Cracks in Functionally Graded Materials

Materials Science and Engineering A362 (2003) 2–16

Cracks in functionally graded materials�

H.-A. Bahra, H. Balkea, T. Fettb, I. Hofingera, G. Kirchhoffc, D. Munzb,∗,A. Neubrandd, A.S. Semenova, H.-J. Weissc, Y.Y. Yangb

a Department of Mechanical Engineering, Dresden University of Technology, Dresden 01062, Germanyb Institut für Materialforschung, Forschungszentrum Karlsruhe, Karlsruhe 76021, Germany

c Fraunhofer Institute for Material and Beam Technology, Winterbergstrasse 28, Dresden 01277, Germanyd Department of Materials Science, University of Technology Darmstadt, Darmstadt 64297, Germany

Received 18 July 2002; received in revised form 29 November 2002

Abstract

The weight function method is described to analyze the crack growth behavior in functionally graded materials and in particular materialswith a rising crack growth resistance curve. Further, failure of graded thermal barrier coatings (TBCs) under cyclic surface heating by laserirradiation is modeled on the basis of fracture mechanics. The damage of both graded and non-graded TBCs is found to develop in severaldistinct stages: vertical cracking→ delamination→ blistering → spalling. This sequence can be understood as an effect of progressiveshrinkage due to sintering and high-temperature creep during thermal cycling, which increases the energy-release rate for vertical crackswhich subsequently turn into delamination cracks. The results of finite element modeling, taking into account the TBC damage mechanisms,are compatible with experimental data. An increase of interface fracture toughness due to grading and a decrease due to ageing have beenmeasured in a four-point bending test modified by a stiffening layer. Correlation with the damage observed in cyclic heating is discussed. Itis explained in which way grading is able to reduce the damage.© 2003 Published by Elsevier B.V.

Keywords: FGM; Fracture mechanics;R-curve; Residual stresses; TBC

1. Introduction

Replacing the sharp transition of the material propertiesin a bimaterial joint by a continuous transition, i.e. introduc-tion of a graded interlayer, may improve the strength of thejoint. Various arguments have been mentioned in literature,according to which a graded material should be beneficial.These arguments are (see e.g.[1]) that: (i) thermal stressescan be reduced; (ii) thermal stresses at critical locations canbe reduced; (iii) stress jumps at the interface can be avoided;(iv) the driving force for crack extension, the stress intensityfactor, can be reduced; and (v) the strength of the interfacial

� This is an overview of parts of the key results obtained within thepriority program “Functionally Graded Materials” funded by the GermanResearch Foundation (DFG) in the years 1995–2000.

∗ Corresponding author. Present address: Institut fur Zuverlassigkeitim Maschinenbau, Universitat Karlsruhe (TH), Postfach 3640, Karlsruhe76021, Germany. Tel.:+49-7247-82-4894; fax:+49-7247-82-2347.

E-mail address: [email protected] (D. Munz).

bond, which may be characterized by the interfacial fracturetoughness, can be increased.

Not all these arguments are convincing. The reduction ofthermal stresses depends strongly on the thickness of theinterlayer and the mechanical boundary conditions. A thininterlayer between two different materials or a thin coatingcontributes only negligibly to the equilibrium conditions offorce and moment and therefore leads to a small change inthe stress distribution only. On the other hand, a continuouslinear gradation in a plate leads to zero thermal stress, iffree bending of the plate is possible. If, however, the plateis constrained against bending, a linear gradation leads toa change in the thermal stresses, but not to a general re-duction in the stresses. This is shown inFig. 1, where thedimensionless stressσ/E1α1�T is plotted versus the coor-dinatey in the direction of the gradation.E1 andα1 are theYoung’s modulus and the thermal expansion coefficient fory = 0, E2 andα2 are the corresponding values fory = H ,and�T is the change of the temperature from the stress freestate.

0921-5093/$ – see front matter © 2003 Published by Elsevier B.V.doi:10.1016/S0921-5093(03)00582-3

H.-A. Bahr et al. / Materials Science and Engineering A362 (2003) 2–16 3

0 1-1

-0.5

0

0.5

1

1.5

2

(b)

y/H

σ/E1α1∆T

0 1

0

(a)

y/H

E,αE2/E1=5, α2/α1=0.2

0 1-10

-8

-6

-4

-2

0

2

4

0 1

-1

-0.5

0

0.5

1

1.5

2(c) (d)

y/H y/H

E2/E1=0.2, α2/α1=5E2/E1=5, α2/α1=5

σ/E1α1∆Tσ/E1α1∆T

Fig. 1. Stress distribution in a plate constraint against bending after a change in temperature (linear gradation and bimaterial).

The jump of the stress in a bimaterial is not necessarilydangerous, because failure is caused by stresses and not bychanges in the stresses. If there are flaws, such as voids orinclusions or an irregular boundary at the interface, the jumpin the stress may lead to locally high stresses[2].

In a 6-year program of the German Science Foundation,different aspects of functionally graded materials were inves-tigated. One aspect was the mechanical properties of thesematerials. Results were published in different papers[3–6].

This paper concentrates on cracks in functionally gradedmaterials. There exist many publications on cracks in func-tionally graded materials under mechanical and thermalloading [7–35]. Section 2of this paper deals with the ap-plication of the weight function method to calculate stressintensity factors, with the analysis of the rising crackgrowth resistance and with residual stresses. InSection 3,the effect of grading on delamination crack propagationin TBCs under heating by means of laser irradiation isanalyzed.

