Availableonlineatwww.sciencedirect.com IONAL JOURNAL Science Direct and RES ELSEVIER International Journal of Solids and Structures 44(2007)3328-3343 www.elsevier.com/locate/ijsolstr Crack bifurcation in laminar ceramics having large compressive stress K. Hbaieb a. *, R. M. McMeeking a. b, F.F. Lange a Materials Department, Unicersity of California, Santa Barbara, CA 93106, US.A Department of Mechanical Engineering, Unicersity of California, Santa Barbara, CA 93106, US.A Received 13 March 2006: received in revised form 16 September 2006: accepted 25 September 2006 Available online 30 September 2006 Abstract Crack bifurcation is observed in laminar ceramics that contain large residual compressive stress. In such composites Ternating material layers have tensile and compressive residual stress, due to thermal expansion mismatch or other sources. The compressive stress ensures that crack growth leading to failure in the laminar system is mediated by thresh- old strength, but, in some cases, it also leads to bifurcation of the propagating flaw. The phenomenon of bifurcation takes place when the crack tip is propagating in the compressive layer, and occurs typically at a distance equal to few laminate thicknesses below the free surface and beyond. The observation of this phenomenon is usually associated with the presence of edge cracking in the compressive layers of the laminar ceramic, although it can also occur in the absence of such edge cracks. In the few cases where bifurcation occurs without edge cracks, the residual stresses and layer thicknesses are close to the condition in which edge cracks will occur. In addition, in this case the bifurcation is confined to near the specimen free surface, and below the bifurcation plane, the cracks are straight. The energy release rates for the straight and bifurcated cracks are calculated from the results of finite element computations and compared When edge cracking is ignored, the crack is simulated as a through-thickness crack in an infinite body, and the energy elease rate is used to predict crack deviation and bifurcation. Based on this, the finite element model successfully pre- dicts bifurcation in only one material combination that was investigated in experiments. However, the experimental bifurcation takes place in two additional material combinations. When the effect of edge cracking is incorporated into the finite element simulations, the energy release rate calculations successfully predict the phenomenon of bifurcation in three material combinations, as observed in the experiments. Since no edge cracks are present in the fourth material combination tested experimentally, its lack of bifurcations is automatically predicted by the model. The presence of edge cracking, or its incipience, is thus concluded to be critical to the occurrence of crack bifurcation in laminar ceramic o 2006 Elsevier Ltd. All rights reserved Keywords: Crack bifurcation; Finite element modeling: Ceramic matrix composites nding author.Tel:+16568747168;fax:+16567744657 Present address: Institute of Materials Research and Engineering (IMrE), 3 Research link, Singapore 117602, Singapore. 0020-7683/S. see front matter 2006 Elsevier Ltd. All rights reserved doi:10.1016
Crack bifurcation in laminar ceramics having large compressive stress K. Hbaieb a,*,1, R.M. McMeeking a,b, F.F. Lange a a Materials Department, University of California, Santa Barbara, CA 93106, USA b Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA Received 13 March 2006; received in revised form 16 September 2006; accepted 25 September 2006 Available online 30 September 2006 Abstract Crack bifurcation is observed in laminar ceramics that contain large residual compressive stress. In such composites, alternating material layers have tensile and compressive residual stress, due to thermal expansion mismatch or other sources. The compressive stress ensures that crack growth leading to failure in the laminar system is mediated by threshold strength, but, in some cases, it also leads to bifurcation of the propagating flaw. The phenomenon of bifurcation takes place when the crack tip is propagating in the compressive layer, and occurs typically at a distance equal to a few laminate thicknesses below the free surface and beyond. The observation of this phenomenon is usually associated with the presence of edge cracking in the compressive layers of the laminar ceramic, although it can also occur in the absence of such edge cracks. In the few cases where bifurcation occurs without edge cracks, the residual stresses and layer thicknesses are close to the condition in which edge cracks will occur. In addition, in this case the bifurcation is confined to near the specimen free surface, and below the bifurcation plane, the cracks are straight. The energy release rates for the straight and bifurcated cracks are calculated from the results of finite element computations and compared. When edge cracking is ignored, the crack is simulated as a through-thickness crack in an infinite body, and the energy release rate is used to predict crack deviation and bifurcation. Based on this, the finite element model successfully predicts bifurcation in only one material combination that was investigated in experiments. However, the experimental bifurcation takes place in two additional material combinations. When the effect of edge cracking is incorporated into the finite element simulations, the energy release rate calculations successfully predict the phenomenon of bifurcation in three material combinations, as observed in the experiments. Since no edge cracks are present in the fourth material combination tested experimentally, its lack of bifurcations is automatically predicted by the model. The presence of edge cracking, or its incipience, is thus concluded to be critical to the occurrence of crack bifurcation in laminar ceramic composites. 2006 Elsevier Ltd. All rights reserved. Keywords: Crack bifurcation; Finite element modeling; Ceramic matrix composites 0020-7683/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2006.09.023 * Corresponding author. Tel.: +1 65 6874 7168; fax: +1 65 67744657. E-mail address: hb-kais@imre.a-star.edu.sg (K. Hbaieb). 1 Present address: Institute of Materials Research and Engineering (IMRE), 3 Research link, Singapore 117602, Singapore. International Journal of Solids and Structures 44 (2007) 3328–3343 www.elsevier.com/locate/ijsolstr
K Hbaieb et al/ International Journal of Solids and Structures 44 (2007)3328-3343 3329 1. Introduction It has been demonstrated that laminar ceramics having residual stresses that are compressive and tensile in alternating layers exhibit threshold strength when subjected to bending(Rao et al. (1999)). The stresses in the compressive layers inhibit the otherwise unstable growth of cracks until the applied load exceeds a threshold. As a consequence, these systems possess the desirable characteristic of reliability under load (i.e. the absence of failure at low applied load) that is absent from ordinary ceramics. In some laminates(Oechsner et al. (1996) Sanchez-Herencia et al. (1999), Sanchez-Herencia et al. (2000), Rao(2001), Rao and Lange (2002), Pontin and (2005)). the cracks that cause failure propagate straight and the measured threshold strength is predict by theoretical and computational models for such cracks(Rao et al. (1999), Hbaieb and McMeeking 2)). In other laminates, containing thick compressive layers and/or having high stress magnitudes in the compressive layers, crack bifurcation within these layers takes place(Oechsner et al. (1996), Sanchez-Herencia et al. (1999), Sanchez-Herencia et al.