Availableonlineatwww.sciencedirect.cor ° Science Direct CERAMICS INTERNATIONAL ELSEVIER Ceramics International 34(2008)1857-1865 www.elsevier.com/locate/ceramint Study of the resistance to crack propagation in alumina by acoustic emission S. Bouras . l. zerizer. f gheldane. M.t. bouazza. B. bouzabata Laboratoire des Materiaux Avances, Universite de Annaba, B P. 12, 23000 Annaba, algeria Received 7 March 2006: received in revised form 4 February 2007; accepted 29 June 2007 Available online 19 August 2007 Abstract The resistance to crack propagation or R-curve effect is still extensively studied in polycrystalline ceramic materials. This effect is shown in the literature by the increase of the stress intensity factor K versus crack extension, or by the decrease of crack velocity versus Ki under flexure tests. We have studied this fracture resistance by using the acoustic emission and the hertzian indentation technique. It was found that the acoustic emission count rate decreases, like crack velocity in conditions of sub-critical crack growth before the failure is reached. Two specimen of alumina, respectively with small and coarse grains, were tested. a high crack extension resistance is obtained for coarse-grained alumina Mechanisms of crack bridging and crack shielding are responsible for the R-curve effect. These mechanisms are also operating during cyclic loading. The resistance to crack propagation increases more and more after each cycle. Also we obtain R-curve effect during static loading On the other hand, toluene or silicone oil corrosive environment delays the crack propagation. But water medium has a reverse effect and the apparent toughness is reduced in this case. C 2007 Elsevier Ltd and Techna Group S r.l. All rights reserved. Keywords: Hertzian indentation; Acoustic emission; Crack extension; R-curve; Sub-critical growth; Alumina 1. Introduction roughness of the fracture surfaces permits physical contact between them as long as the crack does not open too much 1.I. Resistance to crack propagation: R-curve effect 1. 2. Sub-critical crack growth Ceramic brittleness is due to pre-existent micro-cracks which propagate unstably until the failure occurs. Som The sub-critical crack growth is one of the most frequent to stable crack propagation. In this case, the material toughness propagation modes in ceramics. It occurs under stress lower is no longer constant but increases with the crack extension than the critical stress for fracture. Slow crack growth is most The increase of fracture resistance with crack extension suitably described by a relation between the crack velocity V under stable crack growth is due to energy-consuming effects and the stress intensity factor K, in a KrV diagram(Fig. 1).In such as micro-cracking [ crack ramification, bridging by closely on K. For intermediate stresses or stage I, the grains (21, crack shielding by a damaged zone, plastic propagation continues with constant speed. At high stresses, the called r-curve behavior and is associated with the stress intensity factor KI: additional energy is necessary for a crack to depends on the chemical environment, on the loading mode 3] propagate until failure occurs, because of residual compressive and on the initial flaw size [4], but is independent of the stresses appearing in the crack wake and tending to close it. The specimen geometry The catastrophic fracture of a component occurs at the critical value Kic. Klo is a threshold limit [5] below which no crack growth appears. It permits us to define a safety domain for the specimen 2-8842/34.00@ 2007 Elsevier Ltd and Techna Group S.r.L. All rights reserved 10.1016 1-ceramint.2007.06015
Study of the resistance to crack propagation in alumina by acoustic emission S. Bouras *, I. Zerizer, F. Gheldane, M.T. Bouazza, B. Bouzabata Laboratoire des Mate´riaux Avance´s, Universite´ de Annaba, B.P. 12, 23000 Annaba, Algeria Received 7 March 2006; received in revised form 4 February 2007; accepted 29 June 2007 Available online 19 August 2007 Abstract The resistance to crack propagation or R-curve effect is still extensively studied in polycrystalline ceramic materials. This effect is shown in the literature by the increase of the stress intensity factor KI versus crack extension, or by the decrease of crack velocity versus KI under flexure tests. We have studied this fracture resistance by using the acoustic emission and the Hertzian indentation technique. It was found that the acoustic emission count rate decreases, like crack velocity in conditions of sub-critical crack growth before the failure is reached. Two specimen of alumina, respectively with small and coarse grains, were tested. A high crack extension resistance is obtained for coarse-grained alumina. Mechanisms of crack bridging and crack shielding are responsible for the R-curve effect. These mechanisms are also operating during cyclic loading. The resistance to crack propagation increases more and more after each cycle. Also we obtain R-curve effect during static loading. On the other hand, toluene or silicone oil corrosive environment delays the crack propagation. But water medium has a reverse effect and the apparent toughness is reduced in this case. # 2007 Elsevier Ltd and Techna Group S.r.l. All rights reserved. Keywords: Hertzian indentation; Acoustic emission; Crack extension; R-curve; Sub-critical growth; Alumina 1. Introduction 1.1. Resistance to crack propagation: R-curve effect Ceramic brittleness is due to pre-existent micro-cracks which propagate unstably until the failure occurs. Some technical ceramics exhibit a crack growth resistance that leads to stable crack propagation. In this case, the material toughness is no longer constant but increases with the crack extension. The increase of fracture resistance with crack extension under stable crack growth is due to energy-consuming effects such as micro-cracking [1], crack ramification, bridging by grains [2], crack shielding by a damaged zone, plastic deformation, or phase transformation. This phenomenon is called R-curve behavior and is associated with the stress intensity factor KI: additional energy is necessary for a crack to propagate until failure occurs, because of residual compressive stresses appearing in the crack wake and tending to close it. The roughness of the fracture surfaces permits physical contact between them as long as the crack does not open too much. 1.2. Sub-critical crack growth The sub-critical crack growth is one of the most frequent propagation modes in ceramics. It occurs under stress lower than the critical stress for fracture. Slow crack growth is most suitably described by a relation between the crack velocity V and the stress intensity factor KI in a KI–V diagram (Fig. 1). In stage I, at low applied stresses, the crack velocity depends closely on KI. For intermediate stresses or stage II, the propagation continues with constant speed. At high stresses, the velocity becomes again very sensitive to KI in the stage III. V depends on the chemical environment, on the loading mode [3] and on the initial flaw size [4], but is independent of the specimen geometry. The catastrophic fracture of a component occurs at the critical value KIC. KI0 is a threshold limit [5] below which no crack growth appears. It permits us to define a safety domain for the specimen. www.elsevier.com/locate/ceramint Ceramics International 34 (2008) 1857–1865 * Corresponding author. 0272-8842/$34.00 # 2007 Elsevier Ltd and Techna Group S.r.l. All rights reserved. doi:10.1016/j.ceramint.2007.06.015
