Tectonophysics, 49 (1978) 97-118 97 @ Efsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

CRACK EXTENSION FROM FLAWS IN A BRITTLE MATERIAL SUBJECJED TO COMPRESSION

MARC ADAMS and GEORGE SINES

Jet Propulsion Laboratory, Pasadena, Calif. 91103 (U.S.A.) Materials Dept., School of Engineering, U.C.L.A., Los Angeles, Calif 90024 (U.S.A.)

(Received at May 17,1977; revised version accepted December 29, 1977)

ABSTRACT

Adams, M. and Sines, G., 1978. Crack extension from flaws in a brittle material subjected to compression. Tectonophysics, 49: 97-118.

The mechanisms by which cracks extend from flaws in brittle materials subjected to compressive loads are presented. Although it is recognized that most geological materials are neither dense nor single-phase, this experimental study and analysis are restricted to single-phase, dense materials in order to provide a model amenable to analysis and exper- imental confirmation. The flaws which occur in dense, single-phase materials are divided into three types, each type having different characteristics of crack extension. Experi- mental studies of crack extension from flaws introduced into blocks of polymethylme- thacrylate plastic are described. The studies show that crack extension from three- dimensional flaws is more complex than two-dimensional theories predict. The extension of secondary cracks may result in more damage than that which would be predicted by considering only the primary crack extension treated by current theories. The importance of the dynamic behavior of flaws which stick and then suddenly slip is shown.

INTRODUCTION

Studies on crack formation under biaxial compression in multiphase, brit- tle materials (Bieniawski, 1967a, b; Peng and Johnson, 1972; Adams and Sines, 1976) are difficult to perform: the test sample must be large to be representative of the material, the tests must be replicated many-fold to get statistical confidence on the, properties of the natural material, the details of the micro-mechanics of crack extension are difficult to observe, and the models of crack formation difficult to analyze. Even for the much simpler case of single-phase dense, brittle materials under uniaxial compression, the mechanisms of crack initiation, crack growth, and compressive failure are incompletely understood.

The theory proposed. by Griffith (1921) and revised by him (1924) has remained basic to ‘considerations of brittle fracture. Babel and Sines (1968) have generalized the theory to treat open flaws; McClintock and Walsh

98

(1962), to flaws that close. These theories give only criteria for the initiation of a crack at a flaw, do not consider its extension, and the analyses are only two-dimensional. These theories, except the first of Griffith (1921), assume that crack extension starts when a localized tensile stress tangential to the surface of an idealized flaw exceeds a critical value. An alternate approach (Erdogan and Sih, 1963; Sih and Cha, 1974) assumes that the crack will ini- tiate in a direction to maximize the release of strain energy. This approach has been applied to three-dimensional flaws in compression (Sih, 19?3), but it too treats only the initiation of cracking.

Studies on single-phase, dense materials are in geologic context when the flaws are large compared to the inhomogeneities in the multi-phase material; thus, behavior deduced from laboratory studies may be pertinent to seismic behavior and large scale jointing. On the other extreme, when flaws are small compared to the particle size of a single-phase component, the studies are also applicable.

It has been established that the process of compressive failure involves the accumulation of large amounts of cracking (Brace et al., 1966; Adams, 1975). To explain this damage, the phenomenon of crack extension from flaws must be understood. A categorization of the types of flaws and a description of the mechanisms by which cracking occurs in compression will be presented. The results of experiments which demonstrate the nature of crack initiation and extension will be described.

FLAW TYPES

The geometry of flaws observed in rocks and ceramics is usually very complex, the flaws being triple-point voids, cracked large grains, inclusions, irregular voids in grams, etc. The models of flaws amenable to analysis are great simplifications of the observed flaws. The twodimensional model varies from circular holes to infinitely flat ellipses; the three-dimensional model varies from spheroids to oblate or prolate ellipsoids. However, these models may represent the geometry in the critical regions of the flaw better than one at first might think (Evans and Tappin, 1972; Evans and Langdon, 1976, pp. 194, 195). For example, the stresses at the tip of an open flaw depend primarily on the smallest distance on the plane of the flaw between the sharp tips and on the radius of the tips, the stresses at the tips are rela- tively insensitive to the shape of the void in the central region. Thus, an inscribed ablate ellipsoid having the appropriate axes and tip radius can rea- sonably represent the flaw because it has nearly the same stresses in the criti- cal region.