(2000), Rao(2001), Rao and Lange (2002), Pontin and Lange(2005)), and the measured threshold strength lies above the level predicted for cracks that grow straight(Rao et al (1999), Hbaieb and McMeekin(2002)) In this context, Rao(2001)and Rao and Lange(2002)investigated four composites. Each composite con- sisted of 5 alumina layers of thickness 550 um having tensile residual stress, with 4 thin compressive layers in between them of thickness 55 um and composed of mixtures of mullite and alumina. The residual stresses ar caused by thermal expansion differences between the alumina and the alumina mullite mixture. These compos- ites are designated MXX according to the amount of mullite present where XX is the weight percentage of it in the compressive layer. In the experiments, pre-cracks approximately 300 um in length were created in the mid dle of the central tensile layer by a vickers indenter with a load of 5 kg. Specimens were then subjected to 4- point flexural bending, causing the cracks to propagate unstably and without bifurcation through the entire central tensile layer, and through about 10% of the two neighboring compressive layers, where the tips arrest- ed. The load was then increased gradually until propagation of the cracks re-commenced, and the flaws then grew stably through the compressive layers until the tip(or tips)reached the next tensile layer or near to it. At that stage, unstable propagation of the cracks set in, and the specimens failed Three of the composites, M40, M55 and M70, i.e. those with higher residual stress magnitue crack bifurcation during their growth under load, as shown in Fig. 1. The fourth, M25, is absent such crack bifurcation. In those laminates exhibiting crack bifurcation during loading, there are also shallow edge cracks at the surfaces of the compressive layers that are produced spontaneously upon cooling after processing and firing of the composite. During cooling, the spontaneous edge cracks propagate down the mid-planes of the compressive layers to arrest in the interior, as depicted schematically in Fig. 2. In these specimens, crack bifur- cation during loading does not take place at the free surface of the component, but instead occurs some dis- tance below, close to the depth corresponding to the tip of the spontaneous edge crack. As can be seen in Fig. l, the angle that each branch makes with the prior crack path is approximately 60(actually 57-67%) The experiments have also confirmed that the propagation of the crack growing under load is stable and not dynamic, so that a monotonically increasing load is required to force the crack progressively throug the compressive layer In the composite designated M25, neither bifurcation of cracks growing under load nor spontaneous edge cracking takes place, and the crack growing under load is also stable. Thus experimental observation in the laminates composed of the materials designated M25, M40, M55 and M70 suggests that crack bifurcation under load occurs in a stable manner under the same conditions that favor the spontaneous creation of edge cracks during cooling, whereas the conditions that do not lead to spontaneous edge cracking during cooling also do not favor crack branching under load To investigate whether crack bifurcation under load is always associated with spontaneous edge cracks, Pontin and Lange(2005)prepared laminar ceramics consisting of thick, tensile layers composed of a mixture of alumina and yttria-stabilized tetragonal zirconia between thinner, compressive layers consisting of a mix- ture of alumina and unstabilized monoclinic zirconia. The residual stresses in this case are generated by a dila tational transformation strain when the unstabilized zirconia transforms from tetragonal to monoclinic. The layer thicknesses were chosen so that spontaneous edge cracking would not occur; i. e the compressive layers were thinner than the critical level for edge cracking. However, the conditions were such that the elimination of the edge cracks was marginal, and in some specimens, in which the compressive layers were slightly thicker
1. Introduction It has been demonstrated that laminar ceramics having residual stresses that are compressive and tensile in alternating layers exhibit threshold strength when subjected to bending (Rao et al. (1999)). The stresses in the compressive layers inhibit the otherwise unstable growth of cracks until the applied load exceeds a threshold. As a consequence, these systems possess the desirable characteristic of reliability under load (i.e. the absence of failure at low applied load) that is absent from ordinary ceramics. In some laminates (Oechsner et al. (1996), Sanchez-Herencia et al. (1999), Sanchez-Herencia et al. (2000), Rao (2001), Rao and Lange (2002), Pontin and Lange (2005)), the cracks that cause failure propagate straight and the measured threshold strength is predicted well by theoretical and computational models for such cracks (Rao et al. (1999), Hbaieb and McMeeking (2002)). In other laminates, containing thick compressive layers and/or having high stress magnitudes in the compressive layers, crack bifurcation within these layers takes place (Oechsner et al. (1996), Sanchez-Herencia et al. (1999), Sanchez-Herencia et al. (2000), Rao (2001), Rao and Lange (2002), Pontin and Lange (2005)), and the measured threshold strength lies above the level predicted for cracks that grow straight (Rao et al. (1999), Hbaieb and McMeeking (2002)). In this context, Rao (2001) and Rao and Lange (2002) investigated four composites. Each composite consisted of 5 alumina layers of thickness 550 lm having tensile residual stress, with 4 thin compressive layers in between them of thickness 55 lm and composed of mixtures of mullite and alumina. The residual stresses are caused by thermal expansion differences between the alumina and the alumina/mullite mixture. These composites are designated MXX according to the amount of mullite present where XX is the weight percentage of it in the compressive layer. In the experiments, pre-cracks approximately 300 lm in length were created in the middle of the central tensile layer by a Vickers indenter with a load of 5 kg. Specimens were then subjected to 4- point flexural bending, causing the cracks to propagate unstably and without bifurcation through the entire central tensile layer, and through about 10% of the two neighboring compressive layers, where the tips arrested. The load was then increased gradually until propagation of the cracks re-commenced, and the flaws then grew stably through the compressive layers until the tip (or tips) reached the next tensile layer or near to it. At that stage, unstable propagation of the cracks set in, and the specimens failed. Three of the composites, M40, M55 and M70, i.e. those with higher residual stress magnitudes, exhibit crack bifurcation during their growth under load, as shown in Fig. 1. The fourth, M25, is absent such crack bifurcation. In those laminates exhibiting crack bifurcation during loading, there are also shallow edge cracks at the surfaces of the compressive layers that are produced spontaneously upon cooling after processing and firing of the composite. During cooling, the spontaneous edge cracks propagate down the mid-planes of the compressive layers to arrest in the interior, as depicted schematically in Fig. 2. In these specimens, crack bifurcation during loading does not take place at the free surface of the component, but instead occurs some distance below, close to the depth corresponding to the tip of the spontaneous edge crack. As can be seen in Fig. 1, the angle that each branch makes with the prior crack path is approximately 60 (actually 57–67). The experiments have also confirmed that the propagation of the crack growing under load is stable and not dynamic, so that a monotonically increasing load is required to force the crack progressively through the compressive layer. In the composite designated M25, neither bifurcation of cracks growing under load nor spontaneous edge cracking takes place, and the crack growing under load is also stable. Thus experimental observation in the laminates composed of the materials designated M25, M40, M55 and M70 suggests that crack bifurcation under load occurs in a stable manner under the same conditions that favor the spontaneous creation of edge cracks during cooling, whereas the conditions that do not lead to spontaneous edge cracking during cooling also do not favor crack branching under load. To investigate whether crack bifurcation under load is always associated with spontaneous edge cracks, Pontin and Lange (2005) prepared laminar ceramics consisting of thick, tensile layers composed of a mixture of alumina and yttria-stabilized tetragonal zirconia between thinner, compressive layers consisting of a mixture of alumina and unstabilized monoclinic zirconia. The residual stresses in this case are generated by a dilatational transformation strain when the unstabilized zirconia transforms from tetragonal to monoclinic. The layer thicknesses were chosen so that spontaneous edge cracking would not occur; i.e. the compressive layers were thinner than the critical level for edge cracking. However, the conditions were such that the elimination of the edge cracks was marginal, and in some specimens, in which the compressive layers were slightly thicker K. Hbaieb et al. / International Journal of Solids and Structures 44 (2007) 3328–3343 3329
