Rigid perfectly plastic two-dimensional polycrystals

We consider rigid–perfectly–plastic polycrystals in the two–dimensional anti– plane shear context. The yield sets of the grains are identified with rectangles in the plane centered at the origin whose sides have length 2 and 2M . The limit M → ∞ corresponds to the grains being rigid in one direction and ductile in the orthogonal direction. We show that for large values of M , there exist polycrystals whose effective yield sets are large in all directions. More precisely, for each value of M , we construct a polycrystal whose yield set contains the set [−f, f ]×[−f, f ], where f = √M−O(1). We also show that the yield set of any isotropic polycrystal is contained in the ball of radius 4 √ M/π centered at the origin. This bound results as an application of the div–curl lemma. The new component of our analysis, that allowed us to obtained sharper results, is that we consider simultaneously not only two, but an infinite number of admissible stress fields whose averages have different directions.