It is convenient to divide the flaws from which cracks extend in brittle materials into three types. Type I flaws are open, Type II flaws may close and the sliding of the contacting faces being resisted by the friction between them, and Type III flaws are closed and their faces locked by mechanical interference or weak bonding.

99

The mathematical expressions that we use for stresses around flaws are based on their being in a homogeneous, isotropic, elastic continuum. Elastic anisotropy of the grains and their heterogeneity have influence, but for very large flaws extending over many grains, the quasi-isotropy, quasi-homogene- ity may make the idealized continuum predictions reasonably good. For flaws the size of a grain and smaller, the isotropic solution will have to serve as a first’approximation.

Type I - open flaw

The two-dimensional, Type I flaw can be in one extreme a circular hole and in the other extreme an infinitely sharp crack. The expression for the tensile stresses along the margin of an elliptical hole, subjected to applied stresses transverse to it was first derived by Inglis (1913). Griffith (1924) equated the tensile stresses at the tip of an infinitely flat ellipse to the intrin- sic material strength in this second theory. Babel and Sines (1968) general- ized this for elliptical flaws of any ratio of major or minor axes. Both the Griffith theory and its generalization have been successful in explaining strengths measured in tension-tension and tension-compression for a vari- ety of brittle materials. If one of the principal stresses applied to the brittle material is tensile, crack initiation and unstable crack growth coincide or almost coincide (Bieniawski, 1967b) and structural failure comes from one of the first flaws to initiate a crack. The nature of crack extension is very different when the principal stresses are all compressive; experimental studies

6

FLAW

_ ?

2

EXTENDED CRACK

Fig. 1. Crack extension from a flaw subjected to compression.

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(Brace and Bombolakis, 1963; Adams, 1975) have shown that cracks which extend from flaws in compressive states do so stably, i.e., the crack length is a function of the applied stress. These extending cracks tend to align them- selves so that the normal to the crack plane (8) is perpendicular to the direc- tion of the principal compressive stress as shown in Fig. 1. This has been ob- served in polymethylmethacrylate plastic (Adams, 1975), glass (Bieniawski, 1967b) rocks (Brace and Bombolakis, 1963), and cement (Kobayashi, 1971). Stresses at crack initiation may be determined by using the elliptical flaw models, but analytical expressions for the stresses at the extending crack tip have not been formulated yet.

If an open flaw closes, the stress normal to the plane of the flaw in excess of that required for closure is transmitted across the flaw faces and thus is not concentrated at the tips. As shown in Fig. 2, the circumferential stress around the tip of the flaw (urrp) arises from the intensification of the shear stress in the plane of the flaw, minus the frictional shear resistance (7 - rf ) and from intensification of the normal stress up to the value of the closure stress (a,). The normal stress un acting across the faces of a closed flaw is equal to the applied normal stress uXX reduced by the value of the normal stress u, needed to close it.

Flaws which close and slide under applied stresses may exhibit several kinds of behavior according to the analysis by ~cclintock and Walsh (1962).

(1) When the flaw is open, the compressive stress u,, normal to the plane of the flaw is concentrated at the tips of the flaw and this concentrated compressive stress, when added to the concentrated tensile stress caused by the applied shear r,., results in a lower maximum tensile value of up0 in the vicinity of the tips. If the flaw closes under a very small applied compressive stress, u, 1 0, and if there is a very low coefficient of friction between the flaw faces, P = 0, crack initiation depends solely on the resolved shear stress on the crack plane. This type of closed flaw will initiate cracks at lower applied stresses than its open counterpart, because the compressive stress CI,, is no longer concentrated at the tips of the flaw after it reaches the value o, needed to close it, thus the local tensile stress upp is higher than when the flaw is open.

(2) If u, = 0 but the friction is high, cr. 2 1, crack initiation will occur at applied stresses higher than its open counterpart, because the resolved shear is reduced by the opposing, frictional shear traction on the flaw faces.