3330 K Hbaieb et al. International Journal of Solids and Structures 44(2007)3328-3343 50 um Fig 1. Optical micrographs of crack branching for the materials: (a)M40, (b)M55 and (c)M70. The crack propagating under load grows vertically downward and is arrested just inside the compressive layer, which has a slightly darker shade than the tensile layers. Under rising load, the crack then bifurcates and the branches grow stably as the load is increased. The branching does not occur at the free surface but some distance below it and has been revealed by removal of material. The horizontal dark line is the spontaneous edge crack, which grow down into the material as the surface is removed by grinding, causing some of the branched crack to open up so that it becomes visible Tractions to compensate res/ stress at free surface \Spontaneous edge Residual stress in infinite Fig. 2. A spontaneous edge crack along the mid- plane of a compressive layer of a laminar ceramic extending from the surface to terror of the layer. than in others, spontaneous edge cracks did occur. In all specimens tested by Pontin and Lange(2005), whether there were edge cracks or not, bifurcation of the crack growing under load did occur, and this took in the alumina/mullite specimens tested by Rao(2001)and Rao and Lange(2002), some distance
than in others, spontaneous edge cracks did occur. In all specimens tested by Pontin and Lange (2005), whether there were edge cracks or not, bifurcation of the crack growing under load did occur, and this took place, as in the alumina/mullite specimens tested by Rao (2001) and Rao and Lange (2002), some distance Fig. 1. Optical micrographs of crack branching for the materials: (a) M40, (b) M55 and (c) M70. The crack propagating under load grows vertically downward and is arrested just inside the compressive layer, which has a slightly darker shade than the tensile layers. Under rising load, the crack then bifurcates and the branches grow stably as the load is increased. The branching does not occur at the free surface but some distance below it and has been revealed by removal of material. The horizontal dark line is the spontaneous edge crack, which grows down into the material as the surface is removed by grinding, causing some of the branched crack to open up so that it becomes visible in the micrograph. Fig. 2. A spontaneous edge crack along the mid-plane of a compressive layer of a laminar ceramic extending from the surface to the interior of the layer. 3330 K. Hbaieb et al. / International Journal of Solids and Structures 44 (2007) 3328–3343
K Hbaieb et al/ International Journal of Solids and Structures 44 (2007)3328-3343 below the free surface. However, deep into the specimens, in contrast to flaws in the specimens tested by rao (2001)and Rao and Lange(2002), the fronts of the cracks propagating under load continued to grow straight Thus, the branching segment of the crack was sandwiched between two sectors of flaw that continued to grow straight, one at the free surface and the other deep in the specimen away from the free surface. Consequently the evidence from the experiments of Pontin and Lange(2005)is ambiguous, with bifurcation of cracks grow- ing under load being observed in the absence of spontaneous edge cracks, but with less of the crack front being subject to bifurcation than the cases where it takes place in the presence of spontaneous edge cracks a purpose of the current paper is to investigate the reason why the cracks growing under load in some of the specimens tested by Rao(2001)and Rao and Lange(2002)(i.e M40, M55 and M70) bifurcated, and why others (i.e. M25) did not. There are various hypotheses on why cracks will deviate from a straight path, and why they might split into two branches. One point of view is that a crack will follow the path that produces the largest energy release rate G( Griffith(1920)and Griffith(1924), and when a deviation will produce the largest energy release rate, the crack will curve to follow the line that maximizes this parameter. Other hypotheses suggest that crack stability is assured when T(i.e. the non-singular stress component parallel to the crack sur- face)is negative (i.e. compressive), so that the tendency for branches to grow is inhibited( Cotterell and rice (1980), Rice(1968), Kfouri (1986). In contrast, deviation of the path will then occur if the T-stress is positive Another hypothesis is that flaws propagate so that the Mode II stress intensity factor associated with in-plane shear relative to the crack is zero( Cotterell and Rice (1980). In addition, Lugovy et al. (2002) have suggested that the free surface of a compressive layer, exposed by a crack propagating through it, will produce a bifur- cation of the crack due to the tendency for edge cracks to form at the surface of compressive layers above a critical thickness. Such an argument is persuasive, since straight propagation entirely through the compressive layer must lead to edge cracking in laminae of sufficient thickness. Furthermore, their idea is backed up by agreement between their model and their data for crack bifurcation(Lugovy et al. (2002) The behavior of the cracks observed by rao(2001), Rao and Lange(2002)and Pontin and Lange(2005) will be analyzed to assess them in the light of these hypotheses. For this purpose, we carry out finite element release rate hypothesis for crack deviation and bifurcation is consistent with the observed behavior IeEp.s simulations of cracks in laminar ceramics to understand their behavior, and to see if the maximal energ avior takes place in bending specimens containing surface cracl ce cracks, and thus having a 3-dimensional geometry and loading. However, as a first step, and to avoid very large 3-dimensional computations, we inves- tigate the situation by solving 2-dimensional plane strain problems designed to model the conditions at and near the free surface at which the tensile bending stresses are greatest. We present results that compare the energy release rates for bifurcated and straight cracks in an attempt to establish the conditions that cause crack branching. We first consider a crack propagating in a plane strain laminar ceramic strip subjected to the stress that arises at the tensile surface of a 4-point flexural specimen. We note that this configuration is sufficient to address the hypothesis of Lugovy et al.(2002), since some free surface of the compressive layer is exposed by the penetration without branching of the crack into the compressive layer. In the absence of a satisfactory explanation of crack bifurcation and branching based on the energy release rate for cracks in plane strain laminar strips, including the hypothesis of Lugovy et al. (2002)which fails to survive the scrutiny of our computations, we then consider the effect of stresses arising in a finite thickness laminar ceramic body due to the presence of specimen free surfaces, and the spontaneous formation of edge cracks in the compres- sive layers. We find that when these stresses are taken into account in an approximate, 2-dimensional model the energy release rate at the tip of the bifurcating crack growing under load can be correlated in a suggestive manner with the tendency for such crack branching 2. Model description e: Fig 3 depicts a geometry used in the analysis that contains a straight crack without branching or bifurca- on. Due to symmetry, only one quarter of the whole specimen is modeled, specifically the segment illustrated in Fig 3. The model is used to simulate the experiments of Rao(2001)and Rao and Lange(2002)carried out for the alumina-alumina/mullite composites, and, therefore, the layer dimensions and properties are the same as in the experiment. As in the experiment, the quadrant analyzed contains two compressive layers with thick ness f,=55 um and two and a half tensile layers of thickness t2=550 um. The height of the model, h, is
