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Crack growth resistance under thermal shock loading of alumina
Materials Science and Engineering A247 (1998) 142 – 151
Crack growth resistance under thermal shock loading of alumina M. Saaˆdaoui a, G. Fantozzi b,* a
b
EMI, BP 765 Rabat, Morocco GEMPPM-UMR CNRS 5510, INSA Lyon, 20 A6enue Albert Einstein, 69621 Villeurbanne, France Received 21 April 1997; received in revised form 6 November 1997
Abstract Thermal shock experiments, conducted in an apparatus in which all the parameters can be controlled, are modelled by a two dimensional cooling model, allowing a precise determination of the induced stress intensity factors (SIF). Fracture mechanics analysis in terms of stress intensity factors is applied to determine R-curve behaviour of indentation cracks in alumina materials subjected to thermal shock. The instant of unstable crack growth was obtained by acoustic emission (AE). As in bending tests, the coarse grained material showed a more pronounced R-curve behaviour than the fine grained material. The results are discussed considering the influence of the R-curve behaviour on the retained strength after thermal shock. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Thermal shock; R-curve; Acoustic emission; Alumina
1. Introduction Significant advances in the understanding of thermal shock behaviour of ceramic materials have been made by Hasselman [1,2]. He derived a model which describes strength degradation for brittle materials subjected to thermal shock. No strength degradation occurs when the applied temperature difference, DT, is smaller than a critical value DTc, then, a steep drop of the retained strength, sr, occurs with DT \DTc. The model leads to the dilemma that materials with high initial strength exhibit catastrophic strength degradation at DT \ DTc and does not take into account materials with rising fracture resistance curves (R-curves). A refine analysis of the crack propagation under thermal loading can be made by a fracture mechanics approach in terms of stress intensity factor (SIF) or energy release rate [3 – 7]. The crack extension during thermal shock can be described by a diagram representing the induced SIF, KTS, as a function of crack length and time (Fig. 1). The envelope of all the KTS curves is important for explaining crack propagation and arrest. Materials exhibiting R-curve behaviour are generally characterised by three consecutive stages of crack prop* Corresponding author. Tel.: + 33 4 72438382; fax: +33 4 72438528. 0921-5093/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S0921-5093(97)00768-5
agation, under thermal shock loading (Fig. 1, curve 1). The onset of crack propagation occurs when, KTS = KRi
(1)
where KTS is the stress intensity factor generated during thermal shock and KRi is the initial value of the R-curve. If the R-curve is steep enough, the crack extension is first stable as, dKTS/da BdKR/da
(2)
where a is the crack size and becomes unstable when,
Fig. 1. Illustration of crack growth under thermal shock loading for a material with rising (curve 1) and flat (curve 2) R-curve behaviour.
M. Saaˆdaoui, G. Fantozzi / Materials Science and Engineering A247 (1998) 142–151
dKTS/da \dKR/da
(3)
For large crack lengths, the extension again becomes stable as dKTS/da B0, thus the condition in Eq. (2) is always satisfied. For a material with a flat R-curve (Fig. 1, curve 2), the crack propagation is first unstable and becomes stable for larger crack sizes. For large initial cracks, the propagation is always stable, independently of the R-curve behaviour. Pompe [8] has applied this approach to provide a qualitative description of the consequence of a rising R-curve on the thermal shock behaviour of ceramics. It can be seen (Fig. 1) that a strong R-curve behaviour diminishes the crack propagation during thermal shock loading and thus enhances retained strength. However, few works were published on the thermal shock behaviour of R-curve materials [6,9 – 12]. Swain [6] was the first to apply the R-curve concept to the thermal shock analysis of transformation toughened zirconia. Lutz and Swain [9] have shown the correlation between R-curve behaviour and the thermal shock strength of duplex ceramics. Schneider et al. [11] have conducted in situ observation of the predicted stages of the crack growth for R-curve material under thermal shock loading. Applying the R-curve measured under mechanical loading at room temperature may lead to unrealistic predictions as the R-curve is not an intrinsic property of a material, but depends also on the loading situation and the initial crack shape. Bahr et al. [12] have shown particularly the difference between macro-crack Rcurve from controlled bending test and that predicted for thermal shock test. In this study, alumina is chosen as a model material known to exhibit R-curve behaviour due to grain bridging [13,14], the importance of which depends on the microstructure, particularly the grain size of the material. The crack growth resistance under thermal shock conditions is determined for short indentation cracks by using in situ measurements of acoustic emission and compared to the R-curves obtained from bend tests.
2. Experimental procedure Three alumina ceramics with different microstructures were used for this study. An homogenous fine grained material with an average grain size of 3 mm called A3, was obtained by natural sintering at 1550°C for 2 h of high purity (\ 99.9) powder (SM8 Baikowski). An alumina called AH, with a bimodal microstructure constituted of a matrix with an average grain size of 5 mm containing elongated grains up to 300 mm, was obtained by additional heat treatment at 1600°C for 12 h. A commercial alumina (AF997, Desmarquest) exhibiting pronounced long-crack R-curve behaviour [15] was also used. This material, character-
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Table 1 Crack sizes and temperature differences Nuance
Pi (N)
c (mm)
DTa (K)
DTc (K)
A3 AH A25
30 – 400 30 – 300 100 – 400
70 – 350 90 – 380 150 – 500
940 940 840
\1000 \1000 845
ised by an average grain size of 25 mm is designated A25. Thermal shock tests were carried out on indented rectangular bars with dimensions 3× 4 × 40 mm3 for A3 and AH and 4× 6× 40 mm3 for A25. The largest face of each specimen was polished to 1 mm with diamond paste and the samples were annealed at 1300°C for 2 h to release machining residual stresses. Controlled flaws were then generated by the introduction of indentation cracks using a Vickers hardness tester, with various indentation loads Pi, between 30 and 400 N. An indentation was made in the centre of the polished surface with its diagonals parallel to the specimen edges. The residual stresses generated during indentation were removed by further annealing of the samples at 1300°C for 2 h. The length of the surface radial crack, c, was measured from the centre of the indentation by optical microscopy. The range of the crack sizes obtained for the three materials are indicated in the Table 1. For the A25 alumina, a minimum of 100 N indentation load was necessary to obtain single cracks from the corners of the indent, which gave a crack radius of : 150 mm. Details of the thermal shock testing have been described previously (see [16]). The specimens were initially heated in a resistance furnace at different temperatures for 10 min and cooled symmetrically by the application of two jets of pulsed air to the largest faces. The acoustic emission (AE) of the sample was monitored by a piezoelectric transducer connected to a wave guide in discreet contact with the specimen. The emissive response during the first 6 sec of cooling was recorded, allowing the determination of the instant of the onset of unstable crack propagation. The applied temperature difference, DTa, was chosen below the critical value for non indented specimen, DTc (Table 1) in order to propagate only the indentation-induced cracks.
