《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_residual stress-1

J. Am. Ceram. Soc., 91[4] 1218-1225 DOl:10.1l11.1551-29162008.02 c 2008 The American Ceramic urna Nondestructive measurement of the residual stress profile in Ceramic laminates Matteo Leoni, Matteo Ortolani, Massimo bertoldi, Vincenzo M. sglavo, and Paolo Scardi Department of Materials Engineering and Industrial Technologies, University of Trento, Mesiano, Trento 38100, Italy Millimeter-thick symmetric ceramic laminates, designed to The proposed design procedure points out the importance of ossess a specific through-thickness residual stress profile, the knowledge of the residual stresses within the material. At were produced by tape casting from blends of alumina, zircon- tempts have been made to validate the calculated data by using ia, and mullite powders. The residual stress profile was checked fluorescence piezo-spectroscopy on Cr+, which is always pres- nondestructively by X-ray energy-dispersive diffraction using ent as impurity in alumina. The ob d results could repro- synchrotron white-beam radiation Measurement of the average duce the step profile as obtained by the calculations, although stress on very small volumes(ca. 10 um along the specimen the intensity of stresses was always much lower than expected hickness) provided results in good agreement with the design The discrepancy was related to the fact that piezo-spectroscol crystalline phase composing the laminate allowed inferences to not within the material be made on stress partition and grain-grain coupling in the a nondestructive measurement of residual stresses within the laminate can be performed by using neutron or X-ray diffrac tion techniques. The limited thickness of the specimen and the need for a high spatial resolution are, however, a challenge for both neutrons and laboratory X-rays: the former cannot be L. Introduction focused to a very small size (a few micrometers size would be HE most important limitation to the use of ceramic materials required in practical cases), and the latter have a shallow pen- n structural applications is inherent in their scarce mechan tration in alumina and in ceramic materials in general. more- ical reliability. Their brittleness is directly related to the low over, traditional measurement techniques(sin" v, n-rotation varlous olve rotation or ated both during production and in service. The consequence is specimen(thus a change in the sampled volume)during data a scatter of strength data too large to allow safe design, unless acquisition. This is deleterious when a pointwise resolution statistical approaches embodying acceptable minimum failure sought, like in the case analyzed here. Synchrotron radiation in risk are used many efforts have been made in the last decades this case offers an optimal solution, as it couples the possibility to overcome such problems to work with a narrow probe and to tune the wavelength of the The fracture behavior of ceramics can be improved by using radiation to increase the penetration within the material. A well- the reinforcing action of grain anisotropy or second phases, by hickness residual stress mapping is therefore the promotion of crack shielding effects by phase transforma- ndently of the particular geometry of the tion or microcracking, and by introducing low-energy paths for rack propagation in porous- or within weak interlayers in lam In this work, residual stresses of three symmetric ceramic inates As an alternative laminated structures characterized laminates designed and produced according to the procedure by the presence of thin layers in residual compression alternated proposed by Sglavo and colleagues were measured by minimum failure stress(threshold stress)or fracture toughness CCLRC Daresbury Laboratory Synchrotron Radiation Source values as high as 17 MPa-m (DL-SRS). Results are compared with calculated profiles and Recently, Sglavo and colleagues -16 have proposed the pos- discussed in terms of the microstructure of the single ceramic sibility of improving the mechanical behavior of ceramic lami- laminae ates by introducing a residual stress profile originating from constituting layers. By controlling the development of residual Il. Experimental Procedure tresses in ceramic multilayers, materials characterized by high fracture resistance and limited strength scatter have been de- (1) Specimens gned and produced. A specific procedure has been devel- Symmetric ceramic laminates were produced from laminae obtained knowledge of elastic properties (Young's modulus and Poissons Alpha-alumina (A- 6SG, ALCOA Corp, Pittsburgh, PA) was ratio) and thermal expansion coefficients of the constituent ma- terials. Residual stresses are included in the apparent fracture mullite(KM101, KCM Corp, Nagoya, Japan) and yttria(3 strength once the size of the flaws is known. ion of the final toughness of the material allowing pred mol%)-stabilized zirconia(TZ-3YS. TOSOH Corp, Tokyo, good agreement has been observed between theoretical failure tresses and experimental mechanical strengt amina/zirconia composite layers were prepared. Such compos- Ites were labeled as AMy and AZy, where"A, ""M, "Z, "and C.H. Hsuch-contnibuting editor y stand for alumina, mullite, zirconia and the volume percent content of mullite or zirconia, respectively. For the productio of the laminates, rectangular cards around 50 mm x 50 mm 2007: approved October 12, 200 cut from different green laminae, stacked together, and thermo- author to whom correspondence should be addressed. e-mail: Matteo. Leoni unitnit compressed at 70 C under a pressure of 30 MPa for 15 min. Bars 218

Nondestructive Measurement of the Residual Stress Profile in Ceramic Laminates Matteo Leoni,w Matteo Ortolani, Massimo Bertoldi, Vincenzo M. Sglavo, and Paolo Scardi Department of Materials Engineering and Industrial Technologies, University of Trento, Mesiano, Trento 38100, Italy Millimeter-thick symmetric ceramic laminates, designed to possess a specific through-thickness residual stress profile, were produced by tape casting from blends of alumina, zircon￾ia, and mullite powders. The residual stress profile was checked nondestructively by X-ray energy-dispersive diffraction using synchrotron white-beam radiation. Measurement of the average stress on very small volumes (ca. 10 lm along the specimen thickness) provided results in good agreement with the design data. Moreover, the possibility of independently measuring each crystalline phase composing the laminate allowed inferences to be made on stress partition and grain–grain coupling in the laminas. I. Introduction THE most important limitation to the use of ceramic materials in structural applications is inherent in their scarce mechan￾ical reliability. Their brittleness is directly related to the low value of fracture toughness and to the presence of flaws gener￾ated both during production and in service. The consequence is a scatter of strength data too large to allow safe design, unless statistical approaches embodying acceptable minimum failure risk are used1 : many efforts have been made in the last decades to overcome such problems. The fracture behavior of ceramics can be improved by using the reinforcing action of grain anisotropy or second phases, by the promotion of crack shielding effects by phase transforma￾tion or microcracking,1 and by introducing low-energy paths for crack propagation in porous2 or within weak interlayers in lam￾inates.3–6 As an alternative, laminated structures characterized by the presence of thin layers in residual compression alternated with thicker layers in tension7–12 have been shown to possess a minimum failure stress (threshold stress) or fracture toughness values as high as 17 MPa
m1/2. Recently, Sglavo and colleagues13–16 have proposed the pos￾sibility of improving the mechanical behavior of ceramic lami￾nates by introducing a residual stress profile originating from differences in thermal expansion coefficients of the different constituting layers. By controlling the development of residual stresses in ceramic multilayers, materials characterized by high fracture resistance and limited strength scatter have been de￾signed and produced.13–16 A specific procedure has been devel￾oped to design symmetric laminate structures starting from the knowledge of elastic properties (Young’s modulus and Poisson’s ratio) and thermal expansion coefficients of the constituent ma￾terials. Residual stresses are included in the apparent fracture toughness of the material allowing prediction of the final strength once the size of the flaws is known.13–15 In most cases, good agreement has been observed between theoretical failure stresses and experimental mechanical strength. The proposed design procedure points out the importance of the knowledge of the residual stresses within the material. At￾tempts have been made to validate the calculated data by using fluorescence piezo-spectroscopy on Cr31, 17 which is always pres￾ent as impurity in alumina.18 The obtained results could repro￾duce the step profile as obtained by the calculations, although the intensity of stresses was always much lower than expected. The discrepancy was related to the fact that piezo-spectroscopy allows the measurement of stresses only on external surfaces and not within the material.19 A nondestructive measurement of residual stresses within the laminate can be performed by using neutron or X-ray diffrac￾tion techniques.20–22 The limited thickness of the specimen and the need for a high spatial resolution are, however, a challenge for both neutrons and laboratory X-rays: the former cannot be focused to a very small size (a few micrometers size would be required in practical cases), and the latter have a shallow pen￾etration in alumina and in ceramic materials in general. More￾over, traditional measurement techniques (sin2 c, Z-rotation, and the various modifications20,21) involve rotation or tilt of the specimen (thus a change in the sampled volume) during data acquisition. This is deleterious when a pointwise resolution is sought, like in the case analyzed here. Synchrotron radiation in this case offers an optimal solution, as it couples the possibility to work with a narrow probe and to tune the wavelength of the radiation to increase the penetration within the material. A well￾defined through-thickness residual stress mapping is therefore possible, independently of the particular geometry of the specimen under analysis. In this work, residual stresses of three symmetric ceramic laminates designed and produced according to the procedure proposed by Sglavo and colleagues13–17 were measured by synchrotron radiation X-ray diffraction on station 16.3 at the CCLRC Daresbury Laboratory Synchrotron Radiation Source (DL-SRS). Results are compared with calculated profiles and discussed in terms of the microstructure of the single ceramic laminae. II. Experimental Procedure (1) Specimens Symmetric ceramic laminates were produced from green laminae obtained by tape casting water-based slurries.13–15,17 Alpha-alumina (A-16SG, ALCOA Corp., Pittsburgh, PA) was considered as the fundamental starting material. High-purity mullite (KM101, KCM Corp., Nagoya, Japan) and yttria (3 mol%)-stabilized zirconia (TZ-3YS, TOSOH Corp., Tokyo, Japan) powders were chosen as the second phases. The exper￾imental procedure used to produce the green tapes is described in detail in previous works.13–15,17 Alumina/mullite and al￾umina/zirconia composite layers were prepared. Such compos￾ites were labeled as AMy and AZy, where ‘‘A,’’ ‘‘M,’’ ‘‘Z,’’ and ‘‘y’’ stand for alumina, mullite, zirconia and the volume percent content of mullite or zirconia, respectively. For the production of the laminates, rectangular cards around 50 mm 50 mm were cut from different green laminae, stacked together, and thermo￾compressed at 701C under a pressure of 30 MPa for 15 min. Bars C.-H. Hsueh—contributing editor w Author to whom correspondence should be addressed. e-mail: Matteo.Leoni@unitn.it Manuscript No. 22917. Received March 12, 2007; approved October 12, 2007. Journal J. Am. Ceram. Soc., 91 [4] 1218–1225 (2008) DOI: 10.1111/j.1551-2916.2008.02260.x r 2008 The American Ceramic Society 1218

