Variational Micro‐Macro Model and Deformation Textures Predicted for Steels

Abstract

This variational micro-macro model is presented in the context of the "pure" and "relaxed" Taylor models, from which it evolved, but also in relation to the Sachs-type models and, more generally, in the context of volume-fraction or "one-point" models. It will indeed be shown that any one-point model must set statistical assumptions and that, to this general frame, the present model adds only one essential assumption-namely a plausible principle of minimal inhomogeneity. The numerical implementation is summarized. Predictions for the strain distribution are discussed. Predicted deformation textures are compared with measurements on steel sheets.

Physical Meaning and Experimental Check of a Variational Principle for Macro-to-micro Transition

“…the actual distribution (D k ) has the least heterogeneity h. The PMH allows successful predictions of deformation textures for steels (Arminjon & Imbault, 1996), but it cannot be derived from deterministic mechanics. Instead, we shall establish below a link between the PMH and the maximum entropy principle.…”

“…( 9). From these, we may calculate the current average rotation rates and update the orientations after a small deformation step, i.e., we may calculate the texture evolution in the same way as in Arminjon & Imbault (1996). It should be clear, however, that the two new models may be used also in much more general situations.…”

“…The prediction of the crystallographic texture of low-C steels after cold-rolling is a severe test for polycrystal models, as it has been discussed in some detail in previous works (see Arminjon & Imbault, 1996, and references therein). On Fig.…”

“…The general figure of the texture being very well predicted by both models, the precise comparison is based on the "skeleton lines", the (standard) definition of which may be found e.g. in Arminjon & Imbault (1996), and that contain the main preferred orientations. Both models overestimate the sharpness of the texture.…”