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Kigure N.I8. R curves with same j and decreasing /( for increasing thickness value of a (see figure N.IO) and a decreasing /i. Much further work is still required before the effect of thickness on the R curve can be quantified.

Apart from a theoretical approach, experimental determinations of R curves are still too few. The measurements of R from the energy release rate in a slow crack-growth test may be complemented by a more direct method. The latter can be achieved by establishing the plastic energy consumption from the measured plastic strains. Some preliminary work in this direction was performed by Rooke and Bradshaw [33]- who found an R curve resembling the ones determined by other methods.

### 8.5 Plane stress testing

Plane stress fracture testing requires large specimens. No standard specimens have yet been proposed. It is obvious that the bend specimen and the compact tension specimen are not suitable for thin sheets. Most in-

figure S.I9. Plane stress fracture test with punched tape and teletype output of load. COD. and compliance. Note anti-buckling guides, and film camera for slow-growth records vestigators make use of center-cracked panels. An example is presented in figure 8.19. Usually, moving pictures are taken during each test in order to have a record of slow crack growth. If the tests should yield useful values for Ku or Kw, they wil have to be performed satisfying the screening criteria 2a< W/3 and trr < 2/3ervs.

Unlike plane strain specimens, the plane stress specimens need not always be fatigue cracked. When the toughness is high enough for the cracks to show significant amounts of slow growth, the crack can be simulated by a sharp saw cut. A saw cut will slightly increase the stress for onset of crack growth, but once crack growth has started the saw cut has changed into a real crack and the residual strength will be equal to the strength in the case of a fatigue crack of the same initial length. This has been demonstrated by tests [34], When fracture of sheets with saw cuts is not preceded by slow crack growth the simulation of the crack by means of a saw cut is not permissible. In that case the bluntness of the tip of the saw cut may be sufficient to raise the stress cx, for crack initiation above the fracture stress ac.

This behaviour can be explained on the basis of the energy balance concept, as in figure 8.20. A sharp crack starts slow growth at tr, and gives failure at the stress ac. For a saw cut-simulated crack the stress for crack initiation can be raised to a2■ The energy balance will now be reached at B, and fracture will occur at ac as before. Saw cuts can be made so blunt that crack initation does not occur until a stress <r3 larger than ac. In that case no energy balance can be reached and immediate fracture instability occurs, not preceded by slow growth. The permissible

Figure 8.20. Blunt crack tips in plane stress testing [I. 34] (courtesy McMillan)

Figure 8.20. Blunt crack tips in plane stress testing [I. 34] (courtesy McMillan)

bluntness depends upon the material properties. Sharper cracks are required for materials of lower toughness: for H-ll steel an increase of residual strength was already observed [35] at root radii in the order of 20 microns. The magnitude of permissible bluntness has to be determined from experiments.

The non-singular term -a in the stress field equatons(3.16) for a cracked plate in uniaxial tension indicates that the stress ax along the edges of the crack is compressive and of the order of the applied longitudinal tensile stress. Especially in thin sheets this compressive stress can cause buckling of the plate segment adjacent to the crack (figure 8.21). One can easily demonstrate this buckling by manually pulling a sheet of paper with a central transverse tear. Since buckling may affect slow crack growth and residual strength, it has been the subject of several investigations [2,36-39],

Carlson et al. [38] have treated buckling formally as a plate stability problem. Usually however, a simple column buckling formula is used [12, 36, 37], Since the transverse compressive stress along the crack edge is equal to the nominal uniform stress a. buckling will commence when a = ab defined by

Eq (8.26) is the Euler formula for buckling of a column of thickness B. modulus E and effective length /,„ with hinged ends. The length of the column le will be related to the crack length a by le = aa . (8.27)

There is a difference of opinion as to the most realistic value of a, which is in the order of 0.5 [36], but most probably depends upon sheet thickness [2, 38],