April 2008 Nondestructive Measurement of the Residual Stress Profile 1219 AM AMZ AzO, 41 um A40.35pm 730.36 AM20.44m A20.35pm AZO, 90 um AM40,93 AM20, 44 um AZ20, 35 um AMIO, 42 um AZ40,522 AZ40, 522 um AZ0.540 symmetry axis Fig. 1. Architecture and composition of the laminates. Dimensions are not in scale of nominal size 10 mm x 50 mm were cut from the green tion of the beam would not completely solve the problems laminates and subjected to the same thermo-compression pro- short wavelength would mean working at small 20 angles to cedure. All samples were then sintered in air at 1600.C for 2 h have a sufficient intensity and well-separated refections, i.e. to different composition and architecture, were prepared as sho s Three different laminates labeled as am. az. and amz. wi work with an elongated, diamond-shaped gauge volume.* Tra- ditional measurement strategies are also not suitable for the An additional set of ca l-mm-thick monolithic laminates was used sin2 y technique, 20-22 it is necessary to collect 20 data at lso produced by stacking together laminae of the same com- various orientations of the specimen with respect to the incom- position, taken from the tapes from which the AM, AZ, and ing beam: the gauge volume would be rotated and ultimately AMZ specimens were made. bathe regions with different stress, composition, and depth, thus making extremely difficult the decoupling of all effects (2) Residual Stress Measurement There is therefore the need for a technique able to perform an The determination of residual stresses necessarily involves the absolute residual strain measurement within a fixed gauge vol measurement of a residual strain. Stresses can be obtained from ume, sliding along the specimen thickness Synchrotron radia strains when a suitable grain interaction model, compatible tion at high-energy beamlines provides a possible solution to the ith the microstructure of the specimen under analysi problem. Two major techniques allow for a pointwise strain available mapping at high-energy synchrotron beamlines, namely (i n this case, the residual strain profile was measured nonde- train scanning based on focusing geometry and a bidimension structively by X-ray diffraction. The large thickness of the al detector(3D strain microscopy; see, e.g., Lienert et al.2)and specimen(with respect to X-ray penetration depth) and the si- D white beam Laue diffraction in energy-dispersive mode(see ation of strai g, Steuwer et al. ). The second one will be used here. By pose serious challenges for traditional laboratory-based tech- ombining Bragg and de broglie equations, a relationship be niques. Choosing a shorter wavelength to increase the penetra- tween the measured energy and the interplanar spacing d can be found once the incidence angle of the incoming beam 0, is defined. In fact one can write Table L. Laminates Design Data: Composition, Thickness, and Calculated Residual Stress of Half lar (Fig. 1) =2dsm0=2m→E=-l1 L Thickness (um) Residual stress(MPa) E 2 sin ei d where 2 is the radiation wavelength. h is the planck constant. 2AM20 and c is the speed of light in vacuum. For practical purposes, 3AM30 data from a large portion of the reciprocal space can be collected when the incidence angle is of the order of a few degrees. The 4AM40 AM20 same measurement can be made both on the stressed specimen to obtain d and on a reference specimen to obtain dr, in order to 7AMO calculate the residual strain(E= d/dr-1). a reference value d is used in place of the traditionally used strain-free do as strains are AZ expected to be present in the composite laminae. By using dr AZ40 433 only the contribution due to interaction between laminae should 2AZ20 130 be obtained 3 AZO By changing the position of the gauge volume along the 4AZ20 thickness of the specimen, a residual strain map can be obtained. 5AZ40 The three main problems of this technique are () the need of AMz high accuracy, as absolute values of the lattice spacing are re- I AZ30 32 quired, (ii) the need for a proper reference which is seldom 2 AZO 41 available, and (iii) the choice of proper elastic constants in order 3AM40 5AZ40 outgoing bear

of nominal size 10 mm 50 mm were cut from the green laminates and subjected to the same thermo-compression pro￾cedure. All samples were then sintered in air at 16001C for 2 h. Three different laminates labeled as AM, AZ, and AMZ, with different composition and architecture, were prepared as shown schematically in Fig. 1 and specified in Table I. An additional set of ca. 1-mm-thick monolithic laminates was also produced by stacking together laminae of the same com￾position, taken from the tapes from which the AM, AZ, and AMZ specimens were made. (2) Residual Stress Measurement The determination of residual stresses necessarily involves the measurement of a residual strain. Stresses can be obtained from strains when a suitable grain interaction model, compatible with the microstructure of the specimen under analysis,23,24 is available. In this case, the residual strain profile was measured nonde￾structively by X-ray diffraction. The large thickness of the specimen (with respect to X-ray penetration depth) and the si￾multaneous variation of strain and composition with the depth pose serious challenges for traditional laboratory-based tech￾niques. Choosing a shorter wavelength to increase the penetra￾tion of the beam would not completely solve the problems: short wavelength would mean working at small 2y angles to have a sufficient intensity and well-separated reflections, i.e. to work with an elongated, diamond-shaped gauge volume.z Tra￾ditional measurement strategies are also not suitable for the proposed problem. For instance, in the well-known and widely used sin2 c technique,20–22 it is necessary to collect 2y data at various orientations of the specimen with respect to the incom￾ing beam: the gauge volume would be rotated and ultimately bathe regions with different stress, composition, and depth, thus making extremely difficult the decoupling of all effects. There is therefore the need for a technique able to perform an absolute residual strain measurement within a fixed gauge vol￾ume, sliding along the specimen thickness. Synchrotron radia￾tion at high-energy beamlines provides a possible solution to the problem. Two major techniques allow for a pointwise strain mapping at high-energy synchrotron beamlines, namely (i) strain scanning based on focusing geometry and a bidimension￾al detector (3D strain microscopy; see, e.g., Lienert et al. 25) and (ii) white beam Laue diffraction in energy-dispersive mode (see, e.g., Steuwer et al. 26). The second one will be used here. By combining Bragg and de Broglie equations, a relationship be￾tween the measured energy and the interplanar spacing d can be found once the incidence angle of the incoming beam yi is defined. In fact one can write l ¼ 2d sin yi ¼ hc E ) E ¼ hc 2 sin yi 1 d (1) where l is the radiation wavelength, h is the Planck constant, and c is the speed of light in vacuum. For practical purposes, data from a large portion of the reciprocal space can be collected when the incidence angle is of the order of a few degrees. The same measurement can be made both on the stressed specimen to obtain d and on a reference specimen to obtain dr, in order to calculate the residual strain (e 5 d/dr1). A reference value dr is used in place of the traditionally used strain-free d0 as strains are expected to be present in the composite laminae. By using dr only the contribution due to interaction between laminae should be obtained. By changing the position of the gauge volume along the thickness of the specimen, a residual strain map can be obtained. The three main problems of this technique are (i) the need of high accuracy, as absolute values of the lattice spacing are re￾quired, (ii) the need for a proper reference, which is seldom available, and (iii) the choice of proper elastic constants in order AZ0, 41 µm AM20, 44 µm AM30, 48 µm AM40, 43 µm AM20, 44 µm AM10, 42 µm AZ0, 540 µm AZ30, 36 µm AZ0, 41 µm AM40, 93 µm AZ0, 41 µm AZ40, 522 µm AZ40, 35 µm AZ20, 35 µm AZ0, 90 µm AZ20, 35 µm AZ40, 522 µm AM AZ AMZ symmetry axis Fig. 1. Architecture and composition of the laminates. Dimensions are not in scale. Table I. Laminates Design Data: Composition, Thickness, and Calculated Residual Stress of Half Laminate (Fig. 1) Layer Thickness (mm) Residual stress (MPa) AM 1 AM0 41 76 2 AM20 44 178 3 AM30 48 265 4 AM40 43 373 5 AM20 44 178 6 AM10 42 4 7 AM0 540 76 AZ 1 AZ40 35 89 2 AZ20 35 130 3 AZ0 90 448 4 AZ20 35 130 5 AZ40 522 89 AMZ 1 AZ30 36 32 2 AZ0 41 336 3 AM40 93 717 4 AZ0 41 336 5 AZ40 522 178 z Actually the shape of the gauge volume is that of a prism with rhombic base, with the prism axis parallel to the goniometer axis. The sides of the prism are defined by the incoming and the outgoing beam, whereas the height is equal to the primary (and secondary) beam width. April 2008 Nondestructive Measurement of the Residual Stress Profile 1219