below the free surface. However, deep into the specimens, in contrast to flaws in the specimens tested by Rao (2001) and Rao and Lange (2002), the fronts of the cracks propagating under load continued to grow straight. Thus, the branching segment of the crack was sandwiched between two sectors of flaw that continued to grow straight, one at the free surface and the other deep in the specimen away from the free surface. Consequently, the evidence from the experiments of Pontin and Lange (2005) is ambiguous, with bifurcation of cracks growing under load being observed in the absence of spontaneous edge cracks, but with less of the crack front being subject to bifurcation than the cases where it takes place in the presence of spontaneous edge cracks. A purpose of the current paper is to investigate the reason why the cracks growing under load in some of the specimens tested by Rao (2001) and Rao and Lange (2002) (i.e. M40, M55 and M70) bifurcated, and why others (i.e. M25) did not. There are various hypotheses on why cracks will deviate from a straight path, and why they might split into two branches. One point of view is that a crack will follow the path that produces the largest energy release rate G (Griffith (1920) and Griffith (1924)), and when a deviation will produce the largest energy release rate, the crack will curve to follow the line that maximizes this parameter. Other hypotheses suggest that crack stability is assured when T (i.e. the non-singular stress component parallel to the crack surface) is negative (i.e. compressive), so that the tendency for branches to grow is inhibited (Cotterell and Rice (1980), Rice (1968), Kfouri (1986)). In contrast, deviation of the path will then occur if the T-stress is positive. Another hypothesis is that flaws propagate so that the Mode II stress intensity factor associated with in-plane shear relative to the crack is zero (Cotterell and Rice (1980)). In addition, Lugovy et al. (2002) have suggested that the free surface of a compressive layer, exposed by a crack propagating through it, will produce a bifurcation of the crack due to the tendency for edge cracks to form at the surface of compressive layers above a critical thickness. Such an argument is persuasive, since straight propagation entirely through the compressive layer must lead to edge cracking in laminae of sufficient thickness. Furthermore, their idea is backed up by agreement between their model and their data for crack bifurcation (Lugovy et al. (2002)). The behavior of the cracks observed by Rao (2001), Rao and Lange (2002) and Pontin and Lange (2005) will be analyzed to assess them in the light of these hypotheses. For this purpose, we carry out finite element simulations of cracks in laminar ceramics to understand their behavior, and to see if the maximal energy release rate hypothesis for crack deviation and bifurcation is consistent with the observed behavior. The experimental behavior takes place in bending specimens containing surface cracks, and thus having a 3-dimensional geometry and loading. However, as a first step, and to avoid very large 3-dimensional computations, we investigate the situation by solving 2-dimensional plane strain problems designed to model the conditions at and near the free surface at which the tensile bending stresses are greatest. We present results that compare the energy release rates for bifurcated and straight cracks in an attempt to establish the conditions that cause crack branching. We first consider a crack propagating in a plane strain laminar ceramic strip subjected to the stress that arises at the tensile surface of a 4-point flexural specimen. We note that this configuration is sufficient to address the hypothesis of Lugovy et al. (2002), since some free surface of the compressive layer is exposed by the penetration without branching of the crack into the compressive layer. In the absence of a satisfactory explanation of crack bifurcation and branching based on the energy release rate for cracks in plane strain laminar strips, including the hypothesis of Lugovy et al. (2002) which fails to survive the scrutiny of our computations, we then consider the effect of stresses arising in a finite thickness laminar ceramic body due to the presence of specimen free surfaces, and the spontaneous formation of edge cracks in the compressive layers. We find that when these stresses are taken into account in an approximate, 2-dimensional model, the energy release rate at the tip of the bifurcating crack growing under load can be correlated in a suggestive manner with the tendency for such crack branching. 2. Model description Fig. 3 depicts a geometry used in the analysis that contains a straight crack without branching or bifurcation. Due to symmetry, only one quarter of the whole specimen is modeled, specifically the segment illustrated in Fig. 3. The model is used to simulate the experiments of Rao (2001) and Rao and Lange (2002) carried out for the alumina–alumina/mullite composites, and, therefore, the layer dimensions and properties are the same as in the experiment. As in the experiment, the quadrant analyzed contains two compressive layers with thickness t1 = 55 lm and two and a half tensile layers of thickness t2 = 550 lm. The height of the model, h, is K. Hbaieb et al. / International Journal of Solids and Structures 44 (2007) 3328–3343 3331
K Hbaieb et al. International Journal of solids and Structures 44(2007)3328-3343 Fig. 3. Layered ceramic body analyzed in the plane strain calculations(not drawn to scale). Due to symmetry, only a quarter of a specimen is modeled. The crack is located along the bottom edge from the bottom left corner and its tip is in the compressive layer 5000 um, and is therefore many times greater than the thickness of the tensile layers. The crack has its tip in the first compressive layer adjacent to the central tensile one (i.e. the leftmost layer in Fig. 3). The bending of the specimen causes tensile loading parallel to the vertical direction in Fig. 3, and the problem is analyzed as if it were in plane strain with tension parallel to the layers. The Youngs modulus, E, and the coefficient of ther mal expansion(CTE), o, of the tensile alumina and compressive alumina mullite layer materials are presented in Table l, with the different values given for the compressive layers depending on their mullite content. In the simulations, we ignore any difference or variation in Poissons ratio, v, and set its value equal to 0. 