Stress hydrostatic pressure

In Chapter III, surface free energy and surface stress were treated as equivalent, and both were discussed in terms of the energy to form unit additional surface. It is now desirable to consider an independent, more mechanical definition of surface stress. If a surface is cut by a plane normal to it, then, in order that the atoms on either side of the cut remain in equilibrium, it will be necessary to apply some external force to them. The total such force per unit length is the surface stress, and half the sum of the two surface stresses along mutually perpendicular cuts is equal to the surface tension. (Similarly, one-third of the sum of the three principal stresses in the body of a liquid is equal to its hydrostatic pressure.) In the case of a liquid or isotropic solid the two surface stresses are equal, but for a nonisotropic solid or crystal, this will not be true. In such a case the partial surface stresses or stretching tensions may be denoted as Ti and T2-... 

The situation is more complex for rigid media (solids and glasses) and more complex fluids that is, for most materials. These materials have finite yield strengths, support shears and may be anisotropic. As samples, they usually do not relax to hydrostatic equilibrium during an experiment, even when surrounded by a hydrostatic pressure medium. For these materials, P should be replaced by a stress tensor, <3-j, and the appropriate thermodynamic equations are more complex. 

Extensional Viscosity. In addition to the shear viscosity Tj, two other rheological constants can be defined for fluids the bulk viscosity, iC, and the extensional or elongational viscosity, Tj (34,49,100—107). The bulk viscosity relates the hydrostatic pressure to the rate of deformation of volume, whereas the extensional viscosity relates the tensile stress to the rate of extensional deformation of the fluid. Extensional viscosity is important in a number of industrial processes and problems (34,100,108—110). Shear properties alone are insufficient for the characterization of many fluids, particularly polymer melts (101,107,111,112). 

In the large-diameter vertical cylindrical tanks, because hoop stress is proportional to diameter, the thickness is set by the hydrostatic hoop stresses. Although the hydrostatic forces increase proportionally with the depth of Hquid in the tank, the thickness must be based on the hydrostatic pressure at the point of greatest depth in the tank. At the bottom, however, the expansion of the shell owing to internal hydrostatic pressure is limited so that the actual point of maximum stress is slightly above the bottom. Assuming this point to be about 1 ft (0.305 m) above the tank bottom provides tank shells of adequate strength. The basic equation modified for this anomaly is... 

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... 

Note that the solution predicts that simple shear produces normal stresses. In fact, although simple shear occurs at constant volume, the normal stresses in general give rise to a hydrostatic pressure. Determination of the normal stresses in the case of a hypoelastic equation of grade one could, at least in principle, determine the coefficients a, Ug, and Ug individually. 

There are four commonly occurring states of stress, shown in Fig. 3.2. The simplest is that of simple tension or compression (as in a tension member loaded by pin joints at its ends or in a pillar supporting a structure in compression). The stress is, of course, the force divided by the section area of the member or pillar. The second common state of stress is that of biaxial tension. If a spherical shell (like a balloon) contains an internal pressure, then the skin of the shell is loaded in two directions, not one, as shown in Fig. 3.2. This state of stress is called biaxial tension (unequal biaxial tension is obviously the state in which the two tensile stresses are unequal). The third common state of stress is that of hydrostatic pressure. This occurs deep in the earth s crust, or deep in the ocean, when a solid is subjected to equal compression on all sides. There is a convention that stresses are positive when they pull, as we have drawn them in earlier figures. Pressure,... 

Fig. 3.2. Common slates of stress, tension, compression, hydrostatic pressure and shear.

Australia, is the doyen of materials scientists who study the elastic and plastic properties of minerals under hydrostatic pressure and also phase stability under large shear stresses (Paterson 1973). J.-P. Poirier, in Paris, a professor of geophysics, was trained as a metallurgist one of his special skills is the use of analogue materials to help understand the behaviour of inaccessible high-pressure polymorphs, e.g., CaTi03 perovskite to stand in for (Mg, FelSiOi in the earth s mantle (Poirier 1988, Besson el al. 1996). 

Mechanical properties of hydrogenated titanium alloys are strongly dependent on the applied stress tensor, especially on its hydrostatic component. This was illustrated by the high-pressure tensile and extrusion tests on the Ti-6Al-2.5Mo-2Cr alloy and the same alloy hydrogenated to a = 0.15 wt.%H. Tests were carried out using the apparatus at the Institute of Metal Physics UD RAS operating at hydrostatic pressures of machine oil to 15 kbax and temperatures to 250°C. 

