THEORETICAL & APPLIED MECHANICS LETTERS 3, 051001 (2013)

Modeling and numerical investigation of slow crack growth and crackarrest in ceramic polycrystals

Bassem El Zoghbi,1, 2, a) Rafael Estevez,1, b) and Christian Olagnon2, c)

1)Laboratoire SIMAP, UMR CNRS 5266, Grenoble-INP, UJF, Universite de Grenoble, 38402, France2)Laboratoire MATEIS, UMR CNRS 5510, INSA Lyon, Universite de Lyon, 69621, France

(Received 30 June 2013; accepted 19 July 2013; published online 10 September 2013)

Abstract Intergranular slow crack growth in zirconia polycrystal is described with a cohesive zonemodel that simulate mechanically the reaction-rupture mechanism underlying stress and environ-mentally assisted failure. A 2D polycrystal is considered with cohesive surfaces inserted along thegrain boundaries. The anisotropic elastic modulus and grain-to-grain misorientation are accountedfor together with an initial stress state related to the processing. A minimum load threshold is shownto originate from the onset of the reaction-rupture mechanism to proceed where a minimum tractionis reached locally and from the magnitude of the initial compression stresses. This work aims atproviding reliable predictions in long lasting applications of ceramics. c© 2013 The Chinese Societyof Theoretical and Applied Mechanics. [doi:10.1063/2.1305101]

Keywords zirconia, ceramic, slow crack growth, cohesive zone, intergranular failure

Zirconia has been one of the most important ceram-ics used in various domains as thermal barriers, coat-ings, and in medical applications. Their intrinsic advan-tages are wear chemical inertness and resistance. How-ever, zirconia is prone to a delayed damage mechanismthat results in slow crack growth (SCG). This mecha-nism is environmentally assisted by the presence of wa-ter at the crack tip and along the grain boundaries,which reduces the energy necessary for failure. Char-acterization of SCG is achieved by studying the varia-tion of the crack velocity under different load level. Itis show experimentally,1–3 that beyond a load thresh-old K0, SCG takes place at a velocity that increaseswith load (regime I). Regime I depends on tempera-ture (T ) and water concentration (rH2O).

2 Increasingthe water concentration induced a shift in the V –KI

curve with an increase in the crack velocity at a givenload KI and a decrease in the magnitude of K0. Thesame trend is observed when increasing the temperaturewith an additional decreasing of the threshold K0. Theslope of the regime I in the V –KI curve is not affectedby the environment but the kinetics of SCG (velocity,K0) are strongly dependent on the magnitude of themechanical load, water concentration and temperature.The present study aims at predicting the load thresh-old K0 and the regime I of the V –KI curve and at pro-viding insight of how the crack arrest originates. Themechanism underlying SCG is described with a cohesivemodel that represents the reaction-rupture mechanism.A thermally activated formulation is adopted for the co-hesive model which is presented by Romero de la Osa etal.4,5 It is shown that the formulation can capture theregime I and the load threshold that originates in the

a)Corresponding author.Email: Bassem.El-Zoghbi@simap.grenoble-inp.fr.b)Correponding author.Email: Rafael.Estevez@simap.grenoble-inp.fr.c)Corresponding author. Email: Christian.Olagnon@insa-lyon.fr.

lgV

KI

KI

0

I

0

T, rII

III

H O2

Fig. 1. Schematic description of SCG in terms of crack ve-locity V versus load level KI.

presence of initial stress related to the processing.In this section, a cohesive zone model (CZM) for

the reaction-rupture mechanism underlying SCG in sin-gle and polycrystal ceramics is presented based on thedescription of SCG by Michalske and Freiman.6 Basi-cally, the breakdown of the siloxane bond under tensionis assisted by the presence of water molecule for theformation of two silanol groups. This description hasbeen confirmed by atomistic calculations presented byZhu et al.7 who investigate the reaction-rupture in asilica nanorod. Their analysis shows that once a stressthreshold is reached locally, the reaction-rupture is ener-getically favorable, and the activation energy decreasedwith the applied stress. These observations are formu-lated with a cohesive zone methodology.4,5 A thermallyactivated cohesive model is proposed where the openingrate describing the damage kinetics is taken as

