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Application of linear elastic fracture mechanics to the quantitative evaluation of fluid-inclusion decrepitation Alfred Lacazette Department of Geosciences, Pennsylvania State University University Park, Pennsylvania 16802

ABSTRACT Linear elastic fracture mechanics quantitatively relates the driving stress of fracture propagation to the material properties of the fractured solid and may be used to model the decrepitation (fracturing) of fluid inclusions under the influence of internal pressure. Existing experimental data on decrepitation can be explained by the methods of fracture mechanics. The decrepitation pressure vs. size relation may therefore be predicted for fluid inclusions in minerals that have not been the subject of quantitative decrepitation studies. It may be possible to determine the fracture toughness of minerals by decrepitation studies utilizing synthetic fluid inclusions.

INTRODUCTION Fluid inclusions are samples of ancient geologic fluids that are trapped inside single crystals of quartz, calcite, or other minerals. During fluid-inclusion studies, a mineral sample is warmed or cooled, and phase changes within fluid inclusions are observed. The composition and density of the inclusion fluid at the time of entrapment can be deduced from knowledge of the temperatures at which these phase transitions occur, provided that the inclusion has not been damaged or otherwise changed since its formation. This information can be used to yield estimates of the fluid temperature and pressure at the time either of mineral growth or postgrowth microcracking of the host mineral when subsequent crack healing traps fluid inclusions. Fluid-inclusion data are commonly used to delimit or determine metamorphic or diagenetic conditions and the circumstances of petroleum and ore formation. Fluid inclusions are routinely heated during laboratory study. Heating of inclusions beyond their formation temperature can occur naturally within the earth. Thermally induced pressure changes within the fluid inclusion can lead to fracturing (decrepitation) of inclusions and a concomitant decrease in the bulk density of the fluid inclusion. Unrecognized decrepitation can thereby lead to the misinterpretation of fluidinclusion data. It is also possible that decrepitation effects could be exploited to yield information about the postentrapment thermal history of a sample if the effects of decrepitation can be recognized and quantitatively evaluated. Understanding the controls of decrepitation is thus a key problem in fluid-inclusion research. The studies of Leroy (1979), Swanenburg (1980), Gratier and Jenatton (1984), Prezbindowski and Larese (1987), and Bodnar et al. (1989) provide extensive quantitative and semiquantitative data on decrepitation, yet the literature at 782

the physical processes of cracking. The fundamental concept is the formulation of the fracture process in terms of a thermodynamic energy balance in which the work of cracking is supplied by the release of mechanical energy. This concept was first formulated by Griffith (1924) and is known as the Griffith criteria. Pure tensile fracture is referred to as mode I fracturing or simply as "cracking," in contrast to shear fracturing (mode II) and tearing (mode III). The discussion below considers only pure mode I failure because decrepitation is a pure mode I phenomena.

present lacks a general theory of decrepitation. While this paper was in review, I was made aware of an abstract (Wanamaker et al., 1987) and a work in press (Wanamaker et al., 1990) that apply fracture mechanics to the problem of fluid-inclusion decrepitation. Linear elastic fracture mechanics has been used successfully to analyze a wide variety of engineering and geologic problems over the past 30 years. Fracture mechanics provides a way of quantitatively relating the driving stress of a fracture to its size and shape and to the material properties of the fractured solid. In this paper, I show that linear elastic fracture mechanics provides a general way to quantify the decrepitation behavior of fluid inclusions and to demonstrate this application with some example calculations. I will show that the existing experimental data on decrepitation can be explained by the principles of fracture mechanics and that these same methods may therefore be used to predict the decrepitation behavior of minerals for which no experimental data exist. A short, elementary introduction to fracture mechanics is provided here. Roedder (1984) provided a comprehensive introduction to fluid-inclusion studies.

where a is the driving stress (compression considered positive); b is the crack size (see Fig. 1); 7 is a geometric and load factor that is characteristic of the crack shape, location, and loading condition (Tada et al., 1973; Murakami, 1987). K^ (i.e., K- one) is the mode I stress intensity factor and is material independent. When a crack is in unstable equilibrium, so that it is neither propagating nor closing, but is on the verge of propagation, then K| = Kic, where K| C is the critical stress intensity factor or fracture toughness, which is a measurable material property. A given crack will extend when K[ exceeds K l c and will close (but not necessarily heal) when K[ drops below K.^. Measurements of K| C for many minerals and for a range of representative rock types have been tabulated by Atkinson (1984) and Atkinson and Meredith (1987b).

