- Email: [email protected]
Flaw Characterization
7.06 Flaw Characterization T. L. ANDERSON Structural ReliabilityTechnology, Boulder, CO, USA 7.06.1 INTRODUCTION
227
7.06.2 FLAW SHAPE AND DIMENSIONS
227
7.06.2.1 7.06.2.2 7.06.2.3 7.06.2.4 7.06.2.5
227 229 229 230 230
Crack Shape Crack Dimensions Solutions for Idealized Crack Shapes Complex Flaw Shapes Flaw Tip Radius
7.06.3 FLAW ORIENTATION
230 230 231 231 234 235
7.06.3.1 Simple Projection Method 7.06.3.2 Maximum Energy Release Rate Model 7.06.3.2.1 Theoretical background 7.06.3.2.2 Defining an equivalent mode I crack 7.06.3.2.3 Comparison with the projection method 7.06.4 RECATEGORIZATION
236 236 237
7.06.4.1 Why Recategorize? 7.06.4.2 Standard Recategorization Rules 7.06.5 INTERACTION OF MULTIPLE FLAWS
238 239 239 242 242 242
7.06.5.1 Principles of Interaction 7.06.5.1.1 Coplanar cracks 7.06.5.1.2 Parallel cracks 7.06.5.1.3 Elastic versus elastic–plastic deformation 7.06.5.2 Interaction Rules in Standard Assessment Documents 7.06.6 SUMMARY
242
7.06.7 REFERENCES
242
7.06.1
The engineer performing such an assessment must always be mindful of these and other uncertainties in the input data. Refer to Chapter 7.15 for a discussion of sensitivity and reliability analysis.
INTRODUCTION
Key inputs to a structural integrity assessment include the size, location, and orientation of the flaw(s) being assessed. Nondestructive examination (NDE) to detect and size cracks is a crucial first step in obtaining such information. The present chapter focuses on the proper interpretation and application of NDE data to the structural integrity assessment. It should be noted that uncertainties in flaw detection and sizing often overshadow the sophisticated analytical models available for assessing the structural significance of cracks.
7.06.2 7.06.2.1
FLAW SHAPE AND DIMENSIONS Crack Shape
Most cracks and planar flaws fall into one of five categories: (i) Surface-breaking flaws. (ii) Through-wall flaws. 227
228
Flaw Characterization
(iii) Buried (subsurface) flaws. (iv) Corner flaws. (v) Edge flaws. Figure 1 illustrates these flaw shapes. Note that the actual flaw may have an irregular shape, but it is typically idealized with a simple shape. The shapes of surface, buried, and corner cracks are usually idealized as semielliptical, elliptical, and quarter-elliptical, respectively. Through-wall and edge cracks are
normally assumed to have a rectangular shape, at least in flat plates. The idealization of crack shape is necessary for pragmatic reasons. Most conventional NDE techniques are incapable of characterizing a detailed crack-front profile. Even if the true crack-front shape were known, standard fracture mechanics solutions are typically available only for simple shapes, as discussed in Section 7.06.2.3 below.
Figure 1 Typical flaw shapes considered in structural integrity assessments.
Flaw Shape and Dimensions 7.06.2.2
Crack Dimensions
The dimensions of an idealized crack are defined by the maximum extent of the actual flaw. Stated another way, the idealized crack shape completely inscribes the actual shape, as Figure 2 illustrates. In the case of part-through cracks, the key dimensions, defined in Figure 1, are the length and depth. The buried crack has an additional dimension, d, to indicate the distance from the center of the flaw to the nearest free surface. Note that some of the dimensions in Figure 1 use symbols multiplied by a factor of 2. For example, the elliptical buried crack is 2c long by 2a deep. This convention has been adopted in much of the published literature to maintain consistency between crack dimensions for full-, half-, and quarter-elliptical shapes. As a rule of thumb, a ‘‘2’’ in front of a crack dimension indicates that there are two crack tips in that direction. For example, a surface crack, with dimensions 2c by a, has two crack tips in the length direction (along the free surface) and one tip in the depth direction. In addition to dimensions, the relative position along the crack front can be important. In the case of surface, buried and corner cracks, the crack front position is characterized by the elliptic angle, f, which is defined in Figure 3(a). For through-wall and edge cracks, fracture mechanics parameters such as the stress intensity factor typically vary along the crack front, so a local crack front coordinate (z) can be defined, as Figure 3(b) illustrates. However, many of the standard solutions for through-wall and edge cracks are based on
Figure 2 The idealized flaw shape should completely inscribe the actual flaw.
229
two-dimensional models, and consequently do not account for through-thickness variations in relevant parameters. 7.06.2.3
Solutions for Idealized Crack Shapes
As stated above, most published fracture mechanics solutions assume an idealized crack shape. For example, expressions for the mode I stress intensity solutions for surface, buried, and corner cracks assume an elliptical shape and typically have the following form: rffiffiffiffiffiffi pa KI ¼ Gs Q
ð1Þ
where G is a dimensionless geometrical factor, s is a characteristic stress, and Q is a flaw shape parameter: Q ¼ 1 þ 1:464
a1:65 c
for arc
ð2Þ
The geometrical factor, G, is a function of crack shape (surface, buried, or corner), crack dimensions, crack front location, structural dimensions, and loading (e.g., membrane, bending, polynomial, etc.). The form of Equation (1) stems from the classical solution for an elliptical buried crack in an infinite body subject to a uniform tensile stress normal to the crack plane. Equation (2) is actually a curve-fit approximation of the square root of an elliptic integral of the second kind: 2
Q ¼
Z 0
p=2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 a2 2 1 sin ðfÞ df c2
ð3Þ
Figure 3 Definition of crack front location. (a) Parametric angle on elliptical cracks and (b) crack front position on a through-wall crack.
230
Flaw Characterization
Equation (3) does not have a known closedform solution, but Equation (2) provides a good approximation. Equations (1)–(3) and the accompanying discussion serve to illustrate the point that standard fracture mechanics solutions are inextricably linked to the assumed flaw shape. Consequently, these solutions are not rigorously correct for real flaws that deviate from the ideal. As long as the departure from the ideal crack shape is not too severe, the standard solutions should be reasonable engineering approximations. In principle, standardized structural integrity assessment procedures do not forbid the use of more rigorous solutions for the precise flaw geometry of interest. However, obtaining adequate inspection data, let alone the solution itself, is usually not practical. 7.06.2.4
Complex Flaw Shapes
The published literature contains a limited amount of information on complex crack shapes. For example, Bezensek and Hancock (2001) consider two coalescing surface cracks and provide experimental and numerical analysis. Annex F of BS 7910 (2000) contains a simplified procedure for handling complex cracks. Such cracks are treated by breaking them into simpler shapes for which standard solutions exist. 7.06.2.5
Flaw Tip Radius
Assessments of crack-like flaws normally do not consider out-of-plane flaw dimensions or tip radius. An implicit assumption of classical fracture mechanics theory is that the flaws in question are planar (as opposed to volumetric) with sharp tips. In reality, many flaws that are treated as sharp cracks have a finite tip radius. Lack-of-fusion defects in welds are a prime example of such flaws. Although the sharpcrack assumption is very pessimistic in some cases, standardized structural integrity procedures do not provide a rational means to quantify the effect of tip radius on the risk of brittle fracture. Suitable models may eventually find their way into standard engineering practice, but are not available as of early 2000s. Even with such models, measuring the tip radius of relatively sharp planar flaws may be problematic. Fatigue life prediction is one area where tip radius has been explicitly considered (Suresh, 1991). For relatively modest notches, various stress concentration factors such as Neuber’s factor, have been routinely applied. For more
severe notch-like features, a pseudo-fracture mechanics model has been moderately successful. In this latter case, the cycles to initiate a sharp crack at the tip of a notch with tip radius r has pffiffiffi been correlated with the parameter DK= r; where DK is the cyclic stress intensity factor inferred by treating the notch-like feature as a crack. 7.06.3
FLAW ORIENTATION
Most standardized structural integrity procedures are based on mode I fracture mechanics, where a principal stress acts normal to the crack plane such that the crack faces open. For cracks that are not aligned with the principal stress axes, one must either perform a mixed-mode analysis or apply a suitable approximation to reduce the problem to pure mode I. Codified structural integrity procedures typically choose the latter approach, although there is advice on mixed mode loading in R6 (British Energy, 2001) as discussed in Chapter 7.03. Mode I approximations for off-axis cracks can entail either a simple projection of the crack onto a principal plane or a more sophisticated model based on mixed-mode fracture mechanics. The projection method is described below, followed by an example of a more sophisticated approach. Several recognized flaw assessment procedures, including BS 7910 (2000) use the projection method. API 579 (2000) defines an equivalent mode I crack based on a more rigorous model described in Section 7.06.3.2 below. The latest release of the R6 procedure (British Energy, 2001) adopts the API method for characterizing off-axis flaws. 7.06.3.1
Simple Projection Method
Figure 4 illustrates the projection method. An equivalent mode I crack is defined by projecting the flaw onto a principal plane. For biaxial loading, there are two possible choices for the projection. In such cases the more conservative option should be taken, i.e., ( c¼
c1 ; c2 ;
pffiffiffiffiffi pffiffiffiffiffi s1 c1 Zs2 c2 pffiffiffiffiffi pffiffiffiffiffi s1 c1 os2 c2
ð4Þ
The symbols in Equation (4) are defined in Figure 4. Although the projection method has been used in various well-respected flaw evaluation codes over the years, it does not have a sound theoretical basis. It can actually be nonconservative in some instances, as discussed below.
