Mixed-mode fracture in an R-curve material

Engineering Fracture Mechanics 79 (2012) 380–388

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Mixed-mode fracture in an R-curve material Karthik Gopalakrishnan ⇑, John J. Mecholsky Jr. Materials Science and Engineering Department, University of Florida, Gainesville, FL 32611-6400, USA

a r t i c l e

i n f o

Article history: Received 8 July 2011 Received in revised form 15 November 2011 Accepted 23 November 2011

Keywords: Ceramics Mixed-mode fracture R-curves Surface flaw Crack growth

a b s t r a c t Mixed-mode fracture of a mica glass ceramic, an R-curve material, was analyzed. Vickers indentation cracks of varying sizes were introduced on the tensile surface of glass ceramic bars fractured in three point flexure. Mixed-mode fracture from indentation cracks was governed by a rising crack growth resistance behavior. Minimum Strain Energy Density theory based on the singular stress terms was effective in describing mixed-mode fracture in the glass ceramic and in previously studied non R-curve ceramics in flexure. This agreement implies that the effect of non singular stress terms in mixed-mode fracture from relatively small surface cracks in flexure is negligible. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Engineering applications of ceramics often involve mixed-mode loading conditions, where both tensile and shear stresses act simultaneously. Thus, the study of fracture of ceramics under the complex mixed-mode (I/II) loading conditions is of considerable practical interest. Mixed-mode (I/II) fracture has been studied on various ceramic specimens using different test geometries. Some of the test geometries used is flexure bars [1–4], angled internal cracked plates [5] and circular discs in diametral compression [6–8]. Previous studies on mixed-mode fracture were focused on materials which have a constant value for fracture toughness and have no crack growth resistance behavior. To the authors’ knowledge there is no literature on mixed-mode fracture studies which vary the crack sizes on materials which show rising crack growth resistance (R-curve) behavior with crack extension. This research aims to study the behavior of R-curve materials in mixed-mode (I/II) loading by varying the crack sizes. A mica glass ceramic (MGC) was chosen as the representative R-curve material. The MGC used has wide applications because of its high machinability, high electrical resistivity, zero porosity and capability to withstand high temperatures [9–11]. The areas of application include ultra high vacuum environments such as electric insulators, the aerospace industry, nuclear related environments, welding nozzles and industrial high heat electrical cutting operations such as electrode supports. Materials with R-curve behavior i.e., increased crack growth resistance with increased crack size, are being used in many engineering applications due to their improved damage tolerance and improved thermal shock resistance [12]. Previous studies have shown that an R-curve effect is observed in brittle materials which have elongated grains with high aspect ratio [13,14]. Silicon nitride used in hybrid ball bearings [13], toughened zirconia used as dental ceramics [15] and glass ceramics like Pyroceram [16] and barium silicate [13] are good examples of R-curve materials. Mechanisms such as crack bridging [17,18], crack deflection [19], microcracking [20,21] and transformation toughening [22] (e.g., in zirconia) shield the propagating crack, thus toughening the material.

⇑ Corresponding author. Address: 277 Corry Village, Apt 3, Gainesville, FL 32603, USA. Tel.: +1 352 870 7891. E-mail address: gaka1umt@ufl.edu (K. Gopalakrishnan). 0013-7944/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2011.11.019

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Nomenclature a depth of critical crack a11, a12, a22 functions of crack turning angle (used in the equation for effective fracture toughness, KC evaluated from Non Coplanar Strain Energy Release Rate theory) b semi-elliptical half-crack length b11, b12, b22 functions of crack turning angle and Poisson’s ratio (used in the equation for effective fracture toughness, KC evaluated from Minimum Strain Energy Density theory) c critical crack size d specimen thickness KI mode I stress intensity factor KII mode II stress intensity factor KIC critical mode I stress intensity factor KC effective mixed-mode fracture toughness L support span free surface correction factor for surface cracks in expression for KI M1 M2 free surface correction factor for surface cracks in expression for KII P failure load r distance from the crack tip Scrit critical strain energy density w specimen width Y1 geometric constant for KI equation Y2 geometric constant for KII equation c crack turning angle, i.e. angle between the initial crack plane and the plane of propagation. h crack orientation m Poisson’s ratio p 3.14 r failure stress rc critical failure stress for pure mode I loading CSERR Coplanar Strain Energy Release Rate MGC Mica Glass Ceramic MSED Minimum Strain Energy Density MTS Maximum Tangential Stress NCSERR Non Coplanar Strain Energy Release Rate SEM Scanning Electron Microscopy SERR Strain Energy Release Rate SIF Stress Intensity Factor

