- Email: [email protected]
Crack extension under compressive loading
0013-79ul&(53.00+ .oo RigamonPressLtd.
Engirvcring FractureMmhanicsVol. 20,No. 3, pp. 463473,19&o Flinti in UK U.S.A.
CRACK. EXTENSION
UNDER
COMPRESSIVE
LOADING
PAUL S. STEIF Department of Mechanical Engineering, Carnegie-Mellon University, Schenley Park, Pittsburgh, PA 15206, U.S.A. Abatrae-Non-planar extension (kinking) of a crack subjected to remote biaxial compression is studied to gain insight into compressive failure of brittle solids. Approximate expressions are developed for the stress intensity factors at the tips of the resulting wing cracks. Assuming the wing crack propagates at a constant value of KL, one can calculate wing crack extension vs load. The significance of the dependence on various parameters is discussed.
1. INTRODUCTION INTERFBT in the mechanical response of geological materials such as rocks to natural loadings motivates the study of compressive fracture in brittle solids. Based on extensive mechanical testing of rocks, the magnitude of the compressive stress at which overall failure occurs is generally said to exceed, at times greatly, the tensile failure stress. One explanation for this stress differential effect is that, in contrast to the tension case, the initiation of brittle crack growth under compression is not tantamount to overall failure (see Paterson[l]). Wing crack growth, i.e. out-of-plane extension of preexisting cracks, characteristic of compressive loading, occurs stably: for at least some range of wing crack lengths, such growth takes place only with increasing load. Measurements of acoustic emission and elastic wave speeds confirm that considerable micro-cracking occurs prior to final failure. In this paper, the out-of-plane extension of a single crack is studied. A solution to this problem is a first step towards an in-depth understanding of compressive failure, further steps being the problems of crack interaction and linkage. The more relevant three-dimensional problem is bypassed in favor of a more mathematically tractable two-dimensional one. The two-dimensional problem also has the advantage of being directly comparable to experiments on cracked plates, essentially a two-dimensional configuration. We consider an inlinite body deforming in plane strain subjected to a uniform stress state at infinity with principal axes aligned with the coordinate axes x1 - x2. The body contains a line crack of length 2a oriented at an angle 0 with respect to the x1-axis. Assuming the far-field stresses are not equal, the crack faces will slide parallel to themselves, inducing a non-zero Mode II stress intensity factor (SIF) Kn. Because both far-field principal stresses are taken to be compressive, the crack is closed and Ki is zero, i.e. the singular part of the near-tip stress field involves only stresses that are anti-symmetric with respect to the crack plane. Closure of the crack can also introduce frictional resistance to sliding. When the maximum tensile hoop stress at some fixed small radial distance from the crack tip reaches a critical value, wing cracks develop as the initial propagation occurs at an angle a from the original plane of the crack (see Fig. 1). Since the prevailing singular held is pure Mode II, a will be Seen to be approx. 70”, consistent with experiments. This problem has been considered by Nemat-Nasser and Horii [2], who find the stress intensity factors at the wing crack tips via the numerical solution to a singular integral equation. Their results provide a comparison for the equations derived below. In the following section, approximate expressions for the in-plane SIF’s K;kiwand Kg* at the wing crack tip are developed. If one assumes crack propagation occurs at a constant critical SIF, i.e. the material is brittle, the expressions for K;“‘” and K$w allow one to calculate wing crack extension as a function of applied stress. The influence of parameters such as confining pressure, coefficient of friction and initial crack orientation is examined, and some possible implications for failure are cited. 2. ANALYSIS The SIFs at the tips of the wing cracks may be written formally as
464
PAUL S. STEIF
I
02
Fig. 1. Wing crack configuration and definition of coordinate axes.
where K$’ and @“I are the SII?s that woufd prevail at a wing crack were it an isolated crack of the same length in an infinite body, and Ki* and Kiy are just the differences between the actual intensities K;u’” and eg and the contributions Kim’ and ti,r’. In terms of the far-field stresses resolved along the n’ - t’ axes (see Fig. 1), Kiy’ and Kt’ are given by
where 1 is the length of each wing crack. The quantities Kiti and Kip reflect the influence of the main crack on the stress field experienced at the wing tips. Approximate expressions are now developed for kTe and Kip in terms of relevant parameters. Since we are assuming the faces of the main crack remain in contact due to the ~ornpr~~ve loading, the far-field stresses still effectively “see” a wing crack of length i, i.e. the main crack does not “lengthen” the wing crack. But the relative sliding of the faces of the main crack, caused by the far-field loading, serves to wedge open the wing cracks. We take this wedging action, which is a driving force for wing crack growth, to represent the total influence of the main crack on the wings. Let d be the relative sliding across the main crack faces where they meet the wings cracks; d depends on the relative values of 1 and a, as well as on other parameters. If one further assumes that all the influence of the main crack on the wing cracks can be incorporated into the wedging displacement d, then, by dimensional considerations and linearity, the stress intensity factors Kpfi and e must be of the form
GW where E is Young’s modulus and j3i and & are dimensionless functions of Poisson’s ratio and the angle a. We now determine A by considering a straight crack of half-length a + I subjected to far-field shearing (see Fig. 2). Note that this is the special case of wing cracks aligned with the main crack, i.e. a = 0. This elasticity problem is readily solved; one finds that the relative tangential
Crack extension under compressive loading
T
I
t.s?!---
I T
2 0 +-.&-I
Fig. 2. Crack configuration for calculating sliding displacement.