l858 S. Bouras et al./Ceramics Intemational 34(2008)1857-1865 Fig. 1. Crack velocity as a function of applied stress intensity factor (Krv diagram). In the stage I, the growth rate is controlled by a corrosion K,(MPam) reaction at the crack tip. The variation of v versus KI Fig. 2. Sub-critical crack growth under constant load with natural cracks or a described by the following law: macro-crack 191 V=AK ( where Kappl=oYais applied by external load and AKI is the contribution of the reinforcement mechanisms The R-curve where A and n are constants that characterize the material and effect compensates the effect of the sub-critical growth. This its environment respectively has been shown on alumina and zirconia by fett and munz In the stage Il, the growth rate is controlled by the diffusion [10, 11)using flexure tests under constant load. The authors of corrosive species towards the crack tip [6,7] reported the KrV diagram(Fig. 2)that shows a minimum In the stage Ill, governed also by a similar law as relation(1), velocity when K,increases he driving force is higher and the failure is imminent This Observations of brittle crystalline solids in the transmission stage is due to different mechanisms: thermally activated electron microscope by Lawnet al [14], Hockey and Wiederhom growth, growth controlled by a dislocation process, etc. [7, 8]. [15], Hockey and Lawn [16] have revealed inter-facial misfit Different sub-critical growth behaviors due to an R-curve dislocations which form when crack interfaces rebond effect have been observed in the case of a macro-crack imperfect registry, but no evidence for crack tip plasticity. It is (initiated from a notch in bending) or of natural flaws(pores, this absence of plasticity which explains why the static fatigue inclusions, etc 3,]. This R-curve effect, with a decrease limit of glasses and ceramics corresponds to equilibrium cracks in velocity, is well observed at the start of the macro-crack and to the true Griffith criterion growth and tends towards saturation(Fig. 2). When the driving force becomes closer to KIc and the propagation rate quickly 1.3. Acoustic emission increases. the behavior becomes similar to that of natural cracks The acoustic emission(A E )designates all manifestation of The r-curve effect in alumina was studied by Fett and Munz acoustic waves whose source is within a material undergoing a [10-12]. The KrV curves for macro-cracks show an initial structural modification induced by an applied mechanical load decrease followed by an increase of velocity up to a When an irreversible energy release occurs, a part is catastrophic failure. A similar behavior has been observed transformed emitted stress wave that can be detected by Steinbrech [13]. The driving force at the sharp crack tip [14], with an appropriate sensor from th In the polycrystalline ceramic materials, the sub-critical stress intensity factor"Kappl, by the stress shielding effect due crack growth, essentially intergranular, is one of the sources of to the crack surface interaction In the coarse-grained alumina A.E. Because of the microstructure (rigid grain, porosity studied by the authors(grain size is about 20 um), the shielding inclusions, etc. the crack tip is irregular and the growth occurs process is essentially bridging of the crack surfaces by by sequential steps along the tip. An A E event corresponds at each step. The A E. method must be highly sensitive and the The effective stress intensity factor at the crack tip, Ktip, may sensor, even distant from the source, detects the source event be written as Therefore, it is not necessary to localize the source The interest is to acquire data about emissive events and to YV-ldde (2) relate them to their origin mechanisms
In the stage I, the growth rate is controlled by a corrosion reaction at the crack tip. The variation of V versus KI is described by the following law: V ¼ AKn I (1) where A and n are constants that characterize the material and its environment respectively. In the stage II, the growth rate is controlled by the diffusion of corrosive species towards the crack tip [6,7]. In the stage III, governed also by a similar law as relation (1), the driving force is higher and the failure is imminent. This stage is due to different mechanisms: thermally activated growth, growth controlled by a dislocation process, etc. [7,8]. Different sub-critical growth behaviors due to an R-curve effect have been observed in the case of a macro-crack (initiated from a notch in bending) or of natural flaws (pores, inclusions, etc.) [3,9–11]. This R-curve effect, with a decrease in velocity, is well observed at the start of the macro-crack growth and tends towards saturation (Fig. 2). When the driving force becomes closer to KIC and the propagation rate quickly increases, the behavior becomes similar to that of natural cracks. The R-curve effect in alumina was studied by Fett and Munz [10–12]. The KI–V curves for macro-cracks show an initial decrease followed by an increase of velocity up to a catastrophic failure. A similar behavior has been observed by Steinbrech [13]. The driving force at the sharp crack tip [14], ‘‘tip stress intensity factor’’ Ktip, differs from the ‘‘applied stress intensity factor’’ Kappl, by the stress shielding effect due to the crack surface interaction. In the coarse-grained alumina studied by the authors (grain size is about 20 mm), the shielding process is essentially bridging of the crack surfaces by interlocking grains. The effective stress intensity factor at the crack tip, Ktip, may be written as Ktip ¼ Kappl DKI (2) where Kappl = sYa1/2 is applied by external load and DKI is the contribution of the reinforcement mechanisms. The R-curve effect compensates the effect of the sub-critical growth. This has been shown on alumina and zirconia by Fett and Munz [10,11] using flexure tests under constant load. The authors reported the KI–V diagram (Fig. 2) that shows a minimum in velocity when KI increases. Observations of brittle crystalline solids in the transmission electron microscope by Lawn et al.[14], Hockey andWiederhorn [15], Hockey and Lawn [16] have revealed inter-facial misfit dislocations which form when crack interfaces rebond in imperfect registry, but no evidence for crack tip plasticity. It is this absence of plasticity which explains why the static fatigue limit of glasses and ceramics corresponds to equilibrium cracks and to the true Griffith criterion. 1.3. Acoustic emission The acoustic emission (A.E.) designates all manifestation of acoustic waves whose source is within a material undergoing a structural modification induced by an applied mechanical load. When an irreversible energy release occurs, a part is transformed in an emitted stress wave that can be detected with an appropriate sensor. In the polycrystalline ceramic materials, the sub-critical crack growth, essentially intergranular, is one of the sources of A.E. Because of the microstructure (rigid grain, porosity inclusions, etc.) the crack tip is irregular and the growth occurs by sequential steps along the tip. An A.E. event corresponds at each step. The A.E. method must be highly sensitive and the sensor, even distant from the source, detects the source event. Therefore, it is not necessary to localize the source. The interest is to acquire data about emissive events and to relate them to their origin mechanisms. Fig. 1. Crack velocity as a function of applied stress intensity factor (KI–V diagram). Fig. 2. Sub-critical crack growth under constant load with natural cracks or a macro-crack [9]. 1858 S. Bouras et al. / Ceramics International 34 (2008) 1857–1865