Journal of the American Ceramic Society-Leoni et al. Vol 91. No 4 to calculate a residual stress profile from the measured strain Table Il. Linear Shrinkage (%), Density(p), Porosity ( profile(Hooke's law is used Youngs Modulus(E), and Poissons Ratio(v)of the Hom Monolithic laminates, i.e of uniform composition, were used geneous Laminates After Sintering as reference samples for the measurement of dr. To reconstruct the residual stress profile, elastic constants also have to be cho- Laminate rinkage p(g/em,) P(%) sen properly; however, as Hooke's law is linear, the residual 3.95 ress profile matches the residual strain profile just a scale fac- 394±140.230 AZIo 4.17 373+110.237 tor is present), provided that a single phase or a uniform com- Az20 4.37 342+170.24 osition is considered AZ30 4.59 330+5 For a quantitative determination of the stresses, a plane state 0.2 AZ40 19.5 4.79 of strain is considered in the laminae. Nothing is said about the 303+90.257 ansverse direction, not thoroughly investigated in the present AMIO 3. 8.2304+130.231t work(e.g, 033 is expected in the vicinity of the interfaces). Un- AM20 3.32 13.1264+60.232 3.13 der the hypotheses of planar state of strain, the Voigt model can AM40 12.0 2.90 20.6 168+4 0.229 ca hes strain to be transferred in a plane from grain to grain.The Estimated by numerical analysis. ty was taken into account as proposed by Tanak et al.: mechanical anisotropy of the chosen materials is low thus limited error would be made if average macros constants would be used in place of the voigt average width) was chosen, respectively, for the transversal and the lon- values of the elastic constants used in the calculation itudinal modes. An incidence angle of 5(half of the internal vided in Table II(obtained from Sglavo and colleagues a angle of the gauge volume rhomb) was used in all cases. The size Bertoldi") f the primary beam was selected by means of motor-controlled Diffraction measurements were conducted using the white- crossed slits, whereas the diffracted beam was shaped by a dou- eam setup available at the Daresbury Laboratory Srs ble crossed-slits assembly(50 cm distance between the slits). The (Daresbury, U. K )on the 16.3 beamline. As shown in Fig. 2, ame size chosen for the primary beam was also selected on the everal possible orientations of the specimen with respect to the diffracted arm, for both secondary slit assemblies beam are possible, leading to the measurement of stress com- Even if the synchrotron beam is virtually parallel, a residual ponents along different directions in space. Despite the great divergence is nevertheless always present in the direct beam care taken in the alignment, the actual setup did not allow the the actual beam size on the specimen is thus larger than the pecimen to be rotated about the center of the gauge volume theoretical one. This would implicitly add a smoothin with a sufficient accuracy, better than the smallest size of the effect on the result due to the convolution between the actua beam. Just one operating mode (ie, one component of the probe size function and the true strain/stress curves. To rain tensor) can thus be measured at a time for each specimen: limit possible cross-talk effects, the spacing between adjacent provides the less biased estimate(albeit limited just to one points was chosen as double with respect to the beam size ction) for the residual stress in the gauge volume (ie.,20m) planes T口 Measured planes 口 Measured planes E33 transversal longitudinal mode ll longitudinal mode i Fig 2 imen and beam orientations(3D representation and side view). For the longitudinal modes, laminae are supposed to be stacked perpen- dicular to the viewing direction in the side view drawings)

to calculate a residual stress profile from the measured strain profile (Hooke’s law is used). Monolithic laminates, i.e., of uniform composition, were used as reference samples for the measurement of dr. To reconstruct the residual stress profile, elastic constants also have to be cho￾sen properly; however, as Hooke’s law is linear, the residual stress profile matches the residual strain profile (just a scale fac￾tor is present), provided that a single phase or a uniform com￾position is considered. For a quantitative determination of the stresses, a plane state of strain is considered in the laminae. Nothing is said about the transverse direction, not thoroughly investigated in the present work (e.g., s33 is expected in the vicinity of the interfaces). Un￾der the hypotheses of planar state of strain, the Voigt model can be used to describe the grain interaction in the system, as it as￾sumes strain to be transferred in a plane from grain to grain. The effect of porosity was taken into account as proposed by Tanaka et al. 27: mechanical anisotropy of the chosen materials is low; thus limited error would be made if average macroscopic elastic constants would be used in place of the Voigt average.28 Actual values of the elastic constants used in the calculation are pro￾vided in Table II (obtained from Sglavo and colleagues13–16 and Bertoldi 17). Diffraction measurements were conducted using the white￾beam setup available at the Daresbury Laboratory SRS (Daresbury, U.K.) on the 16.3 beamline. As shown in Fig. 2, several possible orientations of the specimen with respect to the beam are possible, leading to the measurement of stress com￾ponents along different directions in space. Despite the great care taken in the alignment, the actual setup did not allow the specimen to be rotated about the center of the gauge volume with a sufficient accuracy, better than the smallest size of the beam. Just one operating mode (i.e., one component of the strain tensor) can thus be measured at a time for each specimen: this provides the less biased estimate (albeit limited just to one direction) for the residual stress in the gauge volume. A beam size of 10 mm 4 mm and 4 mm 10 mm (height by width) was chosen, respectively, for the transversal and the lon￾gitudinal modes. An incidence angle of 51 (half of the internal angle of the gauge volume rhomb) was used in all cases. The size of the primary beam was selected by means of motor-controlled crossed slits, whereas the diffracted beam was shaped by a dou￾ble crossed-slits assembly (50 cm distance between the slits). The same size chosen for the primary beam was also selected on the diffracted arm, for both secondary slit assemblies. Even if the synchrotron beam is virtually parallel, a residual divergence is nevertheless always present in the direct beam: the actual beam size on the specimen is thus larger than the theoretical one. This would implicitly add a smoothing effect on the result due to the convolution between the actual probe size function and the true strain/stress curves. To limit possible cross-talk effects, the spacing between adjacent points was chosen as double with respect to the beam size (i.e., 20 mm). Table II. Linear Shrinkage (%), Density (q), Porosity (P), Young’s Modulus (E), and Poisson’s Ratio (m) of the Homo￾geneous Laminates After Sintering Laminate Shrinkage r (g/cm3 ) P (%) E (GPa) n AZ0 17.0 3.95 o1 394714 0.230 AZ10 4.17 o1 373711 0.237 AZ20 18.0 4.37 o1 342717 0.244 AZ30 4.59 o1 33075 0.251 AZ40 19.5 4.79 o1 30379 0.257 AM10 3.58 8.2 304713 0.231w AM20 14.5 3.32 13.1 26476 0.232w AM30 3.13 16.4 20875 0.232w AM40 12.0 2.90 20.6 16874 0.229w w Estimated by numerical analysis.17 Measured planes Measured planes Measured planes transversal longitudinal mode II longitudinal mode I ε33 ε22 ε11 1 2 3 2 1 3 2 3 1 Fig. 2. Specimen and beam orientations (3D representation and side view). For the longitudinal modes, laminae are supposed to be stacked perpen￾dicular to the viewing direction in the side view drawings). 1220 Journal of the American Ceramic Society—Leoni et al. Vol. 91, No. 4