3. Modelling and stress intensity factors Non-indented specimens of the A25 alumina (4× 6× 40 mm3) were used to model the thermal shock tests and to compare experimental results with theoretical predictions.
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Fig. 2. AE response for non indented A25 samples at DTc =845 K and at DT= 1000 K.
3.1. AE analysis As it has been previously outlined [17], a direct correlation was observed between the number of unstable propagated cracks and the number of AE events, for specimens with natural flaws. No acoustic event appears until the applied temperature difference reaches the critical value DTc =845 K. At this temperature difference, the acoustic emission response shows only one event at 980 ms (Fig. 2a), corresponding to the propagation of a single longitudinal crack observed at the centre of the frontal face of the test specimen. Above DTc, the number of acoustic peaks rises continuously with increasing applied temperature difference. The time to initiate the first acoustic peak, ti (Fig. 2b), corresponding to the onset of the unstable crack propagation, decreases rapidly when the applied temperature difference increases (Table 2), as it was suggested by Bahr et al. [18] in a qualitative description of thermal shock cracking. Table 2 Time to initiate an AE event as a function of applied DT DT (K) ti (ms)
845 980
855 910
900 550
960 350
1000 260
Fig. 3. Illustration of heat transfer model (a) and the normal stresses sxx at the plane x =0 (b), for A25 sample cooled at DTc.
3.2. SIF diagrams For the fracture mechanics analysis, the thermal shock experiments were analysed using a 2-D cooling model in which the heat exchange was defined by both frontal and lateral surface heat coefficients (Fig. 3a) having the same measured value hF = hL = 600 W m − 2 K − 1 [16]. The temperature field distribution at any cooling time was determined using a finite element method, solving the heat transfer system in which all the thermophysical properties (conductivity, specific heat capacity and density) were taken as a function of temperature. The transient stress field was then computed by finite element calculation from the previous temperature field, and the thermoelastic properties of the material (Young’s modulus and thermal expansion coefficient), taken as a function of temperature. The maximum tensile stress is obtained on the largest face of the specimen, precisely in the middle of this face were the first crack propagation was observed. In the following, only the stresses sxx (y, t) along the line x=0 (Fig. 3a) which generates an opening mode for the first propagated crack will be taken into account. An illustration of these stresses is given in Fig. 3b. Considering a semi-elliptical surface crack (Fig. 4), the average stress intensity factors KA, corresponding to the deepest point, and KC, corresponding to the surface
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Fig. 4. Geometrical parameters of a semi-elliptical surface crack.
point, were calculated by the weight function method [19], using the symmetrical stress field sxx (y, t). At a given cooling time, KA and KC are dependent on the crack depth a and the ratio of the semi-axes a/c. Fig. 5 shows the variations of KA and KC with the crack depth for the transient stress distributions represented in Fig. 3b. The SIF were computed for a/c =0.5 and for semi-circular surface cracks (a/c= 1). We note that in the later case, the KC curve runs above the KA curve at any cooling time, which suggests that for this crack geometry, the initiation of the crack propagation occurs at the surface of the specimen. Thus, it is sufficient to determine KC if only the onset of unstable crack propagation has to be considered.
Fig. 5. Applied thermal shock SIF KC (continuous lines) and KA (dashed lines) corresponding, respectively, to the surface and the deepest point of semi elliptical surface cracks with aspect ratio a/c= 0.5 and a/c =1.
Fig. 6. Envelope of the KTS curves for monodimensionnal and semi-circular surface cracks. The horizontal line represents the bending KRi value at 800°C and the vertical dashed lines are the limits of domain of the pre-existing flaws in the A25 material.
The envelopes of the KA and KC curves correspond to the maximum of these curves which are reached after a cooling time of 1020 ms. It is interesting to note that, for both KA and KC, the maximum curve is not different from that computed for the instant of the apparition of AE event, tAE = 980 ms.