S. Bouras et al. /Ceramics Intemational 34(2008)1857-1865 859 200 > lV Time(s) 200400600800100012001400160018002000 uL「Luu Fig. 3. Burst type signal: amplitude crest to crest. When the source liberates an energy pulse, the correspond ing signal of A E is of burst type(Fig 3). This signal is easily Fig 4. Numbering of arks of an acoustic emission salvo nguished from the ground noise because of its high amplitude. The signal contains some information concerning is loaded by a spherical indenter of radius R, a circular crack of the source phenomenon. radius ro, forms a minimum critical load Pcro is slightly Crack propagation is the result of successive jumps of superior to the radius a of the contact zone between the ball and localized parts of the crack front. Each crack jump(an ark) is the material. As the load increases, the crack propagates both at the source of acoustic emission. A group of arks represents a the surface over a circle around the contact zone, and in depth as crack that emits an event(a salvo). Each salvo is represented by a conical shape crack of length c [20]. " See Electronic Annex 1 an acoustic emission peak in the curves amplitude-time. The in the online version of this article". This method needs the events count rate, dN/dt, is a correct image of the average crack presence of natural initial flaws at the surface of the sample growth rate. It is related to the stress intensity factor by a similar Upon attaining a critical""Griffith configuration", a favourably equation as relation(1). located flaw runs around the contact to form a surface ring Previous works [17, 18] have already shown that the events crack. The faws, introduced by polishing, are Griffith elliptic count rate exhibits the same functional dependence on K as defects with a front curve and become sharp crack after a does the slow crack velocity. Evans and Linzer [17]showed that lifetime [21-24 the relationship between the events count rate dN/dt and the Using spheres, flat punches and peeling, Maugis and time-to-failure can be effectively used for the failure prediction. Barquins [25] and Maugis [26] studied the equilibrium, healin However, this prediction needs a well-characterized material and the propagation of sub-critical Griffith cracks according to and a well-defined low-level emission. Khuri et al. [ 19] used the loss of the elasticity at the crack tip in elastic solids, glasses acoustic surface wave techniques on polished ceramics for and ceramics. The edge of the contact area can be considered as detecting and characterizing surface cracks and also to predict an interface crack tip in mode I such that the system is the same failure. Our experimental conditions of A E are close to those for a notched solid with an imposed crack path Maugis has also used by Evans and Linzer. In fact, the time-to-failure can be widely evoked the Griffith criterion at the crack tip local measured by AE In the present work, the acoustic emission is detected by a Bruel Kjaer piezoelectric transducer of frequency range 200- band filter >200 kHz, gain 40 dB), and an amplifier(variable gain from 0 to 50 dB with a filter of 200 kHz to 2 MHz)are used. After amplification, the signal is memorised in a home made counter. The number of times that it overpasses a given fixed threshold V* above the noise level (Fig. 4), is recorded as counts N. The counts can be cumulative(cumulative counts: M) Y initial crack or initialized at regular intervals(count rate: dN/dt: number of arks by unit of time) 14. Hertzian indentation In our work, the A E technique is used to follow the changes in cracks propagation dynamics in a Hertzian indentation loading. The Hertzian indentation(Fig. 5) is a mechanical Fig. 5 ics of the Hertzian contact: a contact radius; ro crack radiu haracterization method. When a brittle material plane surface the surface; c crack length in depth; R ball radius; P indentation load
When the source liberates an energy pulse, the corresponding signal of A.E. is of burst type (Fig. 3). This signal is easily distinguished from the ground noise because of its high amplitude. The signal contains some information concerning the source phenomenon. Crack propagation is the result of successive jumps of localized parts of the crack front. Each crack jump (an ark) is the source of acoustic emission. A group of arks represents a crack that emits an event (a salvo). Each salvo is represented by an acoustic emission peak in the curves amplitude-time. The events count rate, dN/dt, is a correct image of the average crack growth rate. It is related to the stress intensity factor by a similar equation as relation (1). Previous works [17,18] have already shown that the events count rate exhibits the same functional dependence on K1 as does the slow crack velocity. Evans and Linzer [17] showed that the relationship between the events count rate dN/dt and the time-to-failure can be effectively used for the failure prediction. However, this prediction needs a well-characterized material and a well-defined low-level emission. Khuri et al. [19] used acoustic surface wave techniques on polished ceramics for detecting and characterizing surface cracks and also to predict failure. Our experimental conditions of A.E. are close to those used by Evans and Linzer. In fact, the time-to-failure can be measured by A.E. In the present work, the acoustic emission is detected by a Bru¨el & Kjær piezoelectric transducer of frequency range 200– 300 kHz with a 2–3 mV ground noise. A preamplifier (passband filter >200 kHz, gain 40 dB), and an amplifier (variable gain from 0 to 50 dB with a filter of 200 kHz to 2 MHz) are used. After amplification, the signal is memorised in a home made counter. The number of times that it overpasses a given fixed threshold V* above the noise level (Fig. 4), is recorded as counts N. The counts can be cumulative (cumulative counts: N) or initialized at regular intervals (count rate: dN/dt: number of arks by unit of time). 1.4. Hertzian indentation In our work, the A.E. technique is used to follow the changes in cracks propagation dynamics in a Hertzian indentation loading. The Hertzian indentation (Fig. 5) is a mechanical characterization method. When a brittle material plane surface is loaded by a spherical indenter of radius R, a circular crack of radius r0, forms a minimum critical load PCr0 is slightly superior to the radius a of the contact zone between the ball and the material. As the load increases, the crack propagates both at the surface over a circle around the contact zone, and in depth as a conical shape crack of length c [20]. ‘‘See Electronic Annex 1 in the online version of this article’’. This method needs the presence of natural initial flaws at the surface of the sample. Upon attaining a critical ‘‘Griffith configuration’’, a favourably located flaw runs around the contact to form a surface ring crack. The flaws, introduced by polishing, are Griffith elliptic defects with a front curve and become sharp crack after a lifetime [21–24]. Using spheres, flat punches and peeling, Maugis and Barquins [25] and Maugis [26] studied the equilibrium, healing and the propagation of sub-critical Griffith cracks according to the loss of the elasticity at the crack tip in elastic solids, glasses and ceramics. The edge of the contact area can be considered as an interface crack tip in mode I such that the system is the same for a notched solid with an imposed crack path. Maugis has also widely evoked the Griffith criterion at the crack tip local conditions. Fig. 3. Burst type signal: amplitude crest to crest. Fig. 4. Numbering of arks of an acoustic emission salvo. Fig. 5. Schematics of the Hertzian contact: a contact radius; r0 crack radius at the surface; c crack length in depth; R ball radius; P indentation load. S. Bouras et al. / Ceramics International 34 (2008) 1857–1865 1859
