By Francisco Chinesta, Elias Cueto
This booklet teams the most advances in fabric forming, contemplating varied strategies, either traditional and non-conventional. It focuses
on polymers, composites and metals, that are analyzed from the cutting-edge. particular emphasis is dedicated to the contributions of the ecu medical organization for fabric Forming (ESAFORM) over the last decade and particularly those coming from its annual foreign conference.
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106 (1984) 295-298 17. : Impact of the grain size distribution on the yield stress of heterogeneous materials. Int J Plasticity 23 (2007) 114-142 18. : Berechnung der elastischen konstanten des vielkristalls aus den konstanten des einskristalls. Z. Phys. (1958) 151 19. : A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: application to zirconium alloys. Acta Metall. Mater. 41 (1993) 2611-2624 20. : Tuning a self-consistent viscoplastic model by finite element results I: Modelling.
2 Theories of linear transformations and tensor representations The procedure used to introduce anisotropy by Barlat et al.  and Karafillis and Boyce  is equivalent with the application of a fourth order linear transformation operator on the stress tensor ı = Lı or on the stress deviator s = Ls , for pressure independent plasticity. Thus, an anisotropic yield function is obtained from an isotropic function by substituting the principal value of the stress tensor (or deviator) by the principal value of ı or s .
For isotropic materials, yield criteria have the same form in any reference frame. For anisotropic materials, yielding properties are directional, thus the expression of the yield criterion depends on the reference frame. The simplest form of the yield criterion is with respect to a coordinate system associated with the axes of symmetry of the material. For instance, due to the symmetry of their thermo-mechanical processing history, sheet metals exhibit orthotropic symmetry characterized by three mutually orthogonal planes of symmetry.