FUNDAMENTALS OF LINEAR ELASTIC FRACTURE MECHANICS Many textbooks and review articles provide an introduction to fracture mechanics (Lawn and Wilshaw, 1975; Broek, 1987; Atkinson, 1987). Atkinson (1984) and Atkinson and Meredith (1987a) discussed the velocity of fracture propagation and the role of chemical environment. Stresses and displacements around ellipsoidal fractures were considered in detail by Pollard and Segall (1987). Linear elastic fracture mechanics quantitatively relates the driving stress of a crack to the material properties of the cracked solid, to the shape and loading condition of the crack, and to

Values of Y and formulas for K[ for various crack configurations are available in handbooks such as Tada et al. (1973) and Murakami (1987). The crack configurations relevant to decrepitation studies are internal cracks, which are entirely contained within the material of interest and thus do not intersect a surface. Solutions for internal cracks assume that the crack is embedded in an infinite medium, usually with applied stresses at infinity. Although this is an idealization, the effect of a surface is small if the crack is at least a few radii from a surface. Classic linear elastic solutions for cracks assume that the medium is homogeneous, isotropic, perfectly brittle (i.e., no plastic deformation), and linearly elastic until fracture, and that the crack is a sin-

The fundamental relation for a mode I fracture derives directly from Griffith theory and is given by Ki = -Yab\

(1)

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gle, atomically sharp crack. All of these idealizations have been shown to be excellent approximations for cracks in single crystals and for microscopic cracks in polycrystalline materials at scales above that of the crystal lattice (Lawn, 1983; Atkinson, 1987). DECREPITATION OF FLUID INCLUSIONS Decrepitation of fluid inclusions occurs when an inclusion develops internal overpressure either naturally or within the earth, so that the internal pressure of the inclusion increases until the wall cracks. Bodnar et al. (1989) presented extensive new and previously published quantitative data on decrepitation of synthetic fluid inclusions containing pure water in quartz (Fig. 2). Decrepitation of inclusions is clearly volume dependent, as was concluded by Bodnar et al. (1989). This behavior can be predicted in principal by evaluation of equation 1 for quartz by using representative inclusion shapes. Equation 1 may be evaluated for the data presented by Bodnar et al. (1989) as follows. In the case of laboratory overheating in the fluidinclusion stage, the confining pressure is 1 atm (0.1 MPa), so-that the internal pressure of a fluid inclusion is approximately equal to - a . In the case of natural overheating within the earth or of overheating in a hydrothermal bomb during an experiment, - a is given by the effective stress (CTe), which is defined as oe = O c - P ,

(2)

where oq is the confining pressure and P is the fluid pressure in the inclusion. Deviatoric stresses are an additional concern for confined grains during natural overheating within the

P e n n y - s h a p e d (circular d i s k ) Y=

1.13

- length = 2b

i

Infinite flat tunnel crack Y = 1.77

Sphere y = 2.98

Infinite cylindrical hole Y = 3.98

G E O L O G Y , August 1990

earth, but these will not be considered here. Deviatoric stresses will not affect crystals that have grown outward into open vugs or veins. The internal pressure required to decrepitate a fluid inclusion in a given mineral may be determined by combining equations 1 and 2 and solving for the pressure required to make K ( equal to K I C for the mineral. Decrepitation results from the inclusion itself acting as an internally pressurized crack. A Y factor of 1.13 is appropriate for evaluation of a penny-shaped crack, whereas Y = 1.17 for an infinitely long tunnel crack (Lawn and Wilshaw, 1975; Broek, 1987). Tada et al. (1973) and Murakami (1987) provided a value of 3.98 that is appropriate for an internally pressurized, infinitely long circular cylindrical hole. Baratta and Parker (1983) and Murakami (1987) provided general procedures for estimating Y for spherical voids. For an internally pressurized spherical void, Y = 2.98. Although using any of these Y factors is clearly an oversimplification for the sometimes complex inclusions used in the study of Bodnar et al. (1989), these factors provide approximate geometric analogs to the shapes of many fluid inclusions and should bracket the values appropriate for real inclusions. The choice of b presents some difficulties, given the range of actual inclusion shapes. The effective flaw size of an inclusion may be approximated by a sphere of radius b with an equal included volume that will most accurately represent an inclusion that approximates a spherical shape. The shape factor for an infinitely long cylindrical hole should accurately represent an elongate tubular or negativecrystal-shaped inclusion (i.e., a crystal-shaped void); however, the correct b value for such a

Figure 1. Crack geometries used to model fluid inclusions. Sphere and cylinder are assumed to contain infinitesimally short and sharp annular ring and longitudinal cracks, respectively, in their walls.

