# Static and Cyclic Fatigue of Zirconia Materials
## Measured by Double Torsion

Jérôme Chevalier, Christian Olagnon, Gilbert Fantozzi

► To cite this version:

Jérôme Chevalier, Christian Olagnon, Gilbert Fantozzi. Static and Cyclic Fatigue of Zirconia Materials Measured by Double Torsion. R. C. Bradt; D. P. H. Hasselman; D. Munz; M. Sakai; V. Ya. Shevchenko. Fracture Mechanics of Ceramics. Fatigue, Composites, and High-Temperature Behavior, 12, Springer, pp.45-60, 1996, Fracture Mechanics of Ceramics, 978-1-4613-7683-5. 10.1007/978-1-4615-5853-8_4. hal-01677828

HAL Id: hal-01677828

https://hal.science/hal-01677828

Submitted on 5 Apr 2023

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
STATIC AND CYCLIC FATIGUE OF ZIRCONIA MATERIALS MEASURED BY
DOUBLE TORSION

J. Chevalier, C. Olagnon, and G. Fantozzi

INSA-G MPPM, RA 341

F6962 1 Villeurbanne Cedex France

ABSTRACT

A study of the static and cyclic fatigue behaviour of 3Y-TZP has been conducted with the Double Torsion (DT) method. In order to obtain precise results, we have first focused on the DT specimen itself. An evolution of the stress intensity factor with crack length has been observed, and taken into account in a modified K₁ expression. A strong crack accelerating effect has been observed in cyclic fatigue compared to static fatigue. Power laws were shown to describe well data for both static and cyclic measurements with an exponent lower in cyclic loading. Thresholds under which no propagation occured were measured. They were lower under cyclic solicitation, and depended on loading conditions.

# I. INTRODUCTION

The use of Yttria-Stabilized Zirconia Polycrystal (Y-TZP) ceramics for hip prosthesis applications is growing because of their unique advantages at room temperature¹. Y-TZP presents a high toughness associated with an high strength, as consequences of transformation toughening. Moreover, its fine grain microstructure leads to very good wear properties. For hip prosthesis applications, the evaluation of the lifetime is of major interest and must integrate crack propagation laws under service conditions. Hip protheses are subjected to both static and cyclic solicitations. Therefore, subcritical crack growth laws under static and cyclic loadings are required.

Subcritical crack growth (SCG) under static loads has been extensively studied in the past decades. Theoretical and experimental investigations have shown that SCG was due to stress corrosion at the crack tip². Crack extension rate (V) is controlled by the reaction between the ceramic and the corrosive molecules and by the applied stress intensity factor K₁. It has been observed for a wide range of ceramic materials that three propagation stages were present at ambient temperature³. The first stage is mainly governed by the reaction rate at the crack tip, while the second is ruled by the diffusion of the corrosive molecules to
the crack tip. The third stage stage takes place near the toughness KIC and leads to fast fracture. Figure (1) shows a typical V-KI diagram in a log-log scale, where the three stages are present. In the first propagation stage, i.e for crack rates less than 10⁻⁴ m/s², V can be related to KI by the expression:

$$V = A * K_I^n \quad (1)$$

where A and n are material properties.

KIO is a threshold value under which no propagation occurs. It has sometimes been observed in ceramic materials⁴ and is of prime importance in practice because it defines unlimited lifetime for ceramic components.

In the past decades, the traditional view of ceramics as brittle materials prohibited the admittance of specific cyclic mechanisms (as observed in metals) operative under cyclic loading conditions. The refuted existence of mechanical fatigue was based on the absence of plasticity. It was shown for glass and porcelain that SCG under cyclic loading could be explained by the same environmental and corrosive processes responsible for SCG under steady loads⁵. However, recently, following the studies of Dauskardt et al. for Mg-PSZ⁶, several investigations have shown reduced lifetimes for both transforming¹¹,¹² and nontransforming ceramics¹³⁻¹⁶ under cyclic loading conditions.

<figure><img src="image_2.png" /><figcaption>Figure 1: Typical V-KI diagram in ceramics (log-log scale).</figcaption></figure>

