《复合材料 Composites》课程教学资源（学习资料）第五章 陶瓷基复合材料_Al2O3-SiC-34

f MATERIALIA Pergamon Acta mater.49(2001)3553-3563 www.elsevier.com/locate/actamat TEMPERATURE DEPENDENCE OF CRACK WAKE BRIDGING STRESSES IN A SIC-WHISKER-REINFORCED ALUMINA G.R. SARRAFl-NOUR and T w. COYLET Department of Materials Science and Engineering, University of Toronto, 184 College St, Toronto, oN, Canada M5S 3F4 Received 12 July 2000: received in revised form 8 June 2001: accepted 8 June 2001) Abstract-The crack face bridging behavior of a SiC-whisker-reinforced alumina composite was charac rized between room temperature and 1400oC in air. The bridging relations of the material were deconvoluted from the chevron-notched specimen R-curves by employing the fracture mechanics weight function method and assuming that pullout bridging was the dominant bridging mechanism. The results indicated that the maximum bridging stress decreased linearly with increasing temperature while the distribution of the bridging tresses appeared to shift towards larger crack opening displacements. The former trend was found to good agreement with literature data on the temperature dependence of residual stresses measured in sim omposites, while the latter agreed with the observation of an increasing whisker pullout length in such s with increasing temperature. In agreement with previous studies of this material, some crack- one between 1000C and 1300C, no influence from the microcrack zone on the bridging stresses was detected. 200/ Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Fracture fracture toughness; Crack wake bridging stress 1 INTRODUCTION operating in the crack wake zone T>1200.C. In contrast to these studies It is now well-known that when incorporated in a cer- made in other research works [10,11] imic matrix, strong ceramic whiskers can improve the strong toughening component due tor of as high as four relative to the monolithic matrix microcrack zone, and crack branching [12], in the material[1-3]. Various toughening mechanisms con- composite at T>1200 C with no mention of any R- sidered as contributing to the improved fracture curve behavior. One reason for the inconsistency of resistance of the composite material include crack these results may be the sensitivity of the fra acture deflection, microcracking, and crack-wake bridging behavior of the composite and the operative toughen mechanisms the latter has widely been accepted to be of the material: predominantly effects associated with composite [4-7 of the whisker/matrix interface [15-18] Previous studies of the fracture behavior of Sic. Crack-wake-bridging behavior of monolithic and whisker-reinforced alumina based on R-curve reinforced ceramics is frequently studied by measurements concluded a strong crack wake bridg- employing R-curve measurements, with the observed e hg component in the toughening of the material at behavior commonly discussed based on the rate and that crack wake toughen- [46,8,9, 19-21]. However, a major deficiency ing mechanisms dominated the fracture resistance of inherent to such inherent to such evaluations is the strong dependence the composite; however, the active mechanism was the r suggested to change from a predominantly frontal specimen/crack and the type of loading, making the zone(microcracking) type at low temperatures to one results obtained by using different crack geometries essentially incomparable. In spite of numerous studie on the fracture and R-curve behavior of sic-whisker addressed. reinforced alumina composites, there are only few Coyle) studies dealing directly with the characterization of 1359-6454/01/20.00 O 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved P:S1359-6454(01)00244-0

Acta mater. 49 (2001) 3553–3563 www.elsevier.com/locate/actamat TEMPERATURE DEPENDENCE OF CRACK WAKE BRIDGING STRESSES IN A SiC-WHISKER-REINFORCED ALUMINA G. R. SARRAFI-NOUR and T. W. COYLE† Department of Materials Science and Engineering, University of Toronto, 184 College St., Toronto, ON, Canada M5S 3E4 ( Received 12 July 2000; received in revised form 8 June 2001; accepted 8 June 2001 ) Abstract—The crack face bridging behavior of a SiC-whisker-reinforced alumina composite was charac￾terized between room temperature and 1400°C in air. The bridging relations of the material were deconvoluted from the chevron-notched specimen R-curves by employing the fracture mechanics weight function method and assuming that pullout bridging was the dominant bridging mechanism. The results indicated that the maximum bridging stress decreased linearly with increasing temperature while the distribution of the bridging stresses appeared to shift towards larger crack opening displacements. The former trend was found to be in good agreement with literature data on the temperature dependence of residual stresses measured in similar composites, while the latter agreed with the observation of an increasing whisker pullout length in such composites with increasing temperature. In agreement with previous studies of this material, some crack-tip toughening due to a microcrack damage zone formed around the crack-tip by diffusional cavitation could be observed at T
