Implementation of incremental variational formulations based on the numerical calculation of derivatives using hyper dual numbers

Abstract

In this paper, novel implementation schemes for the automatic calculation of internal variables, stresses and consistent tangent moduli for incremental variational formulations (IVFs) describing inelastic material behavior are proposed. IVFs recast inelasticity theory as an equivalent optimization problem where the incremental stress potential within a discrete time interval is minimized in order to obtain the values of internal variables. In the so-called Multilevel Newton–Raphson method for the inelasticity theory, this minimization problem is typically solved by using second derivatives with respect to the internal variables. In addition to that, to calculate the stresses and moduli further second derivatives with respect to deformation tensors are required. Compared with classical formulations such as the return mapping method, the IVFs are relatively new and their implementation is much less documented. Furthermore, higher order derivatives are required in the algorithms demanding increased implementation efforts. Therefore, even though IVFs are mathematically and physically elegant, their application is not standard. Here, novel approaches for the implementation of IVFs using HDNs of second and higher order are presented to arrive at a fully automatic and robust scheme with computer accuracy. The proposed formulations are quite general and can be applied to a broad range of different constitutive models, which means that once the proposed schemes are implemented as a framework, any other dissipative material model can be implemented in a straightforward way by solely modifying the constitutive functions. These include the Helmholtz free energy function, the dissipation potential function and additional side constraints such as e.g. the yield function in the case of plasticity. Its uncomplicated implementation for associative finite strain elasto-plasticity and performance is illustrated by some representative numerical examples.