Critical crack lengths

Critical loads and critical crack length for some materials calculated from equations (6.2.15)—(6.2.18) (Hagan, 1979)... 

When some critical crack length (between 200 pm and 400 pm, i.e. about 15% of the contact diameter) is reached, a brittle propagation stage is observed which is associated with a sudden and drastic drop in the lateral stiffness, K. The measured crack width in the plane of the contact is then of the order of magnitude of the contact diameter. Post-mortem microscope observation of specimen cross sections in the contact zone (Fig. 8) indicates that the depth of the cracks is of the order of magnitude of the contact radius (i.e. about 900 pm). The two deep cracks induced at the edge of the contact may thus be viewed as some kind of half-penny cracks whose radii are approximately equal to the radius of the contact. In the subsequent part of this paper, the two deep cracks will be referred to as primary cracks . 

Interestingly, the ductile-brittle transition observed for the MIM system provided an opportunity to assess the material fracture toughness, which was not possible using classical fracture mechanics tests due to the intrinsic brittleness of the MIM system. The measurement of the critical crack length, Lc, in the contact plane at the onset of brittle propagation allows estimation of a fracture toughness K C = a x+JnLc in the order of 0.85 MPa m1/2, i.e. much less than that of a poly(methylmethacrylate) homopolymer (1.20 MPa m1/2). 

Note the inverse square dependence on the stress (squared). As the load on a material increases, the critical crack length gets shorter. 

At room temperature, PP is close to its Tg(0-25°C) and well above its normal brittle-ductile transition temperature ( -30°C). However the presence of surface cracks in the photo-oxidized film is apparently sufficient to promote brittle failure at room temperature. According to the Griffith crack theory, once a critical crack length has been exceeded, a critical crack velocity is required to propagate the crack. If this velocity is not exceeded, cold drawing of the amorphous zones ensues. 

The energies involved are illustrated in Figure 17.7b, and it is seen that a maximum in free energy occurs at a crack length Lcr, beyond which the crack will grow spontaneously if the test piece remains under stress. From Formula (17.8) it follows that the critical crack length is given by... 

The classic Griffith-Orowan theory describes the relationship between strength and toughness of brittle materials such as ceramics (Griffith, 1920 Orowan, 1949). In the simple basic equation of the theory, the stress to fracture <7f is related to Youngs elastic modulus E, the fracture energy y and the critical crack length c by... 

Figure 11.5 (a) Uniformly stressed solid, (b) Relaxed volume in vicinity of crack of length c. (c) Plot of Eq. (11.7) as a function of c. The top curve represents the surface energy term, and the lower curve represents the strain energy release term. Curve labeled L ot is sum of the two curves. The critical crack length Ccric t which fast fracture will occur corresponds to the maximum, (d) Plot of Eq. (11.7) on the same scale as in part (c) but for y/2 times the applied stress applied in (c). Increasing the applied stress by that factor reduces Ccni by a factor of 2. 

The starting point of the fracture theory is the Griffith-Irwin theory of cohesive fracture. It has been extended to the adhesive fracture According to this theory the fracture strength a of an adhesive bond is related to the fracture energy e and the critical crack length 1. It is... 

The variation in the free energy with the number of loading cycles is presented in Figure 17.4. The graph shows that the more the stress amplitude the smaller the size of the crack embryo. The critical crack length is achieved in 170 cycles under an amplitude of 100 MPa, but under a stress amplitude of 75 MPa, 350 cycles are necessary. There is a critical number of cycles at which the free energy has a maximum and where the system becomes unstable, that is, when... 

The intersections of yield and fracture lines mark the transitions between modes. Most usually, the crack length a is a small fraction of the width of the bar. The critical crack length is then a (see Figure 5.17). As stress is increased from zero, the event which occurs first, depending on the size a of crack present, is... 

The third curve represents the difference between released energy and rupture energy for the various lengths of crack the quantity Z,g represents the critical crack length, that is the value for which the increase of length of the crack releases more energy than is consumed in the creation of new rupture surfaces. 

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