K Hbaieb et al/ International Journal of Solids and Structures 44 (2007)3328-3343 Table 2 Residual stresses in alumina and alumina/mullite layers Alumina G-(MPa) Alumina/mullite a/(MPa) 282 Mullite/alumina(M40 440.4 Mullite/alumina(M55) 5944 Mullite/alumina(M70) Tensile laver Compressive layer Fig. 4. Detail of the model geometry at the crack tip. ed with its tip in the compressive layer so that its half length is a, slightly greater than the half width of the central tensile layer. Thereafter, under stable growth, the crack either extends straight so that its half length extends to a t b, or it bifurcates symmetrically in such a way that two branches of length b and included angle 20 extend from the straight crack, as shown in Fig. 4. Computations are carried out using the finite element code ABAQUS (2004). Various checks were carried out to ensure accuracy of the results, both in terms of the fineness of the finite element model and the reliability of the results for the crack tip energy release rate, computed by the domain integral method in ABAQUS. Thi exercise included comparison of the numerical results with those for a branched crack in an infinite body of homogenous material calculated by Vitek(1977)using a dislocation model, and a check against results for a kinked crack provided by Cotterell and Rice(1980). It was concluded from these studies that models and methods of sufficient accuracy are being employed for the computations described in this paper 3. Results for cracks subject to tension and thermal stress Fig 5a shows results for the energy release rate for straight and bifurcated cracks where 0=600 for the crack branches. These results are obtained with a thermal stress present along with an applied tensile stress representing the effect of bending. No other external loads are present. The applied stress for the straight crack is chosen for each crack half length a t b and for each compressive layer(M25, M40, M55 or M70 )so that the energy release rate, G, at the crack tip is exactly equal to the toughness, Ge. This is achieved by adjusting the applied stress in each calculation for the straight crack until the Mode I crack tip stress intensity factor due to the combination of applied load and thermal stress is exactly equal to 2, MPaym which is assumed to be the
ed with its tip in the compressive layer so that its half length is a, slightly greater than the half width of the central tensile layer. Thereafter, under stable growth, the crack either extends straight so that its half length extends to a + b, or it bifurcates symmetrically in such a way that two branches of length b and included angle 2h extend from the straight crack, as shown in Fig. 4. Computations are carried out using the finite element code ABAQUS (2004). Various checks were carried out to ensure accuracy of the results, both in terms of the fineness of the finite element model and the reliability of the results for the crack tip energy release rate, computed by the domain integral method in ABAQUS. This exercise included comparison of the numerical results with those for a branched crack in an infinite body of homogenous material calculated by Vitek (1977) using a dislocation model, and a check against results for a kinked crack provided by Cotterell and Rice (1980). It was concluded from these studies that models and methods of sufficient accuracy are being employed for the computations described in this paper. 3. Results for cracks subject to tension and thermal stress Fig. 5a shows results for the energy release rate for straight and bifurcated cracks where h = 60 for the crack branches. These results are obtained with a thermal stress present along with an applied tensile stress representing the effect of bending. No other external loads are present. The applied stress for the straight crack is chosen for each crack half length a + b and for each compressive layer (M25, M40, M55 or M70) so that the energy release rate, G, at the crack tip is exactly equal to the toughness, Gc. This is achieved by adjusting the applied stress in each calculation for the straight crack until the Mode I crack tip stress intensity factor due to the combination of applied load and thermal stress is exactly equal to 2, MPa ffiffiffiffi mp which is assumed to be the Table 2 Residual stresses in alumina and alumina/mullite layers Alumina r2 r (MPa) Alumina/mullite r1 r (MPa) Mullite/alumina (M25) 22.5 282 Mullite/alumina (M40) 35.2 440.4 Mullite/alumina (M55) 47.5 594.4 Mullite/alumina (M70) 59 739 Fig. 4. Detail of the model geometry at the crack tip. K. Hbaieb et al. / International Journal of Solids and Structures 44 (2007) 3328–3343 3333
3334 K Hbaieb et al. International Journal of Solids and Structures 44(2007)3328-3343 Strai 60° Branched crack Branched crack(M25/- 0.2 Crack length 2(a+b A Branched crack G x' Branched crack(M70 Branched crack Straight cracl Branched crack(M55 0.6 0.4 2a/t2 Crack Length 2(a+b)/tz Fig. 5.(a)The energy release rate versus crack length for straight and bifurcated cracks in composites M25, M40, M55 and M70 subject to thermal stress and applied load simulating the effect of bending. The branched crack having branches of length b extending from a crack of half length a is subjected to the same applied stress as a straight crack of half length a t b. The common value of a for the branched cracks is shown by the dashed vertical line. (b) Energy release rate results for straight and bifurcated cracks in the composite systems under consideration, M40. M55 and M70, where the stresses imposed on the cracks are due to applied loads, thermal stress, and the effect of a ee surface and spontaneous edge cracks. plane strain fracture toughness K=VGE for the compressive layer no matter its composition. Note that E=E/(I-v). It should be noted that, due to the compressive thermal stress near the flaw tip, the required stress to propagate it actually increases with the length of the crack, confirming that growth at this stage is stable( Rao et al. (1999), Hbaieb and McMeeking(2002). The same stress as used for a straight crack of half length a t b to cause the crack tip energy release rate to equal the toughness is also applied to the bifurcated crack having the same value of a+b(see Fig 4). All branched cracks are assumed to emanate from a straight crack having the same half-length a Fig. 5a shows the energy release rate results for different values of a+ b. The results for all straight cracks are represented by the single straight line at G/G=l. As expected, the energy release rate at the tips of bifur- cated cracks is smaller then that for a straight crack in all cases when the length of the branch, b, is very small. This feature is consistent with the results for branched cracks in homogeneous materials free of thermal stress where the energy release rate for the branched cracks is always smaller than that for the straight crack(Vitek
plane strain fracture toughness Kc ¼ ffiffiffiffiffiffiffiffiffi GcE0 p for the compressive layer no matter its composition. Note that E0 = E/(1 m 2 ). It should be noted that, due to the compressive thermal stress near the flaw tip, the required stress to propagate it actually increases with the length of the crack, confirming that growth at this stage is stable (Rao et al. (1999), Hbaieb and McMeeking (2002)). The same stress as used for a straight crack of half length a + b to cause the crack tip energy release rate to equal the toughness is also applied to the bifurcated crack having the same value of a + b (see Fig. 4). All branched cracks are assumed to emanate from a straight crack having the same half-length a. Fig. 5a shows the energy release rate results for different values of a + b. The results for all straight cracks are represented by the single straight line at G/Gc = 1. As expected, the energy release rate at the tips of bifurcated cracks is smaller then that for a straight crack in all cases when the length of the branch, b, is very small. This feature is consistent with the results for branched cracks in homogeneous materials free of thermal stress, where the energy release rate for the branched cracks is always smaller than that for the straight crack (Vitek, Fig. 5. (a) The energy release rate versus crack length for straight and bifurcated cracks in composites M25, M40, M55 and M70 subject to thermal stress and applied load simulating the effect of bending. The branched crack having branches of length b extending from a crack of half length a is subjected to the same applied stress as a straight crack of half length a + b. The common value of a for the branched cracks is shown by the dashed vertical line. (b) Energy release rate results for straight and bifurcated cracks in the composite systems under consideration, M40, M55 and M70, where the stresses imposed on the cracks are due to applied loads, thermal stress, and the effect of a free surface and spontaneous edge cracks. 3334 K. Hbaieb et al. / International Journal of Solids and Structures 44 (2007) 3328–3343