Large Plastic Deformations and Ductile Fracture of Polycrystals: Multiscale Modeling 
Very far, in the Land of High Pressures, behind the
Severe Plastic Deformations Hill, lies a mysterious country called
Ideal Plasticity. Some think that there is only one StrainStrength
Curve leading there, some believe that there are several,
while others think that there is no such a country at all.
Many went there; only a few reached it, most fractured on the way.
People tell wonders about it: As if metals live there in a complete idyll,
deforming without fracturing and hardening. As if those who lived there,
became superplastic, their grains became small and pure, and they started to
react to external influences in a completely different way. People
admire those who returned from this country, and respectfully add "Nano"
to their titles.
From a legend told by an old alloy Ti64 who survived a severe plastic deformation to a young inexperienced alloy. Below are some of my contributions in this area. Proposing a continuum model of grain refinement and damage of polycrystalline materials under severe plastic deformation (SPD). The model makes two natural assumptions: (1) the selfsimilarity of grain refinement and microdamage, and (2) the complementarity of these two processes. The model uses a system of kinetic equations to explain the formation of structure during an SPD process. Analyzing these equations gives the following conclusions:
Y. Beygelzimer. Grain refinement versus voids accumulation during severe plastic deformations of polycrystals: A mathematical simulation, Mechanics of Materials, Vol. 37, No. 7, pp. 753767 (2005). Grain Refinement and Viscous Fracture of Metals during Severe Plastic Deformation: Mathematical Simulation (ppt, pdf), 2004. Developing a continuum theory of plastic deformation in structurallyinhomogeneous porous bodies. Based on this theory, proposing a mathematical model of metal forming of porous and powder materials. Developing a new model for predicting the ductility of materials under deformation. Y. Beygelzimer, V. Varyukhin, B. Efros. Physical mechanics of hydrostatic treatment of materials. Donetsk: DonFTI NANU, 192 pages, 2000 (in Russian). Y. Beygelzimer. Constitutive Equations of a Porous Body with a StructurallyInhomogeneous Matrix, Proceedings of International Workshop on Modelling of Metal Powder Forming Processes, Grenoble, France, July 2123, 1997, pp. 5767. Y. Beygelzimer, B. Efros, V. Varyukhin, A. Khokhlov. A continuum model of a structurallyinhomogeneous porous body and its application for the study of stability and viscous fracture of materials deformed under pressure. Engineering Fracture Mechanics, Vol. 48, No. 5, pp. 629640 (1994). On some models in metal forming: From continuum mechanics to mircomechanics and back (ppt, pdf), 2002. Proposing a computational model of elasticplastic deformation of microinhomogeneous materials, which accounts for rotations of structural elements. Introducing new geometrical structures into plasticity theory ("thick yield surface", "cloud of internal stress") which effectively describe elasticplastic deformation of microinhomogeneous materials. The picture below demonstrates a cloud of internal stress. Publications: Y. Beygelzimer, A. Spuskanyuk. The thick yield surface: An idea and approach for investigating its structure. Philosophical Magazine A, Vol. 79, No. 10, pp. 24372459 (1999). Y. Beygelzimer, A. Spuskanyuk, V. Varyukhin. On The Loading Surface of Microinhomogeneous Materials, Resent Development in Computer Modeling of Powder Metallurgy Processes, IOS Press, pp. 1728, 2001.
