P. Vena et al/Mechanics Research Communications 35(2008)576-582 Table 1 Kc(MPa mIP) 7.75×10- 010 9492 222 8.37×10 AM40 688×106 0.2450 868×106 ment analyses at increasing crack length from a minimum of o um up to a maximum length of 250um ries of finite ele- The effective toughness of the symmetric laminate subjected to residual stress was obtained through a A two-step procedure is used to determine the effective toughness(Vena et al, 2005): ( i)a series of analysis with increas- ing crack length in which external loads simulating the four-point bending test (spans S1= 20 mm and sz=40 mm accord- ng to the experimental procedure reported in Sglavo and bertoldi, 2006)with no residual stress is simulated and ii)a series of analysis with increasing crack length in which thermal expansion without external load is simulated. The crack propagation along the symmetry plane was simulated by progressively releasing the nodal constraints. Each crack propagation step represented an increase of crack length Aa=5 um; refined analyses with Aa=2.5 um from 150 um to 220 um crack lengths were also performed. In the first series of analysis a unit force is used and a stress intensity factor due to external force is found by calculat the energy release rate g(a)(load controlled test): an aU G(a)= in which n is the total potential energy, U is the elastic energy of the system and a the current crack length. Assuming a mode I crack propagation, the stress intensity factor k for a unit force is K"(a)=VG(a-n2 The above expounded finite element procedure has been validated through a quantitative comparison with results obtained through contour integrals presented on a different ceramic laminate in Chen et al. (2007)and Bertarelli(2007) In the second series of analysis, the stress intensity factor in the laminate subjected to residual stress Kres(a)is calculated through the same procedure based on nodal constraint release as expounded above Due to linearity of the constitutive laws and to the linearity of the strain -displacements relationships, the principle of superposition of the solution from the thermal loading and that from the force loading can be used to determine the total stress intensity factory applied to the laminate subjected to the residual stress field and to the four-point bending external loads k(a K(a=Kres(a)+K(a)P in which P is the force applied in the four-point bending loading condition( Fig. 2). 0.=693MPa g =342MPa 2A230AZ0:AM40Az0A240 Crack length2 [um 21 Fig. 2. Effective fracture toughness of the AMZ laminate Dashed lines represent the applied stress intensity factor to determine laminate strength(slope 693 MPa)and lower threshold(slope 342 MPa).The effective toughness of the symmetric laminate subjected to residual stress was obtained through a series of finite ele￾ment analyses at increasing crack length from a minimum of 0 lm up to a maximum length of 250 lm. A two-step procedure is used to determine the effective toughness (Vena et al., 2005): (i) a series of analysis with increas￾ing crack length in which external loads simulating the four-point bending test (spans S1 ¼ 20 mm and S2 ¼ 40 mm accord￾ing to the experimental procedure reported in Sglavo and Bertoldi, 2006) with no residual stress is simulated and (ii) a series of analysis with increasing crack length in which thermal expansion without external load is simulated. The crack propagation along the symmetry plane was simulated by progressively releasing the nodal constraints. Each crack propagation step represented an increase of crack length Da ¼ 5 lm; refined analyses with Da ¼ 2:5 lm from 150 lm to 220 lm crack lengths were also performed. In the first series of analysis a unit force is used and a stress intensity factor due to external force is found by calculating the energy release rate GðaÞ (load controlled test): GðaÞ¼ oP oa ¼ oU oa ð2Þ in which P is the total potential energy, U is the elastic energy of the system and a the current crack length. Assuming a mode I crack propagation, the stress intensity factor Kr for a unit force is Kr ðaÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi GðaÞ E 1 m2 r ð3Þ The above expounded finite element procedure has been validated through a quantitative comparison with results obtained through contour integrals presented on a different ceramic laminate in Chen et al. (2007) and Bertarelli (2007). In the second series of analysis, the stress intensity factor in the laminate subjected to residual stress KresðaÞ is calculated through the same procedure based on nodal constraint release as expounded above. Due to linearity of the constitutive laws and to the linearity of the strain–displacements relationships, the principle of superposition of the solution from the thermal loading and that from the force loading can be used to determine the total stress intensity factory applied to the laminate subjected to the residual stress field and to the four-point bending external loads KðaÞ: KðaÞ ¼ KresðaÞ þ Kr ðaÞP ð4Þ in which P is the force applied in the four-point bending loading condition (Fig. 2). Table 1 Material properties Material E (GPa) a (C1 ) KC (MPa m1/2) m AM0/AZ0 394 7:75  106 3.6 0.2300 AZ100 204 – – 0.2900 AM100 229 – – 0.2700 AZ30 322.5 8:37  106 3.9 0.2465 AM40 317 6:88  106 2.4 0.2450 AZ40 302.5 8:68  106 4.5 0.2525 Fig. 2. Effective fracture toughness of the AMZ laminate. Dashed lines represent the applied stress intensity factor to determine laminate strength (slope 693 MPa) and lower threshold (slope 342 MPa). 578 P. Vena et al. / Mechanics Research Communications 35 (2008) 576–582