Flaw Orientation 7.06.3.2
Maximum Energy Release Rate Model
Figure 5 shows a plate with a crack that is oriented at an angle b from the plane normal to the applied stress. If this crack propagates by brittle fracture or fatigue, it will tend to align itself perpendicular to the applied stress. That is, an initially mixed-mode crack will tend to become a mode I crack when it propagates, assuming the material is relatively homoge-
231
neous. If there are no planes of weakness in the material, the propagating crack will follow the path where the driving force is highest, which turns out to be approximately normal to the maximum principal stress in most cases. The preferred initial direction of propagation for a mixed-mode crack can be determined by evaluating the local energy release rate as a function of propagation direction. The critical conditions for propagation can then be determined by comparing the maximum energy release rate to a critical value for the material. Finally, an equivalent mode I crack can be defined that will propagate given the same principal stresses as in the mixed-mode case. The details of the energy release model are outlined below. This is followed by a derivation of the equivalent mode I crack. The analysis that follows was first presented in chapter 2 of Fracture Mechanics: Fundamentals and Applications (Anderson, 1995) and stems from the work of Erdogan and Sih (1963), Williams and Ewing (1972), and Cottrell and Rice (1980). 7.06.3.2.1
Theoretical background
Figure 6 illustrates the three ways in which a cracked body can be loaded. Mode I loading results from stresses normal to the crack plane and tends to open the crack, while modes II and III correspond to in-plane and out-ofplane shear, respectively. The stress fields near the crack tip, in terms of polar coordinates, are given as follows: Mode I: Figure 4 The simple projection method for cracks not oriented on a principal plane.
KI 5 y 1 3y p ffiffiffiffiffiffiffi cos cos srr ¼ 2 4 2 2pr 4
Figure 5 Propagation of a crack oriented at an angle b from the applied stress.
ð5aÞ
232
Flaw Characterization
Figure 6 Three modes of loading a crack. KI 3 y 1 3y þ cos syy ¼ pffiffiffiffiffiffiffi cos 4 2 4 2 2pr
ð5bÞ
KI 1 y 1 3y þ sin try ¼ pffiffiffiffiffiffiffi sin 4 2 4 2 2pr
ð5cÞ
Mode II: KII 5 y 3 3y p ffiffiffiffiffiffiffi þ sin srr ¼ sin 2 4 2 2pr 4
ð6aÞ
KII 3 y 3 3y syy pffiffiffiffiffiffiffi sin sin 2 4 2 2pr 4
ð6bÞ
KII 1 y 3 3y try ¼ pffiffiffiffiffiffiffi cos þ cos 2 4 2 2pr 4
ð6cÞ
Figure 7 Infinitesimal kink at the tip of a crack.
Mode III: KIII y trz ¼ pffiffiffiffiffiffiffi sin 2 2pr KIII y tyz ¼ pffiffiffiffiffiffiffi cos 2 2pr
ð7aÞ
ð7bÞ
When a cracked body is subject to a system of applied forces, one, two, or all three modes of loading may be present. The crack will propagate along the most favorable path, which depends on the driving force relative to the material resistance. For a perfectly isotropic and homogeneous material, the optimal crack propagation direction corresponds to the direction where the driving force is a maximum. Consider a crack with an infinitesimal kink at the tip, as Figure 7 illustrates. This kink represents the initial propagation. The local stress intensity factors at the tip of this kink differ from the nominal K values of the main crack. If we define a local x–y coordinate
system at the tip of the kink and assume that Equations (5)–(7) define the local stress fields, the local mode I, II, and III stress intensity factors at the tip are obtained by summing the appropriate stress components at a: pffiffiffiffiffiffiffi kI ðaÞ ¼ syy 2pr ¼ C11 KI þ C12 KII
ð8aÞ
pffiffiffiffiffiffiffi kII ðaÞ ¼ txy 2pr ¼ C21 KI þ C22 KII
ð8bÞ
kIII ðaÞ ¼ C33 KIII
ð8cÞ
where kI and kII are the local stress intensity factors at the tip of the kink and KI and KII are the stress intensity factors for the main crack. The coefficients Cij are given by C11
a 1 3 3a ¼ cos þ cos 4 2 4 2
ð9aÞ
3 a 3a sin þ sin 4 2 2
ð9bÞ
C12 ¼
C21 ¼
1 a 3a sin þ sin 4 2 2
ð9cÞ
Flaw Orientation C22
a 3 1 3a þ cos ¼ cos 4 2 4 2 C33 ¼ cos
a 2
ð9dÞ
ð9eÞ
The energy release rate for the kinked crack in plane strain is given by GðaÞ ¼
1 n2 2 k2 ðaÞ 2 kI ðaÞ þ kII ðaÞ þ III 1n E
ð10Þ
Consider a case where there is no out-of-plane shear, such that KIII ¼ kIII(a) ¼ 0. Figure 8 is a plot of G(a) normalized by G(a ¼ 0) for various ratios of KII/KI. It turns out that the peak in G(a) at each KII/KI ratio corresponds to the point where kI exhibits a maximum and kII ¼ 0. Thus, the maximum energy release rate is given by Gmax ¼
ð1 n2 ÞkI2 ða Þ E
ð11Þ
where a* is the angle at which both G and kI exhibit a maximum and kII ¼ 0. Crack growth in a homogeneous material should initiate along a*. The configuration in Figure 5 involves mode I and mode II loading. The global stress intensity factors are given by pffiffiffiffiffiffiffi KI ¼ s cos2 ðbÞ pc0
ð12aÞ
pffiffiffiffiffiffiffi KII ¼ s sinðbÞ cosðbÞ pc0
ð12bÞ
Substituting the above expressions into Equation (8a) gives the following relationship for the local mode I stress intensity factor for a
233
kinked crack: pffiffiffiffiffiffiffi kI ðaÞ ¼ s pc0 ½cos2 ðbÞC11 þ sinðbÞ cosðbÞC12
ð13Þ
For a given crack orientation angle (b), Equation (13) can be solved for the optimum kink angle, a*, at which kI reaches a maximum. Figure 9 is a plot of the predicted propagation direction as a function of crack orientation. The diagonal line represents propagation normal to the applied stress. The energy release rate model predicts a propagation direction that differs slightly from the principal stress plane. Note, however, that a* is the initial propagation angle, and the crack will eventually align with the principal plane after propagating a finite distance. That is, KII will eventually be zero with continued crack propagation. The plate in Figure 5 can be generalized to biaxial loading, as Figure 10 illustrates. The mode I and II stress intensity factors for this configuration are given by pffiffiffiffiffiffiffi KI ¼ s1 pc0 ðcos2 b þ B sin2 bÞ
ð14aÞ
pffiffiffiffiffiffiffi KII ¼ s1 pc0 ðsin b cos bÞð1 BÞ
ð14bÞ
where B is the biaxiality ratio, defined as B¼
s2 s1
ð15Þ
The local mode I stress intensity for the biaxial case is as follows: pffiffiffiffiffiffiffi kI ðaÞ ¼ s1 pc0 f½cos2 ðbÞ þ B sin2 ðbÞC11 þ sinðbÞ cosðbÞð1 BÞC12 g
Figure 8 Local energy release rate as a function of kink angle.
ð16Þ
234
Flaw Characterization
Figure 9 Predicted propagation angle for the configuration in Figure 5.
into the above expression: pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi s1 pceff ¼ s1 pc0 f½cos2 ðbÞ þ B sin2 ðbÞC11 þ sinðbÞ cosðbÞð1 BÞC12 g
ð18Þ
Solving for ceff gives ceff ¼ f½cos2 ðbÞ þ B sin2 ðbÞC11 ða Þ c0 þ sinðbÞ cosðbÞð1 BÞC12 ða Þg2
Figure 10 Angled crack subject to biaxial loading.