Figure 2. Data of fluidinclusion decrepitation pressure in quartz as function of inclusion volume, after Leroy (1979), Swanenburg (1980), and Bodnar et al. (1989). Data of Swanenburg (1980), after Bodnar et al. (1989). Curves represent decrepitation pressure as calculated from equation 1 for various rvalues and K l c = 0.85 MPam % to is taken as radius of sphere of equivalent volume to that shown. See text for further discussion.

geometry will vary both with its volume and aspect ratio, because b is the radius of the cylinder. Leroy (1979) found that the decrepitation pressure of negative-crystal-shaped inclusions with a volume greater than 2.2 x 104 (jm 3 was independent of the degree of elongation of the inclusion. This result is expected, given the relative insensitivity of equation 3 in this volume range to both b and shape factor (Fig. 2). Although they did not account for the inclusion volume in their analysis, Bodnar et al. (1989) attempted to estimate the influence of inclusion shape on decrepitation pressure and found that inclusions with well-developed negative crystal shapes generally decrepitate at higher pressures than amoeboid inclusions. They also noted significant deviations from this shape-pressure relation, which may be a result of the complexities of stress analysis for irregular cracks (Westmann, 1966). In spite of these difficulties, most applicable shape factors have values between 1.0 and 4.0, so that the values selected above should adequately bracket the correct shape factors. Atkinson and Meredith (1987b) reviewed the available experimental measurements of K I C for quartz. Measured values lie between 0.31 and 2.4 MPa-m'A depending on the temperature, crystallographic fracture orientation, and testing method. Atkinson and Meredith (1987b) showed that Kjc is relatively low in the (lOlO) orientation. Decrepitation fractures should tend to initiate and propagate in the crystallographic orientation along which Kjc has the lowest value. The observation of Gratier and Jenatton (1984) that ellipsoidal inclusions with long axes parallel to the c axis tend to form decrepitation fractures parallel to the (llOO), (lOlO), and (0110) planes may be explained by this mechanism in conjunction with the expected stress distribution around a pressurized ellipsoid (Pollard and Segall, 1987). However, in a more general case, other factors, such as the presence of dislocations or larger geometric irregularities in the

log inclusion v o l u m e (|im ) 783

inclusion wall and/or a complex stress state in the host mineral due either to the inclusion geometry or to the presence of other nearby pressurized inclusions, could cause a fracture to propagate in a less favorable orientation. It is therefore unclear how crystallographic orientation should be accounted for in choosing a value of K j o Atkinson (1979) found that Kic for synthetic quartz varies from 0.85 to 1.0 MPa-m ^ normal to the z and r faces, respectively, at 20 °C. Ferguson et al. (1987) and Norton and Atkinson (1981) found that natural Brazilian quartz has higher values of K[C than does synthetic quartz. Norton and Atkinson (1981) reported a value of 3.7 MPa-m 'h for Brazilian quartz normal to the c axis, although they considered this to be an overestimate. Meredith and Atkinson (1982) found essentially no variation in K I C at 20 °C between water and vacuum. This result is expected on theoretical and experimental grounds because K 1C is essentially independent of chemical environment (Atkinson and Meredith, 1987a). Meredith and Atkinson (1982) provided reliable measurements of K I C over a range of temperatures and reported that Kic measured along the a plane in a direction normal to the z face falls from 1.04 MPa-m14 at 20 °C to 0.89 MPa-m* at 200 °C to 0.84 MPa-m14 at 300 °C where it levels out. K I C = 0.85 MPa-m14 will therefore be used as an estimate of Kic, because the data of Bodnar et al. (1989) was collected between 250 and 675 °C and because data on variations of K, c as a function of temperature in different crystallographic orientations are lacking. Pressure has little or no effect on Ki C (Atkinson, 1984). The results of evaluating equation 1 after solving for pressure as described above are plotted in Figure 2. While Y= 3.98 provides a good lower pressure bound to the decrepitation pressure data, Y = 1.13 provides a conservative upper bound. The empirical best-fit equation of Bodnar et al. (1989) is matched quite well by Y =

