Criterion Coulomb

Abstract. Classical regular and chaotic dynamics of the particle bound in the Coulomb plus linear potential under the influence of time-periodical perturbations is treated using resonace analysis. Critical value of the external field at which chaotization will occur is evaluated analytically based on the Chirikov criterion of stochasticity. 

Coulomb correlation energy, U. The energy gain due to band formation is of the order of the bandwidth, W. Provided U can be calculated Table 1 can be used to examine the criterion UAV = 1 for Mott-localization ... 

In the case of compressive loading, the Coulomb yield criterion is often utilized in the form [20] ... 

Mohr-Coulomb Failure Criterion and Coulomb Powders... 

The most common failure criterion for granular materials is the Mohr-Coulomb failure criterion. Mohr introduced his theory for rupture in materials in 1910. According to his theory, the material fails along a plane only when a critical combination of normal and shear stresses exists on the failure plane. This critical combination, known as the Mohr-Coulomb failure criterion, is given by... 

The Mohr-Coulomb failure criterion can be recognized as an upper bound for the stress combination on any plane in the material. Consider points A, B, and C in Fig. 8.4. Point A represents a state of stresses on a plane along which failure will not occur. On the other hand, failure will occur along a plane if the state of stresses on that plane plots a point on the failure envelope, like point B. The state of stresses represented by point C cannot exist since it lies above the failure envelope. Since the Mohr-Coulomb failure envelope characterizes the state of stresses under which the material starts to slide, it is usually referred to as the yield locus, YL. 

So far, we understand that the flowability of powders depends on their failure stresses from the Mohr-Coulomb failure criterion. Therefore, analyses of powder flows... 

Eq. (8.24), and the modified Mohr-Coulomb yield (or failure) criterion, Eq. (8.27). It should be noted that other yield criteria, such as the von Mises criterion, are used to model the flow of bulk solids in hoppers, and more conditions may need to be imposed, such as the Levy flow rule, in order to close the system of equations [Cleaver and Nedderman, 1993],... 

A yield criterion similar to Coulomb s criterion for the friction between solid bodies has long been used in soil mechanics. It will be discussed in the next section. Later, an account will be given of Jenikes modification to this criterion for bulk solids. 

The general form of the expression for the Coulomb yield criterion used in soil mechanics is... 

In soil mechanics, the pressures lie in a range up to and over 2 MPa, whereas in powder mechanics they are usually below 0.1 MPa. For this range of pressure, the Coulomb criterion is generally not applicable. Only cohesionless solids exhibit Coulomb yield behavior at low pressures. 

In general, this Ck)ulomb yield criterion can be used to determine what stress will be required to cause a ceramic powder to flow or deform. All that is needed are the two characteristics of the ceramic powder the angle of friction, 8, and the cohesion stress, c, for each particular void fraction. With these data, the effective yield locus can be determined, from which the force required to deform the powder to a particular void fraction (or density) can be determined. This Coulomb yield criterion, however, gives no information on how fast the deformation will take place. To determine the velocity that occurs durii flow or deformation of a dry ceramic powder, we need to solve the equation of motion. The equation of motion requires a constitutive equation for the powder. The constitutive equation gives the shear and normal states of stress in terms of the time derivative of the displacement of the material. This information is unavailable for ceramic powders, and the measurements are particularly difficult [76, p. 93]. 

Let us assume that we are considering a cylindrical green body being die pressed, as is shown in Figure 13.33. Under load the particles in a volume element will densify if the Coulomb yield criterion... 

At a particular point, this force balance shows that the radial normal stress, Trr, or applied pressiure is related to the angular normal stress, Tgg, and the two radial shear stresses, and r g. Under load, the particles in a volume element will density, if the Coulombs yield criterion Tij = Tii tan 8 + c has been exceeded. Therefore, we find that... 

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