Figure 12 is a plot of Equation (19) as a function of crack angle and biaxiality ratio. This expression assumes that ceff is defined on the s1 plane. An equivalent crack on the s2 plane can be computed as follows: ceffðs2 planeÞ ¼
Figure 11 is a plot of predicted propagation angle as a function of crack orientation and biaxiality ratio. 7.06.3.2.2
Defining an equivalent mode I crack
Let us now define an effective crack size, ceff, that when oriented on a principal plane, gives the same fracture driving force as the angled crack, i.e.,
ð19Þ
ceffðs1 planeÞ B2
ð20Þ
provided 0oBr1. Applying Equation (19) is cumbersome because it requires solving for a* and evaluating the functions C11.and C12. The following simplified expression provides a good approximation of Equation (19):
ð17Þ
ceff ¼ cos2 ðbÞ þ 0:5ð1 BÞ sinðbÞ cosðbÞ c0 þ B2 sin2 ðbÞ ð21Þ
For the biaxial loading case in Figure 10, ceff can be defined by substituting Equation (16)
for flaws projected on the maximum principal stress (s1) plane. For cracks on the s2 plane,
KI ðceff Þ ¼ kI ða ; b; c0 Þ
Flaw Orientation
235
Figure 11 Effect of biaxiality ratio on propagation angle.
Figure 12
Equivalent mode I crack length as a function of orientation and biaxiality ratio.
applying Equation (20) gives ceffðs2 planeÞ cos2 ðbÞ þ 0:5ð1 BÞ sinðbÞ cosðbÞ ¼ B2 c0 2 þ sin ðbÞ ð22Þ
Figures 13 and 14 compare Equation (21) with the more complex expression for B ¼ 0 and B ¼ 0.5, respectively. API 579 (2000) uses Equations (21) and (22) to define equivalent mode I cracks for off-axis flaws.
7.06.3.2.3
Comparison with the projection method
The simple projection method described above implies the following equivalent mode I crack: cproj ¼ cosðbÞ c0
ð23Þ
Figure 15 compares this simple method with the energy release rate model for uniaxial loading. Note that the projection method can
236
Flaw Characterization
Figure 13 Comparison of equivalent mode I crack lengths for uniaxial loading using the energy release rate model and the simple approximation.
Figure 14 Comparison of equivalent mode I crack lengths for B ¼ 0.5 using the energy release rate model and the simple approximation.
be nonconservative by as much as 20%. Moreover, the projection method does not account for biaxial loading. 7.06.4
free surface might be recategorized as a surface crack, and a deep surface crack may be recategorized as a through-wall crack, as Figure 16 illustrates.
RECATEGORIZATION
When a part-through flaw is close to a free surface, it may be advantageous to recategorize the flaw by assuming the remaining ligament is not present. For example, a buried crack near a
7.06.4.1
Why Recategorize?
In many cases, a flaw whose tip is close to a free surface may be deemed unacceptable by a particular flaw assessment method, but the
Recategorization
237
Figure 15 Comparison of equivalent mode I crack length estimates from the energy release rate model and the simple projection method.
recategorized flaw (which has larger dimensions than the actual flaw) may be acceptable according to the same assessment method. This apparent anomaly is due in part to peculiarities in standardized flaw assessment procedures. For example, in procedures based on the failure assessment diagram (FAD), the reference stress solution which defines the horizontal axis of the FAD (see Chapter 7.03 for a description of FAD methods) is often based on a local ligament yielding criterion. Such expressions tend to be very conservative, particularly when the ligament is small relative to section thickness. When the flaw is recategorized, a different, less conservative reference stress solution may apply. The larger recategorized flaw will tend to have a slightly higher toughness ratio (Kr) on the FAD, but the load ratio (Lr) may be significant lower. The net result is that the recategorized flaw may fall inside the FAD whereas the original flaw was outside the FAD. Idiosyncrasies and conservatisms of standard assessment methods notwithstanding, recategorization can have a physical basis. When plasticity is highly concentrated in the remaining ligament, it can fail, resulting in the crack breaking through to the free surface. Ligament rupture is typically ductile, even if the material is below the upper shelf of toughness. A small ligament has low constraint (stress triaxiality), which tends to result in a ductile fracture mode. A shear lip B451 from the applied stress is often seen
when a ligament breaks through to the free surface. In principle, one might argue that ligament yielding criteria, which often lead to the need for recategorization, are physically based. That is, when a flaw with a small remaining ligament is deemed unacceptable, it is because the assessment is predicting ligament failure. In practice, however, simple ligament yielding models tend to be very conservative when used to predict local failure. A more rigorous (but more complicated) approach to predicting ligament failure would be to perform an elastic–plastic stress analysis on the flaw in question and compare the applied J-integral to the fracture toughness. 7.06.4.2
Standard Recategorization Rules
Most recognized structural integrity procedures do not require recategorization. Rather, they offer it as an option when a given flaw does not pass the initial assessment. In addition to removing the ligament, the assumed flaw length should be increased to account for possible growth in this direction if the ligament were to fail. Figure 16 shows typical recommendations for the dimensions of recategorized flaws when ductile ligament failure is expected. If the ligament failure is potentially brittle, a dynamic or arrest toughness should be used to assess the recategorized flaw dimensions in order to account for the effects of a rapidly running crack tip. Also, the effect of inertia of a
238
Flaw Characterization
Figure 16 Recategorization procedure. (a) Buried flaw recategorized as a surface flaw and (b) surface flaw recategorized as a through-wall crack.
dynamically propagating crack should be taken into account in some manner. The R6 procedure (British Energy, 2001) recommends accounting for inertia effects simply by doubling the assumed length of the recategorized flaw. Annex F of BS 7910 (2000), which covers leak-before-break assessments, provides alternative rules for recategorizing surface flaws as through-wall cracks. These rules are less conservative than Figure 16 and depend on the through-wall stress distribution (i.e., bending versus membrane loading). If the recategorized flaw passes the assessment, the original flaw dimensions are considered acceptable. If neither the original flaw nor the recategorized flaw pass
the assessment, the flaw is deemed unacceptable. 7.06.5
INTERACTION OF MULTIPLE FLAWS
Sometimes it is necessary to assess two or more flaws in close proximity to one another. Standardized structural integrity methodologies include flaw spacing criteria and interaction rules for such situations. The various flaw interaction rules are primarily based on published literature that addressed the effect of flaw proximity on crack driving force. For example, see Murakami and Nemat-Nasser (1982) and O’Donoghue et al. (1986), Leek and Howard
Interaction of Multiple Flaws (1994), Kim and Lo (1995), Bezensek and Hancock (2001), and Moussa et al. (2002). 7.06.5.1
Principles of Interaction
The local stress field and crack driving force for a given flaw can be significantly affected by the presence of one or more neighboring cracks. Depending on the relative orientation
239
of the neighboring cracks, the interaction can either magnify or diminish the crack driving force. The relative magnitude of the interaction may be influenced by the load level. 7.06.5.1.1
Coplanar cracks
Figure 17 illustrates two identical coplanar cracks in an infinite plate. The lines of force
Figure 17 Co-planar cracks. Interaction between cracks results in a magnification of KI.
Figure 18 Interaction of two identical co-planar through-wall cracks in an infinite plate (source Murakami, 1986).
240
Flaw Characterization
represent the relative stress concentrating effect of the cracks. As the ligament between the cracks shrinks in size, the area through which the force must be transmitted decreases. Consequently, the mode I stress intensity factor, KI, is magnified for each crack as the two cracks approach one another. Figure 18 is a plot of the KI solution for the configuration in Figure 17 (Murakami, 1986). As one might expect, the crack tip closest to the
neighboring crack experiences the greater magnification in KI. The KI solution at tip B increases asymptotically pffiffiffi as s-0. At tip A, the solution approaches 2 as s-0 because the two cracks become a single crack with twice the original length of each crack. Figures 17 and 18 illustrate the general principle that multiple cracks in the same plane have the effect of magnifying KI in one another.
Figure 19 Parallel cracks. A mutual shielding effect reduces KI in each crack.
Figure 20 Interaction between two identical parallel through-wall cracks in an infinite plate (source: Murakami, 1986).
