M. Lugory et al. Acta Materialia 53 (2005)289-296 as a function of the crack length(or the crack length the crack with the crack length parameter A2 will have parameter). The calculated increase of Kapp is confirmed an unstable growth from the point B2 to the point C by the experimental data in the laminates with the com- on the Kapp-d plot(Fig. 6(a)). The stable growth of this pressive outer layer(Fig. 4(a). As one can see from crack will occur from the point C to the point D For all ig. 6(a), the cracks that have the crack length parame- cracks with a crack length parameter d from the point ter from O to the point A, will demonstrate an unstable A, to the point A4, the failure occurs at the stress equal crack growth. In this once the crack started to to the slope of the OD straight line, which is a threshold propagate at a certain stress, it cannot be arrested; this stress. The threshold stress thr is determined by the results in complete failure of the sample, since the ap- maximum value of Kapp at the interface between the first plied stress intensity factor is always higher than a frac- (compressive) and the second( tensile) layers, and no ture resistance of the laminate. The cracks that have the failure can occur below the othr if the sample contains crack length parameter between the point Al and the the surface cracks located only in the first layer. The cur point As will propagate in two stages. For example, vature of the Kapp plot is a function of a value of the residual stress. The higher residual stress, the more con- cave the curvature of Kapp is. At a certain small value of a residual compressive stress, the line OD can have only (a)K one intersection point with Kapp plot, and therefore no stable crack growth stage can occur. The conditions of the stable/unstable crack growth tress intensity factor the laminate with residual tensile stress in top layer are shown in Fig. 6(b). The crack with a crack length parameter A1 for such laminates will propagate only unstably at the stress level above oB/. The crack with the crack length parameter A grows unstably at the tress oB. This unstable growth occurs between points B and C(Fig. 6(b)), because the points belonging to the bC segment lie above the Kapp plot. At point C, opressive the condition of the Eq.(6)is violated and the crack growth becomes stable between points C and D, which A, Az A3 Ag means that any crack advancement requires an increase of the applied stress. Point D is a maximum value of Kapp at the interface between the second(compressive) and the third(tensile)layers. This point determines a stress Gop= Othr. Above aop, the crack propagates unstably up to a complete failure. In such a way all ini- tial cracks in the first(tensile)and the second(compres Threshold sive)layers with a crack length parameter greater thar applied stress A,(Fig. 6(b)) will initiate the specimen failure at the same dop=Othr stress value. The initial cracks with tips in the third and the fourth layers will initiate specimen failure at the different stress value that is determined by the maximum value of the Kapp at the interface be- tween the fourth and the fifth layers. This stress is othr for cracks with tips located in the third and the fourth layers. It should be noted that points the B or B3 in Fig. 6(b)correspond to the measured Kapp values(using pop-in"stress), while the points Bor B3 belonging to the OD straight line are determined by the initial notch ensile Compressive Tensile length and the failure stress of the sample As implied by the above analysis, the surface cracks which have sufficient length to fall into the region of a stable crack growth will all cause a failure at the same Fig. 6. Conditions for stable/unstable crack growth in a structure:(a)a range of crack length parameters for stable Othr stress. At the same time, if a residual compressive growth in a lamin a residual compressive stress in a to stress in the top layer is not high enough, the small (b) stable/unstable crack growth in a laminate with a residual cracks can cause catastrophic failure once they start stress in a top layer. grow. Therefore, it might be that different mechanismsas a function of the crack length (or the crack length parameter). The calculated increase of Kapp is confirmed by the experimental data in the laminates with the com￾pressive outer layer (Fig. 4(a)). As one can see from Fig. 6(a), the cracks that have the crack length parame￾ter from O to the point A1 will demonstrate an unstable crack growth. In this case, once the crack started to propagate at a certain stress, it cannot be arrested; this results in complete failure of the sample, since the ap￾plied stress intensity factor is always higher than a frac￾ture resistance of the laminate. The cracks that have the crack length parameter between the point A1 and the point A3 will propagate in two stages. For example, the crack with the crack length parameter A2 will have an unstable growth from the point B2 to the point C on the Kapp–a˜ plot (Fig. 6(a)). The stable growth of this crack will occur from the point C to the point D. For all cracks with a crack length parameter a˜ from the point A1 to the point A4, the failure occurs at the stress equal to the slope of the OD straight line, which is a threshold stress. The threshold stress rthr is determined by the maximum value of Kapp at the interface between the first (compressive) and the second (tensile) layers, and no failure can occur below the rthr if the sample contains the surface cracks located only in the first layer. The cur￾vature of the Kapp plot is a function of a value of the residual stress. The higher residual stress, the more con￾cave the curvature of Kapp is. At a certain small value of a residual compressive stress, the line OD can have only one intersection point with Kapp plot, and therefore no stable crack growth stage can occur. The conditions of the stable/unstable crack growth in the laminate with residual tensile stress in top layer are shown in Fig. 6(b). The crack with a crack length parameter A1 for such laminates will propagate only unstably at the stress level above rOB1. The crack with the crack length parameter A grows unstably at the stress rOB. This unstable growth occurs between points B and C (Fig. 6(b)), because the points belonging to the BC segment lie above the Kapp plot. At point C, the condition of the Eq. (6) is violated and the crack growth becomes stable between points C and D, which means that any crack advancement requires an increase of the applied stress. Point D is a maximum value of Kapp at the interface between the second (compressive) and the third (tensile) layers. This point determines a stress rOD = rthr. Above rOD, the crack propagates unstably up to a complete failure. In such a way all ini￾tial cracks in the first (tensile) and the second (compres￾sive) layers with a crack length parameter greater than A2 (Fig. 6(b)) will initiate the specimen failure at the same rOD = rthr stress value. The initial cracks with tips in the third and the fourth layers will initiate specimen failure at the different stress value that is determined by the maximum value of the Kapp at the interface be￾tween the fourth and the fifth layers. This stress is rthr for cracks with tips located in the third and the fourth layers. It should be noted that points the B or B3 in Fig. 6(b) correspond to the measured Kapp values (using ‘‘pop-in’’ stress), while the points B0 or B0 3 belonging to the OD straight line are determined by the initial notch length and the failure stress of the sample. As implied by the above analysis, the surface cracks which have sufficient length to fall into the region of a stable crack growth will all cause a failure at the same rthr stress. At the same time, if a residual compressive stress in the top layer is not high enough, the small cracks can cause catastrophic failure once they start to grow. Therefore, it might be that different mechanisms Tensile layer Compressive layer Compressive layer Kapp Kc Km Threshold stress Stress intensity factor at constant applied stress a ~ B1 D A1A2 A3 A4 A1 B3 C B2 Compressive layer Tensile layer Kapp Kc Km A B Threshold stress σthr a ~ Tensile layer C A2 A3 σ0B1 σ0B (a) (b) factor Stress intensity at constant applied stress B1 B2 B B3 Fig. 6. Conditions for stable/unstable crack growth in a layered structure: (a) a range of crack length parameters for stable crack growth in a laminate with a residual compressive stress in a top layer; (b) stable/unstable crack growth in a laminate with a residual tensile stress in a top layer. M. Lugovy et al. / Acta Materialia 53 (2005) 289–296 295