Approximation-based implicit integration algorithm for the Simo-Miehe model of finite-strain inelasticity

We propose a simple, efficient, and reliable procedure for implicit time stepping, regarding a special case of the viscoplasticity model proposed by Simo and Miehe (1992). The kinematics of this popular model is based on the multiplicative decomposition of the deformation gradient tensor, allowing for a combination of Newtonian viscosity and arbitrary isotropic hyperelasticity. The algorithm is based on approximation of precomputed solutions. Both Lagrangian and Eulerian versions of the algorithm with equivalent properties are available. The proposed numerical scheme is non-iterative, unconditionally stable, and first order accurate. Moreover, the integration algorithm strictly preserves the inelastic incompressibility constraint, symmetry, positive definiteness, and w-invariance. The accuracy of stress calculations is verified in a series of numerical tests, including non-proportional loading and large strain increments. In terms of stress calculation accuracy, the proposed algorithm is equivalent to the implicit Euler method with strict inelastic incompressibility. The algorithm is implemented into MSC.MARC and a demonstration initial-boundary value problem is solved.