2. Weight function analysis

2.1. Weight functions for the calculation of stressintensity factors

The stresses at the tip of cracks in a graded material aredescribed in the same way as in homogeneous materials witha square root singularity. For cracks parallel to the gradationdirection an external mode I loading leads to a mode I stressintensity factor, whereas for a crack perpendicular to thedirection of the gradation an external mode I loading leadsto mixed modeKI /KII loading. The stress intensity factordepends on the applied stress but also on the distribution ofthe elastic constants, especially the Young’s modulus.

A crack of length “a” parallel to the gradation direction(y-direction) is considered. The stress intensity factors canbe calculated applying the weight function method:

KI =∫ a

0σ(y)hI(y, a)dy (1a)

4 H.-A. Bahr et al. / Materials Science and Engineering A362 (2003) 2–16

KII =∫ a

0τ(y)hII (y, a)dy (1b)

whereσ(y) andτ(y) are the normal and shear stress distri-butions in the uncracked component along the crack plane.ThehI andhII are the mode I and mode II weight functions.For a graded material these weight functions depend on thedistribution of the elastic constantsE(y) andν(y).

Rice [36] developed a relation to calculate the weightfunction from the crack opening displacementu(y, a), thestress intensity factorKr(a) of a reference stress distributionσr(y):

hI(y, a) = E

KIr

∂u(y, a)

∂a· (2)

It can be shown[37] that the same relation holds for a crackin a graded material, whereE(a) at the tip of the crackhas to be introduced for the Young’s modulus. To obtainthe crack opening displacement, a procedure developed forhomogeneous materials (see[38]) can be applied. This wasshown in[37]. The “direct adjustment method”[39] startswith the general type of the weight function

h(y, a) =√

2

πa

[1√

1 − ρ+

∞∑n=1

Dn(1 − ρ)n−1/2

](3)

with ρ = y/a. The expansion is truncated after a certainnumberN. Dn can be obtained fromN conditions. One set ofconditions can be obtained by applyingEq. (1)for referenceloadingσr(y) with known stress intensity factorsKr(a). An-other condition was obtained in[37,40]. It could be shownthat the second derivative of the crack opening displacementat the crack mouth (x = 0) disappears. As a direct conse-quence ofEq. (2) it follows

∂2h

∂y2

∣∣y=0 = 0 (4)

As an example, a component with an external crack is con-sidered. Two reference loadings are considered: a constantcrack face pressureσ(y) = constant and a pair of concen-trated forcesP acting at the crack mouth, leading to the stressintensity factorsKσ andKP . With these three conditions, afour-term weight function can be obtained

h(y, a) =√

2

πa

[1√

1 − ρ+ D1(1 − ρ)1/2 + D2(1 − ρ)3/2

+D3(1 − ρ)5/2]

· (5)

The three conditions lead to

Kσ

σ=

√2a

π(2 + 2

3D1 + 25D2 + 2

7D3) (6a)

KP

P=

√2

πa(1 + D1 + D2 + D3) (6b)

3 − D1 + 3D2 + 15D3 = 0· (6c)

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

a/W=0.4

y/a

0.6

0.7

0.8

h√W

A

B

C

Fig. 2. Weight function for a fully graded plate with an external crack: (A)transition functionEq. (8a); (B) homogeneous material; and (C) transitionfunction Eq. (8b).

From these three equations, the coefficients follow as

D1 = 35

16

√2π

a

Kσ

σ− 15

16

√2πa

KP

P− 7 (7a)

D2 = −35

12

√2π

a

Kσ

σ+ 15

8

√2πa

KP

P+ 25

3(7b)

D3 = 35

48

√2π

a

Kσ

σ− 7

16

√2πa

KP

P− 7

3· (7c)

In Fig. 2, weight functions are given for a homogeneousmaterial and for two gradation functions: homogeneous ma-terial, part C is transition function inEq. (8b).

A : E(y) = E0

(1 − y

2W

)(8a)

C : E(y) = E0

(1 + y

W

)(8b)

where W is the length of the plate iny-direction. It canbe seen that for this linear change in the Young’s modulusby a factor of two, only a small deviation from the weightfunction occurs for a homogeneous material.

The crack opening displacementu(a, y) for given stressdistribution follows fromEq. (2)as

u(y, a) =∫ a

y

h(y, a′)K(a′)E(a′)

da′· (9)

IntroducingEq. (1) leads to

u(y, a) =∫ a

x

h(y, a′)E(a′)

[∫ a′

0σ(y)h(y, a′)dy

]da′· (10)


h1 h2 h3

a

y

f1

f2

h1 h1+h2y

M1 FGM M2

(a) (b)

f

Fig. 3. (a) Crack in a plate with a graded interlayer; and (b) gradedfunctions in the interlayer; M1, M2: homogeneous materials.

