Uniaxial state of stress

Calculations for the minimum performance properties of drill pipe are based on formulas given in Appendix A of API RP 7G. It must be remembered that numbers in Tables 4-80-4-83 have been obtained for the uniaxial state of stress, e.g., torsion only or tension only, etc. The tensile stress resistance is decreased when the drill string is subjected to both axial tension and torque a collapse... 

The reduction in the tensile load capacity of the drill pipe is 311,400 -260,500 = 50,900 lb. That is about 17% of the tensile drill pipe resistance calculated at the minimum yield strength in uniaxial state of stress. For practical purposes, depending upon drilling conditions, a reasonable value of safety factor should be applied. 

From Table 4-84, the collapse pressure resistance in uniaxial state of stress, P - 6,010 psi. Reduced wall thickness for class 2 drill pipe = (0.65)(0.337) = 0.219 in. Reduced D for class 2 drill pipe = 3.826 + (2)(0.219) = 4.264 in. Reduced cross-sectional area of class 2 drill pipe equals ... 

Uniaxial state of stress State of stress in which two of the principle stresses are zero. 

The expression for curvature in (2.7) is the famous Stoney formula relating curvature to stress in the film (Stoney 1909). Stoney s original analysis of the stress in a thin film deposited on a rectangular substrate was based on a uniaxial state of stress. Consequently, his expression for curvature did not involve use of the substrate biaxial modulus Mg. Consequently, (2.7) can be applied in situations in which mismatch derives from inelastic effects. However, the relationship (2.7) is based on Stoney s concept as outline in this section, and it has become known as the Stoney formula. It has the important property that the relationship between curvature k and membrane force / does not involve the properties of the film material. The elastic mismatch strain Cm corresponding to the stress <7 given in (2.8) is... 

One of the most common types of tests for determining the strength properties of adhesives is the tensile test on bulk specimens. The specimens and the test methods are comparable to those used for plastic materials. The properties determined are intrinsic to the material they are obtained under a uniform and uniaxial state of stress, with no influence of the adherends. 

The procedure described above is straightforward in principle. However, in practice, great care must be taken in the test fixture design to assnre that applied loads cause a uniform state of stress in the test specimen. Two types of tests that have been developed for this purpose include (1) the simple tensile test for uniaxial states of stress, (2) the thin-walled tube subjected to combined torsion and internal pressure, for biaxial states of stress. Some theoretical aspects of the simple tensile test are developed in the sample problem which follows. For a more detailed discussion on experimental procedures for characterizing the material properties of composite materials, see Caisson and Pipes and Whitney et al. ... 

For most practical purposes, the onset of plastic deformation constitutes failure. In an axially loaded part, the yield point is known from testing (see Tables 2-15 through 2-18), and failure prediction is no problem. However, it is often necessary to use uniaxial tensile data to predict yielding due to a multidimensional state of stress. Many failure theories have been developed for this purpose. For elastoplastic materials (steel, aluminum, brass, etc.), the maximum distortion energy theory or von Mises theory is in general application. With this theory the components of stress are combined into a single effective stress, denoted as uniaxial yielding. Tlie ratio of the measure yield stress to the effective stress is known as the factor of safety. 

In a recent attempt to bring an engineering approach to multiaxial failure in solid propellants, Siron and Duerr (92) tested two composite double-base formulations under nine distinct states of stress. The tests included triaxial poker chip, biaxial strip, uniaxial extension, shear, diametral compression, uniaxial compression, and pressurized uniaxial extension at several temperatures and strain rates. The data were reduced in terms of an empirically defined constraint parameter which ranged from —1.0 (hydrostatic compression) to +1.0 (hydrostatic tension). The parameter () is defined in terms of principal stresses and indicates the tensile or compressive nature of the stress field at any point in a structure —i.e.,... 

It is emphasized that defines the state of stress and should not be confused with the orientation angle p. For example, — 45° and p = 45° represents equal biaxial tension acting on a sheet whose ellipse is at 45° to one of the chosen external principal stress axes. This is not equivalent to — 0 and ft = 0, which represents uniaxial tension parallel to the elliptic major axis (which is equivalent to = 90° and p — 90°). Because of symmetry considerations, all possible combinations of elliptic... 

More general, bi-axial states of stress in the plane can be readily considered similarly. Since our aim here is to develop the connection between microstructure and uniaxial tensile toughness, such cases will not be developed. 

Consider two cases of spherical particles, one of KRO-1 morphology of a volume fraction of 0.23 of randomly wavy PB rods, and the other pure PB — both occupying a volume fraction of 0.22 in PS. We are interested in the craze initiation condition for these two particles at room temperature under a uniaxial tensile stresso. We consider the state of stress at a typical equatorial point A along the particle interface, on the PS side of the particle, as shown in Fig. 33. We determine first by standard methods the elastic properties of the particles and their thermal expansion coefficients, together with the elastic properties and thermal expansion coefficients of the composite matrix as a whole consisting of particles and the majority phase of PS. 

