Pressure anisotropy

A closer look at the atomic configurations reveals structural transitions which are triggered preferentially by pressure anisotropy. The most frequent transition in the x compression involves of course molecule translations, rotations of the methyl groups (which occur even for V/Vo close to 1) and reorientations of the molecules, primarily via relative translations and rotations aroimd their C-N axes. Two such abrupt transitions are observed when V/Vo = 67-70% and the estimated pressure is 25-30 GPa, and V/Vo = 59-62% with an estimated pressure of 45-55 GPa. In the z compression the methyl groups seem to undergo smoother transitions as each molecule s center-of-mass is translated in z. In contrast, the y compression causes abrupt transitions when V/Vo - 76 81% and 63-66% and the pressure is estimated to lie in the ranges 2-5 and 13-16 GPa, respectively. 

Figure I. Classification of finite fractured states of rocks at high PT-conditions under pressure anisotropy...

In the balance of torces it is not actually the radial pressure (prad), but the axial pressure (pax) that counts, because pressure anisotropy is to be expected (like in bulk mechanics). The value needs to be defined in order to quantify the pressure anisotropy ... 

For an ideal elastic solid, pdV must be replaced in Eq. (4.4) by the product of the pressure tensor and the differential of the deformation tensor. In the limiting case of a fluid phase, pressure anisotropies disappear. In the treatment that follows, we will neglect pressure anisotropies and assume sufficient mobilities of the components [91,92], which then allows the introduction of a scalar chemical potential for the components without any problem. We will return to this point again in Section 4.3.7 and in Section 5.4.4. 

So far we considered phases with sufficient atomic mobihties and vanishing pressure anisotropies such that we could use the term -pdV to describe the mechanical energy increment (see also footnote 6). Generally, for elastic deformations (i.e. usually small deformations), this increment has to be expressed in terms of the stress tensor components Sy and the differential strain tensor components dey ... 

Another important special case of a homogeneous deformation (i.e. y is positionally constant) in which now, however, pressure anisotropies are effective, is the uniaxially stressed cubic crystal. Let us assume this time that there is tensile stress in the x-direction. There it holds for small effects (i.e. Hooke s law fulfilled) that s = P. dcxx/dsxx const = xx/ xx nnd also dsyy/dsxx — ds z/dsxx yy/sxx zz/ xx const. 

Heterogeneity, nonuniformity and anisotropy are terms which are defined in the volume-average sense. They may be defined at the level of Darcy s law in terms of permeability. Permeability, however, is more sensitive to conductance, mixing and capillary pressure than to porosity. 

Sinee our system in the slit is anisotropie and inhomogeneous, it makes sense to eonsider the loeal pressure tensor, Eq. (42), whieh depends on the distanee z from the adsorbing wall. If one defines Px = Pxx Py = Pyy ... 

In crystals with the LI2 structure (the fcc-based ordered structure), there exist three independent elastic constants-in the contracted notation, Cn, C12 and 044. A set of three independent ab initio total-energy calculations (i.e. total energy as a function of strain) is required to determine these elastic constants. We have determined the bulk modulus, Cii, and C44 from distortion energies associated with uniform hydrostatic pressure, uniaxial strain and pure shear strain, respectively. The shear moduli for the 001 plane along the [100] direction and for the 110 plane along the [110] direction, are G ooi = G44 and G no = (Cu — G12), respectively. The shear anisotropy factor, A = provides a measure of the degree of anisotropy of the electronic charge... 

The orientation of an anisodiametric filler which results from pressure molding is known to bring about anisotropy of properties in different directions, the extent of which is determined by the matrix polymer [367], This is not always a plausible result from the technological standpoint, if only because orientation, frequently nonuniform, may build up to cause buckling of articles [368-370]. 

In addition, intensity changes under increasing pressure have been observed. For example, the most intense Raman line at STP conditions is the flg component of v ( 220 cm ), but at about 2 GPa the intensity decreases in favor of the ag component of Vi ( 475 cm ) which on further compression gains more intensity (about a factor of 2 at 5 GPa) [120]. This behavior was explained by the anisotropy of the crystal s compressibihty [139] and differences in the components of the Raman tensor of the two modes [87] with respect to the crystal axes [109]. 

The flow velocity, pressure and dynamic viscosity are denoted u, p and fj and the symbol (...) represents an average over the fluid phase. Kim et al. used an extended Darcy equation to model the flow distribution in a micro channel cooling device [118]. In general, the permeability K has to be regarded as a tensor quantity accounting for the anisotropy of the medium. Furthermore, the description can be generalized to include heat transfer effects in porous media. More details on transport processes in porous media will be presented in Section 2.9. 

Shale stability is an important problem faced during drilling. Stability problems are attributed most often to the swelling of shales. It has been shown that several mechanisms can be involved [680,681]. These can be pore pressure diffusion, plasticity, anisotropy, capillary effects, osmosis, and physicochemical alterations. Three processes contributing to the instability of shales have to be considered [127] ... 

---