(3) If the flaws close partially, the open portions can behave like open flaws. The normal stress on the flaw plane will be concentrated only at the ends of the open portion and the effective flaw size will be 2c’ as shown in Fig. 3. The shear stress r creates a tensile stress at the tips of the open por- tion (position A on Fig. 3), when r < rf and it is concentrated by the effec- tive length 2~‘. When r > rf, the ciosed portions slide and the shear, dimin-

101

il f 7 w

\ I

u‘pp a: (“xx n 0 0 - u ) 201 /(o? +$I

FOR OPEN FLAW (la,,l~,). 0, = 0

FOR CLOSED FLAW fl~xxi~~cf, on = oxx - UC

WHER,E ao, fl ARE ELLIPTICAL COORDINATES ON THE FREE SURFACE OF THE FLAW

CT

@pp * bxr - 7ff 2#vc$ + 021

FOR OPEN FLAW f~uxx~Suc), rf = 0

FOR CLOSED, SLIDING FLAW

(bxx k,, Txz > Tf), ?f = -PUN

WHERE /J IS THE COEFFICIENT OF FRICTION

Fig. 2. Normal stress at surface of elliptical open and closed flaws, MeClintock and Walsh (1962).

ished by the frictional shear, is concentrated by the full flaw length at the tips (position B). Crack initiation may occur either at A or B depending on the values of the several variables. This has been observed by Hoek and Bien- iawski (1965) in glass plates.

A flaw will close from the center out, or from the tips in, depending on its shape. The flaw in Fig. 3 is shown as closing from the tips inward which is characteristic of an open flaw+ with -sharp tips. Flaws which have flat faces and rounded tips will close down in the center first leaving open segments at the tips. In Appendix II, it is shown that for the idealized elliptic flaw, clo- sure occurs s~ult~eously over all the surface. A precise statement of clo- sure for a planar flaw is - if a flaw is modelled by an ellipse matched to the length ;of the flaw. and to the tip radius, the flaw surface that protrudes the

Cl5SED P5RlWN

TENSILE

OF FLAW

CL5SE5 PORTION

TENSILE am FR5M ?xz

Fig. 3. Stresses around a partially closed flaw.

farthest inside the ellipse will close fii, and conversely the surface that is the farthest outside of the ellipse will close last.

Type III - Clcxwd flaws which are locked

Type III flaws are simikr to Type II except the sliding d~~~e~~ are prevented either by weak bon&g or by rn~h~i~ ~~10~~ between the faces. The stress ~o~~e~~tions around a bonded flaw are the bond is shmmd. The &ms concentq&ion around a mea flaw can be q&e conq3lex uz-ztil it43 faces break free. V&en th stress intensification is the same as for a Type II effect occurs mm&g

4% be ttiee 44lmse.irazm slow lmding. of behavior dependjs-rg on f&e -to ovmxme t&e &x& (7,) tai

103

the shear stress necessary to initiate a crack at an equivalent Type II flaw

(7 )* crit

(I) If ru < Tcritr the applied stress level for crack initiation will be con- trolled by 7, and crack initiation will occur at stresses lower than for Type II flaws. The dynamic effect will cause the crack to extend farther than for Type II flaws. Thus because of earlier crack initiation and greater crack extension, structural collapse may occur at a lower stress for material with Type III flaws than for material with Type II flaws.

A Type II flaw which has a dynamic coefficient of friction much less than the static coefficient of friction will have a dynamic behavior approaching that of the Type III flaw. (Dynamic aspects were not considered in the paper by McClintock and Walsh, 1962.)

Flaws with a large compressive stress normal to the flaw plane will have a tendency to behave as Type III flaws. In parts of the earth’s crust where the weight of overburden rock produces nearly hydrostatic compression, com- pressive stresses should act on flaws of all orientation. However, near the

APPLIED COMPRESSION

CRACK EXTENSION FROM SMALL FLAW

SMALL FLAW

TIP OF LARGE FLAW

ON OF INTENSIFIED COMPRESSION FROM LARGE FLAW

Fig. 4. Flea-on-a-flea flaw crack extension.

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FREE CHIP

EXTENDED CRACK

Fig. 5. Persimmon seed chip formation.

ground surface, we might expect that only those flaws whose planes are per- pendicular to the surface experience compressive stresses.

Flea-~~-F~ea Fiaws

Very small flaws in the vicinity of a large flaw may extend cracks at applied stresses lower than those needed at an isolated large flaw. The stress effective on the small flaw could be almost the full magnitude of the intensi- fied stress at the tip of the large flaw upp (see Fig. 4). The intensified stress decreases as the inverse square root of the distance from the tip of the large flaw.

Persim man seed model

If two open flaws intersect the free surface of a body and are oriented as shown in Fig, 5, the ~oncent~~ stress fields at their tips will overlap where the flaws are close enough. Cracks extending from each flaw will tend to link up and this can be thought of as “squeezing out” the particle between them. It will also be seen below that single surface flaws may extend cracks in a direction to intersect the surface and produce a free chip.

