  Mechanics of Solids A Journal of Russian Academy of Sciences   Founded
in January 1966
Issued 6 times a year
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Yu. N. Bakhareva and Yu. N. Radaev, "A generalization of Shield's selfsimilar solutions of the axially symmetric problem of plasticity," Mech. Solids. 40 (2), 8696 (2005) 
Year 
2005 
Volume 
40 
Number 
2 
Pages 
8696 
Title 
A generalization of Shield's selfsimilar solutions of the axially symmetric problem of plasticity 
Author(s) 
Yu. N. Bakhareva (Samara)
Yu. N. Radaev (Samara) 
Abstract 
A generalization of selfsimilar 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 selfsimilar
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 nonangular coordinate being taken
into account. It is established that the axially symmetric problem
of plasticity has a number of selfsimilar solutions in which the products
of powers of the isostatic coordinates play the role of selfsimilar
variables. For some specific values of the exponents occurring in the representation
of a selfsimilar 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 timevarying firstorder 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 selfsimilar 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. 267287, 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. 9096, 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. 8694, 1990. 
6.  Yu. N. Radaev, "To the theory of 3D equations of mathematical
plasticity," Izv. AN. MTT [Mechanics of Solids], No. 5, pp. 102120, 2003. 
7.  Yu. N. Radaev and Yu. N. Bakhareva, "To the theory of the
axially symmetric problem of mathematical plasticity,"
Vestnik Samarsk. Gos. Unta. Estest. Ser.,
No. 4(30), pp. 125139, 2003. 

Received 
05 March 2003 
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