Engineering Fracture Mechanics 76 (2009) 1800–1810

Contents lists available at ScienceDirect

Engineering Fracture Mechanics

journal homepage: www.elsevier .com/locate /engfracmech

Experimental investigation of the mixed-mode crack propagationin ZrO2/NiCr functionally graded materials

Xin Jin, Linzhi Wu *, Licheng Guo, Hongjun Yu, Yuguo SunCenter for Composite Materials, Harbin Institute of Technology, Harbin 150080, PR China

a r t i c l e i n f o

Article history:Received 15 October 2008Received in revised form 18 March 2009Accepted 8 April 2009Available online 21 April 2009

Keywords:Functionally graded materialsMixed-mode fractureDigital image correlationFracture toughnessCrack propagation

0013-7944/$ - see front matter Crown Copyright �doi:10.1016/j.engfracmech.2009.04.003

* Corresponding author. Tel.: +86 451 86402376;E-mail address: wlz@hit.edu.cn (L. Wu).

a b s t r a c t

Mixed-mode fracture response of ZrO2/NiCr functionally graded materials (FGMs) is inves-tigated using fracture test, digital image correlation technique and extended finite elementmethod. It is found that: (1) prior to crack initiation, the increasing elastic modulus aheadof crack-tip can enhance the mode mixity of crack-tip; (2) the crack with the increasingelastic modulus ahead of crack-tip kinks less than the one with the decreasing elastic mod-ulus, which is caused by the elastic gradient and the local fracture toughness; (3) the het-erogeneity of microstructure can cause the local perturbation but has no obvious effect onthe overall crack propagation path.

Crown Copyright � 2009 Published by Elsevier Ltd. All rights reserved.

1. Introduction

Functionally graded materials (FGMs) are new generation of engineering materials wherein the material composition var-ies gradually in some directions to achieve unique mechanical, thermal and electrical performances different from those ofhomogeneous or joined dissimilar materials. FGMs are ideal candidates for the applications requiring multifunctional per-formance. For example, the ceramic/metal FGMs can be designed to reduce thermal stresses and take advantage of the heatand corrosion resistances of ceramic and the mechanical strength of metal. Due to the variation of properties, the fracturebehavior in FGMs is very complicated.

In contrast to the extensive investigations on the mode-I fracture of FGMs [1–10], few studies on the mixed-mode frac-ture of FGMs have been reported. Konda and Erdogan [11] demonstrated the dependence of the stress intensity factors onthe relative angle between crack and material gradient, and predicted an asymmetry stress field around the crack-tip, poten-tially resulting in the crack kink. Gu and Asaro [12] studied the mixed-mode fracture of FGMs with consideration of a crackperpendicular to the material gradient. Considering local homogeneity and maximum energy release rate criterion, theyfound that the crack kinked towards the compliant region due to the greater release of elastic potential energy. Guo et al.[13] studied the plane crack problem for a functionally graded coating-substrate system under a concentrated load. Andthe mixed-mode stress intensity factors and strain energy release rates are obtained. Among the numerical studies relatedto the mixed-mode fracture of FGMs, Kim and Paulino [14] simulated the crack propagation path using finite element meth-od (FEM); Dolbow and Gosz [15] considered the plane problem of an arbitrarily oriented crack using extended finite elementmethod (XFEM).

Comparing with the theoretical and numerical efforts, the experimental investigations about the mixed-mode fracture ofFGMs are rare. Rousseau and Tippur [16] used coherent gradient sensing (CGS) technique to investigate the fracture behavior

2009 Published by Elsevier Ltd. All rights reserved.

fax: +86 451 86402386.

Nomenclature

a crack lengthr radial coordinate of polar coordination at crack-tipux displacement component along x-directionuy displacement component along y-directionu0x rigid body translation along x-directionu0y rigid body translation along y-directionA1 rigid body rotationB specimen thicknessE elastic modulusEtip elastic modulus of crack-tipKI mode-I stress intensity factorKIC fracture toughnessKII mode-II stress intensity factorP applied loadS span length of three-point-bending specimenW specimen widthY correction factor of the stress intensity factora kinking angleh angular coordinate of polar coordination at crack-tipmtip Poisson’s ratio of crack-tiprs yielding strengthw mode mixity of crack-tipDP load increment in the straight part of load–deflection curveDd deflection increment in the straight part of load–deflection curve

