A generating function for the yield criterion of isotropic and anisotropic polycrystalline materials

Summary

A yield criterion for elastic pure-plastic polycrystalline materials is generated under simplified conditions by assuming that for yielding a certain fraction Q _c of the total number of slip planes in the material has to be active. This fraction Q _c is called the critical active quantity. We suppose Q _c to be independent of the state of stress. The yield criterion is mathematically expressed as an integral, which is a function of Q _c. This criterion can also be used for anisotropic materials.

For isotropic materials the ratio (r) of the yield stress in torsion to that in tension is calculated as a function of Q _c. We find 0.5≤r≤0.61.

The value r=0.5 (Tresca's criterion) is obtained for Q _c=0 and Q _c=1. The value r=0.577 (von Mises criterion) is obtained for Q _c=0.34 and Q _c=0.79. The difference between two criteria with the same r is the magnitude of the yield stress. We think the value Q _c=0.79 corresponds to the experiments for f.c.c. materials, since a rough estimation gives Q _c>0.75 for yielding.

The independence of Q _c on the state of stress brings on that r>0.5 is more probable. This is caused by the slower increase to Q _c in torsion compared with the case of tension.

From the theory follows that in the general case (Q _c≠0) the middle principal stress has influence on yielding.

In this paper we don't determine Q _c, but adapt its value to the experimental results. However, a rough estimation of Q _c is given for isotropic materials.