3.3. Comparison between theory and experiment Theoretical fracture analysis of thermal shock predicts that for the critical applied temperature difference, DTc, the following conditions are satisfied: 1. the applied temperature difference is just high enough to activate the largest of the pre-existing cracks with a depth ac. 2. the onset of the crack propagation occurs when the induced thermal shock SIF, KTS, calculated for ac, reaches the toughness of the material. This condition can be reformulated in the way that the envelope of the KTS curves (Fig. 1) must intersect the horizontal straight line representing the toughness at ac. As the first condition corresponds to the experimental observations, the second must be verified in order to validate the thermal shock model used and the AE measurements. For this purpose, the natural surface flaws are assumed to be semi-circular cracks of depth a and the envelope of the KC curves (corresponding to the surface point) is computed for the critical thermal shock condition, DTc = 845 K (Fig. 6). The crack resistance curve (KR curve) of the material was determined in bending at 800°C, which corresponds to the surface temperature at the instant of the apparition of the AE event. The horizontal line, representing the starting value of this curve KRi =2.3 MPa m1/2, is plotted in Fig. 6. It is observed that this line intersects the envelope curve for a crack depth of 78
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mm, which is inside the domain of pre-existing flaws determined from strength bending tests. The limits of the domain are indicated by the vertical dashed lines in Fig. 6. For comparison, the envelope curve computed for a monodimensional (1-D) crack is also reported in Fig. 6. On this curve, the KRi value is reached for a crack depth of 30 mm, which is less than the lower limit of the flaw domain. This means that, if we assume that the natural surface flaws are monodimensional cracks, it leads to an overestimation of the induced thermal shock stress intensity factors and to erroneous predictions when the SIF diagrams are used. In the following, only semi-circular cracks will be considered. Different SIF curves corresponding to thermal shock tests performed at various temperature differences above DTc, were also computed (Fig. 7) at the onset of crack propagation ti (Table 2), determined by AE. For all the testing conditions, the intercept (indicated by an arrow) of the KC curve with the initial KRi value occurs for a defect size which is upon the domain of pre-existing flaws. For clarity, the intercepts are represented for only two applied temperature differences 900 and 1000 K. For each applied temperature difference, the KRi (DT) considered is that determined in bending at the calculated surface temperature at the onset of cracking ti (DT). Although the multiple crack propagation is observed at DT \ DTc, the analysis was based on single crack propagation as only the onset of crack propagation (corresponding to the propagation of the critical flaw at ti) was considered. The above results show that the used thermal shock model gives a good agreement between theoretical predictions and the acoustic emission measurements by assuming that the flaws are semi-circular surface cracks. The fracture behaviour is controlled by the crack propagation resistance of the material. In the following, the acoustic emission and the SIF diagrams will be used to evaluate the crack growth resistance curves under thermal shock loading.
Fig. 7. KTS curves computed for various applied temperature differences at the onsets of unstable crack propagation ti.
Fig. 8. Optical micrographs showing the longitudinal propagation of the indented crack.
4. Thermal shock KR curves
4.1. Qualitati6e analysis from AE response Detailed examination of the tested specimens showed that only the indentation-induced cracks running parallel to the longitudinal direction of the specimens had propagated (Fig. 8). The acoustic emission response of the materials (Fig. 9) showed only one event corresponding to the onset of unstable crack growth. However, there was a limit of crack size (350, 380 and 500 mm for A3, AH and A25, respectively) beyond which no AE event was detected, despite the presence of a large longitudinal crack observed by the projection of
Fig. 9. AE response with the time of unstable crack propagation tAE, for indented specimens of A3 alumina with crack radius of 340 mm (1) and 70 mm (2).
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Fig. 10. Comparison of measured tAE times (symbols) as a function of crack size and the theoretical curve (dashed line) corresponding to a constant toughness equal to 2.5 MPa m1/2.
dye penetrant on the indented surface. Indeed, for crack sizes greater than the limiting value, only stable crack propagation, undetected by the AE system, occurred. Qualitative analysis of the crack growth resistance under thermal shock can be done by comparing the time tAE with a theoretical curve representing a constant toughness material. The measured time tAE as a function of the crack size c is shown in Fig. 10 for the three aluminas (symbols). The dashed line represents a theoretical behaviour expected for a constant toughness equal to 2.5 MPa m1/2. This curve was determined from the SIF diagram computed for the thermal shock test conditions, considering semi-circular surface cracks and reporting, for each crack radius c, the instant at which the induced SIF reaches the above toughness value. For the fine grained material A3, a good agreement can be seen between the measured values and the theoretical curve for crack sizes \ 100 mm. For the AH and the A25 materials, the measured values deviates significantly from the theoretical curve and the scatter increases with the crack size. The observed deviation from the theoretical curve can be interpreted as an R-curve behaviour. Notice the analogy between the
present analysis and the indentation strength in bending (ISB) technique for R-curve evaluation [20], the instant of the unstable crack growth replacing the bending strength in our method. However the theoretical curve for a constant toughness material is not universal like in bending tests (the logarithmic plot of the bending strength versus the indentation load is a line with a slope of − 1/3 for a constant toughness material), but depends on the material and the test conditions including the specimen geometry and the heat transfer. Thus, direct comparison of the materials cannot be made from Fig. 10.
4.2. KR cur6es During the thermal shock experiments, the time of the acoustic emission signal tAE corresponds to the onset of unstable crack propagation defined by the point of contact (Fig. 1) where: KTS = KR
(4)
and dKTS/da =dKR/da
(5)
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where KTS is the induced thermal shock SIF and KR is the fracture resistance of the material. So, calculation of KTS at the acoustic emission time, for controlled crack sizes, allows the determination of the R-curve under thermal shock conditions. Regarding the crack indent as a half-penny crack, the applied average stress intensity factor at the surface point was calculated for initial crack radius. The small initial stable crack propagation before the AE time is neglected since the indentation residual stresses have been released by heat treatment. Fig. 11 shows the KTS curves (solid lines) computed at the AE times as a function of the crack radius, c, and the resulting TS– KR curves (symbols) for the three materials (each data point corresponds to one specimen with an initial crack size c; this size varies from one specimen to another one). For the fine grained alumina, A3, a nearly constant value KR = 2.7 MPa m1/2 is observed for crack sizes \100 mm. The initial increase of the KR curve can be attributed to an underestimation of the measured small crack sizes because it is difficult to observe and determine the total length by optical microscopy. This leads to an important underestimation of the induced SIF as the KTS curves rise rapidly in the range of small crack sizes. A significant increase in the KR value with crack size is observed for both the bimodal (AH) and the homogeneous coarse grained (A25) aluminas. For the AH material, the KR value is :2.1 MPa m1/2 for c= 100 mm and approaches 3 MPa m1/2 for c= 350 mm. For the A25 material, KR increases from 2.5 to nearly 3.5 MPa m1/2 when the crack size varies from 100 to 450 mm. For both materials, the plateau toughness is not attained in the investigated range of crack sizes, indicating even higher toughness values may be achieved with extended crack growth.
mal shock degradation. A similar difference has been observed in the case of macro-crack R-curves in alumina [12] and it can be attributed to the variation with the temperature of the bridging parameters responsible for the R-curve behaviour. Another reason for this difference is the temperature dependence of the material properties taken into account in the thermal shock analysis. For the coarse grained material A25, the KR
5. Discussion A comparison of the KR curves under thermal shock conditions and those obtained from bending tests at ambient temperature can be seen in Fig. 12. The bending KR curves were determined by the indentation strength bending (ISB) method, using identical indented specimens than for the thermal shock tests. The results obtained under thermal loading are in agreement with those obtained from the mechanical tests: the fine grained material A3 has a nearly flat R-curve while the bimodal and the coarse grained materials (AH and A25, respectively) exhibit rising R-curve behaviour. However, the KR curves determined under thermal shock conditions are below those obtained from ISB tests at room temperature. This confirms that the room temperature KR curves are not suitable to predict ther-
Fig. 11. Family of KTS curves (lines) computed at AE times and resulting TS – KR curves (symbols) for the three aluminas.
