Isotropic stress

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. 

Note that both force and area are vectors, whereas pressure is a scalar. Hence the directional character of the force is determined by the orientation of the surface on which the pressure acts. That is, the component of force acting in a given direction on a surface is the integral of the pressure over the projected component area of the surface, where the surface vector (normal to the surface component) is parallel to the direction of the force [recall that pressure is a negative isotropic stress and the outward normal to the (fluid) system boundary represents a positive area]. Also, from Newton s third law ( action equals reaction ), the force exerted on the fluid system boundary is of opposite sign to the force exerted by the system on the solid boundary. 

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 ... 

Experimental support on the validity of Eq. 4.7-2 was presented by Spencer et al. (32), who also proposed a theoretical derivation based on considering a discrete number of contact points between solids and containing walls. They assumed isotropic stress distribution (K = 1) and obtained an expression identical to Eq. 4.7-2... 

Next we proceed with the force and torque balances. Since pressure builds up in the down-channel direction, the force and torque balances are made on a differential increment in the down-channel direction this is illustrated in Fig. 9.29, where the various forces acting on the element are also depicted. These forces can be expressed in terms of the coefficients of friction, local geometry, and the differential pressure increment, which compensate for the other forces and torques. For an isotropic stress distribution, these are... 

Perhaps the most severe assumption in the Darnell and Mol model is the isotropic stress distribution. Recalling the discussion on compaction in Section 4.5, the stress distribution in the screw channel is expected to be complex. The first attempt to account for the nonisotropic nature of the stress distribution was made by Schneider (36). By assuming a certain ratio between compressive stresses in perpendicular directions and accounting for the solid plug geometry, he obtained a more realistic stress distribution, where the pressure exerted by the solids on the flights, the root of the screw, and the barrel surface are all different and less than the down-channel pressure. The ratio between the former and the latter is of the order of 0.3-0.4. 

Tensors that are proportional to 8 are sometimes called isotropic tensors. For an incompressible material, gradients of p, but not p itself, can affect fluid motion. Thus, a uniform isotropic tensor of arbitrary magnitude can be added to T (or or) without consequence to the flow behavior. Adding such an isotropic tensor is equivalent to adding a constant to each diagonal component of the stress tensor. Thus, if the fluid is incompressible, a is determined only up to an additive isotropic tensor, and the stress-free state is synonymous with the state of isotropic stress. 

The loading of the structure is mechanically very simple since it consists in an initial isotropic stress field related to the dead weight. Concerning the mechanical boundary conditions, a zero normal stress is prescribed on the free boundaries of the pattern (wall of the drifts and well, and ground surface), and the symmetry planes are characterized by a zero normal displacement. Before excavation, the rock mass is supposed to be in a compressive stress state, and the principal minor stress 03, indicator of maximum compression, is equal to CTh (and 0i=O2=o =ayy). This stress increases (in absolute value) with depth, from -1.1 MPa at the top of the wells to -1.6 MPa at its base, and to -3.1 MPa in lower limit of the model. The excavation of drifts and wells causes a disturbance of this initial stress field (see fig.3). It is noticed that, apart... 

The constitutive equations are expressed in terms of the rates of the following variables the effective isotropic stress p = J i the deviator stress q = V(3/2sijSij) the volumetric strain = s,i and the deviator strain = V(2/3e,je,), where Sjj and e, refer to the stress and strain deviator, respectively. Compression is taken positive. The material is assumed to be isotropic and its deformation can be decomposed into elastic and plastic parts. The chemical effects on the material behaviour are described in terms of the contaminant mass concentration c, the ratio of contaminant mass to total fluid mass. 

During isotropic loading, plastic deformation takes place when the isotropic stress p reaches the preconsolidation pressure pf. The pressure pf is a measure of the size of the yield surface on the isotropic axis and can be viewed as an hardening/softening parameter (the specific shape of the yield surface is described in the next section). An essential feature of the proposed model is the decrease of pf with respect to an increase in contaminant concentration. This can be expressed as... 

Figure 2. Chemical softening (CHS) curves for various values of the constant a, in the diagram concentration vs effective isotropic stress.

This relation contains two competing terms the first term represents plastic hardening as a function of the volumetric part of plastic strain, the second term describes chemical softening due to an increase in contaminant concentration. Let us consider the plastic response to an increase in contaminant concentration at constant isotropic stress. The condition p =pc =0 in equation (7) implies... 

