material fabrication even though some new experimental techniques have been
developed recently (see Lambros et al., 1999; Butcher et al., 1999; Parameswaran and Shukla,1998).
The focus of this paper is on the calculation and comparison of the stress intensity fac-
tors obtained for a cracked FGM plate by using the several different numerical techniques.
However, in the next section we first, briefly discuss finite element solutions for uncracked
FGM plates to validate the approximations and assumptions used. Subsequently, a technique
similar to that of Gu et al. (1999) is used to evaluate the J-integral numerically. The strain
energy release rate G and the J-integral are calculated for an edge-cracked FGM subject to
far field loading. A modified path independent integral, J similar to that proposed by Honein
and Herrmann (1997) is computed for the nonhomogeneous case. The results are compared to
the analytical solutions of Erdogan and Wu (1997). The relationship between the accuracy of
the finite element method (FEM) and mesh refinement is also investigated. To our knowledge,
this is the first numerical implementation and assessment of the path independent J-integral
for FGMs.
2. Study of uncracked FGM plate
The finite element method has been used extensively in solving problems involving homoge-
neous materials. The error introduced by geometric discretization of the domain, necessary in
FEM, has also received close scrutiny. However, when modeling an inhomogeneous material
(in this context, defined as one which possesses a continuous spatial variation of E and/or ) a
material property discretization is introduced in addition to the geometric one. In this section,
we investigate the accuracy of the FEM, when a material property discretization is introduced,
by comparing FEM results to simple analytical solutions. (Note that it is possible to avoidmaterial discretization by allowing a specific material property variation when formulating
the stiffness matrix. This case is being investigated in a separate work.)
7/28/2019 anlas00
3/13
Numerical Calculation of Stress Intensity Factors 133
Figure 1. Edge-cracked plate under uniform traction or displacement loading.
2.0.1. Finite element solution
The finite element solutions of a rectangular plate made of a functionally graded material
under uniform traction or constant displacement loadings are obtained using ABAQUS (1997).
Assume that the material gradient is in the x-direction, and is exponential according to
(x) = 1ex (1)where is the shear modulus of the material, 1 the shear modulus at x = 0, and is amaterial constant which represents the length scale over which the properties change (the units
for are 1/length). The Poissons ratio, , is taken to be constant. The normal stress values are
calculated on the line of symmetry. The results evaluated on the line of symmetry correspond
to the results of an infinite strip, so they can be compared to those obtained analytically in the
paper by Erdogan and Wu (1997).
2.1. UNIFORM TRACTION LOADING
The plate considered has height twice its width, and is symmetric with respect to its midline,
the y = 0 axis. The geometry of the problem is given in Figure 1 with b = 0. Using symmetry,only the upper half is considered in the finite element model. The upper edge is loaded by a
uniform traction, yy = 0. The lower edge has a zero displacement boundary condition
7/28/2019 anlas00
4/13
134 G. Anlas, N. H. Santare and J. Lambros
Table 1. Percent error in FEM results for uniform traction loading, 40
elements used in the x-direction.
x/ h E2/E1 = 2 E2/E1 = 5 E2/E1 = 10 E2/E1 = 20
0.0 0.37 1.31 2.38 3.86
0.1 0.04 0.14 0.52 1.24
0.2 0.05 0.08 0.35 0.87
0.3 0.05 0.15 0.15 0.06
0.4 0.09 0.36 0.61 0.84
0.5 0.08 0.47 0.94 1.55
0.6 0.05 0.43 0.96 1.76
0.7 0.02 0.18 0.59 1.35
0.8 0.08 0.28 0.27 0.04
0.9 0.18 0.8 1.81 3.65
1.0 0.16 3.18 21.28 50.07
in the y-direction to account for symmetry. The plate is divided into a uniform mesh with
40 elements along the x-direction and 10 elements along the y-direction, resulting in 400
elements (900 deg of freedom (dof)).The material is functionally graded, and the modulus of elasticity changes exponentially
along the x-axis as, E(x) = E1ex , where E1 is the elastic modulus at x = 0. From equation(1), it can be shown that E1 = 21(1 + ). For the finite element solution, this change ismodeled discretely by assigning each of the 40 regions along the x-axis the value of E at
the centroid of the region, calculated according to the exponential relation given above. The
Poissons ratio is taken as = 0.3. Note that the use of a uniform rectangular mesh makesthe assignment of material properties quite straightforward. For a radially focused mesh, as
is commonly used in fracture problems, the material property discretization may be more
involved.