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149
Fig. 12. Comparison between TS– KR curves (symbols) and the KR curves from ISB tests (lines). The dashed lines represent the KRi values at high temperatures.
curve under thermal loading is more rising than that obtained in bending tests. This may be attributed to the assumption of the constant semi-circular shape of the cracks which overestimates the KR values in bending tests [21]. Indeed, the cracks become semi-elliptical during the propagation and for the ISB tests, the variation of the crack geometry has been taken into account, using the effective geometrical coefficient proposed by Krause [22]. However, the KRi values obtained for notched bend specimens at about the temperatures of thermal shocks (800 and 900°C for the A25 and AH materials, respectively) are shown by the dashed lines in Fig. 12 (the KRi value was not measured for the A3 alumina). It can be seen that for small crack sizes, the initial TS –KR values are similar to the KRi ones. This implies that, for the onset of thermal shock induced crack propagation prediction, fracture mechanics analysis can be achieved by using a KRi value determined from bending test at a temperature representative of the thermal shock. Such analysis gives a good agreement between theoretical predictions and experimental observations for the A25 and AH materials and for other alumina and zirconia based ceramics [23].
To investigate the influence of the R-curve behaviour on the strength degradation by thermal shock, a set of non indented specimens (with dimensions 4×6 ×40 mm3) were thermally shocked at different applied temperature differences and their retained strength, sr, was measured in 4-point bending with a 35/10 mm span. The relative residual strength sr/s0, where s0 is the initial strength, is plotted as a function of the applied temperature difference in Fig. 13. For the three studied aluminas, a typical catastrophic strength loss can be observed. The strength is constant up to a critical applied temperature and it decreases in a very narrow temperature interval the width of which is B 15 K. The critical applied temperature differences are 85595, 8459 5 and 825915 K for the A3, A25 and AH aluminas, respectively and the relative catastrophic drops of the retained strength reaches 80, 45 and 62%, respectively. At DT = 1000 K, the relative loss of the strength is nearly 80% for the three materials. The significant strength degradation observed for the AH and A25 aluminas implies that, for these materials, the R-curves are not rising enough to induce a gradual decrease in retained strength with increasing tempera-
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Fig. 13. Relative retained strength as a function of the applied temperature difference.
ture. This can be seen in Fig. 14 which shows the envelope of the KTS curves representing the crack driving force for the critical thermal shock conditions (DTc = 845 K) for the A25 alumina, and the TS– KR curve obtained at DT = 840 K. It can be considered that this TS–KR curve corresponds to the critical condition according to the low difference between DT and DTc. The plot of the mean size of the natural flaws ( :65 mm) on the envelope curve gives a TS –KR value of : 2.1 MPa m1/2 (solid symbol in Fig. 14). Above this value, the TS–KR curve rises less than the envelope curve (dKR/dc BdKTS/dc). So at the onset of cracking, unstable crack propagation occurs, which leads to a large final crack size and thus to an important strength loss as for a constant toughness material. However, the final crack length decreases and the residual strength increases with increasing steepness of the R-curve as observed for the AH and A25 aluminas compared to the A3 alumina. If the steepness of the R-curve becomes sufficient, unstable crack growth disappears dur-
Fig. 14. Comparison between the envelope of the KTS curves (line) and the TS –KR curve (symbols) at DTc. The solid symbol denotes the initial value TS – KRi corresponding to the initiation of crack propagation from natural flaws.
ing thermal shock as it has been observed by Hoffmann et al. for Si3N4 based materials [24]. The influence of the microstructure on the strength degradation of thermally shocked alumina was previously studied by Gupta [25]. He observed a gradual decrease in the retained strength with increasing quenching temperature for a coarse grained alumina with a grain size of 85 mm. But in this case, the gradual strength degradation can be attributed to the presence of large flaws which exhibit only stable propagation rather than to a R-curve behaviour. Indeed, an estimation of the KTS curves, in the case of the water quench used by the author, indicates that the crack size above which the stable propagation occurs is :200 mm, which is certainly reached in the coarse grained material.
6. Conclusion A method has been developed to determine the Rcurve behaviour of short indentation-induced surface cracks under thermal shock conditions. The method is based on in situ measurements by acoustic emission of the time of unstable crack growth and fracture mechanics analysis in terms of stress intensity factors. The acoustic emission measurements allow qualitative analysis of the R-curve behaviour using an approach similar to the indentation strength bending method. The TS–KR curves obtained for alumina ceramics with different microstructures are compatible with bending results: the fine grained material (A3) shows a flat R-curve while the bimodal (AH) and the homogeneous coarse grained (A25) materials exhibit rising R-curve behaviour. However, the KR curves determined under thermal shock loading are below those resulting from indentation strength bending tests at room temperature, which can be attributed to the temperature effect on the material properties and the grain bridging parameters. But, the KR values obtained for notched bend specimens at the temperature of thermal shocks are similar to the thermal shock KR values. So, the fracture mechanics analysis leads to a good agreement between theoretical predictions and experimental data. Significant strength degradatations after thermal shock are observed for the three aluminas but the retained strength is higher for the AH and A25 aluminas which present a steeper R-curve than the A3 alumina. Concerning the crack arrest investigation, the TS–KR values must be determined for long crack sizes. For this purpose, future work will be focused on the detection of the stable propagation which occurs above the limiting value of the crack size imposed by the AE system.
