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Induced flow anisotropy and its application to sheet forming processes with complex strain path changes

Introduction and motivation


Metal forming processes involve large plastic strains and severe strain path changes. In many materials large plastic strains lead to strong flow anisotropy. This induced anisotropic behavior manifests itself in the case of a strain path change by very different stress-strain responses depending on the mode of the strain path change. While many metals exhibit a drop of the yield stress (Bauschinger effect) after a load reversal, some metals show an increase of the yield stress after an orthogonal strain path change (i.e., so-called cross hardening) followed by work-softening at the beginning of the second stage. After softening, further hardening takes place (Fig 1). The reason for this induced flow anisotropy is the development of persistent dislocation structures during large deformations. These deformations consist of walls of high dislocation density separating low-density cells (Fig 2). One side of each wall consists of excess dislocations of the same sign, and the other side of such dislocations of the opposite sign. After a load reversal, plastic deformation takes place due to the slip on the same slip systems but in opposite direction. Excess dislocations contribute to this slip, resulting in the Bauschinger effect. After an orthogonal strain path change, new slip systems are activated and the existing dislocation walls act as obstacles, resulting in the cross hardening. Work-softening at the beginning of the second loading stage is attributed to deformation localization in micro-bands and to breakdown dislocation walls formed during pre-straining. Further hardening is due to formation of new dislocation walls.

To model the Bauschinger effect, kinematic hardening has been successfully used for years. Accordingly, forming processes where load reversal is the dominant strain path change as in the case of draw-bending test (Fig 3) can be simulated using the concept of the combined isotropic-kinematic hardening. The relaxed shape of the sheet strips can be predicted very well using FE analysis and combined isotropic-kinematic hardening model for two different sheet thicknesses (Fig 4 and 6).

However, the usage of kinematic hardening results in a drop of the yield stress after an orthogonal strain path change. This contradicts tests results on materials exhibiting the cross hardening effect (Fig 6).

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Fig. 1: Simple shear tests at 45°with respect to the RD after simple shear along RD (FeP06)       Fig. 2: Persistent dislocation structures       Fig. 3: Draw-bending testing machine

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Fig. 4: Experimental and simulated results for the final shape of 2 mm thick metal strips subjected to draw-bending       Fig. 5: Experimental and simulated results for the final shape of 1 mm thick metal strips subjected to draw-bending       Fig. 6: Orthogonal test simulation using combined hardening



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Fig. 7: Shape of the yield surface after 10% equivalent plastic strain in simple shear and in tension       Fig. 8: Evolution of the yield surface based on H

Peeters et. al. have formulated a polycrystal model that makes an effort to take the evolution of the microstructure on the grain level into account. The microstructure is represented by three dislocation densities: density of the dipolar dislocations between the walls, density of dipolar immobile dislocations in the walls, and density of polar dislocations at wall boundaries. Each density contributes then to the critical resolved shear stress on a corresponding slip system. To extract the macroscopic quantities the fully constraint Taylor model is used. Using this approach Peeters et. al. were also able to simulate the cross hardening in an orthogonal test. Fig 8 shows the yield surface after ten percent shear pre-deformation (bold line) and ten percent tensile pre-deformation (thin line). It is evident that the cross hardening should be attributed to the change of the yield surface shape (distortional hardening). Accordingly, to model cross hardening, we have to take the change of the yield surface shape into account. We base the model on the standard elasto-plastic framework with the yield function as well as isotropic and kinematic hardening. The only difference is that in the expression for the equivalent stress (1) we use besides the Mises operator Idev, an additional forth order tensor, denoted here by H, representing the distortional hardening. The evolution equation for the distortional hardening has an Armstrong-Frederick form whereby H is split into directional HD and latent part HL (2).

(1) RTEmagicC_ba4cf6b1eb.jpg
(2) RTEmagicC_5e54655c3c.jpg

Using (2) and appropriate values of material parameters it is possible to achieve that the material becomes stronger in the direction orthogonal to that of preloading just as in the case of Peeters microstructural model (Fig 8). So for example in the case of tensile pre-loading the yield surface evolves from a circle to an ellipse stretched into the direction of shear stress and in the case of simple shear vice versa.




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Fig. 9: Ring splitting test       Fig. 10: Contours of the residual stress (Mises) after splitting (only half of the ring is shown)

In the ring splitting test (Fig 9) several rings are cut out of the frame of a cylindrical deep drawn cup. Due to the fact that the material points are subjected mainly to tangential compression as long as they are in the flange area followed by either radial tension or radial compression during the bending stage complex strain path changes appear in material points in the sheet. Due to the closed shape and accordingly high structural stiffness the drawn cup contains high residual stresses which are strongly dependent on the material hardening behavior. The ring splitting method represents a very effective method to estimate them. Due to the complexity of the strain path changes taking place during deep drawing the ring splitting test seems to be an appropriate method to investigate the influence of the distortional hardening on the residual stresses in the cup. To this end a FE analysis was performed using the commercial program ABAQUS (Fig 10). We did a quantitative study using an exemplary set of material constants. First results suggest that the ring openings and residual stresses are strongly influenced by the presence of distortional hardening.


Contact/Author information


Dr.-Ing. Vladislav Levkovitch