24 for the entire model throughout all analyses Displacement boundary constraints are applied to the segment analyzed and shown in Fig 3 so that, other than on the crack surface, no displacement occurs normal to the left and bottom edges. All edges of the model are free of shear traction and the right hand edge is free of normal trac- tion as well. in addition the crack surfaces are free of all traction Prior to any other calculations, thermal residual stresses are induced in the model by simulation of the cool- ing of the specimen to 1200C below the stress-free temperature. Note that this produces a slight discrepancy from the experimental case, since plane strain is used in this calculation and the equal biaxial thermal residual tress in the experimental specimen is not reproduced. The resulting stresses near the crack plane in the absence of the crack are uniform in each layer, but near the topmost edge they are nonuniform in a given layer because of the lack of constraint at the free surface. However, the top edge is sufficiently far from the crack plane that the nonuniform stresses there have negligible effect on the calculations of the energy release rate for the crack. The uniform in-plane stress near the crack plane in each layer that arises in the absence of the crack is given in Table 2. After the thermal residual stresses have been induced in the model, a uniform traction is applied along the top edge to simulate the applied tension The models analyzed contain either a straight crack as depicted in Fig 3 or a branched crack. Both geom etries are depicted in detail of the crack tip region shown in Fig. 4. The crack is considered to have first arrest Table I Material properties of alumina and alumina/mullite layers Youngs modulus, E(GPa) Coefficient of thermal expansion, a(10C) mullite/alumina a(M25) a(M40) a(M55) Mullite/alumina(M70) Mullite
5000 lm, and is therefore many times greater than the thickness of the tensile layers. The crack has its tip in the first compressive layer adjacent to the central tensile one (i.e. the leftmost layer in Fig. 3). The bending of the specimen causes tensile loading parallel to the vertical direction in Fig. 3, and the problem is analyzed as if it were in plane strain with tension parallel to the layers. The Young’s modulus, E, and the coefficient of thermal expansion (CTE), a, of the tensile alumina and compressive alumina/mullite layer materials are presented in Table 1, with the different values given for the compressive layers depending on their mullite content. In the simulations, we ignore any difference or variation in Poisson’s ratio, m, and set its value equal to 0.24 for the entire model throughout all analyses. Displacement boundary constraints are applied to the segment analyzed and shown in Fig. 3 so that, other than on the crack surface, no displacement occurs normal to the left and bottom edges. All edges of the model are free of shear traction and the right hand edge is free of normal traction as well. In addition, the crack surfaces are free of all traction. Prior to any other calculations, thermal residual stresses are induced in the model by simulation of the cooling of the specimen to 1200 C below the stress-free temperature. Note that this produces a slight discrepancy from the experimental case, since plane strain is used in this calculation and the equal biaxial thermal residual stress in the experimental specimen is not reproduced. The resulting stresses near the crack plane in the absence of the crack are uniform in each layer, but near the topmost edge they are nonuniform in a given layer because of the lack of constraint at the free surface. However, the top edge is sufficiently far from the crack plane that the nonuniform stresses there have negligible effect on the calculations of the energy release rate for the crack. The uniform in-plane stress near the crack plane in each layer that arises in the absence of the crack is given in Table 2. After the thermal residual stresses have been induced in the model, a uniform traction is applied along the top edge to simulate the applied tension. The models analyzed contain either a straight crack as depicted in Fig. 3 or a branched crack. Both geometries are depicted in detail of the crack tip region shown in Fig. 4. The crack is considered to have first arrest- 2t 1t 1t 2t h 2 2t Fig. 3. Layered ceramic body analyzed in the plane strain calculations (not drawn to scale). Due to symmetry, only a quarter of a specimen is modeled. The crack is located along the bottom edge from the bottom left corner and its tip is in the compressive layer. Table 1 Material properties of alumina and alumina/mullite layers Young’s modulus, E (GPa) Coefficient of thermal expansion, a (106 C1 ) Alumina 401 8.3 Mullite/alumina (M25) 347 7.75 Mullite/alumina (M40) 319 7.37 Mullite/alumina (M55) 293 6.94 Mullite/alumina (M70) 268 6.46 Mullite 220 5.3 3332 K. Hbaieb et al. / International Journal of Solids and Structures 44 (2007) 3328–3343
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
K Hbaieb et al/ International Journal of Solids and Structures 44 (2007)3328-3343 977). However, it can be seen in Fig. 5a that the energy release rate at the tips of bifurcated cracks in heter- ogeneous composites, subject to applied loading and thermal stress, increases with the length of the branch relative to the value of the energy release rate for the straight crack. Indeed, for branches such that b/t2 is greater than about 0.01, the energy release rate at the tips of the branched cracks in the M70 composite is larger than the energy release rate for a straight crack with the same value of at b and the same value of applied stress. It can be deduced from Fig. Sa that bifurcated cracks in a heterogeneous composite having ther- mal stress require a larger applied stress to cause the flaw to propagate compared to the stress required to propagate a straight crack in almost all cases studied. A lower stress for propagation of the bifurcated crack only occurs for long branches, as can be seen in Fig 5a for the M70 composite, and as can be deduced from Fig. Sa by extrapolation of the results in the case of the other composites, M25, M40 and M55 4. Discussion of the results for cracks subject to tension and thermal stress As noted above, the theories regarding the directional growth of a crack in a brittle material involve con- cepts such as deviation to maximize the crack tip energy release rate, deviation from the established path when there is a positive T-stress, and cracks that choose the path that gives a zero value of the Mode II stress inten- sity factor. Unfortunately, none of these concepts is in agreement with the observed behavior in the layered composites as interpreted through the computational results presented in the previous section. When the length of the branches is short, the energy release rate for the straight cracks is always higher than the energy release rate at the tips of the branches. Indeed, if the results in Fig. 5a are extrapolated to where b is zero (i.e infinitesimal branches), the energy release rate for the straight cracks is considerably higher than that for the branches. For Mode I loading of a straight crack, this feature is well known( Cotterell and Rice(1980)). Thus, according to the notion that cracks should follow the path of maximum energy release rate, there is no evi- dence in the results of Fig 5a that explains why cracks deviate from the straight path in composites M40, M55 and M70. Similarly, symmetry dictates that the straight cracks modeled have zero Mode II stress inten- sity factors, whereas infinitesimal branches extending from a Mode I crack will have a non-zero Mode Il stress intensity. Thus, the concept that cracks follow the path that gives a zero Mode II stress intensity factor also predicts that all composites tested by rao(2001)and Rao and lange(2002)will have straight cracks, in con- tradiction of the evidence. Although we do not present results for T computed from the finite element results just described, the results for the energy release rate in Fig. Sa give an indication of the sign of the T-stress for straight cracks of length a. The fact that Fig. 5a demonstrates that the energy release rate for branched cracks emanating from a straight one of length a all increase as b becomes larger shows clearly that the T-stress for the straight crack of length a is positive( Cotterell and Rice(1980)), and is so in all cases studied, i.e. M25 M40, M55 and M70. Thus, according to the idea that a