Experimental and Computational Study of Fracturing in an Anisotropic Brittle Solid

Mechanics of Materials 28Ž. 1998 247±262

Experimental and computational study of fracturing in an anisotropic brittle solid

Abbas Azhdari 1, Sia Nemat-Nasser ) Center of Excellence for AdÕanced Materials, Department of Applied Mechanics and Engineering Sciences, UniÕersity of California-San Diego, La Jolla, CA 92093-0416, USA Received 21 November 1996; revised 26 August 1997

Abstract

For isotropic materials, stress- and energy-based fracture criteria lead to fairly similar results. Theoretical studies show that the crack-path predictions made by these criteria are not identical in the presence of strong material anisotropy. Therefore, experiments are performed to ascertain which criterion may apply to anisotropic materials. Notched specimens made from sapphire, a microscopically homogeneous and brittle solid, are used for the experiments. An attempt is made to locate the notch on different crystallographic planes and thus to examine the fracture path along different cleavage planes. The experimental observations are compared with the numerical results of the maximum tensile stress and the maximum energy-release rate criteria. It is observed that most of the notched specimens are fractured where the tensile stress is maximum, whereas the energy criterion fails to predict the fracture path of most of the specimens. For the tensile-stress Žn. s p s r g s fracture criterion, a dimensionless parameter, A ''2 R0 nn a Enn, is introduced, where nnand E nn, respectively, are the tensile stress and Young's modulus in the direction normal to the cleavage plane specified by the angle a Žthe fracture surface energy of this plane is ga . and R0 is a characteristic length, e.g. the notch radius. Five out of six notched specimens which were tested, fractured at the point and along a cleavage plane where AŽn. is maximum. This parameter takes into account the effects of the surface energy of the corresponding cleavage plane, as well as the strength of the atomic bonds in the direction normal to the cleavage plane. It is suggested that for a specimen with a pre-existing crack, the Žc. s r g maximum value of the parameter, A K nn ' a Enn , should be used as the measure of the fracturing condition, where K nn is the hoop stress intensity factor in the direction normal to the cleavage plane. For the notched sapphire specimens, Laue-diffraction analysis, along with the considerations of the geometry of the unit cell of sapphire, shows that the weakest family of cleavage planes,Ä4 1012 and Ä4 1010 , are the fractured planes. q 1998 Elsevier Science Ltd. All rights reserved.

Keywords: Experimental fracture mechanics; Fracture criterion; Brittle fracture; Anisotropic; Sapphire; HoopŽ. normal stress; Energy-release rate; Finite-element method

1. Introduction whereas the analysis of cracks in anisotropic solids has received much less attentionŽ for references, see For isotropic materials, fracture has been exten- Azhdari, 1995; Azhdari and Nemat-Nasser, 1996a,b. . sively studied both theoretically and experimentally, Under quasi-static loading, the often-used fracture

criteria are:Ž. 1 maximum K III, Ž. 2 zero K ,3 Ž. ) Corresponding author. E-mail: [email protected]. maximum hoop stress andŽ. 4 maximum energy-re- 1 E-mail: [email protected]. lease rate. For isotropic cases, these criteria lead to

0167-6636r98r$19.00 q 1998 Elsevier Science Ltd. All rights reserved. PII: S0167-6636Ž. 97 00062-8 248 A. Azhdari, S. Nemat-NasserrMechanics of Materials 28() 1998 247±262 fairly similar results. Theoretical studies, however, view of the above considerations and also of material show that the crack-path predictions made by these availability, sapphire was chosen for our experi- criteria are not identical in the presence of anisotro- ments. py, e.g. each criterion predicts a different angle for The crystallographic structure of sapphire is the crack propagation from a pre-existing crack. hexagonal±rhombohedral. Its unit cell is shown in There are other fracture criteria which stand on their Fig. 1. In a single crystal, different crystallographic own merit. Among these are the criteria of the planes have different surface energies which have an strain±energy±density functionŽ. Sih, 1972 , the nor- important influence on the fracture process. For alu- mal-stress ratioŽ. Buczek and Herakovich, 1985 and minaŽ. polycrystalline sapphire , the tensile strength the tensor polynomialŽ Tsai and Wu, 1971; Wu, is approximately 0.20 GPa, while the tensile strength 1974. . The last two fracture criteria have been pro- of sapphire is 7±15 GPa which is fairly close to the posed to assess the fracture resistance of structures theoretical strength. Because of the very high surface fabricated from composite materials. The inconsis- energy of theŽ. 0001 -plane, the fracture of sapphire tency in the predictions of various fracture criteria is usually does not occur along the basal plane. Separa- demonstrated by, e.g. Obata et al.Ž. 1989 , Azhdari tion along this plane is very difficult because it is not Ž.1995 and Azhdari and Nemat-Nasser Ž 1996a,b . , electrostatically neutral. The unit cell can be visual- where additional references are also found. Fracture ized as aluminum ions stacked between the basal criteria should be based upon physical models and planes of oxygen atoms. Thus, the creation of a experimental observations, established from the fracture surface on the basal plane,Ž. 0001 , would knowledge of the fracturing mechanisms. Therefore, experiments should be performed to ascertain which criterion is applicable to anisotropic materials, e.g. when fracture occurs and in which direction it propagates. In addition, if none of the existing criteria can predict the fracture, then other alternatives must be sought, based on theoretical and experimental studies. The experimental work reported here is a step in this direction. Theoretical studies have focused on homogeneous anisotropic materials. Highly heterogeneous composite materials tend to complicate the experimental results. Although each constituent of a composite materialŽ. e.g. matrix, fibers, whiskers may be considered separately as a homogeneous material, the composite itself is inhomogeneous at a length-scale which is large enough to complicate the fracture process. A class of materials which can be used for our purposes is that of single crystals which are microscopically homogeneous and anisotropic. In the process of material selection, in addition to being single-crystal, material brittleness is also important; in ductile materials a plastic region is invariably formed around the crack tip. In addition to the homogeneity and brittleness, it is preferable to per- form the experiments on thin plates, since theoretical studies often consider plane problemsŽ plane stress Fig. 1. Coordinate systems in the crystallographic structure of or plane strain. . This provides closer connection sapphire; ABCD is theŽ. 1120 -plane Ž coordinate systems are not between the theoretical work and the experiments. In related to those shown in Figs. 2, 3, 5, 6, 7, and 8. . A. Azhdari, S. Nemat-NasserrMechanics of Materials 28() 1998 247±262 249