April 2008 Nondestructive Measurement of the Residual Stress Profile To achieve high precision and accuracy, the goniometer and pecimen were carefully aligned. The goniometer was aligned (4) using the X-ray beam and checked against the NIST SRM640a silicon standard at the chosen incidence angle. the surface of the specimen was aligned both optically (using a telescope aligned 0z =-7oz (5) with the center of rotation of the goniometer)and by using the X-rays, by moving, tilting, and rocking the specimen in order to reach the condition where the direct beam intensity is cut half- In the formulae, A and Z symbols designate, respectively, al way by the specimen itself. A thick aluminum block was placed mina and zirconia; fA and fz are the volume fractions, aa and in the primary beam path to attenuate the intensity and avoid az the linear thermal expansion coefficients (8. x 10- and detector breakage. It is well known that alumina darkens under 10.5 x 10-K for alumina and zirconia), and shear (G)and an X-ray beam: such a phenomenon was used to check for pos- bulk(K) moduli can be obtained from Young,'s modulus E and sible misalignment of the specimen, as a clear persistent trace Poissons ratio v as was visible on the samples after exposure to the X-ray beam. Movements were always imposed in the EA E EA avoid backlash effects (1+vA) (3) Detector Calibration K= The energy-dispersive detector does not directly provide readout 3(1-2vz) in terms of intensity versus interplanar spacing (or something uivalent to it). Conversely, it gives a signal from 1024 char AT is the temperature interval in which stresses are supposed to develop. As the temperature interval can be large, stresses in unknown)energy. a proper energy versus channel calibration is the minoritary phase can be quite important: just as an example, herefore necessary. Once the incidence angle is known, this in- values as high as ca 500 MPa are expected for zirconia in the formation can be directly related to the interplanar spacing(ac- AZ20 laminate when a AT= 1000 K(reasonable if sintering is cording to Eq. (1). Calibration was performed by collecting the involved)is chosen. A completely analogous reasoning holds X-ray emission patterns of Cu, Tb, Mo, and Ag stimulated by a k). hen zirconia is replaced by mullite(for which aM=5. 1 x 10 radioactive--Am source. Each emission peak was fitted with a pseudo-Voigt curve to obtain the position of the peak centroid Figure 3 shows the bounding values expected on the basis of A linear relationship was found between channel and emission Eqs.(2)5) for the AZ and AM composites used in the present energy taken from the literature, at least in the wavelength ork. Data were obtained by assuming a AT of 1125"C and the range useful for the present measurement. final composites being fully dense, the temperature interval was nosen by assuming specimen contraction to start at 1150.C, in II. Results and discussion Expected values are also compared with experimental data. Ex (1) Reference Interplanar spacing and Intralaminar Stresses perimental strains were calculated from the measured dr by us- g the literature strain-free interplanar spacing do(e= dr/do-1). geneous laminates allowed the determination, in all measured and converted into stresses assuming the voigt model of grain patterns, of the reference interplanar spacing d for the visible peaks of alumina, zirconia, and mullite. Only the most intense he various phases, i.e., yttria-stabilized zirconia, 48-0224 reflections were characterized in detail. Interplanar data mea- alumina, 46-1212: mullite, 79-1275. Cell parameters of sured in single phases of a multiphase solid depart from litera- ttria-stabilized zirconia were properly corrected by means of ture data measured on uniform strain-free powder specime the formulae of Toraya(a=0.35963+0.000227x nm (do). Such a discrepancy can have different origins: it can be due c=0.51892-0000256x nm, where x is the mol% of YO,s)to (do). Such a discrepan, the presence of impurities(diluted sys- take the actual content of the stabilizer into account. Corr to instrumental effects tems obeying, e.g., Vegard's law), or possible interaction among were modified accordingly. It should be noted that grains (of the same or different composition)in the sintered data relative to mullite are intrinsically less accurate due to structure In our case, the last mechanism can probably com- the large stoichiometry variations possible for that material re pletely explain the observed trends. Grains in a sintered compact flected into variations in cell parameter are in fact forced to cool down in a constrained environment. Experimental results are compared in Fig 3 with the kreher local intralaminar residual stresses can develop owing to differ- Pompe model (errors in Fig. 3 were calculated on the basis of the ences, e.g., in orientation(even in a homogeneous system. if the exist von Mises criterion is not fulflled)or in thermal expansion and Trends are well matched for the lower stabilizer contents, where mechanical properties(when different crystalline phases are sin- as departures are observed with increasing content of the second tered together) of the grains. Several models exist for the pre- diction of the macroscopic and microscopic behavior of a Several ar ents can be used to justify the experimental composite. In the case oto wo-phase particulate composites, the percolation limit for zirconia in alumina is around 16%(see of such stresses. Supporting literature data exist for the case e.g., Pecharroman et al.): above that limit, the hypotheses on alumina-zirconia composites such as the az ones used here which the Kreher-Pompe model is based (particles embedded in According to the Kreher-Pompe model, the bounding values a matrix) may not be verified there is no for the stress in Alumina(oa and oA)and zirconia(oz and the Az40(and, in lesser part, the Az20) laminate would follow oi)in an Azy homogeneous composite laminate can be eval he Kreher-Pompe model predictions. The maximum expected uated by considering the Hashin-Shtrikman bounds for the stress in the zirconia phase for AT=1125 K is of the order of 500 MPa, about two times the observed data. The sintering be havior of zirconia and alumina is influenced by the mutual pres- ence and by the simultaneous action of an external (or residual fz/KA+fA/KZ+3/(4GA) A az)AT(2) load. The temperature at which the plastic flow limit for zirconia and alumina starts being higher than the constraint residual stresses is thus expected to be influenced by the composition, Uz/KA +fA/KZ+3/(4Gz) (xA-xz)△T(3) and to be different for the two phases. It cannot therefore be excluded that an effective AT, lower than 1125 K, is active on