positive T-stress at the crack tip will favor crack prop- agation instability on the existing path( Cotterell and rice(1980)), all composites tested by rao(2001)and Rao and Lange(2002)should experience deviation in the crack growth direction and branching, in contrast to the experimental results. One feature of the T-stress results does, however, suggest a possible explanation The results shown in Fig. 5a for M25, where the energy release rate rises rather slowly as b is increased, indi cate that the T-stress for the straight crack, while positive, is rather low in magnitude in the M25 material com- pared to the other cases Perhaps T in the m25 composite is below a threshold level that may be required for crack deviation and branching. This possibility is persuasive, but it has not been mooted so far in the literature of this subject, and there is as yet no other experimental evidence that this might be However, the preceding paragraph appeals to a rather simplistic set of models regarding the deviation of cracks propagating in a brittle manner. Instead, we can consider a richer possibility discussed by Cotterell and Rice(1980) in the context of homogeneous materials, and also used by He and Hutchinson (1989)to explain and model the deviation and non-deviation of brittle cracks approaching an interface between dissim- ilar materials. In this model, there are a large number of micron the brittle material, and they have a variety of sizes and orientations, or, as the crack grows, a number of branches are thrown out by the crack ip in random orientations and, within limits, to random extents. In either case, we can consider a straight crack with a number of incipient branches along its front with a variety of orientations and lengths Given the results in Fig 5a, the longer branches will have a larger energy release rate than segments of the crack that are still straight. Therefore, such long branches will extend rapidly, and propagation of the straight sectors will
1977). However, it can be seen in Fig. 5a that the energy release rate at the tips of bifurcated cracks in heterogeneous composites, subject to applied loading and thermal stress, increases with the length of the branch relative to the value of the energy release rate for the straight crack. Indeed, for branches such that b/t2 is greater than about 0.01, the energy release rate at the tips of the branched cracks in the M70 composite is larger than the energy release rate for a straight crack with the same value of a + b and the same value of applied stress. It can be deduced from Fig. 5a that bifurcated cracks in a heterogeneous composite having thermal stress require a larger applied stress to cause the flaw to propagate compared to the stress required to propagate a straight crack in almost all cases studied. A lower stress for propagation of the bifurcated crack only occurs for long branches, as can be seen in Fig. 5a for the M70 composite, and as can be deduced from Fig. 5a by extrapolation of the results in the case of the other composites, M25, M40 and M55. 4. Discussion of the results for cracks subject to tension and thermal stress As noted above, the theories regarding the directional growth of a crack in a brittle material involve concepts such as deviation to maximize the crack tip energy release rate, deviation from the established path when there is a positive T-stress, and cracks that choose the path that gives a zero value of the Mode II stress intensity factor. Unfortunately, none of these concepts is in agreement with the observed behavior in the layered composites as interpreted through the computational results presented in the previous section. When the length of the branches is short, the energy release rate for the straight cracks is always higher than the energy release rate at the tips of the branches. Indeed, if the results in Fig. 5a are extrapolated to where b is zero (i.e. infinitesimal branches), the energy release rate for the straight cracks is considerably higher than that for the branches. For Mode I loading of a straight crack, this feature is well known (Cotterell and Rice (1980)). Thus, according to the notion that cracks should follow the path of maximum energy release rate, there is no evidence in the results of Fig. 5a that explains why cracks deviate from the straight path in composites M40, M55 and M70. Similarly, symmetry dictates that the straight cracks modeled have zero Mode II stress intensity factors, whereas infinitesimal branches extending from a Mode I crack will have a non-zero Mode II stress intensity. Thus, the concept that cracks follow the path that gives a zero Mode II stress intensity factor also predicts that all composites tested by Rao (2001) and Rao and Lange (2002) will have straight cracks, in contradiction of the evidence. Although we do not present results for T computed from the finite element results just described, the results for the energy release rate in Fig. 5a give an indication of the sign of the T-stress for straight cracks of length a. The fact that Fig. 5a demonstrates that the energy release rate for branched cracks emanating from a straight one of length a all increase as b becomes larger shows clearly that the T-stress for the straight crack of length a is positive (Cotterell and Rice (1980)), and is so in all cases studied, i.e. M25, M40, M55 and M70. Thus, according to the idea that a positive T-stress at the crack tip will favor crack propagation instability on the existing path (Cotterell and Rice (1980)), all composites tested by Rao (2001) and Rao and Lange (2002) should experience deviation in the crack growth direction and branching, in contrast to the experimental results. One feature of the T-stress results does, however, suggest a possible explanation. The results shown in Fig. 5a for M25, where the energy release rate rises rather slowly as b is increased, indicate that the T-stress for the straight crack, while positive, is rather low in magnitude in the M25 material compared to the other cases. Perhaps T in the M25 composite is below a threshold level that may be required for crack deviation and branching. This possibility is persuasive, but it has not been mooted so far in the literature of this subject, and there is as yet no other experimental evidence that this might be so. However, the preceding paragraph appeals to a rather simplistic set of models regarding the deviation of cracks propagating in a brittle manner. Instead, we can consider a richer possibility discussed by Cotterell and Rice (1980) in the context of homogeneous materials, and also used by He and Hutchinson (1989) to explain and model the deviation and non-deviation of brittle cracks approaching an interface between dissimilar materials. In this model, there are a large number of microflaws in the brittle material, and they have a variety of sizes and orientations, or, as the crack grows, a number of branches are thrown out by the crack tip in random orientations and, within limits, to random extents. In either case, we can consider a straight crack with a number of incipient branches along its front with a variety of orientations and lengths. Given the results in Fig. 5a, the longer branches will have a larger energy release rate than segments of the crack that are still straight. Therefore, such long branches will extend rapidly, and propagation of the straight sectors will K. Hbaieb et al. / International Journal of Solids and Structures 44 (2007) 3328–3343 3335