X. Jin et al. / Engineering Fracture Mechanics 76 (2009) 1800–1810 1801

of the glass particle/epoxy FGMs with a crack perpendicular to the elastic gradient and proved experimentally that the crackkinked towards the compliant region. And they pointed out that both maximum hoop stress [17] and vanishing KII criteria[18] can be applied to predict the kinking angle for FGMs. Similar problem was also studied by Tilbrook et al. [19] for thealumina/epoxy FGMs using crack growth experiment, phase-shifted Moiré interferometry technique and FEM. They foundthat the elastic gradient had a dominant influence on the mode mixity of the crack-tip and the crack propagation direction.Abanto-Bueno and Lambros [20] experimentally studied the fracture response of the FGMs (ultraviolet-irradiation-hardenedpolymer) under different mixed-mode loading conditions which were generated by applying an inclined external in-planeloading, inclining the crack at an angle to the material gradient, and a combination of both cases, respectively. In addition,they extracted the fracture parameters including the stress intensity factors KI, KII and T-stress using digital image correlation(DIC) technique, and discussed the effect of T-stress on the crack kink. For the dynamic fracture of FGMs, Kirugulige and Tip-pur [21] studied the influence of material gradient on the crack propagation during a mixed-mode dynamic fracture test.They applied CGS technique coupled with a high-speed photography to map the transient crack-tip deformation beforeand after the crack initiation, and found that the crack situated on the compliant side of the specimen kinked less thanthe one situated on the stiff side.

In this paper, a detailed experimental investigation is conducted to understand the influences of the varied mechanicalproperties (elastic modulus E and fracture toughness KIC) on the fracture response of a ceramic/metal FGMs under the qua-si-static mixed-mode loading condition, which has not been found in previous reports. The FGMs consisting of ZrO2 and NiCralloy is fabricated by powder metallurgy. The distribution curves of E and KIC in the ZrO2/NiCr FGMs are obtained from thetests conducted on the homogeneous specimens with different volume fractions of NiCr. The mixed-mode loadings are gen-erated by offsetting the loading-point relative to the crack line. The deformation around the crack-tip prior to the crack ini-tiation is measured using DIC technique. The influences of the gradients of the elastic modulus and fracture toughness on thecrack initiation and propagation are analyzed. In addition, according to the maximum hoop stress criterion, the crack initi-ation and propagation are predicted by DIC and XFEM, respectively.

2. Materials and experimental procedure

2.1. Processing and materials

The ZrO2/NiCr FGMs used in the present investigation is fabricated by powder metallurgy. Commercially available ZrO2

(manufactured by Shanghai Institute of Ceramics) and Ni–20 wt%Cr alloy powder (NiCr, manufactured by General ResearchInstitute for Nonferrous Metals) are chosen as the raw materials. The ZrO2 and NiCr powders are mixed in volume ratios of10-0, 9-1, 8-2, 7-3, 6-4 and 5-5, respectively, and each mixture is suspended in ethanol, milled for 3 h by a ball mill and thendried. All mixed powders are stacked sequentially so as to form a green compact with graded composition in a steel die and

1802 X. Jin et al. / Engineering Fracture Mechanics 76 (2009) 1800–1810

compressed at 20 MPa. Then the green compacts are sintered under the hot-pressing condition of 1300 �C and 5 MPa for1.5 h. The relative density of the ZrO2/NiCr FGMs is 97.2%. In the same process, the homogeneous composites with the mate-rial composition corresponding to the each layer of the ZrO2/NiCr FGMs are also fabricated.

Fig. 1 shows the microstructure of the ZrO2/NiCr FGMs. The white and gray regions exhibit the NiCr and ZrO2 phases,respectively. The microstructure of each layer in the FGMs is characterized by the NiCr particles dispersed in the ZrO2 matrix.In addition, the stepwise gradient in the microstructure becomes comparatively unclear due to the sintering process.

2.2. Mechanical properties

The variations of elastic modulus (E) and fracture toughness (KIC) in the FGMs are determined directly from the three-point-bending test (Fig. 2a) and the fracture test (Fig. 2b) on the homogeneous specimens, respectively. Elastic modulus Eis calculated by the equation

E ¼ S3DP

4WB3Ddð1Þ

where S, W and B are the span length, specimen width and specimen thickness, respectively; DP and Dd are the load incre-ment and deflection increment as measured from the straight part of the load–deflection curves [22]. Bending tests are con-ducted on an Instron 5569 universal testing machine at crosshead speed of 0.5 mm/min. And three test pieces are tested toget the average value of E for each material composition. According to ASTM E399-90, an edge crack with the length 4 mmand the width 0.3 mm is cut along the mid-span of the fracture specimen using a high-speed diamond impregnated circularsaw. Fracture tests are conducted on the Instron 5569 universal testing machine at the crosshead speed of 0.05 mm/min.Fracture toughness KIC is calculated by the equation

K IC ¼3PSY

ffiffiffiap

2WBð2Þ

where P, S, a, W and B are the fracture load, span length, crack length, specimen width and specimen thickness, respectively,and Y is a correction factor of the stress intensity factor and given as a function of a/W [23]. The average fracture toughnessfor each material composition is taken from three tests.