Table 1 SaphikonŽ. located in Milford, New Hampshire has Fracture surface energies of different cleavage planes in sapphire supplied the sapphire sheets. According to this com- Fracture planeŽ.Ä4Ä4Ä4 0001 1126 1010 1012 pany's catalog, some of the physicalrmechanical EnergyŽ. Jrm2 )40 24.4 7.3 6 properties of sapphire are as follows: Crystallographic structure: Hexagonal, rhombohedral single crystal necessitate the separation between planes of oppo- Tensile strength: 60,000 psiŽ. at 258C sitely charged ions and would require a great deal of Compressive strength: 350,000 psi Ž. energy. Wiederhorn 1969 has measured the surface Young's modulus: 60=106 psiŽ.Ž 414 GPa at energies for sapphire along some crystallographic 258C. planes using the double-cantilever technique which Poisson's ratio: 0.28±0.33 Ž. was originally devised by Gilman 1960 ; see Table Hardness: 9 on MOHS scaleŽ quartz is 7, diamond 1 for fracture surface energies of the cleavage planes is 10. Ž.Ä4Ä4Ä4 0001 , 1126 , 1010 and 1012 . Table 1 shows Melting point: 2,0538C Ä4 Ä4 that the 1010 - and 1012 -planes would be pre- We establish the intersection of the as-received ferred planes for fracturing. These relatively high sapphire sheet with the cleavage planes recorded in values of the surface energy also explain why frac- Table 1. The intersection of this sheet with the four ture in polycrystals tends to be intergranular. crystallographic planesŽ. 0001 ,Ä4Ä4 1126 , 1010 and Experiments are performed on notched sapphire Ä41012 , can be found by considering the geometry of specimens. An attempt has been made to locate the the unit cellŽ. Fig. 1 . The six lines of intersection are Ž notch on different crystallographic planes see Table shown in Fig. 2, which displays the directions on the . 1 and, thus, to examine the fracture path along Ž.1120 -plane in which fracture would possibly occur. different cleavage planes. In order to support and For example, considering Table 1, the horizontal line verify the experimental results, the finite-element in Fig. 2 has the minimum chance of being the code ABAQUS is used to analyze the stress field in the samples. This code is capable of handling elasticity problems with any degree of anisotropy. The calculation of the material constants of sapphire, finite element methodŽ. FEM pre-processing and FEM simulation are presented. Then the post- processing of the FEM results and a discussion are given. The experimental observations are compared with the results of the maximum-tensile-stress and maximum-energy-release-rate fracture criteria, calculated by FEM. The results are compared and discussed. Finally, a summary and then the conclusions are given.

2. Experimental procedure

2.1. Material selection() sapphire

Sapphire, the single-crystal aluminaŽ. Al23 O , which is very brittle and has a high modulus, was chosen for the experiments. Sapphire plates with the Ž.1120 -orientation and 127=127=2 mm size Ž 127 s mm 5 inch. were used. This crystallographic plane Fig. 2. Intersection ofŽ. 1120 -plane with other cleavage planes of is shown in Figs. 1 and 2 as the rectangle ABCD. sapphire. 250 A. Azhdari, S. Nemat-NasserrMechanics of Materials 28() 1998 247±262 fracture direction because this line is the intersection anisotropic solid. However, in highly brittle materi- of the strongest plane, theŽ. 0001 -plane, with the als such as sapphire, it is very difficult to extend an sapphire sheet. All the other lines in Fig. 2 are existing crack in a stable manner and arrest it. There- potential fracture directions, depending upon the fore, the simpler case of a sample with a pre-existing loading direction, geometry, etc. Fig. 2 is a key for notchŽ. instead of a crack is considered. the specimen design, showing the direction in which Specimens of the size of 24=24=2 mm are the specimen should be cut out from the as-received used. A semi-circular notch with a radius of 2 mm is sapphire sheet. produced on one side of the specimen. At first glance, a 2 mm notch may seem too wide to simulate 2.2. Specimen design a crack, but as will be shown later, when the notch The ultimate goal of this study is to examine the radius is large, points on the notch surface are more kinking direction of a pre-existing crack in an distinguishable for measurement. As a result, the

