## A yield criterion

A yield criterion is a statement of the conditions under which deformation will occur. If the condition is not met, there is no deformation, and conversely. The simplest imaginable yield criterion is that of Tresca (1864):

or when the difference between any two principal stresses exceeds a material constant, k (determined experimentally for any given material),

Figure 9.4. Variation of stain rate, s, with applied stress, a, in perfectly plastic and viscoplastic materials.

Figure 9.4. Variation of stain rate, s, with applied stress, a, in perfectly plastic and viscoplastic materials.

- s yielding will occur. An alternative, the von Mises yield criterion, is:

In this case, each of the three principal stresses contributes.

Let us investigate the relation between the von Mises criterion and J2. After some manipulation we can obtain:

(ai - a2)2 + (a2 - as)2 + (a3 - ai)2 = 2 (of + a2'2 + a3'2) + 2J2 (9.ii)

where the primes denote deviatoric stresses as before. Note that we started with total stresses on the left side. Had we started with deviatoric stresses, we would have obtained the same result, as P drops out. Thus, the yield criterion is unchanged if we use deviatoric stress instead of total stress. From Equation (9.i0), noting that the shear stresses vanish because we are here dealing with principal stresses, we find that the term in brackets on the right-hand side of Equation (9.ii)is equal to 2J>. Thus, the von Mises yield criterion reduces to 6J > k, or since J = a2, we have ae > ^Jk/6. In other words, when ae equals or exceeds Vk/6, yielding will occur.

Yield criteria are often associated with perfect plasticity. A perfectly plastic material does not deform at stresses below its yield strength, k. However, once the applied stress reaches k, the material begins to deform, and it deforms at a rate such that the stress does not exceed k (Figure 9.4). In terms of Glen's flow law, a perfectly plastic material would be represented by n ^to so there would be no strain until ae equaled B, where B would be the equivalent of Vk/6. Viscoplastic or Bingham materials also exhibit a yield stress, but once the yield stress is reached, the material deforms at a rate that depends on the amount by which the applied stress exceeds the yield stress (Figure 9.4). Inasmuch as there may, indeed, be a stress below which ice does not deform, it resembles a nonlinear viscoplastic material. Glen's flow law does not recognize this yield stress, however, but approximates it by predicting very small strain rates at low stresses.

Thus, by using ae in the flow law, we are not incorporating a yield stress per se. Rather, we are simply saying that the strain rate in any given direction is likely to be a function of all of the stresses acting on the material, not just the stresses in that direction. For example, the flow law states that ice will shear faster under a stress axy if there is also a deviatoric normal stress, cx'x, on it. Experiments by Li et al. (1996) firmly support this concept.

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