Linear elastic fracture mechanics provides a crack growth criterion that has been ... link" theory (Weibull 1939) for brittle materials; however, the dimensionless .... rate into two components due to opening and sliding and assuming failure.
PROBABILITY OF C R A C K G R O W T H IN POISSON FIELD O F PENNY CRACKS By S. Mesarovic, 1 D. Gasparini, 2 S. Muju, 3 and M. McNelis4 ABSTRACT: An expression is derived for the probability of crack growth in a material with penny cracks of random size and orientation. It is assumed that crack growth occurs when the maximum local Mode I stress intensity factor, K'h exceeds K,c; the criterion is generalized for the three-dimensional problem. The location along the crack front and the magnitude of K', are determined conditional on a penny crack of prescribed size and orientation. The probability distribution function of K'I is then computed using a derived probability density function for the crack size. The probability of crack growth is expressed as a function of four dimensionless parameters: The expected number of cracks in a volume; the expected value of the Mode I stress intensity factor for a crack normal to the direction of stress, divided by Klc; the probability density function of a normalized crack size; and the Poisson ratio. A study that quantifies the effects of the four dimensionless parameters is presented.
INTRODUCTION
Linear elastic fracture mechanics provides a crack growth criterion that has been experimentally verified, at least for brittle materials with prescribed, deterministic crack morphologies and simple, two-dimensional stress conditions. However, which crack growth criterion is most appropriate for general stress states, and how to apply a crack growth criterion to a material with flaws of random type, size, and orientation remain unresolved issues. The latter issue is examined herein. An expression is derived for the probability of crack growth in a material with penny cracks of random size and orientation. The formulation uses the following assumptions: 1. The material is subject to a uniaxial tensile stress, a. 2. The material is linear elastic, brittle, isotropic, and homogeneous with a prescribed, deterministic, critical Mode I stress intensity factor, K,c. 3. A Poisson field of penny cracks of random size and orientation exists in the material. \ is the mean spatial occurrence rate of cracks. 4. \ is such that there is no interaction between cracks. 5. For any point on the crack front, a local (i.e., depending on the plane of crack propagation) Mode I stress intensity factor, K„, can be defined. Crack growth occurs when the maximum (over the entire crack front) local Mode I stress intensity factor, K',, exceeds K/c. Such a failure criterion is a three-dimensional generalization of the maximum hoop-stress criterion. 'Grad. Student, Dept. of Civ. Engrg., Case Western Reserve Univ., Cleveland, OH 44106. 2 Assoc. Prof., Dept. of Civ. Engrg., Case Western Reserve Univ., Cleveland, OH. 3 Grad. Student, Dept. of Civ. Engrg., Case Western Reserve Univ., Cleveland, OH. "Res. Engr., NASA, Lewis Res. Ctr., Cleveland, OH 44135. Note. Discussion open until October 1, 1992. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on December 3, 1990. This paper is part of the Journal of Engineering Mechanics, Vol. 118, No. 5, May, 1992. ©ASCE, ISSN 0733-9399/92/0005-0961/$1.00 + $.15 per page. Paper No. 1005. 961 Downloaded 30 Oct 2010 to 134.121.161.15. Redistribution subject to ASCE license or copyright. Visit
To apply the aforementioned crack growth criterion, the location along the crack front and the magnitude of K', are determined for a penny crack of prescribed size and orientation. The probability distribution function of K', is then computed using a derived probability density function of the crack direction angle and an assumed density function for the crack size. The probability distribution function of K', and the Poisson field assumption for the occurrence of cracks are sufficient for computing the probability of crack growth as a function of four dimensionless parameters. The form of the solution for the probability of crack growth is the same as that of the "weak link" theory (Weibull 1939) for brittle materials; however, the dimensionless parameters show how physical variables affect the probability of crack growth. A study that shows the effects of each parameter is presented. Extension of the formulation to elliptical cracks and to more general stress states is discussed. CRACK GROWTH CRITERIA AND STATISTICAL STRENGTH THEORIES