Figure 3. Yield curve and plastic strain rate vector in the plane deviatoric stress vs effective isotropic stress.

A back pressure is applied at the bottom of the specimen and the top drainage line is closed (valve (7) in figure 1) while total isotropic stress is increased in order to maintain a constant effective stress. The saturation time corresponds to the moment when the pressure at the top of the specimen equals the back pressure. This time is physically the time for the pressure to diffuse it through the specimen. Theoretically, if the system is saturated there is no flow during this stage. Practically we ensured saturation measuring the Skempton coefficient which has to be near to I. It can be noted that, in this kind of materials, saturation is never perfectly ensured. Moreover saturation phase can induce damaging and it would be of interest to quantify this effect. 

The first term in the right-hand side of (8.101) represents the isotropic stress similar to pressure and caused by the thermal (Brownian) motion of particles... 

Hk = iXa- (A and a being the longitudinal saturation magnetostriction and isotropic stress constant, respectively)... 

In contrast Qo withstands hydrostatic pressure up to 20GPa [81]. However, the football molecules seem to be unstable towards uniaxial or shear stresses whereas they are stable under isotropic stress where the spherical molecules are homogeneously deformed. In a dense arrangement of Cgo spheroids, 48 of the 60 carbon atoms have a quasi-tetrahedral coordination which is required in the diamond structure. Only small structural rearrangements are then necessary for the transformation into diamond [78],... 

Nose-Hoover thermostat [40] and Shinoda barostat [41] are applied at MD runs. As a fluid medium is simulated, the external pressure is established by changing only the size of the simulation box to fit the pressure tensor component to the target value. The sizes Lx and Ly remain the same during the simulation, and the isotropic stress tensor is maintained hydrostatically by the fluid phases. The simulations are carried out for 1.5 million timesteps, or 6 ns, and component densities are then averaged over the last 500,000 timesteps in 100 bins along the z-axis to obtain the profiles. 

To determine a constant and isotropic stress rate, only a minimal surface area is required for the calculation nevertheless it is necessary to consider differing strengths in the warp and weft directions of the fabric and, to some extent, of the foil. This should preferably be taken into account in creating the cutting pattern when additional pre-stresses have to be induced in the membrane to create structural shapes which could not otherwise be developed. 

Chu, T.Y. and Hashin, Z. (1971) Plastic behavior of composites and porous media under isotropic stress. International Journal of Engineering Science, 9, 971-994. 

Delamination of the MPL from the GDL substrate has not been widely reported but may occur during freeze-thaw cycles, as occurs with catalyst-layer delamination from the membrane [131, 132]. A different situation occurs in the GDL/MPL, where the pore diameters are on the order of a micron or larger and the water is not hydrating the sulfraiic acid of the ionomer. The volume expansion caused by ice formation can produce large isotropic stresses that can damage the structure of the catalyst layer, the MPL, or the GDL. 

The underlined term gives the isotropic stress kgT6 p by integration by parts, and can be dropped in the incompressible fluid (see Section 3.7.2). 

Note that eqn (7.4) is valid even if the excluded volume effect is accounted for since, as shown in eqn (4.135), the pseudo-potential described by the delta function does not change the expression for the stress tensor (apart from the isotropic stress, i.e., the pressure). 

Deformation tests are started at a certain isotropic stress state, the axial strain is increased with a constant strain rate (Fig. 11.40). The related evolution of volume, AE rate, ultrasonic wave velocities, and permeability is... 

In case of isotropic stress, the entire soil structure is compacted without any noticeable redirection of structural elements. Compression forms a homogenous isotropic microstructure with sufficient strength and moderate compressibility. 

Lintilhac (1977) supported this hypothesis by showing that in Coleus stems an imposed force induces a pattern of cell wall orientation corresponding to the principal stresses generated by the force. Stress-induced alterations in the plane of cell division were also demonstrated in Jerusalem artichoke callus tissue by Yeoman and Brown (1971). Lintilhac points out that this model may explain the selective development of a single cell among a mass of apparently identical cells. The cell in the very center of the nucellus, for example, is unique in its exposure to isotropic stress, and it is this cell which divides and develops into the megasporangium. 

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