Evaluation of bounds to the effective yield surface for face-centered cubic polycrystalline materials

We discuss the macroscopic yielding behavior of rigid-perfectly plastic polycrystalline materials and the dependence of such behavior on the microstructure. We then describe a new upper bound for the effective yield surface of such a polycrystal. This new bound is based on a variational principle which is derived by adding and subtracting the energy of a linear heterogeneous reference material, and is written in terms of the local and effective moduli of this heterogeneous material. Such a bound is achieved essentially by bounding the effective modulus of the reference material. Such a theory guarantees a bound at least as tight as Taylor's (i.e. 1-point statistics) if the arithmetic mean bound is used to obtain the effective reference modulus and certain restrictions are placed on the eigenvalues of the local moduli. If the effective modulus is bounded using two-point statistics then we are guaranteed a bound which is tighter than that of Taylor. The two-point statistics are obtained using Orientation Imaging Microscopy. We show that for ideally textured materials some reduction from the Taylor bound is obtained. However for materials which have even a small degree of randomness, this reduction is negligible. We then compare these bounds with universal bounds which are functions only of the yield surface of the reference crystallite.; We show that the distance between the universal upper bound and the universal lower bound is quite large for the case of cubic polycrystals. We also show that that the Taylor bound, for the case of completely random polycrystals, lies nearly at the average of the uniform stress lower bound and the universal upper bound.