580 P. Vena et al/Mechanics Research Communications 35(2008)576-582 The linearity of the governing equations allows one to calculate the stress field due to residual stresses and applied exter- nal forces by making use of the superposition principle in which ols and og are the Cauchy components of the stress field due to the temperature change and to a unit external force (for a four-point bending configuration), respectively For the sake of computational accuracy, a round notch is considered at the tip of the crack, with notch radius of 4 um. this notch size is assumed to be consistent with the microstructures of the materials. As discussed in Lei et al. (1998). the size of the finite elements should be based on the requirement of statistical independence thus requiring that the smallest element be of the order of a few grain size. However, mathematically, this seems to be an unnecessary requirement as in the case of a non-singular stress field the computed Weibull stress value should be independent of the finite element mesh used. The Weibull stress ow(di) calculated for the critical crack length acr =169 um, and m=35(referring to Sglavo and Ber- toldi, 2006 )is reported in Fig. 4a. The values for different sizes of the process zone(radius from 5 um to 100 um)are pre- nted. In this figure ai is the nominal applied stress which is linearly related to the It can be observed that for ai greater than 330 MPa a linear relationship between the Weibull stress and the external the oretical stress is found. This linear relationship is independent of the process zone size. Whereas, for of lower than 330 MPa, the Weibull stress is not linearly related to the external loads and an appreciable dependence on the process zone size found. For the largest process zone size (rmax=100 um), a bilinear ow(oi)relationship is found. This response is owed to the residual stress field; indeed, for low magnitude external loads(Gi 330 MPa), the tensile esidual stress acting in the direction perpendicular to the interface planes into the wake of the crack gives the predominant contribution into the integral 8), see Fig 4b. The high value of m makes this effect particularly remarkable When the external load increases, the most relevant contribution to the integral 8)is the combination of residual stress and the stress due to the external forces acting along the direction perpendicular to the crack propagation ahead of the crack tip, see Fig 4c. This type of stress is the one that leads to the failure mode experimentally observed (ie crack propagate along the direction perpendicular to the interfaces). For di >330 MPa the relationship between the Weibull stress and the applied theoretical stress is linear and can be writ ten in the following form in which a is dependent on the parameter m used in the integral()and do is dependent on the residual stress field. For the AMZ laminate studied in this paper one has a= 20.3 and oo=283 MPa. The probability of failure for a given set of Weibull parameters m and awo, is expressed as( beremin, 1983: Esposito et a 2007) Taking logarithms twice the function f[In(oi)] can be defined mdh|(以)=m[(=) (12) which represents the Weibull plot used to determine the Weibull parameters for a series of experimental data. Note that, in case of no residual stress field, the Weibull plot( 12)is a linear function with slope m. a4000 3000 AZ40 1000 AM40 AM40 100200300400500 1 Fig 4. Weibull stress versus nominal stress at different radii (acr= 169 um. m=35)(a). Sketch of the dominant stress components for low (b)and high(c)The linearity of the governing equations allows one to calculate the stress field due to residual stresses and applied exter￾nal forces by making use of the superposition principle: rijmaxðPÞ ¼ rres ij þ Prr ij   max ð9Þ in which rres ij and rr ij are the Cauchy components of the stress field due to the temperature change and to a unit external force (for a four-point bending configuration), respectively. For the sake of computational accuracy, a round notch is considered at the tip of the crack, with notch radius of 4 lm. This notch size is assumed to be consistent with the microstructures of the materials. As discussed in Lei et al. (1998), the size of the finite elements should be based on the requirement of statistical independence thus requiring that the smallest element be of the order of a few grain size. However, mathematically, this seems to be an unnecessary requirement as in the case of a non-singular stress field, the computed Weibull stress value should be independent of the finite element mesh used. The Weibull stress rWðrT 1Þ calculated for the critical crack length acr ¼ 169 lm, and m ¼ 35 (referring to Sglavo and Ber￾toldi, 2006) is reported in Fig. 4a. The values for different sizes of the process zone (radius from 5 lm to 100 lm) are pre￾sented. In this figure, rT 1 is the nominal applied stress which is linearly related to the applied force. It can be observed that for rT 1 greater than 330 MPa a linear relationship between the Weibull stress and the external the￾oretical stress is found. This linear relationship is independent of the process zone size. Whereas, for rT 1 lower than 330 MPa, the Weibull stress is not linearly related to the external loads and an appreciable dependence on the process zone size is found. For the largest process zone size ðrmax ¼ 100 lmÞ, a bilinear rWðrT 1Þ relationship is found. This response is owed to the residual stress field; indeed, for low magnitude external loads ðrT 1 < 330 MPaÞ, the tensile residual stress acting in the direction perpendicular to the interface planes into the wake of the crack gives the predominant contribution into the integral (8), see Fig. 4b. The high value of m makes this effect particularly remarkable. When the external load increases, the most relevant contribution to the integral (8) is the combination of residual stress and the stress due to the external forces acting along the direction perpendicular to the crack propagation ahead of the crack tip, see Fig. 4c. This type of stress is the one that leads to the failure mode experimentally observed (i.e. crack propagation along the direction perpendicular to the interfaces). For rT 1 > 330 MPa the relationship between the Weibull stress and the applied theoretical stress is linear and can be writ￾ten in the following form: rW ¼ a rT 1 r0   ð10Þ in which a is dependent on the parameter m used in the integral (8) and r0 is dependent on the residual stress field. For the AMZ laminate studied in this paper one has a ¼ 20:3 and r0 ¼ 283 MPa. The probability of failure, for a given set of Weibull parameters m and rW0, is expressed as (Beremin, 1983; Esposito et al., 2007): F ¼ 1 exp rW rW0  m ð11Þ Taking logarithms twice the function f½lnðrT 1Þ can be defined: f lnðrT 1Þ   ¼ ln ln 1 1 F  ¼ m ln a rW0 eln rT 1 r0    ð12Þ which represents the Weibull plot used to determine the Weibull parameters for a series of experimental data. Note that, in case of no residual stress field, the Weibull plot (12) is a linear function with slope m. Fig. 4. Weibull stress versus nominal stress at different radii ðacr ¼ 169 lm; m ¼ 35Þ (a). Sketch of the dominant stress components for low (b) and high (c) external loads. 580 P. Vena et al. / Mechanics Research Communications 35 (2008) 576–582