-135

-125

-115

2.98, which indicates that a spherical geometry provides a good overall mechanical approximation of inclusion shape. Equation 1 also predicts the convergence of the experimental data at volumes above 10 4 (xm3, because the shape factor steadily loses significance in this region. The observed leveling of decrepitation pressure with increasing size is also predicted. Scatter of the data may result from variations in the inclusion shapes and crystallographic fracture orientation and from difficulties in the choice of b, as discussed above. Nevertheless, fracture mechanics is clearly applicable to the evaluation of decrepitation data. DISCUSSION A fracture mechanics approach presents two distinct advantages to purely empirical equations such as that of Bodnar et al. (1989). First, the linear elastic fracture mechanics equations can be extrapolated to inclusion volumes well outside of the range of experimental data. Second, linear elastic fracture mechanics allows the estimation of decrepitation pressures of fluid inclusions in any mineral. For example, experimental data relating decrepitation pressure to inclusion volume is not available for calcite. If we assume that l v a l u e s of 1.13 and 3.98 are also appropriate for fluid inclusions in calcite and we choose Kic = 0.19 MPa-mlfl for the ( l O l l ) plane of calcite (Atkinson and Meredith, 1987b), the range of decrepitation pressures of inclusions in calcite can be predicted. The relative weakness of calcite is readily apparent. For example, a 100 /im 3 inclusion in calcite should hold a maximum of about 99 MPa, whereas an equal-sized inclusion in quartz can hold up to 443 MPa. This explains the low overpressures that inclusions in calcite will support when subjected to experimental overheating and also the ease with which such inclusions can be reset by natural thermal events (Prezbindowski and Larese, 1987).

-105

Quartz, CH 4 T h (°C)

-95

-85

-135

-125

The estimates derived above are also in accord with data shown in Figure 3 on methane fluid inclusions that are trapped in calcite and quartz veins at Ganister, Pennsylvania (D. C. Srivastava, T. Engelder, and A. Lacazette, unpublished). These data also provide an illustration of the application of fracture mechanics to a natural fluid-inclusion problem. Both vein sets formed during the same tectonic event, and both minerals contain 6-8-^m-diameter, primary, 98%-99% pure methane fluid inclusions in growth zones in crystals that grew freely outward into veins. The equations of Angus et al. (1978) and equation 1 here were used to determine the homogenization temperatures (Th) that inclusions of this size would have in each mineral if they contained methane at the maximum earth surface temperature (35 °C) and at the bounding pressures obtained from Y= 1.13 and Y= 3.98. These temperatures are plotted in Figure 3. Inclusions at a lower pressure would be less dense and would thus show a higher T^. Both data sets have similar maximum T h values, and the interval -100 to - 9 5 °C is the mode for each set. Temperatures in calcite reach to just above the minimum possible temperature for this mineral, whereas quartz contains inclusions that homogenize no lower than -130.7 °C, well above the minimum temperature. Inclusions in quartz showing T^ below the mode are common, whereas in calcite they are rare. Although it is possible that methane densities were different during calcite growth than during quartz growth, the inclusions in calcite may originally have been similar to those in quartz, but highdensity (low 7h) inclusions in calcite may have decrepitated during uplift or at the surface. In any case, the highest density inclusion in calcite may only be used to provide a lower bound on the maximum density that prevailed during calcite growth because of the likelihood of decrepitation of denser inclusions. On the other hand, the highest density inclusion in quartz should

-115

-105

-95

-85

Calcite, CH 4 T h (°C)

Figure 3. Histograms showing 7"h (liquid + vapor) to liquid for methane fluid inclusions in vein quartz and calcite from Ganister, Pennsylvania. Arrows show temperatures that mark upper and lower limits of decrepitation. Maximum density (low-temperature) limit for quartz is at about -158 °C on solidus, so that nucleation of vapor phase would not be observed. See text for discussion. 784