In nontransforming ceramics such as Al₂O₃ and Si₃N₄, crack-bridging occurs in static loading conditions and imparts for a rising toughness (R-curve)¹⁷,¹⁸. The R-curve results from the accumulation of shielding by crack bridging ligaments with crack extension. A decrease of crack-tip shielding has been observed in cyclic fatigue from in situ observations of the crack-bridging ligaments degradation, due to repeated abrasion¹⁹. Gilbert et al.²⁰ and more recently Jacobs et al.²¹ have modelled cyclic crack propagation in this way, and a good agreement with experimental results was obtained. In transformation-toughened zirconia ceramics, crack-tip plasticity occurs due to the martensitic transformation from tetragonal to monoclinic symmetry at the crack front²²,²³. A compressive transformed zone surrounds the crack and results in crack-tip shielding, thus in a rising toughness. It is to note that crack bridging can superimpose to transformation toughening. Recently, Tsai et al.²⁴ have investigated the origin of cyclic fatigue in a Ce-TZP/Al₂O₃ ceramic. From their results, it appears that cyclic fatigue effects are due to the transformation zone shielding degradation. Transformation zones formed during cyclic loading showed lower compressive residual stresses at the crack tip than the zones formed in static loading. Observations of compressive transformed zones revealed a different morphology in cyclic loading as compared to sustained loading conditions. As for nontransforming ceramics, cyclic fatigue
effects are therefore linked to the degradation of crack-tip shielding. In our mind, cyclic fatigue can therefore be expressed in terms of stress corrosion or/and crack-tip shielding degradation.

SCG under sustained loading in 3Y-TZP has been studied by Knechtel et al.<sup>25</sup> and by Chevalier et al.<sup>26</sup> and was shown to be due to stress corrosion. Knechtel et al.<sup>25</sup> suggested the presence of a threshold stress intensity factor $K_{I0}$, but the few results presented are to be confirmed. Independently, cyclic fatigue of the same material has been investigated by Liu et al.<sup>7</sup>. The authors proposed a propagation law in cyclic fatigue for short cracks, strongly dependent on the maximum stress intensity factor of the cycles. Cyclic fatigue effects were attributed to microcraking around the crack front, but no microstructural observations addressed this issue. No threshold $K_{I0}$ under cyclic loads was noted, because of the relatively high crack rates investigated ($V>10^{-9}$ m/s). The purpose of the present investigation was to determine precise SCG laws in both static and cyclic fatigue, and to compare them in terms of shielding degradation. Threshold intensity factors were investigated from the characterisation of low crack rates ($<10^{-10}$ m/s). Macroscopic flaws were characterized with the Double Torsion (DT) method, and results were compared with those reported by Liu et al.<sup>7</sup> for microscopic cracks (Vickers-induced cracks).

## II. EXPERIMENTAL PROCEDURE

### II.1. Material

The material studied in this work was a 3%mol Yttria-stabilized Zirconia (3Y-TZP) sintered at 1450°C (PROZIR, Ceramiques Techniques Desmarquest). A density of 6.06 g/cm³ was measured by Archimed method and the average grain size was 0.58 µm. 3Y-TZP exhibits a steeply rising R-Curve over few microns (~ 10 µm), and a subsequent plateau toughness<sup>7</sup>. Therefore, DT study using specimens containing large cracks in the present investigation was not perturbed by a rising toughness.

### II.2. Specimen geometry

The geometry of the DT specimen is shown in figure (2). It consists of a plate of dimensions 20*40*2 mm³.

<figure><img src="image_3.png" /><figcaption>Figure 2: Double Torsion specimen and loading configuration.</figcaption></figure>
The tensile surface was polished down to 1 µm to observe the cracks with a precision of ± 2µm with an optical microscope. The notch of dimensions : length $a_0=10$ mm and root $ρ=0.1$mm was machined with a diamond saw. Subsequent precracking induced a sharp 'natural' crack of length of ~13mm in each test specimen. A good alignment in the jig, associated with a good control of the specimen parallelism, allowed to obtain a straight propagation through the specimen.

Most of the studies conducted by Double Torsion published in the literature were carried out on specimens with a groove to guide the crack. However, it has been shown that such a groove has a strong influence on the results³, leading to significant errors on crack velocities. Therefore, no groove was machined on the tensile side in order to avoid any residual stress intensity factor.

### II.3. Testing methods

Tests were conducted using an universal electromechanic testing machine (Model 8562, Instron) in ambient air, water, Ringer solution (characteristic of the human body) or Silicon oil. Cyclic fatigue frequency was limited to 1 Hz by the testing machine.

For the determination of the evolution of $K_I$ with crack length and for the study of SCG under static loading, experimental work consisted of constant loading and relaxation tests. In constant loading tests, specimens were subjected to static loads (P) for varying durations. Crack speed in each case was calculated as a ratio of crack increment to the loading time. The load (P) was related to the stress intensity factor. One test therefore allowed us to obtain a "crack growth rate-applied stress intensity factor" couple. SCG laws were recorded in the graphical form of well-known (V-K₁) relation. In the relaxation tests the pre-cracked specimens were subjected to fast loading, followed by subsequent stopping of the crosshead at a given load. The crack propagation induced a load relaxation. The load versus time curve allowed the determination of the V-K₁ curve²⁸. Cyclic fatigue tests consisted in submitting the DT specimens to sinusoidal loads at constant $P_{max}$ and $ΔP$. Data were presented in terms of V-K<sub>max</sub> or V-ΔK (where K<sub>max</sub> and ΔK are respectively the maximum and the range of the applied stress intensity factor during a cycle).