Interaction of Multiple Flaws
241
Table 1 Comparison of flaw interaction criteria in various structural integrity assessment procedures. Effective dimensions after interaction
Configuration
Assessment procedure
Criteria for interaction
1. Coplanar surface flaws
BS 7910 (2000) SINTAP (1999) API 579 (2000)
sr2c1 for a1/c141 or a2/c241 s ¼ 0 for a1/c1 and a2/c2o1 (c1oc2) src1 þ c2
ASME (1998)
srmax[2a1, 2a2]
BS 7910 (2000) SINTAP (1999) API 579 (2000)
sr2c1 for a1/c141 or a2/c241 s ¼ 0 for a1/c1 and a2/c2o1 (c1oc2) src1 þ c2
ASME (1998)
srmax[2c1, 2c2]
BS 7910 (2000) SINTAP (1999) API 579 (2000)
sra1 þ a2
ASME (1998)
srmax[2a1, 2a2]
BS 7910 (2000) SINTAP (1999)
s1ra1 þ a2 s2r2c1 for a1/c141 or a2/c241 s2 ¼ 0 for a1/c1 and a2/c2o1 (c1oc2) s1ra1 þ a2 and s2rc1 þ c2 s1rmax[2a1, 2a2] and s2rmax[2a1, 2a2]
2a ¼ 2a1 þ 2a2 þ s1 2c ¼ 2c1 þ 2c2 þ s2
sra1 þ a2
a ¼ a1 þ 2a2 þ s 2c ¼ max[2c1, 2c2] a ¼ a1 þ 2a2 þ s 2c ¼ max[2c1, 2c2] a ¼ a1 þ 2a2 þ s 2c ¼ max[2c1, 2c2]
2. Coplanar embedded flaws
3. Coplanar embedded flaws
4. Coplanar embedded flaws
API 579 (2000) ASME (1998) 5. Coplanar surface and embedded flaws
6. Coplanar surface and embedded flaws
sra1 þ a2
BS 7910 (2000) SINTAP (1999) API 579 (2000)
sra1 þ a2
ASME (1998)
srmax[a1, 2a2]
BS 7910 (2000) SINTAP (1999)
s1ra1 þ a2 s2r2c1 for a1/c141 or a2/c241 s2 ¼ 0 for a1/c1 and a2/c2o1 (c1oc2) s1ra1 þ a2 and s2rc1 þ c2 s1rmax[a1, 2a2] and s2rmax[2a1, 2a2]
API 579 (2000) ASME (1998)
a ¼ max[a1, a2] 2c ¼ 2c1 þ 2c2 a ¼ max[a1, a2] 2c ¼ 2c1 þ 2c2 a ¼ max[a1, a2] 2c ¼ 2c1 þ 2c2 2a ¼ max[2a1, 2a2] 2c ¼ 2c1 þ 2c2 þ s 2a ¼ max[2a1, 2a2] 2c ¼ 2c1 þ 2c2 þ s 2a ¼ max[2a1, 2a2] 2c ¼ 2c1 þ 2c2 þ s 2a ¼ 2a1 þ 2a2 þ s 2c ¼ max[2c1, 2c2] 2a ¼ 2a1 þ 2a2 þ s 2c ¼ max[2c1, 2c2] 2a ¼ 2a1 þ 2a2 þ s 2c ¼ max[2c1, 2c2]
2a ¼ 2a1 þ 2a2 þ s1 2c ¼ 2c1 þ 2c2 þ s2 2a ¼ 2a1 þ 2a2 þ s1 2c ¼ 2c1 þ 2c2 þ s2
a ¼ a1 þ 2a2 þ s1 2c ¼ 2c1 þ 2c2 þ s2
a ¼ a1 þ 2a2 þ s1 2c ¼ 2c1 þ 2c2 þ s2 a ¼ a1 þ 2a2 þ s1 2c ¼ 2c1 þ 2c2 þ s2
242 7.06.5.1.2
Flaw Characterization Parallel cracks
Figure 19 illustrates two parallel cracks. In this case, the cracks tend to shield one another, which results in a decrease in KI relative to the single-crack case. Figure 20 shows the KI solution for this geometry (Murakami, 1986). This is indicative of the general case where two or more parallel cracks have a mutual shielding interaction when subject to mode I loading. Consequently, multiple cracks that are parallel to one another are of less concern than multiple cracks in the same plane. 7.06.5.1.3
Elastic versus elastic–plastic deformation
The forgoing examples consider only elastic behavior. Flaw interaction rules in most standardized structural integrity methods (see below) are based on elastic interaction effects. One exception to this trend is the SINTAP (1999) procedure, which has elastic-based interaction rules for ‘‘fracture-controlled’’ situations and a separate set of interaction rules for ‘‘collapse-controlled’’ cases. Some studies, including the work of Kim and Lo (1995), indicate that the load level has an effect on the magnitude of the interaction between multiple flaws. In a series of elastic– plastic J-integral analyses, Kim and Lo showed that the magnitude of the interaction effect increases with plastic deformation. 7.06.5.2
Interaction Rules in Standard Assessment Documents
One way to account for the effect of multiple flaws is to apply a rigorous KI solution (or Jsolution, in the case of elastic–plastic loading) for the configuration of interest. Such solutions exist only for a few cases, however, so this more rigorous approach is not particularly practical. Standard structural integrity assessment procedures typically apply a much simpler approach. First of all, interaction of parallel flaws is not normally considered in standard assessments because any such interaction is beneficial, in that it decreases the crack driving force (Figures 19 and 20). For multiple cracks that are coplanar (or nearly coplanar), the separation distance is compared to a minimum allowable value. If the separation distance is greater than the minimum allowable, the flaws can be treated individually without considering interaction. If the spacing between two flaws is less than the minimum value, the two flaws are combined into a single flaw for the purposes of the structural integrity assessment.
Table 1 summarizes interaction rules in various assessment procedures for several examples of part-through flaws. Note that the minimum allowable separation distance differs significantly among the standard methods. It would appear that further work is necessary to reconcile these differences and to determine which set of rules are the most appropriate. 7.06.6
SUMMARY
Cracks and crack-like flaws are characterized through their planar dimensions. For the sake of simplicity, in most cases, flaws are assumed to have an idealized shape. More complex descriptions of crack shape are usually not practical due to the unavailability of fracture mechanics solutions and the difficulty of precise crack mapping from inspection results. The overwhelming majority of fracture mechanics analyses, particularly those covered by standardized structural integrity procedures, assume mode I loading. When mixedmode loading conditions are present, most codified structural integrity assessments define an equivalent mode I crack. When a crack tip is close to a free surface, it may be advantageous to recategorize such a crack. For example, a deep surface crack may be recategorized as a through-wall crack, and an embedded flaw may be recategorized as a surface crack. Two or more cracks that are in close proximity can affect one another. The interaction tends to be detrimental (increased fracture driving force) when cracks are co-planar, and the effect can be beneficial for parallel cracks. 7.06.7
REFERENCES
T. L. Anderson, 1995, ‘‘Fracture Mechanics: Fundamentals and Applications,’’ 2nd edn., CRC Press, Boca Raton, FL. API 579, 2000, API Recommended Practice 579. ‘‘Fitnessfor-Service,’’ American Petroleum Institute, Washington, DC. ASME, 1998, ‘‘ASME Boiler and Pressure Vessel Code, Section XI—Division 1,’’ American Society of Mechanical Engineers, New York. B. Bezensek and J. W. Hancock, 2001, Brittle fracture from surface breaking defects. In: ‘‘ASME PVP Vol. 423,’’ American Society of Mechanical Engineers, pp. 25–31. British Energy, 2001, ‘‘Assessment of the Integrity of Structures Containing Defects,’’ R6 Revision 4, British Energy Generation, Gloucester, UK. BS 7910, 2000, BS7910:1999, Incorporating Amendment No. 1. ‘‘Guide on Methods for Assessing the Acceptability of Flaws in Metallic Structures,’’ BSI, London. B. Cottrell and J. R. Rice, 1980, Slightly curved or kinked cracks. Int. J. Fract., 16, 155–169. F. Erdogan and G. C. Sih, 1963, On the crack extension in plates under plane loading and transverse shear. J. Basic Eng., 85, 519–527.
References
243
D. S. Kim and K. H. Lo, 1995, Crack interaction criteria in pressure vessels and pipe. J. Offshore Mech. Arctic Eng., 117, 260–264. T. H. Leek and I. C. Howard, 1994, Estimating the elastic interaction factors of two coplanar surface cracks under mode I load. Int. J. Pres. Ves. Piping, 60, 307–321. W. A. Moussa, R. Bell and C. L. Tan, 2002, Investigating the effect of crack shape on the interaction behavior of noncoplanar surface cracks using finite element analysis. J. Pres. Ves. Techol., 124, 234–238. Y. Murakami, 1986, ‘‘Stress Intensity Factors Handbook,’’ Pergamon, New York.
Y. Murakami and S. Nemat-Nasser, 1982, Interacting dissimilar semi-elliptical surface flaws under tension and bending. Eng. Fract. Mech., 16, 373–386. P. E. O’Donoghue, T. Nishioka and S. N. Atluri, 1986, Analysis of interaction behavior of surface flaws in pressure vessels. J. Pres. Ves. Technol., 108, 24–32. SINTAP, 1999, ‘‘Structural Integrity Assessment Procedures for European Industry,’’ European Union. S. Suresh, 1991, ‘‘Fatigue of Metals,’’ Cambridge University Press, Cambridge. J. G. Williams and P. D. Ewing, 1972, Fracture under complex stress—the angled crack problem. Int. J. Fract., 8, 441–446.
Copyright 2003, Elsevier Ltd. All Rights Reserved. No part of this publication may be reproduced, stored in any retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publishers.