A two dimensional continuum element with four nodes and four integration points is used
for the plane strain problem. Stresses are calculated at the nodes of the 40 elements on y = 0(the axis of symmetry). Since each row of elements has a discrete modulus, there will be
significantly different nodal stresses for each node shared by adjacent elements. The reported
results for yy /0 are the numerical averages in these cases. The results for E2/E1 = 2, 5,10 and 20 are compared to the analytical results from Erdogan and Wu (1997) in Table 1
where E2 is the modulus of elasticity at x = h. For an assumed exponential gradient as inEquation (1), there are two independent parameters needed to establish the material property;
either the modulus at two locations, as used here, or the modulus at one point and the length
scale .
The table shows that in this case, the error involved is less than 1% for most of the nodes. Ingeneral, the average error increases with increasing E2/E1. The errors at the end points x = 0and x = h are larger due to the discretization of the material property in the element. Thereis no adjacent element to mitigate this discrepancy as there is at the other points (recall that
the material property assignment is made at the centroid of the elements, not at the nodes).
Table 2 shows the same comparison for a 100 by 25 element mesh. It is seen that these results
are even closer to the analytical results.
7/28/2019 anlas00
5/13
Numerical Calculation of Stress Intensity Factors 135
Table 2. Percent error in FEM results for uniform
traction loading, 100 elements used in the x-direction.
x/ h E2/E1 = 2 E2/E1 = 5 E2/E1 = 10
0.0 0.10 0.70 1.35
0.1 0.04 0.31 0.76
0.2 0.03 0.22 0.55
0.3 0.05 0.05 0.05
0.4 0.04 0.26 0.48
0.5 0.08 0.43 0.82
0.6 0.09 0.43 0.93
0.7 0.01 0.23 0.63
0.8 0.04 0.19 0.13
0.9 0.09 0.54 1.39
1.0 0.04 1.43 10.40
2.2. UNIFORM DISPLACEMENT LOADING
For the case of uniform displacement loading, the boundary condition at the top and the bottom
of the plate is an applied constant displacement, V0, in the y-direction. The numerical results
for normalized stress, yy h(1 2)/E1V0 for uniform displacement loading are compared toanalytical results from Erdogan and Wu (1997), and the percent error is tabulated in Table 3.
The comparison of the FEM results to analytical results shows that the error is less than
0.1% in most of the cases, when 400 elements are used to discretize the domain. With the
same discretization, the results of constant displacement loading are more accurate than the
corresponding ones for uniform traction loading (compare with Table 1). One reason is that the
analytical solution for the uniform traction problem used in the comparison, is exact only on
the line of symmetry, y = 0. The displacement solution on the other hand, is exact everywherein the domain. The comparison of analytical and FE results in Table 4 shows that with 100
elements used in the direction of property change, instead of the 40, FE results are almost
identical to the analytical solution.
3. Study of Edge-cracked FGM Plate
In this section, the problem of a nonhomogeneous finite plate with an edge crack is stud-
ied. The problem geometry is shown in Figure 1 for either a uniform traction or a uniform
displacement applied in the y-direction.
The domain is discretized into uniform meshes of 20 by 20 elements (about 880 dof), 100
by 25 elements (about 5250 dof) and 200 by 50 elements (about 20800 dof) respectively. Afour-node, two dimensional plane strain element is used. Stress intensity factors are calculated
using the strain energy release rate, G, the J-contour integral as r 0 (we will denote thislimiting value as J (1)), and the contour integral J which will be defined subsequently. The
numerically evaluated stress intensity factors are then compared to analytical results given by
Erdogan and Wu (1997). The effect of mesh size, crack length, , and the relation among G,
J (1), and J are discussed.