Several crack growth criteria have been proposed for brittle materials in the planar mixed-mode condition shown in Fig. 1. Thoughtful comparisons of alternate criteria are given by Ghosn (1986) and Maccagno and Knott (1989). The three basic criteria are: 1. Maximum hoop (or tangential) stress criterion that can be stated equiv-
FIG. 1. Crack in Mixed Mode Condition 962 Downloaded 30 Oct 2010 to 134.121.161.15. Redistribution subject to ASCE license or copyright. Visit
alently in terms of the nonsingular part of the stress—the maximum local Mode I stress intensity factor, K',. 2. Minimum strain energy density criterion. 3. Maximum energy release rate of crack extension force criterion. The first criterion may have different forms depending on whether one actually solves the kinked crack problem or simply assumes that, initially, crack growth is determined by the initial stress distribution. Although it is more accurate to solve the kinked crack problem, the criterion becomes difficult to apply. The second approach, as proposed by Erdogan and Sih (1963), is simpler to apply. It states that a crack will propagate in the direction perpendicular to that of the maximum local Mode I stress intensity factor when K!, > KIC; all quantities are determined from analyzing the initial crack configuration. The criterion assumes that brittle materials fail in tension in the opening mode. An implication of the criterion is that for pure Mode II, a crack will propagate at an angle of about - 70°. Another implication is that the smallest critical applied stress does not occur when a crack is perpendicular to the direction of the stress (6 = 0°) but rather for a crack at an angle 9 between 20° and 30°. Experiments on glass by Panasyuk et al. (1965), on plexiglass by Erdogan and Sih (1963), and on polymethylmethacrylate (PMMA) by Williams and Ewing (1972) and Maccagno and Knott (1989) largely support the criterion that crack growth occurs when K1, > Klc. The experiments show that the minimum critical applied load occurs for cracks with 0 between 20°-30°. Data on the crack extension angle show considerable scatter. Williams and Ewing (1972), who used specimens of the type shown in Fig. 1, observed that the direction of crack growth did not approach -70° for pure Mode II (0 —> 90°). A possible explanation may be that as 8 —> 90°, not only does K, —> 0 and KH —» 0 but also K,IKU —» 0, thus suggesting that for such crack orientations the behavior should approach that of pure Mode II. But 9 = 90° is a nonsingular case, meaning no stress concentration occurs, and classical stress analysis is sufficient. What actually happens is that the effect of a crack diminishes as 8 —> 90°. It is then logical to expect that the relative importance of microcracks, especially those close to the crack tip as shown in Fig. 2, will increase. Although much smaller, such microcracks may produce the singularity the relative importance of which increases as 9 —* 90°, since the singularity produced by the principal crack becomes insignificant. This explanation is
t Microcrack Principal crack
FIG. 2. Potentially Significant Microcracks for 9 -» 90°
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supported by experimental results in the region 0 -» 90° that show more scatter than for other crack orientations, perhaps because of randomly sized and orientated microcracks. Maccagno and Knott (1989), who used symmetric and antisymmetric four-point bending mechanisms, did observe that crack propagation occurred at an equivalent angle of -70° as Mode II became dominant. Thus the Erdogan and Sih (1963) criterion predicts observed phenomena well. Motivated partially by the data of Williams and Ewing (1972) and partially by the fact that the first criterion includes only one stress, Sih (1974) suggested that a crack will grow in the direction of the minimum strain-energy density factor (the nonsingular part of the strain-energy density) when that quantity reaches a critical value for the material. Sih's (1974) criterion, however, does not predict the experimental observation that the smallest critical applied load occurs for 8 between 20°-30°. The third criterion is perhaps the most physical one. It states that a crack will grow when the energy release rate reaches a critical value for the material. The direction of growth is the