displa~m~t
A, of the crack faces at x = a is given by
A,=4(1;v2)l&iTjGz
(3)
At this point one might be inclined to identify A with A,. If A, were applied at x = a, then, according to equation (2bj, the resulting SIF would be roughly? that prevailing at the tip of a crack of length 2u + 21in a shear field T. We defined Kp and Kifl, however, to correspond to the singular stresses that are otter and abwe the singular stresses that would prevail were the wing crack isolated in an infinite body, Equation (3) implicitly includes the “isolated” contribution to the SIF. Since the wing crack is of length 1, the isolated contribution (in Mode II) is: f$‘=z
al
J 2.
Invoking a relation such as (2b), we say that luiyl corresponds to the application at x = a df a sliding displacement AimI given by
where c, which is independent of a .and 1, is determined below. We retain z to denote the effective shearing stress experienced by the main crack and define A by A SAf-Aiml=
4(1 - 9) TJ~ E
Therefore, by equations (2), @’ and @
- cl.
(4)
are given by
Equations (5) are now shown to give the correct dependence on 1 and a for l/a $- 1 provided c is chosen properly. Consider the wing crack configuration in Fig. 3(a) in which the far-field stress state is one of uniaxial compression parallel to the wings which are taken to be much longer than the main crack. Were it not for the main crack no stress singularity would exist at the wing tips: Pi-’ = @i = 0, and so the contributions KY and Kp are equal to the total wing tip SIFs. For tIf B,, is chosen such that KIK,, is cm-rectfor I/a-+0, then K,, is incorrect by at most a factor ,,h for the entire range of i/a.
466
PAUL S. STEIF
this limit of l/a-,o~, as far as can be seen from the wing tips, the wedging apart of the wings by sliding of ’ the main crack has the same effect as a point displacement has on a crack of length 21 (see Fig. 3b). This configuration has a Mode I SIF at the wing tips of
where & is the effective wedging displacement. Now, as I/a becomes large, the sliding that occurs on the main crack becomes independent of 1. Thus, by dimensional considerations, we have
and consequently, K1 =f(cl, v) JZ JI
for f % 1
where f(a, v) is a dimensionless function of c1and Poisson’s ratio v. We therefore require that the contributions Kid and e, which derive from sliding alone, have the same dependence on 1 and a as in eqn (6) in the limit as Z/a -+ co. This is accomplished by giving c the value c=
4(1 - 9) E ‘*
Expanding eqns (5) for l/u % 1 with c as given above, one can show that the correct dependence ‘on 1 and u results. ,8t and flu are now determined by considering the limit l/a @ 1. Explicit forms for Kj”h and J@‘@ for vanishingly small wing cracks have been developed by Cotterell and Rice[3] using a perturbation expansion for slightly curved cracks. Although only strictly valid for c1+O, the expressions given in [3] are shown to be very accurate over a wide range of angles a. When converted to the present situation and notation, their results are: K,= -i(sinq+sinF)r,/&
Ku=;
( cosqf3cos~
0
t (a)
(b)
Fig. 3(a). Long wing crack limit with @‘I = K#” = 0. Fig. 3(a). Main crack is replaced by an effective wedging &.
) r,/&.
Crack extension under compressive loading
467
In the limit as l-0, the stress intensity factors at the wings are simply given by the parts Kp and e; thus from eqns (5) we have
By choosing 4(1 -vz~~~=~(~os~+3eos%)~
4(1 - vZ)/?* = -i(sinr+sin:)$
we can force our appro~mate expressions for the stress intensity factors to give the same values as calculated by Cotterell and Rice[3] in the limit as t+O. Finally, friction at the sliding main crack is modelled by defining the effective shear stress r experienced by the main crack to be
where (mr and a;,,, are the shearing and normal components of the far-field stress resolved on the main crack axes, and p is the Coulomb friction coefficient. The sign of the term A ,&rmn]is always chosen so as to oppose the applied shearing u,,~. In terms of the applied stresses q and a,, the wing tip SIFs are
a,) cos 28 +gu2-
Kf”= [fb
II
[Jzi?
I$ T
II 1 (6 - a,)sin28+p >J[ z‘z
cos;+3cos$ (
+;(% - a,) cos 28
00
d
-o,)sin(2@+2a)
+;
- Ji]
II
[&X
f(q+aJ
- Ji].
It is assumed that a, < rr, < 0, and thus the wings initiate at an angle tl> 0. The driving shear stress r,, = f (az - a,) sin 26 is negative, and thus the fsictional resistance
is chosen to be positive. The solution is valid only if
is negative, otherwise the main crack is locked and cannot slide.