1000°C. Although the material exhibited toughening by both a microcrack zone and a bridging zone between 1000°C and 1300°C, no influence from the microcrack zone on the bridging stresses was detected.  2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Fracture & fracture toughness; Crack wake bridging stress 1. INTRODUCTION It is now well-known that when incorporated in a cer￾amic matrix, strong ceramic whiskers can improve the fracture resistance of the resulting composite by a fac￾tor of as high as four relative to the monolithic matrix material [1–3]. Various toughening mechanisms con￾sidered as contributing to the improved fracture resistance of the composite material include crack deflection, microcracking, and crack-wake bridging by the reinforcement. Amongst these toughening mechanisms the latter has widely been accepted to be responsible for the observed R-curve behavior in the composite [4–7]. Previous studies of the fracture behavior of SiC￾whisker-reinforced alumina based on R-curve measurements concluded a strong crack wake bridg￾ing component in the toughening of the material at temperatures beyond 1000°C [8]. Further studies [6, 9] arrived at the conclusion that crack wake toughen￾ing mechanisms dominated the fracture resistance of the composite; however, the active mechanism was suggested to change from a predominantly frontal zone (microcracking) type at low temperatures to one † To whom all correspondence should be addressed. E-mail address: coyle@ecf.utoronto.ca (T. W. Coyle) 1359-6454/01/$20.00  2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S13 59-6454(01)00244-0 operating in the crack wake zone (bridging) at T
1200°C. In contrast to these studies, observations made in other research works [10, 11] supported a strong toughening component due to a frontal microcrack zone, and crack branching [12], in the composite at T
1200°C with no mention of any R￾curve behavior. One reason for the inconsistency of these results may be the sensitivity of the fracture behavior of the composite and the operative toughen￾ing mechanisms to the details of the microstructure of the material: predominantly effects associated with whisker surface morphology [13, 14] and the nature of the whisker/matrix interface [15–18]. Crack-wake-bridging behavior of monolithic and reinforced ceramics is frequently studied by employing R-curve measurements, with the observed behavior commonly discussed based on the rate and the amount of the rise of the R-curve, for example [4–6, 8, 9, 19–21]. However, a major deficiency inherent to such evaluations is the strong dependence of the R-curves on the geometry of test specimen/crack and the type of loading, making the results obtained by using different crack geometries essentially incomparable. In spite of numerous studies on the fracture and R-curve behavior of SiC-whisker￾reinforced alumina composites, there are only few studies dealing directly with the characterization of

3554 SARRAFI-NOUR and COYLE- CRACK WAKE BRIDGE bridging behavior in this material [22, 23]. A main objective in this work was to characterize the bridging stress and its variation with temperature in a Sic- whisker-reinforced alumina between room tempera ture and 1400C. For this purpose, a hybrid experimental/numerical scheme was employed which utilized the R-curves obtained from chevron-notched xure test specimens of the composite to deconvol ute the distribution of the bridging stresses as function of crack opening displacement at various tempera 2. EXPERIMENTAL PROCEDURES 2. 