2.2. Stress intensity factor in a material with a gradedinterlayer under thermal loading [41]

A joint with a graded interlayer between two materials isconsidered, which is subjected to a change in temperature�T from the stress-free state (Fig. 3). The thickness of thetwo materials ish1 andh3, the thickness of the interlayerh2.It is assumed that the Young’s modulusE and the thermalexpansion coefficientα follow the same gradation function,whereas the Poisson’s ratio is assumed to be constant. Therelation

f(x) = f2 − (f2 − f1)

(h2 − y

h2

)n

(11)

is used forE andα with n = 1/2, 1 and 2. The normal-ized stressσx in the uncracked joint is shown inFig. 4 forα1/α2 = 0.5,E1/E2 = 2,h3/H = 0.05 andh1/H = 0.125(h1 + h2 + h3 = H). There is a continuous transition inthe stresses. The stresses in the top layer are higher than inthe ungraded joint. InFig. 5, the normalized stress intensityfactor is plotted versus the crack length. As can be expectedfrom the stress distribution in the uncracked joint, the stress

0 0.1 0.2-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

y/H

two layers joint

n=1/2

n=2 n=1

σx/α E ∆T1 1

Fig. 4. Normalized stress distribution in the uncracked joint.

0 0.05 0.1 0.150

0.2

0.4

0.6

0.8

1KI/α1E1∆T H

two layers joint

n=1/2

n=1n=2

a/H

interface

√

Fig. 5. Normalized stress intensity factor for joints with a graded interlayer.

intensity factor in the joint with a graded interlayer is higherfor small cracks compared with the ungraded joint. If thecracks are growing into the graded layer the stress intensityfactor decreases, whereas the stress intensity factor in theungraded joint increases as the crack approaches the inter-face. In Fig. 6, the normalized stress intensity factors arecompared for three joints: A, bimaterial withh1/H = 0.05;B, joint with a graded interlayer of thicknessh3/H = 0.05andh1/H = 0.05 (grading function withn = 1); C, bima-terial joint with h1/H = 0.1. For the joints A and C, thestress intensity factor increases as the crack approaches theinterface. When comparing the joints A and B, it can be seenthat in the top layer the stress intensity factor is loweredby introducing a graded interlayer. In the graded interlayeritself, however, the stress intensity factor is higher.

These examples show that a detailed discussion is neces-sary for an assessment of the benefit of a graded interlayer.

0 0.02 0.04 0.06 0.08 0.10

0.2

0.4

0.6

0.8

1

a/H

A

B

C

KI/α1E1∆T√H

Fig. 6. Normalized stress intensity factors for three joints.


In a bimaterial withE1/E2 > 1, the stress intensity will ex-ceed the fracture toughness, if the crack approaches the in-terface. Then, the crack may arrest at the interface becauseof the lowerK in material 2 or the crack may kink and ex-tend along the interface. In the graded material the stress in-tensity factor may always be below the fracture toughness,if a suitable grading function is selected. The gradation alsoaffects subcritical crack extension. Under thermal fatigueloading, the crack growth rate da/dN depend on the rangeof the stress intensity factor�K, which can be described bythe Paris-relationda

dN= C(�K)n (12)

The parametersC and n depend on the material and are,therefore, a function ofy. By integration ofEq. (12)froman initial to a critical crack size, the number of cycles tofailure can be obtained. A possible beneficial effect dependson the distribution ofK and the material parameters alongthe crack path.

Based on these considerations, a beneficial effect of agraded interlayer on the crack growth behavior may be pos-sible, but has to be evaluated for each specific problem.

2.3. Rising crack growth resistance curve ingradient materials

In ceramic materials the resistance against crack extensionincreases with increasing crack extension. For most materi-als this effect is caused by crack border interaction behindthe crack tip due to frictional contact of the crack bordersor unbroken islands. This crack border interaction can bedescribed by compressive stressesσbr(y) acting along thecrack border and leading to a negative stress intensity factorKbr(a). The stress intensity factor at the crack tipK0 is thesuperposition ofKbr(a) and

Kappl(a) = σ√aY

( a

W

)· (13)

For homogeneous materials, the crack extends at constantK0 and, therefore,Kappl increases to compensate the increaseof the negativeKbr. The “bridging stress” of a homogeneousmaterial is a unique function of the crack opening displace-mentu:

σbr = σbr(u)

The crack growth resistance curve—Kappl = KR as a func-tion of the crack extension�a is not an unique curve. Therelation depends on the initial crack size and the stress dis-tribution in the uncracked component.

In a gradient material, the critical stress intensity factorat the onset of crack extensionK0 and the relation betweenσbr and the crack opening displacement depend ony:

K0 = K0(y), σbr = σbr(u, y)

The crack growth resistance curve can be obtained from therelation measured between the external load or stress applied

and the crack extension by usingEq. (11)with Y(a/W) for thespecific gradient functionE(y). On the other hand, the crackgrowth resistance curve can be calculated from the knownrelationsK0(y) andσbr(y, u) by applying the relations

utotal = uappl + ubr

σtotal = σappl + σbr

andEq. (10):

utotal(y, a) =∫ a

y

h(y, a′)E(a′)

[∫ a′

0σbr(utotal, y)h(y, a

′)dy

]da′

+∫ a

y

h(y, a′)E(a′)

[∫ a′

0σappl(y)h(y, a

′)dy

]da′

(14)

Kappl= σappl√aY

( a

W, y

)= K0(y) + Kbr(y) = K0(y) + σbr

√aY

( a

W, y

)·

(15)