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Acknowledgements The authors acknowledge their colleagues Drs J. Dubois and Y. Jorand for helpful discussions and assistance for materials processing.
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Crack growth resistance under thermal shock loading of alumina M. Saaˆdaoui a, G. Fantozzi b,* a
b
EMI, BP 765 Rabat, Morocco GEMPPM-UMR CNRS 5510, INSA Lyon, 20 A6enue Albert Einstein, 69621 Villeurbanne, France Received 21 April 1997; received in revised form 6 November 1997
Abstract Thermal shock experiments, conducted in an apparatus in which all the parameters can be controlled, are modelled by a two dimensional cooling model, allowing a precise determination of the induced stress intensity factors (SIF). Fracture mechanics analysis in terms of stress intensity factors is applied to determine R-curve behaviour of indentation cracks in alumina materials subjected to thermal shock. The instant of unstable crack growth was obtained by acoustic emission (AE). As in bending tests, the coarse grained material showed a more pronounced R-curve behaviour than the fine grained material. The results are discussed considering the influence of the R-curve behaviour on the retained strength after thermal shock. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Thermal shock; R-curve; Acoustic emission; Alumina
1. Introduction Significant advances in the understanding of thermal shock behaviour of ceramic materials have been made by Hasselman [1,2]. He derived a model which describes strength degradation for brittle materials subjected to thermal shock. No strength degradation occurs when the applied temperature difference, DT, is smaller than a critical value DTc, then, a steep drop of the retained strength, sr, occurs with DT \DTc. The model leads to the dilemma that materials with high initial strength exhibit catastrophic strength degradation at DT \ DTc and does not take into account materials with rising fracture resistance curves (R-curves). A refine analysis of the crack propagation under thermal loading can be made by a fracture mechanics approach in terms of stress intensity factor (SIF) or energy release rate [3 – 7]. The crack extension during thermal shock can be described by a diagram representing the induced SIF, KTS, as a function of crack length and time (Fig. 1). The envelope of all the KTS curves is important for explaining crack propagation and arrest. Materials exhibiting R-curve behaviour are generally characterised by three consecutive stages of crack prop* Corresponding author. Tel.: + 33 4 72438382; fax: +33 4 72438528. 0921-5093/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S0921-5093(97)00768-5
agation, under thermal shock loading (Fig. 1, curve 1). The onset of crack propagation occurs when, KTS = KRi
(1)
where KTS is the stress intensity factor generated during thermal shock and KRi is the initial value of the R-curve. If the R-curve is steep enough, the crack extension is first stable as, dKTS/da BdKR/da
(2)
where a is the crack size and becomes unstable when,
Fig. 1. Illustration of crack growth under thermal shock loading for a material with rising (curve 1) and flat (curve 2) R-curve behaviour.
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dKTS/da \dKR/da
(3)
For large crack lengths, the extension again becomes stable as dKTS/da B0, thus the condition in Eq. (2) is always satisfied. For a material with a flat R-curve (Fig. 1, curve 2), the crack propagation is first unstable and becomes stable for larger crack sizes. For large initial cracks, the propagation is always stable, independently of the R-curve behaviour. Pompe [8] has applied this approach to provide a qualitative description of the consequence of a rising R-curve on the thermal shock behaviour of ceramics. It can be seen (Fig. 1) that a strong R-curve behaviour diminishes the crack propagation during thermal shock loading and thus enhances retained strength. However, few works were published on the thermal shock behaviour of R-curve materials [6,9 – 12]. Swain [6] was the first to apply the R-curve concept to the thermal shock analysis of transformation toughened zirconia. Lutz and Swain [9] have shown the correlation between R-curve behaviour and the thermal shock strength of duplex ceramics. Schneider et al. [11] have conducted in situ observation of the predicted stages of the crack growth for R-curve material under thermal shock loading. Applying the R-curve measured under mechanical loading at room temperature may lead to unrealistic predictions as the R-curve is not an intrinsic property of a material, but depends also on the loading situation and the initial crack shape. Bahr et al. [12] have shown particularly the difference between macro-crack Rcurve from controlled bending test and that predicted for thermal shock test. In this study, alumina is chosen as a model material known to exhibit R-curve behaviour due to grain bridging [13,14], the importance of which depends on the microstructure, particularly the grain size of the material. The crack growth resistance under thermal shock conditions is determined for short indentation cracks by using in situ measurements of acoustic emission and compared to the R-curves obtained from bend tests.
2. Experimental procedure Three alumina ceramics with different microstructures were used for this study. An homogenous fine grained material with an average grain size of 3 mm called A3, was obtained by natural sintering at 1550°C for 2 h of high purity (\ 99.9) powder (SM8 Baikowski). An alumina called AH, with a bimodal microstructure constituted of a matrix with an average grain size of 5 mm containing elongated grains up to 300 mm, was obtained by additional heat treatment at 1600°C for 12 h. A commercial alumina (AF997, Desmarquest) exhibiting pronounced long-crack R-curve behaviour [15] was also used. This material, character-
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Table 1 Crack sizes and temperature differences Nuance
Pi (N)
c (mm)
DTa (K)
DTc (K)
A3 AH A25
30 – 400 30 – 300 100 – 400
70 – 350 90 – 380 150 – 500
940 940 840
\1000 \1000 845
ised by an average grain size of 25 mm is designated A25. Thermal shock tests were carried out on indented rectangular bars with dimensions 3× 4 × 40 mm3 for A3 and AH and 4× 6× 40 mm3 for A25. The largest face of each specimen was polished to 1 mm with diamond paste and the samples were annealed at 1300°C for 2 h to release machining residual stresses. Controlled flaws were then generated by the introduction of indentation cracks using a Vickers hardness tester, with various indentation loads Pi, between 30 and 400 N. An indentation was made in the centre of the polished surface with its diagonals parallel to the specimen edges. The residual stresses generated during indentation were removed by further annealing of the samples at 1300°C for 2 h. The length of the surface radial crack, c, was measured from the centre of the indentation by optical microscopy. The range of the crack sizes obtained for the three materials are indicated in the Table 1. For the A25 alumina, a minimum of 100 N indentation load was necessary to obtain single cracks from the corners of the indent, which gave a crack radius of : 150 mm. Details of the thermal shock testing have been described previously (see [16]). The specimens were initially heated in a resistance furnace at different temperatures for 10 min and cooled symmetrically by the application of two jets of pulsed air to the largest faces. The acoustic emission (AE) of the sample was monitored by a piezoelectric transducer connected to a wave guide in discreet contact with the specimen. The emissive response during the first 6 sec of cooling was recorded, allowing the determination of the instant of the onset of unstable crack propagation. The applied temperature difference, DTa, was chosen below the critical value for non indented specimen, DTc (Table 1) in order to propagate only the indentation-induced cracks.
