Yield criteria Tresca

It can be proved from Eq. (2.156) that, for materials with Poisson s ratio of 0.3 (which is true for most solids), the maximum shear stress oz — or occurs at z/rc = 0.48. Consequently, according to Tresca s criterion, the yield stress Y in a simple compression is 0.62 p0. Therefore, when the hardness or the yield stress Y of the particle material is less than 0.62 times the maximum contact pressure, the sphere will, most likely, undergo plastic deformation. From the elastic collision of two solid spheres, the maximum contact pressure is given by Eq. (2.134). Thus, the relation between the critical normal collision velocity, Ui2Y. and the yield stress is given by... 

The tensile yield stress variation as a function of W for a material which has a von Mises-type yield locus is illustrated schematically in Figure 5. This variation is caused by the fact that as the width of the specimen increases, the biaxiality also increases toward the asymptotic value at plane strain. If the material obeys the von Mises yield criterion exactly, the plane strain yield stress should be 15% higher than it would be for simple tension. On the other hand, if the material obeys the Tresca yield criterion, the plane strain yield stress should be identical... 

The value of a can be determined from a yield criterion. The criterion of Tresca yields a = 2 (This has been used in fig.l). The criterion of v. Mises yields a = V3. A more refined analysis of the Taylor type yields... 

The Mohr circle representation (Fig. 9.6c) is a graphical method of relating stress components in different sets of axes. When the axes in the material rotate by an angle B, the diameter of the circle rotates by an angle 2 B. If the material yields, the circle has radius k, the constant in the Tresca yield criterion. The axes of the Mohr diagram are the tensile and shear stress components. Thus, in the left-hand circle, representing the stresses at A in Fig. 9.6b, the ends of the horizontal diameter are the principal stresses. The principal axes are parallel and perpendicular to the notch-free surface. There is a tensile principal stress Ik parallel to the surface, and a zero stress perpendicular to the surface. The points at the ends of the vertical diameter represent the stress components in the a)3 axes, rotated by 45° from the principal axes. In the a/3 axes, the shear stresses have a maximum value k, and there are equal biaxial tensile stresses of magnitude = k (the coordinate of the centre of the circle). 

In the second form the von Mises criterion expresses directly the fact that the yield depends equally on the three shear stresses (cTj — oy)/2. A somewhat simpler criterion, the Tresca yield criterion, makes the slightly different assumption that yield takes place when the largest of these three shear stresses reaches a critical value. The surface in cr-space that represents the criterion is therefore defined by the six equations... 

One simple criterion for yielding under multiaxial stresses is known as the Tresca yield criterion. This approach recognizes that the maximum shear stress is one-half the difference between the maximum and minimum principal stresses. In terms of the uniaxial yield stress [Pg.187]

Tresca Yield Criterion. The earliest proposal for a yield criterion in metals is due to Tresca (24), and it stated that yield occurs when the maximum shear stress reaches a critical value. With <7i > <72 > <73 the criterion can be written as... 

Compare this to the prediction of ay/2 from the Tresca criterion. The yield criteria for both the Tresca and Von Mises theories are shown graphically in Figure 6. For simplicity, the plots are shown for conditions of plane stress (ie 03 = 0). We can see that the Von Mises criterion describes an ellipse in stress space, with the Tresca criterion consisting of a series of straight lines bounded by the Von Mises limits. 

As shown earlier, a simple criterion for yield is that the maximum shear stress reaches a critical value given by t = Oy/2, where Oy is the tensile yield stress (ie the Tresca yield criterion). Substituting and rearranging equation 29 gives... 

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. 

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)...

A very simple explanation of the effect of notching has been given by Orowan [95], For a deep, symmetrical tensile notch, the distribution of stress is identical to that for a flat frictionless punch indenting a plate under conditions of plane strain [102] (Figure 12.31). The compressive stress on the punch required to produce plastic deformation can be shown to be (2 + 7t)K, where K is the shear yield stress. For the Tresca yield criterion the value is l.Sloy and for the von Mises yield criterion the value is 2.82oy, where 0 is the tensile yield stress. Hence for an ideally deep and sharp notch in an infinite solid the plastic constraint raises the yield stress to a value of approximately 2>Oy which leads to the following classification for brittle-ductile behaviour first proposed by Orowan [95] ... 

The reason for this is that the specimen yields everywhere in the cross section. The so-called yield criterion, defined in section 3.3.1 below, has to be fulfilled. If the Tresca yield criterion is used, the difference between smallest and largest stress has to be identical for the specimen to yield everywhere. 

The Tresca yield criterion or maocimum shear stress criterion is not directly based on the considerations of the previous sections, but it fulfils them nevertheless. It states that the maximum shear stress in the material point determines yielding. This maximum shear stress can be determined graphically using Mohr s circle, see figure 2.3 on page 34. The maximum principal stress is denoted as a, the intermediate value as au, and the smallest as crni- The maximum shear stress is... 

Since the Tresca yield criterion can be easily evaluated using Mohr s circle, it is often used in heuristic explanations. Using it in the calculation of plastic deformations, for example, with the method of finite elements, is problematic, though, as we will see on page 96. 

Furthermore, the relation between stresses and plastic strain rates must be unique. From this, it can be seen that the yield surface must be strictly convex and continuously differentiable to allow the formulation of a flow rule. The Tresca yield criterion is not continuously differentiable (there is no unique normal vector at its corners), and on the surfaces, different stress states fulfil equation (3.42) for a given Therefore, a flow rule cannot be derived using this criterion. 

In reality, the stress state is biaxial at the notch root (the radial stress at the surface is zero), so that there is no difference to the uniaxial case if the Tresca yield criterion is used. If the von Mises yield criterion is used, there is a slight difference which is neglected here. 

If we use the Tresca yield criterion, yielding occurs exactly at cri = Rp. With the von Mises yield criterion, the result is /i/2 [(tri — ctc) + cr f + a ] = Rp. Depending on the value of the circumferential stress, the axial stress at which yielding starts may be up to 15.5% larger than with the Tresca criterion (see... 

Because the yield criterion is independent of the hydrostatic component of stress, we can replace ai, an and am by a I + p,a + pandam + p, respectively, without affecting the material s state with regard to yield. Thus, if the point in principal stress space (ai, an, am) lies on the yield surface, so does the point (ai + p, an + p, am + p). This shows that the yield surface must be parallel to the 111 direction, and has the appearance as sketched in Figure 12.11. The material isotropy implies equivalence between o, ou and am and hence that the section has a threefold symmetry about the 111 axis. The assumption that the behaviour is the same in tension and compression implies an equivalence between ai and —ai and so on and hence we have finally sixfold symmetry about the 111 direction. This is most clearly shown by the Tresca yield surface in Figure 12.11. 

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