The mechanism of cone crack initiation and propagation is Physique et de Physique des Materiaux)laboratory ell understood by using the Hertz-Huber stress tensor and from an alumina powder(99.9% purity). Sintered at 1550C fracture mechanics [24, 27-30). Sneddon [31] has given the under an applied uniaxial load of 40 MPa, the resulting stress field grain size is between 2 and 10 um. The second material In brittle materials, the Hertzian crack is catastrophically (Fig. 6b), an industrial alumina provided by Desmarquet produced in a very short time. However, in materials exhibing Ceramiques et Techniques(code AF997,99.8%purity),has an R-curve effect, the crack occurs progressively as loading been sintered at 1600C ending in coarse grains with sizes proceeds: small cracks are initiated from random flaws to form between 5 and 20 um. relatively deep circle arc that coalesce in a circular crack at the urface, and in a conical crack in depth 32] The critical value Pc corresponding to the final unstable crack extension is determined by the A.E. event of the highe amplitude. The crack jumps occurring before reaching P( oduce acoustic emission events of small amplitudes The A.E. monitoring is then a tool for following the resistance to crack propagation step by step. It can also be used to determine the fracture surface energy and the toughness by using classical theories of the Hertzian indentation [29, 30, 33-41 Initial flaws of a chosen size are introduced by using a polishing paper of a specific abrasion grit size[21,32]. The balls used in the present work have the following characteristics emented carbide 94WC-6Co, density: 15 g/cm, hardness 1550 Hv, E=630 GPa, U=0.22, radius from I to 14 mm. They have been provided by CIMAP-France 2. Description of the materials We studied two alumina ceramics. The first material has been processed by G.E. MPP.M( Groupe d' Etudes de Metallurgie y200 Fig. 7. Stages of a circular fissure formation. Laboratory alumina(small Fig. 6. Microstructure of alumina:(a) elaborated in laboratory (small grains) grains); grit size 3 um: (a) Pmax=700N, R=4 mm; (b)Pmax=950N, and(b) industrial alumina(coarse grains R=8mm:(c)Pmax= 1350N, R=3 mm
The mechanism of cone crack initiation and propagation is well understood by using the Hertz–Huber stress tensor and fracture mechanics [24,27–30]. Sneddon [31] has given the stress field. In brittle materials, the Hertzian crack is catastrophically produced in a very short time. However, in materials exhibing an R-curve effect, the crack occurs progressively as loading proceeds: small cracks are initiated from random flaws to form relatively deep circle arc that coalesce in a circular crack at the surface, and in a conical crack in depth [32]. The critical value PC corresponding to the final unstable crack extension is determined by the A.E. event of the highest amplitude. The crack jumps occurring before reaching PC produce acoustic emission events of small amplitudes. The A.E. monitoring is then a tool for following the resistance to crack propagation step by step. It can also be used to determine the fracture surface energy and the toughness by using classical theories of the Hertzian indentation [29,30,33–41]. Initial flaws of a chosen size are introduced by using a polishing paper of a specific abrasion grit size [21,32]. The balls used in the present work have the following characteristics: cemented carbide 94WC–6Co, density: 15 g/cm3 , hardness: 1550 Hv, E = 630 GPa, v ¼ 0:22, radius from 1 to 14 mm. They have been provided by CIMAP-France. 2. Description of the materials We studied two alumina ceramics. The first material has been processed by G.E.M.P.P.M (Groupe d’Etudes de Me´tallurgie Physique et de Physique des Mate´riaux) laboratory (Fig. 6a) from an alumina powder (99.9% purity). Sintered at 1550 8C under an applied uniaxial load of 40 MPa, the resulting grain size is between 2 and 10 mm. The second material (Fig. 6b), an industrial alumina provided by Desmarquet Ce´ramiques et Techniques (code AF997, 99.8% purity), has been sintered at 1600 8C ending in coarse grains with sizes between 5 and 20 mm. Fig. 6. Microstructure of alumina: (a) elaborated in laboratory (small grains) and (b) industrial alumina (coarse grains). Fig. 7. Stages of a circular fissure formation. Laboratory alumina (small grains); grit size 3 mm: (a) Pmax = 700 N, R = 4 mm; (b) Pmax = 950 N, R = 8 mm; (c) Pmax = 1350 N, R = 3 mm. 1860 S. Bouras et al. / Ceramics International 34 (2008) 1857–1865
S. Bouras et al. /Ceramics intemational 34(2008)1857-1865 3. Results The acoustic activity recorded before reaching Pc(Fig 9) similar to the A.E. activity shown on the left of the higher event Under Hertzian indentation loading, the circular and conical observed in Fig. 8. These acoustic signals obtained during racks do not occur immediately. They are formed gradually by ding below the critical load are linked to sub-critical crack progression and propagation of the initial crack(Fig. 7) until propagation. In fact, this A.E. activity has already been shown the formation of a complete circle whichever the used loading by Nadeau [42] as caused by the interaction between the node(monotonous, static, or cyclic). This effect is due to an moving crack and defects on surfa increase of the crack propagation resistance(R-curve effect When a crack is initiated at a favourable flaw, it propa Fig. 7 shows the various steps of formation of a circular along a certain proportion of the length of a circle of radius ro fissure resulting from loading up to the loads of 700 N(Fig. 7a), (partial circular crack), at the surface. Its radius does not change 950N(Fig. 7b), and 1350N(Fig. 7c) followed by an with the increase of the load And by further loading, the partial unloading. The AE activity for the load of 1450n higher circular crack will grow by following the same path of radius ro. than Pc is given in Fig. 8. The crest amplitude value of the in conformity with the Hertzian stress field highest A.E. peak corresponding to the critical load Pc, given The number n of a.e. events increases under the effect of by the relation dB=20 logIo V+ 60 [32], where V is the the increasing applied load, first slowly, then quickly as the recorded voltage, is 57 dB. This event peak recorded at 580 s unstable propagation and the failure become imminent corresponds to the closing of the partial circle as observed (Fig. 10). And below Pc, the events count rate decreases ( Fig. 7c) because of small A.E. activity(Fig. 11)revealing an increase of the R-curve effect. This change in events count rate qualitatively similar to the variation of crack velocity if one assumes that each AE to a step of a crack 1200 900 0 0.3 20040060080010001200 Time(s) Fig 8. Acoustic emission during loading up to a load Pmax(1450 N)higher than Fig. 10. Typical A.E. events count as a function of load(Pmax=750N) Pc. Laboratory alumina(small grains); grit Laboratory alumina(small grains); grit size 3 um; R=3 mm. Alumina with small grains 10001200 Time(s) Laboratory alumina(small grains): grit size 3 um; R=3mm.wer than Pc. Fig. 11. Events count rate until a load below the critical load Pc Fig. 9. Acoustic emission for loading up to a load of 820N lowe (Pmax=750N