The variations of E and KIC with the material composition are illustrated in Fig. 3a. The elastic modulus decreases monot-onously from 201 to 119 GPa with increasing the NiCr content. The fracture toughness increases monotonously from3.2 MPa m1/2 to 12.0 MPa m1/2 when the NiCr volume fraction increases from 0% to 50%. It should be noted that KIC is thefracture toughness of each homogeneous specimen under the plane strain condition. The geometries of all specimens satisfythe criterion of ASTM E399-90. That is

B; a � 2:5ðK IC=rsÞ2 ð3Þ

where B, a and rs are the specimen thickness, the crack length and the yielding strength of the homogeneous specimens,respectively. In addition, according to the relationship between the material composition and the position of the each layerin the ZrO2/NiCr FGMs, the distribution curves of E and KIC along the width of the FGM specimen are obtained and shown inFig. 3b.

2.3. Mixed-mode fracture test

To exclude the influences of specimen geometry and loading condition on the fracture behavior, all specimens with an iden-tical geometry are prepared, and all specimens are tested under the same mixed-mode loading condition by loading at an offset

Fig. 1. Microstructure of the ZrO2/NiCr FGMs.

(a)

S=32

Load

W=3

B=4

(b)

S=32

a=4

Load

W=8

B=4

Fig. 2. Schematics of the three-point-bending test (a) and the fracture test (b) (Dimensions in mm.)

0 1 2 3 4 5 6 70

50

100

150

200

0

2

4

6

8

10

12

EE (

GPa

)Width of the FGM specimen (mm)

KIC

KIC

(M

Pa m

1/2 )

(b) (a)

0 10 20 30 40 500

50

100

150

200

0

2

4

6

8

10

12

E (

GPa

)

Volume fraction of NiCr (%)

EK

ICK

IC (

MPa

m1/

2 )

Fig. 3. Elastic modulus and fracture toughness as functions of the material composition (a) and width of the FGM specimen (b) in the ZrO2/NiCr FGMs.

S=32

a=2

x

y

r θ

δ =4

Load

W=7.2

B=4

Fig. 4. Schematic of the mixed-mode fracture test on the FGM specimens. (Dimensions in mm.)

X. Jin et al. / Engineering Fracture Mechanics 76 (2009) 1800–1810 1803

distance (d = 4 mm) from the initial crack line, as shown in Fig. 4. An edge notch (root radius�150 lm and length 2 ± 0.01 mm)is cut along the mid-span of each specimen. For the FGM specimens, the edge notch is parallel to the material gradient. Themixed-mode fracture test is conducted on the Instron 5569 universal testing machine at the crosshead speed of 0.05 mm/min.

2.4. Digital image correlation (DIC) technique

To observe the deformation around the crack-tip, some specimens are examined using DIC technique during the mixed-mode fracture test. DIC is a robust technique used to measure the full-field deformation accurately. To determine the defor-mation of the specimen surface, two images containing the random speckles on the specimen surface, one prior to deforma-tion and the other after deformation, are compared. Details of the method can be found in Refs. [24–26].

A schematic of the DIC set-up for the mixed-mode fracture test is shown in Fig. 5. Prior to the fracture test, one surface ofthe specimen is sprayed with black paint using an airbrush to create a random speckle pattern. An optical microscope (Olym-pus, SZX12) coupled with a CCD camera (PixeLINK, PL-A741) is focused on the region around the crack-tip to record thespeckle pattern with one frame per second during the testing. In addition, the crack kinking angle and crack trajectoryare measured from the recorded images.

2.5. Maximum hoop stress criterion

The crack propagation direction is predicted by the maximum hoop stress criterion [17] which has been accepted to pre-dict the crack propagation path in the homogeneous materials. According to this criterion, the crack will propagate along the

White Light CCDComputer

Load

Crack

Speckle pattern

Fig. 5. Schematic of the DIC set-up for the mixed-mode fracture test.

1804 X. Jin et al. / Engineering Fracture Mechanics 76 (2009) 1800–1810

direction of the maximum hoop stress around the crack-tip. Therefore, the kinking angle a can be computed uniquely bysolving the equation

K I sinaþ K IIð3 cos a� 1Þ ¼ 0 ð4Þ

where KI and KII are mode-I and mode-II stress intensity factors, respectively. And the crack propagation will occur under thecondition of

cosa2

K I cos2 a2� 3

2K II sina

� �� K IC ð5Þ

where KIC is the fracture toughness at the crack-tip. Once KI and KII have been measured, the kinking angle a in Eq. (4) can bepredicted. Then substituting a, KI, KII and KIC into Eq. (5), if Eq. (5) is satisfied, the crack will propagate along the predicteddirection.