To achieve high precision and accuracy, the goniometer and specimen were carefully aligned. The goniometer was aligned using the X-ray beam and checked against the NIST SRM640a silicon standard at the chosen incidence angle. The surface of the specimen was aligned both optically (using a telescope aligned with the center of rotation of the goniometer) and by using the X-rays, by moving, tilting, and rocking the specimen in order to reach the condition where the direct beam intensity is cut half￾way by the specimen itself. A thick aluminum block was placed in the primary beam path to attenuate the intensity and avoid detector breakage. It is well known that alumina darkens under an X-ray beam: such a phenomenon was used to check for pos￾sible misalignment of the specimen, as a clear persistent trace was visible on the samples after exposure to the X-ray beam. Movements were always imposed in the same direction, to avoid backlash effects. (3) Detector Calibration The energy-dispersive detector does not directly provide readout in terms of intensity versus interplanar spacing (or something equivalent to it). Conversely, it gives a signal from 1024 chan￾nels, each one providing information on a well-defined (a priori unknown) energy. A proper energy versus channel calibration is therefore necessary. Once the incidence angle is known, this in￾formation can be directly related to the interplanar spacing (ac￾cording to Eq. (1)). Calibration was performed by collecting the X-ray emission patterns of Cu, Tb, Mo, and Ag stimulated by a radioactive 235Am source. Each emission peak was fitted with a pseudo-Voigt curve to obtain the position of the peak centroid. A linear relationship was found between channel and emission energy taken from the literature,29 at least in the wavelength range useful for the present measurement. III. Results and Discussion (1) Reference Interplanar Spacing and Intralaminar Stresses Diffraction measurements conducted on B1-mm-thick homo￾geneous laminates allowed the determination, in all measured patterns, of the reference interplanar spacing dr for the visible peaks of alumina, zirconia, and mullite. Only the most intense reflections were characterized in detail. Interplanar data mea￾sured in single phases of a multiphase solid depart from litera￾ture data measured on uniform strain-free powder specimens (d0). Such a discrepancy can have different origins: it can be due to instrumental effects, the presence of impurities (diluted sys￾tems obeying, e.g., Vegard’s law), or possible interaction among grains (of the same or different composition) in the sintered structure. In our case, the last mechanism can probably com￾pletely explain the observed trends. Grains in a sintered compact are in fact forced to cool down in a constrained environment: local intralaminar residual stresses can develop owing to differ￾ences, e.g., in orientation (even in a homogeneous system, if the von Mises criterion is not fulfilled) or in thermal expansion and mechanical properties (when different crystalline phases are sin￾tered together) of the grains. Several models exist for the pre￾diction of the macroscopic and microscopic behavior of a composite. In the case of two-phase particulate composites, the Kreher–Pompe model30,31 can be used for the estimation of such stresses. Supporting literature data exist for the case of alumina–zirconia composites such as the AZ ones used here.31 According to the Kreher–Pompe model, the bounding values for the stress in Alumina (sþ A and s A) and zirconia (sþZ and s Z ) in an AZy homogeneous composite laminate can be eval￾uated by considering the Hashin–Shtrikman bounds for the composite moduli30–32 and imposing stress balance: sþ A ¼ 3fZ ð Þ fZ=KA þ fA=KZ þ 3=ð4GAÞ ð Þ aA aZ DT (2) s A ¼ 3fZ ð Þ fZ=KA þ fA=KZ þ 3=ð4GZÞ ð Þ aA aZ DT (3) sþ Z ¼ fA fZ sþ A (4) s Z ¼ fA fA s Z (5) In the formulae, A and Z symbols designate, respectively, al￾umina and zirconia; fA and fZ are the volume fractions, aA and aZ the linear thermal expansion coefficients (8.1 106 and 10.5 106 K1 for alumina and zirconia), and shear (G) and bulk (K) moduli can be obtained from Young’s modulus E and Poisson’s ratio n as GA ¼ EA 2 1ð Þ þ nA ; GZ ¼ EZ 2 1ð Þ þ nZ ; KA ¼ EA 3 1ð Þ 2nA ; KZ ¼ EZ 3 1ð Þ 2nZ DT is the temperature interval in which stresses are supposed to develop. As the temperature interval can be large, stresses in the minoritary phase can be quite important: just as an example, values as high as ca. 500 MPa are expected for zirconia in the AZ20 laminate when a DT 5 1000 K (reasonable if sintering is involved) is chosen. A completely analogous reasoning holds when zirconia is replaced by mullite (for which aM 5 5.1 106 K1 ). Figure 3 shows the bounding values expected on the basis of Eqs. (2)–(5) for the AZ and AM composites used in the present work. Data were obtained by assuming a DT of 11251C and the final composites being fully dense; the temperature interval was chosen by assuming specimen contraction to start at 11501C, in accordance with the dilatometric measurements of Bertoldi17 Expected values are also compared with experimental data. Ex￾perimental strains were calculated from the measured dr by us￾ing the literature strain-free interplanar spacing d0 (e 5 dr/d01), and converted into stresses assuming the Voigt model of grain interaction. Literature d0 were fetched from the PDF2 cards for the various phases, i.e., yttria-stabilized zirconia, 48–0224; alumina, 46–1212; mullite, 79–1275. Cell parameters of yttria-stabilized zirconia were properly corrected by means of the formulae of Toraya33 (a 5 0.3596310.000227x nm, c 5 0.518920.000256x nm, where x is the mol% of YO1.5) to take the actual content of the stabilizer into account. Corre￾sponding d0 were modified accordingly. It should be noted that d0 data relative to mullite are intrinsically less accurate due to the large stoichiometry variations possible for that material re- flected into variations in cell parameters. Experimental results are compared in Fig. 3 with the Kreher– Pompe model (errors in Fig. 3 were calculated on the basis of the estimated standard deviations of the fit). Discrepancies exist. Trends are well matched for the lower stabilizer contents, where￾as departures are observed with increasing content of the second phase. Several arguments can be used to justify the experimental observation. Focusing on the AZ composites, it is known that the percolation limit for zirconia in alumina is around 16% (see, e.g., Pecharroma´n et al. 34): above that limit, the hypotheses on which the Kreher–Pompe model is based (particles embedded in a matrix) may not be verified. Hence, there is no guarantee that the AZ40 (and, in lesser part, the AZ20) laminate would follow the Kreher–Pompe model predictions. The maximum expected stress in the zirconia phase for DT 5 1125 K is of the order of 500 MPa, about two times the observed data. The sintering be￾havior of zirconia and alumina is influenced by the mutual pres￾ence and by the simultaneous action of an external (or residual) load. The temperature at which the plastic flow limit for zirconia and alumina starts being higher than the constraint residual stresses is thus expected to be influenced by the composition, and to be different for the two phases. It cannot therefore be excluded that an effective DT, lower than 1125 K, is active on April 2008 Nondestructive Measurement of the Residual Stress Profile 1221

1222 Journal of the American Ceramic Society-Leoni et al. Vol 91. No 4 600 400 AM40 AM20 AZO AZ AZ40 20 mm Fig 4. Comparison of the length of different strips after sintering. The tarting size of the green sample (It The presence of residual porosity strongly influences the irconia content(wo‰%) macroscopic elastic properties of the AM set of homogeneous composites(cf Table ID). As a consequence, as shown in Fig. 5 elastic moduli for alumina-mullite composites are far from the theoretical limits(Voigt-Reuss and Hashin-Shtrikman 2.37)for an equivalent fully dense homogeneous specimen. Nevertheless porosity should not influence the strain measurements carried 0 out in the present work as they involve the interaction stresses among crystalline grains, keeping in mind that diffraction is in -200·9 sensitive to voids. This does not mean that porosity plays no role in determining the actual stress level state in the grains Figure 5 also reports the elastic modulus trend for the al- mullite content (vols s250 nia and or alumina, cont ues. This aspect is not consid nor by viable analytical alter ues for zirconia are approached when a AT of ca 700 K is used An analogous reasoning is valid for the AM 100 composites. Again, the trend is matched for the guest phase and mullite content(vol%) the absolute values are lower than the expectations (cf. Fig. 3) Similitude between Az and am composites can also be evi- denced king at the sintering behavior. a good sintering behavior of homogeneous laminates has already been reported orks 13,15 but some further information can be useful here. Figure 4 shows the final length of several sintered tapes of homogeneous composition compared with the starting length(100 mm)of a green sample Linear shrinkage values after sintering are included in the range between 12% and 20%, the exact values being reported in Table Il. Although differences 200 among various compositions do not appear to be very large, the corresponding final densities can be very dissimilar because the volume reduction is approximately three times the linear con- traction: this is confirmed by the density results proposed in Table ii where elastic modulus and poissons ratio are also from independent measurements previously performed One can observe that in alumina-mullite composites a resid- ual porosity is always present, its amount increasing with mullite zirconia content(vol%) content. Such behavior can be accounted for by the relatively Fig. 5. Avert 'expermental values are represented by points.The lower sintering temperature used in the present work(1600C)in macroscopic Youngs modulus(E) for(a) AM and(b mparison with higher temperature schedules usually used for Voigt and Reuss limits (dotted lines)and upper and lower Hashin mullite consolidation .3 bounds(continuous lines) for a fully dense composite are also shown

