Anisotropic stress

The shock-compression pulse carries a solid into a state of homogeneous, isotropic compression whose properties can be described in terms of perfect-crystal lattices in thermodynamic equilibrium. Influences of anisotropic stress on solid materials behaviors can be treated as a perturbation to the isotropic equilibrium state. ... 

Figure 2-5 Physical Significance of the Anisotropic Stress-Strain Relations...

Thus the total stress, cry, at any point within a fluid is composed of both the isotropic pressure and anisotropic stress components, as follows ... 

Using the model described here, the anisotropic stress field inside the silicon ridge waveguides shown in Figure 9 were calculated. Applying Equation 1 to evaluate the anisotropic index of refraction, the FEM mode solver was used to determine the effective index values Nje and Njm for the... 

The original solids-conveying model developed by Darnel and Mol [7] assumed that the pressure (or stress) in the solid bed is isotropic. This assumption was made to simplify the mathematics and because of the lack of stress data for solid bed compacts. Previous research, however, showed that stresses in solid compacts are not isotropic [8]. Anisotropic stresses can be represented by the lateral stress ratio. It is defined as the ratio of the compressive stress in the secondary direction to the compressive stress in the primary direction, as shown in Fig. 4.7 and Eq. 4.1. 

Unlike the previous models by Darnell and Mol [14] and Tadmor and Klein [1], which are based upon the assumption of isotropic stress conditions, Campbell s model [20] considered anisotropic stress conditions, as suggested by Schneider [15], but it was assumed to be 1.0 due to the lack of published experimental data on the subject. Variations on the model set forth by Campbell and Dontula [20] include a modification to incorporate the lateral stress ratio [19, 22], and other modifications discussed by Hyun et al. [21, 23]. A modified Campbell-Dontula model with a homogeneous lateral stress is as follows ... 

The next advance in the understanding of the forces which underlie the topochemical principle was due to McBride (3). He introduced the concept of local stress to explain the details of the mechanisms by which diacyl peroxides decompose in the solid state. McBride showed that least motion can be overcome in these cases by anisotropic stresses equivalent to many tens of kilobars of pressure exerted by the product carbon dioxide molecules trapped in unfavorable lattice poitions. 

Note that there are some variations in the literature about the definition of these quantities sometimes the definitions of 4> and 4 are swapped, and their sign is also sometimes different. Here we choose the convention that and 4 are equal in the absence of anisotropic stress (see below), and that is the quantity that appears in the Laplacian term of the 00 part of the Einstein equations (the general relativistic analog of the Poisson equation), thus following the Newtonian convention to note the gravitational potential by. ) It is of course possible to define other scalar gauge invariant quantities. For example one can define... 

Finally, the anisotropic stress I jU, is already gauge invariant (because it has non unperturbed counterpart), and we also decompose it into... 

If we neglect the expansion term, we see that the first equation reduces to the usual Poisson equation. The last equation insures that the two Bardeen potentials are similar since in general the anisotropic stresses are small (they are negligible for non relativistic matter as well as for a scalar field). Note that in the absence of any form of matter all the scalar metric perturbations are 0. In addition to the scalar perturbations, there exists one equation for the tensor modes ... 

Although this solution has been derived in the case of a scalar field, on can in fact show that it is valid as long as there are no large scale anisotropic stresses. If we suppose the the Universe undergoes a series of eras of constant w, then the above equation can be solved in the hypothesis where the last era of constant w lasts much longer than the previous ones. One obtains... 

Rheometers are currently under development that will enable the anisotropic stress tensor of anisotropic complex fluids such as block copolymer melts and solutions to be probed, even during large amplitude shear. Here, a small amplitude probe waveform is applied orthogonal to the primary large amplitude shear flow. This could provide the linear dynamic modulus of an anisotropic system under nonlinear flow. 

Ramberg, H., 1959. The Gibbs free energy of crystals under anisotropic stress, a possible cause for preferred mineral orientation. Anais da Escola de Minas de Ouro Preto 32, 1-12. 