For long cracks buckling occurs well before the specimen is ready to fail, and therefore it may affect the residual strength. For this reason buckling is usually prevented in residual strength tests by the application of rigid bars (figures 8.19 and 8.21), known as anti-buckling guides. Photoelastic studies by Dixon and Strannigan [36] have shown that the maximum stress at the tip of a slit in an unrestrained model was about 30 per cent greater than under the application of anti-buckling guides. This will of course affect the residual strength. Reductions in residual strength of about 10 per cent were reported by Walker [2] and Trotman [39], and up to 40 per cent by Forman [37]. Of course, the reduction must depend upon crack length. Some of the many data of Walker [2] are presented here in figure 8.22, to illustrate the effect of buckling on slow stable crack growth and residual strength.

Although buckling guides are usually considered a prerequisite for a

Figure 8.22. Lffcct of buckling on slow crack growth and residuat strength [2]

Figure 8.22. Lffcct of buckling on slow crack growth and residuat strength [2]

useful residual strength test, it is questionable whether they are always necessary. In practical sheet structures, buckling will often not be fully restrained. F.ven in stiffened structures buckling can sometimes be restrained only by the in-plane bending stiffness of the stringers. Only when there is a stringer across the crack is buckling fully prevented by the out-of-plane bending stiffness of the stringer. A calculation of the residual strength of a sheet structure in which buckling is not restrained should, of course, be based on tests without buckling guides, since Klc is lower for that case. The anti-buckling guides applied in an actual test can be observed in figure 8.19.

The question sometimes arises whether it is permissible to establish the critical crack size by cutting a slit while the specimen is under constant load. Tests have shown [28] that this would provide false information. Specimens containing an initial central slit were loaded to the point where slow growth initiation was about to occur. From then on the load was kept constant, and the two ends of the crack were propagated simultaneously by means of two jeweller's saws until fracture occurred. Some test results are presented in figure 8.23.

At first glance one would expect failure during sawing to occur when the critical crack length (associated with the applied stress) is reached: i.e. at the passage of the upper curve in figure 8.23. However, sawing could be continued far beyond this point. It might be argued that the discrepancy could be due to the bluntness of the saw cut as compared to a fatigue crack, since the upper curve in figure 8.23 is valid for a slowly-growing actual crack. However, the data points are so far off the curve in a vertical direction that the bluntness of the saw cut cannot be the sole cause of the discrepancy.

When the results are considered with the R curve concept, they appear to be more rational. This is outlined in figure 8.24. The lower part of this figure shows the residual strength diagram. A crack of initial length a0 can be loaded to a stress n, (point A) where slow crack growth commences. When the stress is raised to ac the crack will have propagated to if,, where fracture instability occurs. The upper part of figure 8.24 shows the corresponding energy-balance diagram in terms of the energy release rate G and the crack growth resistance R. Slow growth begins when the stress is raised to <r,. Then G = a0/Ii. represented by point A. During further increase of the stress the R curve is followed. Finally, when the stress has reached a, and the crack has grown to ac (point B), the crack can

Figure 8.23. Artificial slow crack growth under constant stress [28] (courtesy Pergamon)

Figure 8.23. Artificial slow crack growth under constant stress [28] (courtesy Pergamon)

propagate under constant stress. G will increase and remain larger than R, and fracture instability occurs.

Consider now a crack of initial length a0 loaded to aThe crack is extended artificially by sawing, while the stress is kept constant at <r,. The energy release rate G will increase proportionally to a, according to G = na}a/E, and it will follow the straight line A-D. Finally at C, the condition G^R is fulfilled and fracture instability occurs (point C, in the lower part of figure 8.24). If fracture were to occur at D as assumed first.

the R curve should shift during sawing to the position of the dashed curve in figure 8.25, where fracture would occur at D. A shift of the R curve during the lest is unlikely [28], When in the test described, sawing is stopped at H, followed by continuous loading, failure should occur at K. Equivalent points in the lower diagram are indicated. This behaviour was confirmed by tests [28].

In conclusion it can be stated that the execution of a fail-safe lest by means of extending the crack by sawing under load leads to an over-estimation of the critical crack length.

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