1. Material and specimen preparation Fig. 1. Schematic of the fracture plane of a chevron notch The SiC-whisker-reinforced composite material ed in this study was a commercial cutting tool gra material( Grade wG-300, Greenleaf Corp, Saeger Series) with a cross-head speed of 50 um/min. The town, PA)and contained 33 vol. of Sic-whiskers flexure test fixture was made from various grades of spersed in an alumina matrix. The composite pow- SiC conforming to the specifications given in ASTM cess. The material is hot-pressed uniaxially in the minimum of three specimens(most of the tests were shape of plates, which are virtually of full density. conducted using four or five specimens per test Most of the SiC-whiskers have a diameter between condition)was used at each of the test temperatures I and I um and aspect ratios of 10-100. The alum- ambient, 800oC, 1000C, 1200 C, 1300C and ina matrix grain size ranges between I and 5 Hm. 1400C under air atmosphere. The heating schedule The microstructure of the material shows preferred of the fracture tests at elevated temperatures consisted orientation of the Sic-whiskers within the plane nor- of heating the specimen to the test temperature at mal to the hot-pressing axis. In addition, whisker-rich 15C/min, soaking for 20 min to achieve temperature and whisker-poor regions, typically 30-50 um in size, equilibrium in the furnace and loading the samples to due to reinforcement clustering typical of whisker- failure followed by cooling to room temperature at reinforced composites can be observed in the 15C/min. The loading-point compliance of the speci material. The composite material is of high purity and mens during the fracture test was determined by mea- erain boundaries and whisker/matrix interfaces [11, of the specimens and correlating it with the load-point 4]. Such a clean interface between the SiC-whiskers displacement through an analytical relation[25, 26) and the alumina matrix makes this composite an The differential displacement was measured between excellent model material to study the temperature dependence of the bridging stresses. The absence of points, each located at 15 mm from either side of the an(amorphous)second phase at the interface in this central measuring point on the specimen, using a sin- from a boundary phase with increasing temperature. [27. The test setup arrangement and the displacement The composite test specimens were received from measuring device were schematically shown else- manufacturer in the form of machined rectangular where[26]. The accuracy of the loading-point co flexure specimens with length(L)45-47 mm, width pliance measurements on the chevron-notched speci (W)5.00 mm and depth(B)of 4.00 mm. The width mens by employing this method was verified both numerI ally using three-dimensional finite element, pressing axis. A chevron-notch was cut into the test FE, analysis and experimentally by conducting con specimens using a 300-um-thick diamond cut-off trolled fracture tests on test specimens of materials wheel such that both the resulting crack-plane and the with a flat R-curve behavior at room and elevated crack propagation direction were parallel to the hot- temperatures [26] The chey hematically in Fig. 1, had an initial depth RESULTS adw=0.32 and a final depth a /W=l 3.1.R-cu 2. 2. Controlled fracture test The were calculated from the load-dis- fracture tests were conducted under thre raceme es using standard fracture mechanics nding load and displacement control conditions on procedure on the compliance measurement ervo-hydraulic testing machine (Instron 8500 method [4]. For this purpose, the required cor

3554 SARRAFI-NOUR and COYLE: CRACK WAKE BRIDGE bridging behavior in this material [22, 23]. A main objective in this work was to characterize the bridging stress and its variation with temperature in a SiC￾whisker-reinforced alumina between room tempera￾ture and 1400°C. For this purpose, a hybrid experimental/numerical scheme was employed which utilized the R-curves obtained from chevron-notched flexure test specimens of the composite to deconvol￾ute the distribution of the bridging stresses as function of crack opening displacement at various tempera￾tures. 2. EXPERIMENTAL PROCEDURES 2.1. Material and specimen preparation The SiC-whisker-reinforced composite material used in this study was a commercial cutting tool grade material (Grade WG-300, Greenleaf Corp., Saeger￾town, PA) and contained 33 vol.% of SiC-whiskers dispersed in an alumina matrix. The composite pow￾der preparation is a proprietary powder blending pro￾cess. The material is hot-pressed uniaxially in the shape of plates, which are virtually of full density. Most of the SiC-whiskers have a diameter between 0.1 and 1 µm and aspect ratios of 10–100. The alum￾ina matrix grain size ranges between 1 and 5 µm. The microstructure of the material shows preferred orientation of the SiC-whiskers within the plane nor￾mal to the hot-pressing axis. In addition, whisker-rich and whisker-poor regions, typically 30–50 µm in size, due to reinforcement clustering typical of whisker￾reinforced composites can be observed in the material. The composite material is of high purity and generally