The major toughening mechanism in the MGC which is responsible for the R-curve effect is crack bridging [16,23]; which exerts a closure stress on the crack, thus inhibiting crack propagation and catastrophic failure. There is a critical size beyond which crack bridging, which is mainly responsible for the R-curve behavior, does not increase the resistance to crack propagation. This critical size is defined as the bridging zone size. The bridging zone sizes are usually in the micron range. For example, the bridging zone size for silicon nitride (TSN-03H) [13] is 335 lm and for canasite glass ceramic [24] is 382 lm. Thus, it is important to use small precracks for evaluating R-curve behavior. There was an attempt by Singh and Shetty [6] to determine the effect of mixed-mode loading in alumina and CeO2-TZP ceramic disk specimens which show R-curve behavior. However, because of the large chevron notch type cracks introduced in the specimens, no crack growth resistance effect could be observed. The objective of the present study was to determine the effects of mode I and combined mode (I/II) loading on a commercially available MGC1 using controlled surface cracks in flexure. The glass ceramic studied contains interlocking mica flakes of high aspect ratio randomly oriented in a borosilicate glass matrix [12,13,25]. 2. Experimental procedures The MGC used has a composition of 46% SiO2, 17% MgO, 16% Al2O3, 10% K2O, 7% B2O3 and 4% F [11]. The elongated mica crystals are interlocked and dispersed in the glass matrix, which accounts for the easy machinability of MGC bars [9,10]. MGC bars of dimensions 40 mm  4 mm  3 mm were used for the study. Vickers indentations were introduced at the center of the tensile surface of the specimens using 0.05, 0.1, 0.2, 0.3, 0.5, 1 and 2 kg loads in order to create varying crack sizes. Multiple indentations were also introduced to create larger cracks. The as-indented bars were tested in three point flexure by 1

Macor, Corning Glass Works, Corning, NY.

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Fig. 1. Schematic of three point flexure specimen with the inclined crack at the center of the tensile surface. h represents the crack orientation and c represents the crack turning angle.

aligning the cracks at various angles, h (90, 80, 70, 55, 45) with respect to the tensile axis, as shown in Fig. 1. h = 90represents the crack loaded in pure mode I. Mode mixity is introduced at the crack tip for cracks oriented at h < 90. The flexure tests were conducted at room temperature on a span of 25 mm at a loading rate of 150 N/min. In order to determine the influence of residual stress due to indentation, a few MGC bars with different sizes of precracks were annealed at 750C for 2 h [26,27] and tested in pure mode I, i.e. h = 90. The fracture surfaces were examined using an optical microscope to determine the failure origins. The flexural strength, i.e., the stress at failure, r was calculated from the following equation only for those samples that failed from the indentations.

r ¼ 3PL=2wd2

ð1Þ

where P is the failure load, L is the support span, w is the specimen width and d is the specimen thickness. Samples that failed from edges or from other inherent flaws in the material were rejected. The semi minor axis or critical crack depth, a and major axis or the semi-elliptical crack length, 2b were measured for the samples that failed from the indentations. The indented cracks can be approximated as semi-circular surface cracks, and the equivalent semi-circular crack length or the critical crack size, c is given by [28]

c¼

pffiffiffiffiffiffi ab

ð2Þ

Fig. 2 shows representative micrographs of two fracture surfaces of MGC showing semi-elliptical surface cracks demonstrating the sizes expected in service. Semi-elliptical surface cracks are representative of the type of cracks that will be observed in most applications. The strengths of the annealed samples were compared to those of the as indented samples with similar crack sizes. 3. Expressions for stress intensity factors For similar crack sizes, the strengths for annealed and indented samples were seen to be comparable (Table 1). A major toughening mechanism seen in glass ceramics is microcracking near the crack tip; this microcracking relieves the stress near the crack tip and shields the crack [29]. The local residual stress from indentation gets relieved due to the microcracking, and hence, as-indented samples have similar strengths to the annealed samples. A similar phenomenon was seen to occur in Li2O–2SiO2 glass ceramics [20] and canasite glass ceramics [24]. The analytical expressions for the stress intensity factors (SIF) for surface cracks under mixed-mode loading conditions are given by [3,7]:

pffiffiffi 2 K I ¼ Y 1 M 1 r c sin h

ð3Þ

pffiffiffi K II ¼ Y 2 M 2 r c sin h cos h

ð4Þ

Fig. 2. SEM micrographs of two different fracture surfaces in the MGC showing the semielliptical cracks (The arrows border the critical crack).

