Elastic stress concentration

The stress intensity factor is a means of characterising the elastic stress distribution near the crack tip but in itself has no physical reality. It has units of MN and should not be confused with the elastic stress concentration factor (K,) referred to earlier. 

Impact strength also increases as the notch depth is decreased. The variation of impact strength with notch depth and radius may be rationalised for some materials by use of the linear elastic stress concentration expression. 

For the purposes of performing an impact test on a material it is proposed to use an elastic stress concentration factor of 3.5. If the notch tip radius is to be 0.25 mm estimate a suitable notch depth. 

Flaw activity and the resultant stress concentration factors can also be expected to depend on the material s rheology. The sample loading history and path play a major role in determining the behavior of a given flaw as described elsewhere (6,11,12,13), and these ideas are currently being extended to account for recent developments in constitutive equation theory for solid polymers and the idea of a flaw spectrum. In this paper, time and path dependence are not considered further, and the calculations are based on elastic stress concentration factors associated with elliptic flaw geometries. 

Stress Bate at Particles. The stress component, avv, acting parallel to the boundary between rubber particles and matrix is important for the initiation of crazes. It reaches a maximum value (which can be about twice the outer stress, a0) at the equatorial regions of the particles. Besides depending on the shape of the particles and Poisson s ratio, the elastic-stress concentration at the rubber particles depends mainly on the ratio x = Gp/GM, where GP and GM are the Youngs modulus of the particles and the matrix, respectively. This ratio has been calculated by Michler (14) on the basis of the solution obtained by Goodier for an isolated particle embedded in a matrix and subjected to uniaxial tension (15) (see Figure 9). 

Figure 17. Schematic representation of the three-stage mechanism of toughening in PA6 blends (a) elastic stress concentration, between particles and the for-...

Elastic stress concentrations cannot explain most product failures, since yielding nearly always occurs before crack initiation. However, they indicate locations where yielding is likely to occur first. Therefore, the failure stress in Charpy impact tests (Section 9.5.1) should not be calculated using the notch q value. Craze formation is another form of (localised) yielding, which also modifies the stress distribution in the product. Section 9.4.4 shows that craze breakdown may occur at a critical opening displacement, rather than at a critical stress. Hence, elastic-plastic analyses must be used for most polymer product failures. 

Here, the main interest is in the opening-mode response, so we discuss only the mode I form of linear elastic stress concentration in polar coordinates, r, 0, and z see Fig. 12.2. References to texts giving more complete coverage for modes II and III are given at the end of the chapter, as stated above. 

Both Eqs. (11.1) and (11.2) account for the effect of transverse strain on plastic strain intensity factor characterized by the modified Poisson s ratio, V. In Eq. (11.1), this is accounted for by the ratio Sy/Sa, whereas in Eq. (11.2) the ratio Eg/E serves the same purpose as will be shown later. The modified Poisson s ratio in each case is intended to account for the different transverse contraction in the elastic-plastic condition as compared to the assumed elastic condition. Therefore this effect is primarily associated with the differences in variation in volume without any consideration given to the nonlinear stress-strain relationship in plasticity. Instead the approaches are based on an equation analogous to Hooke s law as obtained by Nadai. Gonyea uses expression (rule) due to Neuber to estimate the strain concentration effects through a correction factor, K, for various notches (characterized by the elastic stress concentration factor, Kj). Moulin and Roche obtain the same factor for a biaxial situation involving thermal shock problem and present a design curve for K, for alloy steels as a function of equivalent strain range. Similar results were obtained by Houtman for thermal shock in plates and cylinders and for cylinders fixed to a wall, which were discussed by Nickell. The problem of Poisson s effect in plasticity has been discussed in detail by Severud. Hubei... 

Chattopadhyay incorporated the peak stress effects addressing them as notches and employed a local strain approach to provide an appraisal of the ASME and MlTl Codes with reference to the fCg factor. In that work, the ratio Sp/is treated as a strain concentration and is used as a square of the elastic stress concentration using Neuber s approach. Thus... 

These parameters are defined in the scheme of Fig. 6.5(a). For a constant particle volume content Vp, the distance A depends directly on the diameter D (A D). Under load, the elastic stress concentration is enhanced up to after void formation around the particle. In the case of small particles and small distances, only very thin matrix strands exist between the voids, which can be easily stretched (distance A below a critical size see Fig. 6.5(b)). In the case of larger particles,... 

ELASTIC STRESS CONCENTRATIONS AND JOINT EFFICIENCIES FOR ALUMINIUM-... 

As explained above, the existence of a sharp notch can strengthen the ductile metal due to the triaxiality of stress. The ratio of notched-to-unnotched yield stress is referred to as the plastic constraint factor, q. In contrast to the elastic stress concentration factor that can reach values in excess of 10, the value of q does not exceed 2.57 (Orowan, 1945). However, brittle metals could prematurely fail due to stress increase at the notch before plastic yielding occurs. 

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