exhibits no or very little glass phase at the grain boundaries and whisker/matrix interfaces [11, 24]. Such a clean interface between the SiC-whiskers and the alumina matrix makes this composite an excellent model material to study the temperature dependence of the bridging stresses. The absence of an (amorphous) second phase at the interface in this material avoids complicating behavior that may arise from a boundary phase with increasing temperature. The composite test specimens were received from the manufacturer in the form of machined rectangular flexure specimens with length (L) 45–47 mm, width (W) 5.00 mm and depth (B) of 4.00 mm. The width (W) of the test specimens was parallel to the hot￾pressing axis. A chevron-notch was cut into the test specimens using a 300-µm-thick diamond cut-off wheel such that both the resulting crack-plane and the crack propagation direction were parallel to the hot￾pressing axis. The chevron notch geometry, shown schematically in Fig. 1, had an initial depth a0/W = 0.32 and a final depth a1/W = 1. 2.2. Controlled fracture test Fracture tests were conducted under three-point￾bending load and displacement control conditions on a servo-hydraulic testing machine (Instron 8500 Fig. 1. Schematic of the fracture plane of a chevron notch. Series) with a cross-head speed of 50 µm/min. The flexure test fixture was made from various grades of SiC conforming to the specifications given in ASTM C1211-92 and had a lower-span-distance of 40 mm. A minimum of three specimens (most of the tests were conducted using four or five specimens per test condition) was used at each of the test temperatures: ambient, 800°C, 1000°C, 1200°C, 1300°C and 1400°C under air atmosphere. The heating schedule of the fracture tests at elevated temperatures consisted of heating the specimen to the test temperature at 15°C/min, soaking for 20 min to achieve temperature equilibrium in the furnace and loading the samples to failure followed by cooling to room temperature at 15°C/min. The loading-point compliance of the speci￾mens during the fracture test was determined by mea￾suring a differential displacement on the tensile face of the specimens and correlating it with the load-point displacement through an analytical relation [25, 26]. The differential displacement was measured between the center-point of the specimen and two reference points, each located at 15 mm from either side of the central measuring point on the specimen, using a sin￾gle linear variable displacement transducer, LVDT [27]. The test setup arrangement and the displacement measuring device were schematically shown else￾where [26]. The accuracy of the loading-point com￾pliance measurements on the chevron-notched speci￾mens by employing this method was verified both numerically using three-dimensional finite element, FE, analysis and experimentally by conducting con￾trolled fracture tests on test specimens of materials with a flat R-curve behavior at room and elevated temperatures [26]. 3. RESULTS 3.1. R-curves The R-curves were calculated from the load–dis￾placement curves using standard fracture mechanics procedure based on the compliance measurement method [4]. For this purpose, the required com-

SARRAFI-NOUR and COYLE- CRACK WAKE BRIDGE 3555 pliance-crack length relation for the chevron-notched lowing a similar methodology, a procedure was specimen was calculated by employing a three- recently developed to obtain the distribution of brid dimensional FE solution [28, 29] obtained for the ing tractions from R-curves measured using chevron- same specimen geometry under four-point flexure notched flexure specimens [28]. The method allows load condition and modifying the solution to allow the deconvolution of a bridging relation from th for the change in the span-to-width ratio. (The com- measured R-curve by performing an iterative analysis pliance of the chevron-notched specimen under three- on the R-curve data which incorporates the inter- point flexure condition and without any crack was rela lations between the fracture mechanics weight func- obtained by using a simplified three-dimensional tion, stress intensity factor, and crack opening dis finite element model composed of -600 linear placement, and an implicit assumption of a dominant elements. pullout bridging mechanism. The analysis procedure, Typical R-curves calculated from the load-dis- as described in detail previously [28], involves fittir placement curves of the specimens fractured at vari- the right-hand side of the following equation to the ous temperatures are shown in Fig. 2. A fully con- R-curve obtained using a chevron-notched