The existing studies on the behavior of cracks in continuously graded materials assume the elastic properties to vary in the plane of the crack. In the case of a plate graded along the thickness and having a crack in its plane, the elastic properties will vary along the crack front. The present study aims at investigating the effect of elastic gradients along the crack front on the structure of the near-tip stress fields in such transversely graded materials. The first four terms in the expansion of the stress field are obtained by the eigenfunction expansion approach (Hartranft and Sih, 1969, “The Use of Eigen Function Expansion in the General Solution of Three Dimensional Crack Problems,” J. Math. Mech., 19(2), pp. 123–138) assuming an exponential variation of the elastic modulus. The results of this part of the study indicated that for an opening mode crack, the angular structure of the first three terms in the stress field expansion corresponding to $r(−1∕2)$, $r0$, and $r1∕2$ are identical to that given by Williams’s solution for homogeneous material (Williams, 1957, “On the Stress Distribution at the Base of a Stationary Crack,” ASME J. Appl. Mech., 24, pp. 109–114). Transversely graded plates having exponential gradation of elastic modulus were prepared, and the stress intensity factor (SIF) on the compliant and stiffer face of the material was determined using strain gauges for an edge crack subjected to pure bending. The experimental results indicated that the SIF can vary as much as two times across the thickness for the gradation and loading considered in this study.

1.
Erdogan
,
F.
, 1995, “
Fracture Mechanics of Functionally Graded Materials
,”
Composites Eng.
0961-9526,
5
(
7
), pp.
753
770
.
2.
Gu
,
P.
, and
Asaro
,
R. J.
, 1997, “
,”
Int. J. Solids Struct.
0020-7683,
34
(
1
), pp.
1
17
.
3.
Gu
,
P.
, and
Asaro
,
R. J.
, 1997, “
Cracks Deflection in Functionally Graded Materials
,”
Int. J. Solids Struct.
0020-7683,
34
(
24
), pp.
3085
3098
.
4.
Rousseau
,
C.-E.
, and
Tippur
,
H. V.
, 2001, “
,”
Int. J. Solids Struct.
0020-7683,
38
, pp.
7839
7856
.
5.
Anlas
,
G.
,
Lambros
,
J.
, and
Santare
,
M. H.
, 2002, “
Dominance of Asymptotic Crack Tip Fields in Elastic Functionally Graded Materials
,”
Int. J. Fract.
0376-9429,
115
, pp.
193
204
.
6.
Parameswaran
,
V.
, and
Shukla
,
A.
, 2002, “
Asymptotic Stress Fields for Stationary Cracks Along the Gradient in Functionally Graded Materials
,”
ASME J. Appl. Mech.
0021-8936,
69
, pp.
240
243
.
7.
Chalivendra
,
V. B.
,
Shukla
,
A.
, and
Parameswaran
,
V.
, 2003, “
Quasi-Static Stress Fields for a Crack Inclined to the Property Gradation in Functionally Graded Materials
,”
Acta Mech.
0001-5970,
162
, pp.
167
184
.
8.
Jain
,
N.
,
Shukla
,
A.
, and
Rousseau
,
C.-E.
, 2004, “
Crack Tip Stress Fields in Materials With Linearly Varying Properties
,”
Theor. Appl. Fract. Mech.
0167-8442,
42
, pp.
155
170
.
9.
Shim
,
D. J.
,
Paulino
,
G. H.
, and
Dodds
,
R. H.
, 2006, “
Effect of Material Gradation on K-Dominance of Fracture Specimens
,”
Eng. Fract. Mech.
0013-7944,
73
, pp.
643
648
.
10.
Abanto-Bueno
,
J.
, and
Lambros
,
J.
, 2006, “
An Experimental Study of Mixed Mode Crack Initiation and Growth in Functionally Graded Materials
,”
Exp. Mech.
0014-4851,
46
(
2
), pp.
179
196
.
11.
Walters
,
M. C.
,
Paulino
,
G. H.
, and
Dodds
Jr.,
R. H.
, 2004, “
,”
Int. J. Solids Struct.
0020-7683,
41
, pp.
1081
1118
.
12.
Yildirim
,
B.
,
Dag
,
S.
, and
Erdogan
,
F.
, 2005, “
,”
Int. J. Fract.
0376-9429,
132
, pp.
369
395
.
13.
Ayhan
,
A. O.
, 2007, “
Stress Intensity Factors for Three-Dimensional Cracks in Functionally Graded Materials Using Enriched Finite Elements
,”
Int. J. Solids Struct.