EXPERIMENTS ON CRACK EXTENSION

An analysis of crack extension from flaws in materials subjected to cam- pression has not been formulated yet; therefore, to obtain an understanding, the following experimental study was carried out. Models of cracks were ma-

105

chined into blocks of polymethylmethacrylate plastic (PMMA) and loaded in compression. The plates used for the two-dimensional study were 1.27 X 8.89 X 8.89 cm with a 1.90 cm long slit cut through with a jeweler’s saw. (See Fig. 6.) The three-dimensional study was performed on blocks 2.54 X 5.08 X 8.26 cm. (See Fig. 7.) The following procedure was used to introduce “penny-shaped” flaws. Semi-circular slits were machined at an angle into a face of the block using a 1.90 cm diameter by 0.15 mm thick milling saw. A 0.51 cm radius relief cut had to be made first to provide clearance for the saw arbor. Cementing two blocks together, each containing a semi-circular slit produced an imbedded penny-shaped crack as shown.

The blocks were loaded in uniaxial and biaxial compression as shown in Fig. 7. A 100,000 lb. Riehle beam machine was used to apply the load. Biaxial compression was obtained by using a machinist’s vise to constrain the transverse displacement of the specimen. A focused light projected the out- line of the flaw and the extending crack onto tracing paper mounted on the back of the specimen. The crack extension which occurred at each increment of load was traced on the paper, The plastic is visco-elastic and acts brittlely if loaded rapidly. The specimens were loaded at 8,900 N/s. and then imme-

W&BEDDED

Fig. 6. Two-dimensional slit specimen (one slit per plate).

J 1.27

-EDGE SLIT

GREATEST CG~PRE~lV~ LOAD

COMPRESSIVE

_---_-_- _-

1.90 DIAMETER 0.015 THICK

\ \

\ \

\

DIMENS+ONS: CM LJ

Fig. 7. Penny-shaped flaw in plastic block.

diately unloaded. Crack extension was recorded and then the specimen was loaded to the next higher stress.

Two-Dimensional Slit Flaws

The results of the two~~ension~ studies are shown in Figs. 8a-i. The numbers give the loads applied to extend the crack to the length marked. The applied stress for crack initiation at edge flaws of most orientations did not appear significantly different from that of imbedded flaws; however, the edge flaws extended longer cracks at a g&en stress and they did not become parallel to the direction of the Ioad but prowar to the edge of the plate.

The behavior of the slit fIaws is coqnplicated by locking under load so that the crack extension does not always irmawe directly with the appbed stress. If the crack closes but the faces slide, slow, stable extension is observed. If the faces lock, no extension is observed until they break free and shdei then

107

sudden growth occurs and arrests. Either behavior can be observed at some time during loading. An extreme example was the plate of Fig. 8c with an imbedded flaw, t = 70” (the angle g is defined on Fig. 6) which was loaded to 102.4 MPa and extended a crack too small to be shown. When reloaded again from zero, the large extension shown occurred suddenly at 55.2 MPa. No further extension was observed until the stress reached 98.5 MPa when the second sudden extension occurred. The point of nucleation of the large extension at 55.2 MPa during the second loading was at a different position on the slit than the very small extension observed during the first loading. The extension at 98.5 MPa appears to have initiated from the 55.2 MPa extension as though it was acting as the originating flaw. Perhaps the ma- chined slit locked after the 55.2 MPa extension and never unlocked leaving the 55.2 MPa extension to function as the only flaw. Stick-slip behavior is p~icularly prevalent for flaws with large [, because these flaws have a greater resolved compressive stress normal to their faces.

Surface roughness is also important to stick-slip behavior. The roughness of the two-dimensional slits was greater than that for the three-dimensional, 130 versus 0.8 (urn r.m.s., respectively. Even so, a little stick-slip behavior was observed with some three-dimensional ones. All stick-slip behavior was accompanied by noise.

As shown in Figs. 8b and c at a stress of about 98.5 PMa cracks suddenly extended almost t0 the ends of the test plate in a direction parallel to the

LOAD

u

(a) IMBEDDED FLAW (b) IMBEDDED FLAW (c) IMBEDDED FLAW ( = 450 5 = 200 [ = 700

NUMBERS INDICATE VALUE OF APPLIED STRESS (MPal AT INDICATED

LIMIT OF CRACK EXTENSION

FLAW LENGTH IS 1.90CM EXCEPT FOR (hl AND (ik WHERE IT IS 1.27 CM.