specimen trolled crack propagation regime could not be achieved in the tests conducted at room temperature due to the strong run-arrest 14, 20] nature of the crack owth behavior of the composite at room tempera- I+/=. ture. As can be seen from the r-curve results at room ∑c(a-xy temperature(Fig. 2, left), on many occasions an unstable crack growth/re-initiation preceded the crack arrest point. Since unstable crack propagation is not a condition in favor of formation of bridging where Kr is the fracture resistance of the material whiskers/ligaments in the crack wake, a continuous measured by an external observer during the course R-curve representing the true shielding contribution of stable crack propagation, Ktip is the stress intensity of the crack wake could not be obtained for the test factor at the crack tip which would be equal to K specimens at room temperature. The run-arrest the initial value of the fracture resistance or the tough- behavior was particularly visible around the ness of the material in the absence of any crack wake maximum load of the load-displacement curves at contributions, and Kbr is the stress intensity factor due room temperature and significantly diminished with to crack wake bridging. The function h of the inte- increasing temperature above 800C grand is the fracture mechanics weight function for 3. 2. Deconvolution of the bridging stresses from the the wake tractions in the appropriate chevron-notched specimen geometry [33] and the polynomial function within the square brackets is the expansion of the ett and collaborators have described a method- unknown bridging stress, o in the crack length, a ology for analysing R-curves arising from the crack and position, x, co-ordinates. Iterative analysis of th wake bridging process in ceramics based on the frac- R-curve data using equation(1) yields the coef- ture mechanics weight function method [30-32]. Fol- ficients, Ci, of the unknown bridging stress function 105 9.5 95 8.5 E8.5 7.5 7.5 6.5 5.5 040.50.60.70.80.9 0.3040.50.60.70.80.9 a/ in equation(1)

SARRAFI-NOUR and COYLE: CRACK WAKE BRIDGE 3555 pliance–crack length relation for the chevron-notched specimen was calculated by employing a three￾dimensional FE solution [28, 29] obtained for the same specimen geometry under four-point flexure load condition and modifying the solution to allow for the change in the span-to-width ratio. (The com￾pliance of the chevron-notched specimen under three￾point flexure condition and without any crack was obtained by using a simplified three-dimensional finite element model composed of
600 linear elements.) Typical R-curves calculated from the load–dis￾placement curves of the specimens fractured at vari￾ous temperatures are shown in Fig. 2. A fully con￾trolled crack propagation regime could not be achieved in the tests conducted at room temperature due to the strong run–arrest [4, 20] nature of the crack growth behavior of the composite at room tempera￾ture. As can be seen from the R-curve results at room temperature (Fig. 2, left), on many occasions an unstable crack growth/re-initiation preceded the crack arrest point. Since unstable crack propagation is not a condition in favor of formation of bridging whiskers/ligaments in the crack wake, a continuous R-curve representing the true shielding contribution of the crack wake could not be obtained for the test specimens at room temperature. The run–arrest behavior was particularly visible around the maximum load of the load–displacement curves at room temperature and significantly diminished with increasing temperature above 800°C. 3.2. Deconvolution of the bridging stresses from the R-curve Fett and collaborators have described a method￾ology for analysing R-curves arising from the crack wake bridging process in ceramics based on the frac￾ture mechanics weight function method [30–32]. Fol￾Fig. 2. R-curves from the composite specimens at room temperature (left) and at elevated temperatures (right). The solid curves included with the elevated temperature R-curve data are plotted using the best-fit coefficients in equation (1). lowing a similar methodology, a procedure was recently developed to obtain the distribution of bridg￾ing tractions from R-curves measured using chevron￾notched flexure specimens [28]. The method allows the deconvolution of a bridging relation from the measured R-curve by performing an iterative analysis on the R-curve data which incorporates the inter￾relations between the fracture mechanics weight func￾tion, stress intensity factor, and crack opening dis￾placement, and an implicit assumption of a dominant pullout bridging mechanism. The analysis procedure, as described in detail previously [28], involves fitting the right-hand side of the following equation to the R-curve obtained using a chevron-notched specimen: KR Ktip Kbr (1) Ko