PAUL S. STEIF
468
Equations (7) for the wing tip SIF’s are compared with the results of Nemat-Nasser and Horii[2] for the case of trl = 0, ,u = 0.3, and 8 = 54”. In Fig. 4, the normalized stress intensity factors are plotted as a function of wing crack angle IX Results from Nemat-Nasser and Horii[2] are reproduced in Fig. 5.t The following symbols are respectively equivalent: 8, 3/, c, ul in [2] and (Y, 90-8, a, cr2 in the present work; thus Figs. 4 and 5 may be directly compared. Though the magnitudes of the SIF’s calculated here differ somewhat from those given in [2], particularly for larger wing cracks, the trends with wing angle a and wing length are very similar. The favorable comparison between Figs. 4 and 5 suggests that the conclusions that are reached below regarding the dependence of wing crack growth on various parameters are at least qualitatively correct. Crack extension vs load is now calculated for a load history that is a two-dimensional analog of the triaxial compression test frequently performed on rock or soil samples. We consider 0, = o, both to decrease from zero (negative stresses denoting compression) until (r, = o2 = - 0,; subsequently u, is held fixed at c, = -u,,, and c2 continues to decrease with c2 = -oO - s. Both a, and s are inherently positive: co is the confining pressure and s is the differential stress.
)(
;w
lw= 0.4 r
0.3 0.2 -
0.10,
alx7r)
-0.1 -0.2 -
Fig. 4. Normalized wing-tip stress intensity factors as a function of wing orientation for various wing crack lengths as given by the present approximate calculation. Parameter values are 0 = 54 and p = 0.3.
K,
y= 0.2-T
Kll
mm
loilrn
y=o.z7r
.4
.4
.2
.2
0
0
-.2
-.2
Fig. 5. Normalized wingtip stress intensity factors as a function of wing orientation for various wing crack lengths as calculated by Nemat-Nasser and Horii[Z] via a singular integral equation. Note: the symbols 6, y, c and CT,,in [2] and the symbols a, 90” - 0, n and a2 in the present work are equivalent. tFigure 5, an improved version of the original tigure appearing in [2], was kindly supplied by Prof. Nemat-Nasser.
449
Crack extension under compressive loading
Crack initiation is taken to occur at the angle a, denoted by CQ,at which the hoop stress o, of the singular field is a maximum for fixed r. With only a Mode II singular field, ol, satisfies
or 4(1, x 70.53”. The angular dependence of J$” and Kp8 for l+O as calculated approximately by Cotterell and Rice[3J derives from the angular dependence of crggand a, at the tip of the main crack. Since CT@ achieves a maximum at TV = cql,so does #Tw (also o, and K;H;j*vanish at u = or,,).Thus, a maximum hoop stress i~tiation criterion is ~n~mount to a criterion based on the rna~rn~ Kfuilrg, within the approximation inherent in the results given in Ref. [3]. With a propagation criterion appropriate to brittle solids, a calculation of wing crack initiation and growth as a function of load can now be based simply on K;*= = K,, where Kit is the fracture toughness. For the loading history prescribed above, one obtains from eqn (7a)
cw=
[-fro-s(1
+cos(2# +2a))]
t J
-A
J[ ;
-~sin28+p[00+~s(l+cos2B)]][JGYl-Ji]
(8)
where A =i(sini+sinq). If the same crack of length 2a was subjected to a tensile stress normal to the crack plane, then, within the context of linear elastic fracture mechanics, the tensile stress or which initiates fracture must satisfy
The body under compression is now assumed to respond in an ideally brittle fashion, that is, wing crack growth occurs whenever the Mode I SIF Ky‘Qis equal to the initial fracture toughness Ki,. This implicitly relates the loading to a given length of wing crack. Combining equations (8) and (9), one finds the differential stress s to be related to the wing crack length I by
s
2+~[A&,&%~)
oT= A[sin 28 - ~(1 + cos 20)][JiTZ
-I- fl] - &I - 11+ COS (3 + 2a)IJL
where L = l/20. A special case of eqn (10) is that of I = 0; the co~es~nding is the stress needed to initiate the wing crack:
$1 <=Afsin28
(10)
differential stress, denoted by s,,
2+2;rAp -p(l
+cos28)]’
According to eqn (lo), which was derived for straight wing cracks and included only the effect of w” on crack growth, wing cracks approach a finite, often very short, length as the applied load becomes unbounded, unless the wing crack has a certain orientation. Wing cracks are exposed in general, however, to a non-zero K$Q. This causes the wing crack to deviate from its initial path of a, z 70” and eventually align itself such that the tangent at the wing tip is normal to the direction
410
PAUL S. STEIF
of minimum remote compression (see Fig. 6). Curving towards the axis of maximum compression is observed experimentally, and we attempt to account for this change in wing crack orientation. Proceeding approximately, we calculate the wedging contribution Kpe on the basis of a straight wing crack of length 1 oriented at an angle ~6, the angle at which the curved wing crack meets the main crack. For calculating the isolated contribution to K;“‘“g,the wing crack is taken to be oriented at an angle a = (x/2) - 0, i.e. normal to the direction of minimum compression. These modifications mean that, in eqn (lo), A is calculated with a = h, and cos (28 + 2a) equals - 1. All results presented below are based on eqn (10) with these modifications. 3. RESULTS AND DISCUSSION In Fig. 7, the normalized initiation stress s,/or is plotted as a function of crack angle 8 for various values of friction coefficient p (a0 = 0). For a given p, there is a crack angle at which the initiation stress is a minimum. As expected, the initiation stress for a particular angle increases with friction coefficient. More significantly, the minimum in the initiation stress versus 8 curves becomes narrower as p increases. That is, while cracks at many angles are likely to initiate at a stress slightly above the minimum initiation stress in the case of p = 0, only cracks in a narrow range of angles can be initiated when p is high. Note the particular tendency for cracks at small angles 8 to have their initiation stress raised by increasing p; such cracks not only have a small resolved shear stress, they also have a large resolved normal stress.