Fig. 8. Crack extension from twodimensional flaws. (To be continued on next page.)

108

LOAD

u

;I:

67.0 51.2 39.4

SLIT

@I EDGE FLAW [ = 300

NO CRACKING

PLATE

(eb EDGE FLAW t =‘450

SLIT

----. 39.4

(f) EDGE FLAW [ = 000

(9) EDGE FLAW (h) EDGE FLAW E = 700 E = 450

(i) DOUBLE EDGE FLAW E = 450

NUMBERS IXDlCATE VALUE OF APPLIED STRESS Ma) AT INDICATED LIMIT OF CRACK EXTENSION

Fig. 8 (cont.).

load. The lengths of these cracks were no longer a function of stress, the extended crack orientation, or the slit orientation. explanation of this is in Appendix I.

the applied A p&&le

Figure 8i shows the behavior of the double edge flaw for t = 45” in which a crack did not extend directly between the slits. Cracks extended indepen- dently from the slits q& this is at&but& to each slit alte~&Gy and inter- mittently locking and extending cracks. From double slits in which t was less than 45’) cracks did extend direct@ between the slits as shown in Fig. 5.

109

Three-Dimensional, Penny-Shaped Flaws

Figures 9a-c show the crack extension from the penny-shaped flaw model by giving the front projection of the cracks under increasing load. These “primary cracks” initiate at the flaw tips closest to the applied compression. Their initial direction was normal to the plane of the flaw in agreement with the prediction of Sih and Cha (1974) which is based on local strain energy release. This is not to say that it may not also agree with predictions (as yet not made) based on other criteria and models, such as crack initiation on the plane normal to the maximum local tensile stress on an interior elliptical slit or on an oblate spheriod, or on flaws that close and have friction between their faces. (McClintock, 1963). Sih and Cha did not predict the other two modes of cracking which were observed and are presented below. Comparing the crack initiation loads for a full circular imbedded flaw at 31.0 MPa with the semi-circular imbedded flaw at 48.3 MPa {Figs. 9a and b) one can see the inverse square root dependence of the flaw size on the initiating stress if one assumes the effective size of the semicircular flaw is half that of a circular one. The crack initiation stresses for the imbedded half flaw (Fig. 9b) and the surface half flaw (Fig. 9d) do not differ by much; however, the crack for the surface flaw is 50% longer than that for the imbedded half-flaw after crack initiation for all stresses. The crack from the surface half-flaw inter- sects the surface to form a chip.

The behavior of the full-flaw which intersects the surface at one tip (Fig. 10 and Fig. 9c) is more complex than when it is in the center of the block. Cracks initiate near the free surface at slightly lower stresses. Cracking at the interior tip requires slightly higher stresses for equivalent crack exten- sion than for the fully imbedded flaw.

The imbedded flaw initiates cracks at lower stress under uniaxial loading than biaxial loading (Figs. 9a and e). The length of the crack in the direction of the greatest compression is about the same at any stress. The impo~ant distinction is that “wings” form on the crack in the direction of the trans- verse compression.

Several aspects of the threedimensional crack growth from penny-shaped flaws in compression can be seen in Fig. 11. The primary cracks initiate at the tips of the flaw nearest the applied compression and with increasing load extend away from the tip in a smooth curve to become parallel to the applied compression and to spread around the perimeter of the flaw. Two other kinds of cracks have been observed. We call them “petal” cracks and “fish fin” cracks. The petal cracks form on the flaw perimeter near the edge of the primary crack, Perhaps they are initiated by the concentrated shear on the plane of the crack. Their lengths remain a fraction of that of the pri- mary crack. Fish fin cracks form near the center of the flat surface of the penny-shaped flaw as shown in the sketch, Fig. 11, and visible in the photo- graph, Fig. 10. Although these cracks initiate after the primary crack does, they can extend to a greater length. The stresses around the three-dimen-

110

FLAW

I LTJ 31.0

I? . m 51.7

f---J 58.6

a 65.5

SURFACE

F

lzl 20.7 I gg 61.7

m 27.6 cl%6

f3 34.5 0 66.5

m44.8 a 72.4

a 82.8

(c) IMBEDDED FLAW INTERSECTtt’&3 SURFACE $=460

L FLAW

0 48.3

m 69.0

m 82.8

m 103.0

(b) ;N_E4iQALF FLAW IMBEDDED

ktt

I m 56.6

0 a 65.5

gg 75.9

m 82.8

a 89.6

OAlE HALF t=lAW IMPER&ECTthK3 SURFACE 6 = 450

NlJt’itEtERS tWQtCATE VALUE OF APPLIED STRESS (MFI) AT INDtCATtiD LIMtT Of CRACK EXTEDIWbN

Fig. 9. Craek extension from three-dimensional flaws.

