Finite deformation plasticity in principal axes: from a manifold to the euclidean setting

Abstract

This contribution presents a link between early developments in finite deformation plasticity, where the notion of a covariant formulation is introduced and employed, and more recent developments, where rediscovery of a fundamental work of Hill on the method of principal axes led to a very efficient implementation scheme. More precisely, we demonstrate how to develop a covariant theory of finite deformation plasticity in an invariant form, by making use of the elastic principal stretches. We also show how to implement principal axis formulation in the framework of manifold, to carry out all the necessary manipulations by exploiting the Lie derivative formalism and eventually to simplify the final result to the Euclidean setting. Much of our work on numerical implementation reflects the fruitful cross-fertilization of ideas with those from theoretical formulation. In particular, we show how the operator split method, which is typically used to simplify the plastic flow computation, can also be used to reduce the computational cost related to the special finite element interpolation schemes based on incompatible modes. The latter proves to be an indispensable ingredient for accommodating the near-incompressibility constraint arising in the finite deformation deviatoric plasticity. An important advantage of the proposed formulation as opposed to alternative remedies (e.g. B-bar method) is that the basic structure of the governing equations need not be modified.