Hydrostatic axis

Figure 24. Parabolic failure surfaces coaxial to the hydrostatic axis...

This implies that, by spherical decomposition, cylindrical polar coordinates are introduced in terms of the hydrostatic axis. 

Experimentally, it has been found that hydrostatic stresses, characterised mathematically by ai =0-2 =0-3, do not cause plastic deformations. Therefore, it can be assumed that the deviation of the stress state from a state of hydrostatic stress determines whether the material yields. In the space of principal stresses, the yield surface thus is a surface (cylindrical or prismatic, for example) that encloses the hydrostatic space diagonal ai = a2 = <73. If we change the position on the hydrostatic axis (the space diagonal), the surface enclosing the axis changes neither its shape nor its size. Mathematically, this can be represented by subtracting the hydrostatic part of the stress ... 

The yield surface for the von Mises yield criterion (occasionally called distor-tional strain energy criterion) is cylindrical in the space of principal stresses, with its centre coinciding with the hydrostatic axis = [Pg.90]

J2 measures the distance from the hydrostatic axis in the principal stress space. 

The parabolically modified yield criterion assumes that the yield surface is a paraboloid centred on the hydrostatic axis (figure 3.24(a)). Its radius depends on the hydrostatic stress [Pg.92]

Annealing drawn PE hydrostatically at high pressure, generates a wide spectrum of crystal thicknesses varying from the common oriented chain folded to the chain-extended structures — where folds and ties tend to disappear63 —. This range of crystal thicknesses coupled with the chain axis orientation, offers a suitable model in... 

As is well known, the earth is mainly a fluid the upper crust is an exception, but it is extremely thin layer with respect to the earth s radius. For this reason it is natural to expect that rotation around its axis makes the shape of the earth practically the same as if it was a fluid, and we will follow this conventional point of view. Suppose that during this motion the mutual position of all elementary volumes of the earth remains the same, and correspondingly each of them is involved only in rotation with angular velocity m. This means that the effect of different types of currents inside the earth is neglected and we deal with hydrostatic equilibrium. 

At the instant of contact between a sphere and a flat specimen there is no strain in the specimen, but the sphere then becomes flattened by the surface tractions which creates forces of reaction which produce strain in the specimen as well as the sphere. The strain consists of both hydrostatic compression and shear. The maximum shear strain is at a point along the axis of contact, lying a distance equal to about half of the radius of the area of contact (both solids having the same elastic properties with Poisson s ratio = 1/3). When this maximum shear strain reaches a critical value, plastic flow begins, or twinning occurs, or a phase transformation begins. Note that the critical value may be very small (e.g., in pure simple metals it is zero) or it may be quite large (e.g., in diamond). 

Another well-known phenomenon is the Weissenberg effect, which occurs when a long vertical rod is rotated in a viscoelastic liquid. Again, the shearing generates a tension along the streamlines, which are circles centred on the axis of the rod. The only way in which the liquid can respond is to flow inwards and it therefore climbs up the rod until the hydrostatic head balances the force due to the normal stresses. 

Figure 4. Plot of the full-field solution for the normalized hydrostatic stress

Fig. 16.4 (a) The crystal ceiis of anatase Ti02. (b) View of the octahedra-packing of anatase Ti02 the thick red lines indicate the axis in representative octahedra which, under stress, results in a change of band gap. (c) Band gap variation in anatase Ti02 under hydrostatic, epitaxial, and uniaxial stress. Adapted from Yin etal. [72] with kind permission from American Institute of Physics (2010). 

With this idea in mind, the horizontal surface in Figure 6.3b can be taken as a reference level at which Ap = 0. Just under the meniscus in the capillary the pressure is less than it would be on the other side of the surface owing to the curvature of the surface. The fact that the pressure is less in the liquid in the capillary just under the curved surface than it is at the reference plane causes the liquid to rise in the capillary until the liquid column generates a compensating hydrostatic pressure. The capillary possesses an axis of symmetry therefore at the bottom of the meniscus the radius of curvature is the same in the two perpendicular planes that include the axis. If we identify this radius of curvature by b, then the Laplace equation applied to the meniscus is Ap = 2y/b. Equating this to the hydrostatic pressure gives... 

