P. Vena et al /Mechanics Research Communications 35(2008) 576-582 he crack in the layer n propagates when the condition is met, in which Kc is the intrinsic fracture toughness of the composite material in the layer n. Combining the relationships (4)and(5), taking the equality sign, the effective toughness Keff(a) is obtained Keff(a)=ke-Kres(a) (6) nd the relation(5)can be written as lich Per is the force at whi As shown in Fig. 2, the effective stress intensity factor exhibited an increasing trend in the layers characterized by a com- ressive residual stress After having reached a maximum va hen the crack tip is located beyond the interface between he azo and az40 layers the curve decreases as a conseque the tensile residual stress. The critical crack length found through this approach is approximately 169 um which is the result found through the analytical procedures(see Sglavo and Bertoldi, 2006) Through a direct comparison between the effective toughness and the stress intensity factor due to external loads (k(a)=K P). it is possible to determine the stability condition for crack growth In particular, for small magnitude of external loads P, the k(a) curve may intercept the Keff(a) curve for two values of a, ay a1 and a2. This indicates that any crack shorter than al does not propagate and cracks with an a a propagate until they reach the length Through this procedure, a critical value of the external load Pcr for which a2 @cr can be identified. The value of an for Pcr will denote the almax which represents the maximum length of a non-propagating crack. The critical value of external load for the laminate under study is Pcr=47.3 N, which corresponds to a tensile bending strength of 693 MPa. The average strength obtained through the four-point experimental tests is 692+ 25 MPa( Sglavo and bertoldi, 2006). For external loads P> Pcr, all cracks longer than a, will propagate unstably. Moreover, a lower threshold for the external oad Pmin may be identified: the minimum value of P for which al For P< Pmin no crack propagation is expected except for very long cracks(a>acr)which have no practical interest. The theoretical applied stress at Pmin is 342 MPa. 3. Statistical analysis of laminates with residual stress As opposed to the energy approach, a local approach based on the weibull theory is presented here. In particular, the stress fields obtained through the finite element analyses were post-processed with the purpose to determine the Weibull stress defined (P) (omax))"dv (omax))"sin addr 8 (omax)being the positive part of the maximum principal stress, m the Weibull modulus, vo a reference volume, v the volume of the process zone and t the thickness of the sample (beremin, 1983: Esposito et al. 2007). The Weibull modulus is, incidentally, a stress exponent that describes the relation between the weibull stress(directly related to the failure probability) and the relevant applied stress. The reference volume Vo should be chosen consistently with the material microstructure; nevertheless, from a numerical point of view, it can be considered as a scale factor and it can be hosen arbitrarily (potentially the unit volume) but kept constant for all the analyses (munz and fett, 1999; Lei et al, 1998). In this application, the process zone is assumed to be circular with radius Imax and Vo=tL L being a characteristic length. a polar coordinate system was used to compute the volume integral(see Fig 3). Due to the symmetry of the model, the inte- gral over half process zone was calculated roces F/2 Fig 3. Graphical representation of the process zone: increasing radii are shown(not to scale).The crack in the layer n propagates when the condition KðaÞ P Kn c ð5Þ is met, in which Kn c is the intrinsic fracture toughness of the composite material in the layer n. Combining the relationships (4) and (5), taking the equality sign, the effective toughness KeffðaÞ is obtained: KeffðaÞ ¼ Kn c KresðaÞ ð6Þ and the relation (5) can be written as Kr ðaÞPcr P Kn c KresðaÞ ð7Þ in which Pcr is the force at which crack propagates. As shown in Fig. 2, the effective stress intensity factor exhibited an increasing trend in the layers characterized by a com￾pressive residual stress. After having reached a maximum value when the crack tip is located beyond the interface between the AZ0 and AZ40 layers the curve decreases as a consequence of the tensile residual stress. The critical crack length found through this approach is approximately 169 lm which is the same result found through the analytical procedures (see Sglavo and Bertoldi, 2006). Through a direct comparison between the effective toughness and the stress intensity factor due to external loads ðKextðaÞ ¼ Kr PÞ, it is possible to determine the stability condition for crack growth. In particular, for small magnitude of external loads P, the KextðaÞ curve may intercept the KeffðaÞ curve for two values of a, say a1 and a2. This indicates that any crack shorter than a1 does not propagate and cracks with a1 < a < a2 propagate until they reach the length a2. Through this procedure, a critical value of the external load Pcr for which a2 ¼ acr can be identified. The value of a1 for Pcr will denote the a1max which represents the maximum length of a non-propagating crack. The critical value of external load for the laminate under study is Pcr ¼ 47:3 N, which corresponds to a tensile bending strength of 693 MPa. The average strength obtained through the four-point experimental tests is 692  25 MPa (Sglavo and Bertoldi, 2006). For external loads P > Pcr, all cracks longer than a1 will propagate unstably. Moreover, a lower threshold for the external load Pmin may be identified: the minimum value of P for which a1 ¼ a2. For P < Pmin no crack propagation is expected except for very long cracks ða > acrÞ which have no practical interest. The theoretical applied stress at Pmin is 342 MPa. 3. Statistical analysis of laminates with residual stress As opposed to the energy approach, a local approach based on the Weibull theory is presented here. In particular, the stress fields obtained through the finite element analyses were post-processed with the purpose to determine the Weibull stress defined as rWðPÞ ¼ 1 V0 Z V ð Þ hrmaxi mdV 1=m ¼ 2t V0 Z rmax 0 Z p 0 ð Þ hrmaxi m sin hdhdr 1=m ð8Þ hrmaxi being the positive part of the maximum principal stress, m the Weibull modulus, V0 a reference volume, V the volume of the process zone and t the thickness of the sample (Beremin, 1983; Esposito et al., 2007). The Weibull modulus is, incidentally, a stress exponent that describes the relation between the Weibull stress (directly related to the failure probability) and the relevant applied stress. The reference volume V0 should be chosen consistently with the material microstructure; nevertheless, from a numerical point of view, it can be considered as a scale factor and it can be chosen arbitrarily (potentially the unit volume) but kept constant for all the analyses (Munz and Fett, 1999; Lei et al., 1998). In this application, the process zone is assumed to be circular with radius rmax and V0 ¼ tL2 , L being a characteristic length. A polar coordinate system was used to compute the volume integral (see Fig. 3). Due to the symmetry of the model, the inte￾gral over half process zone was calculated. Fig. 3. Graphical representation of the process zone: increasing radii are shown (not to scale). P. Vena et al. / Mechanics Research Communications 35 (2008) 576–582 579