On the extremal stress and displacement in Hencky plasticity (Q799408)

Some earlier studies have established the existence of an extremal displacement field \(\bar u\) for an elasto-perfectly plastic material employing Hencky's law. The construction of the extremal stress field \({\bar\sigma }\) from \(\bar u\) and duality between the spaces of stresses and strains have also been investigated. In the present paper, considering a pair of extremal fields \(\bar u\), \({\bar\sigma }\), it is proved that the singular density of the deformation \(\epsilon^ D(u)\) is determined by the absolutely continuous tensor \({\bar\sigma }\)(D) via a relation stated in the paper. The proof has been presented with the help of several theorems and lemmas and requires a good knowledge of mathematical analysis.