Self-consistent method applied to quasi-rate-independent polycrystals

Abstract

Geometric and kinematic aspects of intragranular, as well as intergranular plastic deformation of polycrystals, are discussed. Elastic strain is covered by the effective field homogenization method inside a representative volume element (RVE). Evolution equation formed by tensor representation having incremental form is postulated to model inelastic metals. The rate dependence takes place by means of stress rate-dependent value of the initial yield stress. Extending concept of Zorawski [private commun., 1974] deformation geometry is based on constrained micro and free macro rotations in intermediate reference configuration. This has as a consequence that evolution equation for plastic spin of RVE is an outcome of evolution equation for plastic stretching. The macroscopic evolution equation is based on the Vakulenko's concept of thermodynamic time [Izv. AN SSSR Mekhanika Tverdogo Tela 1 69 (1970)]. Eshelby 4-tensor [Proc. R. Soc. A 241 376 (1957)] is determined numerically taking account of cubic anisotropy and the theory is applied to slightly disordered fcc polycrystals. For some characteristic-given stress histories (leading to low, medium and high strain rates) number of active slip systems for RVE with 125 grains are found and compared with the so-called J2-approach.

†This paper is devoted to Gerard Maugin – friend and scientist.

Acknowledgements

Thanks are due to Dr C. Albertini and the late Dr M. Montagnani of JRC for long-term fruitful collaboration, which made possible multiaxial dynamic tests on AISI 316H stainless steel specimens. The author is also thankful to referees whose comments have substantially contributed to the paper; all footnotes, section 5, as well as necessary references, have been written in answer to their questions.

Notes

†This paper is devoted to Gerard Maugin – friend and scientist.

†Such a definition of dislocation density has been extensively used in numerous papers devoted to continuum theory of dislocations 7, 11–14. It covers only geometrically necessary dislocations (GND) and does not take into account statistically stored dislocations (SSD) with opposite signs of dislocations appearing at dislocation loops and dipoles. For this reason, in the paper by Kröner 15, a more precise definition is given by infinite number of correlation functions composed by fundamental dyadic of Burgers vector and dislocation line tangent vector. Kröner's two-point autocorrelation function is here especially convenient since it is proportional to total dislocation line length inside RVE (see also 16). It is worthy of note that diverse non-Euclidean interpretations of F _P and its gradient may cover not only GND but implanting Eshelbian strains as well 7.

†The wording ‘matrix’ is compatible with Levin's notation. On the other hand, in the paper by Drugan and Willis 19, the authors use, perhaps more appropriate, name ‘homogeneous comparison medium’. According to the considerations of the second section elasticity tensor of such a matrix is taken as the average of all the inclusions i.e. grains belonging to the considered RVE. If material does not posses voids, as it is tacitly assumed here, then the average over the orientations given at RHS of (14) is the same as the spatial average throughout the RVE.

†In componenal form this reads .