3. Modelling and stress intensity factors Non-indented specimens of the A25 alumina (4× 6× 40 mm3) were used to model the thermal shock tests and to compare experimental results with theoretical predictions.
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Fig. 2. AE response for non indented A25 samples at DTc =845 K and at DT= 1000 K.
3.1. AE analysis As it has been previously outlined [17], a direct correlation was observed between the number of unstable propagated cracks and the number of AE events, for specimens with natural flaws. No acoustic event appears until the applied temperature difference reaches the critical value DTc =845 K. At this temperature difference, the acoustic emission response shows only one event at 980 ms (Fig. 2a), corresponding to the propagation of a single longitudinal crack observed at the centre of the frontal face of the test specimen. Above DTc, the number of acoustic peaks rises continuously with increasing applied temperature difference. The time to initiate the first acoustic peak, ti (Fig. 2b), corresponding to the onset of the unstable crack propagation, decreases rapidly when the applied temperature difference increases (Table 2), as it was suggested by Bahr et al. [18] in a qualitative description of thermal shock cracking. Table 2 Time to initiate an AE event as a function of applied DT DT (K) ti (ms)
845 980
855 910
900 550
960 350
1000 260
Fig. 3. Illustration of heat transfer model (a) and the normal stresses sxx at the plane x =0 (b), for A25 sample cooled at DTc.
3.2. SIF diagrams For the fracture mechanics analysis, the thermal shock experiments were analysed using a 2-D cooling model in which the heat exchange was defined by both frontal and lateral surface heat coefficients (Fig. 3a) having the same measured value hF = hL = 600 W m − 2 K − 1 [16]. The temperature field distribution at any cooling time was determined using a finite element method, solving the heat transfer system in which all the thermophysical properties (conductivity, specific heat capacity and density) were taken as a function of temperature. The transient stress field was then computed by finite element calculation from the previous temperature field, and the thermoelastic properties of the material (Young’s modulus and thermal expansion coefficient), taken as a function of temperature. The maximum tensile stress is obtained on the largest face of the specimen, precisely in the middle of this face were the first crack propagation was observed. In the following, only the stresses sxx (y, t) along the line x=0 (Fig. 3a) which generates an opening mode for the first propagated crack will be taken into account. An illustration of these stresses is given in Fig. 3b. Considering a semi-elliptical surface crack (Fig. 4), the average stress intensity factors KA, corresponding to the deepest point, and KC, corresponding to the surface
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Fig. 4. Geometrical parameters of a semi-elliptical surface crack.
point, were calculated by the weight function method [19], using the symmetrical stress field sxx (y, t). At a given cooling time, KA and KC are dependent on the crack depth a and the ratio of the semi-axes a/c. Fig. 5 shows the variations of KA and KC with the crack depth for the transient stress distributions represented in Fig. 3b. The SIF were computed for a/c =0.5 and for semi-circular surface cracks (a/c= 1). We note that in the later case, the KC curve runs above the KA curve at any cooling time, which suggests that for this crack geometry, the initiation of the crack propagation occurs at the surface of the specimen. Thus, it is sufficient to determine KC if only the onset of unstable crack propagation has to be considered.
Fig. 5. Applied thermal shock SIF KC (continuous lines) and KA (dashed lines) corresponding, respectively, to the surface and the deepest point of semi elliptical surface cracks with aspect ratio a/c= 0.5 and a/c =1.
Fig. 6. Envelope of the KTS curves for monodimensionnal and semi-circular surface cracks. The horizontal line represents the bending KRi value at 800°C and the vertical dashed lines are the limits of domain of the pre-existing flaws in the A25 material.
The envelopes of the KA and KC curves correspond to the maximum of these curves which are reached after a cooling time of 1020 ms. It is interesting to note that, for both KA and KC, the maximum curve is not different from that computed for the instant of the apparition of AE event, tAE = 980 ms.
3.3. Comparison between theory and experiment Theoretical fracture analysis of thermal shock predicts that for the critical applied temperature difference, DTc, the following conditions are satisfied: 1. the applied temperature difference is just high enough to activate the largest of the pre-existing cracks with a depth ac. 2. the onset of the crack propagation occurs when the induced thermal shock SIF, KTS, calculated for ac, reaches the toughness of the material. This condition can be reformulated in the way that the envelope of the KTS curves (Fig. 1) must intersect the horizontal straight line representing the toughness at ac. As the first condition corresponds to the experimental observations, the second must be verified in order to validate the thermal shock model used and the AE measurements. For this purpose, the natural surface flaws are assumed to be semi-circular cracks of depth a and the envelope of the KC curves (corresponding to the surface point) is computed for the critical thermal shock condition, DTc = 845 K (Fig. 6). The crack resistance curve (KR curve) of the material was determined in bending at 800°C, which corresponds to the surface temperature at the instant of the apparition of the AE event. The horizontal line, representing the starting value of this curve KRi =2.3 MPa m1/2, is plotted in Fig. 6. It is observed that this line intersects the envelope curve for a crack depth of 78
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mm, which is inside the domain of pre-existing flaws determined from strength bending tests. The limits of the domain are indicated by the vertical dashed lines in Fig. 6. For comparison, the envelope curve computed for a monodimensional (1-D) crack is also reported in Fig. 6. On this curve, the KRi value is reached for a crack depth of 30 mm, which is less than the lower limit of the flaw domain. This means that, if we assume that the natural surface flaws are monodimensional cracks, it leads to an overestimation of the induced thermal shock stress intensity factors and to erroneous predictions when the SIF diagrams are used. In the following, only semi-circular cracks will be considered. Different SIF curves corresponding to thermal shock tests performed at various temperature differences above DTc, were also computed (Fig. 7) at the onset of crack propagation ti (Table 2), determined by AE. For all the testing conditions, the intercept (indicated by an arrow) of the KC curve with the initial KRi value occurs for a defect size which is upon the domain of pre-existing flaws. For clarity, the intercepts are represented for only two applied temperature differences 900 and 1000 K. For each applied temperature difference, the KRi (DT) considered is that determined in bending at the calculated surface temperature at the onset of cracking ti (DT). Although the multiple crack propagation is observed at DT \ DTc, the analysis was based on single crack propagation as only the onset of crack propagation (corresponding to the propagation of the critical flaw at ti) was considered. The above results show that the used thermal shock model gives a good agreement between theoretical predictions and the acoustic emission measurements by assuming that the flaws are semi-circular surface cracks. The fracture behaviour is controlled by the crack propagation resistance of the material. In the following, the acoustic emission and the SIF diagrams will be used to evaluate the crack growth resistance curves under thermal shock loading.
