Stress concentrations

When a specimen is stressed, one can think of continuous lines of force running through the material the stronger the force, the more the lines. If the specimen has constant cross section, the stress, which can be represented as the number of line of force per area, will be constant throughout the specimen. If the specimen narrows, the lines of force become more concentrated and the stress increases in that region. If there is a hole in the specimen, the lines will tend to flow aroimd the hole and become more concentrated in its vicinity. These lines become even more concentrated in the vicinity of a sharp discontinuity such as a square hole or a crack. (This is why portholes and hatches in the bulkheads of ships are roimd or oval. The windows in the Comet aircraft were square.) 

In 1898, G. Kirsch showed that stress concentration in the vicinity of a round hole increased by a factor of 3. G.V. Kolosoff solved the problem of stress concentration in the vicinity of an elliptical hole in 1910, but his paper was published in Russian and was virtually unknown in the West. In the West, the seminal paper on the subject was published by C.E. Inglis in 1913 in which he showed that the stress concentration in the vicinity of the tip of an elliptic hole was given by 

One can appreciate the effectiveness of the stress concentration in a crack whose tip radius is on the order of an atomic dimension. No doubt the reader is familiar with how much easier it is to tear open a plastic packet if there is a small indentation or crack to start the tear. It is also a common practice to drill a small hole at the end of a crack in a metal plate to blimt the crack by increasing the tip radius to relieve the stress concentration. 

One can make a simple model for fracture by estimating the stress at the tip of the crack by multipl)nng Equation 9.6 by the applied stress and equating this to the theoretical tensile strength, estimated to be E/10, 

Basically, in the vicinity of a sharp comer all fringes converge toward the apex. Having a high density of lines at this point indicates the presence of high stress level. At a rounded corner there will be considerably less concentration. Besides the molding problems, sharp corners often cause premature failure because of the stress concentration. To avoid these problems, inside comer radii should be equal to one-half the nominal wall 

Many injection molded products will influence the final product s performance, dimensions, and other characteristics. The mold includes the cavity shape, gating, parting line, vents, undercuts, ribs, hinges, and so on (Table 3-17). The mold designer must take all these factors into account to eliminate problems. At times, to provide the best design 

Water channels Side (actuated by cams, gears, or hydraulic cylinders) 

Ejector mechanism (pins, blades, stripper plate) Ejector return pins 

Hold cavity (cavities) in fixed, correct position relative to machine nozzle 

A detailed account of the development and of the present state of the theory of fracture mechanics is given in the series Mechanics of Fracture [2]. From the extensive general literature on this subject only a few works may be cited which deal with the deformation and fracture of engineering materials [3] or specifically polymers [4—6] and standards [7, 8]. For the determination of the material functions R and Kc three experimental methods of controlled crack propagation are predominantly used (Fig. 9.1)  

For details of these methods, their evaluation, and possible influences of sample geometry on the resulting material functions the reader is referred to the general literature cited [2—8]. In this Section the most frequently used crack opening mode, namely mode (I), will be considered (Figs. 9.1 and 9.2). 

For a uniaxially stressed isotropic, elastic, thin plate containing an infinitely sharp edge crack a state of plane stress arises which has a singularity at the crack tip [3-8], for r the components of stress are expressed as 

Tensile specimen with single edge crack. 

In the case of a thick plate the lateral contraction of the material at the crack tip is hindered a state of plane strain prevails and in addition to and Oy a normal stress Uz exists  

---

In Figure 5.24 the predicted direct stress distributions for a glass-filled epoxy resin under unconstrained conditions for both pha.ses are shown. The material parameters used in this calculation are elasticity modulus and Poisson s ratio of (3.01 GPa, 0.35) for the epoxy matrix and (76.0 GPa, 0.21) for glass spheres, respectively. According to this result the position of maximum stress concentration is almost directly above the pole of the spherical particle. Therefore for a... 

Figure 5.24 The predicted direct stress concentration at different locations within the domain...

A considerable reduction in stress concentration could be achieved by using a cross-bore which is eUiptical in cross-section, provided the major axis of the eUipse is normal to the axis of the main cylinder. A more practical method of achieving the same effect is to have an offset radial hole whose axis is parallel to a radius but not coincident with it (97,98). Whenever possible the sharp edges at the intersection of the main bore with the cross bore are removed and smooth rounded corners produced so as to reduce the stress raising effects. 

One aspect of pressure vessel design which has received considerable attention in recent years is the design of threaded closures where, due to the high stress concentration at the root of the first active thread, a fatigue crack may quickly initiate and propagate in the radial—circumferential plane. Stress intensity factors for this type of crack are difficult to compute (112,113), and more geometries need to be examined before the factors can be used with confidence. 

Division 2. With the advent of higher design pressures the ASME recognized the need for alternative rules permitting thinner walls with adequate safety factors. Division 2 provides for these alternative rules it is more restrictive in both materials and methods of analysis, but it makes use of higher allowable stresses than does Division 1. The maximum allowable stresses were increased from one-fourth to one-third of the ultimate tensile stress or two-thkds of the yield stress, whichever is least for materials at any temperature. Division 2 requkes an analysis of combined stress, stress concentration factors, fatigue stresses, and thermal stress. The same type of materials are covered as in Division 1. 

As the laminate industry grew, this anisotropic behavior was accepted and fabrication techniques adapted to it. For example, expansion and contraction space was left between wall panels, very strong adhesives were developed for bonding the product to substrates, special substrates were qualified, and where it was necessary to cut holes into the laminates the corners were radiused to prevent cracking from stress concentration. 

Tests using a constant stress (constant load) normally by direct tension have been described in ISO 6252 (262). This test takes the specimen to failure, or a minimum time without failure, and frequently has a flaw (drilled hole or notch) to act as a stress concentrator to target the area of failure. This type of testing, as well as the constant strain techniques, requires careful control of specimen preparation and test conditions to achieve consistent results (263,264). 

Stress concentration Kis defined (1) as local stress/mean stress in a particle and calculated according to if = 1 + 2 LR) length and R is the radius of the crack tip. 

Fracture mechanics (qv) affect adhesion. Fractures can result from imperfections in a coating film which act to concentrate stresses. In some cases, stress concentration results in the propagation of a crack through the film, leading to cohesive failure with less total stress appHcation. Propagating cracks can proceed to the coating/substrate interface, then the coating may peel off the interface, which may require much less force than a normal force pull would require. 

The code provides no guidance for analysis but requires that external and internal attachments be designed to avoid flattening of the pipe, excessive locahzed bending stresses, or harmful therm gradients, with further emphasis on minimizing stress concentrations in cyclic service. 

Under cyclic or repeated stress conditions, rupture of protective oxide films that prevent corrosion takes place at a greater rate than that at which new protec tive films can be formed. Such a situation frequently resiilts in formation of anodic areas at the points of rupture these produce pits that serve as stress-concentration points for the origin or cracks that cause ultimate failure. 

An important but frequently overlooked condition that can result in SCC involves concentration effects. These effects are of two types— stress concentration and corrodent concentration. 

---