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.
In accordance with the great importance given to the subject of stiffness degradation, in particular with regard to metal forming, this work experimentally investigates the anisotropic elastic properties of plastically prestrained coldrolled sheet metal (stainless steel EN 1.4301, also AISI 304). From the experiments performed, two main conclusions regarding stiffness degradation can be extracted. First, since under specific stretching the intensity of the normalized Young's moduli degradation in both directions remains approximately similar, it may be concluded that the potential initial elastic anisotropy tends to be preserved during loading. Second, as the evidenced stiffness degradation has proved to be strongly correlated with the stretching direction of the sheet metal, it can be concluded that the stiffness evolution in the cold rolled sheet steel is path dependent. These interesting discoveries also provide some answers for modelling the kinetic damage evolution laws in damage mechanics.
The paper focuses on the modelling of springback within a formed stainless steel sheet. The main subject of this work is the construction of a constitutive model which simultaneously considers sheet anisotropy, damage evolution, and stiffness degradation in material during forming. The developed model is based on the Gurson-Tvergaard-Needleman damage model, which is adequately extended by the implementation of the anisotropic Hill48 plasticity and Mori-Tanaka's approach to stiffness degradation. Considering the established relationships, some material parameters that are included in the model are characterised by the corresponding measurements. The experimental validation of the developed constitutive model is performed on a springback test, which consists of bending and releasing rectangular stainless steel specimens that were previously plastically prestrained to a different degree, either in the rolling or transverse direction. A comparison of the proposed modelling approach to the classical approach by using the Hill48 model clearly indicates that the simultaneous modelling of material phenomena, especially the coupling of stiffness degradation with anisotropic plasticity, can be the true key to obtaining a more accurate prediction of the springback in sheet-metal-forming applications.
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