Computational visco-plasticity for crystals lacking five independent slip systems

For a crystal structure to undergo any arbitrary volume-invariant plastic deformation by the mechanism of crystallographic slip, a set of five linearly independent slip systems is needed. While high symmetry crystals possess a relatively large number of physically distinct slip systems of which a set of five independent slip systems can be composed, in case of low symmetry crystals, such as for example many polymer crystals, less than five independent slip systems may be present, in which case these materials have limited deformation possibilities. For these crystals, lacking five independent slip systems, besides incompressibility, some extra kinematic constraints need to be satisfied. The kinematic deficiencies of the material structure are identified as inextensible directions within the crystal lattice. Each kinematic constraint is encountered for by a restriction on the deformation rate tensor. A finite element formulation is presented for low symmetry crystal structures lacking five independent slip systems. The formulation is based on the decomposition of the deviatoric stress tensor into a part associated with crystallographic slip and a part related to the extra kinematic constraints. These constraints are taken into account by additional weighting expressions. The present formulation of the single-crystal visco-plasticity model allows for an implementation of the extra kinematic constraints into a finite element context by either a Lagrange multiplier method or a penalty method. Since large plastic deformation of low symmetry crystalline structures generally induces significant crystallographic texture, affecting the mechanical behaviour of these materials, the lattice orientations are incrementally updated.