Comprehensive Structural Integrity ISBN (set): 0-08-043749-4 Volume 7; (ISBN: 0-08-044152-1); pp. 227–243
227
7.06.2 FLAW SHAPE AND DIMENSIONS
227
7.06.2.1 7.06.2.2 7.06.2.3 7.06.2.4 7.06.2.5
227 229 229 230 230
Crack Shape Crack Dimensions Solutions for Idealized Crack Shapes Complex Flaw Shapes Flaw Tip Radius
7.06.3 FLAW ORIENTATION
230 230 231 231 234 235
7.06.3.1 Simple Projection Method 7.06.3.2 Maximum Energy Release Rate Model 7.06.3.2.1 Theoretical background 7.06.3.2.2 Defining an equivalent mode I crack 7.06.3.2.3 Comparison with the projection method 7.06.4 RECATEGORIZATION
236 236 237
7.06.4.1 Why Recategorize? 7.06.4.2 Standard Recategorization Rules 7.06.5 INTERACTION OF MULTIPLE FLAWS
238 239 239 242 242 242
7.06.5.1 Principles of Interaction 7.06.5.1.1 Coplanar cracks 7.06.5.1.2 Parallel cracks 7.06.5.1.3 Elastic versus elastic–plastic deformation 7.06.5.2 Interaction Rules in Standard Assessment Documents 7.06.6 SUMMARY
242
7.06.7 REFERENCES
242
7.06.1
The engineer performing such an assessment must always be mindful of these and other uncertainties in the input data. Refer to Chapter 7.15 for a discussion of sensitivity and reliability analysis.
INTRODUCTION
Key inputs to a structural integrity assessment include the size, location, and orientation of the flaw(s) being assessed. Nondestructive examination (NDE) to detect and size cracks is a crucial first step in obtaining such information. The present chapter focuses on the proper interpretation and application of NDE data to the structural integrity assessment. It should be noted that uncertainties in flaw detection and sizing often overshadow the sophisticated analytical models available for assessing the structural significance of cracks.
7.06.2 7.06.2.1
FLAW SHAPE AND DIMENSIONS Crack Shape
Most cracks and planar flaws fall into one of five categories: (i) Surface-breaking flaws. (ii) Through-wall flaws. 227
228
Flaw Characterization
(iii) Buried (subsurface) flaws. (iv) Corner flaws. (v) Edge flaws. Figure 1 illustrates these flaw shapes. Note that the actual flaw may have an irregular shape, but it is typically idealized with a simple shape. The shapes of surface, buried, and corner cracks are usually idealized as semielliptical, elliptical, and quarter-elliptical, respectively. Through-wall and edge cracks are
normally assumed to have a rectangular shape, at least in flat plates. The idealization of crack shape is necessary for pragmatic reasons. Most conventional NDE techniques are incapable of characterizing a detailed crack-front profile. Even if the true crack-front shape were known, standard fracture mechanics solutions are typically available only for simple shapes, as discussed in Section 7.06.2.3 below.
Figure 1 Typical flaw shapes considered in structural integrity assessments.
Flaw Shape and Dimensions 7.06.2.2
Crack Dimensions
The dimensions of an idealized crack are defined by the maximum extent of the actual flaw. Stated another way, the idealized crack shape completely inscribes the actual shape, as Figure 2 illustrates. In the case of part-through cracks, the key dimensions, defined in Figure 1, are the length and depth. The buried crack has an additional dimension, d, to indicate the distance from the center of the flaw to the nearest free surface. Note that some of the dimensions in Figure 1 use symbols multiplied by a factor of 2. For example, the elliptical buried crack is 2c long by 2a deep. This convention has been adopted in much of the published literature to maintain consistency between crack dimensions for full-, half-, and quarter-elliptical shapes. As a rule of thumb, a ‘‘2’’ in front of a crack dimension indicates that there are two crack tips in that direction. For example, a surface crack, with dimensions 2c by a, has two crack tips in the length direction (along the free surface) and one tip in the depth direction. In addition to dimensions, the relative position along the crack front can be important. In the case of surface, buried and corner cracks, the crack front position is characterized by the elliptic angle, f, which is defined in Figure 3(a). For through-wall and edge cracks, fracture mechanics parameters such as the stress intensity factor typically vary along the crack front, so a local crack front coordinate (z) can be defined, as Figure 3(b) illustrates. However, many of the standard solutions for through-wall and edge cracks are based on
Figure 2 The idealized flaw shape should completely inscribe the actual flaw.
229
two-dimensional models, and consequently do not account for through-thickness variations in relevant parameters. 7.06.2.3
Solutions for Idealized Crack Shapes
As stated above, most published fracture mechanics solutions assume an idealized crack shape. For example, expressions for the mode I stress intensity solutions for surface, buried, and corner cracks assume an elliptical shape and typically have the following form: rffiffiffiffiffiffi pa KI ¼ Gs Q
ð1Þ
where G is a dimensionless geometrical factor, s is a characteristic stress, and Q is a flaw shape parameter: Q ¼ 1 þ 1:464
a1:65 c
for arc
ð2Þ
The geometrical factor, G, is a function of crack shape (surface, buried, or corner), crack dimensions, crack front location, structural dimensions, and loading (e.g., membrane, bending, polynomial, etc.). The form of Equation (1) stems from the classical solution for an elliptical buried crack in an infinite body subject to a uniform tensile stress normal to the crack plane. Equation (2) is actually a curve-fit approximation of the square root of an elliptic integral of the second kind: 2
Q ¼
Z 0
p=2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 a2 2 1 sin ðfÞ df c2
ð3Þ
Figure 3 Definition of crack front location. (a) Parametric angle on elliptical cracks and (b) crack front position on a through-wall crack.
230
Flaw Characterization
Equation (3) does not have a known closedform solution, but Equation (2) provides a good approximation. Equations (1)–(3) and the accompanying discussion serve to illustrate the point that standard fracture mechanics solutions are inextricably linked to the assumed flaw shape. Consequently, these solutions are not rigorously correct for real flaws that deviate from the ideal. As long as the departure from the ideal crack shape is not too severe, the standard solutions should be reasonable engineering approximations. In principle, standardized structural integrity assessment procedures do not forbid the use of more rigorous solutions for the precise flaw geometry of interest. However, obtaining adequate inspection data, let alone the solution itself, is usually not practical. 7.06.2.4
Complex Flaw Shapes
The published literature contains a limited amount of information on complex crack shapes. For example, Bezensek and Hancock (2001) consider two coalescing surface cracks and provide experimental and numerical analysis. Annex F of BS 7910 (2000) contains a simplified procedure for handling complex cracks. Such cracks are treated by breaking them into simpler shapes for which standard solutions exist. 7.06.2.5
Flaw Tip Radius
Assessments of crack-like flaws normally do not consider out-of-plane flaw dimensions or tip radius. An implicit assumption of classical fracture mechanics theory is that the flaws in question are planar (as opposed to volumetric) with sharp tips. In reality, many flaws that are treated as sharp cracks have a finite tip radius. Lack-of-fusion defects in welds are a prime example of such flaws. Although the sharpcrack assumption is very pessimistic in some cases, standardized structural integrity procedures do not provide a rational means to quantify the effect of tip radius on the risk of brittle fracture. Suitable models may eventually find their way into standard engineering practice, but are not available as of early 2000s. Even with such models, measuring the tip radius of relatively sharp planar flaws may be problematic. Fatigue life prediction is one area where tip radius has been explicitly considered (Suresh, 1991). For relatively modest notches, various stress concentration factors such as Neuber’s factor, have been routinely applied. For more
severe notch-like features, a pseudo-fracture mechanics model has been moderately successful. In this latter case, the cycles to initiate a sharp crack at the tip of a notch with tip radius r has pffiffiffi been correlated with the parameter DK= r; where DK is the cyclic stress intensity factor inferred by treating the notch-like feature as a crack. 7.06.3
FLAW ORIENTATION
Most standardized structural integrity procedures are based on mode I fracture mechanics, where a principal stress acts normal to the crack plane such that the crack faces open. For cracks that are not aligned with the principal stress axes, one must either perform a mixed-mode analysis or apply a suitable approximation to reduce the problem to pure mode I. Codified structural integrity procedures typically choose the latter approach, although there is advice on mixed mode loading in R6 (British Energy, 2001) as discussed in Chapter 7.03. Mode I approximations for off-axis cracks can entail either a simple projection of the crack onto a principal plane or a more sophisticated model based on mixed-mode fracture mechanics. The projection method is described below, followed by an example of a more sophisticated approach. Several recognized flaw assessment procedures, including BS 7910 (2000) use the projection method. API 579 (2000) defines an equivalent mode I crack based on a more rigorous model described in Section 7.06.3.2 below. The latest release of the R6 procedure (British Energy, 2001) adopts the API method for characterizing off-axis flaws. 7.06.3.1
Simple Projection Method
Figure 4 illustrates the projection method. An equivalent mode I crack is defined by projecting the flaw onto a principal plane. For biaxial loading, there are two possible choices for the projection. In such cases the more conservative option should be taken, i.e., ( c¼
c1 ; c2 ;
pffiffiffiffiffi pffiffiffiffiffi s1 c1 Zs2 c2 pffiffiffiffiffi pffiffiffiffiffi s1 c1 os2 c2
ð4Þ
The symbols in Equation (4) are defined in Figure 4. Although the projection method has been used in various well-respected flaw evaluation codes over the years, it does not have a sound theoretical basis. It can actually be nonconservative in some instances, as discussed below.