3. Results Under Hertzian indentation loading, the circular and conical cracks do not occur immediately. They are formed gradually by progression and propagation of the initial crack (Fig. 7) until the formation of a complete circle whichever the used loading mode (monotonous, static, or cyclic). This effect is due to an increase of the crack propagation resistance (R-curve effect). Fig. 7 shows the various steps of formation of a circular fissure resulting from loading up to the loads of 700 N (Fig. 7a), 950 N (Fig. 7b), and 1350 N (Fig. 7c) followed by an unloading. The AE activity for the load of 1450 N higher than PC is given in Fig. 8. The crest amplitude value of the highest A.E. peak corresponding to the critical load PC, given by the relation dB = 20 log10 V + 60 [32], where V is the recorded voltage, is 57 dB. This event peak recorded at 580 s corresponds to the closing of the partial circle as observed (Fig. 7c). The acoustic activity recorded before reaching PC (Fig. 9) is similar to the A.E. activity shown on the left of the higher event observed in Fig. 8. These acoustic signals obtained during loading below the critical load are linked to sub-critical crack propagation. In fact, this A.E. activity has already been shown by Nadeau [42] as caused by the interaction between the moving crack and defects on surface. When a crack is initiated at a favourable flaw, it propagates along a certain proportion of the length of a circle of radius r0 (partial circular crack), at the surface. Its radius does not change with the increase of the load. And by further loading, the partial circular crack will grow by following the same path of radius r0, in conformity with the Hertzian stress field. The number N of A.E. events increases under the effect of the increasing applied load, first slowly, then quickly as the unstable propagation and the failure become imminent (Fig. 10). And below PC, the events count rate decreases because of small A.E. activity (Fig. 11) revealing an increase of the R-curve effect. This change in events count rate is qualitatively similar to the variation of crack velocity if one assumes that each A.E. event corresponds to a step of a crack Fig. 8. Acoustic emission during loading up to a load Pmax (1450 N) higher than PC. Laboratory alumina (small grains); grit size 3 mm; R = 4 mm. Fig. 9. Acoustic emission for loading up to a load of 820 N lower than PC. Laboratory alumina (small grains); grit size 3 mm; R = 3 mm. Fig. 10. Typical A.E. events count as a function of load (Pmax = 750 N). Laboratory alumina (small grains); grit size 3 mm; R = 3 mm. Fig. 11. Events count rate until a load below the critical load PC (Pmax = 750 N). S. Bouras et al. / Ceramics International 34 (2008) 1857–1865 1861
S. Bouras et al./Ceramics Intemational 34(2008)1857-1865 . Alumina with coarse grains 400 A-Alumina with small grains 450 命命命自命 80010001200 0100200300400500600700 Time(s Fig. 12. Events count rate until a load close to the critical load Pc Fig. 14. A.E. Activity under cyclic loading. Laboratory alumina(small grains): increment. Besides, the amplitude of the A.E. peaks decreases (Figs. 8 and 9)indicating that the crack increment steps become smaller. Above Pc, as the load becomes closer to the unstable -A cycle 3 propagation condition, a very fast rise in count rate is observed (Fig. 12)with peaks of higher amplitudes As observed in Fig. Il, the resistance to crack propagation is higher in coarse-grained alumina. This figure points out also that the starting of fracture occurs earlier for coarse-grained alumina. but because of the r-curve effect. the final fracture 000000 o cycle 4 will occur later than in the case of fine-grained alumina as Fig. 12 shows. However, the R-curve effect in coarse-grained alumina is often depending on the material microstructure type nd on testing conditions [43-46. Bradt and Scott [43] showed also that the effect of the reinforcement is important as the grain size increases. But, for a lower grain size of 2 or 3 um, the R curve is flat and independent of the crack length. Fig. 15. Detail of events count rate during each of the cycles shown in Fig. 14 We have also studied the R-curve effect under a constant applied load at about PN0.80 Pc(reached with a very fast We also carried out successive cycles of loading-unloading g) as it is shown in ite, as well as the amplitude of the signals, decrease from the zones(crack shielding) that are taken in consideration by the resistance to crack propagation following cycle. The last cycle continues until the failure of the Before this critical event the count rate decreases rapidly during each cycle as Fig. 15 shows, indicating 0.8 again an increase in crack growth resistance associated with more efficient crack shielding. At the approach of the fracture (cycle 4), the events count rate increases: the crack reaches a 号0 critical length provoking the failure 乙 The resistance to crack propagation was also studied by changing the medium (water, toluene, silicone oil). The environment modifies the crack behavior and shows clearly a difference in apparent resistance to crack propagation therefore in apparent toughness. The tests are carried out at Pc. As it is seen in Fig. 16, the presence of the toluene or the silicone oil delays the unstable event and requires a higher critical load than the one required in air. The presence of water has an inverse role and provokes the critical event earlier. During loading under 10001200 aggressive medium like water, the size of the initial flaw increases by sub-critical crack growth and becomes unstable 13.AEactivity under constant load as a function of time. Laboratory with a lower Pc value. So, the apparent toughness is reduced in na(small grains); grit size 3 um; R=3mm
increment. Besides, the amplitude of the A.E. peaks decreases (Figs. 8 and 9) indicating that the crack increment steps become smaller. Above PC, as the load becomes closer to the unstable propagation condition, a very fast rise in count rate is observed (Fig. 12) with peaks of higher amplitudes. As observed in Fig. 11, the resistance to crack propagation is higher in coarse-grained alumina. This figure points out also that the starting of fracture occurs earlier for coarse-grained alumina. But, because of the R-curve effect, the final fracture will occur later than in the case of fine-grained alumina as Fig. 12 shows. However, the R-curve effect in coarse-grained alumina is often depending on the material microstructure type and on testing conditions [43–46]. Bradt and Scott [43] showed also that the effect of the reinforcement is important as the grain size increases. But, for a lower grain size of 2 or 3 mm, the Rcurve is flat and independent of the crack length. We have also studied the R-curve effect under a constant applied load at about P 0.80 PC (reached with a very fast loading) as it is shown in Fig. 13. As depicted, the events count rate, as well as the amplitude of the signals, decrease from the resistance to crack propagation. We also carried out successive cycles of loading–unloading (Fig. 14). After every cycle, the structure presents damaged zones (crack shielding) that are taken in consideration by the following cycle. The last cycle continues