A unified stress update algorithm for explicit transient shell dynamics with combined isotropic-kinematic hardening in Eulerian rate-type phenomenological finite elasto-plasticity models

This paper discusses applications of the objective stress rates to a unified stress update algorithm for transient shell dynamic analysis within the context of explicit time integration. We propose a unified stress update algorithm via establishing the Eulerian rate-type constitutive equations of hypoelastic–plastic materials. The constitutive equations are often given on the additive decomposition of the rate of deformation tensor under the assumption that the elastic part of the rate of deformation tensor may be characterized to be hypoelastic with grade zero, i.e., the fourth-rank constant isotropic tensor. Several objective stress rates are derived through the Lie derivative of the Cauchy or Kirchhoff stress tensor. Those stress rates cover, for instance, the Jaumann, Green–Naghdi, Truesdell, Oldroyd, and Cotter–Rivlin stress rates. Further, the unified stress update algorithm is implemented along with the so-called radial return method and the phenomenological plasticity theory with combined isotropic/kinematic hardenings. Among the objective stress rates employed here, we emphasize especially the stress update procedure of the Green–Naghdi stress rate for transient shell dynamics, because this algorithm developed in this work is not shown even in the representative commercial codes. In the implementation of the unified stress update algorithm, we applied our algorithms to the Belytschko–Lin–Tsay shell theory to validate its accuracy and effectiveness. Moreover, a slightly modified secant algorithm for the zero transverse normal stress of the shell is also developed. The ultimate objectivity of this work is to provide a guideline for the best choice of an appropriate objective stress rate for phenomenological elastoplastic models. Several numerical examples are demonstrated, including non-contact transient shell dynamics examples and contact–impact examples such as a ball-plate impact problem, by which the accuracy and effectiveness of the unified stress update algorithm, developed in this work, are addressed.