Tresca yield condition

Our geometric model of the crystal is most appropriate for polycrystals since we have hypothesized that any and all planes and slip directions are available for slip (i.e. the discrete crystalline slip systems are smeared out) and hence that slip will commence once the maximum shear stresses have reached a critical value on any such plane. This provides a scheme for explicitly describing the yield surface that is known as the Tresca yield condition. In particular, we conclude that yield occurs when... 

Consider first the geometrical character of the stress. A multiaxial stress can be characterized by the three principal stresses Sj, S2, and S3, listed in descending value. Shear yielding depends primarily upon the difference between Sj and S3. The Tresca yield condition, Sj - S3 = Y (applicable to metals), has been modified for polymers (for which the shear yield stress Y increases with hydrostatic pressure) (24). 

In spite of the relative simplicity of the Tresca criterion, conditions for shear yielding in isotropic polymers are best summarized by the von Mises criterion (11),... 

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 initial yield of the pressure vessel occurs at the bore when the following yield condition according to Tresca/Guest is fulfilled [lOj ... 

A yielding criterion gives critical conditions (at a given temperature and strain rate) where yielding will occur whatever the stress state. Two main criteria, originally derived by Tresca and von Mises for metals, can be applied to polymers (with some modifications due to the influence of hydrostatic pressure) ... 

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. 

For t 00 follows (pu 0, that means, the time yield locus is a parallel line to the a -axis, i.e. failure criterion of ideal plasticity by Tresca. The bulk material is hardening to a complete solid state with plastic failiue conditions as a limitation. 

Although the relative simplicity of the Tresca criterion is rather attractive it is found that the criterion suggested by von Mises gives a somewhat better prediction of the yield behaviour of most materials. The criterion corresponds to the condition that yield occurs when the shear-strain energy in the material reaches a critical value and it can be expressed as a symmetrical relationship between the principal stresses of the form... 

In three-dimensional principal stress space the modified Tresca cylinder becomes a hexagonal pyramid and the pressure-dependent von Mises cylinder becomes a cone. The significance of the apices of the pyramid and cone is that they define the conditions for which there can be yielding under the influence of hydrostatic stress alone. This is something which cannot happen for materials which obey the unmodified criteria (Section 5.4.3). 

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