Fig. 3. Notched specimens of different orientations are obtained from the as-received sapphire sheet. A. Azhdari, S. Nemat-NasserrMechanics of Materials 28() 1998 247±262 251 fracture-initiation point may be measured more accu- rately. Location of the fracture initiation point is a prime parameter in the testsŽ see the results and discussion in Section 5. . For each specimen, the direction of the notch aligns with one of the six lines shown in Fig. 2. As a result, different specimens have different crystallographic directions in the same crystallographic plane. Fig. 3 shows how the six specimens and their corresponding notches relate to the as-received sapphire sheet. Each specimen makes some angle, u, with respect to the coordinate system of the sheet Žu is the angle between the x1- and x-directions, measured counterclockwise. . The procedure to calculate the elastic±stiffness constants with respect to the specimen coordinate system Ž.x±y system in Fig. 3 is presented in Section 3.1.

2.3. Loading deÕice

Due to the brittleness and hardness, machining and drilling of sapphire are difficult. For example, making a compact-tension specimen requires drilling two holes in the specimen, which is relatively difficult. Secondly, a displacement-controlled test is de- sirable because it provides stable crack growth. For a highly brittle material such as sapphire, the energy required for crack extension is relatively small. Therefore, once a crack is initiated, it may continue Fig. 4.Ž. a Overall view of specimen setup, Ž. b magnified front view andŽ. c magnified top view. to propagate even under theŽ. nominally displacement-controlled conditions. Specimens are therefore designed such that they can be made with minimal the specimen does not affect these quantities; see machining and they are loaded by a displacement- Sections 3 and 4. controlled mechanism. The loading device uses the thermal expansion of a small block of steelŽ. 4=4=10 mm ; see Fig. 4. 3. FEM simulation The steel block with a thermal-expansion coefficient of 20=10y6r8C is fitted inside the notch, and then 3.1. Calculation of elastic±stiffness matrix heated by a soldering iron. The effective length of the block of steel is about 4 mm. It is observed that No closed-form solution exists for anisotropic the specimens are fractured when the steel block notched specimens of finite size. Therefore, compari- reaches a temperature of around 150 to 2008C. The son of the experimental and theoretical results re- objective of the test is to measure the point on the quires a numerical approach using, for example, a notch surface at which the fracture is initiated and, finite-element method. The commercial FEM code, also, to determine which cleavage plane is the frac- ABAQUS which can handle elasticity problems of tured plane. Due to the linearity of the system, the any degree of anisotropy is used. In general, any magnitude of the load exerted by the steel block on FEM code requires that the input elastic±stiffness 252 A. Azhdari, S. Nemat-NasserrMechanics of Materials 28() 1998 247±262 matrix be expressed with respect to the body-coordi- used for the Miller±Bravais indices. For sapphire, nate systemŽ e.g. the x±y coordinate system in the symmetry of point group 3m reduces the number Fig. 3. . The stress±strain relation for a generally of independent elastic stiffnessŽ. or compliance con- s anisotropic linearly elastic body, sijs ijmn´ mn Ži, stants from 21 to 6. For example, the elements of the j, m, ns1, 2, 3. , is expressed in matrix notation as elastic±stiffness matrix for sapphire, referred to the swx swx wxs wxy1 Ä4t S Ä4g or Ä4g C Ä4t , where C S ; the coordinate system x123±x ±x , of Fig. 1, reported by vectors t and g are the compact forms of the stress Wachtman et al.Ž. 1960 , are: Ž. s s s s and strain engineering tensors and S and C are the S114.968, S 121.636, S 131.109, S 14 0.235, stiffness and compliance matrices, respectively. s s s s S150, S 160, S 22S 11, S 23S 13 , Energy and symmetry considerations reduce the S syS , S s0, S s0, S s4.981, number of elastic constants to 21. Fig. 1 shows the 24 14 25 26 33 S s0, S s0, S s0, S s1.474, S s0, rectangular Cartesian coordinate system, x ±x ±x , 34 35 36 44 45 123 s s s s y r to which the elastic constants are referred, and the S460, S 55S 44, S 56S 14, S 66Ž.S 11S 22 2. hexagonal coordinate system, x±y±z±c, which is Note that the units are 10y12 dynrcm2 where 1

Fig. 5. FEM mesh for the notched specimenŽ. 2984 elements and 3047 nodes . A. Azhdari, S. Nemat-NasserrMechanics of Materials 28() 1998 247±262 253

r 2 s = y6 s = dyn cm 14.503 10 psi, e.g. S11 4.968 components of the same vector given in the primed 1012 dynrcm2 s72.05=106 psi. systemŽ.Ž. e.g. x±y in Fig. 3 ; see Azhdari 1995 for As mentioned earlier, the experiments are per- more details. formed on samples obtained from aŽ. 1120 -sapphire 3.2. Mesh generation and external loading simula- plate. For FEM calculations, the elastic±stiffness X tion matrix, sijkl, in theŽ. 1120 -plane Žx±y coordinate system in Fig. 3. is required. This is given by Finite-element calculations are performed using X s sijklRRRRs im jn kp lq mnpq, where R ij is the corre- 2984 elements and 3047 nodesŽ. Fig. 5 . Note that sponding orthogonal rotation tensor which trans- this mesh considers 48 nodes on the inner surface of forms the components of an arbitrary vector given in the notch, and thus the numerical investigation in the unprimed system Ž.x123±x ±x in Fig. 1 to the regard to the stresses on the notch surface is per-