There is also in some bombs, the transverse fuze which is fitted into a cavity of the body at right angles to its axis. The latter type of fuze, when used in depth bombs, is activated by pressure of water(hydrostatic, also called thwartship fuze)... 

The behaviour of the phonons can be related to pressure induced structural modifications. Unfortunately, for the Y123 compounds, the variation of the bond lengths as a function of pressure has been studied only for low pressures, up to 0.56 GPa [23] and there are no data available for an independent verification of the predictions of local structural modifications. The structural hydrostatic pressure measurements have found a similar pressure dependence of the a- and c-axes for the underdoped and overdoped Y123, while, for the b axis, the underdoped exhibits a larger compression ( 20%) than the overdoped sample [23], The Raman data for low pressures... 

For the c-BN formation a stress threshold was observed in the deposited layers. The h-BN intermediate layer shows a preferred orientation, where the c-axis of the h-BN is parallel to the substrate. Both effects are explained by the compressive biaxial stress induced by the ion bombardment. The mechanism for the conversion of h-BN into c-BN is explained by rather high temperatures originated during thermal spikes (direct h-BN —> c-BN transformation). The stress caused by the bombardment with high energetic ions is considered to be a reason for the growth of the c-BN crystals [190, 191]. A stress within the layer of up to 10 GPa has been observed. This biaxial stress causes a hydrostatic pressure up to the values usual in HP-HT synthesis. 

Studies involving fluid shear, hydrostatic compression, biaxial and uniaxial stretch, or a combination of two or more of these factors indicate that fluid shear is a major factor affecting bone cell metabolism and cells subjected to mechanical stress reshape and align themselves with their long axis perpendicular to the axis of force. Cells also exhibited remodeling of the actin cytoskeleton and increases in PKC levels, processes thought to be involved in the early phase of mechanochemical transduction. 

Now, we may evaluate the surface reconstruction generated by the suppression of a few kilobars of stress along the d axis, using the deformation coefficients under hydrostatic pressure. The authors of Ref. 136 have determined, in neutron-scattering experiments, the variation of the crystallographic parameters of anthracene ... 

Figures 20A and B show the PL spectra, recorded at 290 K, at 600 nm, and as a function of pressure, for Cs9(SmW10O36) and SmWi0O36-LDH, respectively (Park et al., 2002). For the sake of comparison, the line shapes are normalized and displaced along the vertical axis. In both cases, the peak position is red-shifted by 4—5 nm when the hydrostatic pressure increases from 1 bar to 61 kbar. It was shown that the red-shift from A to A lies solely in the deformation of the samarium complexes by the uniaxial stress exerted by the host layers, whereas the shift from B to B is also influenced by the change in the cation environment. Under the same conditions, B is not at the same position for the non-intercalated (HN (n -b u t y 1) 3) 9 (SmW10O3e) and Cs9(SmWi0O36) compounds (Park et al., 2002). Thus only peak A is available to measure the unixial stress. This observation can be used to determine the uniaxial stress, when the external pressure is zero. For the SmW10O36—LDH system, the uniaxial stress varies significantly from 75 at 28 K to 140 kbar at 290 K (Park et al., 2002).

Normally the coordinate system is chosen in such a way that T13 = T31 = T23 = T32 = 0 In general, use is made of normal stress differences, N1 and N2, because they do not include undetermined hydrostatic pressures that are always present but not affect the material properties (as long as they are not too high). In Table 15.1, also the possibilities to determine the normal stress differences or combinations are depicted. In the modem rheogoniometers also normal stress differences can be determined but. They follow from measurements of normal forces, Fn, or normal stresses, T22, as is also depicted in Table 15.1. For the measurements of the normal stresses T22 pressure gauges have to be mounted in the Couette cylinders, in the capillary of the capillary rheometer (in both cases quite difficult to mount) and in the plate of a cone and plate instrument at several distances from the axis (not that difficult). Sometimes use is made of a slit rheometer instead of a capillary rheometer, because pressure gauges are much easier to mount (Te Nijenhuis, 2007, Chap. 9.1.2). 

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