I
c2
Fig. 6. Schematic of wing crack curving with a definition of initial wing angle h.
0’
1
IO
20
30
40
50
60
70
00
908
Fig. 7: Normalized differential stress to initiate wing crack growth as a function of main crack angle for various values of friction coefficient (zero confining pressure).
Crack extension under compnssive loading
471
A variation with 0 which is similar to that displayed in Fig. 7 is shown in Fig, 8 for a variety of wing crack lengths (again a0 = 0). Stability of wing crack growth, i.e. the need to raise the load to extend the wing crack, for zero confining pressure (a, = 0), is attributed to the nature of the wedging contribution: at a constant load e decreases as L increases. Thus an increasing load is required to drive wing crack growth. When the conGning pressure cr,is non-zero, the term Kim’is non-zero: since the remote loading is compressive, wing crack growth is further stabilized. In Fig. 9, the influence of ~1on the relation between load and wing crack length is shown for the case of zero confining pressure and for a large confining pressure. The initiation stress is raised sharply by confining pressure when the friction is high, whereas it is unaffected by confining pressure in the absence of friction. As the wing crack subjected to Corning pressure grows, however, the ratio of driving stress at high friction to that at low friction diminishes. Hence, stability of crack growth is greater the smaller the friction, in the sense that
the equality holding when a0 = 0. A very simple criterion for crack coalescence might be that the wing cracks reach a critical length, related to, say, the initial crack spacing. In Fig. 10, the stress s, needed to achieve an arbitrarily chosen wing crack length Z/&z= 2.0 is shown as a function of confining pressure for various angles 8. If all cracks were initially oriented at the same value of 6, then a Mohr~o~omb failure criterion would emerge: the differential failure stress increases linearly with confining pressure, the rate of increase varying with 8. From eqn (lo), this can be seen to be true irrespective of the choice 1/2a = 2.0. The results shown above serve to identify some of the limitations to very simple approaches to compressive failure of brittle materials. One such approach considers a single elliptical flaw subjected to remote stressing. Calculation of the failure stress involves m~imi~ng the tensile stress with respect to position about the flaw, and then maximizing with respect to the orientation of the
Fig. 8. Norm&&
differential stress to achieve the indicated amounts of wing crack growth as a function of main crack angle (zero confining pressure).
472
PAUL S. STEIF
Q”
0.4
0.8
1.6
t.2
2.0 20
Fig. 9. Normalized differential stress as a function of wing crack length for various values of friction coeflbient. Lower plot is for zero confining press&e, upper plot is for a confining pressure u&r = 10.0.
SC
T 800r &=
2.0
p = 0.5
cd=300
700-
600/ 500-
400-
300."
Fig. 10. Normalized differential stress required to achieve a wing crack length &!G = 2.0 as B function of confining pressure for’%arious main crack angles.
Crack extension under compressive loading
473
to the remote stress field. That is, failure occurs when conditions become critical at the “worst’‘-oriented flaw. First, this approach ignores the distinction between crack initiation and overall failure. Indeed, the present work gives theoretical support to the proposition, long suggested by experiments, that growing wing cracks to lengths at which they can interact and coalesce requires considerably higher differential stresses than those needed solely to initiate them. Secondly, using the “worst’‘-flaw calculation (even accounting for the increase in load with wing crack extension) haphazardly with an initial crack density of unspecified orientation distribution is likely to result in an underestimation of the failure stress. Since cracks of exactly the “worst” orientation are apt to be very far apart, cracks at other angles are forced to participate in the failure process. Such cracks, it has been shown here, may have considerably higher initiation and propagation stresses depending on the amount of friction. flaw relative
Acknowledgements-It&
a pleasure to thank M. F. Ashby and S. D. Cooksley for many useful discussions in connection with compressive fracture. This work was carried out while the author was a National Science Foundation NATO Postdoctoral Fellow at the University of Cambridge, England. Additional support by the Department of Mechanical Engineering, Carnegie-Mellon University is gratefully acknowledged.
REFERENCES [l] M. S. Paterson, Experimental Rock Deformation-The Brittlefieid. Springer-Verlag, New York (1978). [2] S. Nemat-Nasser and H. Hot-ii, Compression-induced nonplanar crack extension with application to splitting, exfoliation, and rockburst. J. Geophys. Res. 87(B8), 6805-6821 (1982). [3] B. Cotterell and J. R. Rice, Slightly curved or kinked cracks. Int. J. Fracture 16, 155-169 (1980). (Received 9 September 1983)
Engirvcring FractureMmhanicsVol. 20,No. 3, pp. 463473,19&o Flinti in UK U.S.A.
