V.M. Sglavo, M. Bertoldi/Composites: Part B 37 (2006)481-489 therefore, mutually dependent also for this reason and it is not possible to design the desired mechanical behavior by using a quare-wave stress(single layer)profile, only. Fortunately, almost all these problems can be overcome by considering a multilayered structure Before moving towards a more complex profile, it is useful to analyze another simple case. Consider two stress profiles obtained by the combination of simple square-wave profiles of different (double, for simplicity) amplitude and identical extension(Fig. 5). This situation corresponds to laminates with two layers of different composition and identical thickness. The actual order of the two layers is the only difference between the two examined profiles. It is clear from Fig. 5 that the order of the compressive layers is important either for the final strength and the stability interval. Such consideration is general and the final conclusion can be drawn that the compression intensity in successive layers must be Fig. 4. Square-wave stress profile and relative apparent fracture toughness continuously growing to obtain a properly designed T-curve At this point, the principle of superposition can be used to This special case is useful to discuss an important point. calculate the T-curve for a general multi-step profile. The Depending on the width(x2-x1)of the compressive layer, the proposed approach can be extended in fact to n layers provided tangent point can fall beyond the position x2. In this case, the that n step profiles with amplitude Ao, (Fig. 2), equal to the stress stable growth range is automatically defined by the interval [ri, increase of layer with respect to the previous one, are considered x2] and the maximum stress is lower than the tangent stress. A general equation, which defines the apparent fracture toughness Strength and instability point become, therefore, mutually for layer i in the interval [i-I,xi ( Fig. 2), can be obtained independent within a certain degree. certain depth from the surface is, therefore, suitable to generate T=kc- arcsin a stable growth range for surface defects. Unfortunately, this simple solution is not actually practicable because the forces x;<x<x equilibrium in the component is not satisfied. In addition, in order to achieve elevated strength, the required compressive where i indicates the layer rank and x; is the starting depth of stress is usually very high and localized intense interlaminar layer. Eq (9)represents a short notation of n different equations, shear stresses can be generated; these can be then responsible the sum being calculated for different number of terms for each i for delamination between layers. Edge cracking can also arise This represents a mathematical translation of the ' memory at the interface between highly compressed laminae. This effect of stress history that deeper layers maintain with respect to phenomenon was analyzed in a previous work [19] to occur the layer previously encountered by the propagating crack. when the layer thickness is larger than a critical value, Regardless the layer order, since 2n parameters(r, Ad )are now c=K2/0.341+v)02], oe being the compressive stress and v available and two conditions have to be satisfied(forces the Poisson's ratio Layer thickness and compressive stress are, equilibrium and equivalence between the sum of single layer v√v Fig. 5. Residual stress and corresponding T-curve for two simple square-wave profiles placed in different orderThis special case is useful to discuss an important point. Depending on the width (x2Kx1) of the compressive layer, the tangent point can fall beyond the position x2. In this case, the stable growth range is automatically defined by the interval [x1, x2] and the maximum stress is lower than the tangent stress. Strength and instability point become, therefore, mutually independent within a certain degree. A single compressive layer of proper thickness placed at a certain depth from the surface is, therefore, suitable to generate a stable growth range for surface defects. Unfortunately, this simple solution is not actually practicable because the forces equilibrium in the component is not satisfied. In addition, in order to achieve elevated strength, the required compressive stress is usually very high and localized intense interlaminar shear stresses can be generated; these can be then responsible for delamination between layers. Edge cracking can also arise at the interface between highly compressed laminae. This phenomenon was analyzed in a previous work [19] to occur when the layer thickness is larger than a critical value, tcZK2 C=½0:34ð1CnÞs2 c , sc being the compressive stress and n the Poisson’s ratio. Layer thickness and compressive stress are, therefore, mutually dependent also for this reason and it is not possible to design the desired mechanical behavior by using a square-wave stress (single layer) profile, only. Fortunately, almost all these problems can be overcome by considering a multilayered structure. Before moving towards a more complex profile, it is useful to analyze another simple case. Consider two stress profiles obtained by the combination of simple square-wave profiles of different (double, for simplicity) amplitude and identical extension (Fig. 5). This situation corresponds to laminates with two layers of different composition and identical thickness. The actual order of the two layers is the only difference between the two examined profiles. It is clear from Fig. 5 that the order of the compressive layers is important either for the final strength and the stability interval. Such consideration is general and the final conclusion can be drawn that the compression intensity in successive layers must be continuously growing to obtain a properly designed T-curve. At this point, the principle of superposition can be used to calculate the T-curve for a general multi-step profile. The proposed approach can be extended in fact to n layers provided that n step profiles with amplitude Dsj (Fig. 2), equal to the stress increase oflayerj with respectto the previous one, are considered. A general equation, which definesthe apparent fracturetoughness for layer i in the interval [xiK1, xi] (Fig. 2), can be obtained T Z Ki CKX i jZ1 2Y c p 0:5 Dsres;j p 2 Karcsin xjK1 c h i    xiK1!x!xi (9) where i indicates the layer rank and xj is the starting depth of layer j. Eq. (9)represents a short notation of n different equations, the sum being calculated for different number of terms for each i. This represents a mathematical translation of the ‘memory’ effect of stress history that deeper layers maintain with respect to the layer previously encountered by the propagating crack. Regardless the layer order, since 2n parameters (xi, Dsi) are now available and two conditions have to be satisfied (forces equilibrium and equivalence between the sum of single layer Fig. 4. Square-wave stress profile and relative apparent fracture toughness. Fig. 5. Residual stress and corresponding T-curve for two simple square-wave profiles placed in different order. 484 V.M. Sglavo, M. Bertoldi / Composites: Part B 37 (2006) 481–489