zirconia and/or alumina, contributing to lower actual stress val￾ues. This aspect is not considered by the Kreher–Pompe model nor by viable analytical alternatives to it. The experimental val￾ues for zirconia are approached when a DT of ca. 700 K is used in Eqs. (2) and (3). An analogous reasoning is valid for the AM homogeneous composites. Again, the trend is matched for the guest phase and the absolute values are lower than the expectations (cf. Fig. 3). Similitude between AZ and AM composites can also be evi￾denced by looking at the sintering behavior. A good sintering behavior of homogeneous laminates has already been reported in previous works,13,15 but some further information can be useful here. Figure 4 shows the final length of several sintered tapes of homogeneous composition compared with the starting length (100 mm) of a green sample. Linear shrinkage values after sintering are included in the range between 12% and 20%, the exact values being reported in Table II. Although differences among various compositions do not appear to be very large, the corresponding final densities can be very dissimilar because the volume reduction is approximately three times the linear con￾traction: this is confirmed by the density results proposed in Table II where elastic modulus and Poisson’s ratio are also reported as from independent measurements previously performed.13–17 One can observe that in alumina–mullite composites a resid￾ual porosity is always present, its amount increasing with mullite content. Such behavior can be accounted for by the relatively lower sintering temperature used in the present work (16001C) in comparison with higher temperature schedules usually used for mullite consolidation.35,36 The presence of residual porosity strongly influences the macroscopic elastic properties of the AM set of homogeneous composites (cf. Table II). As a consequence, as shown in Fig. 5, elastic moduli for alumina–mullite composites are far from the theoretical limits (Voigt–Reuss and Hashin–Shtrikman32,37) for an equivalent fully dense homogeneous specimen. Nevertheless, porosity should not influence the strain measurements carried out in the present work as they involve the interaction stresses among crystalline grains, keeping in mind that diffraction is in￾sensitive to voids. This does not mean that porosity plays no role in determining the actual stress level state in the grains. Figure 5 also reports the elastic modulus trend for the al￾umina–zirconia homogeneous laminates. Data are well within 20 mm AZ40 AZ20 AZ0 AM20 AM40 green Fig. 4. Comparison of the length of different strips after sintering. The starting size of the green sample (100 mm) is shown on the top. 0 10 20 30 40 0 10 20 30 40 −800 −600 −400 −200 0 200 400 600 σ (MPa) zirconia content (vol%) (a) −800 −600 −400 −200 0 200 400 σ (MPa) mullite content (vol%) (b) Fig. 3. Residual stress in the phases as a function of composition in (a) alumina–zirconia and (b) alumina–mullite composites. Limiting stress values for ideal composites have been calculated according to the Kreher–Pompe model (see text for details). 0 20 40 60 80 100 0 20 40 60 80 100 0 50 100 150 200 250 300 350 400 450 E (GPa) mullite content (vol%) (a) 0 50 100 150 200 250 300 350 400 450 E (GPa) zirconia content (vol%) (b) Fig. 5. Average macroscopic Young’s modulus (E) for (a) AM and (b) AZ composites. Experimental values are represented by points. The Voigt and Reuss limits (dotted lines) and upper and lower Hashin bounds (continuous lines) for a fully dense composite are also shown. 1222 Journal of the American Ceramic Society—Leoni et al. Vol. 91, No. 4

April 2008 Nondestructive Measurement of the Residual Stress Profile the voigt and Reuss bounds, and are extremely well approxi mated by the Hashin-Shtrikman bounds; however, differences between all those models are small. Porosity does not play an important role in determining the macroscopic properties of the homogeneous AZ composites: elastic moduli for the pure phase are quite close to literature data. A strong (if not dominant) 600 fluence is instead observed for the AM composites (2) Identification of the laminae As thickness and strain-free interplanar spacing of the various laminae in the composite laminates are different, the ability to univocally identify a lamina from its diffraction pattern is of paramount importance. Direct comparison of the patterns of specimens and reference(homogeneous laminates) gives just w一 qualitative information. Quantitative evaluation can be pe 151.61718192.02.1 formed by monitoring the trend of alumina peak intensities in d(A) to identify the transition of the gauge volume from one lamina Fig. 7. Diffraction pattern collected in the middle of the monolithic to the neighboring one. To allow for this, raw data should be AZ40 layer: raw data (points)and fitting results (line). The residual suitably processed in order to take the(fast )decay of the synch (difference between data and model) is reported below rotron beam intensity(very short lifetime) into account: an ab- solute scaling was possible on the basis of an exponential decay cording to the published theory According to the measure- of the beam current monitor with time evaluated by fitting the ment method, which involves diffracting atomic planes to be (measured) beam current data during an entire measurement parallel to the surface, the measured strain should be the tensor component perpendicular to the surface and should be indepen Figure 6 shows as an example the case of the (104)a-Al203 dent of the peak chosen for the analysis(constant strain). To test peak in the AM laminate. The transitions corresponding to the edges of each laminae as shown in Fig. I are well matched when was performed, and gave the same result, within errors, for all both integral intensity of the (104) reflection and the corre- visible peaks (noise being higher for lower intensity peaks) sponding interplanar spacing dr (evaluated on the homogeneou Again, the Voigt model of grain interaction can be assumed laminates) are considered for the determination of residual stress from residual strain data (A) Residual Stress Profile: As a preliminary test, the X- in the composite material: we should, however, keep in mind ay beam was scanned along the entire thickness of the Az lam that the voigt model represents a limiting case of grain intera inate entering from the top surface and working in transversal tion and in real cases a more complex behavior can occur 23.24 mode(as shown in Fig. 2). Both zirconia and alumina signals are The reference interplanar spacing d was used to obtain the data visible in the patterns of the composite laminae, allowing the Fig. 8. Errors associated with single phases and residual strain in both phases to be evaluated. However, the in- weighted average(shown in Fig. 8 and the following figures) tensity of useful (e.g. nonoverlapping) zirconia peaks is quite were calculated on the basis of the estimated standard deviations w in the patterns relative to composite layers, even after long counting times. As an example, Fig. 7 shows the diffraction sidered as an independent variable, was also propagated. It pattern collected in the middle of the AZ40 layer, this being the should be noted that the error on dr almost always dominates one containing the highest zirconia load: the (103) peak can be This is a consequence of the fact that peaks in the monoliths are much broader than the corresponding ones in the laminates. clearly identified and separated, all others being severely over- Errors in the minority phases were observed to be always bigger average than the corresponding ones in alumina, but a much higher am- (isotropic) residual stress can be obtained from the residual plification by the elastic modulus is active in the alumina matrix strain data, leading to the trends presented in Fig. 8 for alumin nd zirconia. Here the determined stresses are compared with the macroscopic interlaminar residual stresses calculated on the basis of the laminate architecture and materials properties ac- 2.555 50 -400 30 2.551 2004006008001000 2.550 Fig 8. Through-thickness residual stress profile in the Az specime he(104)reflection of alumina(dots)and ults for alumina(open dots) and zirconia( squares). The average stress is also shown(stars). The line connecting the points serves as a guide for the eye. For clarity, only the errors associated with the weighted average Vertical lines identify the ideal boundaries of the lami tress are shown

the Voigt and Reuss bounds, and are extremely well approxi￾mated by the Hashin–Shtrikman bounds; however, differences between all those models are small. Porosity does not play an important role in determining the macroscopic properties of the homogeneous AZ composites: elastic moduli for the pure phases are quite close to literature data. A strong (if not dominant) in- fluence is instead observed for the AM composites. (2) Identification of the Laminae As thickness and strain-free interplanar spacing of the various laminae in the composite laminates are different, the ability to univocally identify a lamina from its diffraction pattern is of paramount importance. Direct comparison of the patterns of specimens and reference (homogeneous laminates) gives just qualitative information. Quantitative evaluation can be per￾formed by monitoring the trend of alumina peak intensities in the depth-resolved measurements. Intensity changes allow one to identify the transition of the gauge volume from one lamina to the neighboring one. To allow for this, raw data should be suitably processed in order to take the (fast) decay of the synch￾rotron beam intensity (very short lifetime) into account: an ab￾solute scaling was possible on the basis of an exponential decay of the beam current monitor with time evaluated by fitting the (measured) beam current data during an entire measurement shift. Figure 6 shows as an example the case of the (104) a-Al2O3 peak in the AM laminate. The transitions corresponding to the edges of each laminae as shown in Fig. 1 are well matched when both integral intensity of the (104) reflection and the corre￾sponding interplanar spacing dr (evaluated on the homogeneous laminates) are considered. (A) Residual Stress Profile: As a preliminary test, the X￾ray beam was scanned along the entire thickness of the AZ lam￾inate entering from the top surface and working in transversal mode (as shown in Fig. 2). Both zirconia and alumina signals are visible in the patterns of the composite laminae, allowing the residual strain in both phases to be evaluated. However, the in￾tensity of useful (e.g., nonoverlapping) zirconia peaks is quite low in the patterns relative to composite layers, even after long counting times. As an example, Fig. 7 shows the diffraction pattern collected in the middle of the AZ40 layer, this being the one containing the highest zirconia load: the (103) peak can be clearly identified and separated, all others being severely over￾lapped. Using the average (i.e., isotropic) elastic constants, an average (isotropic) residual stress can be obtained from the residual strain data, leading to the trends presented in Fig. 8 for alumina and zirconia. Here the determined stresses are compared with the macroscopic interlaminar residual stresses calculated on the basis of the laminate architecture and materials properties ac￾cording to the published theory.13–15 According to the measure￾ment method, which involves diffracting atomic planes to be parallel to the surface, the measured strain should be the tensor component perpendicular to the surface and should be indepen￾dent of the peak chosen for the analysis (constant strain). To test this hypothesis, a full pattern fit with independent peak positions was performed, and gave the same result, within errors, for all visible peaks (noise being higher for lower intensity peaks). Again, the Voigt model of grain interaction can be assumed for the determination of residual stress from residual strain data in the composite material: we should, however, keep in mind that the Voigt model represents a limiting case of grain interac￾tion and in real cases a more complex behavior can occur.23,24 The reference interplanar spacing dr was used to obtain the data shown in Fig. 8. Errors associated with single phases and weighted average (shown in Fig. 8 and the following figures) were calculated on the basis of the estimated standard deviations of fitted peak positions. In the calculation, the error on dr, con￾sidered as an independent variable, was also propagated. It should be noted that the error on dr almost always dominates. This is a consequence of the fact that peaks in the monoliths are much broader than the corresponding ones in the laminates. Errors in the minority phases were observed to be always bigger than the corresponding ones in alumina, but a much higher am￾plification by the elastic modulus is active in the alumina matrix phase. 0 100 200 300 400 0 10 20 30 40 50 60 70 80 90 100 2.550 2.551 2.552 2.553 2.554 2.555 relative intensity (a.u.) depth (µm) d (Å) Fig. 6. Integral intensity of the (104) reflection of alumina (dots) and corresponding reference interplanar spacing dr (line) for the AM spec￾imen. The line connecting the circles serves just as a guide for the eye. Vertical lines identify the ideal boundaries of the laminae. 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 0 d (Å) ∆I 0 200 400 600 800 1000 Intensity (a.u.) Fig. 7. Diffraction pattern collected in the middle of the monolithic AZ40 layer: raw data (points) and fitting results (line). The residual (difference between data and model) is reported below. 0 200 400 600 800 1000 −800 −600 −400 −200 0 200 400 σ (MPa) depth (µm) Fig. 8. Through-thickness residual stress profile in the AZ specimen measured in transversal mode: design values (line) and measurement re￾sults for alumina (open dots) and zirconia (squares). The average stress is also shown (stars). The line connecting the points serves as a guide for the eye. For clarity, only the errors associated with the weighted average stress are shown. April 2008 Nondestructive Measurement of the Residual Stress Profile 1223