## III. STUDY OF THE DOUBLE TORSION SPECIMEN

### III.1. Introduction

The Double Torsion method was chosen because of its simple geometry and the possibility of conducting extensive stable crack propagation in a wide range of velocities under constant loading or in the stress-relaxation mode²⁷,²⁹. Moreover, studies based on a compliance analysis show that the stress intensity factor is independent of the crack length²⁹. Therefore, the DT test theoretically offers a great advantage in comparison with other methods. However, the effective constancy of $K_I$ over the specimen length has often been questioned³⁰. From a finite element stress analysis, Trantina³¹ observed a variation of $K_I$ with crack length strongly dependent on the crack profile. Chevalier et al.³² have shown the important role of contact stresses generated on the uncracked compressive side of the specimen. Therefore, in this study, we have first studied the evolution of $K_I$ with crack length, in order to further obtain accurate crack propagation laws. A corrective formula, including the variation of crack length is proposed to obtain SCG parameters during static and cyclic fatigue.
The well-known analytical expression of the stress intensity factor for the DT specimen is based on a compliance analysis²⁹.

A number of assumptions is required for the K₁ determination:

The first is that compliance C, given as the ratio of load-point displacement to the load, varies linearly with crack length. The crack front is assumed to be straight through the thickness of the specimen, and invariant with crack length. Finally, failure is assumed to occur in mode I. Under these restrictive assumptions, K₁ can be calculated with the following expression:

$$K_I = H * P \quad (2)$$

$$\text{with} \quad H = \frac{W_m}{T^2} * \left[ \frac{3 * (1 + \nu)}{\psi(T/W) * W} \right]^{1/2} \quad (3)$$

where W_m, T, W are geometrical parameters defined in figure (2) and ν is the Poisson ratio, ψ(T/W) is a geometrical function²⁸.

We must note that the analytical stress intensity factor is therefore independent of the crack length. However the numerous assumptions for establishing this formula have to be checked. The first work has consisted in verifying the linear dependance of the compliance (C) with crack length (a) in the range of crack lengths investigated (10-30 mm). In this range of crack lengths, C(a) was defined as a straight line, therefore, possible changes in K₁ with crack length could therefore not be attributed to a change in compliance formulation.

### III.2. Constant loading tests

Constant loading tests were performed in air, in order to investigate the relationship between crack length and crack velocity during subcritical crack extension in the linear region of compliance. For this purpose, an unique specimen was loaded alternatively by three loads (100 N, 105 N and 110 N) and crack velocity V was recorded as a function of crack length for each load. The results are presented in a log (V) versus log (a/a₀) plot in figure (3). It appears that crack velocity increases with crack length for the three applied stresses, traducing an increase in K₁. Indeed, slow crack velocity under static loading in the investigated range (10⁻⁹ - 10⁻⁵ m/s) is directly related to K₁ in the first stage of SCG (cf. equation 1). Therefore, expression of K₁ with equation (2) seems to be inappropriate.

<figure><img src="image_4.png" /><figcaption>Figure 3: Crack kinetic versus ratio of crack length to notch length for three applied loads.</figcaption></figure>

■: P=110 N

● and ○: P=105 N (●: a₀=10mm, ○: a₀=21mm)

◆: P=100 N
On a log-log plot, the experimental results define three parallel straight lines.

Therefore, in the range of crack velocities investigated and for a given applied load, the velocity V appears to be linked to the crack length by:

$$V = F(P) * (a / a_0)^m \quad (4)$$

Where $a_0$ is the notch length, F(p) depends on the applied load and m appears to be independent of P.

We therefore defined a corrected velocity Vcorr only function of the applied load P, independent of the crack length. Vcorr is related to V by:

$$V_{corr} = V * (a_0 / a)^m \quad (5)$$

### III.3: Multi relaxation tests

In order to investigate the relationship between P and Vcorr, multi relaxation tests were performed in ambient air on a single specimen for crack velocities in the range of 10⁻⁷ - 10⁻⁵ m/s (again in the first stage of SCG).

The results are presented in the form of log (V) or log (Vcorr) as a function of log (P) in figure (4). The effect of crack length is obvious from the shift of log (V) vs log (P) curves, although all of them show similar slopes. The increase of initial crack length in the relaxation test results in a shift to lower values of P, due to the influence of crack extension on KI. Interestingly, the plot of log (Vcorr) vs log (P) do not present this shift, as the effect of crack length is hereby taken into account. Note that the slope of log (Vcorr) vs log (P) is different from those of the log (V) vs log (P) curves.