a a0 h(x,a)i j3 i,j 0 Cij·aj (ax) i dx where KR is the fracture resistance of the material measured by an external observer during the course of stable crack propagation, Ktip is the stress intensity factor at the crack tip which would be equal to Ko, the initial value of the fracture resistance or the tough￾ness of the material in the absence of any crack wake contributions, and Kbr is the stress intensity factor due to crack wake bridging. The function h of the inte￾grand is the fracture mechanics weight function for the wake tractions in the appropriate chevron-notched specimen geometry [33] and the polynomial function within the square brackets is the expansion of the unknown bridging stress, sbr, in the crack length, a, and position, x, co-ordinates. Iterative analysis of the R-curve data using equation (1) yields the coef- ficients, Cij, of the unknown bridging stress function

3556 SARRAFI-NOUR and COYLE- CRACK WAKE BRIDGE and the initial value of fracture toughness of the curve while simultaneously minimizing the follow- material, Ko(if not determined through indeper experiments) The displacement of the crack walls due to the closure bridging tractions, Sn can be obtained fol ∑(oD)a-(oG))=min lowing Rice's relation between the stress intensity factor, K, crack opening displacement, 8, and th weight function, h[341 where obS)ai is the magnitude of the calculated bridging stress at the crack opening 8 for the crack H do length a; and <obd)> is an average bridging stress (2)calculated over a range of the crack lengths, including 7, at the same crack opening displacement Analysis of the r-curves of the composite speci mens tested at different temperatures by using the where H is an appropriate elastic modulus(equal to procedures described allowed deconvolution of the E and E=E/(1-v), with v being the Poisson's bridging relation and K, from the R-curve results. As ratio, under the plane stress and plane strain con- discussed previously [26], the Ko obtained through the ditions, respectively ) By employing the integral form analysis procedure describes the crack tip toughness from the bridging stresses could be calculated once from crack wake shielding processes. Therefore, it the coefficients Ci were known should be viewed as containing the contributions from any other toughening mechanisms operative in the material, including the toughness of the matrix, Or=F2 co h(x, o h(x', a' a (3) Typical fit curves to the R-curves and the resulting bridging relations obtained from the analysis of th yade R-curves of the composite material at various tem peratures are shown in Figs 2 and 3, respectively. Due to the strong influence of the run-arrest behavior on the shape of the R-curves of the composites at room In equation(3)x represents the co-ordinate at which temperature, these R-curves were not used in the the crack opening displacement is to be determined, analysis. However, the minimum value of kg that x'is the co-ordinate where the(bridging)stress o is occurred after multiple run-arrests and at long crack acting, and a' is a running integration variable. The lengths was assumed as the initial value of fracture total crack surface displacement can be obtained by toughness, Ko. The magnitude of the bridging stress the superposition of the crack surface displacements in the composite at room temperature was estimated The former displacement could be determined The plane-strain elastic modulus of the composite numerically by using finite element analysis [28 The bridging relation is finally obtained by correl displacement fields was determined from the speci- ating the total crack surface displacements and the magnitude of the bridging stress through the crack ength and position co-ordinates, a and x. It should 15 be noted that a bridging relation calculated through 800°c the analysis procedure is valid only if the resulting 1000°C Oh vs. 8 for various crack lengths are self-consistent, respect to COD is independent of the crack length. In oo order to help converge to such solutions during the iteration analysis a number of constraints imposed on the bridging stress function representing the expansion of the bridging stress in the crack co ordinate system [28]. These constraints were selected based on the fundamentals of the(pullout) bridging mechanism. Further experience with this procedure indicated that the constraints introduced could not of the procedure to a dis crack opening displacement, 28 [um] tribution of bridging stresses leading to a self-consist ent bridging relation under all circumstances. In some crack op isplacement obtained from the analysis of the cases, it was necessary to fit equation(I)to the R- R-curve SiC-whisker-reinforced alumina composite