Fig. 6. Tensile stress snn acting normal to the cleavage plane m±m at an arbitrary point P, creating a crack of length L. 254 A. Azhdari, S. Nemat-NasserrMechanics of Materials 28() 1998 247±262

formed at just these 48 distinct nodes. Uniform Gc . For isotropic materials, these measures are not pressure p Ž.Fig. 6 is assumed to act on the contact direction-dependent, whereas in the anisotropic case, area between the steel block and the specimen as they are, i.e. different planes require different ener- shown in Fig. 6. The magnitude of this uniform gies to fractureŽ. see Table 1 . Thus the role of the pressure is not needed if just the location and the surface energies of different cleavage planes should orientation of the planes which carry the maximum be taken into consideration. tensile stresses are of interestŽ this is because the The fracture resistanceŽ. or surface energy of behavior of the system is assumed to be linear, i.e. sapphire for different crystallographic planes is given the response of the system is proportional to p.. in Table 1. Fig. 6 shows a typical point P on the Thus, no effort was made to measure the magnitude notch surface. A typical cleavage plane passing of the applied pressure. Finally, a plane-stress condi- through this point is denoted by mmŽ making the tion is assumed which is suitable for the thin speci- angle a with respect to the x-axis. . Using the FEM mens used. FEM is used to calculate the tensile results for points on the inner surface of the notch, stresses acting on the inner surface of the notch, and the tensile stress, snn, on the cleavage plane, mm,is s y 2 q y also to estimate the energy-release rate for possible obtained from snnŽ.s yys xxcos a s xx cracking along different crystallographic planes on sxycos 2a. This tensile stress, s nn, has the tendency the notch surface. to break the interatomic bonds along the mm-direction. Finally, for each cleavage plane, the point at

which the tensile stress, snn, is maximum is found. 4. FEM results The location of this point on the semi-circle of the notch is denoted by angle b ŽS.; see Fig. 6. It should

be mentioned that even though at this point snn is 4.1. Calculation of the stress field near the notch maximum on the cleavage plane mm, this cleavage surface plane is not necessarily shear-free. All 48 distinct nodes on the notch are examined. At each node the Under the conditions described in Section 3.2, the existing 6 different cleavage planes are studiedŽ Fig. displacement and stress fields are calculated for each 2. . Therefore, for each specimen, snn is calculated specimen shown in Fig. 3, using the FEM code for 48=6s288 different cases. ABAQUS. Since the stress concentration is relatively In general, a cleavage plane makes an angle l high on the inner surface of the notchŽ see Figs. with theŽ. 1120 -plane. The mm-direction represents 4±6. , fracture preferentially initiates from this area the intersection of the corresponding cleavage plane and then propagates into the specimen. Two ques- with theŽ. 1120 -plane. Therefore, snn does not tions are examined in some detail: necessarily represent the tensile stress acting normal Ž.1 The point on the inner surface of the specimen to the actual cleavage plane. For a plane-stress condi- at which fracture may initiate. tion, the tensile stress acting on the actual cleavage 2 Ž.2 The direction of crack propagate. plane isŽ sin ls. nn. This is used in the corresponding Following the assumption that crack extension calculations. starts in the plane normal to the direction of greatest According to the procedure discussed above, the tensile stressŽ the maximum tensile stress criterion; related calculations are carried out for all 6 speci- see, for example, Erdogan and Sih, 1963 and Otsuka mens; see Fig. 3. The results are shown in Table 2. et al., 1975. , assume that the specimen fractures In Table 2, b ŽS. defines the point on the notch from the point and in the direction where the tensile surface at which the calculated snn is maximum on stress first reaches a critical value sc ; this is similar the particular cleavage plane of angle a. The angle to the maximum tensile-stress criterion. This critical b ŽS. Žthe angular position of the point of maximum tensile stress, sc , is associated with the energy re- snn on the notch surface. is positive when measured quired for breaking the interatomic bonds in the from the x-axis in the clock-wise direction. The s ŽS. direction of maximum tension. The quantity c may experimental counterpart of b is denoted by bŽF.; be related to other fracture measures such as K Ic or this is the angle at which the fracture was actually A. Azhdari, S. Nemat-NasserrMechanics of Materials 28() 1998 247±262 255

Table 2 Comparison of FEM and experimental results;Ž.Ž S , G . and Ž. F stand for tensile-stress, energy-release-rate and fractured-by-experiment, respectivelyŽ. see Figs. 2, 3, and 6 Crystallographic planes Angles Ž.8 Specimen a1 Specimen a2 Specimen a3 Specimen a4 Specimen a5 Specimen a6 Ž.1010 a 90 180 32.4 122.4) 52 142 b ŽS. 93.7 22.5 142.5 60 127.5 56.2 b ŽG. 93.7 30 142.5 60 127.5 63.7

Ž.1102 a 57.6 ) 147.6 0 90 19.6 109.6 ) b ŽS. 112.5 41.2 157.7 93.7 142.5 71.2 b ŽG. 101.2 48.7 157.7 86.2 138.7 71.2

