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Ab Initio Calculations of the Critical Values of Elastic Energy Density

Kinga T. Nalepka 2Pawel Nalepka 1Ryszard B. Pecherski 1,2

1. Polish Academy of Sciences, Institute of Fundamental Technological Research (IPPT PAN), Świętokrzyska 21, Warszawa 00-049, Poland
2. Cracow University of Technology, Institute of Structural Mechanics, Warszawska 24, Kraków 31-155, Poland


An energy-based approach to limit states in elastic solids of arbitrary anisotropy was developed by Rychlewski [1]. It is based on the theory of elastic eigen-states and the concept of energy orthogonal stress states, which make it possible to decompose additively the elastic energy density into not more than six disjoint parts. Such an approach can appear also helpful in the field of nanomechanics filling the gap between atomistic calculations and continuum mechanics modelling of the behaviour of different kinds of crystalline nanostructures and macroscopic material properties. Under the limit state in nanostructures, we can understand the limit of linear elastic behaviour. We propose to calculate the critical energy of pertinent elastic eigen-states from quantum mechanical theory of nc-metals [2]. The analysis is confined to the cubic crystals. In such a case, three elastic eigen-states exist. The quantum mechanical model for an ideal single crystal of Cu is studied and the comparison with the results obtained for Al crystal is made. Using the Wigner-Seitz cellular approach and the Slater method the structure of the s, d, f and p energy bands was calculated. This enables obtaining of the internal energy of the crystal volume confined in the deforming Wigner-Seitz cell. Finally the critical energy for the particular elastic eigen-state was determined.
Acknowledgement: The work supported by the Ministry of Science within the framework of the research project PBZ-KBN-095/T08/2003-2006.
1. J. Rychlewski, Elastic energy decomposition and limit criteria, Uspekhi Mekh.-Advances in Mech., 7, 51-80, 1984 (in Russian) and: Unconventional approach to linear elasticity, Arch. Mech., 47, 149-171, 1995.
2. K. Nalepka, R.B. Pęcherski, Energy-based criteria of limit states. How to calculate energy limits from first principles, Rudy Metale, R48, 533-536, 2003 - in Polish.


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Presentation: oral at E-MRS Fall Meeting 2004, Symposium H, by Ryszard B. Pecherski
See On-line Journal of E-MRS Fall Meeting 2004

Submitted: 2004-05-17 09:12
Revised:   2009-06-08 12:55