3556 SARRAFI-NOUR and COYLE: CRACK WAKE BRIDGE and the initial value of fracture toughness of the material, Ko (if not determined through independent experiments). The displacement of the crack walls due to the closure bridging tractions, dbr, can be obtained fol￾lowing Rice’s relation between the stress intensity factor, K, crack opening displacement, d, and the weight function, h [34]: h H K ∂d ∂a (2) where H is an appropriate elastic modulus (equal to E and E = E/(1n2 ), with n being the Poisson’s ratio, under the plane stress and plane strain con￾ditions, respectively). By employing the integral form of equation (2) the crack wall displacements arising from the bridging stresses could be calculated once the coefficients Cij were known: dbr 1 E i j3 i,j 0 Cij
a x h(x,a)
a a0 h(x,a)(a (3) x) i aj dxda In equation (3) x represents the co-ordinate at which the crack opening displacement is to be determined, x is the co-ordinate where the (bridging) stress s is acting, and a is a running integration variable. The total crack surface displacement can be obtained by the superposition of the crack surface displacements due to the applied load and due to the bridging stress. The former displacement could be determined numerically by using finite element analysis [28]. The bridging relation is finally obtained by correl￾ating the total crack surface displacements and the magnitude of the bridging stress through the crack length and position co-ordinates, a and x. It should be noted that a bridging relation calculated through the analysis procedure is valid only if the resulting sbr vs. d for various crack lengths are self-consistent, i.e., the distribution of the bridging stresses with respect to COD is independent of the crack length. In order to help converge to such solutions during the iteration analysis a number of constraints were imposed on the bridging stress function representing the expansion of the bridging stress in the crack co￾ordinate system [28]. These constraints were selected based on the fundamentals of the (pullout) bridging mechanism. Further experience with this procedure indicated that the constraints introduced could not guarantee the convergence of the procedure to a dis￾tribution of bridging stresses leading to a self-consist￾ent bridging relation under all circumstances. In some cases, it was necessary to fit equation (1) to the R￾curve while simultaneously minimizing the follow￾ing relation: (sbr(d)ai sbr(d)) 2 min (4) where sbr(d)ai is the magnitude of the calculated bridging stress at the crack opening d for the crack length ai and sbr(d)
is an average bridging stress calculated over a range of the crack lengths, including ai , at the same crack opening displacement. Analysis of the R-curves of the composite speci￾mens tested at different temperatures by using the procedures described allowed deconvolution of the bridging relation and Ko from the R-curve results. As discussed previously [26], the Ko obtained through the analysis procedure describes the crack tip toughness of the material in the absence of any contributions from crack wake shielding processes. Therefore, it should be viewed as containing the contributions from any other toughening mechanisms operative in the material, including the toughness of the matrix, at the crack tip. Typical fit curves to the R-curves and the resulting bridging relations obtained from the analysis of the R-curves of the composite material at various tem￾peratures are shown in Figs 2 and 3, respectively. Due to the strong influence of the run-arrest behavior on the shape of the R-curves of the composites at room temperature, these R-curves were not used in the analysis. However, the minimum value of KR that occurred after multiple run–arrests and at long crack lengths was assumed as the initial value of fracture toughness, Ko. The magnitude of the bridging stress in the composite at room temperature was estimated differently and will be discussed later on in the paper. The plane-strain elastic modulus of the composite material required for the calculation of crack opening displacement fields was determined from the speci￾Fig. 3. Distribution of the bridging stresses as a function of crack opening displacement obtained from the analysis of the R-curves of the SiC-whisker-reinforced alumina composite.