Ž.Ž.Ž.Ž.1012 , 0112 , 1216 , 2116 a 38 128 y19.6 70.4 0 90 b ŽS. 123.7 67.5 41.2 112.5 153.7 82.5 b ŽG. 112.5 67.5 48.7 112.5 142.5 78.7

Ž.Ž.Ž.0001 , 1126 , 1126 a 090y57.6 32.4 y38 52 b ŽS. 18.7 90 63.7 131.2 45 127.5 b ŽG. 22.5 86.2 67.5 127.5 45 116.2

Ž.Ž.Ž.Ž.1012 , 0112 , 2116 , 1216 a y38 52 y95.6 y5.6 y76 14 b ŽS. 45 112.5 86.2 33.7 67.5 146.2 b ŽG. 60 101.2 78.7 33.7 67.5 142.5

Ž.1102 a y57.6 32.4 I115))y25.2 I95.6 y5.6 b ŽS. 71.2 127.5 116.2 41.2 101.2 33.7 b ŽG. 78.7 112.5 90 45 105 37.5 y y y y Experimental results aŽF. 61, 53 126, 135 116, 110 115, 83 99, 97 112, 99 bŽF. 115, 120 70, 74 120, 115 63, 65 101, 100 63, 68

initiated on the notched specimen. The values of this specimen, the calculation of the energy is straightfor- angle for different specimens are also presented in wardŽ. use the FEM mesh given in Fig. 5 , whereas Table 2. for the cracked specimen, the calculation requires a new FEM mesh to account for the crack. For this, 4.2. Calculation of energy-release rate due to crack- fine discretization is added around the point where ing of the notch surface the crack is placed; see Fig. 7. As Fig. 8Ž. this is a magnified Fig. 7 shows, this Assume, now, that a crack with length L occurs mesh can simulate cracks along 7 different directions in the direction mm, shown in Fig. 6. This changes and this provides the possibility to approximately the total potential energy of the sample. Denote by locate one of these 7 directions on the cleavage GŽmm. this energy change due to cracking. In what plane, mm. As a result, the use of this mesh facili- follows, it is assumed that the crack lengthŽ in any tates the examination of all 6 different cleavage direction.Ž. is always LLf50 mm . Thus, the term planesŽ. Fig. 2 at any typical point on the notch `energy released per L crack length' will be replaced surface. Then, for the examination of points on the by the commonly used term `energy-release rate', notch surface, the fine mesh is located at all 48 using L as the unit of length. To calculate GŽ mm., nodes along the notch surface, as shown in Fig. 7. In find the difference between the work done by the addition, Fig. 8 demonstrates the configuration of the externally applied loads for the two cases of the points around the fine mesh before and after crack un-cracked and cracked specimen. For the un-cracked opening. 256 A. Azhdari, S. Nemat-NasserrMechanics of Materials 28() 1998 247±262

Fig. 7. FEM mesh for the notched throat at a location of possible cracking.

A calculation similar to the one performed for occurs at that point and in one particular cleavage tensile stress is performed to find the point at which plane. Note that, for each specimen, calculation of the maximum energy would be released if a crack the tensile stress requires just one run of ABAQUS, whereas the energy-release-rate calculation involves 48=6s288 runs of ABAQUS. The released energy due to the occurrence of a crack in the mm-direction, GŽmm., is calculated by the following formula:

M Žmm.Žs x.Žc.Žy nc. G Ý Ä Puiiu i is1

q Ž y.Žc.Žy nc. PiiÕ Õ i4 ,1Ž.

Ž x. Ž y. where Piiand P are the x- and y-components of an externally applied point force at the corre- Žc. Žc. sponding ith point, respectively; uiiand Õ are the displacements of the ith point of a cracked notch in Žnc. Žnc. the x- and y-directions, respectively; uiiand Õ are the displacements of the ith point of a non- cracked notch in the x- and y-directions, respectively and, finally, M is the number of the externally applied point forces. Moreover, as mentioned earlier, the magnitude of the resultant load associated with Ž x. Ž y. Piiand P is immaterial. The calculated values Žnc. Žnc. of uiiand Õ are not too sensitive to the fineness Fig. 8. FEM mesh illustrating the configurations before and after of the mesh around the crack tip, because the crack Žnc. Žnc. crack-opening. and the points where uiiand Õ are calculated A. Azhdari, S. Nemat-NasserrMechanics of Materials 28() 1998 247±262 257