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K. Gopalakrishnan, J.J. Mecholsky Jr. / Engineering Fracture Mechanics 79 (2012) 380–388 Table 1 Strengths for as-indented and annealed MGC flexure specimens for different crack sizes. Crack size (lm)

Strength (MPa)

35 54 97 105 122 155 170 210

As-indented

Annealed

115 106 95 94 87 80 74 65

117 105 98 93 87 84 73 65

where r is the applied stress given by Eq. (1), c is the critical flaw size given by Eq. (2), Y1 and Y2 are the geometric constant and M1 and M2 are the free surface correction factors. Similar equations apply for the SIFs for both as-indented and annealed samples in the case of glass ceramics as opposed to non R-curve materials, wherein the effect of residual stress needs to be accounted for in the as-indented samples. For a semi-circular surface crack, the geometric constant, Y1 is given by [30]:

Y 1 ¼ 2=

pffiffiffiffi

p

ð5Þ

Kassir and Sih developed the expression for KII for penny shaped cracks with the geometric constant Y2 given by [31]:

Y 2 ¼ 4=

pffiffiffiffi pð2  mÞ

ð6Þ

The value of Y2 comes out to be 1.32 for the MGC which has a Poissons ratio, m = 0.29 [11]. The critical mode I SIF, KIC can be determined from Eq. (11) by substituting h = 90.

pffiffiffi K IC ¼ Y 1 M 1 rc c

ð7Þ

where rc is the critical stress for pure mode I loading. Studies conducted by Randall [32] and other researchers [7,33] agree that the correction factor is within 10–12% for surface cracks. Based on this fact, the value of M1 is taken to be 1.1 for the current study. Smith et al. [34] have shown that the values of M2 are approximately equivalent to M1 for equivalent crack sizes and shapes. The ratio M2/M1 is thus taken to be unity. Thus, once the applied stress and the flaw dimensions are measured, the SIFs can be calculated from Eqs. (3) and (4). 4. Three point flexure results h = 90corresponds to the case of pure mode I loading. The fracture toughness calculated for the various critical crack sizes is designated as KIC. Fracture toughness, KIC for the MGC is shown to be a function of the crack size (Fig. 3). KIC values increase from 0.7 MPa-m1/2 at c = 15 lm to 1.65 MPa-m1/2 at c = 150 lm, showing the increase in resistance to crack growth as crack length increases. The value of KIC remains constant at 1.65 MPa-m1/2 for c > 150 lm. This value is statistically the same as the toughness value of 1.58 ± 0.09 MPa-m1/2 obtained for chevron notched flexure specimens of the MGC reported by Reddy et al. [35](p > 0.05).

1.8 1.6

KIC (MPa-m1/2)

1.4 1.2 1 0.8 0.6 0.4 0.2 0

0

50

100

150

200

250

Crack size,c (microns) Fig. 3. Fracture toughness of MGC in pure mode I as a function of crack size (The curve is the best polynomial fit).

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Crack bridging is mainly responsible for the R-curve behavior in this material [16,23]. With increasing crack sizes, the bridging force exerts a closure stress leading to an increase in crack growth resistance which results in an increased KIC. The MGC used has a maximum bridging zone size of about 150 lm. The fracture of the specimen at h = 90occurred along the initial plane of the flaw, since the loading corresponds to pure Mode I. However for other crack orientations, the crack turned from its initial plane and propagated in a direction normal to the maximum principal tensile stress. The angle by which the crack deviates from its initial flaw plane is defined as the crack turning angle. The crack turning angles, c, for a particular crack orientation, h, were found to be nearly a constant for all the crack sizes tested. However, c increases with decreasing h (Fig. 4) and varies according to the following relation

