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Jackson Evans
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Ip Vanish Crack [TOP]

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Ip Vanish Crack

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Some new forms of energy components must be incorporated into the energy balance equation, from which the input of energy needed to propagate the crack and subsequently the stress at the onset of fracture can be determined. The additional energy that significantly dominates over the surface energy is the irreversible energy dissipated by the way of the plastic strains that precede the leading edge of a moving crack. For stationary cracks the additional terms within the energy balance equation were introduced by Irwin and Orowan. An extension of these concepts is found in the experimental work of V. Panin, who has shown that the irreversible deformation is primarily confined to the prefracture zones associated with a stationary or a slowly growing crack.

cracks. Griffith crack is either stationary or catastrophic under the positive K-gradient loading configuration. What visibly is missing in the classic theory is the transition period from a stationary state to a moving crack, which is accomplished by the insertion of the period of slow stable crack growth (SCG) and made possible by accounting for the highly non-linear deformation processes preceding fracture.

Here r is the Euler gamma function. We note that for 1

First we shall consider the case of a smooth Euclidean crack represented by a structured cohesive crack model as shown in Fig. 3. Two sets of coordinates are used, the dimensional coordinates are shown in Fig. 3, a, while the nondimensional are explained in Fig. 3, b. The distance measured from the origin of coordinates is denoted by x (or 5 = x/ a), while the distance measured from the tip of the physical crack is denoted by x1 (or X = xjR), and the ratio a/a1 - k. The crack length is a, the length of the extended crack is a1 = a + R, and the profile of the entire crack is described, cf. [4, 11], by the following expression involving the inverse hyperbolic functions

Fig. 4. Shows the opening displacements at the center of a cohesive crack and at the tip of the physical crack as functions of the applied load Q. Shows the ratio of CTOD (crack tip opening displacement) to CCOD (crack center opening displacement). Despite the nonlinear nature of the problem these results show that CTOD is roughly one half of CCOD through the entire range of the loading parameter Q. The ratio depicted in the figure can be very closely approximated by the simple equation CTOD = = 0.504 CCOD

Figure 4, a shows the dependence of the tip displacement and the center displacement on the applied load Q, while Figure 4, b illustrates the fact that the ratio of these two quantities CTOD/CCOD remains constant almost throughout the entire range of loading. The constant is 0.504, which provides a good rule of thumb: the tip displacement of the cohesive crack is roughly one half of the mouth displacement measured at the crack center. This observation provides helpful information for an experimentalist, who utilizing various clip gages can access the center of the crack much easier than the tip of the physical crack. Thus, once CCOD is measured, the tip displacement, CTOD, can be estimated with good accuracy.

The profile of the entire extended crack, normalized by the constant C = 4gy a/(nE), is shown in Fig. 5, a, while Figure 5, b shows the same profile normalized by CTOD. In constructing these figures the following equations were used

Fig. 5. Profiles of the cusp region of the cohesive crack: when the crack opening displacements are normalized by the constant C _ 4gy / (nE), see Eq. (10) (a), and when half of CTOD is used as the normalization constant, see Eq. (11) (b). Nondimensional loading parameter, proportional to g/ gy , is denoted by Q

Figure 6, b shows that the agreement between the exact and the approximate formulae for the cohesive crack opening displacements within the cusp region is indeed good for all values of the loading parameter Q. Therefore, to simplify all further calculations we shall employ formula (14). Of particular interest will be the strains within the Panin zone, which are defined by the derivative


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