TAILIEUCHUNG - Noncommutative deformation and a topological nature of koiter singularity
In this paper we constructed the model of noncommutative plastic deformation and give the proof of Koiter hypothesis. We showed, that the occurrence of Koiter singularity has topological reasons and the number of Koiter singularities - is the topological Pontriagin number. | Vietnam Journal of Mechanics, VAST, Vol. 27, No. 3 (2005) , - 185 NONCOMMUTATIVE DEFORMATION AND A TOPOLOGICAL NATURE OF KOITER SINGULARITY TRINH VAN KHOA ' Hanoi Architectural rJ,niversity Abstract. In this paper we constructed the model of noncommutative plastic deformation and give the proof of Koiter hypothesis. We showed, that the occurrence of Koiter singularity has topological reasons and the number of Koiter singularities - is the topological Pontriagin number. 1. INTRODUCTION The mechanism for planes of atoms shear in a crystal relative to one another is used when describing the process of plastic deformation in a simple case. The elastic limit is defined as Tcr = ~~' where a is the interatomic distance, d is the distance between the glide planes, and therefore the value of T er must be about an order of magnitude less than that of the shear modulus G. The experimental data [15] have shown, however, that for tin G = x 10 11 dyne/cm 2 , while T ~ 13x106 dyne/cm 2 , for silver the corresponding values are x 10 11 and 6 x 10 6 , for aluminium, x 10 11 and 4 x 10 11 dyne/ cm 2 . It should be noted that a more exact account of the arrangement of planes of atoms in shear yields an estimate Tcr rv G /30, and this exceeds the experimentally found value by several orders of magnitude. The reason for such a phenomenon is the coherence of rearrangements of the crystal structure. Therefore to describe plastic deformation, it is worthwhile to use the phenomenological approach accepted in continuum mechanics. From the point of view of phenomenology, till now there are many models of plastic deformation. Almost in all monographs, for example [1-3 , 13], the following models are considered: sliding, regular, current, singular, analytical and deformation plasticity. The determinant ratio in deformation is a major problem, which was submitted for discussion. The most simple kind was presented in the form of Hencky-Nadai: at at T = T, T () k E {
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