Ž mm. Žthe points where the external loads are considered, mum snn and G are distinct and can easily be Fig. 6. are far from each other. compared with the actual failure location. According to the procedure discussed above, for each specimen and for each of the 6 directions 5. Results and discussion shown in Fig. 2, denoted by a in Fig. 6, ABAQUS is run 48 times, corresponding to 48 nodal points on The results of the numerical analysis of the two the notch surface. Then GŽmm. is calculated for all fracture measuresŽ tensile stress and energy-release these 48 different configurations. The maximum, rate. are presented in Table 2. The point and the Ž mm. Gmax , of this set of 48 G 's occurs at a point on direction of fracture Ž.bŽF.Žand a F. are directly ŽG. the notch surface, Gmax , denoted by b ; see Fig. 6 measured on the fractured specimensŽ Fig. 10 shows and Table 2. The angle b ŽG. Žthe angular position on the picture of 6 of the broken specimens. . The the semi-circular notch. at which Gmax occurs is Laue-diffraction analysis is performed on two of the positive when measured from the x-axis in the specimens and the crystallographic indices are deter- clock-wise direction. Note that b ŽG. defines the mined. This determination along with the geometric point on the notch surface where the maximum considerations of the crystallographic planes, facili- energy is released if a crack initiates in the cleavage tate the determination of the crystallographic indices plane of direction a shown in Fig. 6. Since there are of all other planes. The experiments are conducted 6 different a directions, b ŽG. is calculated 6 times on 2 specimens of each configuration shown in Fig. for each specimen. The results are shown in each 3 and, thus, altogether, 12 specimens are used. Table column of Table 2. 2 is a summary of the FEM analysis as well as the Finally, the variations of the normal tensile stress, experimental results on all 6 different specimens. In Ž mm. snn, and the energy released, G , with respect to each column of Table 2, the bold-face numbers and the location on the notch-surface are presented in the numbers with a superscript star indicate that the Fig. 9 for specimen a1 and the cleavage plane a5 corresponding FEM results match the experimental sy Ž.a 388 . This figure shows that the snn- and results. Note that the FEM analysis is performed on GŽmm.-curves have:Ž. 1 fairly sharp maxima; Ž. 2 sim- 48 distinct points on the notch surface, and thus ilar trend andŽ. 3 their maximum points occur at every two consecutive points are about 1808r48f48 different angles. The first result is important from the apart. In view of this, the FEM results such as the experimental viewpoint, since the points of maxi- values of a, b ŽS., and b ŽG., are accurate to within "48. Next, the conclusions drawn from Table 2 are presented. Let us now compare the experimental and numerical results from Table 2; this is done by comparing

a 's and aŽF.Ž's and also b 's and b F.'s. In order to clarify how this comparison is made, consider the column corresponding to the specimen a1 in Table 2. As is seen, the cleavage planeŽ. 1102 is the only plane whose corresponding numerical resultsŽ angles a and b ŽS.. match closely with the experimental

resultsŽ. angles aŽF.Žand b F. ; note that a 's indicate the cleavage plane and b 's are the angular positions of the points on the notch surfaceŽ. Fig. 6 . This shows that specimen a1 is fractured along the Ž.1102 -plane at the point Ž on the notch surface . where the tensile stress is maximum. Fig. 9. Tensile stress and energy-release rate versus notch-throat This method of comparison shows that 5 of the angle b for specimen a1 and cleavage plane a5 Ž.a sy388 ; see specimens are fractured along the weakest cleavage Figs. 2 and 6. family of planesÄ4 1012 and one specimen Ž.a4is 258 A. Azhdari, S. Nemat-NasserrMechanics of Materials 28() 1998 247±262

Fig. 10. Fractured specimens; compare with Fig. 3. fractured along the second weakest family of cleav- tensile-stress and maximum energy-release rate crite- age planesÄ4 1010 ; see Table 1. Moreover, Table 2 ria predict the same crack path. These occasional shows that 5 out of 6 samplesŽ. except for a2 are agreements between the predictions made by the fractured according to the maximum-tensile stress stress- and energy-based criteria are also observed in Ž b ŽS. in bold.Ž . For some of the samples a4 and the analytical studies; see Azhdari and Nemat-Nasser a6. the FEM results show that the maximum Ž.1996b . A. Azhdari, S. Nemat-NasserrMechanics of Materials 28() 1998 247±262 259