This equation has to be solved iteratively.Another problem is the determination of the bridging

stresses as a function of the crack opening displacement. Fora given initial crack length and applied stress, the bridgingstresses can be obtained as a function of the distance behindthe crack tipσbr(y) from measurements of the crack open-ing displacementumeasured(y, a) = utotal(y, a) at an appliedstressσappl(y). Eqs. (10) and (14)can be written in the formof

ubr(y, a) = umeasured(y, a)

−∫ a

y

h(y, a′)E(a′)

∫ a′

0σappl(y)h(y, a

′)dy da′

=∫ a

y

h(y, a′)E(a′)

∫ a′

0σbr(y)h(y, a

′)dy da′ (16)

From one measurement of the crack opening displacementfor one crack lengtha = a0 + �a and one applied load orσappl(y), the unknown functionσbr(y) can be obtained. In adiagram ofσbr versus the crack opening displacement withthe coordinatex as a parameter, this leads to one point foreach curve. The complete relationσbr(y, u) can be obtainedfrom measurements with different external loads or differentcrack extensions�a applied.

As an example results of Neubrand et al.[42] areshown for an alumina/aluminum gradient material. First,σbr–u-relation was determined from measurements of thecrack opening displacement of compact tension specimensof a homogeneous material with 27% Al. InFig. 7, the mea-sured crack opening displacementumeasured, the calculatedone for the applied loaduappl, and the difference of bothubrare plotted versus the distance behind the crack tip. Apply-ing Eq. (16)with a constant Young’s modulus, the bridgingstress as a function of the distance behind the crack tip


0 2 4 6 8 100

2

4

6

8

10

12

a-x (mm)

u

(µm)

uappl

umeas

-ubr

Fig. 7. Measured and calculated crack opening displacement (Al/Al2O3

with 27% Al).

0 2 4 6 80

20

40

60

80

100

-σbr

(MPa)

u (µm)a-x (mm)0 2 4 6 8 100

20

40

60

80

100

(MPa)

(a) (b)-σbr

Fig. 8. Bridging stress vs.: (a) distance from crack tip; and (b) crackopening displacement (Al/Al2O3 with 27% Al).

is obtained as shown inFig. 8a and theσbr–u-relation asshown inFig. 8b.

Fracture mechanics tests were performed using thecompact tension specimens with three different gradingfunctions shown inFig. 9. The stress intensity factor wascalculated from the load and the corresponding crack lengthby applying a relation given in[43]. The results are shown inFig. 10. There is a significant effect of the grading functionon the crack growth resistance curve. In addition predictionsof theKR curves were made from theσbr–u-relation. For thegradient material, it was assumed that the relation between

0 10 20 30

0

10

20

30

p=1/3

p=1

p=3

x

W

cAl

(%)

x (mm)

pWxc )1/2(27.003.0 −+=

Fig. 9. Grading functions of a CT-specimen.

Fig. 10. Measured crack growth resistance curves for three differentgradation functions.

the bridging stress and the crack opening displacement fordifferent compositions is given by

σbr(u, composition) = cAlp(u),

wherecAl is the relative content of aluminum in the mate-rial andp(u) taken fromFig. 8b. Based on the experimentalresults, the change of the stress intensity factor at the initi-ation of crack extension was assumed as to be

K0 = 2.3 + 10.7cAl[MPa

√m

]In Fig. 11, the experimentally determined and the calculatedcrack growth resistance curves are compared. In a secondcalculation the effect of the residual stresses in the gradedplates as determined by a finite element analysis (seeFig. 12)was considered. The stress intensity factor of the residualstresses was calculated according to

Kres =∫ a

0σres(y)h(y, a)dy·

Then the crack growth resistance curve is calculated as

Kappl = K0 − Kres− Kbr·Comparing the different curves inFig. 11shows that there isa fairly good agreement between the directly obtained curves(thick lines) and the curves calculated from the bridgingrelation.

2.4. Determination of residual stresses ingradient materials

Gradient materials contain residual stresses due to the pro-cessing conditions. It is possible to determine these residualstresses from measurements of the crack opening displace-ment. The procedure is as follows: notches of different lengthare cut in the direction of gradation. The crack opening dis-


0 2 4 6 8 10 120

5

10

15

KR

MPa√m

∆a (mm)

p=1/3

including σres

without σres

0 2 4 6 8 10 12 140

5

10

15 p=1

∆a (mm)

KR

MPa√m

including σres

without σres

0 2 4 6 8 10 12 140

5

10

15

KR

MPa√m

∆a (mm)

p=3

including σres

without σres

Fig. 11. Comparison of measured and calculated crack growth resistance curves.

placement is measured at the crack mouth. The crack open-ing displacement is given by

ures =∫ a

y

h(y, a′)E(a′)

[∫ a′

0h(y, a′)σres(y)dy

]da′· (17)

To solve this equation, the residual stresses are described apolynomial expression:

σres =n∑

i=0

Aiyi· (18)

InsertingEq. (18)into Eq. (17)leads to:

ures =n∑

i=0

AiCi (19)

with

Ci(y) =∫ a

y

h(y, a′)E(a′)

[∫ a′

0h(y′, a′)y′idy′

]da′· (20)

Eq. (19) is a linear system of equations, which has to besolved forn measured crack opening displacements with ndifferent notch depths.

