Keywords: numerical modelling, rupture, plastic behaviour implementation, finite strain, dynamic loading, bifurcation theory.

The design of metallic structures subjected to dynamic loading often requires numerous experimental tests. In order to reduce the design cost of such structures and to enable optimisation of high stressed components, efficient numerical programs are necessary. The objective of this paper is to present the capabilities of such a program which uses the standard mechanical properties of the structure under consideration in order to obtain a good prediction of dynamic rupture. The finite elements modelling associated with an explicit time integration (ABAQUS/Explicit here) is the most widely used method to simulate the evolutions of complex structures under dynamic loading. The subject of this paper is limited to the analysis of metallic volumetric parts that cannot be modelled as beams or shells. Since the ABAQUS code do not enable to implement complex criterions for the deletion of elements, describing the rupture phenomena, the user must write its own subroutine, termed "VUMAT" (User-defined Material behaviour subroutine), to integrate the behaviour. Thus, the rupture criterion is implemented in the same subroutine and complete the latter.

On the one hand, the numerical integration of the elastoplastic behaviour of the material are presented. The study of nonlinear problems (finite transformations) using a finite-element method is complex. The corotationnal representation is chosen to describe the kinematics of the problem. The ABAQUS code makes the assumption of the linearity of the displacement field. Consequently, every useful tensors can be easily calculated during the entire increment with the help of the deformation gradient tensor values provided at each bound of the increment by ABAQUS momentum balance module. The elastic loading (or unloading) is described by a first order differential equation where the specific Cauchy stress rate tensor is function of the strain rate tensor via the elastic Hooke tensor. The elastoplastic behaviour is expressed as a non-linear differential equation. Two integration techniques (Forward Euler scheme and Return-Mapping method) of the constitutive law have been studied.

The forward Euler scheme associated with a stability control is used to integrate the elastoplastic behaviour. During the increment of the elastic-plastic transition, a subdivision of this increment leading to the succession of an elastic sub increment and a plastic sub increment is carried out.

The return-mapping method is another possible integration scheme [1]. It consists in an elastic prediction followed by several plastic corrections. The aim of these plastic corrections is to optimise the stress state and the equivalent plastic strain so that the yield surface equation is established at the end of the increment.

Simple tests (tension and simple shear) are used to validate the numerical implementations. The computational times obtained with the different numerical integration schemes are compared throughout the simulation of the impact of an automotive part (steering knuckle). The implicit algorithm (Return-Mapping) is actually easier to implement and requires near the same CPU time than for the forward Euler method.

On the other hand, the introduction and the validation of the studied macroscopic rupture criterion are discussed. An equilibrium bifurcation theory associated to the material behaviour is chosen as a rupture criterion. The main interest of this method suggested by Nême [2] is that the rupture criterion is not an independent parameter totally disconnected from the constitutive laws of the material under consideration and determined according to the experimental dynamic tests of structures. From this point of view, the bifurcation theory used as a rupture criterion should be more general and "cheaper" than standard rupture criteria since no structure tests are necessary and only tensile tests of samples are required for elastic properties and hardening law determination.

Several impacts of steering knuckles owing to an impact tower have been conducted. The rupture-limit velocity of the falling mass and the fractures locations are compared with numerical simulations using our subroutine. The rupture phenomenon seems to be well predicted by the new rupture criterion. At the moment the numerical simulations forecast the rupture-limit velocity with an error around 5%.

Some new developments are planned. The mesh size dependency on the onset of the rupture criterion is going to be analysed throughout the diametrical dynamic crush of a simple ring between two rigid surfaces. The simulation of the crack propagation can be improved. The extension of this work for shell elements is also engaged.

1

J.C. Simo, T.J.R. Hugues, "General return-mapping algorithms for rate- independent plasticity", Constitutive Laws for Engineering Materials: Theory and applications, Ed DESAI et al., Elsevier, vol 1., pp. 221-231, 1987.

2

A. Nême and N. Dahan, "Analysis of the perforation of Elastoplastic Plates with Bifurcation Theory", DYMAT JOURNAL, Vol. 2, no2, pp. 87-103, June 1995.

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