Hydrostatic Stresses

It is shown that solute atoms differing in size from those of the solvent (carbon, in fact) can relieve hydrostatic stresses in a crystal and will thus migrate to the regions where they can relieve the most stress. As a result they will cluster round dislocations forming atmospheres similar to the ionic atmospheres of the Debye- Huckel theory ofeleeti oly tes. The conditions of formation and properties of these atmospheres are examined and the theory is applied to problems of precipitation, creep and the yield point."... 

The bulk modulus is appropriate for situations where the material is subjected to hydrostatic stresses. TTie proof of equations (2.15) and (2.16) is given by Benham et al. 

In the perfectly elastic, perfectly plastic models, the high pressure compressibility can be approximated from static high pressure experiments or from high-order elastic constant measurements. Based on an estimate of strength, the stress-volume relation under uniaxial strain conditions appropriate for shock compression can be constructed. Inversely, and more typically, strength corrections can be applied to shock data to remove the shear strength component. The stress-volume relation is composed of the isotropic (hydrostatic) stress to which a component of shear stress appropriate to the... 

Another loading condition in underwater applications is the application of external hydrostatic stress to plastic structures (also steel, etc.). Internal pressure applications such as those encountered in pipe and... 

FIGURE 9.2 EPR powder pattern of the [2Fe-2S]1+ cluster in spinach ferredoxin. Trace A shows an attempt to fit the spectrum with the diagonal linewidth Equation 9.1. In trace B the spectrum is fitted with the nondiagonal g-strain Equation 9.18. Trace C shows an experiment in which the spectral features are slightly shifted (solid trace) under the influence of an external hydrostatic stress. (Data replotted from Hagen and Albracht 1982.)... 

FIGURE 9.4 Effect of stress on a hyperfine pattern. The four-line parallel hyperfine pattern of the elongated CuOs octahedron in 63Cu(H20)6 is shown in the presence (dotted line) and absence (solid line) of an external hydrostatic stress. (Modified from Hagen 1982a.)... 

An account of the mechanism for creep in solids placed under a compressive hydrostatic stress which involves atom-vacancy diffusion only is considered in Nabarro and Herring s (1950) volume diffusion model. The counter-movement of atoms and vacancies tends to relieve the effects of applied pressure, causing extension normal to the applied stress, and shrinkage in the direction of the applied stress, as might be anticipated from Le Chatelier s principle. The opposite movement occurs in the case of a tensile stress. The analysis yields the relationship... 

ISO/DIS 9080, Thermoplastics pipes for the transport of fluids - Methods of extrapolation of hydrostatic stress rupture data to determine the long-term hydrostatic strength of thermoplastics pipe materials, 1992. 

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

The simulations show that steady-state diffusion conditions throughout the wall thickness are attained in 2 hr for the case when hydrogen outgases through the OD surface and 10 hr when the OD surface is impermeable. We defined an effective time to steady state / as the time at which the hydrogen concentration at NILS at the hydrostatic-stress peak location reaches 98% of the final steady state value. We found the corresponding / values equal to 6.4 min for an outgassing pipeline and 34.25 min for an impermeable pipeline. 

Lastly, we studied the effect of 7-stress on the effective time to steady state and the corresponding magnitude of the peak hydrogen concentration. We found that a negative T -stress (which is the case for axial pipeline cracks) reduces both the effective time to steady state and the peak hydrogen concentration relative to the case in which the T -stress effect is omitted in a boundary layer formulation under small scale yielding conditions. This reduction is due to the associated decrease of the hydrostatic stress ahead of the crack tip. It should be noted that the presented effective non-dimensional time to steady state r is independent of the hydrogen diffusion coefficient D 9. Therefore, the actual time to steady state is inversely proportional to the diffusion coefficient (r l/ ). 

It has therefore been assumed that a virial type of equation can relate the concentration C of the intracrystalline guest to the osmotic pressure (solution thermodynamics) or mean hydrostatic stress intensity (volume filling of pores) (9)... 