111

ggj XI.6

V

m 44.8

m 55.2

m 65.5

(e) IMBEDDED FLAW [ = 450 BIAXIAL COMPRESSION

NUMBERS INDICATE VALUE OF APPLIED STRESS (MPa) AT LIMIT OF CRACK EXTENSION

Fig. 9 (cont.).

S113E F Fig. 10. Penny-shaped flaw tangent to front surface.

112

APPLtED COMPRESSION

Fig. 11. Crack extension from a penny-shaped flaw in a plastic.

sional, crack-extended defect are not known, but in Appendix I a two- dimensional analysis shows that a tensile stress, equal in magnitude to the applied compression, exists at the center of the elliptical surface and is parallel to the surface. It is normal to the plane of the fish fin crack and may be responsible for its formation. It also shows that after this crack forms the tensile stress transverse to its plane at its tip is also equal to the applied com- pression and continues to be equal to it regardless of its length. The applied

113

compressive stress was slightly greater than the nominal tensile strength of PMMA.

Under increasing load, primary cracks initiate and extend until they align with the applied compression and are arrested. Secondary cracks however, may initiate and continue to extend, thus a flaw can create considerably more fracturing in a brittle material than would be predicted by considera- tion of only primary cracks.

DISCUSSION

Although the above studies of crack growth were conducted on models made from an organic polymer in an effort to explain the micro-spalling and crack growth in dense ceramics which are used as engineering structural ma- terials (Adams, 1975; Adams and Sines, 1976; Adams and Sines, 1977), the observations may help explain some features of joints. Some aspects of the morphology of joints are consistent with the crack growth observed in the transparent models. Hodgson (1961) cites Woodwork’s description of small discoid fractures in argillaceous strata of the Boston Basin, “The warped sur- face thus formed may be likened to the rim of a soft hat curled up behind and pulled down in front . . .“. Notice how well this description fits the appearance of the cracks identified as the “primary cracks” extending from the flaw in Figs. 7 and 11.

It is emphasized by Roberts (1961) and by Hodgson that plumose jointing is often observed and they emphasize that the fine structure characteristics of the surface is not obliterated by lateral movements of the mating surfaces. They state this is consistent with a tensile stress field normal to the crack rather than a shear field parallel to the crack. They also observe plumose joints parallel to each other and closely spaced. This is not consistent with a tensile field because of the formation of one joint would interrupt the tensile field so that parallel joints would occur only if they were widely spaced, perhaps at distances several times their length. However, both of the charac- teristics they observed are consistent with the fish fin crack which extends parallel to a compressive field from a flaw transverse to it.

Roberts also states that the presence of plumose markings indicates that fracture occurred at near-sonic velocities and identifies them with the “herring-bone” pattern present on metal fractured at near-sonic velocities. He finds that plumose joints are prevalent in fairly fine-g-rained rocks, thus it might be inferred that some joints may have formed from the sudden un- locking of Type III flaws.

The behavior of brittle materials subjected to compressive loading appears to be, and has been, of more interest to geologists than to materials engi- neers. We have been able to find a few examples to illustrate the modes of crack extension; we welcome information on crack extension in geologic context based on observations or on the literature with which we are only beginning to become acquainted.

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ACKNOWLEDGEMENTS

This work was supported by the NSF under Grant ENG 74-00138 and monitored by C.J. Astill. It is an extension of a study sponsored by the Naval Air Systems Command and monitored by Charles Bersch. The exchange of ideas with Prof. Gerhard Oertel, Geology, UCLA was very help- ful.