until the failure of the sample occurs. Before this critical event, the count rate decreases rapidly during each cycle as Fig. 15 shows, indicating again an increase in crack growth resistance associated with more efficient crack shielding. At the approach of the fracture (cycle 4), the events count rate increases: the crack reaches a critical length provoking the failure. The resistance to crack propagation was also studied by changing the medium (water, toluene, silicone oil). The environment modifies the crack behavior and shows clearly a difference in apparent resistance to crack propagation therefore in apparent toughness. The tests are carried out at PC. As it is seen in Fig. 16, the presence of the toluene or the silicone oil delays the unstable event and requires a higher critical load than the one required in air. The presence of water has an inverse role and provokes the critical event earlier. During loading under aggressive medium like water, the size of the initial flaw increases by sub-critical crack growth and becomes unstable with a lower PC value. So, the apparent toughness is reduced in that case. Fig. 12. Events count rate until a load close to the critical load PC (Pmax = 1000 N). Fig. 13. A.E. activity under constant load as a function of time. Laboratory alumina (small grains); grit size 3 mm; R = 3 mm. Fig. 14. A.E. Activity under cyclic loading. Laboratory alumina (small grains); grit size: 3 mm; R = 3 mm. Fig. 15. Detail of events count rate during each of the cycles shown in Fig. 14. 1862 S. Bouras et al. / Ceramics International 34 (2008) 1857–1865
S. Bouras et al. /Ceramics intemational 34(2008)1857-1865 2500 乙 Initial crack length(um) Fig. 18. Variation of the critical load Pc as a function of initial flaw size hertzian critical load as a function of ball radius for different environments. Laboratory alumina(small grains): grit The effect of slow crack growth, below Pc, is depicted in Fig. 17. It shows schematically, in different mediums, the bridging bridging driving force g as a function of the initial faw size c for applied loads. In the absence of sub-critical crack the initial flaw size is c: and the critical load for an jump is Pci. The unstable jump occurs when the driving force equals the crack growth resistance (G=2 yf= Gc). After this jump, the growth becomes stable, needing an increase of loading for failure. In the case of sub-critical crack growth with water environment, the same flaw of size ci begins to grow in a sub-critical manner under the driving force G4(Pc>Pe, >Pe>Pc, . CCcs Fig. 20. Photo of microstructure showing damaged zones. Laboratory alumina Fig. 17. Schematics of the driving force vs. the crack length 3 um; R-3 mm; Pmax= 1800N
The effect of slow crack growth, below PC, is depicted in Fig. 17. It shows schematically, in different mediums, the driving force G as a function of the initial flaw size c for different applied loads. In the absence of sub-critical crack growth, the initial flaw size is ci and the critical load for an unstable jump is PCi. The unstable jump occurs when the driving force equals the crack growth resistance (G = 2 gf = GC). After this jump, the growth becomes stable, needing an increase of loading for failure. In the case of sub-critical crack growth with water environment, the same flaw of size ci begins to grow in a sub-critical manner under the driving force G4 (<GC) corresponding to the load PC4 (<PCi). So the flaw will grow from ci up to the size c4 where it becomes unstable. Water is more corrosive and the critical driving force is consequently weaker than in the other environment cases as in air, silicone oil and toluene. This diagram also shows the influence of the initial flaw size. Smaller ci implies higher critical loads PC as it is seen in Fig. 18. Table 1 shows the critical crack growth energy gf (or GC) and the apparent fracture toughness KIC values calculated from the Hertzian indentation theory. All gf values are higher for coarse-grained alumina. This grain size effect suggests a crack shielding mechanism involving the bridging of the crack surfaces by sliding grains. The surface crack observations suggest that the R-curve effect is due to crack bridging by grains (Fig. 19) and to the crack shielding by damaged zones (Fig. 20). In any case, the Hertzian indentation provokes a lot of damages, due to the Fig. 16. Hertzian critical load as a function of ball radius for different environments. Laboratory alumina (small grains); grit size 3 mm. Fig. 17. Schematics of the driving force vs. the crack length. Fig. 18. Variation of the critical load PC as a function of initial flaw size. Fig. 19. Vectorisation of the original photo (at the bottom on the left) showing crack bridging sites. industrial alumina 3 mm; R = 4 mm; Pmax = 1600 N. Fig. 20. Photo of microstructure showing damaged zones. Laboratory alumina 3 mm; R = 3 mm; Pmax = 1800 N. S. Bouras et al. / Ceramics International 34 (2008) 1857–1865 1863
S. Bouras et al./Ceramics Intemational 34(2008)1857-1865 Table Results of fracture surface energy (nf) and apparent toughness(Kic) Environment Kic apparent(MPa m) parse grain alumina Small grain alumina Coarse grain alumina Small grain alumina Toluene 43. 5.8 40.2 27.1 Water 20.5 4 4.0 contact with the ball. Most of these damaged zones are the [3] BJ. Pletka, S M. Wiederhom, A comparison of failure predictions by cause of crack slackening trength and fracture mechanics techniques, J Mater. Sci. 17(1982)1247- Comparatively to flexure tests(three-points bending)(45- 47] where, at the beginning of the crack propagation, the [4] A Okada, N. Hirosaki, Subcritical crack growth in sintered silicon nitride exhibiting a rising R-curve, J Am Ceram Soc. 73(7)(1990)2095-2096 bridging interactions between the crack surfaces occur in a zone [5] B R Lawn, Fracture of Brittle Solids, Cambridge University Press, 1993 that spreads over the whole of crack length. When this zone [6] R.J. Charles, Dissolution behavior of macroporous glass, J Am Ceram ncreases, the driving force at the crack tip decreases and Soc.47(1964)154155 consequently an apparent crack growth resistance appears. So, [71 J.R.J.B. Wachtman, Highlights of progress in the science of fracture of ceramics and glass, J. Am. Ceram. Soc. 57(12)(1974)509-519 for a given crack increment, the bridging zone becomes [8]EF. Lange, In Deformation of Ceramic Materials, Pensylvania State stationary and translates in the wake of the advancing crack tip University, Plenum Press, 1975. Pp. 361-381 Then the crack growth resistance becomes constant. [9 T. Fett. D. Munz, Evaluation of subcritical crack extension under constant loading, J. Eur. Ceram. Soc. 6(1990)67-72 [10] T. Fett, D. Munz, Subcritical crack growth of macrocracks in alumina with 4. Conclusions R-curve behavior, J. Am. Ceram Soc. 75 (4)(1992)958-963 [11 T. Fett, D. Munz, Subcritical crack growth of macrocracks in zirconia, J The resistance to crack propagation under Hertzian Mater.Sci.Let.10(1991)1103-1106 indentation was studied on coarse and fine-grained alumina. 2 T. Fett, D. Munz, Subcritical crack growth of macro-and microcracks in With the help of acoustic emission, this mechanical character Plenum Press, New York, 1992, P 604 ization method permits easy detection of the R-curve effect. [13] R.W. Steinbrech, R-curve behavior of ceramics,. in: RC.Bradt, et al The decrease of the events count rate prior to failure, illustrates (Eds ) Fracture Mechanics of Ceramics. 