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 onTo 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