CRACK. EXTENSION
UNDER
COMPRESSIVE
LOADING
PAUL S. STEIF Department of Mechanical Engineering, Carnegie-Mellon University, Schenley Park, Pittsburgh, PA 15206, U.S.A. Abatrae-Non-planar extension (kinking) of a crack subjected to remote biaxial compression is studied to gain insight into compressive failure of brittle solids. Approximate expressions are developed for the stress intensity factors at the tips of the resulting wing cracks. Assuming the wing crack propagates at a constant value of KL, one can calculate wing crack extension vs load. The significance of the dependence on various parameters is discussed.
1. INTRODUCTION INTERFBT in the mechanical response of geological materials such as rocks to natural loadings motivates the study of compressive fracture in brittle solids. Based on extensive mechanical testing of rocks, the magnitude of the compressive stress at which overall failure occurs is generally said to exceed, at times greatly, the tensile failure stress. One explanation for this stress differential effect is that, in contrast to the tension case, the initiation of brittle crack growth under compression is not tantamount to overall failure (see Paterson[l]). Wing crack growth, i.e. out-of-plane extension of preexisting cracks, characteristic of compressive loading, occurs stably: for at least some range of wing crack lengths, such growth takes place only with increasing load. Measurements of acoustic emission and elastic wave speeds confirm that considerable micro-cracking occurs prior to final failure. In this paper, the out-of-plane extension of a single crack is studied. A solution to this problem is a first step towards an in-depth understanding of compressive failure, further steps being the problems of crack interaction and linkage. The more relevant three-dimensional problem is bypassed in favor of a more mathematically tractable two-dimensional one. The two-dimensional problem also has the advantage of being directly comparable to experiments on cracked plates, essentially a two-dimensional configuration. We consider an inlinite body deforming in plane strain subjected to a uniform stress state at infinity with principal axes aligned with the coordinate axes x1 - x2. The body contains a line crack of length 2a oriented at an angle 0 with respect to the x1-axis. Assuming the far-field stresses are not equal, the crack faces will slide parallel to themselves, inducing a non-zero Mode II stress intensity factor (SIF) Kn. Because both far-field principal stresses are taken to be compressive, the crack is closed and Ki is zero, i.e. the singular part of the near-tip stress field involves only stresses that are anti-symmetric with respect to the crack plane. Closure of the crack can also introduce frictional resistance to sliding. When the maximum tensile hoop stress at some fixed small radial distance from the crack tip reaches a critical value, wing cracks develop as the initial propagation occurs at an angle a from the original plane of the crack (see Fig. 1). Since the prevailing singular held is pure Mode II, a will be Seen to be approx. 70”, consistent with experiments. This problem has been considered by Nemat-Nasser and Horii [2], who find the stress intensity factors at the wing crack tips via the numerical solution to a singular integral equation. Their results provide a comparison for the equations derived below. In the following section, approximate expressions for the in-plane SIF’s K;kiwand Kg* at the wing crack tip are developed. If one assumes crack propagation occurs at a constant critical SIF, i.e. the material is brittle, the expressions for K;“‘” and K$w allow one to calculate wing crack extension as a function of applied stress. The influence of parameters such as confining pressure, coefficient of friction and initial crack orientation is examined, and some possible implications for failure are cited. 2. ANALYSIS The SIFs at the tips of the wing cracks may be written formally as
464
PAUL S. STEIF
I
02
Fig. 1. Wing crack configuration and definition of coordinate axes.
where K$’ and @“I are the SII?s that woufd prevail at a wing crack were it an isolated crack of the same length in an infinite body, and Ki* and Kiy are just the differences between the actual intensities K;u’” and eg and the contributions Kim’ and ti,r’. In terms of the far-field stresses resolved along the n’ - t’ axes (see Fig. 1), Kiy’ and Kt’ are given by
where 1 is the length of each wing crack. The quantities Kiti and Kip reflect the influence of the main crack on the stress field experienced at the wing tips. Approximate expressions are now developed for kTe and Kip in terms of relevant parameters. Since we are assuming the faces of the main crack remain in contact due to the ~ornpr~~ve loading, the far-field stresses still effectively “see” a wing crack of length i, i.e. the main crack does not “lengthen” the wing crack. But the relative sliding of the faces of the main crack, caused by the far-field loading, serves to wedge open the wing cracks. We take this wedging action, which is a driving force for wing crack growth, to represent the total influence of the main crack on the wings. Let d be the relative sliding across the main crack faces where they meet the wings cracks; d depends on the relative values of 1 and a, as well as on other parameters. If one further assumes that all the influence of the main crack on the wing cracks can be incorporated into the wedging displacement d, then, by dimensional considerations and linearity, the stress intensity factors Kpfi and e must be of the form
GW where E is Young’s modulus and j3i and & are dimensionless functions of Poisson’s ratio and the angle a. We now determine A by considering a straight crack of half-length a + I subjected to far-field shearing (see Fig. 2). Note that this is the special case of wing cracks aligned with the main crack, i.e. a = 0. This elasticity problem is readily solved; one finds that the relative tangential
Crack extension under compressive loading
T
I
t.s?!---
I T
2 0 +-.&-I
Fig. 2. Crack configuration for calculating sliding displacement.
