G. de portu et al Acta Materialia 53(2005)1511-1520 4(1+vA) B(z)=0(z<0) (-)1+ where p is the probe length parameter(for an unfocused m, p tends to infinity and the right part of Eq. (10) x Re tA+N(tAz +IA tends to unity) and zo represents the location of the focal tA+N(Az +t, plane with respect to the selected origin of the cartesian (7) axes(i.e, 20=0 in the present calculation). According to Eqs.(4 (11), near-edge stress values were calculated as am]2(N)+[m)](N) a function of the abscissa x for different p values. The results of this calculation (for N=9) are presented in Fig. 7 in comparison with experimental data collected +m(N)+ on the A/2AZ specimen with a probe size of I um. In (8) the 2AZ layers, measured stresses are consistent with theoretical profiles calculated for p= 10 um. This is in where N is the number of laminate pairs, E and v are the good agreement with the experimental p= ll um value Youngs modulus and the Poissons ratio of the mate- reported in literature [6]. However, a markedly lower rial, t is the layer thickness and the subscripts A and stress value is recorded nearby the A/2AZ interfaces, AZ refer to the AlO3 and Al_O3/ZrO2 layer, respec- as compared to the calculated value. This trend of stress tively(EA and EAZ= 360 and 300 GPa, respectively). underestimation nearby the interface is also noted in the The superscripts(e)and(b)refer to edge and bulk stres- A layers On the other hand, the magnitude of the com ses, respectively. Cartesian axes x and z were taken per- pressive stress measured at the center of A layers was respectively. The origin of the xz cartesian axes Hap e, significantly higher than that calculated by using an pendicular and parallel to the A/Az interfac experimentally assessed value p=7.5 um, which was re- cted at the interception between the free surface of the ported for polycrystalline Al2O3 [6]. In this context, it sample and the A/AZ interface. As represents the strain can be interesting to comment about the causes of the mismatch between adjacent layers and equals the prod- above discrepancies between experimental and calcu- uct Ax(To- TRT), with Ax-1.0 x 10 K, and lated stress values. It should be noted that the adopted Tos 1200C [10] (TRT is room temperature). According convolution procedure (i.e, Eqs.(9HIl)) takes into ac- to eqs.(4(8), the near-edge residual stress distribution count only the laser penetration depth, while also the within the multilayered specimen can be plotted as a function of x, z, tAz and N, for a given Ia value G.e., 180 Hm). However, in order to obtain a direct comparison with the experimental stress values evalu 2AZ 2AZ ated by fluorescence spectroscopy, ofex), a convolution of the calculated values with the probe depth-response function should be performed [6, 11]. Any point in space within the volume of the probe gives rise to its own local optical scattered intensity spectrum of a given form(e.g Lorentzian). The contribution of this spectrum to the observed spectrum depends on the intensity of incident light and on the probe"sensitivity"at the given loca tion. Both these effects can be represented by an effective intensity (i. e, the probe response function, B(x,y,z, 200-400600-800-1000-1200-1400(m) xo,yo, ]o))of light scattered from the point (x, y, z)when he incident beam is focused on the point(xo, yo, zo). By neglecting variations within the probed volume along the x and y directions(on the focal plane of the measure- ment)and just considering the variation along the z axis, analytical stress predictions can be convoluted accord ing to the following equations oth(x, 2)B(2)dz, p-500 um Fig. 7. Results of theoretical calculations of near-edge residual st (cf. Eqs. (4)(I1))as a function of the laser probe geometry for th ≥ layers A/AZA specimen. A comparison is shown with experimental data collected with a laser beam-diameter of I um.½rðthÞ ii ðbÞ A ¼
4 p ð1 þ mAÞ ð1
mAÞ EADe 1 þ tA tAZ EA EAZ ( )  Re ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 tA
x
1 2 tA þ NðtAZ þ tAÞ     1 2 tA
x
