Issue 41

D. Nowell et alii, Frattura ed Integrità Strutturale, 41 (2017) 197-202; DOI: 10.3221/IGF-ESIS.41.27 199 Since the effect of the wake is clearly important in the characterization of crack tip conditions for a fatigue crack, we decided to investigate a third possible analysis method, using a model with a wake representation built in. The Christopher/James/Patterson (CJP) model [6],[7] was selected because of its relatively simple formulation. D ISCUSSION OF THE CJP MODEL he CJP model attempts to capture some of the additional phenomena generated by a fatigue crack with a plastic wake. This approach leads to a crack description with four parameters as follows: - K F The ‘Forward Stress Intensity Factor’, which is essentially similar to the applied K I in a conventional analysis. - K R A ‘Retardation Stress Intensity Factor’, which arises as a result of the residual stress field set up by the wake, and which might be thought similar to the offset shown in Figure 2. - K S A ‘Shear Stress Intensity describing’ the shear present between the plastic wake and the surrounding elastic material. - T The conventional T-stress or bounded term in the Williams expansion. It is easy to misunderstand some of these terms, so a brief further explanation will be given here with the aid of a diagram (Fig. 3) modified from that presented by James et al. in [7]. First, it is essential to understand that the plastic zone at the tip of the crack creates a wake along the crack faces, as the crack propagates, Hence, there is a plastic enclave of the approximate shape shown in grey in Fig. 3. If one then considers the boundary between this enclave and the surrounding elastic material there may be x, y and shear forces transmitted across this interface and in each case, the force on the enclave by the elastic hinterland will be equal and opposite to that exerted by the enclave on the elastic material. Figure 3 shows the forces exerted on the plastic enclave. Figure 3 : Forces between the plastic enclave (grey), and the surrounding elastic material (after [7]). Starting with Fig. 3a, at maximum there is clearly an applied force opening the crack and causing tensile yield. This is labelled F Ay in the figure. Similarly, there may be shear at both top and bottom of the enclave, and these forces (which for mode I must be in the same direction) are labelled F s in Fig.3. Since the enclave must be in equilibrium, these two shear components must be reacted by a horizontal force, termed F Ax . The authors include a separate term for the T-stress, but this is clearly not a net force across the interface. Hence, it is perhaps more helpful to think of T as being a bounded response to F Ax , which clearly does not cause any stress intensity at the crack tip. At the minimum applied load, James et al consider the crack at least partially closed (Fig. 3b.). They then show F S reversed in sign, which seems plausible, since the crack has reached this state by unloading from the maximum load. The F x force however, is not reversed in sign, so that if Fig. 3b is considered as a free body diagram, then the plastic enclave would not be in equilibrium. What James et al do, however, change is the notation for the x-direction force, calling it now F Px presumably to show that the plastic zone is somehow the entity responsible for the force, rather than the applied load. Of course, in the general case, both applied load, and plastic zone resistance contribute to the force across the interface, and it is not possible (or perhaps helpful) to separate them. In the vertical direction, the sign of the main vertical force is reversed, and the notation is changed. Further, an additional force F C is introduced to represent the contribution to vertical force caused by crack closure. James et al. [6], [7], then go on to collapse the plastic enclave onto the line of the crack, and develop a Muskhelishvili stress function for the surrounding elastic material. This eventually results in the four terms defined above. T (a) (b)