2.6. Modeling by the extended finite element method (XFEM)

The XFEM [27,28] is used to simulate the crack propagation under the mixed-mode loading condition. A typical XFEMmodel is shown in Fig. 6. The model consists of approximately 1600 eight-node isoparametric elements and is analyzed un-der the 2-D plane strain condition. According to Fig. 3b, the varied values of E and KIC along the width of the model are as-signed to the integration points of these elements. The effect of Poisson’s ratio is negligible in FGMs [3]. Therefore, the valueof Poisson’s ratio is taken as 0.3 in the entire model.

3. Results and discussion

To investigate the influence of the varied mechanical properties (E and KIC) on the fracture response of the FGMs underthe quasi-static mixed-mode loading condition, two types of FGM specimens are tested and analyzed: (a) FGM-A corre-sponds to the specimen with a crack situated on the stiff side of the specimen and propagating along the direction of decreas-ing elastic modulus and increasing fracture toughness, (b) FGM-B corresponds to the specimen with a crack situated on thecompliant side of the specimen and propagating along the direction of increasing elastic modulus and decreasing fracturetoughness. Since the specimen geometry and the experimental conditions are same for the two FGM specimens, any differ-ence in the fracture response can be directly attributed to the combined effect of the elastic gradient and the fracture tough-

Loading point

Crack

Fig. 6. Finite element model for simulating the crack propagation path of the FGM specimens.

X. Jin et al. / Engineering Fracture Mechanics 76 (2009) 1800–1810 1805

ness gradient. For comparison, a homogeneous specimen with the material composition of 50% NiCr/50% ZrO2 (named bynon-FGM) is also tested under the same experimental conditions. Three test pieces of each type of specimen are tested toensure the repeatability. The results are discussed in the following sections.

3.1. Crack initiation

The displacement components ux and uy of the FGM-A and FGM-B specimens are measured by the DIC technique justprior to the crack initiation. Fig. 7a and b show the contour plots of ux and uy for the FGM-A specimen under the applied loadof �502 N, respectively, and Fig. 7c and d display the contour plots of ux and uy for the FGM-B specimen under the appliedload of �390 N, respectively. From these contour plots, it can be found that: (1) more fringes are observed in FGM-B thanthose in FGM-A, which implies that the larger deformation forms around the crack-tip when the crack is located on the com-pliant side of the specimen; (2) the asymmetric displacement fields prove the presence of the mixed-mode deformationaround the crack-tip; (3) the prominent difference between FGM-A and FGM-B in these contour plots suggests that the var-ied mechanical properties have a visible influence on the crack initiation. Comparing Fig. 7a with Fig. 7c and Fig. 7b withFig. 7d, it can be found that prior to the crack initiation the crack-tip of FGM-B experiences a larger shear deformation thanthat of FGM-A does, which is caused by the increasing elastic modulus ahead of the crack-tip.

Table 1 lists the measured initial kinking angles of FGM-A and FGM-B. For FGM-A, the crack initiation occurs at an initialkinking angle of 9.1� with respect to the orientation of the edge notch. Whereas for FGM-B, the initial crack propagationdirection is 11.4�. Experimental results show that the crack situated on the stiff side of the specimen kinks less than theone on the compliant side. It should be pointed out that the experimental result obtained under the mixed-mode dynamicloading condition exhibits a different crack propagation pattern [21] in which the crack situated on the compliant side of thespecimen kinks less than the one situated on the stiff side. For comparison, the non-FGM specimen is also tested under thesame experimental condition and its crack initiation shows a kinking angle of 10.6�. The initial kinking angle of non-FGM islarger than the one of FGM-A, but smaller than the one of FGM-B, as shown in Table 1.

The measured displacement fields are used to extract KI and KII. For the linear elastic non-homogeneous materials, it isknown that the asymptotic fields around the crack-tip have the same forms as those in homogeneous materials, but withmaterial properties evaluated at the crack-tip position [1]. Hence, the crack-tip displacement fields of FGMs under the planestrain condition can be written as

0.600

0.700

0.800

0.900 0.9250.950

0.950

0.975

0.925

1.00

0.975 0.900

1.00

1.101.20

0.800

1.10

1.30-1

0

1

2

3

4

5

x (m

m)

y (mm)

(b)

