Coulomb criterion, yield stresses

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. 

Figure 12.10 The yield direction under a compressive stress a, for a material obeying the Coulomb criterion.

In Figure 12.12(b), two states of stress causing yield for a material which satisfies the Coulomb criterion are shown as as and a-j and ag, respectively. In this case, the yield... 

The yield stress of a material is a measured stress level that separates the elastic and inelastic behavior of the material. The magnitude of the yield stress is generally obtained from the results of a uniaxial test. However, the stresses in a structure are usually multiaxial. A measurement of yielding for the multiaxial state of stress is called the yield criterion. For example, the Mohr-Coulomb criterion, the Buyukozturk criterion, and the Drucker-Prager criterion are common yield criteria for concrete, and the von Mises criterion is the widely used yield criterion for steel. 

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. 

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]. 

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... 

This model is based on the mean features of the Mohr-Coulomb model and is expressed with stress invariants [Maleki (1999)] instead of principal stresses. Until plasticity is reached, a linear elastic behaviour is assumed. It is fully described by the drained elastic bulk and shear moduli. The yield surface of the perfectly plastic model is given by equation 7. Function 7i(0) is chosen so that the shape of the criterion in the principal stress space is close to the Lade criterion. 

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. 

The idea of plastic yielding is also applied to granular materials, such as soils and powders. In this case, the shear stress required for deformation depends on the packing density, and the particle shape and surfaee characteristics. The shear resistance is commonly described by the Mohr-Coulomb yield criterion, i.e.. 

For a uniaxial compression test, determine the plane on which the maximum shear stress will occur for a material that obeys the Mohr-Coulomb yield criterion. Suggest an approach for measuring the material parameters (d>, Tq) in the Mohr-Coulomb yield criterion. (Hint construct the Mohr s circle (see Fig. 2.23) for various values of the normal stress.)... 

Most amorphous solids and many crystalline ones, particularly non-metals and polymers, exhibit a Coulomb-Mohr-type (Coulomb 1773 Mohr 1900) yield criterion or plastic-shear resistance such that this resistance on the best shear plane is dependent on the normal stress acting across the plane of shear, resulting in a dependence of the type... 

Overburden of the tunnel 100.0 m The rock mass is considered as the idealized elastoplastic material with the Mohr-Coulomb yield criterion, and all the materials of support system are treated as elastic material. The initial stress field of analysis zone is calculated as self-weight... 

Coulomb Yield Criterion. In 1773, Coulomb (26) identified two components important in the strength of building stone—cohesion and friction. He observed that the shear stress, t, necessary to cause shear failure across a plane is resisted by the cohesion of the material So and by the product, fiorti, across that plane, where the constant /a is called the coefficient of internal friction and force normal to the shear plane ... 

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. 

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). 

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)...

The yield point in compression a was measured for various values of applied tensile stress 02. The results, shown in Figure 11.16, give Oi = —110.0 + 13.65ct2s where both o and 02 are expressed as true stresses in units of MPa. The results therefore elearly do not fit the Tresca criterion, where 0 - 02 = constant at yield neither do they fit a von Mises yield criterion. They are, however, consistent with a Coulomb yield criterion with r = 47.4 — 1.58(/n. 

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. 

Yield in Mohr-Coulomb materials follows a criterion which is the sum of stress terms and the Mises 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 well-known failure criterion according to Jenike [3] was the first, which could describe the behaviour of particulate solids at stress levels relevant in powder technology. It was modified e.g. by Schwedes [4], Molerus [5] and Tomas [6,7]. For Coulomb fnction in the interesting positive compressive stress range, an yield locus can be approximated by a straight line [7] ... 

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