G E O L O G Y , August 1990

reliably record the peak methane density that prevailed during quartz growth because the lowpressure limit of decrepitation is close to the minimum homogenization temperature, and in a data set of this size at least a few dense inclusions should have survived intact. The minimum densities are suspect for both minerals because of the likelihood that some of the inclusions in each mineral decrepitated. Therefore, the lowest density inclusion provides only a minimum bound for the minimum methane density that prevailed during deposition of the vein minerals. Future experimental studies of fluid-inclusion decrepitation should be designed to incorporate the principles of fracture mechanics so that Y, b, and crystallographic effects can be correctly evaluated. This could be accomplished by determining the inclusion shape and dimensions as well as the volume prior to decrepitation and by measuring the crystallographic orientation of decrepitation fractures on a universal stage. In addition to providing greater understanding of inclusion decrepitation, this technique may allow Kjc to be estimated for the experimental mineral by using equation 1. This method may provide a convenient way to determine this parameter over a wide range of temperatures, since the decrepitation temperature can be controlled by varying the density of the synthetic fluid inclusions used in the study (Bodnar et al., 1989). In the above discussion it is assumed that no crack growth can occur at stress intensities below Kjc- When crack growth does occur under such conditions it is referred to as subcritical crack growth (see Atkinson and Meredith, 1987a, for a review). These mechanisms include stress corrosion, dissolution, diffusion, ion exchange, and microplasticity, all of which may be affected by the chemical environment. Subcritical cracking could be an important and overlooked factor in some experimental studies and especially in natural-fluid-inclusion studies, although it is unlikely to have affected most of the data of Figure 2 because the experimental overpressures of Bodnar et al. (1989) persisted for only 5—10 min. Apparent stretching of fluid inclusions (Bodnar and Bethke, 1984; Prezbindowski and Larese, 1987; Ulrich and Bodnar, 1988) and fluid-inclusion reequilibration and density decreases resulting from solution-deposition processes during overheating experiments (Gratier and Jenatton, 1984; Bodnar et al., 1989; Sterner and Bodnar, 1989) may represent subcritical crack growth. In other experiments, Bodnar et al. (1989) reported some observations that may indicate subcritical cracking on a time scale of 40 to 310 days, although in general their results argue against subcritical cracking over time periods of up to 2 yr. The time-dependent nature of slow crack growth could result in decrepitation of inclusions that have been subjected to relatively small amounts of natural overheating on geological time scales. GEOLOGY, August 1990

CONCLUSIONS Fracture mechanics provides a useful theoretical framework for the quantification of fluidinclusion decrepitation behavior. The possibility of natural decrepitation should be considered during studies of inclusions that maintained high internal pressures during uplift or at surface conditions. REFERENCES CITED Angus, S., Armstrong, B., and de Reuck, K.M., 1978, International thermodynamic tables of the fluid state—5. Methane: Oxford, England, Academic Press, 251 p. Atkinson, B.K., 1979, A fracture mechanics study of subcritical tensile cracking of quartz in wet environments: Pure and Applied Geophysics, v. 117, p. 1011-1023. 1984, Subcritical crack growth in geological materials: Journal of Geophysical Research, v. 89, p. 4077-4114. 1987, Introduction to fracture mechanics and its geophysical applications, in Atkinson, B.K., ed., Fracture mechanics of rock: London, Academic Press, p. 1-26. Atkinson, B.K., and Meredith, P.G., 1987a, The theory of subcritical crack growth with applications to minerals and rocks, in Atkinson, B.K., ed., Fracture mechanics of rock: London, Academic Press, p. 111-166. 1987b, Experimental fracture mechanics data for rocks and minerals, in Atkinson, B.K., ed., Fracture mechanics of rock: London, Academic Press, p. 477-526. Baratta, F.I., and Parker, A.P., 1983, Mode I stress intensity factor estimates for various configurations involving single and multiple cracked spherical voids, in Bradt, R.C., Evans, A.G., Hasselman, D.P.H., and Lange, F.F., eds., Fracture mechanics of ceramics, Volume 5: Surface flaws, statistics, and microcracking: New York, Plenum Press, p. 543-567. Bodnar, R.J., and Bethke, P.M., 1984, Systematics of stretching of fluid-inclusions I: Fluorite and sphalerite at 1 atmosphere confining pressure: Economic Geology, v. 79, p. 141-161. Bodnar, R.J., Binns, P.R., and Hall, D.L., 1989, Synthetic fluid inclusions—VI. Quantitative evaluation of the decrepitation behavior of fluid inclusions in quartz at one atmosphere confining pressure: Journal of Metamorphic Geology, v. 7, p. 229-242. Broek, D., 1987, Elementary engineering fracture mechanics (fourth edition): Dordrecht, Netherlands, Sijthoff & Noordhoff International, 437 p. Ferguson, C.C., Lloyd, G.E., and Knipe, R.J., 1987, Fracture mechanics and deformation processes in natural quartz: A combined Vickers indentation, SEM, and TEM study: Canadian Journal of Earth Sciences, v. 24, p. 544-555. Gratier, J.P., and Jenatton, L., 1984, Deformation by solution-deposition and re-equilibration of fluid inclusions in crystals depending on temperature, internal pressure, and stress: Journal of Structural Geology, v. 6, p. 189-200. Griffith, A.A., 1924, The theory of rupture, in Bienzo, C.B., and Burgers, J.M., editors, Proceedings of the First International Congress for Applied Mechanics: Delft, Netherlands, J. Waltman Jr., v. I, p. 55-63. Lawn, B., 1983, Physics of fracture: Journal of the American Ceramic Society, v. 66, p. 83-91.