Vcorr can therefore be expressed as a function of P by:

$$V_{corr} = L * (P)^k \quad (6)$$

<figure><img src="image_5.png" /><figcaption>Figure 4: V (full symbols) and Vcorr (empty symbols) versus P for three successive relaxations on a sample.</figcaption></figure>

■ and □: Final crack length af=15.9mm, ◆ and ◊: Final crack length af=20.0mm,

● and ○: Final crack length af=27.2mm.
### III.4. A modified expression of $K_I$ for the Double Torsion test

Combination of equations (5) and (6) gives:

$$V = L * P^k * \left( \frac{a}{a_o} \right)^m \quad (7)$$

Under the assumptions that (i) both the relaxation tests and the constant loading tests were performed in the first stage of the subcritical crack propagation law (crack velocity typically on the $10^{-9}-10^{-4}$ m/s range), that is to say for velocities given by the power law relationship:

$$V = A * K_I^n \quad (1)$$

and (ii) the stress intensity factor is related to P and a by an empirical relationship of the form:

$$K_I = H * P^x \left( \frac{a}{a_o} \right)^y \quad (8)$$

Replacing the value of $K_I$ from equation (8) in (1) and comparing (8) with (7), we have:

$$n=k \quad \text{and} \quad x=m/n$$

Thus equation (8) can be rewritten in the form:

$$K_I = H * P^x \left( \frac{a}{a_o} \right)^{m/k} \quad (8.b)$$

Hence $K_I$ can be conveniently calculated from constant loading and multirelaxation tests. The values of m and k can be easily obtained using figure (3) and (4), resulting in k=32 and m=6. Therefore, the stress intensity factor is correctly given by:

$$K_I = H * P^x \left( \frac{a}{a_o} \right)^{6/32} \quad (8.c)$$

This expression of $K_I$ will be preferred to the expression (2) for the static and the cyclic fatigue analysis in the following sections.

In a more complete study about the Double Torsion method, Chevalier et al.³² have studied crack geometries in the DT specimen. A curve crack front (already mentioned by a number of authors) was observed, and the compressive side of the specimen remained often uncracked.

Therefore, the authors showed that the increase in $K_I$ was mainly due to the curved crack front, resulting in an uncracked region with increasing residual stresses on the compressive side of the specimen. The classical expression of $K_I$ with equation (2) is therefore not accurate for precise SCG parameters measurements, because these stresses are not taken into account in the standard compliance analysis.
# IV. CRACK PROPAGATION RATES UNDER STATIC LOADING

## IV.1. Static fatigue in air

Tests performed in the preceding section were analysed in terms of $(V-K_I)$ for slow crack growth behaviour of the material in air. Additional stress relaxation and constant loading tests were performed at higher loads in order to determine the complete $V-K_I$ curve of the material (stages II and III of the SCG law), and the presence of a threshold $K_{I0}$ was investigated with the constant load technique at low stress levels. The results of both relaxation and constant loading tests are presented in figure (5). The overall shape of the curve is that of a standart 3-stage graph. The first stage parameters $A$ and $n$ were calculated, and we obtained :

$$A=1,8 \cdot 10^{-26} \text{ and } n=31,5.$$

A well defined threshold $K_{I0}$ is present, under which crack propagation is presumed dormant. The presence of this threshold value at 3,5 MPa√m, already suggested by Knechtel et al.²⁵, is of great importance for hip prosthesis applications.

<figure><img src="image_6.png" /><figcaption>Figure 5 : Complete $V-K_I$ curve of 3Y-TZP.</figcaption></figure>

☐ : Relaxation method, with correction of $K_I$, ○ : Constant load technique, with correction.

## IV.2. Influence of the media on the crack propagation behaviour under static loading

Relaxation tests were performed in water, in a Silicon oil and in Ringer solution to understand the effect of environnement on SCG.

The results are presented in figure (6). In all the liquids, an unique stage is present, because no saturation in corrosive species is reached. The first propagation stage in air is identical to the propagation law in water or Ringer's solution (same slope). The $V-K_I$ law in oil is very different from the others, crack growth appearing only very near $K_{IC}$. The difference between the Silicon oil and the other media is the absence of $H_2O$ molecules. Therefore, SCG is probably due to the stress assisted corrosion of the ceramic by $H_2O$ molecules. It is to note that the different salts present in the Ringer's solution do not promote SCG. This result is of importance since SCG measurements can be conducted in air or water to simulate behaviour in Ringer's solution.
## V. CYCLIC FATIGUE BEHAVIOUR

Liu et al.⁷ and Jacobs et al.²¹ have found that a modified Paris Law of the form:

$$da / dt = A * (K_{max})^m * (\Delta K)^{m'} \quad (9)$$

described well cyclic data for microscopic flaws in a 3Y-TZP, with m=19 and m'=2, and in a Silicon Nitride with m~22 and m'~3.