Δcn = Δ0exp

(−U0 + βσn

kBT

), (1)

where Δcn is the opening rate between two cohesive sur-

051001-2 B. El Zoghbi, R. Estevez, and C. Olagnon Theor. Appl. Mech. Lett. 3, 051001 (2013)

faces, U0 is an activation energy, β has the dimension ofan activation volume, σn is the normal traction of thecohesive surface, Δ0 is the pre-exponential term hav-ing a velocity dimension, kB denotes the Boltzmann gasconstant, and T represents the absolute temperature.

Damage is triggered when the normal traction σn

is larger than σ0n, σ

0n being a local traction threshold.

When the cumulated opening reaches a critical thick-ness Δcr

n , a crack is nucleated locally. The value of Δcrn

is supposed to be about 1 nm in ceramics. Zhu et al.7 ev-idenced a threshold stress σ0

n for the reaction-rupture tobe energetically favorable. For a traction smaller thanthe threshold traction σ0

n neither damage nor reaction-rupture takes place. If the initial traction is larger thanthe threshold σ0

n, the damage process will take place un-til the nucleation of a crack. For a traction smaller thanthe threshold stress σ0

n before the crack nucleation, thereaction-rupture stops.

In a finite element analysis, the reaction-rupture de-scription is incorporated through the traction-openingrelationship as

σn = kn(Δn − Δcn), (2)

where σn represents the normal traction increment,Δn represents the prescribed opening rate, Δc

n is thereaction-rupture opening rate, and kn is a stiffness takenvalue large enough to ensure Δn ≈ Δc

n during the pro-cess of reaction-rupture. In this formulation, contri-bution of the tangential mode to the reaction-ruptureprocess is not considered. A simple elastic response istaken as

σt = ktΔt, (3)

where σt is the tangential traction increment, Δt is theprescribed shear rate on the cohesive surface, and kt isthe tangential stiffness.

A quasi-static finite analysis is considered using atotal Lagrangian description. For this problem, the in-cremental expression of virtual work is4,5∫

V

τ · δηdV +

∫Scz

σα · δΔαdS =

∫∂V

T · δudS, (4)

where V represents the volume in the initial configu-ration, ∂V represents the boundary volume of V , andSCZ represents the cohesive surfaces. The index α corre-sponds to the normal and tangential components in thecohesive formulation. In Eq. (4), τ is the second Piola–Kirchhoff stress tensor, η is the conjugate Lagrangianstrain rate, T is the traction vector, and u is velocityof the conjugate Lagrangian strain. The rate form ofEq. (4) is used to solve the problem at each time incre-ment.

Experimental data are available3 for SCG in zirco-nia single crystals. These are used to calibrate our cohe-sive model. We consider a linear elastic isotropic bulkwith an initial crack that is subjected to mode I andconstant prescribed stress intensity factor KI (as shownin Fig. 2(a)). Cohesive zone is inserted along the crack

symmetry plane, on which the principal stress reachesits maximum value. Small scale damage with a bound-ary layer approach is adopted with the K-displacementfields prescribed along the remote boundary. A naturalcrack is considered, loaded under mode I plane strainconditions. The mesh is refined around the crack prop-agation path (as shown in Fig. 2(b)) and cohesive el-ements are 1 nm long. We prescribe a constant loadin terms of KI (stress intensity factor) and record thecrack advance with time. We extract the velocity inthe steady state regime and get one point in the V –KI

curve. By repeating this procedure, we are able to ad-just the prediction with available experimental data.3

The values of the bulk Young’s modulus Eiso = 315GPa and Poisson’s ratio νiso = 0.24 is derived from thespherical and deviatoric parts of the cubic elastic mod-uli tensor. We adopt Δcr

n = 1nm, and the activationenergy U0 = 160 kJ/mol of the order of the sublimationenergy, based on Zhurkov’s analysis.8 The parametersβ and Δ0 remain to be identified: β controls the slopeof V –KI curve and Δ0 controls its position in the V –KI

graph.4,5 These parameters are adjusted to obtain thebest fit to the experimental data.3 Their values are re-ported in Table 1.