c ffi 90  h

ð8Þ

KI and KII (calculated from Eqs. (3), (4)) for the various crack orientations show increasing stress intensity with crack extension till c ffi 150 lm (Fig. 5). This trend of the SIFs indicates that there is resistance to crack growth with increasing crack length in mixed-mode loading conditions. For a particular crack size, mode I SIF, KI decreases with decreasing h and mode II SIF, KII increases with decreasing h. It should be noted that decreasing h refers to increasing mode mixity. 5. Mixed-mode fracture theories Four widely used mixed-mode fracture theories were examined, namely Non Coplanar Strain Energy Release Rate criterion [36] (NCSERR), Maximum Tangential Stress criterion [37] (MTS), Coplanar Strain Energy Release Rate criterion [1,2,38] (CSERR) and Minimum Strain Energy Density criterion [39] (MSED); using the SIFs computed from analytical expressions

Fig. 4. Crack turning angle for all crack sizes as function of the crack orientation. The error bars represent the standard deviation. h = 90corresponds to pure mode I loading where c = 0.

Fig. 5. Mode I and Mode II SIFs for mixed mode fracture from different crack sizes in MGC. Pure mode I corresponds to h = 90. (The curves are the best fits to aid the eye).

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(Eqs. (3), (4)). The fracture criteria are based on the assumption that crack growth starts at a critical value of a parameter related to the strain energy release rate (SERR) or state of stress near the crack tip and propagates along a plane at which the chosen parameter is at its maximum or minimum. The NCSERR criterion assumes that crack growth starts at a critical value of the SERR in a direction along which the SERR is maximum. An analysis based on the maximum SERR was first performed by Hussain et al [36]. They analyzed the non coplanar crack extension of inclined cracks based on Griffith energy balances. The condition for fracture under the mixedmode loading in this case is given by:

K C ¼ K I ½b11  b12 ðK II =K I Þ þ b22 ðK II =K I Þ2 1=2 ½ð1  c=pÞ=ð1 þ c=pÞc=2p

ð9Þ

b11 ¼ 4ð1 þ 3 cos2 cÞ=ð3 þ cos2 cÞ2

ð10Þ

b12 ¼ 32 sin c cos c=ð3 þ cos2 cÞ2

ð11Þ

b22 ¼ 4ð9  5 cos2 cÞ=ð3 þ cos2 cÞ2

ð12Þ

The combination of KI and KII results in effective mixed-mode fracture toughness and is given by KC as shown in Eq. (9). Erdogan and Sih [37] hypothesized that fracture commences at the tip of the crack in a radial direction and crack growth occurs in the plane which is normal to the direction of maximum tensile stress. The mixed-mode fracture toughness, KC from the conventional MTS criterion is represented by Eq. (13).

K C ¼ K I cosðc=2Þ½cos2 ðc=2Þ  1:5ðK II =K I Þ sin c

ð13Þ

The CSERR criterion states that the crack growth starts along the initial crack plane when the SERR reaches a critical value, leading to the relation below [1,2,38].

K C ¼ K I ½1 þ ðK II =K I Þ2 1=2

ð14Þ

Sih [39] proposed that crack growth starts at a critical value of Strain Energy Density, Scrit, and propagates in the direction of MSED. The Scrit value was postulated as a material constant and hence it could be used as a measure of the fracture toughness of the material under mixed-mode conditions. Based on this concept an effective fracture toughness KC can be derived as

KC ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8ða11 K 2I þ 2a12 K I K II þ a22 K 2II Þ