We therefore hereby analyse our results in terms of V-Kmax and V-ΔK dependence.

Some authors³³,³⁴ have sometimes questioned the validity of results obtained for macrocracks because of the effect of rising crack growth resistance with extension. In 3Y-TZP, however, such effects must be neglectable due to the fact that the plateau toughness is reached at the early stage of propagation. It will be interesting to compare our results on macroscopic flaws with those of Liu et al.⁷ on microcracks.

<figure><img src="image_7.png" /><figcaption>Figure 6. Influence of the media on the stress corrosion of 3Y-TZP.</figcaption></figure>

### V.1. ΔK dependency of da/dt

For a fixed frequency (1 Hz) and for different R ratio (defined as the ratio of Kmax to Kmin in the cycle), we have recorded the crack speed versus the stress intensity range ΔK.

The results are reported on figure (7), in a log (da/dN) versus log (ΔK) diagram. It appears that V is only weakly dependent of ΔK, because of the strong importance of the R ratio.
<figure><img src="image_8.png" /><figcaption>Figure 7: V-ΔK curves for three R ratios, during cyclic fatigue at 1 Hz.</figcaption></figure>

## V.2. Kmax dependency of da/dt

The same results have been analysed in terms of V-K<sub>max</sub> for the different R ratio, and reported in figure (8). This time, the results are more confined, although a slight effect of R is observed. We also note threshold values K<sub>ICmax</sub> for each R ratio. In order to take into account the weak dependence of the V-K<sub>max</sub> curves with R, we have represented the results in a V/(1-R<sup>2</sup>)-K<sub>max</sub> graph. The results reported in figure (9) define a single line of slope m=18 for crack speed superior to 10<sup>-9</sup> m/s. Under 10<sup>-9</sup> m/s we note a strong decrease of the crack rates, and thresholds depending of R are measured.

<figure><img src="image_9.png" /><figcaption>Figure 8: V-K<sub>max</sub> curves for the three R ratios considered, during cyclic fatigue at 1 Hz.</figcaption></figure>
We therefore propose a growth law in cyclic fatigue for macroscopic cracks :

$$da / dt = A * (K_{\max})^m * (\Delta K)^2 \quad (9.\text{b})$$

with m=18, and for $K_{\max} > K_{10max}$.

$K_{10max}$ is dependent of R, and we have :

$K_{10max}=3.2 \text{ MPa}\sqrt{\text{m}} \text{ for } R=0.5, K_{10max}=3.0 \text{ MPa}\sqrt{\text{m}} \text{ for } R=0.3, K_{10max}=2.8 \text{ MPa}\sqrt{\text{m}} \text{ for } R=0.1$

## VI. DISCUSSION

A comparison of the static fatigue and cyclic fatigue results (for R ratio of 0,1) is reported in figure (10) in terms of crack rate versus $K_1$ or $K_{1max}$.

<figure><img src="image_10.png" /><figcaption>Figure 9 : (da/dN)/(1-R²) versus $K_{max}$ curve, comparison with the results of Liu et al on microcracks.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 10 : Comparison of the SCG behaviour of 3Y-TZP in static fatigue (□,○: V-K₁ graph) and in cyclic fatigue (■: V-Kmax for R=0.1).</figcaption></figure>
It appears that:

- Crack propagation rates are higher during cyclic fatigue.

- Stress exponent m = 18 is much lower than exponent n = 32.

- Threshold value K<sub>IOmax</sub> (2,8 MPa√m) in cyclic fatigue is lower than K<sub>IO</sub> in static fatigue (3,5 MPa√m).

The comparison of crack rates under static and cyclic loading in figure (10) is indirect. Indeed, it is more appropriate to compare experimental crack velocities under cyclic fatigue with the integration of the SCG law over the cycles.

If stress corrosion was the unique phenomenon responsible for crack extension, crack rate per cycle would be obtained by:

$$ \frac{da}{dN} = \int_{0}^{T} A * K_{t}^{n} = A * K_{\max}^{n} * \int \left[ \left( \frac{1+R}{2} \right) + \left( \frac{1-R}{2} \right) \sin(\omega t) \right]^{n} = \Omega * K_{\max}^{n} \quad (10) $$

It is to note that the sole presence of stress corrosion during cycling loading would lead to a crack exponent m equal to n. The comparison of the experimentals results of cyclic fatigue for R=0.1 and the simulation by the integration of SCG law over the same conditions is presented on figure (11). The cyclic crack accelerating effect is even pronounced in figure (11) than in (10). Stress corrosion is therefore far from being the sole crack extension factor during cyclic loading.

