Elliptical cracks

The results on contact problems for plates without cracks can be found in (Caffarelli, Friedman, 1979 Caffarelli et al., 1982). Properties of solutions to elliptic problems with thin obstacles were analysed in (Frehse, 1975 Schild, 1984 Necas, 1975 Kovtunenko, 1994a). Problems with boundary conditions of equality type at the crack faces are investigated in (Friedman, Lin, 1996). 

We prove the existence of solutions for the three-dimensional elastoplastic problem with Hencky s law and Neumann boundary conditions by elliptic regularization and the penalty method, both for the case of a smooth boundary and of an interior two-dimensional crack (see Brokate, Khludnev, 1998). It is shown in particular that the variational solution satisfies all boundary conditions. 

We prove an existence theorem for elastoplastic plates having cracks. The presence of the cracks entails the domain to have a nonsmooth boundary. The proof of the theorem combines an elliptic regularization and the penalty method. We show that the solution satisfies all boundary conditions imposed at the external boundary and at the crack faces. The results of this section follow the paper (Khludnev, 1998). 

Fracture mechanics analysis requires the determination of the mode I stress intensity factor for a surface crack having a circular section profile. Here the circular section flaw will be approximated by a semi-elliptical flaw. 

Irwin [23] developed an expression for the mode I stress intensity factor around an elliptical crack embedded in an infinite elastic solid subjected to uniform tension. The most general formulation is given by ... 

Although Griffith put forward the original concept of linear elastic fracture mechanics (LEFM), it was Irwin who developed the technique for engineering materials. He examined the equations that had been developed for the stresses in the vicinity of an elliptical crack in a large plate as illustrated in Fig. 2.66. The equations for the elastic stress distribution at the crack tip are as follows. 

FIG. 13.82 An elliptical, through thickness crack in an elastic sheet subject to a stress a in the x2-direction along Oxl as shown Oxford University Press. 

Besides the maximum tensile stress, also the stress distributions in the tensile direction and the opposite direction are shown in Fig. 13.82. Failure occurs where the stress concentration is highest, i.e. at the tip of the crack. It shows that the stress increases for longer (a increases) and sharper (p decreases) cracks. For thin elliptical cracks, e.g. with a/b = 500, the maximum stress amplification is 1000. Of course cracks are in general not elliptical, but it shows the enormous amplification of the average stress at the crack tip. 

If you ve ever taken any mechanics classes you probably recall that a crack acts as a stress concentrator. In a hypothetical flawless material the lines of stress are uniformly spaced out and a load is evenly borne by all the atoms or molecules in the object But the presence of a hole or a crack requires the stress to go around the opening (Figure 13-31). The stress concentration depends upon the size and shape of the defect. Ing-lis calculated the stress concentration factor for an elliptical hole to be given by Equation 13-23 ... 

The current concentration factor 1 -h I/b) = (1 + / /p), obtained above in (1.21), is also valid for the stress concentration in a stressed (two-dimensional) solid containing an elliptic void with the semi-major and -minor axes of lengths 21 and 26, and having curvature p (= 6 //) at the tips of the major axis. One can therefore easily see that if the void is sharp enough (p 0), or if its length 21 is very large, the stress concentration can increase several levels above the external stress level (far away from the defect) and the solid may break or fracture (or fuse) from the defect or crack tip. 

For quantitative analysis, Inglis considered a uniformly stressed two-dimensional solid like a thin plate, containing an elliptic hole representing the crack (see Fig. 3.4). Let the lengths of the semi-major and -minor axes of the ellipse to be 21 and 2b repectively, and a denote the external (say tensile) stress applied on the sample along the y-direction. We assume the (linear) Hooke law to hold everywhere in the plate and that the boundary surface of the elliptic hole, represented by the equation... 

In a three-dimensional solid containing a single elliptic disk-shaped planar crack perpendicular to the applied tensile stress direction, a straightforward extension of the above analysis suggests that the maximum stress concentration would occur at the two tips (at the two ends of the major axis) of the ellipse. The Griffith stress for the brittle fracture of the solid would therefore be determined by the same formula (3.3), with the crack length 21 replaced by the length of the major axis of the elliptic planar crack. 

Let us consider a thin sheet of width W and thickness B containing an internal elliptical crack, as shown in Figure 14.29. The axes of the ellipse are 2a and 2b, and the laminar sample is assumed to have infinite width, i.e., W > 2a. If a force F is applied at the end surfaces, the sample will be supporting a stress, a ... 

Figure 14.29 Internal elliptical crack in an elastic lamina subjected to a stress a. Distribution of local stresses in the positive x direction.

For an elliptical region surrounding the crack, the particles obeyed the following equations of motion ... 

An instructive two-dimensional calculation that reveals the stress magnifying effects of flaws is that of an elliptical hole in an elastic solid as depicted in fig. 2.12. The crucial idea is that, despite the fact that the specimen is remotely loaded with a stress uq which may be lower than the ideal strength needed to break bonds in a homogeneous fashion, locally (i.e. in the vicinity of the crack-like defect) the stresses at the termination of the major axis of the hole can be enhanced to values well in excess of the remote load. The exact solution to this problem can be found in any of the standard references on fracture and we will content ourselves with examining its key qualitative features. 

The elliptical hole prepares our intuition for the more extreme case that is presented by the atomically sharp crack. For the sharp crack tip, the dominant stresses near the crack tip may be shown to have a singular character with the particular form (see Rice (1968) for example). The realization that the crack tip fields within the context of linear elasticity have such a simple singular form has... 

As the root radius (or radius of curvature) approaches zero, or as the elliptical notch is collapsed to approximate a crack, then the maximum stress should approach infinity (i.e., as p 0, am -> oo). 

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