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IssuesArchive of Issues2005-2pp.86-96

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Yu. N. Bakhareva and Yu. N. Radaev, "A generalization of Shield's self-similar solutions of the axially symmetric problem of plasticity," Mech. Solids. 40 (2), 86-96 (2005)
Year 2005 Volume 40 Number 2 Pages 86-96
Title A generalization of Shield's self-similar solutions of the axially symmetric problem of plasticity
Author(s) Yu. N. Bakhareva (Samara)
Yu. N. Radaev (Samara)
Abstract A generalization of self-similar solutions of the axially symmetric problem that correspond to the yield on a Tresca prism edge is considered. These solutions have been obtained by Shield [1]. The search for new self-similar solutions is based on the relationships of the spatial problem represented in the isostatic coordinates, with the axial symmetry and the possibility of the separation of an additional non-angular coordinate being taken into account. It is established that the axially symmetric problem of plasticity has a number of self-similar solutions in which the products of powers of the isostatic coordinates play the role of self-similar variables. For some specific values of the exponents occurring in the representation of a self-similar solution, the order of the equations of the axially symmetric problem can be reduced by one and the problem can thus be reduced to one nonlinear time-varying first-order equation which then can be reduced to an Abel equation of the first kind. The integral curves of this equation are constructed within the natural domain. In the self-similar solution domain, the maximum (minimum) principal stress is constructed numerically as a function of the polar angle in the meridional plane.
References
1.  R. T. Shield, "On the plastic flow of metals under conditions of axial symmetry," Proc. Roy. Soc. Lond., Vol. 233A, No. 1193, pp. 267-287, 1955.
2.  D. D. Ivlev, "On the general equations of perfect plasticity and statics of a granular medium," PMM [Applied Mathematics and Mechanics], Vol. 22, No. 1, pp. 90-96, 1958.
3.  D. D. Ivlev, Theory of Perfect Plasticity [in Russian], Nauka, Moscow, 1966.
4.  L. M. Kachanov, Fundamentals of Plasticity [in Russian], Nauka, Moscow, 1969.
5.  Yu. N. Radaev, "On Poincaré's canonical transformations and the invariants of the plastic equilibrium equations," Izv. AN SSSR. MTT [Mechanics of Solids], No. 1, pp. 86-94, 1990.
6.  Yu. N. Radaev, "To the theory of 3D equations of mathematical plasticity," Izv. AN. MTT [Mechanics of Solids], No. 5, pp. 102-120, 2003.
7.  Yu. N. Radaev and Yu. N. Bakhareva, "To the theory of the axially symmetric problem of mathematical plasticity," Vestnik Samarsk. Gos. Un-ta. Estest. Ser., No. 4(30), pp. 125-139, 2003.
Received 05 March 2003
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