Shielding degradation during cyclic fatigue can explain this cyclic crack accelerating effect:

Liu et al. have expressed cyclic fatigue behaviour of a Si₃N₄ in air in terms of a mechanical damage of crack tip shielding. It was assumed that a crack would propagate in cyclic fatigue when the stress intensity at the crack tip exceeded the intrinsic toughness of the material K₀. Because of crack tip shielding due to crack bridging, the applied Kᵢ needed to cause crack advance, noted K<sub>lapp1</sub> is given by:

$$ K_{lapp1} = K_{0} + K_{sh} \quad (11) $$

In cyclic fatigue, shielding degradation leads to a decrease of shielding stress intensity factor K<sub>sh</sub> in comparison with the static loading case. The authors showed on the Si₃N₄ that after a

<figure><img src="image_12.png" /><figcaption>figure 11: V-K<sub>max</sub> curve for R=0,1 in the 10⁻⁹-10⁵ m/s crack velocity range.</figcaption></figure>

from experimental data.

from integration of stress corrosion over the cycles.
first decrease of K<sub>sh</sub>, a steady state was reached, so that the crack grew at a constant rate for a constant cycling loading configuration. At this equilibrium, accumulation and degradation of shielding balanced. Cyclic fatigue was therefore expressed in terms of shielding degradation ΔK<sub>sh</sub>=(K<sub>sh,static</sub>-K<sub>sh,cyclic</sub>) between quasi static and cyclic fatigue tests. Therefore, for a Si<sub>3</sub>N<sub>4</sub>, where we remember that stress corrosion is insignificant<sup>16</sup>, SCG in cyclic fatigue was accurately explained by a mechanical equilibrium between accumulation and degradation of crack tip shielding.

For 3Y-TZP, (and other ceramic oxides) sensitive to stress corrosion, one can expect that such an equilibrium exists, but that crack propagation occurs when the stress intensity factor at the crack tip reaches the value needed to cause crack propagation by stress corrosion, i.e below K<sub>Jc</sub>.

Crack velocity could therefore be expressed by :

$$da / dt = A * K_l^n = A * (K_{lapp} + \Delta K_{sh})^n \quad (12)$$

where ΔK<sub>sh</sub> represents the crack tip sheding degradation at equilibrium. Therefore, the crack increment per cycle da/dN can be calculated by :

$$da / dN = \int_{t_1}^{t_2} A * K_l^n * dt = A * \int \left[ K_{max} \left[ \left( \frac{1+R}{2} \right) + \left( \frac{1-R}{2} \right) \sin(\omega t) \right] + \Delta K_{sh} \right]^n dt \quad (13)$$

With the experimental value of da/dN measured in cyclic fatigue (section V), and the SCG parameters A and n measured with constant loading tests, it is possible to determine ΔK<sub>sh</sub> for a given R value and for different K<sub>max</sub> values. This has been conducted by numerical calculations of expression (13) for the three R ratios. For a given K<sub>max</sub> value, expression (13) has been fitted to the experimental da/dN by varying ΔK<sub>sh</sub>. The determination of ΔK<sub>sh</sub> for various K<sub>max</sub> allows the determination of three ΔK<sub>sh</sub> versus K<sub>max</sub> curves, reported in figure (12).

It appears that shielding degradation is decreased with increasing K<sub>max</sub> (increasing crack velocities) and R (decreasing stress intensity range for a given K<sub>max</sub>). The most interesting

<figure><img src="image_13.png" /><figcaption>Figure 12 : ΔK<sub>sh</sub> versus K<sub>max</sub> curves for the three R ratios.
--- R=0,1
--- R=0,3
--- R=0,5.</figcaption></figure>
values of $\Delta K_{sh}$ are the one calculated for $K_{max}=2,8$ MPa√m and $R=0.1$ ($\Delta K_{sh}\sim0.85$), $K_{max}=3$ MPa√m and $R=0.3$ ($\Delta K_{sh}\sim0.65$), $K_{max}=3,1$ MPa√m and $R=0.5$ ($\Delta K_{sh}\sim0.55$). Indeed, they correspond quite well to the difference between $K_{IO}$ and $K_{IOmax}$ (noted $\Delta K_{IO}$) for the three R ratios ($R=0.1, \Delta K_{IO}=0.7 - R=0.3, \Delta K_{IO}=0.5 - R=0.5, \Delta K_{IO}=0.4$). The three thresholds in cyclic fatigue are therefore accurately related to the threshold stress intensity factor in static fatigue $K_{IO}$ and to the crack tip shielding degradation by cyclic loading.

In conclusion, cyclic fatigue of 3Y-TZP seems to be governed by stress corrosion and crack tip shieding degradation, the two processes acting in series. Of course, this has to be confirmed by further studies. We currently work on cyclic fatigue in vacuum and in inert atmospheres, where stress corrosion is inoperant, to confirm this assumption.