1 2 tA þ NðtAZ þ tAÞ         s ; ð7Þ rðthÞ ii ¼ NX¼þ1 N¼
1 ½rðthÞ ii ðeÞ AZðNÞþ½rðthÞ ii ðeÞ A ðNÞ n þ rðthÞ ii h iðbÞ AZ ðNÞ þ rðthÞ ii h iðbÞ A ðNÞ  ; ð8Þ where N is the number of laminate pairs, E and m are the Young
s modulus and the Poisson
s ratio of the mate￾rial, t is the layer thickness and the subscripts A and AZ refer to the Al2O3 and Al2O3/ZrO2 layer, respec￾tively (EA and EAZ @ 360 and 300 GPa, respectively). The superscripts (e) and (b) refer to edge and bulk stres￾ses, respectively. Cartesian axes x and z were taken per￾pendicular and parallel to the A/AZ interface, respectively. The origin of the xz cartesian axes was se￾lected at the interception between the free surface of the sample and the A/AZ interface. De represents the strain mismatch between adjacent layers and equals the prod￾uct Da (T0
TRT), with Da 
1.0 · 10
6 K
1 , and T0  1200 C [10] (TRT is room temperature). According to Eqs. (4)–(8), the near-edge residual stress distribution within the multilayered specimen can be plotted as a function of x, z, tAZ and N, for a given tA value (i.e., = 180 lm). However, in order to obtain a direct comparison with the experimental stress values evalu￾ated by fluorescence spectroscopy, rðexÞ ii , a convolution of the calculated values with the probe depth-response function should be performed [6,11]. Any point in space within the volume of the probe gives rise to its own local optical scattered intensity spectrum of a given form (e.g., Lorentzian). The contribution of this spectrum to the observed spectrum depends on the intensity of incident light and on the probe ‘‘sensitivity’’ at the given loca￾tion. Both these effects can be represented by an effective intensity (i.e., the probe response function, B(x,y, z, x0,y0, z0)) of light scattered from the point (x,y, z) when the incident beam is focused on the point (x0,y0, z0). By neglecting variations within the probed volume along the x and y directions (on the focal plane of the measure￾ment) and just considering the variation along the z axis, analytical stress predictions can be convoluted accord￾ing to the following equations: rðexÞ ii ðx;zÞ ¼ Z þ1
1 rðthÞ ii ðx;z 0 ÞBðz 0 Þ dz 0 ; ð9Þ BðzÞ ¼ p=p ðz
z0Þ 2 þ p2 ðz P 0Þ; ð10Þ BðzÞ ¼ 0 ðz < 0Þ; ð11Þ where p is the probe length parameter (for an unfocused beam, p tends to infinity and the right part of Eq. (10) tends to unity) and z0 represents the location of the focal plane with respect to the selected origin of the cartesian axes (i.e., z0 = 0 in the present calculation). According to Eqs. (4)–(11), near-edge stress values were calculated as a function of the abscissa x for different p values. The results of this calculation (for N = 9) are presented in Fig. 7 in comparison with experimental data collected on the A/2AZ specimen with a probe size of 1 lm. In the 2AZ layers, measured stresses are consistent with theoretical profiles calculated for p = 10 lm. This is in good agreement with the experimental p = 11 lm value reported in literature [6]. However, a markedly lower stress value is recorded nearby the A/2AZ interfaces, as compared to the calculated value. This trend of stress underestimation nearby the interface is also noted in the A layers. On the other hand, the magnitude of the com￾pressive stress measured at the center of A layers was significantly higher than that calculated by using an experimentally assessed value p = 7.5 lm, which was re￾ported for polycrystalline Al2O3 [6]. In this context, it can be interesting to comment about the causes of the above discrepancies between experimental and calcu￾lated stress values. It should be noted that the adopted convolution procedure (i.e., Eqs. (9)–(11)) takes into ac￾count only the laser penetration depth, while also the Fig. 7. Results of theoretical calculations of near-edge residual stresses (cf. Eqs. (4)–(11)) as a function of the laser probe geometry for the 9- layers A/2AZ/A specimen. A comparison is shown with experimental data collected with a laser beam-diameter of 1 lm. 1518 G. de Portu et al. / Acta Materialia 53 (2005) 1511–1520