Virial Isotherm Equation. No isotherm equation based on idealized physical models provides a generally valid description of experimental isotherms in gas-zeolite systems (19). Instead (6, 20, 21, 22) one may make the assumption that in any gas-sorbent mixture the sorbed fluid exerts a surface pressure (adsorption thermodynamics), a mean hydrostatic stress intensity, Ps (volume filling of micropores), or that there is an osmotic pressure, w (solution thermodynamics) and that these pressures are related to the appropriate concentrations, C, by equations of polynomial (virial) form, illustrated by Equation 3 for osmotic pressure ... 

Chemists and physicists must always formulate correctly the constraints which crystal structure and symmetry impose on their thermodynamic derivations. Gibbs encountered this problem when he constructed the component chemical potentials of non-hydrostatically stressed crystals. He distinguished between mobile and immobile components of a solid. The conceptual difficulties became critical when, following the classical paper of Wagner and Schottky on ordered mixed phases as discussed in chapter 1, chemical potentials of statistically relevant SE s of the crystal lattice were introduced. As with the definition of chemical potentials of ions in electrolytes, it turned out that not all the mathematical operations (9G/9n.) could be performed for SE s of kind i without violating the structural conditions of the crystal lattice. The origin of this difficulty lies in the fact that lattice sites are not the analogue of chemical species (components). 

Let us investigate the steady state behavior of multicomponent crystals exposed to uniform but non-hydrostatic stresses. We first introduce some ideas on the thermodynamics of such solids (which will be discussed in more detail in Chapter 14). Solid state galvanic cells can be used to perform the appropriate experiments. 

Gibbs [J.W. Gibbs (1878)] showed that a non-hydrostatically stressed solid surrounded (Fig. 8-7) by a fluid (in which it is soluble) is entirely determined by the nature and state of the solid through the relation... 

Figure 8-7. Solids under non-hydrostatic stress aud the surrounding Gibbs-fluids L.

The surrounding fluid (Fig. 8-7) serves two purposes 1) it transmits the pressure to stress-load the surface and 2) it allows the surface to equilibrate chemically and thus provides juL in Eqn. (8.61) with physical meaning. Ideally, the Gibbs fluid has a vanishing buffer capacity for the solid so that after a change in an, the fluid becomes resaturated with respect to the solid before a noticeable amount of the solid or its surface dissolves. The key to verify Gibbs relation for solids under non-hydrostatic stress is therefore the existence of such an idealized fluid. 

Figure 8-8. Galvanic cell (schematic) for the determination of the chemical potential difference between surfaces 1 and 2 of non-hydrostatically stressed solids. Cross hatched solid electrolyte ...

Let us consider a homogeneously, but not hydrostatically, stressed solid which is deformed in the elastic regime and whose structure elements are altogether immobile. If we now isothermally and reversibly add lattice molecules to its different surfaces (with no shear stresses) from the same reservoir, the energy changes are different. This means that the chemical potential of the solid is not single valued, or, in other words, a non-hydrostatically stressed solid with only immobile components does not have a unique measurable chemical potential [J. W. Gibbs (1878)]. 

Figure 3.8 Edge dislocation in an isotropic elastic body. Solid lines indicate isopotential cylinders for the portion of the diffusion potential of any interstitial atom present in the hydrostatic stress field of the dislocation. Dashed cylinders and tangential arrows indicate the direction of the corresponding force exerted on the interstitial atom.

The flux has two components the first results from the concentration gradient and the second from the gradient in hydrostatic stress.19 The solid circles (cylinders... 

Fj = H j + flj P Accounting for the work against a hydrostatic pressure, P, to move a species with volume Qj (e.g., interstitial diffusion in response to hydrostatic stress gradients). Section 3.5.2. 

The displacements of atoms near a dislocation from their normal lattice positions are the same as the displacements that would be caused by some external stress. Therefore, we can think of the dislocation as causing a stress field around it. Around an edge dislocation, there is a state of hydrostatic stress, crH = ax + <7y + ct-, at a location x, y,... 

Hydrostatic stress field around an edge dislocation. 

In fluid flow, however, it is necessary to split the total stress, deviatoric stress, Tij, and a hydrostatic stress, aH. The deviatoric stress is the one that leads to deformation (Fig. 5.5) and the hydrostatic stress is the one that is described by pressure (Fig. 5.6). 

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