Flaw Orientation 7.06.3.2
Maximum Energy Release Rate Model
Figure 5 shows a plate with a crack that is oriented at an angle b from the plane normal to the applied stress. If this crack propagates by brittle fracture or fatigue, it will tend to align itself perpendicular to the applied stress. That is, an initially mixed-mode crack will tend to become a mode I crack when it propagates, assuming the material is relatively homoge-
231
neous. If there are no planes of weakness in the material, the propagating crack will follow the path where the driving force is highest, which turns out to be approximately normal to the maximum principal stress in most cases. The preferred initial direction of propagation for a mixed-mode crack can be determined by evaluating the local energy release rate as a function of propagation direction. The critical conditions for propagation can then be determined by comparing the maximum energy release rate to a critical value for the material. Finally, an equivalent mode I crack can be defined that will propagate given the same principal stresses as in the mixed-mode case. The details of the energy release model are outlined below. This is followed by a derivation of the equivalent mode I crack. The analysis that follows was first presented in chapter 2 of Fracture Mechanics: Fundamentals and Applications (Anderson, 1995) and stems from the work of Erdogan and Sih (1963), Williams and Ewing (1972), and Cottrell and Rice (1980). 7.06.3.2.1
Theoretical background
Figure 6 illustrates the three ways in which a cracked body can be loaded. Mode I loading results from stresses normal to the crack plane and tends to open the crack, while modes II and III correspond to in-plane and out-ofplane shear, respectively. The stress fields near the crack tip, in terms of polar coordinates, are given as follows: Mode I: Figure 4 The simple projection method for cracks not oriented on a principal plane.
KI 5 y 1 3y p ffiffiffiffiffiffiffi cos cos srr ¼ 2 4 2 2pr 4
Figure 5 Propagation of a crack oriented at an angle b from the applied stress.
ð5aÞ
232
Flaw Characterization
Figure 6 Three modes of loading a crack. KI 3 y 1 3y þ cos syy ¼ pffiffiffiffiffiffiffi cos 4 2 4 2 2pr
ð5bÞ
KI 1 y 1 3y þ sin try ¼ pffiffiffiffiffiffiffi sin 4 2 4 2 2pr
ð5cÞ
Mode II: KII 5 y 3 3y p ffiffiffiffiffiffiffi þ sin srr ¼ sin 2 4 2 2pr 4
ð6aÞ
KII 3 y 3 3y syy pffiffiffiffiffiffiffi sin sin 2 4 2 2pr 4
ð6bÞ
KII 1 y 3 3y try ¼ pffiffiffiffiffiffiffi cos þ cos 2 4 2 2pr 4
ð6cÞ
Figure 7 Infinitesimal kink at the tip of a crack.
Mode III: KIII y trz ¼ pffiffiffiffiffiffiffi sin 2 2pr KIII y tyz ¼ pffiffiffiffiffiffiffi cos 2 2pr
ð7aÞ
ð7bÞ
When a cracked body is subject to a system of applied forces, one, two, or all three modes of loading may be present. The crack will propagate along the most favorable path, which depends on the driving force relative to the material resistance. For a perfectly isotropic and homogeneous material, the optimal crack propagation direction corresponds to the direction where the driving force is a maximum. Consider a crack with an infinitesimal kink at the tip, as Figure 7 illustrates. This kink represents the initial propagation. The local stress intensity factors at the tip of this kink differ from the nominal K values of the main crack. If we define a local x–y coordinate
system at the tip of the kink and assume that Equations (5)–(7) define the local stress fields, the local mode I, II, and III stress intensity factors at the tip are obtained by summing the appropriate stress components at a: pffiffiffiffiffiffiffi kI ðaÞ ¼ syy 2pr ¼ C11 KI þ C12 KII
ð8aÞ
pffiffiffiffiffiffiffi kII ðaÞ ¼ txy 2pr ¼ C21 KI þ C22 KII
ð8bÞ
kIII ðaÞ ¼ C33 KIII
ð8cÞ
where kI and kII are the local stress intensity factors at the tip of the kink and KI and KII are the stress intensity factors for the main crack. The coefficients Cij are given by C11
a 1 3 3a ¼ cos þ cos 4 2 4 2
ð9aÞ
3 a 3a sin þ sin 4 2 2
ð9bÞ
C12 ¼
C21 ¼
1 a 3a sin þ sin 4 2 2
ð9cÞ
Flaw Orientation C22
a 3 1 3a þ cos ¼ cos 4 2 4 2 C33 ¼ cos
a 2
ð9dÞ
ð9eÞ
The energy release rate for the kinked crack in plane strain is given by GðaÞ ¼
1 n2 2 k2 ðaÞ 2 kI ðaÞ þ kII ðaÞ þ III 1n E
ð10Þ
Consider a case where there is no out-of-plane shear, such that KIII ¼ kIII(a) ¼ 0. Figure 8 is a plot of G(a) normalized by G(a ¼ 0) for various ratios of KII/KI. It turns out that the peak in G(a) at each KII/KI ratio corresponds to the point where kI exhibits a maximum and kII ¼ 0. Thus, the maximum energy release rate is given by Gmax ¼
ð1 n2 ÞkI2 ða Þ E
ð11Þ
where a* is the angle at which both G and kI exhibit a maximum and kII ¼ 0. Crack growth in a homogeneous material should initiate along a*. The configuration in Figure 5 involves mode I and mode II loading. The global stress intensity factors are given by pffiffiffiffiffiffiffi KI ¼ s cos2 ðbÞ pc0
ð12aÞ
pffiffiffiffiffiffiffi KII ¼ s sinðbÞ cosðbÞ pc0
ð12bÞ
Substituting the above expressions into Equation (8a) gives the following relationship for the local mode I stress intensity factor for a
233
kinked crack: pffiffiffiffiffiffiffi kI ðaÞ ¼ s pc0 ½cos2 ðbÞC11 þ sinðbÞ cosðbÞC12
ð13Þ
For a given crack orientation angle (b), Equation (13) can be solved for the optimum kink angle, a*, at which kI reaches a maximum. Figure 9 is a plot of the predicted propagation direction as a function of crack orientation. The diagonal line represents propagation normal to the applied stress. The energy release rate model predicts a propagation direction that differs slightly from the principal stress plane. Note, however, that a* is the initial propagation angle, and the crack will eventually align with the principal plane after propagating a finite distance. That is, KII will eventually be zero with continued crack propagation. The plate in Figure 5 can be generalized to biaxial loading, as Figure 10 illustrates. The mode I and II stress intensity factors for this configuration are given by pffiffiffiffiffiffiffi KI ¼ s1 pc0 ðcos2 b þ B sin2 bÞ
ð14aÞ
pffiffiffiffiffiffiffi KII ¼ s1 pc0 ðsin b cos bÞð1 BÞ
ð14bÞ
where B is the biaxiality ratio, defined as B¼
s2 s1
ð15Þ
The local mode I stress intensity for the biaxial case is as follows: pffiffiffiffiffiffiffi kI ðaÞ ¼ s1 pc0 f½cos2 ðbÞ þ B sin2 ðbÞC11 þ sinðbÞ cosðbÞð1 BÞC12 g
Figure 8 Local energy release rate as a function of kink angle.
ð16Þ
234
Flaw Characterization
Figure 9 Predicted propagation angle for the configuration in Figure 5.
into the above expression: pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi s1 pceff ¼ s1 pc0 f½cos2 ðbÞ þ B sin2 ðbÞC11 þ sinðbÞ cosðbÞð1 BÞC12 g
ð18Þ
Solving for ceff gives ceff ¼ f½cos2 ðbÞ þ B sin2 ðbÞC11 ða Þ c0 þ sinðbÞ cosðbÞð1 BÞC12 ða Þg2
Figure 10 Angled crack subject to biaxial loading.