7/28/2019 anlas00
6/13
136 G. Anlas, N. H. Santare and J. Lambros
Table 3. Percent error in FEM results for constant displacement
loading, 40 elements used in the x-direction.
x/ h E2/E1 = 2 E2/E1 = 5 E2/E1 = 10 E2/E1 = 20
0.0 0.9 2 2.9 3.8
0.1 0 0 0.08 0.07
0.2 0 0 0.06 0.05
0.3 0 0 0.1 0.04
0.4 0 0 0.04 0.06
0.5 0 0 0.06 0.09
0.6 0 0 0.05 0.07
0.7 0 0.03 0.04 0.07
0.8 0 0.03 0.05 0.04
0.9 0 0.02 0.05 0.08
1.0 0.85 2 2.84 3.7
Table 4. Percent error in FEM results for con-stant displacement loading, 100 elements used in the
x-direction.
x/ h E2/E1 = 2 E2/E1 = 5 E2/E1 = 10
0.0 0.30 0.80 1.20
0.1 0.00 0.00 0.08
0.2 0.00 0.00 0.00
0.3 0.00 0.00 0.00
0.4 0.00 0.00 0.00
0.5 0.00 0.00 0.03
0.6 0.00 0.00 0.01
0.7 0.00 0.00 0.02
0.8 0.00 0.00 0.00
0.9 0.00 0.00 0.01
1.0 0.35 0.80 1.14
In the general case, when a crack advances at a fixed displacement, the strain energy release
rate is defined as follows:
G = dUdA
(2)
where U is the strain energy, and A is the crack surface area. By using a node release technique
in the finite element program, dU/dA can be approximated by U/A. U is the change
in strain energy as a result of an increment in crack growth, holding the external boundary
conditions constant. The smaller the increment A (A = tb, where t is the thickness ofthe plate), the more accurate this result will be. The stress intensity factor can be calculated
using the following relation for plane strain (Jin and Batra, 1996),
7/28/2019 anlas00
7/13
Numerical Calculation of Stress Intensity Factors 137
Figure 2. J-integral vs. contour, b/ h = 0.2, uniform traction loading.
K
2
l =GEtip
1 2 , (3)where Etip is the Youngs Modulus at the location of the crack tip.
For homogeneous materials, an alternative measure of the strain energy release rate is the
J-integral, which is defined by Rice (1968),
J =
W n1 ijni
uj
x1
ds, (4)
where in a two-dimensional cracked body, is any path beginning at the bottom crack face
and ending on the top crack face. Here W is the strain energy density, uj are the displacement
components and ni are the components of the outward unit normal to . In the absence of
body forces, thermal strains and crack surface tractions, the J-integral is path independent for
homogeneous materials. It is well known that in the case of homogeneous materials, G = J.For a nonhomogeneous material, in general, the J-integral is not path independent. In this
study, the contour integral, J, is calculated using ABAQUS. The change of J with contour
number is shown in Figure 2 for E2/E1 = 2, 5, and 10. In this and subsequent figures, thecontour numbers represent incrementally larger contours around the crack tip where the size
and the increment is governed by the mesh refinement. Each contour is placed symmetrically
around the crack tip and includes an integer number of elements. For a nonhomogeneous
material, G = J because J is path dependent. However, it can be shown that G = J as 0 (Gu et al., 1999). Tohgo et al. (1996) also show that in FGMs the J-integral is pathdependent and its value for a path adjacent to the crack tip is identical to that for homogeneous
materials.