The principle of these testers is that the specimen can be subjected to controlled stresses in two orthogonal directions (biaxial testers) or three orthogonal directions (triaxial testers). In the case of the triaxial testers, two of the orthogonal stresses are usually equal, normally generated by liquid pressure in a pressure chamber. The specimen is placed in a cylindrical rubber membrane and enclosed by rigid end cups. The specimen is consolidated isotropically, i.e. by the same pressure in all three directions which leads to volumetric strain but little or no shear strain. This is followed by anisotropic stress conditions, whereby a greater axial stress is imparted on the specimen by mechanical force through the end cups. In the evaluation of results it is assumed that the principal stresses act on horizontal and vertical planes, and Mohr circles can be easily drawn for the failure conditions. 

The best way to determine the stresses in the scale would be a direct measurement. However, X-ray methods have usually a limited spatial resolution which makes it difficult to measure nonuniform stress fields. The application of OFS for scales consisting of a-Al203 provides a sufficient spatial resolution and permits, in principle, to examine stress variations in the scale. However, only the trace of the stress tensor can be measured for an untextured polycrystalline scale. Thus, anisotropic stress states have to be analysed in combination with a mechanical modelling of the scale loading in order to deduce the stress components from the trace of the stress tensor. 

Figure 1 presents the three basic mechanisms of stress induced permeability change in fractured rock (a) normal closure/opening, (b) shear dilation/contraction and (c) induced anisotropy due to different orientations of fractures and anisotropic stress condition. 

The third mechanism can be seen as a combination of first and second basic mechanisms (Figure 1(c)). In a fractured rock with multiple fractures, anisotropy in fluid permeability may be significant due to the different orientations of fractures and anisotropic stresses. Each fracture is under different contact normal or shear stresses mobilized through deformations and this make the directional permeability anisotropic. 

Anisotropy in the permeability begins to be more significant due to the different orientation of fractures and increasing differential stresses. The maximum anisotropy in permeability has a factor of 2 for the stress ratio 2. The model for this study has near-randomly distributed orientations of fractures. If the model has more unidirectional orientations of fractures, this effect of anisotropic stress will be more significant. 

Prominent channelling effect is observed during the increase of differential stresses. This is due to both anisotropic stress and the dilation of the fractures. Results show that well-connected fractures of larger apertures are the major pathways for fluid flow in fractured rock and pathways can be significantly changed after application of stress. 

They used an anisotropic stress loading on the material and achieved a more anisotropic material response. While an anisotropic, nonlinear elastic-plastic model would be best to model skin, the preceding may be used as an intermediate step in FEA. 

Calculations were performed at 8 volumes for )8-tin and 9 volumes for the sh structure. At each volume two preliminary calculations were performed with different c/a ratios close to the minimum in energy. By linear extrapolation of the anisotropic part of the stress (cr -cr where z is in the c direction and x is in the a direction) it was possible to rapidly find the c/a ratio at which the anisotropic stress is zero. This is the desired equilibrium c/a ratio at which the crystal is in equilrium with an externally applied pressure. With only the linear extrapolation, the residual anisotropy in the stress was typically less than 2 kbars. Thus the true equilibrium structure for each phase was found at each volume. The total energies from calculations performed at the predicted minima are plotted in Fig. 4 together with a curve for diamond Si of comparable accuracy. 

Like the ordinary MD, this is an evolutionary method. However, the physical meaning of the Lagranglan is not clear, hence the time-scale of the motion is also of uncertain meaning. The technique was extended to the case of anisotropic stress, with the MD cell having both shape and volume fluctuations, by Parrlnello and Rahman. They used the procedure to Induce phase changes from one crystal structure to another.Since this theory is so far developed for volume-independent potentials, it is not in principle applicable to metals, because of the strong volume dependence of the potentials in metals. 

The shape of the Bra g reflections reveals that intense and anisotropic stresses are acting in the crystal, probably in the vicinity of the polymer chains in strong extension, during the induction period. During the autocatalytic period, the reduction of the mismatch (see Fig.5) relaxes these stresses. 

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