Mixing first- and second-gradient models in finite element simulations of ductile rupture J.M. Bergheau1, L. Gaubert2, R. Lacroix2, J.B. Leblond3 1
Université de Lyon, ENISE, LTDS, UMR 5513 CNRS, 58 rue Jean Parot, 42023 Saint-Etienne Cedex 02, France 2
3
ESI-France, Le Récamier, 70 rue Robert, 69458 Lyon Cedex 06, France
UPMC Univ Paris 06, UMR 7190, Institut Jean Le Rond d'Alembert, Tour 65-55, 4 place Jussieu, 75252 Paris Cedex 05, France
[email protected]
An interesting second-gradient model for plastic porous solids, extending Gurson’s classical first-gradient model [1] for porous ductile solids, was proposed some years ago by Gologanu et al. [2] in order to settle the issue of unlimited localization of strain and damage and the ensuing mesh sensitivity in finite element calculations. This model was successfully implemented in the finite element code SYSTUS® by Enakoutsa and Leblond [3] by introducing extra nodal degrees of freedom (DOF) representing the components of the strain tensor. One important drawback of this procedure, however, was an awkwardly large number of DOF, notably in 3D, which generated various difficulties, especially with regard to convergence of the global elastoplastic iterations A new implementation solving this problem was very recently proposed by Bergheau et al. [4]. The central point of the new algorithm was a procedure of elimination of the nodal DOF representing the strain components, which permitted to reduce the number of DOF per node to its standard value. This was achieved by writing the equality of the new nodal DOF and the strain components in a weak sense; the “mass matrix” appearing in the lefthand side of the vectorial relation thus obtained was then lumped and inverted straightforwardly, so as to relate the new DOF to the nodal displacements explicitly. The reduction of the number of DOF was found to be quite beneficial to the convergence of the global elastoplastic iterations. However, a mixing of first- and second-gradient models would be desirable in order to reduce computation time and cost, and reasonable at least in simulations of ductile rupture, where second-gradient models are needed only in zones of limited extent, where strain and damage tend to concentrate. Bergheau et al.’s implementation [4] offers an interesting perspective in this context, since the elimination of nodal DOF representing strain components means that exactly the same DOF are used for first- and secondgradient models. This makes the mixing very easy by eliminating the need for development of special transition elements and requiring only elementary, low-level modifications of the code; such modifications essentially reduce to introduction of a
switch at the beginning of the loop over elements, the role of which is to call the appropriate routine to solve the constitutive equations, according to whether the element considered obeys a first- or second-gradient model. This procedure is illustrated in the present paper by comparing full second-gradient and mixed first/second gradient simulations of a CT specimen made of a ductile material and undergoing full rupture. The calculations are performed in 2D and plane strain but extension to 3D cases would be straightforward. The two types of simulations are checked to yield almost identical results, with lower computation time and cost for the second one.
References [1] A.L. Gurson. Continuum theory of ductile rupture by void nucleation and growth: Part I - Yield criteria and flow rules for porous ductile media. ASME Journal of Engineering Materials and Technology, 99, 2-15, 1977. [2] M. Gologanu, J.B. Leblond, G. Perrin, J. Devaux. Recent extensions of Gurson's model for porous ductile metals. In: Continuum Micromechanics, P. Suquet, ed., CISM Courses and Lectures No. 377, Springer, pp. 61-130, 1997. [3] K. Enakoutsa, J.B. Leblond. Numerical implementation and assessment of the GLPD micromorphic model of ductile rupture. European Journal of Mechanics A/Solids, 28, 445-460, 2009. [4] J.M. Bergheau, J.B. Leblond, G. Perrin. A new numerical implementation of a second-gradient model for plastic porous solids, with an application to the simulation of ductile rupture tests. Computer Methods in Applied Mechanics and Engineering, 268, 105-125, 2014.