ð15Þ

where for Poissons ratio of 0.29 [11],

a11 ¼ ð1 þ cos cÞð1:84  cos cÞ=16p a12 ¼

2 sin cðcos c  0:42Þ 16p

a22 ¼ ½2:84ð1  cos cÞ þ ð1 þ cos cÞð3 cos c  1Þ=16p

ð16Þ ð17Þ ð18Þ

6. Evaluation of the mixed-mode fracture theories The mixed-mode fracture theories for non R-curve materials have been evaluated based on the fact that mixed-mode fracture toughness, KC is a material property not related to the conditions of loading and should be equal to mode I fracture toughness, KIC, irrespective of angle h [38]. For the MGC, which is an R-curve material, this criterion should be modified to accommodate the fact that fracture toughness is a function of crack size. The criterion for evaluating the mixed-mode fracture theories in R-curve materials is that the KC–c plots for various h values (evaluated from the theories) should overlap with the KC–c plot at h = 90(pure mode I loading case) or rather, the KC values (calculated from the theories) for a particular crack size should remain constant for all h. The effective mixed-mode fracture toughness, KC was evaluated using the various theories and graphed as a function of crack size for the various crack orientation (Fig. 6). The values of the effective fracture toughness, KC calculated using the NCSERR theory (Eqs. (9)–(12)) corresponding to each crack size, at different levels of mode mixity, is statistically different (p < 0.05). The curves for the effective fracture toughness (from the NCSERR theory) as a function of crack size are thus, different for the various crack orientations which contradicts the fact that the effective fracture toughness value should be a constant for a particular crack size (Fig. 6A). The MTS theory (Eq. (13)) gives negative values for the effective KC at h = 45 for the various crack sizes, and hence cannot be used to describe the mixed-mode fracture in this MGC (Fig. 6B). The predictions of KC for the various crack sizes from the CSERR (Eq. (14)) and MSED (Eqs. (15)–(18)) theories are close. The average values of KC for all crack sizes greater than 150 lm along with the corresponding standard deviations for the two theories are: CSERR (Fig. 6C): 1.53 ± 0.11 MPa-m1/2 and MSED (Fig. 6D): 1.59 ± 0.04 MPa-m1/2 .There is no statistical difference

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Fig. 6. Effective fracture toughness of MGC, estimated from the mixed mode fracture theories for all crack sizes at different mode mixities. h = 90 corresponds to pure mode I loading. Abbreviations: (A) Non Coplanar Strain Energy Release Rate criterion (NCSERR) – Eqs. (9)–(12), (B) Maximum Tangential Stress criterion (MTS) –Eq. (13), (C) Coplanar Strain Energy Release Rate criterion (CSERR) – Eq. (14); and (D) Minimum Strain Energy Density criterion (MSED) – Eqs. (15)–(18).

between these average KC values (p > 0.05). However, CSERR theory cannot be accepted since this theory assumes the crack to propagate in the same plane as that of initial crack, which is contrary to that observed. The mean fracture toughness from the MSED theory is statistically the same as the fracture toughness value for the MGC from previously published results (1.58 ± 0.09 MPa-m1/2 [35]) and the standard deviation is less than the CSERR theory. The MSED criterion, thus, is the most effective mixed-mode fracture criteria in determining the effective fracture toughness in mixed-mode for MGC bars tested using relatively small cracks and fractured in flexure. This result also implies that a crack in MGC subjected to mixed-mode loading conditions starts to grow at a critical value of the Strain Energy Density at the crack tip.

7. Discussion Previous mixed-mode fracture studies on non R-curve ceramic flexure specimens (soda lime glass [1], hot pressed silicon nitride [3]) using surface cracks have shown that the crack turning angles follow the trend similar to Eq. (8). A similar relationship between the crack turning angle and the crack orientation (Eq. (8)) is also seen for ceramic disk specimens with indent cracks in diametral compression (Silicon nitride NBD 300 [40], Pyroceram 9606 [7] and hot pressed alumina [7]). However, the crack turning angles from the predictions of MSED theory or any other mixed-mode fracture theory deviate greatly from the measured angles. The crack turning angles for ceramic disks with larger chevron notch cracks or through-cracks in diametral compression, are greater than, and follow an entirely different trend from, that seen with the crack turning angles for indentation cracks [6,8]. The reason as to why the crack turning angle is a function of the crack type and does not agree with the theories has not been discussed in any detail until recently. The possibility of non singular stress terms in the stress intensity solutions offers a reasonable explanation. This matter will be discussed more fully in a following paragraph. A review of literature on the comparison of mixed-mode fracture theories in non R-curve materials tested in flexure using indentation precracks show that MSED theory provides the best description for mixed-mode failure in ceramic flexure specimens. The mixed-mode fracture theories are again compared based on the criterion that the fracture toughness should