Crack-tip

0.70

0.800.90

1.00

1.10 1.13

1.13

1.15

1.101.15

1.181.20

1.181.20

1.30

1.00

1.40

1.30 0.90

1.50

1.40

-1

0

1

2

3

4

5

x (m

m)

y (mm)

(d)

Crack-tip

3.54

3.58

3.62

3.66

3.70

3.743.78 3.74

3.703.66

3.82

3.62

3.84

3.58

3.54

3.86

3.50

3.88

3.46

3.89

3.903.91

3.42-1

0

1

2

3

4

5

x (m

m)

y (mm)

(c)

Crack-tip

3.18

3.22

3.26

3.30

3.34

3.38

3.40

3.40

3.38

3.34

3.30

3.41

3.26

3.22

3.43

3.18

3.14

3.44

3.41

-3 -2 -1 0 1 2 3

-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3

-3 -2 -1 0 1 2 3

-1

0

1

2

3

4

5

x (m

m)

y (mm)

3.42

3.44

(a)

Crack-tip

Fig. 7. Contour plots of the displacement fields just prior to the crack initiation measured by DIC: (a) ux of FGM-A; (b) uy of FGM-A; (c) ux of FGM-B; and (d)uy of FGM-B. (Displacement contour in 10�2 mm.)

Table 1Measured fracture parameters and corresponding predicted kinking angle based on the maximum hoop stress criterion for FGM-A, FGM-B and non-FGM.

Measured kinkingangle, a (�)

Measured fracture parameters Predicted kinking angle, a (�) Left-hand of Eq. (3) KIC (MPa m1/2)

KI (MPa m1/2) KII (MPa m1/2) W (�)

FGM-A 9.1 ± 0.4� 3.240 0.268 4.76 9.3� 3.241 3.20FGM-B 11.4 ± 0.3� 9.636 1.071 6.33 12.4� 9.640 9.56Non-FGM 10.6 ± 0.4� 12.041 1.158 5.46 10.8� 12.045 12.01

Table 2Fractur

FGM-AFGM-BNon-FG

1806 X. Jin et al. / Engineering Fracture Mechanics 76 (2009) 1800–1810

ux ¼2ð1þ mtipÞ

Etip

ffiffiffiffiffiffiffir

2p

rK I cos h

2 2� 2mtip � cos2 h2

� �þ K II sin h

2 2� 2mtip þ cos2 h2

� �� �� A1r sin hþ u0x ð6Þ

uy ¼2ð1þ mtipÞ

Etip

ffiffiffiffiffiffiffir

2p

rK I sin h

2 2� 2mtip � cos2 h2

� �þ K II cos h

2 2mtip � cos2 h2

� �� �þ A1r cos hþ u0y ð7Þ

where ux and uy are the displacement components parallel and normal to the crack line (see Fig. 4), respectively; KI and KII arethe mode-I and mode-II stress intensity factors, respectively; Etip and mtip are the local elastic modulus and Poisson’s ratio atthe crack-tip, respectively; A1 defines the rigid body rotation; and u0x and u0y denote the rigid body translation along the x-and y-directions, respectively. For FGM-A, FGM-B and Non-FGM, the values of Etip are 201, 136 and 119 GPa, respectively.And the value of mtip is 0.3. The mode-I and mode-II stress intensity factors for the three cases are extracted and listed inTable 1. Furthermore, under the identical experimental condition, the XFEM analysis is carried out for the three cases.The corresponding XFEM results are listed in Table 2.

The intensity of the mixed-mode at the crack-tip is presented in terms of mode mixity. The mode mixity is defined as

w ¼ arctanK II

K I

� �ð8Þ

Prior to crack initiation, the crack-tip has a mode mixity of 4.76� for FGM-A and 6.33� for FGM-B, respectively, as shown inTable 1. The larger mode mixity of the crack-tip can result in a larger initial kinking angle. So the initial kinking angle of FGM-B is larger than the one of FGM-A. Comparing the experimental and XFEM results for FGM-A, FGM-B and Non-FGM, it can befound that prior to the crack initiation the increasing elastic modulus ahead of the crack-tip can enhance the mode mixity ofthe crack-tip.

Table 1 also lists the initial kinking angles predicted by maximum hoop stress criterion. The predicted angles coincidewith the measured ones. Substituting KI, KII, a and KIC into Eq. (5), Eq. (5) is satisfied, which means that the crack will prop-agate along the predicted direction a. This indicates that the crack initiation is mainly affected by the elastic gradient aheadof the crack-tip and the local fracture toughness at the crack-tip. The values of left-hand side of Eq. (5) and KIC for FGM-A,FGM-B and non-FGM are listed in Table 1.