As an example, a result obtained by Neubrand et al.[44] ispresented. In the same material as described inSection 2.3,the crack opening displacement was measured near the sur-face of the specimen for notches with different depths. Theresidual stress distribution calculated from these COD mea-surements are shown inFig. 12. These results are comparedwith finite element calculations considering cooling from600◦C.

3. Reduced delamination in graded TBCs

3.1. Principle of optimizing gradients

Thermal barrier coatings (TBCs) with superior heat re-sistance are applied for gas turbine engine blades with toincrease the energetic efficiency by running at higher oper-ating temperature. The service life of TBCs is limited byspalling fracture[45]. Application of functionally gradedTBCs with improved resistance against delamination andspalling is expected to extend service life. The strategy ofgraded TBC optimization is outlined inFig. 13. It representsan extension of the scheme proposed in[46].


0 10 20 30-150

-100

-50

0

50

100

σres

(MPa)

x (mm)

(a) p=1/3

FEM

COD

(MPa)

0 10 20 30-100

-50

0

50

100

150

σres

x (mm)

(b) bimaterial

FEM

COD

0 10 20 30-100

-50

0

50

100

150p=3(c)

x (mm)

σres

(MPa)

FEM

COD

Fig. 12. Comparison of measured and calculated residual stresses.

Optimization of graded structure

G ≥ GC Delamination crack

Energy

release rate G

Interface

toughness GC

Geometry Load

parameters

Material

parameters

Fig. 13. Principle of optimizing gradients.

The concept of mixed mode crack propagation in layeredmaterials[47] is used in combination with the conventionalcrack propagation criterion based on the strain energy releaserateG and the interface toughnessGC. With the parametersof geometry, loading conditions, and material properties be-ing given, displacements, strains, stresses, and, finally, theenergy release rateG can be calculated for the given con-figuration. The interface toughnessGC which is a given ma-terial constant in simplest fracture mechanical problems isinfluenced by various factors, such as TBC manufacture andoperating conditions, mode mixity of loading, and the tran-sition function in this complex case of a layered structure.

The delamination crack is assumed to obey the conven-tional crack propagation criterion

G ≥ GC, (21)

which means that it propagates if the energy release rateG exceeds its critical valueGC. TBCs modified by gradingin such a way thatG is always belowGC would withstandmore severe conditions without damage. The potential effectof grading on both decreasingG and increasingGC, with


the aim of a reduction of TBC damage is discussed in thepapers[46,48,49].

Two series of experiments have been carried out in con-nection with our investigations aimed at understanding thebasic mechanisms of TBC fracture and checking the use-fulness of the present approach. One series involves cyclicsurface heating by means of a laser beam. Another seriesinvolves various thermal loading regimes and a subsequentfour-point bending test modified by a stiffening layer. Itwas demonstrated previously that cyclic surface heating bylaser irradiation is a suitable test for the performance ofnon-graded and graded TBCs[46,50–52]. Such an experi-ment is suitable for studying the progress of damage and alsofor comparing the performance of non-graded and gradedTBCs. The four-point bending test modified by a stiffeninglayer has proved suitable for investigating the influence ofgrading and aging on interface fracture toughness. These ex-periments and related finite element computations serve tounderstand and quantify the progress of TBC damage witha number of cycles and explain in which way grading mayreduce or prevent this damage.

3.2. Graded TBC with increased interface toughness

Interface fracture toughness was measured on samplesprepared by DLR Köln by means of electron-beam physicalvapor deposition (EB-PVD). The non-graded samples wereflat Nimonic plates with about 100�m NiCoCrAlY bondcoat and 250�m partially stabilized ZrO2. The graded sam-ples contained a 50�m intermediate layer between the bondcoat and ZrO2, with a graded transition from Al2O3 to ZrO2[53].

Interface toughnessGC was measured in a four-pointbending test[54] modified by a stiffening layer[46,48,55,56](Fig. 14). Results of non-graded and graded TBCs are listedin Table 1for three kinds of samples differing by the degreeof aging: as received (without aging); 100 h aged at 1000◦C;and 10 h aged at 1100◦C in air. These results clearly demon-strate two effects: aging at high temperature reducesGC, asreported also in[57] for plasma sprayed coatings; and theintroduction of a graded Al2O3 oxidation barrier betweenzirconia and the bond coat causesGC to increase.

Table 1Results ofGC measurementa

GC (N/m) Non-gradedEB–PVD–TBC

GradedEB–PVD–TBC

Without aging >81b >81b

100 h at 1000◦C 63 >81b

10 h at 1100◦C 37 45

a [51].b In these cases, there was no TBC failure, but a detachment of the

stiffening layer.

Fig. 14. TBC delamination testing equipment.

3.3. Laser-induced TBC damage

Intensive cyclic surface heating by means of laserirradiation is a suitable test for the analysis of TBC dam-age evolution[46,50–52,58]. It can be effectively appliedfor comparing the performance of non-graded and gradedTBCs [51] for a wide range of temperatures and numbersof cycles. In these experiments the sample surface is locallyheated by a 1 kW Nd-YAG laser beam, as shown inFig. 15.The beam is deflected by two synchronized oscillatingmirrors in such a way that a circular area on the samplesurface is heated. The surface temperature is measured bya high-speed pyrometer combined with another oscillatingmirror. The start of the crack can be detected by an acous-tic sensor. A typical cycle consists of two phases: activeheating for 1 and 10 s interval (seeFig. 16). The number

Fig. 15. Set-up of the laser heating experiment.


