G. de Portu et al. Acta Materialia 53(2005)1511-1520 state in the ZrO2 phase is assessed with a high degree of precision using chromophore Al2O3 as a"stress sensor on the other hand. the stress field obtained with using the 460 cm Raman band of ZrO2 is affected by remarkable scattering and, thus, by a high degree of uncertainty. Here, in addition to the causes of low reli- ability already cited for the ZrO, Raman data (cf. be considered, whose impact on the spectral shift of 3 the ZrO2 Raman bands is presently unknown 1500(m) 3.3. Effect of layer geometry on residual stresses 8-100 A comparison between residual stress fields stored in multilayered specimens with different geometry (A/AZ/A vs. A/2AZ/A)is shown in Fig. 6. The thickness ratio, az/ta for specimens A/AZ/A and A/2AZ/A correspond to 1. 4 and 2.8, respectively. From a qualitative point of view. the stress distribution is similar in both cases with the stress profile being parabolic and lacking of symme try nearby the free surfaces of the external A layers. However, the residual stress magnitude significantly in- creased with increasing the taz/ta ratio. The increase in magnitude of stresses stored in the 2AZ layer is particu larly pronounced nearby the junctions with the neigh 100 uring A layers, as compared to the central part of the layers. Fig. 6 also shows that the measured stress val ues strongly depend on the probe configuration adopted. 5 In Fig. 6(a) and(b), data collected on the same specimen are shown with a laser spot size of 5 and I um, respec- 2-100 tively. Changing the laser spot size(i.e, changing the lens of the microscope) involves a major change in the geom- try of the optical probe, both with respect to penetration lepth and lateral resolution. Residual stresses in lar nate materials as a function of the laser penetration depth can be theoretically predicted in areas nearby the Fig. 6. Linear maps collected with a laser beam-diameter of 5 cimen edges, according to the linear superposition (spacing 2 um)(a)and I um(spacing I um)(b)in 13-layers(A/AZ/A) method. A simplified procedure has been proposed in lit nd 9-layers(A/2AZ/A)composite specimens. Note the different trends erature [6,8], which assumes the macrostresses accompa for stress depending on probe configuration. nying lamination result from CtE and elastic (1+)∫EA△ mismatches among individual layers. According to this 1 IA EA procedure, a theoretical estimate of the three-dimen- sional(tensile)stress, ofth), developed (nearby the speci- +N(A +tAZ) men surface) within A and AZ layers can be obtained by solving the following system of equations Kr+N(A+tAZ)+22 2(1+vAz)JEAz△ tan-/x+N(AZ+tA)+tA (-VAz) x+N(tA +tAz)+tA x sin 4(1+vAz)EAz△e VE+N(A+1Az)+1A]+2 p=元(1-)(1+数 号Az-{x-tx+N(A+tA -tan-(x+(N+ 1)(tAZ+tA V蓝a2-kx-1+Na+state in the ZrO2 phase is assessed with a high degree of precision using chromophore Al2O3 as a ‘‘stress sensor’’, on the other hand, the stress field obtained with using the 460 cm
1 Raman band of ZrO2 is affected by remarkable scattering and, thus, by a high degree of uncertainty. Here, in addition to the causes of low reli￾ability already cited for the ZrO2 Raman data (cf. Section 3.1), also the presence of shear stresses should be considered, whose impact on the spectral shift of the ZrO2 Raman bands is presently unknown. 3.3. Effect of layer geometry on residual stresses A comparison between residual stress fields stored in multilayered specimens with different geometry (A/AZ/A vs. A/2AZ/A) is shown in Fig. 6. The thickness ratio, tAZ/tA for specimens A/AZ/A and A/2AZ/A corresponds to 1.4 and 2.8, respectively. From a qualitative point of view, the stress distribution is similar in both cases, with the stress profile being parabolic and lacking of symme￾try nearby the free surfaces of the external A layers. However, the residual stress magnitude significantly in￾creased with increasing the tAZ/tA ratio. The increase in magnitude of stresses stored in the 2AZ layer is particu￾larly pronounced nearby the junctions with the neigh￾bouring A layers, as compared to the central part of the layers. Fig. 6 also shows that the measured stress val￾ues strongly depend on the probe configuration adopted. In Fig. 6(a) and (b), data collected on the same specimen are shown with a laser spot size of 5 and 1 lm, respec￾tively. Changing the laser spot size (i.e., changing the lens of the microscope) involves a major change in the geom￾etry of the optical probe, both with respect to penetration depth and lateral resolution. Residual stresses in lami￾nate materials as a function of the laser penetration depth can be theoretically predicted in areas nearby the specimen edges, according to the linear superposition method. A simplified procedure has been proposed in lit￾erature [6,8], which assumes the macrostresses accompa￾nying lamination result from CTE and elastic mismatches among individual layers. According to this procedure, a theoretical estimate of the three-dimen￾sional (tensile) stress, rðthÞ ii , developed (nearby the speci￾men surface) within A and AZ layers can be obtained by solving the following system of equations: ½rðthÞ ii ðeÞ AZ ¼
2 p ð1 þ mAZÞ ð1
mAZÞ EAZDe 1 þ tA tAZ EAZ EA ( )  sin
1 x þ NðtA þ tAZÞ þ tA ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½x þ NðtA þ tAZÞ þ tA 2 þ z2 q 0 B@ 1 CA 8 >< >:
tan
1 x þ ðN þ 1ÞðtAZ þ tAÞ z ) ; ð4Þ ½rðthÞ ii ðeÞ A ¼ 2 p ð1 þ mAÞ ð1
mAÞ EADe 1 þ tA tAZ EA EAZ ( )  sin
1 x þ NðtA þ tAZÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½x þ NðtA þ tAZÞ2 þ z2 q 0 B@ 1 CA 8 >< >:
tan
1 x þ NðtAZ þ tAÞ þ tA z 9 >= >; ; ð5Þ ½rðthÞ ii ðbÞ AZ ¼ 4 p ð1 þ mAZÞ ð1
mAZÞ EAZDe 1 þ tA tAZ EAZ EA ( )  Re ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 tAZ
x
1 2 tAZ þ NðtAZ þ tAÞ     1 2 tAZ
x
1 2 tAZ þ NðtAZ þ tAÞ         s ; ð6Þ Fig. 6. Linear maps collected with a laser beam-diameter of 5 lm (spacing 2 lm) (a) and 1 lm (spacing 1 lm) (b) in 13-layers (A/AZ/A) and 9-layers (A/2AZ/A) composite specimens. Note the different trends for stress depending on probe configuration. G. de Portu et al. / Acta Materialia 53 (2005) 1511–1520 1517