The Concept of Stress Concentration

Any material which contains a geometrical discontinuity will experience an increase in stress in the vicinity of the discontinuity. This stress concentration effect is caused by the re-distribution of the lines of force transmission through the material when they encounter the discontinuity. Causes of stress concentration include holes, notches, keyways, comers, etc as illustrated in Fig. 2.62. 

The classical equation for calculating the magnitude of the stress concentration at a defect of the type shown in Fig. 2.62(b) is 

The parameter (1 -f l ajr ) is commonly termed the stress concentration factor K,) and for a hole where a = r then K, = 3, i.e. the stresses around the periphery of the hole are three times as great as the nominal stress in the material. 

It should be noted, however, that for a crack-like defect in which r 0 then Kt oo. Obviously this does not occur in practice. It would mean that a material containing a crack could not withstand any stress applied to it. Therefore it is apparent that the stress concentration approach is not suitable for allowing for the effects of cracks. This has given rise to the use of Fracture Mechanics to deal with this type of situation. 

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In the rest of this chapter, we will discuss briefly the theoretical ideas and the models employed for the study of failure of disordered solids, and other dynamical systems. In particular, we give a very brief summary of the percolation theory and the models (both lattice and continuum). The various lattice statistical exponents and the (fractal) dimensions are introduced here. We then give brief introduction to the concept of stress concentration around a sharp edge of a void or impurity cluster in a stressed solid. The concept is then extended to derive the extreme statistics of failure of randomly disordered solids. Here, we also discuss the competition between the percolation and the extreme statistics in determining the breakdown statistics of disordered solids. Finally, we discuss the self-organised criticality and some models showing such critical behaviour. 

As in the case of electrical failure in random conductor-insulator networks in the earlier chapter, we first discuss here the concept of stress concentration in an otherwise perfect solid, which is stressed and contains a single crack inside. Here, the stresses concentrate at the sharp edges of the crack, where it can become much larger compared to the external force. As one increases the external force, the crack starts propagating from such... 

The concept of stress concentration at the crack tip is represented in Figure 10.2. 

A good dispersion of rubber particles appears to favor the nucleation and growth of a large number of thick crazes uniformly distributed in the polystyrene matrix. This is believed to be an efficient source of energy absorption for the material under mechanical loading. The concepts of stress field overlap and critical volume of stress concentration zone for craze initiation were introduced to explain the observed mechanical behavior of HIPS. 

The discussion of stress concentration near a film edge in the next section is followed by a brief review of linear elastic fracture mechanics concepts, a prelude to a discussion of delamination and cracking due to film residual stress. A survey of these topics set in the context of fracture mechanics has been presented by Hutchinson and Suo (1992). The chapter also includes descriptions of various experimental techniques for evaluating the fracture resistance of interfaces between films and substrates. In addition, representative experimental results on the interface fracture resistance, as a function of interface chemistry and environment, are presented for a variety of thin film and multilayer systems of scientific and technological interest. 

To model this, Duncan-Hewitt and Thompson [50] developed a four-layer model for a transverse-shear mode acoustic wave sensor with one face immersed in a liquid, comprised of a solid substrate (quartz/electrode) layer, an ordered surface-adjacent layer, a thin transition layer, and the bulk liquid layer. The ordered surface-adjacent layer was assumed to be more structured than the bulk, with a greater density and viscosity. For the transition layer, based on an expansion of the analysis of Tolstoi [3] and then Blake [12], the authors developed a model based on the nucleation of vacancies in the layer caused by shear stress in the liquid. The aim of this work was to explore the concept of graded surface and liquid properties, as well as their effect on observable boundary conditions. They calculated the hrst-order rate of deformation, as the product of the rate constant of densities and the concentration of vacancies in the liquid. 

A characteristic feature of allelopathy is that the inhibitory effects of allelopathic compounds are concentration dependent. Dose-response curves with known compounds show an inhibition threshold. Below this level either no measurable effect occurs, or stimulation may result. Although the concentration of a compound required to exceed the inhibition threshold varies extensively according to different sensitivities among species and also among phases of the growth cycle for higher plants, the concept of an inhibition threshold seems consistent. Thus, it is reasonable to evaluate how, and if, a subthreshold concentration of an allelochemical may contribute to allelopathic interference. Also in need of evaluation is how environmental conditions may influence the deleterious action of an allelochemical and the concentration required for an effect. Such interactions are especially pertinent for those environmental situations that place some degree of stress on plant functions. 

The explanation of the effect of secondary inclusions on the delocalization of shear banding is based on the concept of modification of the local stress fields and achieving favorable distribution of stress concentrations in the matrix due to presence of inclusions. This leads to a reduction in the external load needed to initiate plastic deformation over a large volume of the polymer. As a result, plastically deformed matter is formed at the crack tip effectively reducing the crack driving force. Above approximately 20 vol% of the elastomer inclusions. 

In this section, the case of a semiinfinite solid with a concentrated force acting on the boundary is introduced. This case was originally solved by Boussinesq (1885). It should be noted that the only difference between this case and the case of a point force in an infinite solid medium is the boundary conditions. Shear stresses vanish on the boundary of the semiinfinite solid. In the following, the concept of a center of compression is introduced. The stress field in a semiinfinite solid with a boundary force can be obtained by superimposing the stress fields from a point force and a series of centers of compression. A center of compression is defined as the combination of three perpendicular pair forces. 

At this point, we have adapted the concept of chemical potential so as to apply it in situations where the stress field is nonhydrostatic as well as nonuniform through space. We have related a material s strain rate or change-of-dimension behavior to its chemical-potential field. But so far we have discussed only problems of continuum mechanics type—the same problems that we were able to discuss effectively in terms of stress. We have used chemical potential in describing continuum-mechanics behavior, the left-hand two boxes in Figure 17.1 but we have not yet used chemical potential in describing change of concentration of an atomic species, which is of course one of the concept s most natural and powerful uses. 

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