The FEM analysis shows that fracture occurs in root of the strength term and also use a characteristic the direction where a high tensile stress acts on a length, e.g. the notch radius R0. The result is weak cleaÕage plane. In other words, it is a compro- snn mise between the surface energyŽ the lower, the AŽN . s 2p R ,2Ž. ( 0 g easier to fracture. and the tensile stress acting normal ( a Enn to this surfaceŽ the higher, the more probable to where the superscriptŽ. N stands for `notch'. Note break the interatomic bonds. . Furthermore, wherever the similarity between the AŽN.-criterion and the a combination of these two is optimized, fracture energy formula of GriffithŽ. 1921, 1925 s occurs. For example, there are cases where the ten- c s Eg r'p a , where s is the critical stress for the sile stress acting on the basal planeŽ.Ž 0001 the ' c extension of a pre-existing crack of length a. strongest cleavage plane. is much higher than the In order to verify the validity of the AŽN.-fracture one acting on the family ofÄ4 1012 -planes, but frac- criterion, the FEM results for the stresses, s , the ture occurs on theÄ4 1012 ones. This is also observed nn values of the corresponding g 's from Table 1, and when the criterion of the energy-release rate is stud- a E obtained from the elastic constants given in ied. This study highlights the role of the cleavage nn Section 3.1 are used. The results are interesting in planes and the differences between their surface the sense that the predictions made by the criterion energies in the fracturing of single crystals. of the maximum-AŽ n. match the experimental results. Note that in a brittle single crystal, it is very The point and the direction of fracturing of 5 speci- unlikely that a crackŽ. or a micro-crack would initi- mens are predicted by this fracture criterion, i.e. for ate and propagate according to the maximum-K I 5 specimensŽ. from 2 , when the 6 different cleavage Ž.zero-K fracture criterion. This is because a weak II planes are compared using criterionŽ. 2 , it is ob- cleavage plane does not always happen to be in the served that the maximum-AŽN. fracture criterion iden- direction of the maximum-K Žor equivalently, zero- I tifies the `cleavage planes' along which the speci- K .. This comment applies to the case of an II mens fractured in the actual tests. already-cracked body. For the case of a notched In contrast to the criterion of maximum-AŽN., the specimen, fracturing does not always occur in the criterion of maximum-GŽ mm.rg fails to predict the direction where the tensile stress is maximum, be- a fracturing of any of the 6 specimens. Therefore, the cause the plane in the direction of the maximum FEM results and the experiments conducted on these tensile stress may be a relatively strong cleavage 6 different specimens show that the maximum-AŽN. plane. Note that the strength of a certain plane in a criterion is capable of predicting the fracture phe- brittle solid is expected to depend on the Young nomenon of such notched specimens. Note that this modulus E and the surface energy g. Thus, the criterion is basically a stress-based fracture criterion, product Ž.Eg may be considered as a measure of the although it contains the parameter g which is an strength of a plane against fracture. a energy-related parameter. Summing up, this criterion can be justified if it is interpreted as follows: The 5.1. Potential fracture criteria maximum-AŽN. may be a suitable fracture criterion because it is based upon the local tensile stress Now, consider a fracture criterion which is based derived from the global external loading and also it upon a combination of the strength of the cleavage contains an energy term which represents the energy plane and the tensile stress acting on that plane. required to break the atomic bonds over the fracture Examine the ratio Ass rŽ.E g and assume that nn nn a plane. fracture initiates at the point and in the direction s r where the ratio A snnŽ.E nnga is maximum. Here, nn is normal to the direction of the cleavage plane 5.2. Comments on the fracture criterion on which crack propagation may occur, snn is the ŽN. tensile stress acting normal to that plane, and ga is The criterion of maximum-A is defined for a the surface energy of that cleavage plane. In order to notched body. Consider applying this criterion to a make this ratio dimensionless, consider the square cracked body to predict possible crack kinking. For a 260 A. Azhdari, S. Nemat-NasserrMechanics of Materials 28() 1998 247±262

notched body, the normal stress snn is finite, whereas surface-energy function, i.e. we consider a criterion if the notch is shrunk to a sharp crack, then this similar to Eq.Ž. 3 for the energy-release rate for an stress becomes infinite at the crack tipŽ singular anisotropic solid having multiple cleavage planes. stress. . Define the hoop stress intensity factor, K nn, First, consider the energy-release rate formula s r through snnK nn '2p r , where r measures the Ž.Irwin's equation s distance from the crack tip. Substitute this for nn X Ž. C into 2 and let r be equal to R0. Then, the charac- s 11 Žk.Žk.Žq k.Žk. G f1II2IIIIŽ.m, v KK f Ž.m, v KK teristic length R0 cancels out and we obtain 2