For macroscopically isotropic polymers, the Tresca and von Mises yield criteria take very simple analytical forms when expressed in terms of the principal stresses cji, form surfaces in the principal stress space. The shear yield surface for the pressure-dependent von Mises criterion [Eqs (14.10) and (14.12)] is a tapering cylinder centered on the applied pressure increases. The shear yield surface of the pressure-dependent Tresca criterion [Eqs (14.8) and (14.12)] is a hexagonal pyramid. To determine which of the two criteria is the most appropriate for a particular polymer it is necessary to determine the yield behavior of the polymer under different states of stress. This is done by working in plane stress (ct3 = 0) and obtaining yield stresses for simple uniaxial tension and compression, pure shear (di = —CT2), and biaxial tension (cti, 0-2 > 0). Figure 14.9 shows the experimental results for glassy polystyrene (13), where the... 

The brittle-ductile transition temperature depends on the characteristics of the sample such as thickness, surface defects, and the presence of flaws or notches. Increasing the thickness of the sample favors brittle fracture a typical example is polycarbonate at room temperature. The presence of surface defects (scratches) or the introduction of flaws and notches in the sample increases Tg. A polymer that displays ductile behavior at a particular temperature can break in the brittle mode if a notch is made in it examples are PVC and nylon. This type of behavior is explained by analyzing the distribution of stresses in the zone of the notch. When a sample is subjected to a uniaxial tension, a complex state of stresses is created at the tip of the notch and the yield stress brittle behavior known as notch brittleness. Brittle behavior is favored by sharp notches and thick samples where plane strain deformation prevails over plane stress deformation. 

Equations (14.10) and (14.12) give the pressure-dependent von Mises criterion. Also, for any state of stresses, P is an invariant given by the expression P = (l/3)(ai-I-Q2-1-cy3). On the basis of this expression, in a uniaxial tension test 02 = a3 = 0)... 

A near-uniform state of stress and strain within a briquette is more difficult to achieve with a roll press than with uniaxial compaction presses (either closed mold or extrusion) because of the more complicated geometry of the pressing chamber (nip plus briquette pockets). Homogeneity (but not necessarily isotropy) could be attained if either ... 

Figure 10.01 State of stress of a cubic glass body under (a) uniaxial stress,(b)pure shear.

As mentioned previously, the boundary condition in DD and FE are different. Periodic boundary condition is used in DD analysis to take into account the periodicity of single crystals whereas confined boundary condition is used in the FE analysis to achieve the uniaxial state of strain. In order for the boundary conditions in FE and DD to be consistent, periodic FE boundary condition is implemented as well. This implementation of periodic FE boundary condition yields a relaxed state of stress with low peak pressure when compared to the experiment as illustrated in Fig. 9(a). Furthermore, both shear and longitudinal waves are generated which is discordant with plane wave characteristics as shown in Fig 9(b). Fig 10 shows the deformed shape when confined and periodic boundary conditions are used. In the confined case there is no distortion in the RVE. However, for the periodic case, considerable... 

Correlation of results from one test to another for a given material becomes difficult because of different stress states of the specimen and the associated strain rates in different tests. In the tensile-impact test, the stress state is uniaxial and it measures the tensile property at a high strain rate. In Izod and Charpy tests, the presence of notch gives a triaxial state of stress. The falling-... 

The strength-differential effect is also reflected prominently in the multi-axial yield criteria which translate the multi-axial stress driving forces for yield into an equivalent uniaxial state of extension (tension) or simple shear <7se that is most relevant to the mechanisms governing plastic flow. In a more mechanistieally relevant statement for polymers, the multi-axial yield criterion of von Mises defines a uniaxial equivalent stress Oe (or a o-se) as... 

To obtain different states of stress and strain, gridded blanks of successively narrower width (but of constant length) are stretched over a punch in a similar manner. A very narrow sheet specimen stretched over a hemispherical punch will therefore approach the state of uniaxial tension. 

This theory asserts that yielding occurs when the largest difference of shear stress equals the shear yield strength. According to this theory, yielding will start at a point when the maximum shear stress at that point reaches one-half of the uniaxial yield strength, Fy. Thus for a biaxial state of stress where ai > 02, the maximum shear stress will be ((Ti - 2)12. 

The experimental set-up can also be used for the testing of stress relaxation in a plane state of stress. In this test a specified quantity of water is pumped rapidly into the device producing a certain arc height and associated multi-axial deformation. Care must be taken to prevent air cushion formation. The gradual decrease in pressure as a function of time can be obtained from the manometer reading. The test procedure and the evaluation of the relaxation test data can be performed in analogy to the stress relaxation test imder an imposed uniaxial deformation. Figure 3.13 shows such a stress relaxation curve at an imposed multi-axial deformation. 

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