Mechanics

2004, The Second Special Issue

Y.N. Radayev, Y.N. Bakhareva

A NUMERICAL METHOD OF SOLUTIONS OF AXIALLY-SYMMETRIC PROBLEM OF THE THEORY OF PERFECT PLASTICITY

The equations of axially-symmetric problems of the mathematical theory of perfect plasticity are considered in the case when a stress state corresponds to an edge of the Tresca prism. The static equilibrium equations are represented in the stress principal lines co-ordinate net (isostatic net). The problem of localization of plastic strains within a neck observed in uniaxially stretched specimen is formulated by the Haar--Karman approach. A numerical method of solution of the problem is proposed for an arbitrary neck free surface. The method based on numerical calculation of the isostatic net. Numerical values of the limit force are obtained in the case of elliptic neck free surface profile and then compared with those known due to Bridgman.

Y.N. Radayev, V.A. Gudkov

THE GROUP-INVARIANT SOLUTIONS OF THE AXIALLY-SYMMETRIC EQUATIONS OF THE MATHEMATICAL PLASTICITY

The Lie algebra and a first order optimal system of subalgebras of the symmetry group of partial differential equations of axially-symmetric problem of the mathematical theory of plasticity represented in the isostatic co-ordinate net are studied. The equations of mathematical plasticity are formulated for stress states corresponding to an edge of the Tresca prism in the Haigh-Westergaard space. By analysing the first order optimal system of subalgebras the group-invariant solutions are determined from systems of ordinary differential equations. A number of new exact solutions are obtained in analytically closed forms. Some of them are represented by the canonical Legendre elliptic integrals.