Table 1. Parameters for the cohesive zone model for SCG ina zirconia single crystal.

U0/(kJ ·mol−1) Δcrn /nm β/nm3 Δ0/(mm · s−1)

160 1 0.027 3.2× 1011

The identified cohesive parameters are used to de-scribe intergranular fracture and SCG in a 2D polycrys-tal. We consider a granular zone made of anisotropichexagonal grains with a random orientation and cohe-sive surfaces inserted along the grain boundaries. Theproblem formulation is depicted in Fig. 3. The polycrys-tal is embedded in a homogeneous equivalent medium.An initial crack emerges in the granular zone. Along theremote boundary, the mode I K-displacement fields areprescribed. Intergranular failure is allowed. In Table2 the cubic elastic constants of zirconia grains are re-ported, and the coefficients of thermal expansion areα1 = α3 = 1.0 × 10−5 K−1, α2 = 1.1 × 10−5 K−1.The surrounding homogenous linear isotropic bulk hasa Young’s modulus E = 315 GPa and Poisson’s coeffi-cient ν = 0.24, and its isotropic coefficient of thermalexpansion is taken as α = (2α1 + α2)/3. In all cases,the polycrystal consists in a 8 × 8 grains with a graindiameter ΦG = 0.8μm. Two types of loading can beconsidered, one is thermal and the other is mechanical.First, the thermal stresses related to the grain-to-grainmisorientation and the cooling between the sintering toroom temperature are not considered. We prescribe aninstantaneous load in terms ofKI which is kept constantin time. The load is relaxed by intergranular failure.We consider a threshold stress σ0

n = 400 MPa corre-sponding to 4% of the athermal stress σc = U0/β. In

051001-3 Modeling and numerical investigation of slow crack growth Theor. Appl. Mech. Lett. 3, 051001 (2013)

(a) (b)

σ ∞

σ ∞

K u displacement fields

Cohesive zoneT =0

i

2 2

1 1

Cohesive zone Cohesive zoneT =0 T =0

I

R

i i i

Fig. 2. (a) Schematic description to model SCG in a zirconia single crystal subjected to mode I, plain strain loading. (b)General view and zoom around the crack tip of the mesh used for the finite element analysis.

(a) (b) (c)

u (K )I

E,v

Crack

αT =0

Crack

αT =0

E,v

GrainsCohesive zone

i

σ∞

σ∞

Fig. 3. Small scale damage configuration used for the analysis of SCG in a 2D polycrystal.

2

1

-1

-2

0

Y/μ

m

0 2 4

2

1

-1

-2

0

Y/μ

m

X/μm X/μm

X/μmX/μm

Y/μ

m

Y/μ

m

0 2 4

2

1

-1

-2

0

0 2 4

2

1

-1

-2

0

0 2 4

4.0

3.0

2.0

1.0

0

σ /GPayy

V/(

m•s

)

-1

(a)

Calibration

Mono X Zirconia

Experimetnal data

Poly X, Experimetnal data (Chevalier et al. )

Ф = 0.5 μmG

Ф = 0.3 μmG

Ф = 1 μmG

• Poly X, Ф = 0.8 μm, Numerical

7

K /(MPa••m )1/2I

0.8 1.3 1.8 2.3 2.8 3.3 3.8 4.3 4.8 5.3 5.8

10

10

10

10

10

10

10

10

10

10

10

10

(b)

G

−2

−3

−4

−5

−6

−7

−8

−9

−10

−11

−12

−13

Fig. 4. (a) Distribution of the stress component σyy during crack propagation in a 2D polycrystal. (b) Comparison of thepredicted V –KI curves for the single and 2D polycrystal. Experimental data are from Ref. 3.

