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