9. Plenum Press, New York. the presence of a sub-critical crack growth mechanism. This 1992.pp.187-208. effect is observed under monotonous, static or cyclic loadings 14] B.R. Lawn, B.J. Hockey, S.M. Wiederhorn, Atomi brittle solids: an electron microscopy study, J. Mater. Sci. 15(5)(1980) It is also found that a coarser grain has high resistance crack 1207-1223. propagation. 15] B.J. Hockey, S.M. Wiederhom, Crack healing in Environments influence crack propagation by stress corro- Bradt, et al.(Eds ) Fracture Mechanics of Ceramics. 6. Plenum Press sion at the crack tip Calculated apparent toughness for four New York, 1983, p. 63 environments leads to classify them regarding the increasing 16] BJ. Hockey, B.R. Lawn, Electron microscopy of microcracking about dentationsin aluminum oxide and silicon carbide. J. Mater. Sci. 10(8) corrosion effect as: toluene. silicone oil. air and water. (1975)1275-1284 The R-curve effect results are in agreement with those [17 A.G. Evans, M Linzer, Failure prediction in structural ceramics using obtained by flexure tests. acoustic emission, J. Am. Ceram Soc. 56(11)(1973)575-581 18] A.G. Evans, M. Linzer, L R. Russel, Acoustic emission and crack pro- pagation in polycrystalline alumina, Mater Sci Eng. 15(1974)253-261 Acknowledgments [19 B.T. Khuri-Yakub, A G. Evans, G.S. Kino, Acoustic surface wave mea ments of surface cracks in ceramics, J. Am. Ceram. Soc. 63(1/2) The authors thank the ceramics group(GEMPPM-INSA (1980)65-71 Lyons) for their help [20] S. Bouras, D. Rouby, G I' Emission Acoustique dans I' Etude de I'Indentation Hertzienne, Revue des Composites et des Materiaux Avances 5(2)(1995)235-252 Appendix A. Supplementary data 1] F.B. Langitan, B.R. Lawn, Hertzian fracture experiments on abraded glass surfaces as definitive evidence for an balance explanation of Auerbach's Law, J. Appl 40(10)(1969)4009-4017 Supplementary data associated with this article can be found 22] B.R. Lawn, D B. Marshall, Indentation fracture and strength degradation in the online version, at doi: 10. 1016/j. ceramint 2007.06.015 in ceramics, in: Proceedings of the International Symposium on Fracture Mechanics of Ceramics, University Park, PA, USA, July 27-29, 1977. (1978),Pp.205-229 References [23] T.R. Wilshaw, The Hertzian fracture test, J. Phys. D. Appl. Phys. 4(1971) 1567-1581 [1] A.G. Evans, K.T. Faber, Crack growth resistance of microcracking brittle [24] B R. Lawn, Partial cone crack formation in a brittle material loaded with a materials. J. Am. Ceram. Soc. 67(4)(1984)255-260 g spherical [2] M.V. Swain, R-curve behavior in a polycrystalline alumina material, J. [25] D. Maugis, M. rmR Soc.A299(1967)307-316. mechanics and the adherence of Mater.Sci.Lett.5(4)(1986)1313-1315 D.:Appl.Phys.l1(1978)1989-2023
contact with the ball. Most of these damaged zones are the cause of crack slackening. Comparatively to flexure tests (three-points bending) [45– 47] where, at the beginning of the crack propagation, the bridging interactions between the crack surfaces occur in a zone that spreads over the whole of crack length. When this zone increases, the driving force at the crack tip decreases and consequently an apparent crack growth resistance appears. So, for a given crack increment, the bridging zone becomes stationary and translates in the wake of the advancing crack tip. Then the crack growth resistance becomes constant. 4. Conclusions The resistance to crack propagation under Hertzian indentation was studied on coarse and fine-grained alumina. With the help of acoustic emission, this mechanical characterization method permits easy detection of the R-curve effect. The decrease of the events count rate prior to failure, illustrates the presence of a sub-critical crack growth mechanism. This effect is observed under monotonous, static or cyclic loadings. It is also found that a coarser grain has high resistance crack propagation. Environments influence crack propagation by stress corrosion at the crack tip. Calculated apparent toughness for four environments leads to classify them regarding the increasing corrosion effect as: toluene, silicone oil, air and water. The R-curve effect results are in agreement with those obtained by flexure tests. Acknowledgments The authors thank the ceramics group (GEMPPM-INSA, Lyons) for their help. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.ceramint.2007.06.015. References [1] A.G. Evans, K.T. Faber, Crack growth resistance of microcracking brittle materials, J. Am. Ceram. Soc. 67 (4) (1984) 255–260. [2] M.V. Swain, R-curve behavior in a polycrystalline alumina material, J. Mater. Sci. Lett. 5 (4) (1986) 1313–1315. [3] B.J. Pletka, S.M. Wiederhorn, A comparison of failure predictions by strength and fracture mechanics techniques, J. Mater. Sci. 17 (1982) 1247– 1268. [4] A. Okada, N. Hirosaki, Subcritical crack growth in sintered silicon nitride exhibiting a rising R-curve, J. Am. Ceram. Soc. 73 (7) (1990) 2095–2096. [5] B.R. Lawn, Fracture of Brittle Solids, Cambridge University Press, 1993. [6] R.J. Charles, Dissolution behavior of macroporous glass, J. Am. Ceram. Soc. 47 (1964) 154–155. [7] J.R.J.B. Wachtman, Highlights of progress in the science of fracture of ceramics and glass, J. Am. Ceram. Soc. 57 (12) (1974) 509–519. [8] F.F. Lange, In Deformation of Ceramic Materials, Pensylvania State University, Plenum Press, 1975, pp. 361–381. [9] T. Fett, D. Munz, Evaluation of subcritical crack extension under constant loading, J. Eur. Ceram. Soc. 6 (1990) 67–72. [10] T. Fett, D. Munz, Subcritical crack growth of macrocracks in alumina with R-curve behavior, J. Am. Ceram. Soc. 75 (4) (1992) 958–963. [11] T. Fett, D. Munz, Subcritical crack growth of macrocracks in zirconia, J. Mater. Sci. Lett. 10 (1991) 1103–1106. [12] T. Fett, D. Munz, Subcritical crack growth of macro- and microcracks in ceramics, in: R.C. Bradt, et al. (Eds.), Fracture Mechanics of Ceramics, 9, Plenum Press, New York, 1992, p. 604. [13] R.W. Steinbrech, R-curve behavior of ceramics, in: R.C. Bradt, et al. (Eds.), Fracture Mechanics of Ceramics, 9, Plenum Press, New York, 1992, pp. 187–208. [14] B.R. Lawn, B.J. Hockey, S.M. Wiederhorn, Atomically sharp cracks in brittle solids: an electron microscopy study, J. Mater. Sci. 15 (5) (1980) 1207–1223. [15] B.J. Hockey, S.M. Wiederhorn, Crack healing in brittle materials, in: R.C. Bradt, et al. (Eds.), Fracture Mechanics of Ceramics, 6, Plenum Press, New York, 1983, p. 637. [16] B.J. Hockey, B.R. Lawn, Electron microscopy of microcracking about indentationsin aluminum oxide and silicon carbide, J. Mater. Sci. 10 (8) (1975) 1275–1284. [17] A.G. Evans, M. Linzer, Failure prediction in structural ceramics using acoustic emission, J. Am. Ceram. Soc. 56 (11) (1973) 575–581. [18] A.G. Evans, M. Linzer, L.R. Russel, Acoustic emission and crack propagation in polycrystalline alumina, Mater. Sci. Eng. 15 (1974) 253–261. [19] B.T. Khuri-Yakub, A.G. Evans, G.S. Kino, Acoustic surface wave measurements of surface cracks in ceramics, J. Am. Ceram. Soc. 63 (1/2) (1980) 65–71. [20] S. Bouras, D. Rouby, G. Fantozzi, Application de la Technique de l’Emission Acoustique dans l’Etude de l’Indentation Hertzienne, Revue des Composites et des Mate´riaux Avance´s 5 (2) (1995) 235–252. [21] F.B. Langitan, B.R. Lawn, Hertzian fracture experiments on abraded glass surfaces as definitive evidence for an energy balance explanation of Auerbach’s Law, J. Appl. Phys. 40 (10) (1969) 4009–4017. [22] B.R. Lawn, D.B. Marshall, Indentation fracture and strength degradation in ceramics, in: Proceedings of the International Symposium on Fracture Mechanics of Ceramics, University Park, PA, USA, July 27–29, 1977, (1978), pp. 205–229. [23] T.R. Wilshaw, The Hertzian fracture test, J. Phys. D.: Appl. Phys. 4 (1971) 1567–1581. [24] B.R. Lawn, Partial cone crack formation in a brittle material loaded with a sliding spherical indenter, Proc. R. Soc. A299 (1967) 307–316. [25] D. Maugis, M. Barquins, Fracture mechanics and the adherence of viscoelastic bodies, J. Phys. D.: Appl. Phys. 11 (1978) 1989–2023. Table 1 Results of fracture surface energy (gf) and apparent toughness (KIC) Environment gf (J m2 ) KIC apparent (MPa m1/2) Coarse grain alumina Small grain alumina Coarse grain alumina Small grain alumina Toluene 49.3 43.1 6.2 5.8 Silicone oil 40.2 33.4 5.6 5.1 Air 32.1 27.1 5.0 4.6 Water 24.8 20.5 4.4 4.0 1864 S. Bouras et al. / Ceramics International 34 (2008) 1857–1865