q m v Žk.Žk. K nn f3IIIŽ., KK ,4 Ž. AŽC . s ,3Ž. g ( a Enn where fi's are known functions of the kink angle v and mmŽ stands for `material parameters', i.e. where the superscriptŽ. C stands for `crack'. Now, functions of material constants Cij's in Section 3.1. , Žk. Žk. consider the condition for crack kinking in single K IIIand K are the stress intensity factors at the crystals of brittle solids having a finite number of tip of the vanishingly small kink of angle v with X cleavage planes, as follows: Among all cleaÕage respect to the pre-existing crack direction and C11 is planes passing through the tip of a pre-existing the compliance in the kink direction; see also Azh- crack, crack extension occurs on the cleaÕage plane dari and Nemat-NasserŽ. 1996b . When the driving ŽC. Žk. which renders A maximum. This can be reworded forceŽ the external loading expressed by K I and Žk. as follows: A pre-existing crack kinks along a cleaÕ- K II . reaches the value where the energy-release rate age plane on which AŽC. is maximum. Note that, is equal to the surface energy of the cleavage plane according to the stress-based fracture criteria of max- lying in the v-direction, then fracturing of this cleav- ŽN. ŽC. imum-A or maximum-A , fracture most likely age plane takes place and one has Gsgv.Ifwe occurs on cleavage planes which carry non-zero seek to find the kink angle v by maximizing the shear stresses. This is not in accordance with the function on the right-hand side of Eq.Ž. 4 , the result- hoop and shear stress intensity factorŽ HSIF and ing kink angle may lie on a strong cleavage plane SSIF. criteria where the maximum-HSIF is accom- which does not fracture under the applied loads and panied by the zero-SSIF, as discussed by Azhdari therefore does not define a physically meaningful and Nemat-NasserŽ. 1996a who, however, do not kink angle. An alternative to this criterion is to address the anisotropy stemming from the material include in the right-hand side the surface energy of resistance to fracturing. the material corresponding to the v-plane and define A related criterion for the crack extension direction in orthotropic composite materials is proposed BŽC . by Buczek and HerakovichŽ. 1985 . They suggest that X C f Ž 11 Žk.Žk.Žk.Žk. a crack kinks in the -direction if the ratio Rr0 , s f Ž.m, v KKqf Ž.m, v KK s r g 1II2IIII f.Ž.Ž.sff r0 , f Tfff is maximum, where s ff 2 v is the stress normal to the f-direction, Tff is the q Žk.Žk. tensile strength of the composite in the f-direction f3IIIŽ.m, v KK ,5 Ž. and r0 is a given distance from the crack tipŽ to be determined experimentally from a baseline test. . Note where BŽC. is a dimensionless quantity and, again, that a drawback of this criterion is its dependence on the superscriptŽ. C stands for `crack'. ŽC. the experimental determination of r0. The fracture Maximizing, now, B with respect to v may criterion in Eq.Ž. 3 has no dependency on distance give the angle at which the pre-existing crack kinks from the crack tip. Žthe occurrence of this kink releases energy at the ŽC. Based on the maximum-A criterion, we seek to rate of gv .Ž.. Note that: 1 the criterion of the maxi- modify the commonly used energy-release rate crite- mum-BŽC. is suitable for the case when the surface rion so as to account for the multi-valuedness of the energy gv is a function of angle v Žan anisotropic A. Azhdari, S. Nemat-NasserrMechanics of Materials 28() 1998 247±262 261 quantity.Ž. and 2 the maximum-AŽC. criterion deals Ž.4 In view of Ž. 1 , the fracture criterion of maxi- ŽN. s p s r g with quantities prior to kinking such as the HSIF mum A ((2 R0 nn a Enn is proposed. The ŽC. Ž.K nn , whereas the maximum-B criterion involves predictions made by this criterion matched the exper- Žk. quantities at the tip of a kinked crack, e.g. K I and imental results for 5 out of 6 specimens. Žk. K II . Ž.5 The stress-based fracture criterion appears to better predict the fracturing of single-crystal brittle solids. This may also be applied to predict crack 6. Summary and conclusions kinking. Ž.6 A crack in a brittle single crystal kinks in the ŽC. s r g The fracture of homogeneous, linearly elastic and direction where A K nn ( a Enn is maximum. anisotropic solids is considered. Sapphire, a single- Here, K nn is the hoop stress intensity factor normal crystal and highly brittle material, is selected for the to a cleavage plane located in the n-direction and ga experiments. Notched specimens with six different is the surface energy of that cleavage plane. Note orientations, Figs. 2 and 3, are made from a sheet of that, the cleavage planeŽ equivalently, the n-direc- Ž. ŽC. s r g sapphire with the crystallographic orientation 1120 . tion. which renders A K nn ( a Enn maximum Specimens are loadedŽ. and eventually fractured by may not, in general, be the zero-shear plane. This is heating up a piece of steel block placed inside the not in accordance with the maximum-hoopŽ zero- notch; see Fig. 4. Two parameters are measured after shear. stress intensity factor; see Azhdari and Ne- each test: the location of the point on the inner mat-NasserŽ. 1996a . surface of the notch at which fracture is initiated and Ž.7 For the notched sapphire specimens, Laue-dif- the direction in which the crack propagates. Laue- fraction analysis, along with the considerations of the diffraction analysis is used to verify the crystallo- geometry of the unit cell of sapphire, shows that the graphic indices of the fractured surfaces. The FEM two weakest families of cleavage planes,Ä4 1012 and code ABAQUS is employed to simulate the loading Ä41010 , happen to be the fractured planes. This study process of the specimens. Table 2 is a summary of highlights the role of the cleavage planes and the the numerical as well as the experimental results. differences between their surface energies in the Assuming that the elastic±stiffness constants and the fracturing of single crystals. surface energies of different cleavage planes reported Ž.8 The commonly used energy-release-rate frac- in the literature for sapphireŽ and used here in the ture criterion is modified such that it can be used for numerical simulations. are reasonably accurate, the a cracked single crystal which possesses a finite following conclusions are obtained: number of cleavage planes; see Eq.Ž. 5 . Ž.1 For the notched specimens, fracture occurs in While experiments on only 12 specimens may not the direction where a high tensile stress acts upon a be sufficient to support the above conclusions, these weak cleavage plane. In other words, there is an conclusions appear to be reasonable on physical interplay between the surface energyŽ the lower, the grounds and they should provide guidance for future easier to fracture. and the tensile stress acting normal work in this area. to this surfaceŽ the higher, the greater chance of breaking the interatomic bonds. . Ž.2 The tensile-stress criterion is compared with Acknowledgements the released-energy criterion. It is observed that al- most all of the notched specimens are fractured This work has been supported in part by ARO where the tensile stress on a weak cleavage plane is under Grant No. DAAL03-92-K-0002 and in part by maximum. In general, the energy criterion failed to the Institute for Mechanics and MaterialsŽ. IMM at predict the observed fracture direction. the University of California, San Diego. Thanks are

Ž.3 The maximum-K III Žzero-K .or the maximum- due to Dr. M. Beizai who helped to develop the G fracture criterion does not seem to apply to graphs of the paper. The computations reported here single-crystal brittle solids where there is a finite were carried out on a CRAY-C90 at the San Diego number of cleavage planes. Supercomputer Center. 262 A. Azhdari, S. Nemat-NasserrMechanics of Materials 28() 1998 247±262

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