4. J-integral
In the case of nonhomogeneous materials, the strain energy density W is not only a function
of (x), but it also depends on x explicitly, i.e., W = W((x),x), due to material propertygradients. This results in an extra term in the classical J-integral which needs to be subtracted
from Equation (4) to obtain the contour independent integral J as
7/28/2019 anlas00
8/13
138 G. Anlas, N. H. Santare and J. Lambros
J =
W n1 ijni
uj
x1
ds
A
W,1q dA, (5)
where is any path beginning at the bottom crack face and ending on the top crack face and A
is the area enclosed by that contour. In Equation (5), W,1 denotes partial differentiation of W
with respect to the explicit x dependence. The
J integral will yield a zero value on any closed
contour in a homogeneous as well as an inhomogeneous material, and therefore when used
for fracture problems will always be path independent. This result was confirmed analytically
in the work of Honein and Herrmann (1997). In the following, we will numerically implement
the path-independent J and compare the calculated results with those derived through other
methods.
The finite element formulation of the contour integral J is constructed as suggested by
Li et al. (1985). In the absence of thermal stresses, and crack-face tractions, the following
discretized form is obtained from Equation (5):
J =
A
Np
=1
(ijui,1 W 1,i )q1,i W,1q1
det
xk
l
p
p, (6)
where p are the weight functions of the corresponding Gauss integration points. The quan-
tities in { } are all evaluated at the integration points for each element within the contourchosen. N is the number of integration points per element and q1 is a device used to facilitate
the evaluation of a contour integral in finite elements. In general, a nodal value of 0 or 1 is
assigned to Q1 and q1 can be defined within an element as follows:
q1 =4
i=1Ni Q1i , (7)
where Ni are the interpolation functions for the elements. In this study, for the calculation
of q1, a plateau function representation of q1 is used (see Li et al., 1985). Recall that W,1 in
equation (12) is not the total derivative but rather the partial with respect to x1 which willonly exist if the material property E is an explicit function of x1, (e.g., E = E(x1)). If thematerial property variation is given explicitly, it is simple to analytically derive an expression
for W/x1. In the case of equation (1), W/x1 = W(,x1)
5. Calculation of the Stress Intensity Factors
We analyze edge-cracked strips with b/ h = 0.4, and E2/E1 = 2.0 and E2/E1 = 0.5 (referto Fig. 1). The stress intensity factors, KI are calculated in three different manners; using the
strain energy release rate G, the contour integral J, and the path independent contour integral
J as follows:
K2I(1 2)Etip
= G = J = J (1) lim0
J. (8)
The graphs ofJ and J are plotted in Figs. 3 and 4 to show their variation with contour number.
Clearly, J is contour dependent as was seen previously in Fig. 2. In contrast, J is contour
independent after the first two contours thus numerically verifying the path independence of
Equation (5).
7/28/2019 anlas00
9/13
Numerical Calculation of Stress Intensity Factors 139
Figure 3. J and J vs. contour, b/ h = 0.4, E2/E1 = 2.0, 20 by 20 mesh.
The values for J and J from the first contour are disregarded since they are generallynot considered accurate for most finite element meshes. Using the results of the J-integrals
calculated from contour 2 to contour 20, and fitting a fourth order polynomial to these data
points, a value for J (1) can be obtained. Extending the curve to larger number of contours
produces very little change. J (1) can be approximated numerically as the intercept of the
polynomial curve fit at n = 1, i.e. the crack tip. From Fig. 3 we can see that the limitingvalue of J, called J (1), is approximately equal to the path independent integral J for a 200
by 50 mesh. In addition, the stress intensity factors can be calculated from energy quantities
using Equation (14). All results are normalized using the procedure outlined by Erdogan and
Wu (1997) and compared to the exact results obtained by them. For uniform traction the
normalized stress intensity factor is KI = KI/0
b where 0 is the applied traction and
b the crack length. For the case of applied displacement, 0,
KI
=KI/0
b where 0
=E1V0/ h(1 2).For b/ h = 0.4, using the 20 by 20 mesh and J, we have computed the normalized stress
intensity factor as 1.9244. The exact result calculated by Erdogan and Wu (1997) is 1.9573.
