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1.Kovalchuk V., Mladenov I.M., Mechanics of infinitesimal gyroscopes on mylar balloons and their action-angle analysis, MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 0170-4214, DOI: 10.1002/mma.6099, Vol.43, No.6, pp.3040-3051, 2020
Kovalchuk V., Mladenov I.M., Mechanics of infinitesimal gyroscopes on mylar balloons and their action-angle analysis, MATHEMATICAL METHODS IN THE APPLIED SCIENCES, ISSN: 0170-4214, DOI: 10.1002/mma.6099, Vol.43, No.6, pp.3040-3051, 2020

Abstract:
Here, we apply the general scheme for description of mechanics of infinitesimal bodies in Riemannian spaces to the example of geodetic and non-geodetic (for two different model potentials) motions of infinitesimal rotators on the Mylar balloon. The structure of partial degeneracy is investigated with the help of the corresponding Hamilton-Jacobi equation and action-angle analysis. In all situations it was found that for any of the sixth disjoint regions in the phase space among the three action variables only two are essential for the description of our models at the level of the old quantum theory (according to the Bohr-Sommerfeld postulates). Moreover, in both non-geodetic models, the action variables were intertwined with the quantum number N corresponding to the quantized value of the radius r of the inflated Mylar balloon.

Keywords:
action-angle analysis, affinely-rigid bodies, Bohr-Sommerfeld quantum theory, Hamilton-Jacobi equation, infinitesimal gyroscopes, mylar balloons, residue analysis

2.Pęcherski R.B., Rusinek A., Frąś T., Nowak M., Nowak Z., Energy-based yield condition for orthotropic materials exhibiting asymmetry of elastic range, ARCHIVES OF METALLURGY AND MATERIALS, ISSN: 1733-3490, DOI: 10.24425/amm.2020.132819, Vol.65, No.2, pp.771-778, 2020
Pęcherski R.B., Rusinek A., Frąś T., Nowak M., Nowak Z., Energy-based yield condition for orthotropic materials exhibiting asymmetry of elastic range, ARCHIVES OF METALLURGY AND MATERIALS, ISSN: 1733-3490, DOI: 10.24425/amm.2020.132819, Vol.65, No.2, pp.771-778, 2020

Abstract:
The aim of the paper is to formulate physically well founded yield condition for initially anisotropic solids revealing the asymmetry of elastic range. The initial anisotropy occurs in material primarily due to thermo-mechanical pre-processing and plastic deformation during the manufacturing processes. Therefore, materials in the "as-received" state become usually anisotropic. After short account of the known limit criteria for anisotropic solids and discussion of mathematical preliminaries the energy-based criterion for orthotropic materials was formulated and confronted with experimental data and numerical predictions of other theories. Finally, possible simplifications are discussed and certain model of isotropic material with yield condition accounting for a correction of shear strength due to initial anisotropy is presented. The experimental verification is provided and the comparison with existing approach based on the transformed-tensor method is discussed.

Keywords:
energy-based yield condition, orthotropic solids, initial anisotropy, strength differential effect, numerical simulation