Keywords: numerical integration, constitutive equations, sheet metal forming, computing methods, finite elements, damage.

One of topics in finite element computations of sheet metal forming is the introduction of new constitutive models, which consider plastic anisotropy, path dependent hardening, damage, path dependency of fracture and other experimentally observed effects. Mostly newly developed constitutive models are not available in commercial finite element codes. To use those models, a user is forced to implement them into a general purpose finite element tool, usually via a user subroutine, which is in fact a programmed numerical integration scheme for an implemented constitutive model. Accurate integration of the nonlinear constitutive laws over the strain path is essential for a precise solution of any nonlinear boundary value problem in continuum mechanics, especially when additional nonlinearities are involved in the problem.

Nowadays, most commonly used schemes for integration of elasto-plastic constitutive laws can be classified into the categories of explicit and implicit schemes. One of the most popular of implicit schemes is the backward-Euler scheme, which is nowadays employed in commercial codes for finite element analysis [1]. Although most recent works prefer to use the backward-Euler scheme in the field of solid mechanics, it is difficult to implement the procedure for complex constitutive relations [2], besides the numerical iteration procedure for root determination consumes a lot of CPU time. On the other hand explicit methods can be generally used to implement any elasto-plastic constitutive law and no iteration procedure is generally required. Yet, if an explicit integration scheme is used the stress state at the end of the increment may not fulfil the yield criterion. Because of the advantages numerous works use the explicit forward-Euler method and deal with the accuracy problems of explicit schemes [2]. Nevertheless, the methods developed increase the computational time.

The objective of this paper is to present a newly developed effective explicit scheme for the integration of constitutive equations and its application to the plane stress elements. The new scheme does not increase CPU time because no additional iterations are needed. It uses the same principles as the backward-Euler scheme, but expressions of this scheme are almost the same as in case of the forward (explicit) Euler scheme. The scheme is then employed via a user subroutine to integrate the GTN model in a sheet metal forming application, studied using continuum C3D8R and shell elements S4R [1]. Results of the case presented are compared with the results, obtained using the backward-Euler scheme (Abaqus/Explicit) and forward-Euler scheme. According to the theory presented and elaborated numerical example it can be concluded that the approach presented, despite being much simpler, reaches the same level of accuracy than the well known backward-Euler scheme, demands less computational operations and generally consumes less CPU time for one stress update operation.

1

ABAQUS Version 6.6, User's Manual. Simulia, Providence, RI, USA, 2006.

2

K.Z. Ding, Q.H. Qin, H.M. Cardew, "Substepping algorithms with stress correction for the simulation of sheet metal forming process", International Journal of Mechanical Sciences, 49(11), 1289-1308, 2007. doi:10.1016/j.ijmecsci.2007.03.010

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