The Coulomb Yield Criterion

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

FIGURE 12 6 Angle of repose of a heap of powder. An acute angle of repose (left) is observed with a powder that does not flow well. An obtuse angle of repose (right) is observed with a powder that flows well. 

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. 

The Tresca yield criterion assumes that the critical shear stress is independent of the normal pressure on the plane on which yield is occurring. Although this assumption is valid for metals, it is more appropriate in polymers to consider the possible applicability of the Coulomb yield criterion [10], which states that the critical shear stress r for yielding to occur in any plane varies linerarly with the stress normal to this plane, i.e. 

The Coulomb criterion was originally conceived for the failure of soils and Tc was termed the cohesion and the coefficient of internal fnction. For a compressive 

For 3deld to occur at the lowest possible value of oi, (cos0sin0 — tan0cos 0) must be a maximum, which gives 

Thus tan0 determines the direction of yield and conversely the direction of yielding can be used to define 0, where tan0 is the coefficient of friction. If the stress CTi is tensile, the angle 0 is given by 

We see that the Coulomb yield criterion therefore defines both the stress condition required for yielding to occur and the directions in which the material will deform. Where a deformation band forms, its direction is one that is neither rotated nor distorted by the plastic deformation, because its orientation marks the direction that establishes material continuity between the deformed material in the deformation band and the undistorted material in the rest of the specimen. If volume is conserved, the band direction denotes the direction of shear in a simple shear (by the definition of a shear strain). Thus for a Coulomb yield criterion the band direction is defined by Equation (11.6). 

The Coulomb criterion was originally conceived for the failure of soils and tc was termed the cohesion and p, the coefficient of internal friction. For a compressive stress, on has a negative sign so that the critical shear stress r for yielding to occur on any plane increases linearly with the pressure applied normal to this plane. 

The shear stress is ti =asin0cos0 and the normal stress on = —o-icos 0. Yield occurs when 

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

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... 

If yielding is to occur by sliding parallel to any plane, it seems reasonable to suppose that there must be a critical shear stress r parallel to that plane. It is also physically reasonable to assume that this critical stress x will be increased if the compressive stress —mean normal stress, as just defined, not nominal stress. Remember also that the usual convention for stress makes - -a the tensile stress.) The simplest assumption is that x depends linearly on Coulomb yield criterion ... 

The compressive stress a applied parallel to the axis of a rod of material that obeys the Coulomb yield criterion with = 10 Pa and fi = 0.4 is gradually increased. Calculate the stress a at which the rod will yield and the angle 0 between the 5neld plane and the axis of the rod. 

Figure 11.12 Mohr circle diagram for two states of stress that produce yield in a material satisfying the Tresca yield criterion (a) and the Coulomb yield criterion (h)...

There have been a number of detailed investigations of the influence of hydrostatic pressure on the yield behaviour of pol3maers. Because it illustrates clearly the relationship between a yield criterion, which depends on hydrostatic pressure, and the Coulomb yield criterion, an experiment will be discussed where Rabinowitz, Ward and Parry [23] determined the torsional stress-strain behaviour of isotropic PMMA under hydrostatic pressures up to 700 MPa. The results are shown in Figure 11.17. 

Mechanical Properties of Solid Polymers 12.23 The Coulomb Yield Criterion... 

We have seen that the Coulomb yield criterion defines both the stresses required for yield and also the directions in which the material deforms. In the case of the von Mises yield criterion, we require a fiu ther development of the theory to predict the directions in which plastic deformation starts. 

The Coulomb yield criterion states that the critical stress, r, for shear deformation to occur in any plane increases Hnearly with the pressure, Un, applied normally to that plane ... 

This equation was found to be valid for a number of polymers (PVC, PC, PMMA, PS, CA) in more or less extended regions of temperature and strain rate [154,156,158]. The (temperature-dependent) activation volumes 7 had at room-temperature values between 1.4 (PMMA) and 17 nm (CA). This means that according to this concept polymer deformation at the yield point is due to the thermally activated displacement of molecular domains over volumes which are between 10 (PMMA) and 120 times (PVC) as large as a monomer unit. It has been indicated by several authors [155—158, 160] that the above criterion (Eq. 8.29) corresponds to the Coulomb yield criterion Tq + MP constant. The coefficient of friction ju is inversely proportional to 7. From an analysis of their experimental data on polycarbonate according to Eq. (8.29) Bauwens-Crowet et al. [158] conclude that two flow processes exist. They relate these to an a-process (jumps of segments of the backbone chains) and to the 3 mechanical relaxation mechanism. 

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