3.2. Crack growth behaviors in FGMs

In the mixed-mode fracture test, the stable crack growth is observed in the FGM-A specimen, but the crack growth of theFGM-B specimen is unstable. Fig. 8 shows the representative load–displacement curves obtained from the mixed-mode frac-ture tests of FGM-A and FGM-B. Fig. 9 shows the stable crack growth in the FGM-A specimen corresponding to Fig. 8. ForFGM-A, when the applied load gets to 505 N, a fast crack growth occurs in the region from the 100% ZrO2 layer to the70% ZrO2 layer, as shown in Fig. 9a and b. Meanwhile, the applied load drops suddenly from 505 N (point A) to 193 N (pointB). As the load decreases from 193 N (point B) to 88 N (point C), the crack propagates stably through the 70% ZrO2 layer intothe 60% ZrO2 layer, as shown in Fig. 9c. When the load decreases to 40 N (point D), the stable crack growth is observed in the50% ZrO2 layer, as shown in Fig. 9d. In the test of FGM-B, however, specimen fails in a brittleness manner. When the appliedload attains the peak value (394 N), the crack initiates and propagates through the graded region in an unstable manner.

The difference in the crack growth behaviors between FGM-A and FGM-B is caused by the fracture toughness gradient(the varied fracture toughness). In the test of FGM-A, the crack propagates through a graded region from the brittle tothe ductile material. The increasing metal content can cause the fracture toughness of the crack-tip to increase with the crackgrowth, which will result in an enhanced R-curve behavior in FGMs. For the mode-I fracture in FGMs, the enhanced R-curve

e parameters calculated by XFEM for FGM-A, FGM-B and non-FGM prior to crack initiation.

KI (MPa m1/2) KII (MPa m1/2) W (�)

3.463 0.283 4.679.442 1.052 6.36

M 11.967 1.098 5.24

0.00 0.02 0.04 0.06 0.08 0.100

100

200

300

400

500

DL

oad

(N)

loading-point displacement (mm)

FGM-A FGM-B

A

B

C

Fig. 8. Load as a function of loading-point displacement in the mixed-mode fracture tests on FGM-A (a) and FGM-B (b).

Fig. 9. Stable crack propagation observed in FGM-A.

X. Jin et al. / Engineering Fracture Mechanics 76 (2009) 1800–1810 1807

behavior can lead to a stable crack growth, which has been proved by Jin and Batra [29] and Tohgo et al. [23,30] using the-oretical and experimental methods. For the present mixed-mode fracture, it is found that the enhanced R-curve behavior alsocan lead to the stable crack growth in FGMs. On the contrary, for the FGM-B specimen, the crack tends to experience adecreasing in the fracture toughness, which would generally result in the unstable crack growth.

3.3. Crack propagation path

The crack propagation paths of the FGM-A, FGM-B and non-FGM specimens are shown in Fig. 10a–c, respectively. InFig. 10a, the notch-tip of the FGM-A specimen exists in the 100% ZrO2 layer and the crack propagates through the gradedregion including the layers with 100%, 90%, 80%, 70%, 60% and 50% ZrO2. In Fig. 10b, the notch-tip of the FGM-B specimenlocates in the 60% ZrO2 layers and the crack propagates through the graded region including the layers with 60%, 70%,80%, 90% and 100% ZrO2. Fig. 10c corresponds to the crack propagation path of the non-FGM specimen. From these micro-graphs, the crack propagation paths of the FGM-A, FGM-B and non-FGM specimens exhibit obvious difference. For FGM-A(Fig. 10a), in the region from the 100% ZrO2 layer to the 70% ZrO2 layer, the crack propagates along a straight line at an angle

Fig. 10. Micrographs of the fractured specimens including: (a) FGM-A; (b) FGM-B; and (c) non-FGM. (The interfaces between the different homogeneouslayers are indicated by the dash line.)

0

1

2

3

4

5

x (m

m)

Expeiment

XFEM

(a)

Notch-tip 0

1

2

3

4

5

x (m

m)

Experiment

XFEM

(b)

Notch-tip

-1.0 -0.5 0.0

0

1

2

3

4

5x

(mm

)

y (mm)-1.0 -0.5 0.0

y (mm)-1.0 -0.5 0.0

y (mm)

Experiment

XFEM

(c)

Notch-tip

Fig. 11. Comparison between observed crack propagation paths and simulated results for FGM-A (a), FGM-B (b) and non-FGM (c).