displa~m~t
A, of the crack faces at x = a is given by
A,=4(1;v2)l&iTjGz
(3)
At this point one might be inclined to identify A with A,. If A, were applied at x = a, then, according to equation (2bj, the resulting SIF would be roughly? that prevailing at the tip of a crack of length 2u + 21in a shear field T. We defined Kp and Kifl, however, to correspond to the singular stresses that are otter and abwe the singular stresses that would prevail were the wing crack isolated in an infinite body, Equation (3) implicitly includes the “isolated” contribution to the SIF. Since the wing crack is of length 1, the isolated contribution (in Mode II) is: f$‘=z
al
J 2.
Invoking a relation such as (2b), we say that luiyl corresponds to the application at x = a df a sliding displacement AimI given by
where c, which is independent of a .and 1, is determined below. We retain z to denote the effective shearing stress experienced by the main crack and define A by A SAf-Aiml=
4(1 - 9) TJ~ E
Therefore, by equations (2), @’ and @
- cl.
(4)
are given by
Equations (5) are now shown to give the correct dependence on 1 and a for l/a $- 1 provided c is chosen properly. Consider the wing crack configuration in Fig. 3(a) in which the far-field stress state is one of uniaxial compression parallel to the wings which are taken to be much longer than the main crack. Were it not for the main crack no stress singularity would exist at the wing tips: Pi-’ = @i = 0, and so the contributions KY and Kp are equal to the total wing tip SIFs. For tIf B,, is chosen such that KIK,, is cm-rectfor I/a-+0, then K,, is incorrect by at most a factor ,,h for the entire range of i/a.
466
PAUL S. STEIF
this limit of l/a-,o~, as far as can be seen from the wing tips, the wedging apart of the wings by sliding of ’ the main crack has the same effect as a point displacement has on a crack of length 21 (see Fig. 3b). This configuration has a Mode I SIF at the wing tips of
where & is the effective wedging displacement. Now, as I/a becomes large, the sliding that occurs on the main crack becomes independent of 1. Thus, by dimensional considerations, we have
and consequently, K1 =f(cl, v) JZ JI
for f % 1
where f(a, v) is a dimensionless function of c1and Poisson’s ratio v. We therefore require that the contributions Kid and e, which derive from sliding alone, have the same dependence on 1 and a as in eqn (6) in the limit as Z/a -+ co. This is accomplished by giving c the value c=
4(1 - 9) E ‘*
Expanding eqns (5) for l/u % 1 with c as given above, one can show that the correct dependence ‘on 1 and u results. ,8t and flu are now determined by considering the limit l/a @ 1. Explicit forms for Kj”h and J@‘@ for vanishingly small wing cracks have been developed by Cotterell and Rice[3] using a perturbation expansion for slightly curved cracks. Although only strictly valid for c1+O, the expressions given in [3] are shown to be very accurate over a wide range of angles a. When converted to the present situation and notation, their results are: K,= -i(sinq+sinF)r,/&
Ku=;
( cosqf3cos~
0
t (a)
(b)
Fig. 3(a). Long wing crack limit with @‘I = K#” = 0. Fig. 3(a). Main crack is replaced by an effective wedging &.
) r,/&.
Crack extension under compressive loading
467
In the limit as l-0, the stress intensity factors at the wings are simply given by the parts Kp and e; thus from eqns (5) we have
By choosing 4(1 -vz~~~=~(~os~+3eos%)~
4(1 - vZ)/?* = -i(sinr+sin:)$
we can force our appro~mate expressions for the stress intensity factors to give the same values as calculated by Cotterell and Rice[3] in the limit as t+O. Finally, friction at the sliding main crack is modelled by defining the effective shear stress r experienced by the main crack to be
where (mr and a;,,, are the shearing and normal components of the far-field stress resolved on the main crack axes, and p is the Coulomb friction coefficient. The sign of the term A ,&rmn]is always chosen so as to oppose the applied shearing u,,~. In terms of the applied stresses q and a,, the wing tip SIFs are
a,) cos 28 +gu2-
Kf”= [fb
II
[Jzi?
I$ T
II 1 (6 - a,)sin28+p >J[ z‘z
cos;+3cos$ (
+;(% - a,) cos 28
00
d
-o,)sin(2@+2a)
+;
- Ji]
II
[&X
f(q+aJ
- Ji].
It is assumed that a, < rr, < 0, and thus the wings initiate at an angle tl> 0. The driving shear stress r,, = f (az - a,) sin 26 is negative, and thus the fsictional resistance
is chosen to be positive. The solution is valid only if
is negative, otherwise the main crack is locked and cannot slide.