Fig. 7. KTS curves computed for various applied temperature differences at the onsets of unstable crack propagation ti.
Fig. 8. Optical micrographs showing the longitudinal propagation of the indented crack.
4. Thermal shock KR curves
4.1. Qualitati6e analysis from AE response Detailed examination of the tested specimens showed that only the indentation-induced cracks running parallel to the longitudinal direction of the specimens had propagated (Fig. 8). The acoustic emission response of the materials (Fig. 9) showed only one event corresponding to the onset of unstable crack growth. However, there was a limit of crack size (350, 380 and 500 mm for A3, AH and A25, respectively) beyond which no AE event was detected, despite the presence of a large longitudinal crack observed by the projection of
Fig. 9. AE response with the time of unstable crack propagation tAE, for indented specimens of A3 alumina with crack radius of 340 mm (1) and 70 mm (2).
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147
Fig. 10. Comparison of measured tAE times (symbols) as a function of crack size and the theoretical curve (dashed line) corresponding to a constant toughness equal to 2.5 MPa m1/2.
dye penetrant on the indented surface. Indeed, for crack sizes greater than the limiting value, only stable crack propagation, undetected by the AE system, occurred. Qualitative analysis of the crack growth resistance under thermal shock can be done by comparing the time tAE with a theoretical curve representing a constant toughness material. The measured time tAE as a function of the crack size c is shown in Fig. 10 for the three aluminas (symbols). The dashed line represents a theoretical behaviour expected for a constant toughness equal to 2.5 MPa m1/2. This curve was determined from the SIF diagram computed for the thermal shock test conditions, considering semi-circular surface cracks and reporting, for each crack radius c, the instant at which the induced SIF reaches the above toughness value. For the fine grained material A3, a good agreement can be seen between the measured values and the theoretical curve for crack sizes \ 100 mm. For the AH and the A25 materials, the measured values deviates significantly from the theoretical curve and the scatter increases with the crack size. The observed deviation from the theoretical curve can be interpreted as an R-curve behaviour. Notice the analogy between the
present analysis and the indentation strength in bending (ISB) technique for R-curve evaluation [20], the instant of the unstable crack growth replacing the bending strength in our method. However the theoretical curve for a constant toughness material is not universal like in bending tests (the logarithmic plot of the bending strength versus the indentation load is a line with a slope of − 1/3 for a constant toughness material), but depends on the material and the test conditions including the specimen geometry and the heat transfer. Thus, direct comparison of the materials cannot be made from Fig. 10.
4.2. KR cur6es During the thermal shock experiments, the time of the acoustic emission signal tAE corresponds to the onset of unstable crack propagation defined by the point of contact (Fig. 1) where: KTS = KR
(4)
and dKTS/da =dKR/da
(5)
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where KTS is the induced thermal shock SIF and KR is the fracture resistance of the material. So, calculation of KTS at the acoustic emission time, for controlled crack sizes, allows the determination of the R-curve under thermal shock conditions. Regarding the crack indent as a half-penny crack, the applied average stress intensity factor at the surface point was calculated for initial crack radius. The small initial stable crack propagation before the AE time is neglected since the indentation residual stresses have been released by heat treatment. Fig. 11 shows the KTS curves (solid lines) computed at the AE times as a function of the crack radius, c, and the resulting TS– KR curves (symbols) for the three materials (each data point corresponds to one specimen with an initial crack size c; this size varies from one specimen to another one). For the fine grained alumina, A3, a nearly constant value KR = 2.7 MPa m1/2 is observed for crack sizes \100 mm. The initial increase of the KR curve can be attributed to an underestimation of the measured small crack sizes because it is difficult to observe and determine the total length by optical microscopy. This leads to an important underestimation of the induced SIF as the KTS curves rise rapidly in the range of small crack sizes. A significant increase in the KR value with crack size is observed for both the bimodal (AH) and the homogeneous coarse grained (A25) aluminas. For the AH material, the KR value is :2.1 MPa m1/2 for c= 100 mm and approaches 3 MPa m1/2 for c= 350 mm. For the A25 material, KR increases from 2.5 to nearly 3.5 MPa m1/2 when the crack size varies from 100 to 450 mm. For both materials, the plateau toughness is not attained in the investigated range of crack sizes, indicating even higher toughness values may be achieved with extended crack growth.
mal shock degradation. A similar difference has been observed in the case of macro-crack R-curves in alumina [12] and it can be attributed to the variation with the temperature of the bridging parameters responsible for the R-curve behaviour. Another reason for this difference is the temperature dependence of the material properties taken into account in the thermal shock analysis. For the coarse grained material A25, the KR
5. Discussion A comparison of the KR curves under thermal shock conditions and those obtained from bending tests at ambient temperature can be seen in Fig. 12. The bending KR curves were determined by the indentation strength bending (ISB) method, using identical indented specimens than for the thermal shock tests. The results obtained under thermal loading are in agreement with those obtained from the mechanical tests: the fine grained material A3 has a nearly flat R-curve while the bimodal and the coarse grained materials (AH and A25, respectively) exhibit rising R-curve behaviour. However, the KR curves determined under thermal shock conditions are below those obtained from ISB tests at room temperature. This confirms that the room temperature KR curves are not suitable to predict ther-
Fig. 11. Family of KTS curves (lines) computed at AE times and resulting TS – KR curves (symbols) for the three aluminas.
