Keywords: finite elastoplasticity, isotropic ductile damage, adaptive remeshing, error estimation, metal forming process.

Most metal forming parts involve complicated geometry and flow characteristics as large (visco)-plasticity flow, heat exchange, ductile damage, evolving contact with friction. The understanding of the role and effect of these phenomena on the final quality of the part, need the use of computer aided numerical simulations (i.e. the virtual metal forming facilities).

Accordingly, to increase the efficiency and the predictive capabilities of the virtual forming tools, an accurate modeling of the micro-defects (i.e. damage) initiation and growth under finite transformations conditions, should be taken into account. This can be achieved by using the coupled approach in the sense that the damage evolution equation is directly incorporated and fully coupled with the thermo-elasto-inelastic constitutive equations. This kind of approach has been employed by many authors using damage models based either on Gurson's theory or on Continuum Damage Mechanics in the Kachanov's sense. These fully coupled approaches allow to predict not only the large transformation of the processed workpiece as large deformations, rotations, and evolving boundary conditions, but also they can indicate where and when the damaged zones can appear inside the formed workpiece.

The damaged thermo-elastoplastic behaviour is described in the framework of the thermodynamics of irreversible processes with state variables. Dealing with the simple first displacement gradient theory, six couples of internal variables are taken into account : the small elastic strain associated to the Cauchy stress tensor; the isotropic and the kinematic hardening variables and the ductile damage variables. Both plastic and damage flows are supposed to be rate independent mechanisms. Using the non-associative framework, both phenomena will be modeled using a yield function together with dissipation potentials [1]. The so-called elastic prediction-return mapping algorithm with an operator splitting methodology is used for the integration of the fully coupled damage elastoplastic constitutive equations. This approach within the coupled problem consists in splitting it into two parts: a damaged elastic prediction (where the problem is assumed to be purely elastic affected by the last damage value), and a damaged-plastic corrector (in which the system of equations includes the damaged elastic relation as well as the damaged-plastic consistency condition). These equations are reduced to only two scalar equations : one governing the plastic flow with damage effect and the other governing the damage growth with plastic flow effect. Newton-Raphson iteration algorithm is then used to solve the discretized constitutive equations in the damaged plastic corrector stage around the current values of the state variables (plasticity, hardening, damage). The numerical procedure has been implemented into the FE code ABAQUS/Explicit via a users subroutine.

An intrinsic difficulty in metal forming process is the constantly changing configuration of the deforming part (finite transformation, thermo- anisotropic plastic flow). When damage is taken into account an accurate remeshing procedure is needed in order to refine the mesh size inside the damaged zones which lead to a macroscopic crack. This allows to remove the fully damaged elements and to define a new free boundary of the deformed part. Also, in metal forming, the mesh size should be adapted to the curvature of complex tools in order to optimize the contact boundaries. These problems can be resolved if an adaptive remeshing scheme is incorporated automatically in the finite element analysis [2]. Remeshing essentially involves a new and better-conditioned mesh replacing the old mesh (distorted and fully damaged), and an accurate transfer of nodal and Gauss points data to the new mesh. Two kinds of error estimates are proposed. The first deals with the gradient of the damage field and the second concerns the curvature of the forming tools. These allow to adapt the mesh in order to improve the geometry of the deformed part and the damage localization leading to a macroscopic crack. To mesh the computational domain, we apply a combined approach which uses a frontal method to define field points and a Delaunay method to construct the connection between these points. This remeshing technique is extended to the case where a prescribed size map is given, for example by means of an a posteriori error estimate. To illustrate how the presented model can be used for the damage initiation prediction during the metal forming processes, and how it can be optimized with respect to damage initiation, some examples (forging, blanking and orthogonal cutting) are given and discussed.

1

K. Saanouni and J.L. Chaboche, "Computational damage mechanics. Application to metal forming", ISBN 0-08-043749-4, Elsevier, 321-376, 2003.

2

H. Borouchaki, A. Cherouat, P. Laug and K. Saanouni, "Adaptive remeshing for ductile fracture prediction in metal forming", C.R. Acd. Sci. Paris, Série. II b, Mécanique des solides et des structures, 330 (10), 709-716, 2002. doi:10.1016/S1631-0721(02)01519-X

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