## VII. CONCLUSION

A study of the 3Y-TZP crack propagation behaviour has been conducted with the Double Torsion method.

1. In order to obtain precise crack propagation laws in both static and cyclic fatigue, the evolution of $K_I$ with crack length has been investigated. The results show that the stress intensity factor is significantly dependent of the crack length, which is not taken into account in the usual compliance analysis. We therefore propose a modified expression that takes into account the evolution of $K_I$ with crack extension.

2. Static fatigue behaviour is due to stress assisted corrosion at the crack tip by the $H_2O$ molecules. Three stages are observed in a V-$K_I$ curve, and the first one is well described by a power law:

$$V = A * K_I^n$$

with $n=32$. A well defined threshold intensity factor $K_{IO}=3,5$MPa√m has been measured.

3. Under cyclic solicitations, a modified Paris law proposed by Liu and Chen⁷ of the form:

$$da / dt = A * (K_{max})^m * (\Delta K)^2$$

with $m=18$ correctly fits the experimental results. Our results on macroscopic flaws are in good agreement with those of Liu and Chen for indentation cracks. Threshold values $K_{IOmax}$ depending on the ($K_{Imin}/K_{Imax}=R$) ratio were also measured.

4. Crack extension rates are much higher and threshold intensity factors lower during cyclic fatigue than during static loading. This has been interpreted in terms of shielding degradation $\Delta K_{sh}$ at the crack tip. $\Delta K_{sh}$ has been calculated by simple numerical simulations for different cyclic loading configurations, and reaches a value on the order of 1 MPa√m for $R=0,1$. Thresholds in cyclic fatigue correspond to $K_{IO}$ in stress assisted corrosion. Their lower values are explained by shielding degradation. These arguments are however to be confirmed by further studies, particularly with tests in inert atmosphere.

## REFERENCES

1. Cales B. "Zirconia ceramic for improved hip prosthesis -a review-" Proceeding of the 6th Biomaterial Symposium, september 1994 - to be published.

2. Wiederhorn S. M. "Subcritical crack growth in ceramics", in: Fracture Mechanics of ceramics, vol.2, Microstructure, Material and applications, 1974, New York, edited by Bradt R.C., Hasselman D.P, Lange FF, 1974, N.Y.
3. Sudreau F. "Methodes d'analyse et de simulation de la fatigue thermique de céramiques thermomécaniques", Ph D. thesis, INSA Lyon, 1992, France, pp 273.

4. Fett T, Germerdonk K, Grossmüller A, Keller K, Munz D "Subcritical crack growth and threshold in borosilicate glass", *Journal of Materials science*, vol.26, 1991, pp 253-257.

5. Evans E. G. & Fuller R. "Crack propagation in ceramic materials under cyclic conditions", *Metallurgical transactions*, 1974, volume 5, pp 27-33.

6. Dauskardt R. H. & Ritchie R. O. "Fatigue crack propagation in transformation toughened zirconia ceramic", *J. Am. Ceram. Soc.*, 1987, volume 70, n°10, pp 248-252.

7. Liu S. Y. & Chen I.W "Fatigue of yttria-stabilized zirconia : II. Crack propagation, fatigue striations, and short-crack behavior", *J. Am. Ceram. Soc.*, 1991, volume 74, n°6, pp 1206-1216.

8. Liu T., Matt R., Grathwohl G. "Static and cyclic fatigue of 2Y-TZP ceramics with natural flaws", *J. Europ. Ceram. Soc.*, 1993, volume 11, pp 133-141.

9. Grathwohl G. & Liu T. "Crack resistance and fatigue of transforming ceramics : I. Materials in the ZrO2-Y2O3-Al2O3 System", *J. Am. Ceram. Soc.*, 1991, volume 74, n°2, pp 318-325.

10. Grathwohl G. & Liu T. "Crack resistance and fatigue of transforming ceramics : II, CeO2-Stabilized Tetragonal ZrO2", *J. Am. Ceram. Soc.*, 1991, volume 74, n°12, pp 3028-3034.

11. Dauskardt H, Marshall D. Ritchie R "Cyclic fatigue-crack propagation in Mg-PSZ ceramics", *J. Am. Ceram. Soc.*, 1990, volume 73, n°4, pp 893-903.

12. Tzai J., Yu C., Shetty D. "Fatigue crack propagation in Ceria-Partially-Stabilized Zirconia (Ce-TZP)-Alumina composites", *J. Am. Ceram. Soc.*, 1990, volume 73, n°10, pp 2992-3001.