PAUL S. STEIF
468
Equations (7) for the wing tip SIF’s are compared with the results of Nemat-Nasser and Horii[2] for the case of trl = 0, ,u = 0.3, and 8 = 54”. In Fig. 4, the normalized stress intensity factors are plotted as a function of wing crack angle IX Results from Nemat-Nasser and Horii[2] are reproduced in Fig. 5.t The following symbols are respectively equivalent: 8, 3/, c, ul in [2] and (Y, 90-8, a, cr2 in the present work; thus Figs. 4 and 5 may be directly compared. Though the magnitudes of the SIF’s calculated here differ somewhat from those given in [2], particularly for larger wing cracks, the trends with wing angle a and wing length are very similar. The favorable comparison between Figs. 4 and 5 suggests that the conclusions that are reached below regarding the dependence of wing crack growth on various parameters are at least qualitatively correct. Crack extension vs load is now calculated for a load history that is a two-dimensional analog of the triaxial compression test frequently performed on rock or soil samples. We consider 0, = o, both to decrease from zero (negative stresses denoting compression) until (r, = o2 = - 0,; subsequently u, is held fixed at c, = -u,,, and c2 continues to decrease with c2 = -oO - s. Both a, and s are inherently positive: co is the confining pressure and s is the differential stress.
)(
;w
lw= 0.4 r
0.3 0.2 -
0.10,
alx7r)
-0.1 -0.2 -
Fig. 4. Normalized wing-tip stress intensity factors as a function of wing orientation for various wing crack lengths as given by the present approximate calculation. Parameter values are 0 = 54 and p = 0.3.
K,
y= 0.2-T
Kll
mm
loilrn
y=o.z7r
.4
.4
.2
.2
0
0
-.2
-.2
Fig. 5. Normalized wingtip stress intensity factors as a function of wing orientation for various wing crack lengths as calculated by Nemat-Nasser and Horii[Z] via a singular integral equation. Note: the symbols 6, y, c and CT,,in [2] and the symbols a, 90” - 0, n and a2 in the present work are equivalent. tFigure 5, an improved version of the original tigure appearing in [2], was kindly supplied by Prof. Nemat-Nasser.
449
Crack extension under compressive loading
Crack initiation is taken to occur at the angle a, denoted by CQ,at which the hoop stress o, of the singular field is a maximum for fixed r. With only a Mode II singular field, ol, satisfies
or 4(1, x 70.53”. The angular dependence of J$” and Kp8 for l+O as calculated approximately by Cotterell and Rice[3J derives from the angular dependence of crggand a, at the tip of the main crack. Since CT@ achieves a maximum at TV = cql,so does #Tw (also o, and K;H;j*vanish at u = or,,).Thus, a maximum hoop stress i~tiation criterion is ~n~mount to a criterion based on the rna~rn~ Kfuilrg, within the approximation inherent in the results given in Ref. [3]. With a propagation criterion appropriate to brittle solids, a calculation of wing crack initiation and growth as a function of load can now be based simply on K;*= = K,, where Kit is the fracture toughness. For the loading history prescribed above, one obtains from eqn (7a)
cw=
[-fro-s(1
+cos(2# +2a))]
t J
-A
J[ ;
-~sin28+p[00+~s(l+cos2B)]][JGYl-Ji]
(8)
where A =i(sini+sinq). If the same crack of length 2a was subjected to a tensile stress normal to the crack plane, then, within the context of linear elastic fracture mechanics, the tensile stress or which initiates fracture must satisfy
The body under compression is now assumed to respond in an ideally brittle fashion, that is, wing crack growth occurs whenever the Mode I SIF Ky‘Qis equal to the initial fracture toughness Ki,. This implicitly relates the loading to a given length of wing crack. Combining equations (8) and (9), one finds the differential stress s to be related to the wing crack length I by
s
2+~[A&,&%~)
oT= A[sin 28 - ~(1 + cos 20)][JiTZ
-I- fl] - &I - 11+ COS (3 + 2a)IJL
where L = l/20. A special case of eqn (10) is that of I = 0; the co~es~nding is the stress needed to initiate the wing crack:
$1 <=Afsin28
(10)
differential stress, denoted by s,,
2+2;rAp -p(l
+cos28)]’
According to eqn (lo), which was derived for straight wing cracks and included only the effect of w” on crack growth, wing cracks approach a finite, often very short, length as the applied load becomes unbounded, unless the wing crack has a certain orientation. Wing cracks are exposed in general, however, to a non-zero K$Q. This causes the wing crack to deviate from its initial path of a, z 70” and eventually align itself such that the tangent at the wing tip is normal to the direction
410
PAUL S. STEIF
of minimum remote compression (see Fig. 6). Curving towards the axis of maximum compression is observed experimentally, and we attempt to account for this change in wing crack orientation. Proceeding approximately, we calculate the wedging contribution Kpe on the basis of a straight wing crack of length 1 oriented at an angle ~6, the angle at which the curved wing crack meets the main crack. For calculating the isolated contribution to K;“‘“g,the wing crack is taken to be oriented at an angle a = (x/2) - 0, i.e. normal to the direction of minimum compression. These modifications mean that, in eqn (lo), A is calculated with a = h, and cos (28 + 2a) equals - 1. All results presented below are based on eqn (10) with these modifications. 3. RESULTS AND DISCUSSION In Fig. 7, the normalized initiation stress s,/or is plotted as a function of crack angle 8 for various values of friction coefficient p (a0 = 0). For a given p, there is a crack angle at which the initiation stress is a minimum. As expected, the initiation stress for a particular angle increases with friction coefficient. More significantly, the minimum in the initiation stress versus 8 curves becomes narrower as p increases. That is, while cracks at many angles are likely to initiate at a stress slightly above the minimum initiation stress in the case of p = 0, only cracks in a narrow range of angles can be initiated when p is high. Note the particular tendency for cracks at small angles 8 to have their initiation stress raised by increasing p; such cracks not only have a small resolved shear stress, they also have a large resolved normal stress.