3336 K Hbaieb et al. International Journal of Solids and Structures 44(2007)3328-3343 die out, with the branches running down the length of the crack front so that the whole flaw deviates and, due to symmetry, bifurcates. As noted above, a version of this model was presented by Cotterell and Rice(1980), who pointed out that when the T-stress is positive for a straight Mode I crack, the lengthening of a deviation will give rise to an increasing energy release rate, thereby causing its unstable growth at the expense of its straight course. In this view, once a deviation becomes established and if it is long enough, the positive T-stress for the original straight crack keeps the branch growing, and in symmetric loadings such as here, bifurcation would take place. The hypothesis presented by Lugovy et al. (2002)is a variant of this model, since their idea, that incipient edge cracking at free surfaces in compressive layers exposed by the crack implies crack bifurca- tion, depends on the same argument-the nucleation spontaneously of a propagating branched crack from a pre-existing, favorably configured flaw. In fact, the stress nucleating the branching under the hypothesis of Lugovy et al.(2002)would be manifested in the form of the T-stress, so their model is essentially the same as that of Cotterell and Rice(1980) These scenarios would predict branching for composites tested by Rao(2001)and Rao and lange (2002)if specimens M40, M55 and M70 all threw out deviations from the straight crack of sufficient length to experi ence an energy release rate larger than that for the undeviated crack, whereas specimen M25 did not. Inspec tion of Fig. 5a indicates that, depending on the way the results are extrapolated, composite M40 would require the spontaneous establishment or the pre-existence of a deviation extending out to around half the thickness of the compressive layer, or about 25 um, to explain crack deviation and bifurcation. Such a length is too large to be credible either as a pre-existing microflaw that links up with the growing straight crack, or as a spontaneous deviation caused by a material heterogeneity influencing the crack growth process. Thus we conclude, that even this more complex model of crack deviation cannot explain the branching observed by Rao(2001) and Rao and Lange(2002), given the results provided in Fig. 5a. We note that the logic of our arguments leads us to dismiss the hypothesis of Lugovy et al. (2002), at least for the specimens under consideration in our com putations, since their model is encompassed by the simulations we have carried out 5. Effect of the spontaneous edge crack As mentioned in the introduction in omposites M40, M55 and M70, the growing crack prop towards an edge crack in the compressive layer that, driven by the thermal residual stresses, has grown in spontaneously from the free surface. The situation is depicted in Fig. 6, which is a sketch of the free surface on the tensile side of the bending specimen showing both the growing crack and the spontaneous edge crack The spontaneous edge crack appears first in specimens M40, M55 and M70 upon cooling after processing. The growing crack is then propagated until it just penetrates into the compressive layer, which is the situation depicted in Fig. 6, where it has not yet started to grow stably in the compressive layer, nor has it yet bifurcated The next step is for the applied load to be increased so that the growing crack in composites M40, M55 and M70 that have these spontaneous edge cracks is propagated towards the edge flaw. Thus the growing crack propagates in the compressive layer, deviates and bifurcates, not only under the influence of the applied load and the thermal residual stress, but also its behavior is dependent on the stress field around the tip of the spon taneous edge flaw After it has spontaneously appeared and grown due to the build up of residual stress during cooling, the edge flaw stops propagating with its crack tip energy release rate exactly equal to the toughness, Ge, so that here are high stresses remaining around its tip. The 3-dimensional state of stress around the tip will affect the growing crack in two different ways. The tensile stress parallel to the y axis(see Fig. 6), present due to plane strain constraint near the tip of the spontaneous edge crack, will assist the propagation of the growing crack in a direct way. However, of more importance to us, the high tensile stress parallel to the x axis associated with he Mode I stress field around the tip of the spontaneous edge crack will augment the T-stress experienced by a growing crack propagating straight. Thus, by the model of Cotterell and rice(1980), the presence of the spon taneous edge crack can destabilize straight propagation of the growing crack, leading it to deviate and, by symmetry, to bifurcate. It is notable that this effect would be strongest a distance c below the free surface, where c is the length of the spontaneous edge crack. This seems to be in accord with the observation of Rao(2001)and Rao and Lange(2002 ) that the growing crack does not deviate or bifurcate at the free surface
die out, with the branches running down the length of the crack front so that the whole flaw deviates and, due to symmetry, bifurcates. As noted above, a version of this model was presented by Cotterell and Rice (1980), who pointed out that when the T-stress is positive for a straight Mode I crack, the lengthening of a deviation will give rise to an increasing energy release rate, thereby causing its unstable growth at the expense of its straight course. In this view, once a deviation becomes established and if it is long enough, the positive T-stress for the original straight crack keeps the branch growing, and in symmetric loadings such as here, bifurcation would take place. The hypothesis presented by Lugovy et al. (2002) is a variant of this model, since their idea, that incipient edge cracking at free surfaces in compressive layers exposed by the crack implies crack bifurcation, depends on the same argument – the nucleation spontaneously of a propagating branched crack from a pre-existing, favorably configured flaw. In fact, the stress nucleating the branching under the hypothesis of Lugovy et al. (2002) would be manifested in the form of the T-stress, so their model is essentially the same as that of Cotterell and Rice (1980). These scenarios would predict branching for composites tested by Rao (2001) and Rao and Lange (2002) if specimens M40, M55 and M70 all threw out deviations from the straight crack of sufficient length to experience an energy release rate larger than that for the undeviated crack, whereas specimen M25 did not. Inspection of Fig. 5a indicates that, depending on the way the results are extrapolated, composite M40 would require the spontaneous establishment or the pre-existence of a deviation extending out to around half the thickness of the compressive layer, or about 25 lm, to explain crack deviation and bifurcation. Such a length is too large to be credible either as a pre-existing microflaw that links up with the growing straight crack, or as a spontaneous deviation caused by a material heterogeneity influencing the crack growth process. Thus we conclude, that even this more complex model of crack deviation cannot explain the branching observed by Rao (2001) and Rao and Lange (2002), given the results provided in Fig. 5a. We note that the logic of our arguments leads us to dismiss the hypothesis of Lugovy et al. (2002), at least for the specimens under consideration in our computations, since their model is encompassed by the simulations we have carried out. 5. Effect of the spontaneous edge crack As mentioned in the introduction, in composites M40, M55 and M70, the growing crack propagates towards an edge crack in the compressive layer that, driven by the thermal residual stresses, has grown in spontaneously from the free surface. The situation is depicted in Fig. 6, which is a sketch of the free surface on the tensile side of the bending specimen showing both the growing crack and the spontaneous edge crack. The spontaneous edge crack appears first in specimens M40, M55 and M70 upon cooling after processing. The growing crack is then propagated until it just penetrates into the compressive layer, which is the situation depicted in Fig. 6, where it has not yet started to grow stably in the compressive layer, nor has it yet bifurcated. The next step is for the applied load to be increased so that the growing