1224 Journal of the American Ceramic Society--Leoni et al Vol 91. No 4 200 -200 -200 -400 -600 50100150200250300 50200250300 depth(um depth(um) hrough-thickness residual stress profile in the AZ specimen. Fig, 10. Thr residual stress profile in the AMZ speci in longitudinal mode I for alumina(open dots) and zirconia mode I for alumina(open dots), zirconia The average stress is also shown (stars). The continuous line monds). The average stress is also shown nds to calculated macroscopic interlaminar stresses. The line stars). The corresponds to calculated macroscopic in- ng the points serves as a guide for the eye. terlaminar stresses. The line connecting the points serves as a guide for the eye. Local deviations due to the noise in the experimental data surement conducted in longitudinal mode I(cf. Fig. 2)are ( the X-ray beam travels over a long distance within the material shown in Fig. 10. the laminae are well identified(as suggested n order to reach the deepest buried layers) are evident. More- by the large phase contrast shown in Table D) and a stress profile over, large differences appear between alumina and zirconia can be reconstructed. Differences are again evident between zir- tresses,with a slight tendency to separation into tensile and conia, alumina, and mullite in the composite laminae. compressive components. The weighted average of zirconia and To confirm that problems can come from the cosintering alumina stresses appears to be closer to the design value. This process, microscopy observations were made on the very same result is partly in contradiction with the expectations: by choos- lamina sintered in a homogeneous laminate and in a multilayer ing the reference interplanar spacing, similar trends would be one. As an example, Fig. ll shows scanning electron micro- expected for alumina and zirconia in the az20 and A z40 layer graphs(SEM) of polished cross sections of the AM40 lamina in The observed differences suggest that d is insufficient or unable the monolithic and in the AMz specimens. Differences both to properly account for the interlaminar residual stresses due the grain size and in the size and shape of the residual porosity the sintering process are clear. Even if sintering temperature and lamina composition A similar although less noisy situation appears when the re- are the same (laminae were taken from the same sheet), the en- sidual stress is measured on the same specimen in longitudinal vironment in which sintering occurred is clearly different as in the mode I(Fig. 2): the result is proposed in Fig 9, to be compared AMZ laminate the AM40 is constrained in between heteroge- with Fig 8. Only the region near the surface is shown, to high- neous laminae. This difference, not accounted for by the kreher- light fine details in the stress trends. a difference in the stresses Pompe or analogous models for the composite, can be respon of alumina and zirconia still appears: zirconia is in tension sible for the apparently inconsistent results found in the various hereas alumina is in compression. The difference increases in- laminates. In fact, dr takes into account just the contribution of side the laminate after an increase in the expected compressive the residual stresses created in a composite in the absence of ex- laminate, and in particular sintering stresses in the component over the whole AT. The residual stress that starts to build phases, thus seem to be highly affected by the environment in during the sintering and the cooling phase is probably sufficier which the cosintering occurs. to change the grain interaction mechanism in the composite. alumina-zirconia-mullite AMZ specimen. Results for a mea- k It appears that the magnitude of the average stress in the The observation is further confirmed by the more complex minate can be correctly predicted by existing models, but not Fig. 11. Scanning electron microscopy images of the AM40 composite: (a)sintered in a homogeneous laminate: (b)cosintered in the aMz multilayered

Local deviations due to the noise in the experimental data (the X-ray beam travels over a long distance within the material in order to reach the deepest buried layers) are evident. More￾over, large differences appear between alumina and zirconia stresses, with a slight tendency to separation into tensile and compressive components. The weighted average of zirconia and alumina stresses appears to be closer to the design value. This result is partly in contradiction with the expectations: by choos￾ing the reference interplanar spacing, similar trends would be expected for alumina and zirconia in the AZ20 and AZ40 layers. The observed differences suggest that dr is insufficient or unable to properly account for the interlaminar residual stresses due to the sintering process. A similar although less noisy situation appears when the re￾sidual stress is measured on the same specimen in longitudinal mode I (Fig. 2): the result is proposed in Fig. 9, to be compared with Fig. 8. Only the region near the surface is shown, to high￾light fine details in the stress trends. A difference in the stresses of alumina and zirconia still appears: zirconia is in tension whereas alumina is in compression. The difference increases in￾side the laminate after an increase in the expected compressive residual stress. Residual stresses within each lamina in a laminate, and in particular sintering stresses in the component phases, thus seem to be highly affected by the environment in which the cosintering occurs. The observation is further confirmed by the more complex alumina–zirconia–mullite AMZ specimen. Results for a mea￾surement conducted in longitudinal mode I (cf. Fig. 2) are shown in Fig. 10: the laminae are well identified (as suggested by the large phase contrast shown in Table I) and a stress profile can be reconstructed. Differences are again evident between zir￾conia, alumina, and mullite in the composite laminae. To confirm that problems can come from the cosintering process, microscopy observations were made on the very same lamina sintered in a homogeneous laminate and in a multilayer one. As an example, Fig. 11 shows scanning electron micro￾graphs (SEM) of polished cross sections of the AM40 lamina in the monolithic and in the AMZ specimens. Differences both in the grain size and in the size and shape of the residual porosity are clear. Even if sintering temperature and lamina composition are the same (laminae were taken from the same sheet), the en￾vironment in which sintering occurred is clearly different as in the AMZ laminate the AM40 is constrained in between heteroge￾neous laminae. This difference, not accounted for by the Kreher– Pompe or analogous models for the composite, can be respon￾sible for the apparently inconsistent results found in the various laminates. In fact, dr takes into account just the contribution of the residual stresses created in a composite in the absence of ex￾ternal residual stresses and assumes the material to be ideally stiff over the whole DT. The residual stress that starts to build up during the sintering and the cooling phase is probably sufficient to change the grain interaction mechanism in the composite. It appears that the magnitude of the average stress in the laminate can be correctly predicted by existing models, but not 0 50 100 150 200 250 300 −800 −600 −400 −200 0 200 400 600 σ (MPa) depth (µm) Fig. 9. Through-thickness residual stress profile in the AZ specimen, measured in longitudinal mode I for alumina (open dots) and zirconia (squares). The average stress is also shown (stars). The continuous line corresponds to calculated macroscopic interlaminar stresses. The line connecting the points serves as a guide for the eye. 0 50 100 150 200 250 300 −800 −600 −400 −200 0 200 σ (MPa) depth (µm) Fig. 10. Through-thickness residual stress profile in the AMZ speci￾men, measured in longitudinal mode I for alumina (open dots), zirconia (squares), and mullite (diamonds). The average stress is also shown (stars). The continuous line corresponds to calculated macroscopic in￾terlaminar stresses. The line connecting the points serves as a guide for the eye. Fig. 11. Scanning electron microscopy images of the AM40 composite: (a) sintered in a homogeneous laminate; (b) cosintered in the AMZ multilayered laminate. 1224 Journal of the American Ceramic Society—Leoni et al. Vol. 91, No. 4