1808 X. Jin et al. / Engineering Fracture Mechanics 76 (2009) 1800–1810

of �9� with respect to the orientation of the edge notch. But in the region of the 50% and 60% ZrO2 layers, the crack curvestowards the right edge of the specimen, which is attributed to the effects of the free-edge and the loading-point. Whereas forFGM-B (Fig. 10b), in the region from the 60% ZrO2 layer to the 90% ZrO2 layer, the crack follows a curved path and the crackkinking angle increases with the crack propagation. The crack in the 90% ZrO2 layer shows an angle of �17� with respect tothe orientation of the edge notch. When the crack propagates into the 100% ZrO2 layer, it kinks towards the right edge of thespecimen. This is caused by the effects of the free-edge and the loading-point. In addition, the crack in the non-FGM spec-imen shows a continuous growth along a straight line except for a little perturbation, as shown in Fig. 10c.

For FGM-A, FGM-B and non-FGM, the crack propagation paths observed in the tests and the ones simulated by XFEM areexhibited in Fig. 11a–c, respectively. It can be seen from Fig. 11a and b that, through the region with gradually varied mate-rial composition, the simulated results agree reasonably with the experimental ones. This indicates that the elastic gradientand the fracture toughness gradient have important influences on the crack propagation path of the FGMs.

X. Jin et al. / Engineering Fracture Mechanics 76 (2009) 1800–1810 1809

3.4. Influence of microstructure on the crack propagation path

There are some local perturbations in the crack propagation paths of the FGM-A, FGM-B and non-FGM specimens, respec-tively, as shown in Fig. 10a–c. And it can be found that the most of local perturbations exhibit in the layers with high NiCrcontent such as the layers with 70%, 60% and 50% ZrO2, respectively. The local perturbations are believed to be caused by thelocal heterogeneity of the microstructure. In the layers with high NiCr content, the dispersion of the metal particles in theceramic matrix is poor. There exist the clusters of the metal particles on the local position, as shown in Fig. 1, which canresult in the local heterogeneity of the microstructure. In the layers with 80% and 90% ZrO2, however, the microstructureis characterized by the single metal particle dispersed homogeneously in the ceramic matrix, which can not result in the lo-cal heterogeneity. Hence, the crack trajectory in the two layers is similar straight without obvious perturbation. In addition,it can be found from Fig. 11 that the local heterogeneity can cause a random perturbation in the local crack propagation pathbut has no obvious effect on the overall crack propagation path.

The interfaces between different homogeneous layers do not have obvious influence on the crack propagation path in theFGMs. The crack always propagates across the interface into the next layer without kinking along the interface (see Fig. 10aand b), which indicates that the bonding strength of the interfaces is high. Tilbrook et al. [19] have pointed out that the poorconnectivity between the different homogeneous layers can result in the crack kinking along the interface.

4. Conclusions

The aim of this experimental investigation is to study the influences of the material gradient on the fracture response ofthe ZrO2/NiCr FGMs under the quasi-static mixed-mode loading condition. Two types of FGM specimens are considered, inwhich FGM-A corresponds to the crack propagating along the direction of decreasing elastic modulus and increasing fracturetoughness, and FGM-B corresponds to the crack propagating along the direction of increasing elastic modulus and decreasingfracture toughness. Following conclusions are drawn from this investigation:

1. Prior to the crack initiation, the increasing elastic modulus ahead of the crack-tip can enhance the mode mixity of thecrack-tip, but the decreasing elastic modulus may reduce the mode mixity of the crack-tip.

2. The elastic gradient ahead of the crack-tip and the local fracture toughness at the crack-tip have dominant influences onthe crack initiation. The initial kinking angle of the crack in FGM-A is less than the one in FGM-B under the quasi-staticloading condition.

3. When the crack propagates along the direction of increasing fracture toughness, the specimen exhibits an enhanced R-curve behavior. A stable crack growth is observed in the FGM-A specimen.

4. The maximum hoop stress criterion is effective to predict the mixed-mode fracture of FGMs. According to this criterion,the crack initiation and propagation path are well predicted by DIC and XFEM, respectively.

5. Local heterogeneity caused by the clusters of the metal particles can result in slight perturbation on the local crack prop-agation paths, but has no obvious influence the overall crack propagation path of the FGM specimens. The interfacesbetween different homogenous layers do not have obvious effect on the crack propagation path due to the high interfacialstrength.

Acknowledgements

The authors are grateful for the financial support by NSFC (10432030, 10502018 and 10672049), National Science Foun-dation for Excellent Young Investigators (10325208).