13. Guiu F., Reece M. J., Vaughan D. A. J. "Cyclic fatigue of ceramics", *Journal of Materials Science*, 1991, volume 26, pp 3275-3286.

14. Fett T., Martin G., Munz G., Thun G. "Determination of da/dN-ΔK1 curves for small cracks in alumina in alternating bending tests", *Journal of Materials Science*, 1991, volume 26, pp 3320-3328.

15. Dauskardt R. James M. Porter J. Ritchie R "Cyclic fatigue crack growth in a SiC-Wiskers-Reinforced Alumina composite : long- and small-crack behavior", *J. Am. Ceram. Soc.*, 1992, volume 75, n°4, pp 759-771.

16. Horibe S "Fatigue of Silicon Nitride Ceramics under cycling loading", *J. Eur. Ceram. Soc.*, 1990, n°6, pp 89-95.

17. Vekinis G. Ashby M. Beaumont P "R-Curve behaviour of Al₂O₃ ceramics", *Acta metall. mater.*, 1990, vol 38, n°6, pp 1151-1162.

18. Rödel J. Kelly J. Lawn B "In situ measurements of bridged crack interfaces in the scanning electron microscope", *J. Am. Ceram. Soc.*, 1990, volume 73, n°11, pp 3313-18.

19. Lathabai S., Rodel S. & Lawn B. R. "Cyclic fatigue from frictional degradation at bridging grains in alumina", *J. Am. Ceram. Soc.*, 1991, volume 74, n°6, pp 1340-1348.

20. Gilbert C. Petrany R, Ritchie R, Dauskardt R, Steinbrech R "Cyclic fatigue in monolithic alumina : mechanisms for crack advance promoted by frictional wear of grain bridges", *Journal of Materials Science*, 1995, volume 30, pp 643-654.

21. Jacobs D. Chen I "Cyclic fatigue in ceramics : a balance between crack shielding accumulation and degradation", *J. Am. Ceram. Soc.*, 1995, volume 78, n°3, pp 513-520.

21. Porter D. Heuer A "Mechanisms of toughening Partially Stabilized Zirconia", *J. Am. Ceram. Soc.*, 1977, volume 60, n°3-4, pp 183-184.

23. Marshall D. Shaw M, Dauskardt R, Ritchie R, Readey M. Heuer A "Crack-tip transformation zones in toughened Zirconia", *J. Am. Ceram. Soc.*, 1990, volume 73, n°9, pp 2659-2666.

24. Tsai J, Shetty D "Cyclic fatigue of Ce-TZP/Al2O3 composites : Role of the degradation of transformation zone shielding", *J. Am. Ceram. Soc.*, 1995, volume 78, n°3, pp 599-608.

25. Knechtel M., Garcia D., Rödel J., Claussen N. "Subcritical crack growth in Y-TZP and Al2O3 - Toughened Y-TZP", *J. Am. Ceram. Soc.*, 1993, volume 76, n°6, pp 2681-2684.

26. Chevalier J, Olagnon C, Fantozzi G, Cales B "Crack propagation behaviour of Y-TZP ceramics", *J. Am. Ceram. Soc.* - accepted for publication.

27. Fuller E "An evaluation of Double Torsion testing - Analysis", in : *Fracture Mechanics Applied to Brittle Materials*, proceedings of the 11th symposium on fracture mechanics, part II. ASTM STP 678, pp 3-19. editor S.W. Freiman.

28. Plekta B, Fuller E, Koepke B "An evaluation of Double Torsion testing - Experimental", in : *Fracture Mechanics Applied to Brittle Materials*, proceedings of the 11th symposium on fracture mechanics, part II. ASTM STP 678, pp 19-38 editor S.W. Freiman.

29. Williams D, Evans A. "A simple method for studying slow crack growth", *Journal of testing and evaluation*, Vol. 1, N°4, July 1973, pp. 264-270.

30. Shetty. D.K., Virkar A.V., Harward M.B. *Journal of The American Ceramic Society- Discussions and Notes*, Vol. 62, 5-6 1979, pp 307-309.
31. Trantina G.G. "Stress analysis of the Double-Torsion specimen", *Journal of The American Ceramic Society*, Vol.60, 7-8 1977, pp 338-341.

32. Chevalier J, Saadaoui M, Olagnon G, Fantozzi G "Experimental investigations on the Double Torsion specimen with a 3Y-TZP", *Ceramics International* - Submitted to publication.

33. Fett T, Munz D, Keller K "determination of subcritical crack growth on glass in water from lifetime measurements on Knoop-cracked specimens", *Journal of Materials Science*, 1988, volume 23, pp 798-803.

34. Fett T, Munz D "Subcritical crack growth of macro- and microcracks in ceramics", in : *Fracture Mechanics of Ceramics*, volume 9, Plenum Press, New York, 1992.