one for which the energy release rate is maximized. To apply the criterion correctly, the kinked crack problem must be solved. However, as pointed out by Ghosn (1986), if it is assumed that the asymptotic displacement field for the straight crack is a good approximation for the kinked crack, the criterion becomes much easier to apply. Moreover, Ghosn (1986) shows that by separating the energy release rate into two components due to opening and sliding and assuming failure in the opening mode, the criterion is equivalent to the maximum hoop stress criterion of Erdogan and Sih (1963). Motivated by the equivalence of the J integral (Rice 1968) to the energy release rate for self-similar crack growth, attempts have been made to define a crack growth criterion in terms of the / integral. However, a proper formulation for non-self-similar crack growth is still to be defined. It has long been understood that fracture-mechanics-based crack growth criteria must be used in a statistical or probabilistic manner for brittle materials with flaws of random type, size, and orientation. Lamon (1988) has recently reviewed the development of statistical strength theories. Such theories are based on the work of Freudenthal (1969) and Weibull (1939) who considered a material as a "series" of "weak-link" arrangement of volume elements having independent and identically distributed strengths. Explicit crack growth criteria were incorporated later. Batdorf and Crose (1974) initially proposed a critical normal stress criterion and postulated that cracking is shear insensitive. Later, Batdorf and Heinisch (1978) included that influence of shear through the concept of an equivalent normal stress. Evans (1978) proposed a maximum, coplanar energy-release rate criterion. Such a criterion assumes in-plane crack growth, which generally is not observed in experiments. Lamon and Evans (1983) also proposed a maximum, total energy-release rate criterion, which predicts that for a crack parallel to the stress direction, the growth occurs in the same direction. In general, for 0 > 45°, this criterion has low correlation with experiments (Hellen and Blackburn 1975). Another direction of research in probabilistic fracture is based on lattice models and computer simulation of random defects and the fracture process (Duxbury and Kim 1990; Srolovitz and Beale 1988; and Bazant et al. 1990). Bazant et al. (1990) has added to the physical base of such models by using an aggregate size distribution to derive an equivalent, nonhomogeneous spring network. The present work differs from such simulation approaches. 964 Downloaded 30 Oct 2010 to 134.121.161.15. Redistribution subject to ASCE license or copyright. Visit
Herein the physical parameters of crack size distribution, orientation distribution, mean occurrence rate of cracks, and the Poisson ratio are used to derive an expression for the probability of crack growth. CRACK GROWTH CRITERION FOR THREE-DIMENSIONAL PROBLEM
The criterion presented and used herein is a generalization of the Erdogan and Sih (1963) criterion for in-plane problems, which states that a crack will grow in the direction normal to the maximum hoop stress. Since the stress field has a singularity at the crack tip, the above criterion actually refers to the regular part of stress; i.e., the stress intensity factor. For the threedimensional case, the stresses are first expressed in a local coordinate system defined by direction s and a plane with normal n K„
(1)
V2TT£
K, cr,„ =
(2)
V2i£
K, (3) V2rrli where K,„ Ks, and K, = local stress intensity factors for normal, sliding, and tearing modes. Coordinate system xu x2, and x3 is defined so that x2 is tangent to the crack front at point A. The assumed criterion for crack growth is: The crack will grow in the plane at which the corresponding local normal mode stress intensity factor is maximized, at the value of the load that produces max K„ = K,c. The chosen criterion is relatively simple and physically meaningful. It agrees with two-dimensional experimental results (2, 3, 5) and it can be directly generalized to three-dimensional problems as follows. This criterion is for crack growth at a particular point on the crack front. At which point the growth will start is a function of a particular loading and crack morphology. Herein, the case of a penny-shape crack in a uniaxial stress field is analyzed. The stress direction is at an angle 0 with the normal to the crack plane as shown in Fig. 3. The local stress field at point A in the coordinate system iji&fe (see Fig. 4) is given by the well-known formulas cos — 1 + sin2 — K,