crack in composites M40, M55 and M70 that have these spontaneous edge cracks is propagated towards the edge flaw. Thus the growing crack propagates in the compressive layer, deviates and bifurcates, not only under the influence of the applied load and the thermal residual stress, but also its behavior is dependent on the stress field around the tip of the spontaneous edge flaw. After it has spontaneously appeared and grown due to the build up of residual stress during cooling, the edge flaw stops propagating with its crack tip energy release rate exactly equal to the toughness, Gc, so that there are high stresses remaining around its tip. The 3-dimensional state of stress around the tip will affect the growing crack in two different ways. The tensile stress parallel to the y axis (see Fig. 6), present due to plane strain constraint near the tip of the spontaneous edge crack, will assist the propagation of the growing crack in a direct way. However, of more importance to us, the high tensile stress parallel to the x axis associated with the Mode I stress field around the tip of the spontaneous edge crack will augment the T-stress experienced by a growing crack propagating straight. Thus, by the model of Cotterell and Rice (1980), the presence of the spontaneous edge crack can destabilize straight propagation of the growing crack, leading it to deviate and, by symmetry, to bifurcate. It is notable that this effect would be strongest a distance c below the free surface, where c is the length of the spontaneous edge crack. This seems to be in accord with the observation of Rao (2001) and Rao and Lange (2002) that the growing crack does not deviate or bifurcate at the free surface, 3336 K. Hbaieb et al. / International Journal of Solids and Structures 44 (2007) 3328–3343
K Hbaieb et al/ International Journal of Solids and Structures 44 (2007)3328-3343 Tensile applied stress due to bending efore bifurcation 4一 due to thernal stress and free surface Alumina (tensile layer) ketch showing both the growing crack and the spontaneous edge crack. Spontaneous edge cracks appear first 55 and M70 are cooled after processing. They occur where the compressive layers have a free surface. The from left to right until it just penetrates into the compressive layer but instead does so some distance below the free surface, arguably adjacent to the tip of the spontaneous edge As illustrated in Fig. 1, the bifurcation of the growing crack takes place quite close to the interface between the tensile and compressive layers of the composite, and therefore quite far from the spontaneous edge crack, which is located approximately in the middle of the compressive slab. a question that arises is whether the influence of the spontaneous edge crack is sufficiently strong at such a distance to cause the deviation and bifurcation of the spontaneous edge crack. This is assessed by simulations as described below. 6. Simulation of the spontaneous edge crack As noted elsewhere(Ho et al. (1995), Lange et al. (2001), Rao(2001), Rao and Lange(2002)and above edge crack emanating from the surface and propagating into the compressive layer(see Figs 2 and 6) is pro- duced due to the triaxial thermal stress state near the surface of the compressive layer. In an infinite, layered uncracked composite, there would be uniform, biaxial residual stresses in each layer as illustrated at the bot tom of Fig. 2, with compressive stresses of magnitude oe parallel to the z and y axes in the compressive layers, and tensile stresses of magnitude a, parallel to the z and y axes in the tensile layers. In a finite body with a free surface, as shown in Fig. 2, these residual stresses are not viable at the surface and there is a transfer of load by shear stress between the compressive and tensile layers. The stress field associated with this transfer can be computed by taking the thermal residual stresses to be uniform in the layers and adding to them the stresses caused by compensating tractions acting at the free surface as shown in Fig. 2. The stress field caused by the compensating tractions has, at the center of a compressive layer, a tensile stress parallel to the x axis that has magnitude ae at the free surface and decays monotonically with distance along the z axis. It is this tensile stress which spontaneously generates the edge crack. The energy release rate G per unit crack area for a very shallow surface flaw extending by propag along the center line of the compressive layer at the free surface is given by(Ho et al. (1995), Lange 0.34G2n1(1-v2 Er where El is Youngs modulus for the material in the compressive layer Eq (1)shows that G depends not only on the magnitude of the compressive residual stress and the elastic properties of the compressive layer, but also
but instead does so some distance below the free surface, arguably adjacent to the tip of the spontaneous edge crack. As illustrated in Fig. 1, the bifurcation of the growing crack takes place quite close to the interface between the tensile and compressive layers of the composite, and therefore quite far from the spontaneous edge crack, which is located approximately in the middle of the compressive slab. A question that arises is whether the influence of the spontaneous edge crack is sufficiently strong at such a distance to cause the deviation and bifurcation of the spontaneous edge crack. This is assessed by simulations as described below. 6. Simulation of the spontaneous edge crack As noted elsewhere (Ho et al. (1995), Lange et al. (2001), Rao (2001), Rao and Lange (2002)) and above, an edge crack emanating from the surface and propagating into the compressive layer (see Figs. 2 and 6) is produced due to the triaxial thermal stress state near the surface of the compressive layer. In an infinite, layered, uncracked composite, there would be uniform, biaxial residual stresses in each layer as illustrated at the bottom of Fig. 2, with compressive stresses of magnitude rc parallel to the z and y axes in the compressive layers, and tensile stresses of magnitude rt parallel to the z and y axes in the tensile layers. In a finite body with a free surface, as shown in Fig. 2, these residual stresses are not viable at the surface and there is a transfer of load by shear stress between the compressive and tensile layers. The stress field associated with this transfer can be computed by taking the thermal residual stresses to be uniform in the layers and adding to them the stresses caused by compensating tractions acting at the free surface as shown in Fig. 2. The stress field caused by the compensating tractions has, at the center of a compressive layer, a tensile stress parallel to the x axis that has magnitude rc at the free surface and decays monotonically with distance along the z axis. It is this tensile stress which spontaneously generates the edge crack. The energy release rate G per unit crack area for a very shallow surface flaw extending by propagation along the center line of the compressive layer at the free surface is given by (Ho et al. (1995), Lange et al. (2001)) G ¼ 0:34r2 c t1ð1 m2Þ E1 ð1Þ where E1 is Young’s modulus for the material in the compressive layer. Eq. (1) shows that G depends not only on the magnitude of the compressive residual stress and the elastic properties of the compressive layer, but also Growing crack before bifurcation x y z Tensile applied stress due to bending c Alumina (tensile layer) Alumina / mullite (compressive layer) Alumina (tensile layer) 1 t 2 1t Spontaneous edge crack due to thermal stress and free surface 2 1t Fig. 6. A 3-dimensional sketch showing both the growing crack and the spontaneous edge crack. Spontaneous edge cracks appear first when specimens M40, M55 and M70 are cooled after processing. They occur where the compressive layers have a free surface. The growing crack propagates from left to right until it just penetrates into the compressive layer. K. Hbaieb et al. / International Journal of Solids and Structures 44 (2007) 3328–3343 3337