Issue34

L.P. Pook, Frattura ed Integrità Strutturale, 34 (2015) 150-159; DOI: 10.3221/IGF-ESIS.34.16 153 shape adjusts itself such that the crack front intersection angle tends to  c . For the symmetric mode and  = 0.3, β c = 100.4  which, as predicted, is approximately the crack front intersection angle in Fig. 4. For the antisymmetric mode and   0.3,  c  67.0  . Crack front intersection angles of about this value have been observed [20]. Figure 5 : Angle crack test piece, definition of crack Figure 6 : Definition of crack front intersection angle, β . surface intersection angle, γ. C RACK PATHS Crack front line tension crack has some analogies with a crystal dislocation. In particular, the elastic stress fields associated with a crack front and with a dislocation are both singularities. The associated energy means that a dislocation has a line tension, which controls its shape under an applied stress field. Similarly [21], a crack front has a line tension which controls its shape, but with the important difference that a crack can grow, but in general cannot contract. At a corner point the corner point singularity provides a point force which balances the line tension in a direction corresponding to the crack front intersection angle. In consequence on a macroscopic scale, a crack front is smooth, and is usually curved as in Fig. 4. Any initial sharp corners rapidly disappear [22]. Further in some circumstances, a growing fatigue crack tends to a particular stable shape [23, 24]. Initial direction of crack growth In general, criteria are needed for the formation of a branch crack, its initial direction, and once formed, whether it will grow [25]. It is important to distinguish between criteria for formation of a branch crack and criteria for its growth. Depending on circumstances, either can dominate behaviour [26, 27]. In two dimensions only modes I and II are possible, the crack front is a point and the crack is a line. The direction of a mode I branch crack is given by the angle θ (Fig. 7). Related experimental work is usually carried out on plates or sheets of constant thickness, which are regarded as quasi two-dimensional. Numerous criteria have been proposed for the initial direction of crack growth [28]. Predicted initial branch crack directions are not sharply defined, so minor deviations from isotropy may have a significant influence on initial branch crack direction. In mixed modes I and II fatigue testing measurement of the initial branch crack direction is of interest. However, the irregularity of actual fracture surfaces and, for curved branch crack paths, the need to estimate a tangent at the start of the branch crack make it difficult to measure crack directions accurately. The directions given in Reference [29] are only repeatable to within 2 or 3 degrees. Experimental results show that under fatigue loading initial crack directions vary widely [30]. The most widely used criterion for the initial crack growth direction, θ , is the Maximum Tensile Stress (MTS) criterion proposed by Erdogan and Sih in 1963 [31]. They considered a circle around the crack tip and took the direction of crack growth to be in the direction of the maximum tangential stress. This is a principal stress so the shear stress is zero, and leads to K I sin  = K II (3cos  - 1) (70.5…     -70.5…  ) where K I and K II are the modes I and II initial crack stress intensity factors. An alternative approach is to find the value of θ for which the mode I branch crack stress intensity factor, k I , has its maximum value and the mode II stress intensity, k II , is zero. Calculation of k I and k II by comparison of stress field components leads to the same result [26, 27], so the two approaches are equivalent. Several criteria have been proposed for the initial direction of crack growth in the general three dimensional case of mixed modes I, II and III loading [28]. In 2001 Schöllmann et al [32] proposed a new criterion, which assumes that crack growth from a point on the crack front is perpendicular to the maximum principal stress. In two dimensions this is equivalent to the MTS criterion. A complication is that, in the presence of mode III on the initial crack, a mode I branch crack only intersects the initial crack front at one point [26, 27, 28]. What happens when a fatigue loading is applied in the presence of mode III on the initial crack is illustrated by the fracture surface of the mild steel angle notch test piece shown in Fig. 8 [33] (cf Fig. 5). The rough area is static fracture where the test piece was broken open for examination. The expected A