On anisotropic flow rules in multiplicative elastoplasticity at finite strains

Hill’s anisotropic formulation of the flow rule is extended so as to fit in the realm of multiplicative finite strain plasticity. The anisotropic Hill-type yield criterion is formulated in terms of purely material quantities. The list of arguments of the flow function includes a non-symmetric material Eshelby-like stress tensor, as well as structural tensors that describe the anisotropy at hand. The formulation is exemplified on orthotropy, where three structural tensors are employed. The fact that the stress tensor is not symmetric necessitates a special treatment of the flow function, where representation theorems of tensor valued function with non-symmetric arguments are invoked. The consequences of such a definition on the resulting inelastic rate are discussed in full. It is shown that the corresponding resulting rate, as defined at the actual configuration, is not symmetric any more. Accordingly, the rate naturally includes a plastic material spin. Moreover, we deal with the theoretically interesting question of how to define spin-free rates. It is also demonstrated that the flow function must depend not only on the stress tensor and on adequate structural tensors, but also on the deformation itself in form of the right Cauchy-Green tensor C. However, this surprising dependency, which must obey a specific form, can be justified as physically meaningful. Various numerical examples of large plastic deformations of structural components are presented, that underpin the capabilities of the formulation.