Micro-macro analysis of steel sheet behaviour in finite element simulations. Application to deep-drawing process

This paper presents a constitutive law based on Taylor’s model implemented in our non-linear finite element code LAGAMINE. The yield locus is only locally described and a particular interpolation method has been developed. This local yield locus model uses a discrete representation of the material’s texture. The interpolation method is presented and a deep-drawing application is simulated in order to show up the influence of the texture evolution during forming processes. 1. STATEMENT OF THE PROBLEM The objective of our research is to integrate the influence of the material’s texture into a finite element code. The constitutive law describing the mechanical behaviour of the studied sample is based on a microscopic approach. The law computation takes place on the crystallographic level. A large number of crystals must be used to represent correctly the global behaviour. The micro-macro transition links the global behaviour to the crystallographic results. The full constraint Taylor’s model is used for the computation of the microscopic behaviour of each crystal and for the micro-macro transition. Unfortunately, this model does not lead to a general law with a mathematical formulation of the yield locus. Only one point of the yield locus corresponding to a particular strain rate direction can be computed. The “direct Taylor’s model” assumes that one macroscopic stress results from the average of the microscopic stresses related to each crystal belonging to a set of representative crystals. The computation of the mechanical behaviour involves a large number of crystals and must be repeated for each integration point of the finite element model, for each iteration of each time step. So, such a micro-macro approach consumes large computation time and seems practically not usable. However, using different simplified approaches, various constitutive laws based on texture analysis have been implemented in the non-linear finite element code LAGAMINE. Our first step in the integration of the texture effects has been the use of a 6 order series yield locus defined by a least square fitting on a large number of points (typically 70300) in the deviatoric stress space (see [4]). Those points were calculated by Taylor’s model based on an assumed constant texture of the material. This fitting is performed once, outside the FEM code. It provides 210 coefficients to describe the whole yield locus. This method, i.e. a global description of the yield locus, is actually used in the FEM code. Unfortunately, taking into account the texture evolution effects with this yield locus would imply the computation of the 210 coefficients of the 6 order series for each integration point, each time a texture updating is necessary. This would require an impressive amount of computation and memory storage (210 coefficients for each integration point) which is only partially useful as generally the stress state remains in a local zone of the yield locus. So, two new approaches, where the whole yield locus is unknown, have been investigated. In the first case, some points in the interesting part of the yield locus are computed with Taylor’s model. This local zone of the yield locus is then represented by a set of hyperplanes which are planes defined in the five-dimensional deviatoric stress space. These planes being fitted on Taylor’s points. As it has been shown in [2], the yield locus discontinuities bred by this very simple interpolation method give rise to convergence problems in the finite element code. That is the reason why a second method has been developed. For that second approach, no yield locus is defined and a direct stress-strain interpolation between Taylor’s points is achieved. In this case, the yield stress continuity conditions are fulfilled but, as there is no yield locus formulation, a particular stress integration scheme has to be used. Both interpolation methods allow us an important computation time reduction with respect to the “direct Taylor’s model” application. Taylor’s model is only used to compute some points in order to achieve the interpolation. These points must be computed in two cases: • When the current part of the yield locus does not content anymore the new stress state and that a new local zone of the yield locus is required. • When the plastic strains significantly deform the material and induce changes in the crystallographic orientations, i.e. when the texture evolves. Indeed, the corresponding mechanical behaviour of the material would no more be correctly represented by the old points. A texture updating must take place. The part yield locus approach presented in this paper can be placed between the microscopic approach (accurate but very slow) and the global yield locus approach (fast but inaccurate and especially not adapted for texture updating). This paper describes the stress-strain interpolation method; interested readers can refer to [5] and [2] for the 6 order and the hyperplanes method. The influence of the texture updating during a forming process has been highlighted by a deep-drawing simulation. 2. STRESS-STRAIN INTERPOLATION 2.1 Local description of a scaled yield locus. This model is particular in the sense that it does not use a yield locus formulation neither for the interpolation nor in the stress integration scheme. We use a linear stress-strain interpolation described by Equation 1.