051001-4 B. El Zoghbi, R. Estevez, and C. Olagnon Theor. Appl. Mech. Lett. 3, 051001 (2013)

Table 2. Cubic elastic constants of zirconia single crystals.9

C11/GPa C22/GPa C33/GPa

430 94 64

Fig. 4(a), we report the distributions of the stress com-ponent (σyy) for different crack advance stages. Thecrack tip is located in the region with the highest stressconcentration. We report the crack velocity for differ-ent load levels in Fig. 4(b). These are compared withthe experimental data of SCG in polycrystals for sin-tered yttrium stabilized zirconia conducted by Cheva-lier et al.3 for comparable grain sizes. We also reportthe curve V –KI corresponding to the calibration of thecohesive zone for the zirconia single crystal. We observethat the predicted V –KI curve is shifted toward largerload values for the polycrystal compared to the case of asingle crystal. The slope of the V –KI curve is compara-ble to the experimental data, but the predicted kineticsof SCG in the 2D simulation are larger than the exper-imental one. This difference is likely to originate fromthe 2D configuration in our simulation that promotesthe crack advance in comparison to the 3D configura-tion. Also, the initial thermal stress is not accountedfor.

The influence of threshold stress σ0n correspond-

ing to the initiation of the reaction-rupture mecha-nism on the SCG is now considered. Zhu et al.7 ev-idenced the existence of a stress threshold σ0

n for thereaction-rupture process to be energetically favorable.The athermal stress σc = U0/β corresponds to a criti-cal stress for the reaction-rupture to proceed. The onsetof the reaction-rupture σ0

n is likely to be a fraction ofσc, of which magnitude is now investigated. Therefore,we report in Fig. 5(a) the V –KI curve for different val-ues of σ0

n = 400, 900, 1 150 MPa corresponding to 4%,9%, 12% of the athermal stress σc. We observe thatthe slope of V –KI curve is not affected by the varia-tion of σ0

n. However, σ0n influences the magnitude of

the threshold load K0 below which a crack is arrested.For the case with σ0

n = 400 MPa, we do not observea load threshold for SCG even for values as small asKI ≈ 0.6MPa and V ≈ 10−16 m/s. For larger value ofσ0n, a crack arrest is observed for K0 = 0.9MPa ·mm1/2

and σ0n = 900 MPa up on initiation of propagation. We

observe that increasing the value of σ0n increases the

magnitude of K0. The value of K0 = 1.1MPa · mm1/2

is observed for σ0n = 1150 MPa. We report in Fig. 5(b)

the cracks paths for arrested cracks corresponding to thethreshold load K0 below which no SCG occurs. Theseresults show that with increasing the threshold stressσ0n, a threshold load of SCG can be observed. The crack

propagation is arrested when the local crystal orienta-tion induces a traction smaller than σ0

n. When consid-ering these ingredients only, we predict a threshold loadof SCG in zirconia around 1 MPa·m1/2 for σ0

n ≈ 0.1σc.The influence of the initial thermal stress state on themagnitude of the threshold loadK0 is investigated next.

10

10

10

10

10

10

10

100.6 1.1 1.6 2.1 2.6

σ = 900 MPa

σ = 400 MPa

σ = 1 150 MPa

0

0

0

21

-1-2

0

Y/μ

m

0 2 4

X/μm

21

-1

-2

0

Y/μ

m

0 2 4

X/μmK /(MPa••m )1/2I

V/(

m•s

)

-1

(a) (b)

1 3

1 3

σ = 900 MPa0

K =0.95 MPa•m 1/2

σ = 1 150 MPa0

0

n

n

K =1.1 MPa•m 1/20

n

n

n

−2

−2

−4

−6

−8

−10

−12

−14

−16

Fig. 5. (a) Influence on the threshold stress of cohesivezone σ0

n and prediction of the V –KI curve for polycrystalof zirconia with ΦG = 0.8μm. (b) Cracks paths for stoppedcracks corresponding to the threshold load K0 for σ0

n = 900,1 150MPa.

E,v

Crack

αT = 0

u = 0y

u = 0y

X

Y

(a)

200

60

-80

-220

σ /MPayy

-360

-500

2

1

-1

-2

0

Y/μ

m

X/μm

6 245 3

(b)

Fig. 6. (a) Boundary conditions for thermal load. (b) Stresscomponent (σyy) distribution in the polycrystal of zirconiawith ΦG = 0.8μm after a thermal cooling ΔT = −1 500K.