0020-7683,
44
, pp.
8579
8599
.
14.
Hartranft
,
R. J.
, and
Sih
,
G. C.
, 1969, “
The Use of Eigen Function Expansion in the General Solution of Three Dimensional Crack Problems
,”
J. Math. Mech.
0095-9057,
19
(
2
), pp.
123
138
.
15.
Hartranft
,
R. J.
, and
Sih
,
G. C.
, 1970, “
An Approximate Three-Dimensional Theory of Plates With Application to Crack Problem
,”
Int. J. Eng. Sci.
0020-7225,
8
, pp.
711
729
.
16.
,
R.
, and
Sih
,
G. C.
, 1975, “
An Approximate Three-Dimensional Theory of Layered Plates Containing Through Thickness Cracks
,”
Eng. Fract. Mech.
0013-7944,
7
, pp.
1
22
.
17.
Sih
,
G. C.
, 1971, “
A Review of Three-Dimensional Stress Problem for a Cracked Plate
,”
Int. J. Fract. Mech.
0020-7268,
7
(
1
), pp.
39
61
.
18.
Benthem
,
J. P.
, 1977, “
State of Stress at the Vertex of a Quarter-Infinite Crack in a Half Space
,”
Int. J. Solids Struct.
0020-7683,
13
, pp.
479
492
.
19.
Benthem
,
J. P.
, 1980, “
The Quarter-Infinite Crack in a Half Space; Alternative and Additional Solutions
,”
Int. J. Solids Struct.
0020-7683,
16
, pp.
119
130
.
20.
Nakamura
,
T.
, and
Parks
,
D. M.
, 1988, “
Three-Dimensional Stress Field Near the Crack Front in a Thin Elastic Plate
,”
ASME J. Appl. Mech.
0021-8936,
55
, pp.
805
813
.
21.
Nakamura
,
T.
, and
Parks
,
D. M.
, 1989, “
Antisymmetrical 3-D Stress Field Near the Crack Front of a Thin Elastic Plate
,”
Int. J. Solids Struct.
0020-7683,
25
, pp.
1411
1426
.
22.
Bazant
,
Z. P.
, and
Estenssoro
,
L. F.
, 1979, “
Surface Singularity and Crack Propagation
,”
Int. J. Solids Struct.
0020-7683,
15
, pp.
405
426
.
23.
Williams
,
M. L.
, 1957, “
On the Stress Distribution at the Base of a Stationary Crack
,”
ASME J. Appl. Mech.
0021-8936,
24
, pp.
109
114
.
24.
Villarreal
,
G.
,
Sih
,
G. C.
, and
Hartranft
,
R. J.
, 1975, “
Photoelastic Investigation of Thick Plate With a Transverse Crack
,”
ASME J. Appl. Mech.
0021-8936,
42
, pp.
9
13
.
25.
Rousseau
,
C.-E.
, and
Tippur
,
H. V.
, 2000, “
,”
Acta Mater.
1359-6454,
48
, pp.
4021
4033
.
26.
Tippur
,
H. V.
,
Krishnaswamy
,
S.
, and
Rosakis
,
A. J.
, 1991, “
A Coherent Gradient Sensor for Crack Tip Deformation Measurements: Analysis and Experimental Results
,”
Int. J. Fract.
0376-9429,
48
, pp.
193
204
.
27.
Dally
,
J. W.
, and
Sanford
,
R. J.
, 1987, “
Strain Gage Methods for Measuring the Opening Mode Stress Intensity Factor KI
,”
Exp. Mech.
0014-4851,
27
, pp.
381
388
.
28.
Berger
,
J. R.
, and
Dally
,
J. W.
, 1988, “
An Error Analysis of Single Strain Gage Determination of the Stress Intensity Factor
,”
Exp. Tech.
0732-8818,
12
(
8
), pp.
31
33
.
29.
McNeill
,
S. R.
,
Peters
,
W. H.
, and
Sutton
,
M. A.
, 1987, “
Estimation of Stress Intensity Factor by Digital Image Correlation
,”
Eng. Fract. Mech.
0013-7944,
28
, pp.
101
112
.
30.
Anderson
,
T. L.
, 1995,
Fracture Mechanics: Fundamentals and Applications
, 2nd ed.,
CRC
,
Boca Raton
.
31.
Joseph
,
P. F.
, and
Erdogan
,
F.
, 1989, “
Surface Crack Problems in Plates
,”
Int. J. Fract.
0376-9429,
41
, pp.
105
131
.