V2i£
a sin — cos2 2 a cosJ
sinf (1
3 sin2
_
2
Ku cos — I 1 — 3 sin2 V2lr£ 2 \ 2
(4)
3 sin — cos2 — 2 2 965 Downloaded 30 Oct 2010 to 134.121.161.15. Redistribution subject to ASCE license or copyright. Visit
N@rmal eraeh
to
FIG. 3. Direction Angles w and 6 and Local Axes xu x2, x3
Crack treat
FIG. 4. Local Coordinate Axes at Point on Crack Front
I 0"i2 I _ Km ,0-32] V2ir€i
. on sin — 2 a ,COS 2j
(5)
CT22 = v ( < T n + CJ33)
(6)
The stress intensity factors as a function of angles 9 and co are 966 Downloaded 30 Oct 2010 to 134.121.161.15. Redistribution subject to ASCE license or copyright. Visit
fCOS20
[K, )
2(T
r" [Km)
2 sin6 cos6 costo
(7) v
2 sin8 cosS sinto
v The local n, s, t system is defined by its unit vectors expressed in the ^ £2, £3 system n = (0; sin(3; cosp) t = (0; cos(3; sin(3) s = (1; 0; 0)
(8)
By substituting (7) into (4), (5), and (6) and performing a coordinate transformation, stresses in the n, s, t system are obtained in the form of (1), (2), and (3) with K„, K„ Ks now as functions of 0, to, a, and p. At a particular point A, defined by angle w, the maximum K„ is defined by K,
0
K
, v)
(li)
or, dividing by 2o-V7rfl/-rr (the value of K', for 0 = 0°)
r^T-W.,)
(12)
The deterministic function I|J(0, v) is shown in Fig. 5 for three values of the Poisson ratio. It was found numerically that the maximum K„ occurs at to = p = 0 for all 0. That is, the maximum occurs when K,H = 0. Therefore, the analysis essentially reduces to the two dimensional case in the sense that 967 Downloaded 30 Oct 2010 to 134.121.161.15. Redistribution subject to ASCE license or copyright. Visit
THETA (Degrees)
FIG. 5. Normalized Maximum Local Mode I Stress Intensity Factor versus 6
only K, and Kn are involved. However, for the penny crack, the magnitude of K',(Q) and the angle of crack propagation are functions of v and are slightly different from those of plane conditions. The largest value of K',(Q) does not occur when 8 = 0°. Fig. 5 represents deterministic K',(Q) values conditional on 8 and a. Since 6 and a are both random variables, K', is also a random variable, defined by a probability distribution function that may be determined from probability density functions of 8 and a. While the density function of a must either be determined experimentally or assumed, the orientation distribution can be derived as follows. Probability Distribution Function of Direction Angle, 8 The angle 8 between the normal to a penny crack of radius a and the direction of applied stress is a random variable. Fig. 6 indicates the geometry from which the distribution function of 0 may be inferred. The basic realization is that the normal may intersect the sphere with equal probability at any two diametrically opposed points on the sphere's surface (i.e., it is uniformly distributed on the sphere). Therefore, the probability distribution function of 8, F@(Q), may be expressed as the ratio of the surface area of two caps, as shown in Fig. 6, to the total surface area of the sphere. That is Fe(8) = P [ Q < 8 ] = ^ - a , C O S 2 9 ) ( 2 ^
(13)
F 0 (e) = l - cose
(14)
Differentiating, the probability density function of 8, / 0 (8), is ZeO) = ^ f
1
= sine
0(6, v)
(21)
Therefore the probability distribution function of ^ can be easily calculated using the integral of (21) once the distribution function of a is estimated. The probability that K'i s K,c for any one crack is, simply vK„ sinBde . . . (22) 2a£(Va) 4*(0, v)
P{K\ < KIC] = FK:(KIC)
Note that P[K', s £ / c ] is a function of iji(8, v), the probability distribution function of a (or a'), and the dimensionless parameters v and 2a£(Va)/ /C/c Vrr . The latter parameter is the expected value of the Mode I stress intensity factor for 9 = 0°, divided by KJC; i.e., 2dE(Va)/Klc VTT = E[K',(Q = FWKIC. Poisson Field of Penny Cracks It is assumed that the random-sized and random oriented penny cracks occur as a homogeneous spatial Poisson process with mean occurrence rate, \ . The number of occurrences of cracks in a volume V, denoted by N(V), is a random variable with a Poisson probability mass function N(V)
(0 =
(\V)'e-
(23)
For each crack occurrence there is a corresponding /^occurrence. Therefore occurrences of K1, is also a spatial Poisson process with mean occurrence rate \. Now define (24)
K,iv — max K'i 0