April 2008 Nondestructive Measurement of the Residual Stress Profile the fine details concerning its repartition between the constituent phases. More variables need to be taken into account in the M. Lewis, "Apparent Fracture Toughness of Si3N4-Based Laminates with Resid micromechanical models describing the stress in a composite al Compressive or Tensile Stresses in Surface Layers, Acta Mater. 53. 289-96 created in nonequilibrium conditions. This further confirms the importance of a direct, experimental measurement of the High Reliability Alumina-Mullite Ceramic Laminates, "J. Am. Ceram Soc. 88 through-the-thickness stresses, which can be properly obtained by synchrotron radiation X-ray diffraction WHy. t Mavo and M. Bertoldi. "Design and Production of Ceramic Laminates with High Mechanical Resistance and Reliability. Acta Mater, 54. 4929-37 16A. Costabile and V. M. Sglavo, "Influence of the Architecture on the Me- The residual stress in a set of millimeter-thick symmetric ceramic chanical performances of Alumina-Zirconit-Mu aminates made of a blend of alumina zirconia and mullite was ci. Technol,45.,11038(2006. analyzed by using synchrotron radiation white-beam diffrac- Bertoldi, "High Reliability Ceramic Laminates by Design, Ph D. Thesis, tion. Using a narrow beam and energy-dispersive mode allowed University of Trento. 2004. Q. Ma and D. R. Clarke, ""Stress Measurement in Single Crystal and Poly a true through-thickness scanning of the specimen with gauge ystalline Ceramics Using Their Optical Fluorescence, J. Am. Ceram Soc. 76[6] plume and shape kept constant from point to point. Average stress results are in good agreement with the design Sergo, D. M. Lipkin, G. de Portu, and David R. Clarke, "Edge Stresses in data. Traditional models for the description of composites are not able to fully describe the partitioning of stresses over the various p V. Hauk. Structural and Residual Stress Analysis by Non-Destructive Methods the percolation limit for zirconia in alumina is overcome The approach proposed here can be used to validate theoret 503 in Diffraction Analysis of the ical residual stress profiles in more general multilayer systems. 38 Microstructure of Materials, Edited by E.J. Mittemeier and P Scardi. Springer 2U. Welzel, P. Lamparter, M. Leoni, and E. J. Mittemeijer, "The Determina ses in Thin Films: Modelling Elastic Grain Interaction philos. 3[36]4045-5702003) The au is, U.K(CCLRC) and the EU ens: On the Concepts of Diffraction Elastic rovision of beamtime at the Daresbury Laboratory SRs. Dr. M. Golsha R. V Martins, and A. Kvick, "A High Energy support. The assoc ope for Local Strain Measurements within Bulk Materials, Mater. Sci. knowledged for the thorough check of the manuscript and the useful discussions M. Turski, P J. Withers and T. Buslap "High- amples Using Full-Profile Analysis of Ener -Dispersive Synchrotron Diffraction Data, "J. Appl. Cryst, 37, 883-9 R Lawn, Fracture of Brittle Solids, 2nd edition, Cambridge University Press, stress and Microstress Using X-Rays, Synchrotron and Neutrons, "JpN. Soc. 四g.Mn.J.A,473252-63(20 UK.1993 nsistent Estimates for Elastic constants vis, A. Kristoffersson. E Carlstrom, and w.J. Clegg, "Fabrication and of Random Polycrystals with Hexagonal, Trigonal, and Tetragonal Symmetries. tion in Ceramic Laminates with Porous Interlayers, " J. An. Ceram. 2369-74(2000) d. Macc. PTho, molson, T. atwood.E.M. Gullikson. M. R Howells. B Bw. J. Clegg. K Kendall, and MeN. Alford, " A Simple Way to Make Tough Ceramics, "Nature(London ), 347, 455-57(1990). W. M. Kriven and D.-H. Kuo. High-Strength Flaw- Tolerant Oxide Ceramic Oxide Ceramic J.H. Scofield A, L. Robinson and). H. Underwood :x-Ray mission Energies Composite U.S. Pat. No lughan. Lawrence Berkeley National Laboratory, Berkeley, CA, 2001 sow Kreher and W Pompe, Internal Stresses in Heterogeneous Solids.Akademie- Growth Inhibitor in Surface and Promoter in Interior U.S. Pat. No. 3. 652.378 ch28,1972. erlag, Berlin, Germany, 1989 G Q. Ma, w Pompe, J. D. French, and D. R. Clarke, "Residual Stresses in Microstructural Engineering with Duplex and Laminar Ceramic Composites, Mater, 42[45] 1673-81( 1994). L M. Chan and G. A. Miller, Design of Lam- 3Z. Hashin and S. Shtrikman, "A Variational Approach to Theory of the Elastic Behavior of Multifaced Materials. " J. Mech. Solids, 11. 127-35 inated Ceramic Composite for Improved Strength and Toughness, "J. Am. Ceram 3H. Toraya, "Effect of YO,s Dopant on Unit Cell Parameters of ZrO, at Low BR. Latkshminarayanan, D K. Shetty, and R.A. Cutler, "Toughening of Lay- Ceram. Soc., 72 [4]662-4 M.P. Rao. A.J. Sanchez-Herencia G.E. Beltz R. M. McMeeking, andF.F and R. Torrecillas, "Percolative M, ec, Moya, S. Deville, J. Chevalier, G. Fantoz Lange, "" Laminar Ceramics That Exhibit a Threshold strengt 3H. Ledbetter, S. Kim, and D. Balzar, "Elastic Properties of Mullite, "J.Am. amn.So,8I10258(199 Orlovskaya, M. Lugovy, V o.J. Adams M. Chheda. J. Shih. J. San nd S. Yarmolenko, ""Robust Design and Man- H. Schneider. K. Okada. and J. A. Pask, Mullite and Mullite Ceramics. ufacturing of Ceramic Laminates with Controlled Thermal Residual Stresses for D J. Green, An Introduction to the Mechanical Properties of Ceramics. Cam IN. Orlovskaya, J Kuebler, V. Subbotin, and M. Lugovy, "Design of Si,Ne- e比比 C de Jonghe and C.S. Lee, "Modeling of Thermal Stresses Based Ceramic Laminates by the Residual Stresses. "J.Mater. Sci- 40, 5443-50

the fine details concerning its repartition between the constituent phases. More variables need to be taken into account in the micromechanical models describing the stress in a composite created in nonequilibrium conditions. This further confirms the importance of a direct, experimental measurement of the through-the-thickness stresses, which can be properly obtained by synchrotron radiation X-ray diffraction. IV. Conclusions The residual stress in a set of millimeter-thick symmetric ceramic laminates made of a blend of alumina, zirconia, and mullite was analyzed by using synchrotron radiation white-beam diffrac￾tion. Using a narrow beam and energy-dispersive mode allowed a true through-thickness scanning of the specimen with gauge volume and shape kept constant from point to point. Average stress results are in good agreement with the design data. Traditional models for the description of composites are not able to fully describe the partitioning of stresses over the various phases in each lamina. Large deviations appear when the percolation limit for zirconia in alumina is overcome. The approach proposed here can be used to validate theoret￾ical residual stress profiles in more general multilayer systems.38 Acknowledgments The authors wish to acknowledge the Council for the Central Laboratory of the Research Councils, U.K. (CCLRC) and the EU under the TMS project for provision of beamtime at the Daresbury Laboratory SRS. Dr. M. Golshan (CCLRC Daresbury Laboratory) is also acknowledged for useful discussion and experimental support. The associate editor and unknown referees are also ac￾knowledged for the thorough check of the manuscript and the useful discussions that followed. References 1 B. R. Lawn, Fracture of Brittle Solids, 2nd edition, Cambridge University Press, Cambridge, U.K., 1993. 2 J. B. Davis, A. 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