Pressure uniaxial tension

Another commonly used elastic constant is the Poisson s ratio V, which relates the lateral contraction to longitudinal extension in uniaxial tension. Typical Poisson s ratios are also given in Table 1. Other less commonly used elastic moduH include the shear modulus G, which describes the amount of strain induced by a shear stress, and the bulk modulus K, which is a proportionaHty constant between hydrostatic pressure and the negative of the volume... 

In bulk material, the resistivity is independent of crystal orientation because silicon is cubic. However, if the carriers are constrained to travel in a very thin sheet, eg, in an inversion layer, the mobility, and thus the resistivity, become anisotropic (18). Mobility is also sensitive to both hydrostatic pressure and uniaxial tension and compression, which gives rise to a substantial piezoresistive effect. Because of crystal symmetry, however, there is no piezoelectric effect. The resistivity gradually decreases as hydrostatic pressure is increased, and then abrupdy drops several orders of magnitude at ca 11 GPa (160,000 psi), where a phase transformation occurs and silicon becomes a metal (35). The longitudinal piezoresistive coefficient varies with the direction of stress, the impurity concentration, and the temperature. At about 25°C, given stress in a (100) direction and resistivities of a few hundredths of an O-cm, the coefficient values are 500—600 m2/N (50—60 cm2/dyn). 

Uniaxial tension testing with superposed hydrostatic pressure has been described by Vernon (111) and Surland et al. (103). Such tests provide response and failure measurements in the triaxial compression or tension-compression-compression octants. 

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.,... 

Figure 14.8 shows stress-strain curves for polycarbonate at 77 K obtained in tension and in uniaxial compression (12), where it can be seen that the yield stress differs in these two tests. In general, for polymers the compressive yield stress is higher than the tensile yield stress, as Figure 14.8 shows for polycarbonate. Also, yield stress increases significantly with hydrostatic pressure on polymers, though the Tresca and von Mises criteria predict that the yield stress measured in uniaxial tension is the same as that measured in compression. The differences observed between the behavior of polymers in uniaxial compression and in uniaxial tension are due to the fact that these materials are mostly van der Waals solids. Therefore it is not surprising that their mechanical properties are subject to hydrostatic pressure effects. It is possible to modify the yield criteria described in the previous section to take into account the pressure dependence. Thus, Xy in Eq. (14.10) can be expressed as a function of hydrostatic pressure P as... 

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... 

A sample of polypropylene tested at 30°C and 10 s shows a yield stress of 35 MPa in uniaxial tension and 38 MPa under uniaxial compression. Calculate the hydrostatic pressure that must be superimposed in order to reach yield stress of 80 MPa. Assume that the material obeys the pressure-dependent von Mises criterion. 

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)... 

The modulus is the most important small-strain mechanical property. It is the key indicator of the "stiffness" or "rigidity" of specimens made from a material. It quantifies the resistance of specimens to mechanical deformation, in the limit of infinitesimally small deformation. There are three major types of moduli. The bulk modulus B is the resistance of a specimen to isotropic compression (pressure). The Young s modulus E is its resistance to uniaxial tension (being stretched). The shear modulus G is its resistance to simple shear deformation (being twisted). 

Which of the following stress fields is (are) statically determinate i) Uniaxial tension, ii) Three-point bending, iii) Thin-walled pressure... 

It is useful to note that eqs. (4.19) and (4.20) for uniaxial tension (or compression), in addition to being applicable to uniaxial strain deformation, as stated above, are also applicable to the case of dilatation responding to negative pressure where the basic symmetry of the deformation is maintained. In that case, however, ffii is replaced with tensile stress (negative pressure), n is replaced with e, the dilatation, and Eq, Young s modulus, is replaced with the bulk modulus K(). Moreover, a must be replaced by fi, which represents the reciprocal of the critical athermal cavitation dilatation. 

MPa in uniaxial tension, and at 31.5 MPa in uniaxial compression. Assuming that the yield stress is a linear function of hydrostatic pressure, calculate a-y under superimposed hydrostatic pressure of SOO MPa. 

Assuming that yielding follows a pressure-dependent von Mises criterion, calculate in uniaxial tension and in uniaxial compression. 

The stress concentration caused by a hole in a plate due to uniaxial tension or biaxial tension will be considered. The biaxial tension would correspond to a cylindrical shell or a spherical shell subject to internal pressure. For the case of a cylindrical shell, the biaxiality is 2 1 corresponding to hoop and longitudinal strain for the case of a spherical shell, the biaxiality ratio is 1 1. 

Mention has already been made that one of the advantages of the controlled pressure molecular dynamics discussed in Section 5.2 is that the form of the apphed pressure tensor P can be used to impart strain to a sample as a function of time in much the same way as in laboratory experiments. Control of appropriate components of the pressure tensor can be used to produce, to take just three examples, uniaxial tension, compression or shear as illustrated in Fig. 5.7... 

The prepared samples were subjected to a gradually increasing uniaxial tension by changing the component of the applied pressure tensor, constant rate... 

The failure characteristics of a food or food material can be measured using compression, tension, or torsion. Of all the available deformation tests, possibly the most common is uniaxial compression (Lelievre et al, 1992). Bulk compression is another type of compression test, but it is seldom used due to the difficulty inapplyingforceby means of hydraulic pressure (Bourne,, 1982). The experimental... 

While the true stress-true strain response is qualitatively similar in both compression and tension, the resulting deformation states are very different. Tensile loading leads to uniaxial molecular orientation along the loading axis. Compression on the other hand results in a biaxial orientation state in a plane perpendicular to the loading direction and so it is expected that quantitatively different stress-strain curves are seen. In addition, as discussed below, the hydrostatic pressure difference between tension and compression leads to differences in yield strength because yield in polymers is pressure dependent. 

A similar relationship as for the uniaxial stress caused by pressure is also true for the shear stress (t) (tension force components... 

The stress distribution given by Eq. 15.1 is shown in Fig. 15.1 for a vessel with r /fj = 2.2, The maximum stress is in the hoop direction and is at the inner surface where r = r. As the pressure is increased, the stresses increase until they reach a maximum limiting stress where failure is assumed to occur. For thin vessels the ASME Code assumes that failure occurs when the yield point is reached. This failure criterion is convenient and is called the maximum principal stress theory. In thick vessels the criterion usually applied for ductile materials is the energy of distention theory. This theory states that the inelastic action at any point in a body under any combination of stresses begins only when the strain energy of distortion per unit volume absorbed at the point is equal to die strain energy of distortion absorbed per unit volume at any point in a bar stressed to the elastic limit under a state of uniaxial stress as occurs in a simple tension test. The equation that expresses this theory is given by... 

Another method for measuring uniaxial extensional viscosity is by bubble collapse. A small bubble is blown at the end of a Ciq>illary tube placed in the test fluid (see Figure 7.6.1). It comes to equilibrium with the surrounding pressure and surface tension. Then at time r = 0 the pressure inside the bubble is suddenly lowered or the surrounding pressure increased. The decrease in bubble radius with time is recorded. If the deformation is reversed (i.e., the pressure inside the bubble is suddenly increased), the growing bubble radius can be used to give the equibiaxial viscosity. This flow appears to be less stable and has not been studied as a rheometer. 

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