I
c2
Fig. 6. Schematic of wing crack curving with a definition of initial wing angle h.
0’
1
IO
20
30
40
50
60
70
00
908
Fig. 7: Normalized differential stress to initiate wing crack growth as a function of main crack angle for various values of friction coefficient (zero confining pressure).
Crack extension under compnssive loading
471
A variation with 0 which is similar to that displayed in Fig. 7 is shown in Fig, 8 for a variety of wing crack lengths (again a0 = 0). Stability of wing crack growth, i.e. the need to raise the load to extend the wing crack, for zero confining pressure (a, = 0), is attributed to the nature of the wedging contribution: at a constant load e decreases as L increases. Thus an increasing load is required to drive wing crack growth. When the conGning pressure cr,is non-zero, the term Kim’is non-zero: since the remote loading is compressive, wing crack growth is further stabilized. In Fig. 9, the influence of ~1on the relation between load and wing crack length is shown for the case of zero confining pressure and for a large confining pressure. The initiation stress is raised sharply by confining pressure when the friction is high, whereas it is unaffected by confining pressure in the absence of friction. As the wing crack subjected to Corning pressure grows, however, the ratio of driving stress at high friction to that at low friction diminishes. Hence, stability of crack growth is greater the smaller the friction, in the sense that
the equality holding when a0 = 0. A very simple criterion for crack coalescence might be that the wing cracks reach a critical length, related to, say, the initial crack spacing. In Fig. 10, the stress s, needed to achieve an arbitrarily chosen wing crack length Z/&z= 2.0 is shown as a function of confining pressure for various angles 8. If all cracks were initially oriented at the same value of 6, then a Mohr~o~omb failure criterion would emerge: the differential failure stress increases linearly with confining pressure, the rate of increase varying with 8. From eqn (lo), this can be seen to be true irrespective of the choice 1/2a = 2.0. The results shown above serve to identify some of the limitations to very simple approaches to compressive failure of brittle materials. One such approach considers a single elliptical flaw subjected to remote stressing. Calculation of the failure stress involves m~imi~ng the tensile stress with respect to position about the flaw, and then maximizing with respect to the orientation of the
Fig. 8. Norm&&
differential stress to achieve the indicated amounts of wing crack growth as a function of main crack angle (zero confining pressure).
472
PAUL S. STEIF
Q”
0.4
0.8
1.6
t.2
2.0 20
Fig. 9. Normalized differential stress as a function of wing crack length for various values of friction coeflbient. Lower plot is for zero confining press&e, upper plot is for a confining pressure u&r = 10.0.
SC
T 800r &=
2.0
p = 0.5
cd=300
700-
600/ 500-
400-
300."
Fig. 10. Normalized differential stress required to achieve a wing crack length &!G = 2.0 as B function of confining pressure for’%arious main crack angles.
Crack extension under compressive loading
473
to the remote stress field. That is, failure occurs when conditions become critical at the “worst’‘-oriented flaw. First, this approach ignores the distinction between crack initiation and overall failure. Indeed, the present work gives theoretical support to the proposition, long suggested by experiments, that growing wing cracks to lengths at which they can interact and coalesce requires considerably higher differential stresses than those needed solely to initiate them. Secondly, using the “worst’‘-flaw calculation (even accounting for the increase in load with wing crack extension) haphazardly with an initial crack density of unspecified orientation distribution is likely to result in an underestimation of the failure stress. Since cracks of exactly the “worst” orientation are apt to be very far apart, cracks at other angles are forced to participate in the failure process. Such cracks, it has been shown here, may have considerably higher initiation and propagation stresses depending on the amount of friction. flaw relative
Acknowledgements-It&
a pleasure to thank M. F. Ashby and S. D. Cooksley for many useful discussions in connection with compressive fracture. This work was carried out while the author was a National Science Foundation NATO Postdoctoral Fellow at the University of Cambridge, England. Additional support by the Department of Mechanical Engineering, Carnegie-Mellon University is gratefully acknowledged.
REFERENCES [l] M. S. Paterson, Experimental Rock Deformation-The Brittlefieid. Springer-Verlag, New York (1978). [2] S. Nemat-Nasser and H. Hot-ii, Compression-induced nonplanar crack extension with application to splitting, exfoliation, and rockburst. J. Geophys. Res. 87(B8), 6805-6821 (1982). [3] B. Cotterell and J. R. Rice, Slightly curved or kinked cracks. Int. J. Fracture 16, 155-169 (1980). (Received 9 September 1983)