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F igure 5.5. Universal representation of energy criterion a, — a, O —A a z a; 1

F igure 5.5. Universal representation of energy criterion

of crack extension under constant stress and under fixed grips. However, this is only so for the onset of crack extension. During crack growth it is not true any more. If the crack extends under constant stress, G develops as indicated by the straight lines in figure 5.5. If crack extension occurs under fixed grips conditions, the stress drops. Since G = na2a/E it follows-that G increases less than proportional to a (figure 5.6). In certain specimen geometries G may even decrease if the crack extends under fixed grip conditions. This problem and its consequences are discussed in chapter 6.

5.3 The crack resistance (R curve)

So far, R was considered independent of crack length. This is approximately true for cracks under plane strain. In the case of plane stress the crack resistance varies with the amount of crack growth, as has been shown by experiments.

Consider a crack in a sheet thin enough for plane stress to occur (chapters 7 and 8). When the specimen is loaded to a stress <r, the crack starts propagating. However, crack growth is stable and fracture does not yet occur. If the stress is kept constant at ct, the crack propagates only over a small distance and stops. A further increase of the stress is required to maintain crack growth: although the crack is longer it can withstand a higher stress. The stress can be increased further with simultaneous crack

growth until at a stress rrc a critical crack size ac. is reached where fracture instability occurs. This is illustrated in figure 5.7.

At the onset of crack propagation the energy criterion must be fulfilled. During stable crack growth the energy release rate is just equal to the crack resistance (if it is lower the crack stops growing, if it is larger fracture instability occurs). The energy release rate is G = na2aiE and both a and a increase during crack growth. This means that G

increases more than proportional to a. Since G = R it must be concluded that R increases, which is shown diagrammatically in figure 5.8.

Suppose a crack of size a, is loaded to a stress ay. If the crack were to extend, the available energy release is given by A. However, this value is too low for crack growth to occur. The stress can be further increased to a,, where the available energy release rate is given by B. Suppose this value is sufficient for crack growth. If the crack were to propagate under constant stress, G would increase according to B-H. This line is lower than the R curve and therefore crack growth under constant stress cannot occur.

Further increase of the stress to <r2 brings about a crack extension Aa2. Both G and R follow the R curve from B to C. Finally, at ac the crack length has become ac and both G and R are at point D. Crack growth at constant stress ac gives an increase of G according to the line DF. This line is above the R curve. Since G remains larger than R, final fracture occurs at point D where

Apparently, eq (5.19) is the energy criterion for fracture under plane stress. An evaluation of this fracture criterion is possible if an analytical expression for the R curve can be derived. Attempts to do this have been made [8-11]. A discussion about the evaluation of this fracture criterion is presented in chapter 8.

Krafft et al. [12] suggested that the R curve is invariant: irrespective of the initial crack size the R curve is .the same. This implies that the fracture condition for cracks of other sizes follows from a construction as in figure 5.9, where tangents are drawn to the unique R curve for various crack lengths.

The R curve represents the energy required for crack growth. In a ductile material this is the work for formation of a new plastic zone at the tip of the advancing crack, plus the work required for the initiation, growth and coalescence of microvoids. The latter is presumably small in comparison with the energy contained in the plastic zone. Therefore the R curve must start from zero as indicated in figure 5.9: at zero stress the size of the plastic zone is zero—it requires no energy to form a plastic zone of zero size. This also means that at any stress different from zero the energy criterion is fulfilled (apart from the work for void formation).

However, crack growth does not occur, because the stresses and plastic strains at the crack tip are still insufficient. The energy criterion is a necessary criterion for crack growth, but not a sufficient criterion.

The material at the crack tip is not ready to separate until the stresses and strains are large enough to make void initiation and coalescence possible. When this is the case a fairly large plastic zone has already formed. Crack growth can then take place only if sufficient energy becomes available upon crack growth to provide the work to form the plastic zone at the new crack tip. Crack growth cannot occur if this energy condition is not satisfied; it need not occur if it is satisfied.

Apparently, all conditions for crack growth are met at point B. This is the same point for cracks of sizes and a2• Therefore:

The onset of stable crack propagation is at a certain value of G, namely G, given by eq (5.20). At G, crack growth occurs from B to Cj or C2 where G = K. This first discrete crack extension is called pop-in, because it is a sudden metastable crack extension, which is often associated with an audible click. After pop-in gradual slow stable crack growth takes place until fracture follows at Dj or D2.

The values of G at fracture are Glcl and Clc2 respectively, for the two cracks considered. This means that the critical energy release rate for plane stress, Glc, is not a constant, and since K\C=EGW it follows that Klc is also not a constant. (The subscript 1 is used to indicate plane stress in mode I in order to distinguish Ku and Glc from the plane strain values Ku and G|C). Contrary to the case of plane strain where G,c and A'k within certain limits are material constants, both Gu and Klc depend upon crack size and they are larger for larger cracks.

It was noted in section 5.1 that a finite width correction can be applied to G through the relation K\ = EGl. By doing this and using the Irwin [3] correction (see chapter 3), it follows that

Eq (5.21) implies that the G lines are curved upward instead of straight, as shown in figure 5.10. For the small crack a, the width correction can be neglected and the G line is approximately straight. For longer cracks the curvature becomes larger and larger. As a result, the Gic value first increases from Glcl to Glc2 and than decreases again to Glc3, as shown once more in figure 5.11.

Various R curve shapes have been proposed [13], some examples, of which are shown in figure 5.12. The horizontal part in these curves is used to explain the pop-in behaviour. It is more likely, however, that the R curve starts at zero as explained earlier in this section. Figure 5.9

Figure 5.10. Gu for panels of finite size

Figure 5.10. Gu for panels of finite size

Figure 5.11. G'lc as a functionW crack size w

Figure 5.11. G'lc as a functionW crack size

Figure 5.12. Alternative R curves shows that pop-in behaviour can still occur if the R curve starts at zero. The shape of the R curve depends upon plate thickness. For thick plates under plane strain it is a straight horizontal line. Thin plates under plane stress show a steep, rising R curve. Plates of intermediate thickness in the transitional.region have an R curve between these two extremes.

The theory of the R curve is not yet well-established. Consequently, the foregoing discussion is somewhat speculative. Chapter 8 presents further information about R curves and plane-stress cracking. Examples of R curves as obtained from experiments are presented there.

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