Using the same mesh, G computed from U/A is not reliable because the mesh is very
coarse. In addition, J (1) is not close to J (see Fig. 3 for a comparison of J and J (1)). When
the 100 by 25 mesh is used however, J and J (1) are almost equal, and G is 4% lower. In this
case, the normalized stress intensity factors are 1.9313, 1.9324 and 1.898 respectively. When
the even finer, 200 by 50 mesh is used, the normalized stress intensity factors are 1.9458,
1.9461 and 1.931, respectively.
A similar computation is carried out for E2/E1 = 0.5. The plots for, J and J (1) are givenin Fig. 5 for a 200 by 50 mesh. Note that in this case J decreases with increasing contour
number since now E2/E1 < 1. The normalized stress intensity factor is 2.231 when calculatedusing J, and 2.229 when calculated using J (1). The exact result is 2.2598 (Erdogan and Wu,
1997). The results show that, computationally J (1) J and the use ofJ (1) gives satisfactoryresults in stress intensity factor calculations with relatively coarse meshes.
7/28/2019 anlas00
10/13
140 G. Anlas, N. H. Santare and J. Lambros
Figure 4. J and J vs. contour, b/ h = 0.4, E2/E1 = 2.0, 200 by 50 mesh.
Figure 5. J and J vs. contour, b/ h = 0.4, E2/E1 = 0.5, 200 by 50 mesh.
6. Comparison of numerically calculated stress intensity factors
We have shown numerically that J is path independent. However, this quantity can be cum-
bersome to compute. We have also shown that, J (1) is a good approximation to J, which is
in agreement with Gu et al. (1999) and Tohgo et al. (1996). In the previous section, we have
also shown that the stress intensity factors can be approximately calculated using J (1) from a
coarse, uniform mesh, when the element size is in the order of
102b, where b is the crack
length. In this section, we study the accuracy of J (1). To generate the numerical results shownin this section, we use a uniform mesh of 100 by 25, with 8-noded elements (15 500 dof).Note that although this mesh is more refined than those used in previous sections, it is still
relatively coarse and does not focus on the crack tip. Figs. 6 and 7 show the normalized stress
intensity factors for different crack lengths and E2/E1 = 2.0 and E2/E1 = 10 respectively.Numerical results are generated using G and J (1), where G is calculated using the node
7/28/2019 anlas00
11/13
Numerical Calculation of Stress Intensity Factors 141
Figure 6. Comparison of the normalized stress intensity factors for uniform traction loading, E2/E1 = 2.0.
Figure 7. Comparison of the normalized stress intensity factors for uniform traction loading, E2/E1 = 10.
release technique described in section3. The exact results are again taken from Erdogan and
Wu (1997).
Even with this relatively coarse mesh, we see that J (1) gives very good results. On average,
the error increases with increasing . However, the interdependence of parameters on , b/ h,
and mesh refinement is rather complex. Ideally, we would like to conduct a full field compar-
ison between the numerical and analytical results. However, the analytical results published
to date do not provide this detail. One local field parameter we do have access to is the crack
opening displacements from Erdogan and Wu (1997). The crack surface displacements foruniform traction loading are calculated for an edge crack with b/ h = 0.2 and plotted in Fig. 8for E2/E1 = ratios of 2.0, 5.0, and 10.0. These results are in very good agreement with theexact results presented by Erdogan and Wu (1997).
7/28/2019 anlas00
12/13
142 G. Anlas, N. H. Santare and J. Lambros
Figure 8. Crack surface displacement for b/ h = 0.2. Edge crack under uniform traction loading,V(s) = E1(x, 0)/[2h0(1 )].
7. Concluding remarks
In this study, stresses in the uncracked plate are calculated under both uniform traction and
uniform displacement loadings. The results are compared to exact solutions in Tables 14. Tra-
ditional finite elements give fairly accurate results for the uncracked case with the assignment
of properties at the centroid of each element.