0,2 0,4 0,6 0,8 1,0 1,2

0

500

1000

1500

2000

Increase oflaser power

Overheating

Tem

pera

ture

[

˚C]

Time [s]

Fig. 16. Thermal cycling: part of an 11 s cycle, heating phase 1 s, andcooling phase 10 s.

of cycles per sample varies between several tens to severalthousands. The maximum surface temperature employed isabout 1600◦C.

TBC damage was investigated by microscopy and metal-lographic cuts. TBC failure was found to proceed in clearlydistinguishable consecutive stages, as shown inFig. 17.Note thatN andTmax have was varied. There is no visibledamage after 11 cycles for all TBCs. Short vertical cracksare observed after 30 cycles withTmax = 1500◦C in thenon-graded TBC (Fig. 17a) and after 300 cycles withTmax =

Fig. 17. Progress of damage in non-graded and graded TBCs[51] N and Tmax denote the number of cycles and maximum surface temperature. Fornon-graded EB-PVD TBC (homogeneous ZrO2): (a) N = 30, Tmax = 1500◦C; (b) N = 1400, Tmax = 1500◦C; and (c)N = 900, Tmax = 1600◦C.For graded EB-PVD TBC (graded transition: Al2O3 → ZrO2): (d) N = 300, Tmax = 1530◦C; (e) N = 1000, Tmax = 1530◦C; and (f) N = 300,Tmax = 1600◦C.

1530◦C in the graded TBC (seeFig. 17d). Long verticalcracks and delamination arise after 1400 cycles at 1500◦Cin the non-graded TBC (seeFig. 17b) and after 300 cycles at1600◦C in the graded TBC (seeFig. 17f). Under the aboveloading conditions, local spalling is observed for non-gradedTBC only.

It can be concluded that TBCs respond to thermal cyclingof increasing severity with the following sequence of phe-nomena:

vertical cracks→ delamination→ blistering→ spalling.

The onset of the stages of damage is delayed by grading.This experimental evidence can serve as a guidance for thefollowing analytical and numerical fracture mechanics anal-ysis aimed at optimizing the effect of grading.

Since, in the present experiments, damage depends on twoparameters,N andTmax only, it can be conveniently repre-sented by damage maps as inFig. 18. Apparently, a gradedinterlayer delays delamination and subsequent further dam-age. Blistering and spalling have been identified visually.Delamination was detected by pyrometry. It is reflected by asudden rise of the surface temperature. The kink in the uppercurve ofFig. 16 is related to a sudden increase of thermalflow resistance due to delamination.

The following fracture mechanics modeling intends to re-produce an essential feature of the damage map, namely, theslope of the line indicating the critical damage threshold, bytaking into account the variability of bothG and GC withN. The same subject was studies in[59] for atmospheric


10 100 1000 10000

1000

1200

1400

1600

no visible damage

vertical cracks

delamination cracks

local blistering

local spalling

Maxim

um

surf

ace t

em

pera

ture

Tm

ax [

oC

]

Number of cycles N(a)

10 100 1000 10000

1000

1200

1400

1600

no visible damage

vertical cracks

delamination cracks

local blistering

Maxim

um

surf

ace t

em

pera

ture

Tm

ax [

oC

]

Number of cycles N(b)

Fig. 18. Damage maps of cyclically heated: (a) non-graded; and (b) gradedTBCs [51].

plasma-sprayed ZrO2 coatings, where time-dependent safetymaps served as an appropriate representation. The TBC fail-ure process was discussed in this sense in[60], where anincrease of interface stress and degrading of bond strengthwith time and number of cycles was analyzed in connectionwith the expected life time before the onset of spalling.

The higher damage resistance of graded TBC as com-pared to non-graded TBC (compare curves inFig. 18bandFig. 18a) is apparently due to the higherGC of graded TBCafter aging under heating (seeTable 1).

3.4. Shrinkage stresses as a driving force

During laser heating (Fig. 16) the TBC-layer is sub-jected to high compressive thermal stress leading tohigh-temperature creep deformation in addition to sintering.Both mechanisms cause a high tensile stress building upwhile cooling down to room temperature after cyclic heating.A similar mechanism was investigated in[61] for metallic(Ti, Al)N coatings under laser shock loading. Considerable

Fig. 19. Shrinkage cracks in thermally cycled TBC.

residual crack opening after cyclic heating (Fig. 19) indi-cates the presence of an average shrinkage strain of about1%. Sintering of TBCs was discussed in[45,51]. Prolongedheating of TBC samples detached from the substrate bychemical dissolution of the NiCoCrAlY bond coat resultedin shrinkage strains ofεS ≈ 1.4% for 100 h at 1200◦C andεS ≈ 0.6% for 250 h at 1050◦C (Fig. 20, curvature dueto inhomogeneity of TBC). Progressive shrinkage due tosintering in thermal cycling increases the energy releaserate for vertical cracks which subsequently turn into delam-ination cracks. (Note that the temperature needed for TBCdamage inFig. 17 is much higher because of the smallertotal duration of laser heating withN cycles.)