APPENDIX I

Tension at an elliptical hole parallel or transverse to a compression

The stress at the surface of an open, elliptical hole (see Figs. 2 and 12) is given by Timoshenko and Goodier (1970, pp. 191-194) in the elliptical coordinates of (Y, 0 as:

+ -

o&3(0, 8 sinh cos

- ezcr” cos

= -0 2a0 2,5 2(e cash 2e0 - cos 20 /I) 1 If the major axis is aligned with the direction of the applied compression, the stress at the crack tip is:

opp(0, 0) = -u sinh 2a0 + 1 - e2 “0

cash 2a. - 1 I

If the major axis is transverse to the direction of the applied compression,

Fig. 12. Elliptic flaw under compressive stress.

115

the stress at the midpoint of the side is:

o~~(~/2, n/Z) = -t7 sinh 2% -1 -ezffO

cash 2eo + 1 1 From the equation of an ellipse:

sinh 2oo = 2ab/c2 ; cash 2a. = (a’ + b2)/c2 ; e2”0 = (a + b)2/c2

where a is the semi-major axis, b is the semi-minor axis and c is the half-focus distance. Substituting these into the above expression, one obtains:

app = +u

Thus, for any ratio of axes, the stress at the tip of the aligned ellipse and the stress at the midpoint of the side of the transverse ellipse is a tensile stress equal in magnitude to the applied compression.

APPENDIX II

The elastic closure of a flat elliptical flaw

If the idealized elliptical open flaw as shown in Fig. 2 is sufficiently flat, elastic displacements from the applied compressive stress oxX can make it close. Griffith (1924) in his eq. 16 shows that the displacement of the free surface of the flaw is:

Kc = F 2C cash a0 sin p (IIa)

The equation of an ellipse in elliptic coordinates given in his reference 19, p. 182 is:

x = C sinh rro sin fl (IIb)

Constant values of p correspond to confocal hyperbolas and constant values of (Y correspond to confocal ellipses, where 2C is the distance between the foci. Coordinate ar = (Y, is the free surface of the ellipse and /3 = 0 is the coor- dinate position of the tip. See Figs. 2 and 12.

The displacement for closure is:

UX =3 (IIc)

Equating these two expressions (IIa) and (IIb) gives the condition for clo- sure :

(J xx =-ftanhe, (IId)

Notice that this condition is not a function of & the position of the ellip- tical flaw; thus, closure occurs simultaneously over the surface of the flaw.

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APPENDIX III

Stresses from suddenly applied tractions

The breaking loose of a Type III flaw can be considered as the sudden removal of restraining tractions on the flaw’s faces. Furthermore, a sudden removal of a traction can. be considered as a sudden superposition of a trac- tion of the opposite sign. We will show that the stress field caused by a sud- denly applied traction is twice that it would be if it had been applied slowly.

A suddenly applied traction does not imply impact of a mass but that the full magnitude of the force is applied from the first instant and is maintained at this value even though the point of application undergoes vibratory dis- placement. (The “first instant” means a time much shorter than the period of vibration.) In static elasticity it is assumed that a traction is slowly increased from zero to its full value so that no such vibration occurs. The effects of tractions can be considered individually and superimposed if the material is linearly elastic and the deflections are small.

The deflection 6, of the body at the point of application of the slowly applied traction 2’ is linearly related by a spring compliance tensor:

&)=cuT (1)

This deflection can be used as a measure of T. The work done on the elastic body by a traction 2’ of the same magnitude

applied suddenly is:

W=T*6, (2)

where 6, is the extreme. deflection produced by T. To find 6, observe that at the extreme deflection the work done is entirely stored as strain energy, because then the kinetic energy is zero. Also the strain energy is only a func- tion of deflection and must be the same whether it is stored suddenly or slowly; thus the strain energy U at the extreme deflection is:

Using the deflection of the slowly applied traction as a measure of T accord- ing to eq. 1:

T = a-‘&j, (4)

which when inserted in eq. 2 gives:

w = (e-l&) - 6, (5)

Equating this expression for the work done by the suddenly applied traction to the strain energy at the extreme deflection, Eq. 3, we obtain:

26, = 6,

117

Thus it is seen that the detection under a suddenly applied traction is twice that which would result from the same traction applied slowly. In a linear elastic system this also means that the stresses in the elastic body from a suddenly applied traction are twice those had it been slowly applied.

The above is not restricted to an isotropic or homogeneous medium. The deflection under sudden loads was studied by Poncelet as early as

1839. Some textbooks in engineering dynamics present a derivation, but from the textbook derivation it might be inferred that a mass must be asso- ciated with the traction, but this is not necessary. The above derivation im- plies that mass is associated only with the elastic body.

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