S. Bouras et al. /Ceramics intemational 34(2008)1857-1865 [26] D. Maugis, Review subcritical crack growth, surface energy, fracture by D.E. Jones, G.A. Schott), Macmillan, London, U. K.(1896)pp toughness, stick-slip and embrittlement, J Mater. Sci. 20(1985)3041-3073. 163-183 I M.T. Huber, On the theorie of contacting solid elastic bodies, Ann. Phys. [38] F. Auerbach, Absolute hardness measurements, Ann. Phys. Chem. 43 14(1904)153-163 (1891)61-100 [28] A.E.H. Love, The stress produced in a semi-infinite solid by pressure [39] F.C. Roesler, Brittle fractures near equilibrium, Proc. Phys. Soc. B69 part of the boundary, Philos. Trans. R Soc. Lond. A 228(1929)377-420 (1956981-992 [29]R. Mouginot, D. Maugis, Fracture indentation beneath flat and spherical [40] P.D. Warren, D.A. Hills, The influence of elastic mismatch between punches, J Mater. Sci. 20(1985)4354-4376 denter and substrate on Hertzian fracture J. Mater. Sci. 29(1994) [30] R. Warren, Measurements of the fracture properties of brittle solids by 2860-2866 Hertzian indentation, Acta Met. 26(1978)1759-1769 [41] J. P.A. Tillet, Fracture of glass by spherical indenters proc, Phys. Soc. B69 [31]N. Sneddon, Boussinesq's problem for a flat-ended cylinder, Proc Camb (1956)47-54 Philos.Soc.42(1946)29-39 [42] J.S. Nadeau, Acoustic emission in the fracture of glass plates, J.Am [32]S Bouras, Etude par Emission de I'Indentation Hertzienne et de I'Indenta- Ceram.Soc.64(10)(1981)585-590. tion Vickers sur une Vitro-ceramique et sur des Lumines, These de [43]RC. Bradt, W.D. Scott, Mechanical properties of alumina, in: L D. Hart Doctorat d'Etat, INSA de Lyon(1993)186. Ed. ), Alumina Chemicals Science and Technologie Handbook, The [33] F.C. Frank, B.R. Lawn, On the theorie of Hertzian fracture Proc. R. Soc. American Ceramic Society, Columbus, Ohio, 1990, pp 23-39 A299(1967)291-306 [34] B.R. Lawn, Indentation crack growth in alumina, in: R C. Bradt, et al. (Eds ) Fracture Mechanics m. Ceran.Soc.81(1998)1977-1994 %图 acenturyafterHertz团4 M.E. Ebrahimi1 alier. M. Saadaoui. G. Fantozzi, Effect可ge Ceramics, 13, Kluwer Academic/Plenum Publishers, 2002pp. 273-286. [35] B.R. Lawn, T.R. Wilshaw, Review-indentation fracture: principles and [45] M. Saadaoui, Contribution a I'etude du comportement thermomecanique applications, J Mater. Sci. 10(6)(1975)1049-1081 des materiaux ceramique a effet de courbe-R: choc et fatigue thermiques, [36] H. Hertz, On the contact of elastic solids, Zeitschrift fur die Reine und These de Doctorat d'Etat, INSA de lyon(1996)175 Angewandte Mathematik 92(1881)156-171, English translation in [46] D Bleise, R w. Steinbrech, Flat R-curve from stable indentation cracks in miscellaneous Papers(translated by D.E. Jones, G.A. Schott), Macmillan, arse-grained alumina, J. Am. Ceram Soc. 77(2)(1994)315-322. don.UK.(1896)p.146-162. [47] G. Fantozzi, J. Chevalier, M. Saadaoui, R-curve effect on slow crack B37] H. Hertz, On the contact of rigide elastic solids and on hardness, growth and thermal shock of ceramics, in: R.C. Bradt, et al Verhandlungen des vereins zur Beforderung des gewerbefleisses 61 Fracture Mechanics of Ceramics. 13. Kluwer Academic/Plenum (1882)449-462, English translation in Miscellaneous Papers(translated ers,2002,pp.213-228
[26] D. Maugis, Review subcritical crack growth, surface energy, fracture toughness, stick-slip and embrittlement, J.Mater. Sci. 20 (1985) 3041–3073. [27] M.T. Huber, On the theorie of contacting solid elastic bodies, Ann. Phys. 14 (1904) 153–163. [28] A.E.H. Love, The stress produced in a semi-infinite solid by pressure on part of the boundary, Philos. Trans. R. Soc. Lond. A 228 (1929) 377–420. [29] R. Mouginot, D. Maugis, Fracture indentation beneath flat and spherical punches, J. Mater. Sci. 20 (1985) 4354–4376. [30] R. Warren, Measurements of the fracture properties of brittle solids by Hertzian indentation, Acta Met. 26 (1978) 1759–1769. [31] N. Sneddon, Boussinesq’s problem for a flat-ended cylinder, Proc. Camb. Philos. Soc. 42 (1946) 29–39. [32] S. Bouras, Etude par Emission de l’Indentation Hertzienne et de l’Indentation Vickers sur une Vitro-ce´ramique et sur des Alumines, The`se de Doctorat d’Etat, INSA de Lyon (1993) 186. [33] F.C. Frank, B.R. Lawn, On the theorie of Hertzian fracture, Proc. R. Soc. A299 (1967) 291–306. [34] B.R. Lawn, Indentation of ceramics wit, spheres: a century after Hertz, J. Am. Ceram. Soc. 81 (1998) 1977–1994. [35] B.R. Lawn, T.R. Wilshaw, Review-indentation fracture: principles and applications, J. Mater. Sci. 10 (6) (1975) 1049–1081. [36] H. Hertz, On the contact of elastic solids, Zeitschrift fu¨r die Reine und Angewandte Mathematik 92 (1881) 156–171, English translation in Miscellaneous Papers (translated by D.E. Jones, G.A. Schott), Macmillan, London, U.K. (1896) pp. 146–162. [37] H. Hertz, On the contact of rigide elastic solids and on hardness, Verhandlungen des Vereins zur Befo¨rderung des Gewerbefleisses 61 (1882) 449–462, English translation in Miscellaneous Papers (translated by D.E. Jones, G.A. Schott), Macmillan, London, U.K. (1896) pp. 163–183. [38] F. Auerbach, Absolute hardness measurements, Ann. Phys. Chem. 43 (1891) 61–100. [39] F.C. Roesler, Brittle fractures near equilibrium, Proc. Phys. Soc. B69 (1956) 981–992. [40] P.D. Warren, D.A. Hills, The influence of elastic mismatch between indenter and substrate on Hertzian fracture, J. Mater. Sci. 29 (1994) 2860–2866. [41] J.P.A. Tillet, Fracture of glass by spherical indenters proc, Phys. Soc. B69 (1956) 47–54. [42] J.S. Nadeau, Acoustic emission in the fracture of glass plates, J. Am. Ceram. Soc. 64 (10) (1981) 585–590. [43] R.C. Bradt, W.D. Scott, Mechanical properties of alumina, in: L.D. Hart (Ed.), Alumina Chemicals Science and Technologie Handbook, The American Ceramic Society, Columbus, Ohio, 1990, pp. 23–39. [44] M.E. Ebrahimi, J. Chevalier, M. Saadaoui, G. Fantozzi, Effect of grain size on crack growth in alumina, in: R.C. Bradt, et al. (Eds.), Fracture Mechanics of Ceramics, 13, Kluwer Academic/Plenum Publishers, 2002, pp. 273–286. [45] M. Saadaoui, Contribution a` l’e´tude du comportement thermomecanique des mate´riaux ce´ramiques a` effet de courbe-R: choc et fatigue thermiques, The`se de Doctorat d’Etat, INSA de Lyon (1996) 175. [46] D. Bleise, R.W. Steinbrech, Flat R-curve from stable indentation cracks in coarse-grained alumina, J. Am. Ceram. Soc. 77 (2) (1994) 315–322. [47] G. Fantozzi, J. Chevalier, M. Saadaoui, R-curve effect on slow crack growth and thermal shock of ceramics, in: R.C. Bradt, et al. (Eds.), Fracture Mechanics of Ceramics, 13, Kluwer Academic/Plenum Publishers, 2002, pp. 213–228. S. Bouras et al. / Ceramics International 34 (2008) 1857–1865 1865