Figure 12 is a plot of Equation (19) as a function of crack angle and biaxiality ratio. This expression assumes that ceff is defined on the s1 plane. An equivalent crack on the s2 plane can be computed as follows: ceffðs2 planeÞ ¼
Figure 11 is a plot of predicted propagation angle as a function of crack orientation and biaxiality ratio. 7.06.3.2.2
Defining an equivalent mode I crack
Let us now define an effective crack size, ceff, that when oriented on a principal plane, gives the same fracture driving force as the angled crack, i.e.,
ð19Þ
ceffðs1 planeÞ B2
ð20Þ
provided 0oBr1. Applying Equation (19) is cumbersome because it requires solving for a* and evaluating the functions C11.and C12. The following simplified expression provides a good approximation of Equation (19):
ð17Þ
ceff ¼ cos2 ðbÞ þ 0:5ð1 BÞ sinðbÞ cosðbÞ c0 þ B2 sin2 ðbÞ ð21Þ
For the biaxial loading case in Figure 10, ceff can be defined by substituting Equation (16)
for flaws projected on the maximum principal stress (s1) plane. For cracks on the s2 plane,
KI ðceff Þ ¼ kI ða ; b; c0 Þ
Flaw Orientation
235
Figure 11 Effect of biaxiality ratio on propagation angle.
Figure 12
Equivalent mode I crack length as a function of orientation and biaxiality ratio.
applying Equation (20) gives ceffðs2 planeÞ cos2 ðbÞ þ 0:5ð1 BÞ sinðbÞ cosðbÞ ¼ B2 c0 2 þ sin ðbÞ ð22Þ
Figures 13 and 14 compare Equation (21) with the more complex expression for B ¼ 0 and B ¼ 0.5, respectively. API 579 (2000) uses Equations (21) and (22) to define equivalent mode I cracks for off-axis flaws.
7.06.3.2.3
Comparison with the projection method
The simple projection method described above implies the following equivalent mode I crack: cproj ¼ cosðbÞ c0
ð23Þ
Figure 15 compares this simple method with the energy release rate model for uniaxial loading. Note that the projection method can
236
Flaw Characterization
Figure 13 Comparison of equivalent mode I crack lengths for uniaxial loading using the energy release rate model and the simple approximation.
Figure 14 Comparison of equivalent mode I crack lengths for B ¼ 0.5 using the energy release rate model and the simple approximation.
be nonconservative by as much as 20%. Moreover, the projection method does not account for biaxial loading. 7.06.4
free surface might be recategorized as a surface crack, and a deep surface crack may be recategorized as a through-wall crack, as Figure 16 illustrates.
RECATEGORIZATION
When a part-through flaw is close to a free surface, it may be advantageous to recategorize the flaw by assuming the remaining ligament is not present. For example, a buried crack near a
7.06.4.1
Why Recategorize?
In many cases, a flaw whose tip is close to a free surface may be deemed unacceptable by a particular flaw assessment method, but the
Recategorization
237
Figure 15 Comparison of equivalent mode I crack length estimates from the energy release rate model and the simple projection method.
recategorized flaw (which has larger dimensions than the actual flaw) may be acceptable according to the same assessment method. This apparent anomaly is due in part to peculiarities in standardized flaw assessment procedures. For example, in procedures based on the failure assessment diagram (FAD), the reference stress solution which defines the horizontal axis of the FAD (see Chapter 7.03 for a description of FAD methods) is often based on a local ligament yielding criterion. Such expressions tend to be very conservative, particularly when the ligament is small relative to section thickness. When the flaw is recategorized, a different, less conservative reference stress solution may apply. The larger recategorized flaw will tend to have a slightly higher toughness ratio (Kr) on the FAD, but the load ratio (Lr) may be significant lower. The net result is that the recategorized flaw may fall inside the FAD whereas the original flaw was outside the FAD. Idiosyncrasies and conservatisms of standard assessment methods notwithstanding, recategorization can have a physical basis. When plasticity is highly concentrated in the remaining ligament, it can fail, resulting in the crack breaking through to the free surface. Ligament rupture is typically ductile, even if the material is below the upper shelf of toughness. A small ligament has low constraint (stress triaxiality), which tends to result in a ductile fracture mode. A shear lip B451 from the applied stress is often seen
when a ligament breaks through to the free surface. In principle, one might argue that ligament yielding criteria, which often lead to the need for recategorization, are physically based. That is, when a flaw with a small remaining ligament is deemed unacceptable, it is because the assessment is predicting ligament failure. In practice, however, simple ligament yielding models tend to be very conservative when used to predict local failure. A more rigorous (but more complicated) approach to predicting ligament failure would be to perform an elastic–plastic stress analysis on the flaw in question and compare the applied J-integral to the fracture toughness. 7.06.4.2
Standard Recategorization Rules
Most recognized structural integrity procedures do not require recategorization. Rather, they offer it as an option when a given flaw does not pass the initial assessment. In addition to removing the ligament, the assumed flaw length should be increased to account for possible growth in this direction if the ligament were to fail. Figure 16 shows typical recommendations for the dimensions of recategorized flaws when ductile ligament failure is expected. If the ligament failure is potentially brittle, a dynamic or arrest toughness should be used to assess the recategorized flaw dimensions in order to account for the effects of a rapidly running crack tip. Also, the effect of inertia of a
238
Flaw Characterization
Figure 16 Recategorization procedure. (a) Buried flaw recategorized as a surface flaw and (b) surface flaw recategorized as a through-wall crack.
dynamically propagating crack should be taken into account in some manner. The R6 procedure (British Energy, 2001) recommends accounting for inertia effects simply by doubling the assumed length of the recategorized flaw. Annex F of BS 7910 (2000), which covers leak-before-break assessments, provides alternative rules for recategorizing surface flaws as through-wall cracks. These rules are less conservative than Figure 16 and depend on the through-wall stress distribution (i.e., bending versus membrane loading). If the recategorized flaw passes the assessment, the original flaw dimensions are considered acceptable. If neither the original flaw nor the recategorized flaw pass
the assessment, the flaw is deemed unacceptable. 7.06.5
INTERACTION OF MULTIPLE FLAWS
Sometimes it is necessary to assess two or more flaws in close proximity to one another. Standardized structural integrity methodologies include flaw spacing criteria and interaction rules for such situations. The various flaw interaction rules are primarily based on published literature that addressed the effect of flaw proximity on crack driving force. For example, see Murakami and Nemat-Nasser (1982) and O’Donoghue et al. (1986), Leek and Howard
Interaction of Multiple Flaws (1994), Kim and Lo (1995), Bezensek and Hancock (2001), and Moussa et al. (2002). 7.06.5.1
Principles of Interaction
The local stress field and crack driving force for a given flaw can be significantly affected by the presence of one or more neighboring cracks. Depending on the relative orientation
239
of the neighboring cracks, the interaction can either magnify or diminish the crack driving force. The relative magnitude of the interaction may be influenced by the load level. 7.06.5.1.1
Coplanar cracks
Figure 17 illustrates two identical coplanar cracks in an infinite plate. The lines of force
Figure 17 Co-planar cracks. Interaction between cracks results in a magnification of KI.
Figure 18 Interaction of two identical co-planar through-wall cracks in an infinite plate (source Murakami, 1986).
240
Flaw Characterization
represent the relative stress concentrating effect of the cracks. As the ligament between the cracks shrinks in size, the area through which the force must be transmitted decreases. Consequently, the mode I stress intensity factor, KI, is magnified for each crack as the two cracks approach one another. Figure 18 is a plot of the KI solution for the configuration in Figure 17 (Murakami, 1986). As one might expect, the crack tip closest to the
neighboring crack experiences the greater magnification in KI. The KI solution at tip B increases asymptotically pffiffiffi as s-0. At tip A, the solution approaches 2 as s-0 because the two cracks become a single crack with twice the original length of each crack. Figures 17 and 18 illustrate the general principle that multiple cracks in the same plane have the effect of magnifying KI in one another.
Figure 19 Parallel cracks. A mutual shielding effect reduces KI in each crack.
Figure 20 Interaction between two identical parallel through-wall cracks in an infinite plate (source: Murakami, 1986).
