Complex step derivative approximation of consistent tangent operators for viscoelasticity based on fractional calculus.

In this contribution, the convergence behaviour of simulations with nonlinear viscoelastic models for rubber-like materials using fractional derivatives is investigated. Based on the complex step derivative approximation, numerical approximation schemes for tangent operators on the local and global algorithmic level are analysed. Material models including fractional derivatives usually exhibit numerical difficulties, since the entire stress history of the material has to be considered for the current stress state. Non-classical methods can help to reduce the numerical effort, but the convergence behaviour is not perfect. In this paper, a classical and a non-classical fractional element based viscoelastic material formulation are analysed. The local and the global convergence rate using analytically and numerically derived tangents are investigated and compared to the convergence behaviour of a standard nonlinear viscoelastic model. Numerical examples on rubber materials are exploited, showing the performance of the proposed methods. It can be proven that the outlined numerical differentiation schemes improve the convergence rate, as well as reduce the computation time for the fractional element based material models. [ABSTRACT FROM AUTHOR]