Elliptical hole

Figure fl.11 An aluminum cooling water conduit severely attacked by a caustic in service. The large circular hole in the machined face is corroded around its edges. The elliptical hole between the 6- and 7-in. ruler markings was formed by corrosion penetrating the conduit wall. 

One of the first solutions to the problem of stresses around an elliptical hole in an infinite anisotropic plate was given by Lekhnitskii [6-7]. A more recent and comprehensive summary of the problem and many others is Savin s monograph [6-8]. Numerous results by Lekhnitskii are shown in his books [6-9 and 6-10]. Two special cases are of particular interest. 

S. G. Lekhnitskii, Stresses in Infinite Anisotropic Plate Weakened by Elliptical Hole, DAN SSSR, Vol. 4, No. 3, 1936. 

Fig. 7.2. Dugdale-Muskhelishvili-model of plastic zones at the ends of a loaded elliptical hole [81].

If a thin solid plate contains an elliptic hole of the half-axes a (the longer one) and b, and the ellipse is oriented so that the longer axis is perpendicular to the direction of the tensile force, then the stress at the two ends of this axis is... 

Valving on a PDMS sheet has also been achieved after an elliptical hole was punched in the PDMS, and an elliptical needle was inserted in the hole. Then valve actuation was achieved by having the elliptical needle rotated for 90°, thus compressing the fluidic channel below for valve closure [1043]. A PDMS membrane was fitted on a glass wafer to create a valve. Three valves placed in series form a diaphragm pump [244]. 

For an elliptical hole (a = long semi axis, b = short semi axis) the radius of curvature at the end of the long axis is b2/a so that... 

In this section the maximum stress concentration developed by an elliptic hole at arbitrary orientation in an infinite elastic sheet subjected to general biaxial stress loading is considered. In Figure 1, consider an elliptic hole of major axis a, minor axis b, and orientation angle fl with respect to an axis of applied uniaxial tension Si. If the sheet is isotropic, elastic, and infinite, then the major principal stress ([Pg.42]

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 ... 

Fig. 1.6. A conducting plate containing an elliptic hole (dielectric) with semi-axes lengths I and 6, subjected to potential difference EqL at its two ends in the vertical direction. Current density in the plate becomes nonuniform and concentrates at the two horizontal tips of the ellipse.

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... 

We now intend to examine the modifying effect of the elliptic hole on the distribution of stress in the plate. A straightforward but tedious solution of the Laplace equation for the displacement vector field leads (See section 1.2.2(a), for the equivalent solution of the scalar potential problem in a two-dimensional conductor with an elliptic dielectric hole inside), to the largest concentration of stress at the point C, where... 

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.

Here the solution of Dugdale will be followed since it clearly shows a significant feature of the model. A flat elliptical hole of length 2c is considered in an infinite plate loaded in tension by a stress a remote from and normal to the ellipse. The ends of the ellipse terminate in small plastic zotees whose boundaries are under uniform internal pressure stresses (see Fig. 2.2). For these internal stresses static equilibrium is achievai by imposing equal and compressive opposite stresses ct,.. This... 

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. 

Consider an elliptical hole characterized by a semimajor axis of length la and semiminor axis of length lb. The result of a detailed elastic analysis of a plane strain geometry with remote loading parallel to the semiminor axis (call this direction y) is an enhancement of the local stresses well above the remote value of (To- The key point that emerges from an analysis of the stresses associated with an elliptical hole are summarized in schematic form in fig. 2.12. In particular, the maximum value of the stress Oyy is given by... 

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... 

Fig. 2.12. Elliptical hole in a material and the resulting stresses in the vicinity of the hole.

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 ... 

The stresses and displacements at the boundaries of the deformable domain (applied or enforced) dictate the boundary conditions, which effectively determine the particular solution to the PDFs. If the problem is truly three-dimensional (because geometry and/or loads vary in all three directions) then numerical solutions are needed for all but the most elementary example cases. FFA is the de facto choice for such scenarios and commercial codes abound ABAQUS and ANSYS (software programs) are arguably the most common. An example of a simple geometry that is nevertheless a truly three-dimensional problem is a film capping an elliptical hole, that is, an elhptical plate or membrane. 

The case of an infinite plate with an elliptical hole was solved by Lekhnitskii [20]. Using anisotropic elasticity and treating the plate as homogeneous, he determined the stresses anywhere in an infinite composite plate with a hole. Due to the stress concentration effect created by the hole, the highest stresses are on the hole boundary. It is important to note that, unlike metals, where the critical location is at 0 = 90° (see Figure 6.8) the critical location for a composite plate is a function of the stacking sequence. It can be shown that Lekhnitskii s solution on the hole edge becomes ... 

Figure 4.21 Small elliptical hole in a plate subjected to uniaxial tension.

The simplest approach to developing a fracture criterion would be to equate the highest tensile stress associated with a crack to the theoretical cleavage stress. For the geometry shown in Fig. 8.3, the crack could be considered to be an elliptical hole and the elastic solution discussed in Section 4.9 can be utilized, i.e.. 

As an example, consider the solution for the maximum stress at the end of an elliptical hole in a large plate under uniaxial tension. Substituting Eq. (8.5) into Eq. (8.43), one finds = which is in agreement with Eq. (8.25). This... 

Figure 8.27 Elliptical hole subjected to a central point force.

Fig. 5.38 A plate with elliptical hole, subjected to the uniform stress...

The interest in stress concentrations in elastic solids for applications to fracture starts with Inglis (1913), who considered the stress-coneentrating effeet of elliptical holes with major and minor axes a and h in a tensile stress field with a aeting across the major axis of the ellipse. His expression for the eoneentrated tensile stress (Ta at point A of the tip of the ellipse is... 

Ultimately, fracture can only occur if all atomic bonds in an area are pulled apart and break. The stress necessary to break a bond (the theoretical strength) is between E/S and E/20, where E is the elastic modulus of the material [51]. Typical tensile stresses applied to highly loaded components are in the order of /1000 or even smaller, and yet fracture of components still occurs. To explain this discrepancy, it is necessary that certain strong local stress concentrations exist these are termed Jlaivs. The action of flaws can be discussed by using the simple example of an elliptical hole in a uniaxial tensile-loaded plate. At the tip of its major semi-axis (which is perpendicularly oriented to the stress direction), the stress concentration is [52] ... 

An elliptical hole must be extremely elongated to create the stress concentration necessary to reach the theoretical strength. To give a simple example, for an applied stress of o = E/1000 the ratio of major to minor semi-axis must be c/h w 50 (c/q > 2500) to create the stress concentration necessary to reach the theoretical strength at a notch tip /10. 

Puncture of the skin may be treated in the same way, except that the damage should be cut out to a round or elliptical hole and new honeycomb of the correct type potted in before fitting a repair plate. Many aircraft manufacturers allow very thin aluminium alloy skins to be repaired with fibreglass overlays and they provide a table relating skin thickness to the number of fibreglass layers, or rather the thickness of fibreglass required. 

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