After the sintering of the polycrystal, the mate-rial is cooled down from approximately 1 500◦C to theroom temperature. This cooling induces second or-der thermal stresses that originate from the grain-to-grain misorientation and the anisotropy of the coeffi-cients of thermal expansion and elastic moduli. We con-sider a uniform cooling temperature ΔT = −1 500K.The corresponding boundary conditions are presentedin Fig. 6(a). The stress component (σyy) distributionis reported in Fig. 6(b), in which region under tractionand compression are observed. From this initial state,an additional mechanical load KI is prescribed and in-tergranular failure is allowed. For σ0

n = 400, 900, 1150MPa, we report the corresponding V –KI curves inFig. 7. For the three cases, it is observed that the pres-ence of initial thermal stresses results in a reductionof the crack velocity at a given load level. The V –KI

curves are shifted towards larger values ofKI. The slopeof the V –KI curves in the regime I is not influencedby the account for the initial thermal stresses. For thecase with σ0

n = 400 MPa (as shown in Fig. 7(a)) a largerthreshold load K0 is observed when the initial thermalstresses are accounted for. For σ0

n = 900, 1 150MPa, the

051001-5 Modeling and numerical investigation of slow crack growth Theor. Appl. Mech. Lett. 3, 051001 (2013)

102

10

10

10

10

10

10

MechanicalThermal+Mechanical

V/(

m••s

)1

σ = 400 MPa0n

K /(MPa•m )1/2I

0.6 1.1 1.6 2.1 2.6 3.1

MechanicalThermal+Mechanical

MechanicalThermal+Mechanical

10

10

10

10

10

10

10

10

V/(

m••s

)1

σ = 900 MPa0n

0.6 1.1 1.6 2.1 2.6 3.1

K /(MPa•m )1/2I

10

10

10

10

10

10

10

10

V/(

m••s

)

1

σ = 1 150 MPa0

0.6 1.1 1.6 2.1 2.6 3.1

K /(MPa•m )1/2I

n

(a) (b) (c)

104

6

8

10

12

14

16

2

4

6

8

10

12

14

16

2

4

6

8

10

12

14

16

Fig. 7. Influence of thermal initial stresses on the prediction of the V –KI curves and K0 for polycrystals of zirconia withΦG = 0.8μm.

2

1

0

Y/μ

m

X/μm

(a)

−6 −5 −4 −3 −1

−2

−1

σ = 400 MPa0n

K =0.63 MPa••m 1/2

0

X/μm

2

1

0

Y/μ

m

(b)

−6 −5 −4 −3 −1

−2

−1

σ = 900 MPa0

K =1.1 MPa•m 1/2

0

n 2

1

0

Y/μ

m

X/μm

(c)

σ = 1 150 MPa0n

K =1.42 MPa•m 1/2

0

−6 −5 −4 −3 −1

−2

−1

n

Fig. 8. Cracks paths for stopped cracks.

magnitude of the threshold load of SCG increases wheninitial stresses are accounted for. We report in Fig. 8 thecrack paths when crack propagation is arrested for thethreshold loadK0 below which the crack is stopped. Weobserved the crack arrest when the crack reaches a re-gion in which the compression is initially high enough toresult in a traction on the cohesive surfaces smaller thanσ0n. The account of thermal initial stress contributes to

a reduction of SCG kinetics. The magnitude of K0 in-creases when the initial thermal stress is accounted for.

A calibration of the cohesive zone parameters ispresented for zironica single crystal based on the co-hesive zone description proposed by Romero de la Osaet al,4,5 in the reaction-rupture mechanism for SCG inceramics. It is shown that the cohesive model can cap-ture realistically experimental data of SCG in a singlecrystal. Then in a 2D polycrystal, intergranular fail-ure is described with this calibrated cohesive model.The influence of the traction threshold σ0

n for the on-set of reaction-rupture and SCG on the prediction ofthe threshold load K0 in a polycrystal is investigated.The initial thermal stresses related to processing have a

beneficial effect on K0 with an increase in the predictedvalue.

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