Uniform Tensile Stress

If it is assumed that uniform tensile stress, like uniform compressive stress (7), has no significant effect on yield, then the yield pressure of a cylinder subjected solely to an internal pressure may be calculated from... 

Sample subjected to uniform tensile stress Tensile strength... 

It is well established that GGA gives a better description of molecular systems, crystal surfaces and surface-molecule interactions. However, there are cases where the GGA results for solids are in much worse agreement with experiment than the LDA ones (e.g., 3-22 jj has been suggested that the effect of using GGA for solids is roughly equivalent to adding uniform tensile stress, and as a result lattice constants are frequently overestimated. 

A uniform tensile stress of 50 MPa acts along the axis of a piezoceramic cylindrical rod of length 12 mm and diameter 6 mm. Calculate the potential difference developed between the ends of the rod. Given that the coupling coefficient is 0.5, calculate the total stored energy in the rod (i 33 — 70(ko d33 = 3 50 pCN-1). [Answer 33.9 kV 25 mJ]... 

Generally, for any dimension therefore, if a crack of length I already exists in an infinite elastic continuum, subject to uniform tensile stress a perpendicular to the length of the crack, then for the onset of brittle fracture, Griffith equates (the differentials of) the elastic energy E with the surface energy E ... 

This approach is the most useful for engineering purposes since it expresses fracture events in terms of equations containing measurable parameters such as stress, strain and linear dimensions. It treats a body as a mechanical continuum rather than an assembly of atoms or molecules. However, our discussion can begin with the atomic assembly as the following argument will show. If a solid is subjected to a uniform tensile stress, its interatomic bonds will deform until the forces of atomic cohesion balance the applied forces. Interatomic potential energies have the form shown in Fig. 1 and consequently the interatomic force, whidi is the differential of energy with respect to linear separation, must pass throt a maximum value at the point of inflection, P in Fig. 1. 

In the case that cracked structure is subjected to the uniform tensile stress total strain of the structure e consists of two components. One is the strain due to the elastic deformation of the matrix b and another is the one due to the crack opening Ec-... 

Figure 8.7 shows how the stress increases near the tips of a crack in a sheet to which a uniform tensile stress is applied normal to the crack. Near the tip of the crack the tensile component of stress must be higher than the applied stress because the stress within the crack is zero and the stress a long way from it must be equal to the applied stress. Near the crack the other components of stress are not all zero and, in the model due to Irwin, it is shown that the whole system of stresses near the tip of a crack loaded in this way can be written in terms of the applied stress, geometrical factors and a single further quantity K called the stress-intensity factor. For an infinite sheet with a central crack of length I, it can further be shown that... 

Fig. 8.7 The tensile stress near a crack in a sheet of polymer in tension, (a) A schematic representation of the stress. The arrows at the top and bottom of the diagram represent a uniform tensile stress applied to the polymer sheet in a direction normal to the crack, represented by the short thick line at the centre. The arrows pointing upwards from the dotted centre line represent the forces exerted by the upper half of the sheet on the lower half, whereas the arrows pointing downwards representthe forces exerted by the lower half of the sheet on the upper half, (b) The form of the tensile stress given by Irwin s equations for a crack of length one unit. See also section 8.3.4.

For mode I loading of an elliptically shaped crack by a uniform tensile stress, the stress intensity factor varies with position along the crack front and is a maximum at the ends of the minor axis. True or False ... 

At first sight, it might seem a simple test to analyse with uniform tensile stress throughout the adhesive layer. In practice, the stress distribution is not uniform the disparity of modulus and Poisson s ratio between the cylinders and the adhesive means that shear stresses are introduced on loading (see Stress distribution Poisson s ratio). Thus, the failure stress is not independent of the dimensions of the joint. 

The uniform tensile stress, cr, is perpendicular to the crack plane and Y is a dimensionless parameter, depending on the geometry and the range of the stress-... 

Fig. 8.18 Schematic representation of a a sharp crack, b a blunt crack with a root radius of po and c a grain suffering a non-uniform tensile stress [15]. With kind permission of Elsevier...

Evans et al. (1961) showed that for a disc acted upon by opposing diametrical loads, there is a uniform tensile stress acting at 90° to the diameter. Under sufficiently high compressive loads, therefore, the resulting tensile stress could exceed the cohesive strength of the material and the disc would split across the diameter. Evans extended the analysis to three-dimensional particles to show that even when particles are stressed compressively, the stress pattern set up by virtue of the shape of the particle may cause it to fail in tension, whether cracks exist or not. 

Only the resolved shear stresses in a small element of the ceramic in front of the slider and at 90 to the P axis are considered. This produces a reasonably uniform tensile stress at right angles to the P axis and lying in the plane of P, P, and F. 

Integration of the crack propagation rate algorithms leads to an appropriate crack depth—time a -1) relationship, which, for a uniform tensile stress situation, is of the form... 

Let us consider a specimen containing a flaw of size 2a loaded under a uniform tensile stress a (Figure 7.4). It feils when stress concentration (stress intensity factor, SIF)... 

The first understanding of why fracture starts from flaws is due to Griffith (1920, 1924) who used the study of Inglis (1913) on the concentration of stresses at an elliptical flaw (Figure 7.7). Let us assume that a flaw of semiaxes a and b a>b) is present in a beam submitted to uniform tensile stress a. The flaw concentrates stresses at the points of minimum curvature, and the maximum stress o is given by... 

Figure 7.7 Elliptical flaw of semi-axes a and 6 in a beam under uniform tensile stress a.

Figure 7.8 Energy of an infinitely narrow elliptical flaw under uniform tensile stress a. Crack equilibrium corresponds to U maximum (f = 1J m", E = 70GPa, o- = 2 MPa).

Under uniform tensile stress (e.g. in a uniaxially tensile stressed fibre), Eq. (7.33) becomes... 

Figure 3. The two elastomers submitted to a uniform tensile stress a.

Griffith applied the Inglis analysis to treat a narrow elliptical crack (b- -0) in a remote, uniform tensile stress field. He also utilized a result from linear elastic theory namely, that for any body under constant applied load during crack propagation. 

Four-point bend testing provides a uniform tensile stress over a relatively large area of the specimen. The elastic stress in the outer layer of the specimen between the two irmer supports can be calculated fiom the following equation ... 

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