In addition, we have used the J contour integral results of ABAQUS in the calculation of
stress intensity factors for an edge cracked plate made of a functionally graded material. We
have numerically demonstrated the path independence of the contour integral J, a modified
J-integral in Equation (11), and shown that J = J (1) where J (1) = lim0 J. We comparedour results for normalized stress intensity factors to the analytical results presented by Erdogan
and Wu (1997). This comparison showed that even using a relatively coarse, uniform mesh,the results obtained from J and J (1) are very close to the analytical ones. However, for
the same mesh, the normalized stress intensity factor results obtained using the node release
technique for G are far less accurate. Clearly, the accuracy of the node release technique can
be improved with mesh refinement. Nevertheless, significant mesh refinement is not needed
to obtain accuracy using J (1), a quantity which can be calculated using many existing finite
element codes. The method presented here is quite simple and easy to implement compared
with the analytical and other computational methods.
Acknowledgement
The authors would like to thank NSF for support of this research through grant CMS-9712831.
References
ABAQUS, Theory Manual, Hibbitt Karlsson and Sorensen, Version 5.7, 1997.
Butcher, R.J., Rousseau, C.E. and Tippur, H.V. (1999). A functionally graded particulate composite: preparation,
measurements and failure analysis. Acta Materialia 47(1), 259268.
7/28/2019 anlas00
13/13
Numerical Calculation of Stress Intensity Factors 143
Delale, F. and Erdogan, F. (1983). The crack problem for a nonhomogeneous plane. Journal of Applied Mechanics
50, 609614.
Eischen, J.W. (1987). Fracture of nonhomogeneous materials. International Journal of Fracture 34, 322.
Erdogan, F. (1995). Fracture mechanics of functionally gradient materials. Composites Engineering 5(7), 753770.
Erdogan, F. and Wu, B.H. (1997). The surface crack problem for a plate with functionally graded properties.
Journal of Applied Mechanics 64, 449456.
Gibson, R.E. (1967). Some results concerning displacements and stresses in a nonhomogeneous elastic half space.
Geotechnique 17, 5867.
Gu, P. and Asaro, R.J. (1997a). Cracks in functionally graded materials. International Journal of Solids and
Structures 34(1), 17.
Gu, P. and Asaro, R.J. (1997b). Crack deflection in functionally graded materials. International Journal of Solids
and Structures 34(24), 30853098.
Gu, P., Dao, M. and Asaro, R.J. (1999). A simplified method For calculating the crack tip field of functionally
graded materials using the domain integral. Journal of Applied Mechanics 66, 101108.
Honein, T. and Herrmann, G. (1997). Conservation laws in nonhomogeneous plane elastostatics. Journal of
Mechanics Physics Solids 45(5), 789805.
Jin, Z.H. and Batra, R.C. (1996). Some basic fracture mechanics concepts in functionally gradient materials.
Journal of Mechanics Physics Solids 44(8), 12211235.
Jin, Z.H. and Noda, N. (1994). Crack tip singular fields in nonhomogeneous materials. Journal of Applied
Mechanics 61, 738740.
Konda, N. and Erdogan, F. (1994). The mixed mode crack problem in a nonhomogeneous elastic plane.Engineering Fracture Mechanics 47, 533545.
Lambros, J., Narayanaswamy, A., Santare, M.H. and Anlas, G. (1999). Manufacture and testing of a model
functionally graded material. Journal of Engineering Materials and Technology 121(4), 488493.
Li, H., Lambros, J., Cheeseman, B.A., Santare, M.H. (2000). Experimental investigation of the quasi-static fracture
of a functionally graded material. International Journal of Solids and Structures (to appear).
Li, F.Z., Shih, C.F. and Needleman, A. (1985). A comparison of methods for calculating energy release rates.
Engineering Fracture Mechanics 21(2), 405421.
Parameswaran, V. and Shukla, A. (1998). Dynamic fracture of a functionally gradient material having discrete
property variation. Journal of Materials Science 33, 33033311.
Rice, J.R. (1968). A path independent integral and the approximate analysis of strain concentration by notches and
cracks. Journal of Applied Mechanics 35, 379386.
Tohgo, K., Sakaguchi, M. and Ishii, H. (1996). Applicability of fracture mechanics in strength evaluation of
functionally graded materials. JSME International Journal Series A 39(4), 479488.