Elliptical defect

The circular nature of the skin defect in living skin, as opposed to the elliptical defect in cadaver skin, underscores the ability of skin fibroblasts... 

The more dangerous defects will be flat defects perpendicular to the current lines. In two dimensions they are lines, but in three dimensions they have the shape of cylinders. Prom the results of Section 1.2.2(a), one can see that the enhancement factor for an elliptic defect (see Fig. 2.4) is k = l/b. Thus a long defect parallel to the current lines b I ) does not affect the current lines, while a long defect perpendicular to the current line b 1) affects them strongly. 

We shall now determine what is the probability of occurrence of a dangerous defect as defined in the preceding section. In the present case of a lattice, it consists in n neighbouring resistors belonging to the same plane perpendicular to the current flow. In d = 2, it is merely a line of n removed bonds and in d = 3, it is an ensemble of n removed bonds forming a hole with approximately the shape of a disc. The current in the immediately adjacent parallel resistor will be in accordance with the enhancement factor in an elliptic defect... 

Figure 4.20. Defects in the form of ellipses and hyperbolae in confocal relationship (a) the elliptical defect is in the plane of the sample him (b) the elliptical defect is normal to the him plane (c) other cases.

Fig. 10.22 Damage progression in the spinel/polycarbonate laminate with 4 % elliptical defects internally inserted along the center line of the spinel after different time durations a 16 ps, b 30.37 ps, c 32 ps and d 52 ps. The damage progression appeared to grow at the sharp edges of the defects until the ceramic was completely failed. Note that the damage did not penetrate the void areas, but was initiated at the boundaries. Reproduced with permission from [166]. Copyright 2009, Elsevier...

This programme was presented last year (ref. 2). The purpose of TERFIS 5 tests carried out in 1996 was to quantify the influence of residual stresses on crack initiation and growth in weldments. They were obtained on stainless steel 316 L tubes where a defect was machined in a welded joint. The tubes were submitted to a low uniform tension (29 Mpa) and to thermal stocks (ex in the case of a circunferential defect AT in the thickness 180°C and 10 cycles in the case of a semi-elliptic defect AT in the thickness 300°C and 1500 cycles). 

Figure 14.9 Fracture surface showing a progressively loaded elliptical defect in which dye penetrant was applied ateach loadingstep, highlightingthe greater growth ofthe crack in the vertical versus horizontal direction, consistent with stress intensity factor calculations. From Ref [7].

Example 4.6. An A302-B material with a yield stress of 50 ksi is to be used in a pressure vessel. The Cv value is 15 ft-lb and an examination of the percent shear in the cross section of tested specimens indicate a temperature in the transition range. Ultrasonic examination of the plate uncovered an elliptical defect under the surface that is 0.375 in. long and 0.25 in. deep. How safe is the vessel if the operating stress is yield ... 

What I would do is the following (a) For low l take the quantum defect matrix elements as they result from the fittings, (b) For high l evaluate them in elliptic coordinates assuming no penetration (the effects of the dipole field are then fully included) in this way a full calculation with an arbitrary number of 1 components can be carried out. 

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. 

Fig. 3.4. A portion of a plate containing an elliptic hole with semi-axes lengths I and 6, under uniform stress a. The stress distribution within the plate becomes nonuniform near the defect (hole) and stresses concentrate at the tips (sharp edges at the horizontal ends) of the hole.

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. 

A crack (or other defect) in a solid will increase the stress in the region of the crack tip by a considerable amount. The stress concentration factor, Ki, is the ratio of the maximum stress due to a crack to the mean stress in the absence of the crack. The value of Ki at the end of the major axis of an elliptical hole or elliptical crack tip in a plane sheet of solid stressed as in Eigure S4.8, was calculated early in the 20th century to be given by ... 

Fig. 8.30 Focal-conic defect structure in SmA A pair of cones with a common elliptical base and a hyperbola connecting cone apices (a) cross-section of the upper cone by plane ABC with gaps between lines (Dupin cyclides) indicating the smectic layers (b) filling the space of the sample by cones of different size (c)...

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