3.5. Model for the analysis of delaminationcrack propagation

A simplified two-dimensional (plain strain) model is con-sidered for the analysis of delamination cracks under ther-mal shock (Fig. 21a). Its symmetry allows the calculationto be restricted to a representative element with one crackonly (Fig. 21b). Elastic isotropy is assumed, withE1, ν1,

Fig. 20. Detached TBC: before (left) and after (right) deformation due tosintering for 100 h at 1200◦C.


Substrate

y

x

TBC

A

A’

(a) (b) (c)

B’

B

h

l h=

H h = 10

a

E , 11

E ,2 2

S

h

pl h=

H h = 10

aE , 11

E ,2 2

Fig. 21. Model for the analysis of delamination crack propagation due to sintering.

E2, andν2 being the elastic constants for the TBC and thesubstrate, respectively. The sintering effect is characterizedby two parameters: average shrinkage strainεS and depthδ(see alsoFig. 21b).

3.6. FE modeling

Delamination crack propagation was computed by meansof the finite element method (FEM). The finite elementmodel corresponding to the problem depicted inFig. 20bisshown inFig. 22. The MARC finite element program pack-age[62] was used for the calculation involving a 2D meshof eight-noded isoparametric finite elements with four inte-gration points for stiffness matrix computation. The numberof elements varied with relative thicknessh/H and dimen-sionless crack lengtha/l and reached about 1500. A highlyrefined self-similarly focused mesh was used in the vicinityof the crack tip. The size of the elements around the crack

Fig. 22. Delamination crack propagation caused by sintering (εs = 0.4%, δ/h = 0.8, a/l = 0.75). Displacements 20 times magnified.

tip was abouth/680. The depth of substrateH was taken as10 h, which was sufficiently large to approximate half space.The representative element (Fig. 21b) was fixed horizontallyon the right. Deviation ofG due to this simplification didnot exceed 8% compared to the case of periodic boundaryconditions.

The energy release rateG is calculated using theJ-integral via differential stiffness technique[62]. Accu-racy of the model and methods considered is checked bycomparison with the analytical result for homogeneous ma-terial within the limits of largel/h and δ/h = 1: GSS =(1− ν2

1)p2h/(2E1) = (1+ ν1)/(1− ν1)E1ε

2Sh/2 (Eq. (22)).

In this model, the effect of sintering is simulated by as-suming a homogeneous initial strainεS in an upper stratumof depthδ of the TBC (seeFig. 21b). As another check ofthe numerical computations, the equivalent problem of ex-ternal pressure:p = E1εS/(1− ν1) applied to the boundary(seeFig. 21c) is calculated as well. The coincidence of the


results turns out to be better than<0.5%, thus proving theconsistency of the calculations.

Delamination is analyzed on the basis of elastic finite el-ement solutions for variable crack lengths and loading con-ditions. The crack configurations fora/l = 0.75 are shownin Fig. 22.

The deformations seen inFig. 22 (20 times magnified)show that the upper layer experiences both opening and slid-ing relative to the substrate, thus indicating “mixed mode”conditions. Variation ofG with crack propagation is given inFig. 23 for various values ofδ/h. The prominent feature ofthe G(a) curves inFig. 23 is the maximum, which is about20–30% of the steady-state valueGSSof an equivalent spec-imen with infinite widthl, which is defined byEq. (22)asfollows:

GSS = 1 + ν1

1 − ν1

E1ε2Sh

2

(δ

h

)2[

1 + 3

(1 − δ

h

)2]

· (22)

The curves inFig. 23start from values ofG|a/l→0 for de-lamination cracks, which are defined by the energy releaserate for a vertical crack. (In case of a homogeneous material,the energy release rate of the infinitesimal delamination kinkis 1/4 of that of the vertical crack). The value ofG|a/l→0may exceedGSS at δ/h = 1, and it depends strongly onthe relative depthδ/h of the sintering zone. With further de-lamination crack propagation,G approaches the steady statevalue GSS, but does not reach it, due to relatively smallvalue of l. For a → l the energy release rateG approacheszero (convergent debonding by[63]). In view of these ten-dencies, the existence of a maximum in theG(a) curves, asshown inFig. 23, is obviously necessary. In contrast toδ/h,the loading parameterεS has no influence onG(a)/GSS.

In reality, the geometry parameterl is more or less ran-dom. The experiments show that the average half distancebetween vertical cracks is about equal to the TBC thicknessh. The valuel = 1.5 h was assumed in[51]. G depends onl/h as shown inFig. 24. (Note also the convergence of thecurves fora → l.) The casel = h is considered here in de-

0,0 0,2 0,4 0,6 0,8 1,0

0,0

0,1

0,2

0,3

G /

GSS

δ/h=0.7

δ/h=0.6

δ/h=0.5

δ/h=0.4

δ/h=0.3

Delamination crack length ratio a/l

Fig. 23. Energy release rate vs. crack length forl = h. Analogous resultsfor l = 1.5 h have been calculated in[51].

0,0 0,2 0,4 0,6 0,8 1,0

0,0

0,1

0,2

0,3

0,4

0,5

l/h =1.0

l/h=1.5

G /

GSS

δ/h=0.2


Fig. 24. Influence of crack net density parameter (l/h) on energy releaserate curves alteration with crack propagation. The mutual unloading ofvertical cracks leads to decreasing ofG for lower l/h.

tail. Natural randomness ofl creates favorable conditions forspalling under certain combinations of loading parametersduring cycling.