Interaction of Multiple Flaws
241
Table 1 Comparison of flaw interaction criteria in various structural integrity assessment procedures. Effective dimensions after interaction
Configuration
Assessment procedure
Criteria for interaction
1. Coplanar surface flaws
BS 7910 (2000) SINTAP (1999) API 579 (2000)
sr2c1 for a1/c141 or a2/c241 s ¼ 0 for a1/c1 and a2/c2o1 (c1oc2) src1 þ c2
ASME (1998)
srmax[2a1, 2a2]
BS 7910 (2000) SINTAP (1999) API 579 (2000)
sr2c1 for a1/c141 or a2/c241 s ¼ 0 for a1/c1 and a2/c2o1 (c1oc2) src1 þ c2
ASME (1998)
srmax[2c1, 2c2]
BS 7910 (2000) SINTAP (1999) API 579 (2000)
sra1 þ a2
ASME (1998)
srmax[2a1, 2a2]
BS 7910 (2000) SINTAP (1999)
s1ra1 þ a2 s2r2c1 for a1/c141 or a2/c241 s2 ¼ 0 for a1/c1 and a2/c2o1 (c1oc2) s1ra1 þ a2 and s2rc1 þ c2 s1rmax[2a1, 2a2] and s2rmax[2a1, 2a2]
2a ¼ 2a1 þ 2a2 þ s1 2c ¼ 2c1 þ 2c2 þ s2
sra1 þ a2
a ¼ a1 þ 2a2 þ s 2c ¼ max[2c1, 2c2] a ¼ a1 þ 2a2 þ s 2c ¼ max[2c1, 2c2] a ¼ a1 þ 2a2 þ s 2c ¼ max[2c1, 2c2]
2. Coplanar embedded flaws
3. Coplanar embedded flaws
4. Coplanar embedded flaws
API 579 (2000) ASME (1998) 5. Coplanar surface and embedded flaws
6. Coplanar surface and embedded flaws
sra1 þ a2
BS 7910 (2000) SINTAP (1999) API 579 (2000)
sra1 þ a2
ASME (1998)
srmax[a1, 2a2]
BS 7910 (2000) SINTAP (1999)
s1ra1 þ a2 s2r2c1 for a1/c141 or a2/c241 s2 ¼ 0 for a1/c1 and a2/c2o1 (c1oc2) s1ra1 þ a2 and s2rc1 þ c2 s1rmax[a1, 2a2] and s2rmax[2a1, 2a2]
API 579 (2000) ASME (1998)
a ¼ max[a1, a2] 2c ¼ 2c1 þ 2c2 a ¼ max[a1, a2] 2c ¼ 2c1 þ 2c2 a ¼ max[a1, a2] 2c ¼ 2c1 þ 2c2 2a ¼ max[2a1, 2a2] 2c ¼ 2c1 þ 2c2 þ s 2a ¼ max[2a1, 2a2] 2c ¼ 2c1 þ 2c2 þ s 2a ¼ max[2a1, 2a2] 2c ¼ 2c1 þ 2c2 þ s 2a ¼ 2a1 þ 2a2 þ s 2c ¼ max[2c1, 2c2] 2a ¼ 2a1 þ 2a2 þ s 2c ¼ max[2c1, 2c2] 2a ¼ 2a1 þ 2a2 þ s 2c ¼ max[2c1, 2c2]
2a ¼ 2a1 þ 2a2 þ s1 2c ¼ 2c1 þ 2c2 þ s2 2a ¼ 2a1 þ 2a2 þ s1 2c ¼ 2c1 þ 2c2 þ s2
a ¼ a1 þ 2a2 þ s1 2c ¼ 2c1 þ 2c2 þ s2
a ¼ a1 þ 2a2 þ s1 2c ¼ 2c1 þ 2c2 þ s2 a ¼ a1 þ 2a2 þ s1 2c ¼ 2c1 þ 2c2 þ s2
242 7.06.5.1.2
Flaw Characterization Parallel cracks
Figure 19 illustrates two parallel cracks. In this case, the cracks tend to shield one another, which results in a decrease in KI relative to the single-crack case. Figure 20 shows the KI solution for this geometry (Murakami, 1986). This is indicative of the general case where two or more parallel cracks have a mutual shielding interaction when subject to mode I loading. Consequently, multiple cracks that are parallel to one another are of less concern than multiple cracks in the same plane. 7.06.5.1.3
Elastic versus elastic–plastic deformation
The forgoing examples consider only elastic behavior. Flaw interaction rules in most standardized structural integrity methods (see below) are based on elastic interaction effects. One exception to this trend is the SINTAP (1999) procedure, which has elastic-based interaction rules for ‘‘fracture-controlled’’ situations and a separate set of interaction rules for ‘‘collapse-controlled’’ cases. Some studies, including the work of Kim and Lo (1995), indicate that the load level has an effect on the magnitude of the interaction between multiple flaws. In a series of elastic– plastic J-integral analyses, Kim and Lo showed that the magnitude of the interaction effect increases with plastic deformation. 7.06.5.2
Interaction Rules in Standard Assessment Documents
One way to account for the effect of multiple flaws is to apply a rigorous KI solution (or Jsolution, in the case of elastic–plastic loading) for the configuration of interest. Such solutions exist only for a few cases, however, so this more rigorous approach is not particularly practical. Standard structural integrity assessment procedures typically apply a much simpler approach. First of all, interaction of parallel flaws is not normally considered in standard assessments because any such interaction is beneficial, in that it decreases the crack driving force (Figures 19 and 20). For multiple cracks that are coplanar (or nearly coplanar), the separation distance is compared to a minimum allowable value. If the separation distance is greater than the minimum allowable, the flaws can be treated individually without considering interaction. If the spacing between two flaws is less than the minimum value, the two flaws are combined into a single flaw for the purposes of the structural integrity assessment.
Table 1 summarizes interaction rules in various assessment procedures for several examples of part-through flaws. Note that the minimum allowable separation distance differs significantly among the standard methods. It would appear that further work is necessary to reconcile these differences and to determine which set of rules are the most appropriate. 7.06.6
SUMMARY
Cracks and crack-like flaws are characterized through their planar dimensions. For the sake of simplicity, in most cases, flaws are assumed to have an idealized shape. More complex descriptions of crack shape are usually not practical due to the unavailability of fracture mechanics solutions and the difficulty of precise crack mapping from inspection results. The overwhelming majority of fracture mechanics analyses, particularly those covered by standardized structural integrity procedures, assume mode I loading. When mixedmode loading conditions are present, most codified structural integrity assessments define an equivalent mode I crack. When a crack tip is close to a free surface, it may be advantageous to recategorize such a crack. For example, a deep surface crack may be recategorized as a through-wall crack, and an embedded flaw may be recategorized as a surface crack. Two or more cracks that are in close proximity can affect one another. The interaction tends to be detrimental (increased fracture driving force) when cracks are co-planar, and the effect can be beneficial for parallel cracks. 7.06.7
REFERENCES
T. L. Anderson, 1995, ‘‘Fracture Mechanics: Fundamentals and Applications,’’ 2nd edn., CRC Press, Boca Raton, FL. API 579, 2000, API Recommended Practice 579. ‘‘Fitnessfor-Service,’’ American Petroleum Institute, Washington, DC. ASME, 1998, ‘‘ASME Boiler and Pressure Vessel Code, Section XI—Division 1,’’ American Society of Mechanical Engineers, New York. B. Bezensek and J. W. Hancock, 2001, Brittle fracture from surface breaking defects. In: ‘‘ASME PVP Vol. 423,’’ American Society of Mechanical Engineers, pp. 25–31. British Energy, 2001, ‘‘Assessment of the Integrity of Structures Containing Defects,’’ R6 Revision 4, British Energy Generation, Gloucester, UK. BS 7910, 2000, BS7910:1999, Incorporating Amendment No. 1. ‘‘Guide on Methods for Assessing the Acceptability of Flaws in Metallic Structures,’’ BSI, London. B. Cottrell and J. R. Rice, 1980, Slightly curved or kinked cracks. Int. J. Fract., 16, 155–169. F. Erdogan and G. C. Sih, 1963, On the crack extension in plates under plane loading and transverse shear. J. Basic Eng., 85, 519–527.
References
243
D. S. Kim and K. H. Lo, 1995, Crack interaction criteria in pressure vessels and pipe. J. Offshore Mech. Arctic Eng., 117, 260–264. T. H. Leek and I. C. Howard, 1994, Estimating the elastic interaction factors of two coplanar surface cracks under mode I load. Int. J. Pres. Ves. Piping, 60, 307–321. W. A. Moussa, R. Bell and C. L. Tan, 2002, Investigating the effect of crack shape on the interaction behavior of noncoplanar surface cracks using finite element analysis. J. Pres. Ves. Techol., 124, 234–238. Y. Murakami, 1986, ‘‘Stress Intensity Factors Handbook,’’ Pergamon, New York.
Y. Murakami and S. Nemat-Nasser, 1982, Interacting dissimilar semi-elliptical surface flaws under tension and bending. Eng. Fract. Mech., 16, 373–386. P. E. O’Donoghue, T. Nishioka and S. N. Atluri, 1986, Analysis of interaction behavior of surface flaws in pressure vessels. J. Pres. Ves. Technol., 108, 24–32. SINTAP, 1999, ‘‘Structural Integrity Assessment Procedures for European Industry,’’ European Union. S. Suresh, 1991, ‘‘Fatigue of Metals,’’ Cambridge University Press, Cambridge. J. G. Williams and P. D. Ewing, 1972, Fracture under complex stress—the angled crack problem. Int. J. Fract., 8, 441–446.
Copyright 2003, Elsevier Ltd. All Rights Reserved. No part of this publication may be reproduced, stored in any retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publishers.
Comprehensive Structural Integrity ISBN (set): 0-08-043749-4 Volume 7; (ISBN: 0-08-044152-1); pp. 227–243