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remain a constant at all crack orientations. A study by Freiman et al.2 [1] on mixed-mode fracture in soda lime glass in four point flexure using Knoop indentation flaws showed that the MSED criterion provided the best fit for the KC values. Petrovic [3] performed similar test on hot pressed silicon nitride and again came to the same conclusion. The normalized SIFs also agreed well with the predictions from the MSED theory. Marshall [2] compared the relative failure strengths in mixed-mode fracture estimated from the fracture theories to the experimental values for as indented hot pressed silicon nitride fractured in flexure, and found that the experimental values agree very well to the theoretical predictions from the MSED theory. Glaesemann et al. [4] also came to a similar conclusion when the experimental relative failure strengths for hot pressed silicon nitride (from Ref. [3]) and for soda lime glass tested in liquid nitrogen were compared with those from the MSED theory. Shetty et al. [7] has shown that ceramic disks with indentation precracks tested in diametral compression, give greater values for KI and KII than those obtained from the four point flexure tests on the same materials. It is seen that centrally cracked square plates give much lesser values for mixed-mode SIFs in comparison to the centrally cracked disks [41]. The comparison of the SIFs from the different test geometries highlights the influence of test geometry on mixed-mode fracture studies. A relatively new approach to fracture criteria, including mixed mode fracture and R-curve materials, is Quantized Fracture Mechanics (QFM) [42]. It accounts for fracture processes at very small length scales. However, QFM coincides with LEFM for larger crack sizes. Since our crack sizes are large compared to atomic dimensions, we cannot distinguish between QFM and LEFM in our studies. Thus, the tenets of LEFM will hold for our discussions. The mixed-mode SIF values for ceramic disks tested with larger chevron notch precracks were found to be greater, in comparison to the mixed-mode SIF estimates from the mixed-mode fracture theories [8]. The influence of the non singular stress terms was suggested to be the possible reason for the overestimation of mixed-mode SIFs in the disks with chevron notch cracks [8]. Recently, there have been discussions on the incorporation of non singular stress (or T-stress) terms at the crack tip, and thus modifying the existing two parameter (KI and KII) fracture criteria into three parameters (KI, KII and T) fracture criteria. Smith et al. [43] modified the MTS criterion to include the effects of both the singular stress terms and the T-stress terms on the tangential stress around the crack tip. Recently, Ayatollahi et al. [44] showed that the use of this modified MTS criterion can explain the greater values of fracture toughness obtained in mixed-mode loading of centrally cracked ceramic disk specimens of soda lime glass, sialon, mullite, silicon carbide and porcelain in diametral compression. The value of the T-stresses estimated for the cracked disk specimens were found to be negative, the magnitude of which is significant [41]. Ayatollahi et al. [41] also showed that the magnitude of the T-stress values for angled center cracked plates of soda lime glass are not as great as in the case of centrally cracked disks of soda lime glass. The magnitude and sign of Tstress is thus, dependent on the test geometry. To the authors’ knowledge, there is no analytical treatment of semi-elliptical cracks in mixed-mode loading that includes the T-stress. The fact that Sih’s two parameter (KI and KII) MSED theory is able to provide constant values for the fracture toughness of the many materials tested in flexure, irrespective of the loading conditions, implies that the effect of the T-stress terms for flexure specimens with relatively small surface cracks is negligible, at least for the cases considered in this paper. However, for disk specimens with indentation cracks, the two parameter mixed-mode fracture theories largely underestimate the values of the mixed-mode SIFs, possibly because of the greater influence of the T-stress in the disk specimens. T-stress calculation for indentation cracks is challenging because of the residual stress associated with indentation and the three dimensional geometry of the indents. Future work should aim to determine the T-stress for semi-elliptical cracks in various test geometries under different loading conditions.

8. Summary and conclusions Mica glass ceramic bars with varying indentation precrack sizes were tested in three point flexure both in pure mode I and under mixed-mode loading states. The results are summarized as follows:  SIF in pure mode I for the mica glass ceramic was found to be a function of crack size, which indicated crack growth resistance behavior with increasing crack extension.  The mixed-mode SIFs were evaluated for the mica glass ceramic and R-curve behavior was exhibited in mixed-mode loading, similar to that seen in pure mode I.  Four popularly used mixed-mode fracture theories based on the singular stress terms were investigated namely, Non Coplanar Strain Energy Release Rate theory, Maximum Tangential Stress theory, Coplanar Strain Energy Release Rate theory and Minimum Strain Energy Density theory. Amongst these, Sih’s Minimum Strain Energy Density theory provided a very convenient and effective way to describe mixed-mode fracture from surface cracks in the mica glass ceramic as well as previously studied non R-curve ceramics in flexure.  Irrespective of the test geometry, crack turning angles for relatively small indentation cracks subjected to mixed-mode loading conditions follow: c ffi 90  h.

2

The angles are mislabeled in the manuscript.

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