References

[1] Eischen JW. Fracture of nonhomogeneous materials. Int J Fract 1987;34(3):3–22.[2] Noda N, Jin ZH. Thermal stress intensity factors for a crack in a strip of a functionally gradient material. Int J Solids Struct 1993;30:1039–56.[3] Jin ZH, Noda N. Crack-tip singular fields in nonhomogeneous materials. ASME J Appl Mech 1994;61:738–40.[4] Parameswaran V, Shukla A. Crack tip stress fields for dynamic fracture in functionally graded materials. Mech Mater 1999;31:579–96.[5] Li H, Lambros J, Cheeseman BA, Santare MH. Experimental investigation of the quasi-static fracture of functionally graded materials. Int J Solids Struct

2000;37:3715–32.[6] Rousseau CE, Tippur HV. Influence of elastic gradient profiles on dynamically loaded functionally graded materials: cracks along the gradient. Int J

Solids Struct 2001;38:7839–56.[7] Kawasaki A, Watanabe R. Thermal fracture behavior of metal/ceramic functionally graded materials. Engng Fract Mech 2002;69:1713–28.[8] Yao XF, Yeh HY, Chen XB. Visualization of crack tip behavior in functionally gradient materials using coherent gradient sensing (CGS). Model Simul

Mater Sci 2005;13:621–32.[9] Guo LC, Noda N. Modeling method for a crack problem of functionally graded materials with arbitrary properties-piecewise-exponential model. Int J

Solids Struct 2007;44:6768–90.[10] Guo LC, Noda N. Fracture mechanics analysis of functionally graded layered structures with a crack crossing the interface. Mech Mater 2008;40:81–99.[11] Konda N, Erdogan F. The mixed mode crack problem in a nonhomogeneous elastic medium. Engng Fract Mech 1994;47:533–45.[12] Gu P, Asaro RJ. Crack deflection in functionally graded materials. Int J Solids Struct 1997;34:3085–98.[13] Guo LC, Wu LZ, Ma L. The interface crack problem under a concentrated load for a functionally graded coating-substrate composite system. Compos

Struct 2004;63:397–406.

1810 X. Jin et al. / Engineering Fracture Mechanics 76 (2009) 1800–1810

[14] Kim JH, Paulino GH. Simulation of crack propagation in functionally graded materials under mixed mode and non-proportional loading. Int J MechMater Des 2004;1:63–94.

[15] Dolbow J, Gosz M. On the computation of mixed-mode stress intensity factors in functionally graded materials. Int J Solids Struct 2002;39:2557–74.[16] Rousseau CE, Tippur HV. Compositionally graded materials with cracks normal to the elastic gradient. Acta Mater 2000;48:4021–33.[17] Erdogan F, Sih GC. On the crack extension in plates under plane loading and transverse shear. ASME J Basic Engng 1963;4:519–25.[18] He MY, Hutchinson JW. Kinking of a crack out of an interface. ASME J Appl Mech 1989;56:270–8.[19] Tilbrook MT, Rozendurg K, Steffler ED, Rutgers L, Hoffman M. Crack propagation path in layered, graded composites. Compos Part B 2006;37:490–8.[20] Abanto-Bueno J, Lambros J. An experimental study of mixed mode crack initiation and growth in functionally graded materials. Exp Mech

2006;46:179–96.[21] Kirugulige MS, Tippur HV. Mixed-mode dynamic crack growth in functionally graded glass-filled epoxy. Exp Mech 2006;46:269–81.[22] Ho WF, Chiang TY, Wu SC, Hsuc HC. Mechanical properties and deformation behavior of cast binary Ti–Cr alloys. J Alloys Compd 2009;468:533–8.[23] Tohgo K, Suzuki T, Araki H. Evaluation of R-curve behavior of ceramic-metal functionally graded materials by stable crack growth. Engng Fract Mech

2005;72:2359–72.[24] Sutton MA, Wolters WJ, Peters WH, Ranson WF, McNeill SR. Determination of displacements using an improved digital image correlation method.

Image Vision Comput 1983;1:133–9.[25] Bruck HA, McNeill SR, Sutton MA, Peters WH. Digital image correlation using Newton–Raphson method of partial–differential correction. Exp Mech

1989;29:261–7.[26] Vendroux G, Knauss WG. Submicron deformation field measurements: Part 2. Improved digital image correlation. Exp Mech 1998;38:86–92.[27] Dolbow JE, Gosz M. On the computation of mixed-mode stress intensity factors in functionally graded materials. Int J Solids Struct 2002;39:2557–74.[28] Huang R, Sukumar N, Prévost JH. Modeling quasi-static crack growth with the extended finite element method Part II: numerical applications. Int J

Solids Struct 2003;40:7539–52.[29] Jin ZH, Batra RC. Some basic fracture mechanics concepts in functionally graded materials. J Mech Phys Solids 1996;44:1221–34.[30] Tohgo K, Iizuka M, Araki H, Shimamura Y. Influence of microstructure on fracture toughness distribution in ceramic–metal functionally graded

materials. Engng Fract Mech 2008;75:4529–41.