The yielding of pipe does not occur provided that the equivalent stress is less than the yield strength of the drill pipe. For practical calculations, the equivalent stress is taken to be equal to the minimum yield strength of the pipe as specified by API. It must be remembered that the stresses being considered in Equation 4-54 are the effective stresses that exist beyond any isotropic stresses caused by hydrostatic pressure of the drilling fluid. 

The hydrostatic pressure has two effects an upward force acting on the wall cross-section of the sub, and a downward stress due to the lateral compression of the subwall. 

The anode effectiveness is only as good as the anode connection and loss of insulation at this point by deep pitting of the HSI or penetration of the anode cable seal will bring about rapid failure. Hydrostatic pressure should be borne in mind when considering the seal required for any depth of water. The useful life of HSI anodes is usually considered at an end after a 33% reduction in diameter, but this depends upon the original diameter, the amount of pitting sustained and the mechanical stresses to be withstood. Thus doubling the cross-sectional area may more than double the effective life of the anode. 

Assuming the appropriate boundary conditions between the internal sphere and any number of spherical layers, surrounding it, in the RVE of the composite, which assure continuity of radial stresses and displacements, according to the externally applied load, we can establish a relation interconnecting the moduli of the phases and the composite. For a hydrostatic pressure pm applied on the outer boundary of the matrix... 

However, even when this is successful there remains a serious problem. Dense carbon is very porous. Even if the surface is not wetted by the electrolyte, below about a 10-cm depth there is enough hydrostatic pressure to push electrolyte into the pores and into the copper/carbon interface. When the electrolyte reaches it, the copper corrodes. Since the corrosion products occupy more volume than the copper, the carbon is put under tensile stress and fails by cracking. 

A formed section would normally be used for the transition between a cylindrical section and conical section except for vessels operating at low pressures, or under hydrostatic pressure only. The transition section would be made thicker than the conical or cylindrical section and formed with a knuckle radius to reduce the stress concentration at the transition, Figure 13.11. The thickness at the knuckle can be calculated using equation 13.46, and that for the conical section away from the transition from equation 13.45. 

The minimum wall thickness required to resist the hydrostatic pressure can be calculated from the equations for the membrane stresses in thin cylinders (Section 13.3.4) ... 

A stress that is describable by a single scalar can be identified with a hydrostatic pressure, and this can perhaps be envisioned as the isotropic effect of the (frozen) medium on the globular-like contour of an entrapped protein. Of course, transduction of the strain at the protein surface via the complex network of chemical bonds of the protein 3-D structure will result in a local strain at the metal site that is not isotropic at all. In terms of the spin Hamiltonian the local strain is just another field (or operator) to be added to our small collection of main players, B, S, and I (section 5.1). We assign it the symbol T, and we note that in three-dimensional space, contrast to B, S, and I, which are each three-component vectors. T is a symmetrical tensor with six independent elements ... 

Rastogi, N.K., Angersbach, A., and Knorr, D. 2000b. Synergistic effect of high hydrostatic pressure pre-treatment and osmotic stress on mass transfer during osmotic dehydration. J. Food Engineer. 45, 25-31. 

For a discussion of the role of non-hydrostatic pressure in phase equilibria investigations, the effects of applied stress and their consequences in metallurgy, geology, etc. see Cahn (1989). 

A. R. Goni and K Syassen, Optical Properties of Semiconductors Under Pressure P. Trautman, M. Baj, and J. M. Baranowski, Hydrostatic Pressure and Uniaxial Stress in Investigations of the EL2 Defect in GaAs... 

The profiles of pendant and sessile bubbles and drops are commonly used in determinations of surface and interfacial tensions and of contact angles. Such methods are possible because the interfaces of static fluid particles must be at equilibrium with respect to hydrostatic pressure gradients and increments in normal stress due to surface tension at a curved interface (see Chapter 1). It is simple to show that at any point on the surface... 

At larger Re and for more marked deformation, theoretical approaches have had limited success. There have been no numerical solutions to the full Navier-Stokes equation for steady flow problems in which the shape, as well as the flow, has been an unknown. Savic (S3) suggested a procedure whereby the shape of a drop is determined by a balance of normal stresses at the interface. This approach has been extended by Pruppacher and Pitter (P6) for water drops falling through air and by Wairegi (Wl) for drops and bubbles in liquids. The drop or bubble adopts a shape where surface tension pressure increments, hydrostatic pressures, and hydrodynamic pressures are in balance at every point. Thus... 

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). 

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