Keywords: localization, limited rate, plasticity, damage, finite strain, mesh sensitivity.

During the last fifteen years, the use of simulations for structural impact analyses in the aircraft industry increased thanks to both the extension of scope of explicit finite element (FE) codes and the increase of computational power. Today it is thus possible to carry out simulations with sufficient conservatism to ensure that an impacted structure is able to withstand the threat, but failure is still difficult to estimate numerically with a high level of accuracy.

One of the most important limitations regarding failure prediction through a simulation with the continuous damage mechanics approach is linked to the mesh dependency of the results [1]. To overcome this drawback spatial localization limiters, classified as integral [2] or differential limiters [3], have been proposed and are now widely used. Those approaches provide a rigorous and systematic framework to handle the problem of mesh dependency but require use of additional boundary conditions (ABC) and code development associated with the non-locality of the constitutive relation is rather complex.

Another possibility first proposed and used in the context of laminated composite materials [4] and C/C materials [5] consists in time localization limiters. More precisely the proposal relies on damage model with bounded rate of damage. The advantages of such approaches is that they lead to standard code developments, the identification of the maximum damage rate relies on standard experiments, for example plate-plate experiments (see [5] for 3D C/C materials and [6] for metallic material). A drawback is that it is necessary to conduct dynamic simulation, even when dealing with external static loading. The fracture phenomenon is in fact seen as a dynamic (even if local) phenomenon. Another difficulty is that this approach has to be adapted depending on the type of materials and the type of phenomenon which lead to fracture.

The aim of the present work is to discuss and present the adaptation of this approach in the case of metallic material prone to locally large plastic deformation induced by necking. The damage is then associated with the development of intense plastic strain [7] . The difficulty of the adaptation of the damage bounded rate concept is due to the fact that two sources of instability are present in the problem, the damage itself and the necking.

That is why a model with a variable at a limited rate dedicated to elasto-plasticity coupled with damage in a finite strain formulation has been proposed [8]. As the damage itself appears as a consequence of a large plastic straining we follow the Lemaitre damage modelling by making use of a damage model associated with plasticity [7].

In order to evaluate the capabilities of the model to be mesh size and mesh orientation independent, simulations results are presented. First, a damageable elasto-plastic beam subjected to dynamic load is considered in order to evaluate the capabilities of the model to be mesh size independent. The material model of "elasto-plasticity with damage and limited rate of plasticity" has been developed in a 2D Matlab finite strain FE code and implemented in an ABAQUS user material law (VUMAT with ABAQUS/Explicit). Two-dimensional examples are finally presented and illustrate the capabilities of the model to be mesh size and mesh orientation independent.

1

Bazant, Z.P. "Instability, ductility and size effect in strain softening concrete", Journal of engineering mechanics, 102, 331-344, 1976.

2

Pijaudier-Cabot, G. and Bazant, Z.P. "Dynamic stability analysis with nonlocal damage" Computers & Structures, 29, 3, 503-507, 1988. doi:10.1016/0045-7949(88)90403-8

3

Lasry, D. and Belytschko, T. "A study of localization limiters for strain-softening in statics and dynamics", Computers & Structures, 33, 3, 707-715, 1989. doi:10.1016/0045-7949(89)90244-7

4

Allix, O. and Deü, J.F. "Delay damage modelling for fracture prediction of laminated composites under dynamic loading" Eng. Transac., 45, 29-46, 1997.

5

Sen-Gupta, J. and Allix, O. and Boucard, P. A. and Fanget, A. and P.L. Hereil, P.L. "Fracture prediction of a 3D C/C under impact" Composite Science and Technology, 65, 375-386, 2005. doi:10.1016/j.compscitech.2004.09.009

6

Suffis, A. "Développement d'un modèle d'endommagement à taux de croissance contrôlé pour la simulation robuste de ruptures sous impacts", PhD Thesis - INSA de Lyon, 2004.

7

Lemaitre, J. "Formulation unifiée des lois d'évolution d'endommagement" C. R. Acad. Sci. Paris, II, 1125-1130, 1987.

8

Court, G. and Allix, O. and Mahé, M. "Modèle à taux de croissance limité pour la prévision de la rupture ductile de structures en grandes déformations" Congrès français de mécanique, 2005.

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