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149
Fig. 12. Comparison between TS– KR curves (symbols) and the KR curves from ISB tests (lines). The dashed lines represent the KRi values at high temperatures.
curve under thermal loading is more rising than that obtained in bending tests. This may be attributed to the assumption of the constant semi-circular shape of the cracks which overestimates the KR values in bending tests [21]. Indeed, the cracks become semi-elliptical during the propagation and for the ISB tests, the variation of the crack geometry has been taken into account, using the effective geometrical coefficient proposed by Krause [22]. However, the KRi values obtained for notched bend specimens at about the temperatures of thermal shocks (800 and 900°C for the A25 and AH materials, respectively) are shown by the dashed lines in Fig. 12 (the KRi value was not measured for the A3 alumina). It can be seen that for small crack sizes, the initial TS –KR values are similar to the KRi ones. This implies that, for the onset of thermal shock induced crack propagation prediction, fracture mechanics analysis can be achieved by using a KRi value determined from bending test at a temperature representative of the thermal shock. Such analysis gives a good agreement between theoretical predictions and experimental observations for the A25 and AH materials and for other alumina and zirconia based ceramics [23].
To investigate the influence of the R-curve behaviour on the strength degradation by thermal shock, a set of non indented specimens (with dimensions 4×6 ×40 mm3) were thermally shocked at different applied temperature differences and their retained strength, sr, was measured in 4-point bending with a 35/10 mm span. The relative residual strength sr/s0, where s0 is the initial strength, is plotted as a function of the applied temperature difference in Fig. 13. For the three studied aluminas, a typical catastrophic strength loss can be observed. The strength is constant up to a critical applied temperature and it decreases in a very narrow temperature interval the width of which is B 15 K. The critical applied temperature differences are 85595, 8459 5 and 825915 K for the A3, A25 and AH aluminas, respectively and the relative catastrophic drops of the retained strength reaches 80, 45 and 62%, respectively. At DT = 1000 K, the relative loss of the strength is nearly 80% for the three materials. The significant strength degradation observed for the AH and A25 aluminas implies that, for these materials, the R-curves are not rising enough to induce a gradual decrease in retained strength with increasing tempera-
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Fig. 13. Relative retained strength as a function of the applied temperature difference.
ture. This can be seen in Fig. 14 which shows the envelope of the KTS curves representing the crack driving force for the critical thermal shock conditions (DTc = 845 K) for the A25 alumina, and the TS– KR curve obtained at DT = 840 K. It can be considered that this TS–KR curve corresponds to the critical condition according to the low difference between DT and DTc. The plot of the mean size of the natural flaws ( :65 mm) on the envelope curve gives a TS –KR value of : 2.1 MPa m1/2 (solid symbol in Fig. 14). Above this value, the TS–KR curve rises less than the envelope curve (dKR/dc BdKTS/dc). So at the onset of cracking, unstable crack propagation occurs, which leads to a large final crack size and thus to an important strength loss as for a constant toughness material. However, the final crack length decreases and the residual strength increases with increasing steepness of the R-curve as observed for the AH and A25 aluminas compared to the A3 alumina. If the steepness of the R-curve becomes sufficient, unstable crack growth disappears dur-
Fig. 14. Comparison between the envelope of the KTS curves (line) and the TS –KR curve (symbols) at DTc. The solid symbol denotes the initial value TS – KRi corresponding to the initiation of crack propagation from natural flaws.
ing thermal shock as it has been observed by Hoffmann et al. for Si3N4 based materials [24]. The influence of the microstructure on the strength degradation of thermally shocked alumina was previously studied by Gupta [25]. He observed a gradual decrease in the retained strength with increasing quenching temperature for a coarse grained alumina with a grain size of 85 mm. But in this case, the gradual strength degradation can be attributed to the presence of large flaws which exhibit only stable propagation rather than to a R-curve behaviour. Indeed, an estimation of the KTS curves, in the case of the water quench used by the author, indicates that the crack size above which the stable propagation occurs is :200 mm, which is certainly reached in the coarse grained material.
6. Conclusion A method has been developed to determine the Rcurve behaviour of short indentation-induced surface cracks under thermal shock conditions. The method is based on in situ measurements by acoustic emission of the time of unstable crack growth and fracture mechanics analysis in terms of stress intensity factors. The acoustic emission measurements allow qualitative analysis of the R-curve behaviour using an approach similar to the indentation strength bending method. The TS–KR curves obtained for alumina ceramics with different microstructures are compatible with bending results: the fine grained material (A3) shows a flat R-curve while the bimodal (AH) and the homogeneous coarse grained (A25) materials exhibit rising R-curve behaviour. However, the KR curves determined under thermal shock loading are below those resulting from indentation strength bending tests at room temperature, which can be attributed to the temperature effect on the material properties and the grain bridging parameters. But, the KR values obtained for notched bend specimens at the temperature of thermal shocks are similar to the thermal shock KR values. So, the fracture mechanics analysis leads to a good agreement between theoretical predictions and experimental data. Significant strength degradatations after thermal shock are observed for the three aluminas but the retained strength is higher for the AH and A25 aluminas which present a steeper R-curve than the A3 alumina. Concerning the crack arrest investigation, the TS–KR values must be determined for long crack sizes. For this purpose, future work will be focused on the detection of the stable propagation which occurs above the limiting value of the crack size imposed by the AE system.
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Acknowledgements The authors acknowledge their colleagues Drs J. Dubois and Y. Jorand for helpful discussions and assistance for materials processing.
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