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Lawn, B.R., and Wilshaw, T.R., 1975, Fracture of brittle solids: Cambridge, England, Cambridge University Press, 204 p. Leroy, J., 1979, Contribution à l'étalonnage de la pression interne des inclusions fluides lors de leur décrépitation: Société Française, Minéralogie et Cristallographie, Bulletin, v. 102, p. 584-593. Meredith, P.G., and Atkinson, B.K., 1982, Hightemperature tensile crack propagation in quartz: Experimental results and application to timedependent earthquake rupture: Earthquake Prediction Research, v. 1, p. 377-391. Murakami, Y., editor-in-chief, 1987, Stress intensity factors handbook: Oxford, England, Pergamon Press, 2 volumes, 1456 p. Norton, M.G., and Atkinson, B.K., 1981, Stressdependent morphological features on fracture surfaces of quartz and glass: Tectonophysics, v. 77, p. 283-295. Pollard, D.D., and Segall, P., 1987, Theoretical displacements and stresses near fractures in rock with application to faults, joints, veins, dikes, and solution surfaces, in Atkinson, B.K., ed., Fracture mechanics of rock: London, Academic Press, p. 277-349. Prezbindowski, D.R., and Larese R.E., 1987, Experimental stretching of fluid inclusions in calcite— Implications for diagenetic studies: Geology, v. 15, p. 333-336. Roedder, E., 1984, Fluid inclusions: Mineralogical Society of America Reviews in Mineralogy, v. 12, 644 p. Sterner, S.M., and Bodnar, R.J., 1989, Synthetic fluid inclusions—VII. Re-equilibration of fluid inclusions in quartz during laboratory-simulated metamorphic burial and uplift: Journal of Metamorphic Geology, v. 7, p. 243-260. Swanenburg, H.E.C., 1980, Fluid inclusions in highgrade metamorphic rocks from S.W. Norway: University of Utrecht, Geologica Ultraiectina, 147 p. Tada, H., Paris, P., and Irwin, G., 1973, The stress analysis of cracks handbook: Hellertown, Pennsylvania, Del Research Corporation. Ulrich, M.R., and Bodnar, R.J., 1988, Systematics of stretching of fluid inclusions II: Barite at 1 atmosphere confining pressure: Economic Geology, v. 83, p. 1037-1046. Wanamaker, B.J., Evans, B., and Wong, T.-F., 1987, Fluid inclusion decrepitation (cracking) in San Carlos olivine: Eos (Transactions, American Geophysical Union), v. 68, p. 1527. Wanamaker, B.J., Wong, T.-F., and Evans, B., 1990, Decrepitation and crack healing of fluid inclusions in San Carlos olivine: Journal of Geophysical Research (in press). Westmann, R.A., 1966, Note on estimating critical stress for irregularly shaped planar cracks: International Journal of Fracture Mechanics, v. 2, p. 561-562.

ACKNOWLEDGMENTS This work was supported by a grant to Terry Engelder from Texaco U.S.A. I thank Terry Engelder, Jack Mecholsky, Art Rose, Steve Mackwell, Tom Mackin, E. K. Graham, Brian Reck, Ian Duncan, and an anonymous reviewer for helpful reviews. Any errors are my responsibility alone.

Manuscript received December 21, 1989 Manuscript accepted March 19, 1990

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