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 shownthe 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