V M. Sglavo, M. Bertoldi Acta Materialia 54(2006 )4929-4937 should be possible to obtain Representing a steep growing ity interval width increases. Since both high strength and a function of c. Moreover, as only surface flaws have been large stable growth interval are desirable, an intermediate onsidered, the T-curve is unique for any defect and, there- value of x has to be adopted in the laminate design. On fore, it can be considered as fixed with respect to the sur- the other hand, an increase of or is useful for increasing face of the body. Consequently, crack length (c) and both the stable growth range and the maximum stres depth from the surface (x) can be regarded as identical Also, if Kc increases, the maximum stress is higher but quantities in the following analysi the stability range decreases; note that Kc is a parameter In order to understand the effect of residual stress inten- that depends on the material selection and is treated as a sity and location on the apparent fracture toughness, it is constant in the design procedure useful to analyse some special cases. First, if the reference A more realistic residual stress shape is the sq model(Fig. 2)corresponds to an edge crack in a semi- profile(Fig 4) defined as infinite body, Eq (2)can be simplified as 0<x<x1 Ores(x) r<x<x (c2-x2)0 x2<x<+∞ where Ya 1.215. One could point out that such a simpli- In this case the T-curve can be calculated both analytically fication is an approximation, as Y maintains a slight depen- and by using the principle of superposition [1, 21]. The dence on x/c[22]. Nevertheless, this allows the calculations square-wave profile can be considered in fact as the sum to be effected in a closed form without the loss of two simple step profiles with stresses of generality tude but opposite sign placed at different depths(xI and One very simple situation corresponds to the step profile x2). The apparent fracture toughness becomes defined 0 0,0<x<x T=Kc-2Y(9 aRIE-arcsin() x<x<x T=Kc-2Y(9 aR[arcsin(2) and shown in Fig. 3a for various Ores values where or is a generic positive stress value. In this case T can be analyti- (8) cally calculated from Eq (4)as A single compressive layer of the appropriate thickness, 0<x<x1 able for ting a stable growth range for surface T=Kc+2Y( oRE-arcsin()),x1<x<+oo defects. Unfortunately, this simple solution is not applica- (6 ble because the force equilibrium in the component is not satisfied. Furthermore, the compressive stress required to by assuming a constant fracture toughness achieve the higher strength is often very high and localised As shown in Fig. 3b, a stability range exists between xi it can generate intense interlaminar shear stresses, which and the tangent point between Kext and T. One can observe can cause delamination between layers. Edge cracking that by increasing x, the strength decreases and the stabil- can occur at the interface between highly compressed x2 a T x√E Fig 3. Step residual stress profile(a) and corresponding apparent fracture toughness(b). The effect of intensity (left) and location (right) of the Fig. 4. Square-wave stress profile and corresponding apparent fracture residual stress is sheshould be possible to obtain T representing a steep growing function of c. Moreover, as only surface flaws have been considered, the T-curve is unique for any defect and, there￾fore, it can be considered as fixed with respect to the sur￾face of the body. Consequently, crack length (c) and depth from the surface (x) can be regarded as identical quantities in the following analysis. In order to understand the effect of residual stress inten￾sity and location on the apparent fracture toughness, it is useful to analyse some special cases. First, if the reference model (Fig. 2) corresponds to an edge crack in a semi￾infinite body, Eq. (2) can be simplified as Kres ¼ Y ðpcÞ 0:5 Z c 0 rresðxÞ 2c ðc2  x2Þ 0:5 dx; ð4Þ where Y  1.1215. One could point out that such a simpli- fication is an approximation, as Y maintains a slight depen￾dence on x/c [22]. Nevertheless, this allows the calculations to be effected in a closed form without the loss of generality. One very simple situation corresponds to the step profile defined as rres ¼ 0; 0 < x < x1; rres ¼ rR; x1 < x < þ1;  ð5Þ and shown in Fig. 3a for various rres values where rR is a generic positive stress value. In this case T can be analyti￾cally calculated from Eq. (4) as T ¼ KC; 0 < x < x1; T ¼ KC þ 2Y c p  0:5 rR p 2  arcsin x0 c     ; x1 < x < þ1; ( ð6Þ by assuming a constant fracture toughness KC. As shown in Fig. 3b, a stability range exists between x1 and the tangent point between Kext and T. One can observe that by increasing x1 the strength decreases and the stabil￾ity interval width increases. Since both high strength and a large stable growth interval are desirable, an intermediate value of x1 has to be adopted in the laminate design. On the other hand, an increase of rR is useful for increasing both the stable growth range and the maximum stress. Also, if KC increases, the maximum stress is higher but the stability range decreases; note that KC is a parameter that depends on the material selection and is treated as a constant in the design procedure. A more realistic residual stress shape is the square-wave profile (Fig. 4) defined as rres ¼ 0; 0 < x < x1; rres ¼ rR; x1 < x < x2; rres ¼ 0; x2 < x < þ1: 8 >< >: ð7Þ In this case the T-curve can be calculated both analytically and by using the principle of superposition [1,21]. The square-wave profile can be considered in fact as the sum of two simple step profiles with stresses of identical ampli￾tude but opposite sign placed at different depths (x1 and x2). The apparent fracture toughness becomes T ¼ KC; 0 < x < x1; T ¼ KC  2Y c p  0:5 rR p 2  arcsin x1 c     ; x1 < x < x2; T ¼ KC  2Y c p  0:5 rR arcsin x2 c    arcsin x1 c     ; x2 < x < þ1: 8 >< >: ð8Þ A single compressive layer of the appropriate thickness, placed at a certain depth from the surface, is therefore suit￾able for generating a stable growth range for surface defects. Unfortunately, this simple solution is not applica￾ble because the force equilibrium in the component is not satisfied. Furthermore, the compressive stress required to achieve the higher strength is often very high and localised; it can generate intense interlaminar shear stresses, which can cause delamination between layers. Edge cracking can occur at the interface between highly compressed x1 x x1 –σ σ –σ σ ψ ψ res R x, c x1 KC Y σR K x x1 res R x, c x1 KC Y K a b T T Fig. 3. Step residual stress profile (a) and corresponding apparent fracture toughness (b). The effect of intensity (left) and location (right) of the residual stress is shown. x1 x, c K ψ σ σ x2 KC ψ x x1 – res x2 R T Fig. 4. Square-wave stress profile and corresponding apparent fracture toughness. V.M. Sglavo, M. Bertoldi / Acta Materialia 54 (2006) 4929–4937 4931