SARRAFI-NOUR and COYLE- CRACK WAKE BRIDGE 3557 400 300 山200 100 400 80012001600 Temperature [C] Fig. 4. Variation of the plane-strain elastic modulus of the SiC-whisker-reinforced alumina composite calculated from the measured initial compliance of the chevron-notched flexure test specimens. The error bars represent the scatter range of the data from three to five individual test specimens This work Yu and Kobayashi[22] Linear fit(data from this work 15 10 40080012001600 1600 TICI Fig. 5. Variation of (a) the maximum value of the bridging stress, and (b) the initial value of the fracture toughness, in the Sic-whisker-reinforced alumina composite as a function of temperature men compliance. For this se,the compliance 4. DISCUSSION values determined from the slope of the linear seg- 4.1. Bridging stress ment of the load-displacement curves from the chev on-notched flexure specimens were compared with The variation of the maximum bridging stress in the calculated compliance from the finite element sol- the composite with temperature is shown in Fig. 5(a) ution to yield the modulus value. These modulus Although the results of R-curve tests at room tem- results are presented in Fig. 4 perature could not be used in the analysis to calculate Variation of the maximum bridging stress from the the bridging relations, a maximum bridging stress of bridging relations(Fig. 3)and the initial value of the --23 MPa estimated through a linear extrapolation of fracture toughness deconvoluted from the R-curve the results from this work to room temperature results are shown in Figs 5(a)and(b), respectively. appeared to be in good agreement with the bridging As described earlier the K, values determined through stress values reported previously for a 30 vol % SIC the analysis include the contribution from any tough- whisker-reinforced alumina [22, 23 ning mechanism operative at the tip of the crack and The linear dependence of the maximum bridging should not be viewed as an indicator of the toughness stress on temperature, suggested by the data shown in of the alumina matrix alone Fig. 5(a), is consistent with the mechanics of pullo

SARRAFI-NOUR and COYLE: CRACK WAKE BRIDGE 3557 Fig. 4. Variation of the plane-strain elastic modulus of the SiC-whisker-reinforced alumina composite calculated from the measured initial compliance of the chevron-notched flexure test specimens. The error bars represent the scatter range of the data from three to five individual test specimens. Fig. 5. Variation of (a) the maximum value of the bridging stress, and (b) the initial value of the fracture toughness, in the SiC-whisker-reinforced alumina composite as a function of temperature. men compliance. For this purpose, the compliance values determined from the slope of the linear seg￾ment of the load–displacement curves from the chev￾ron-notched flexure specimens were compared with the calculated compliance from the finite element sol￾ution to yield the modulus value. These modulus results are presented in Fig. 4. Variation of the maximum bridging stress from the bridging relations (Fig. 3) and the initial value of the fracture toughness deconvoluted from the R-curve results are shown in Figs 5(a) and (b), respectively. As described earlier the Ko values determined through the analysis include the contribution from any tough￾ening mechanism operative at the tip of the crack and should not be viewed as an indicator of the toughness of the alumina matrix alone. 4. DISCUSSION 4.1. Bridging stress The variation of the maximum bridging stress in the composite with temperature is shown in Fig. 5(a). Although the results of R-curve tests at room tem￾perature could not be used in the analysis to calculate the bridging relations, a maximum bridging stress of
23 MPa estimated through a linear extrapolation of the results from this work to room temperature appeared to be in good agreement with the bridging stress values reported previously for a 30 vol.% SiC￾whisker-reinforced alumina [22, 23]. The linear dependence of the maximum bridging stress on temperature, suggested by the data shown in Fig. 5(a), is consistent with the mechanics of pullout