The peculiarities of theG behavior as revealed by thenumerical simulations are the basis for the following analysisof TBC failure.

3.7. Delamination conditions

During cyclic laser beam heating, the shrinkage stress in-creases, and so does the energy release rateG. The numer-ical results ofFig. 25 were obtained under the assumptionof a stress-free shrinkage strainεS = 0.4% andδ/h = 0.8for 100 h at 1000◦C (GSS = 209 N/m), as suggested by theshrinkage evident inFigs. 19 and 20. (This thermal loadingis the same as inTable 1, second line.) The computationswere performed using the following Young’s moduli:E1 =85 GPa,E2 = 215 GPa,ν1 = 0.177,ν2 = 0.3 [64].

After 100 h at 1000◦C, G stays belowGC in the case of agraded TBC (Fig. 25b) andG exceedsGC in the non-gradedTBC so that the delamination crack is driven toa/l ≈ 0.1in Fig. 25a. At higher temperatureTmax and number of cy-clesN, the value ofG rises as a result of increasing shrink-age strainεS and depthδ (seeFig. 26). Furthermore,GCis assumed to be reduced from 63 to 37 N/m (cf.Table 1,1100◦C). If G exceeds the interface toughnessGC, unsta-ble delamination sets in and proceeds to the position indi-cated by the solid arrow inFig. 26at least. Under continu-ing cyclic heating,G increases further and the delaminationcrack is pushed along in stable propagation as indicated bythe dashed arrow. In this way, a delamination crack length ofa/l ≈ 0.65 is reached for the assumed values ofεS = 0.5%andδ/h = 0.9. (Note that the mode mixity angleψ∗ for thedelamination crack inFig. 23 was calculated in[51]. ThedependenceGC(ψ∗) was discussed briefly in[51]).

The described model can explain the delamination crackpropagation with increasingεS and δ due to progressive


0,0 0,2 0,4 0,6 0,8 1,0

0

50

100

150

εS=0.4%

δ/h=0.8

En

erg

y r

ele

ase

rate

G

,

[N/m

]

Non-graded coating

GC=63N/m

Delamination crack length ratio a/l(a)

0,0 0,2 0,4 0,6 0,8 1,0

0

50

100

150

εS=0.4%

δ/h=0.8

En

erg

y r

ele

ase

rate

G

,

[N/m

]

Graded coating

GC>81 N/m

Delamination crack length ratio a/l(b)

Fig. 25. Effect of grading on delamination crack propagation: (a)non-graded TBC (100 h, 1000◦C), l = h; and (b) graded TBC (100 h,1000◦C), l = h.

0,0 0,2 0,4 0,6 0,8 1,0

0

20

40

60

80

100

Stable

delamination

Unstable

delamination

εS=0.3%

δ/h=0.7

εS=0.4%

δ/h=0.8

εS=0.5%

δ/h=0.9

Energ

y r

ele

ase

rate

G

, [N

/m]

GC


Fig. 26. Stable and unstable delamination with increasingεs and δ dueto progressive sintering.

sintering. However, it is not quite obvious how spalling isbrought about in the case of vertical cracks which do notclose under heating. The energy release rate decreases withthe approach of two adjacent delamination cracks so thatdelamination stops (Fig. 26). As a clue to the possible mech-anism, spalling is observed during the heating stage of thecycle only (compare overheating inFig. 16). These obser-vations suggest that bending of the unilaterally attached flapdue to transient heating may be the cause of final detach-ment. Based on this idea, a new model of spalling fractureis developed in[65].

4. Conclusions

The fracture mechanics analysis of cracks in functionallygraded materials leads to the following results.

A graded interlayer does not necessarily lead to a re-duction of the stress intensity factor. The weight functionmethod can be applied to analyze the rising crack growthresistance in graded materials and calculate residual stressesafter cooling from the process temperature.

A fracture mechanics analysis of the performance ofgraded and non-graded TBCs was carried out taking intoaccount the multistage character of the TBC failure process.TBC damage resistance was tested by means of cyclic sur-face heating using laser irradiation. In this way, the influenceof aging and grading on the performance of TBCs couldbe quantified. The observed sequence of TBC damage in-volved several stages: vertical cracking→ delamination→blistering→ spalling.

The fracture mechanics models proposed are aimed atexplaining the essential feature of the damage map inFig. 18, which is the downward slope of the critical dam-age curve. This slope is brought about by both the increaseof shrinkage stress and the decrease of interface fracturetoughness. Accordingly, the principle of graded TBC op-timization is reduced to keepingG below GC as well aspossible.

An increase of interface fracture toughnessGC due tograding and a decrease due to aging was measured in afour-point bending test modified by a stiffening layer. Theresults of finite element modeling of the laser-induced TBCdelamination process are compatible with the experimentaldata. Correlation with the damage observed in cyclic heatingis discussed. It is explained in which way grading may occurto reduce the damage.

Acknowledgements

The authors thank the Deutsche Forschungsgemein-schaft (DFG) for support within the priority program“Gradientenwerkstoffe” and the Deutsches Zentrum fürLuft- und Raumfahrt (DLR) Köln for supplying samplematerial.


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Cracks in Functionally Graded Materials