SUMMARYThe article presents a simple but efficient numerical scheme for the integration of non-linear constitutive equations, in which the principal reason for the inaccuracy of the classical explicit schemes, for example forward-Euler scheme, is effectively eliminated. In the newly developed explicit scheme, where there is no need for iteration, the implementation simplicity of the forward-Euler scheme and accuracy of the approach, which is using the backward-Euler scheme to integrate the constitutive equations, are successfully combined. Computational performance of the proposed next increment corrects error (NICE) integration scheme, particularly regarding the accuracy and the CPU time consumption, is first analysed on a case of complex loading of a material point. When comparing it to the forward-Euler, backward-Euler, trapezoidal and midpoint integration schemes, it turns out that because of its capability of a fast and relatively accurate integration of the constitutive equations, the NICE scheme is very convenient for the integration of constitutive models, where a direct solution technique is used to solve a boundary value problem. Although the deduction of the new integration scheme is general, its implementation for shell applications needs particular care. Namely, in order to satisfy the zero normal stress condition during the whole integration, a through-thickness strain increment has to be adequately chosen in each integration step. The NICE scheme, which was also implemented into ABAQUS/Explicit via User Material Subroutine (VUMAT) interface platform, has been additionally compared with the ABAQUS/Explicit default integration scheme (backward-Euler) and forward-Euler scheme. Two loading case-studies, namely the bending of a square plate and the stretching of a specimen including the onset of necking, are considered with two constitutive models-the von Mises and GTN material model being adopted. Generally, the NICE scheme has demonstrated to be advantageous in cases, where reasonable accuracy and very fast integration of the constitutive model is demanded, which is mostly the case in engineering computations with a direct solution method, for example explicit dynamics and metal forming process simulations.
