V.M. Sglavo, M. Bertoldi/Composites: Part B 37(2006)481-489 483 analyze some special cases. First of all, if the reference model As shown in Fig. 3(b) a stability range exists between x (Fig. 2) is thought to correspond to an edge crack in a semi- and the tangent point between Kext and T. One can observe nfinite body, Eq (2)can be simplified as that for increasing xI the strength decreases and the stability interval width increases. Since, both high strength and large yos ores()T2-2)05 stable growth interval are desirable, an intermediate value of (πc)5 (4) x, has to be considered in the perspective laminate design On the other side an increase of or is useful to increase both the stable growth range and the maximum stress. In where Y=1.1215. One could point out that such simplification addition, if Kc increases, the maximum stress is higher but is not rigorous as Y maintains a slight dependence on x/e [18]. the stability range decreases though one must considered Nevertheless, this allows to perform the calculations in closed that Kc is a parameter that depends on the material form without loosing of generality selection and it is not usually modified in the desi One very simple situation corresponds to the step profil procedure shown in Fig. 3(a) defined as: A more realistic residual stress shape is the square-wave rs=00<x<x1 Ures =-or XI<x<too Ores =00<x<x Ores =or xI<x<x2 In this case, T can be analytically calculated as rs=0x2<x<+∞ 0<x<x1 In this case, the T-curve can be calculated both analytically nd by using the principle of superposition [1, 17]. The square- wave profile can be considered in fact as the sum of two simple T=Kc+2r n[2 arcsin x,<x<+o step profiles with stresses of identical amplitude but opposite sign placed at different depths (xI and x2). The apparent (6) fracture toughness become T=Kc-2y 一 arcsin xI<x<x T=Ko-2y TtR( )-=( x<x<+∞ yx,√C(b) Fig. 3. Step residual stress profile(a) and comesponding apparent fracture toughness(b). The effects of intensity (left)and location(right) of the residual stress areanalyze some special cases. First of all, if the reference model (Fig. 2) is thought to correspond to an edge crack in a semi￾infinite body, Eq. (2) can be simplified as Kres Z Y ðpcÞ 0:5 ðc 0 sresðxÞ 2c c2Kx2  0:5 dx (4) where Yz1.1215. One could point out that such simplification is not rigorous as Y maintains a slight dependence on x/c [18]. Nevertheless, this allows to perform the calculations in closed form without loosing of generality. One very simple situation corresponds to the step profile shown in Fig. 3(a) defined as: sres Z 0 0!x!x1 sres ZKsR x1!x!CN ( (5) In this case, T can be analytically calculated as: T ZKC 0!x!x1 T ZKC C2Y c p 0 @ 1 A 0:5 sR p 2 Karcsin x0 c 0 @ 1 A 0 @ 1 A x1!x!CN 8 >>>>< >>>>: (6) As shown in Fig. 3(b) a stability range exists between x1 and the tangent point between Kext and T. One can observe that for increasing x1 the strength decreases and the stability interval width increases. Since, both high strength and large stable growth interval are desirable, an intermediate value of x1 has to be considered in the perspective laminate design. On the other side, an increase of sR is useful to increase both the stable growth range and the maximum stress. In addition, if KC increases, the maximum stress is higher but the stability range decreases though one must considered that KC is a parameter that depends on the material selection and it is not usually modified in the design procedure. A more realistic residual stress shape is the square-wave profile (Fig. 4) defined as: sres Z0 0!x!x1 sres ZKsR x1!x!x2 sres Z0 x2!x!CN 8 >< >: (7) In this case, the T-curve can be calculated both analytically and by using the principle of superposition [1,17]. The square￾wave profile can be considered in fact as the sum of two simple step profiles with stresses of identical amplitude but opposite sign placed at different depths (x1 and x2). The apparent fracture toughness becomes: T Z KC 0!x!x1 T Z KCK2Y c p 0 @ 1 A 0:5 sR p 2 Karcsin x1 c 0 @ 1 A 2 4 3 5 x1!x!x2 T Z KCK2Y c p 0 @ 1 A 0:5 sR arcsin x2 c 0 @ 1 AKarcsin x1 c 0 @ 1 A 2 4 3 5 x2!x!CN 8 >>>>>>>>>>>< >>>>>>>>>>>: (8) Fig. 3. Step residual stress profile (a) and corresponding apparent fracture toughness (b). The effects of intensity (left) and location (right) of the residual stress are shown. V.M. Sglavo, M. Bertoldi / Composites: Part B 37 (2006) 481–489 483