Optimal Subalgebra Systems of the Symmetry Algebra of Spatial Equations in the Mathematical Theory of Plasticity

This paper considers the natural finite-dimensional (12-dimension) subalgebra of the symmetry algebra associated with the symmetry group of three-dimensional hyperbolic equations of the spatial problem of perfect plasticity, proposed in 1959 by D.D. Ivlev for the states corresponding to an edge of the Coulomb–Tresca prism, represented in the isostatic coordinate net. An algorithm is given for developing the optimal system of one-dimensional subalgebras of this natural finite-dimensional subalgebra of the symmetry algebra, comprising one three-parameter element, 12 two-parameter elements, 66 one-parameter elements, and 108 individual elements (total 187 elements). It was previously demonstrated that the symmetry algebra of the plane problem equations has mathematical dimension 7; the optimal system of one-dimensional subalgebras consists of 1 two-parameter, 11 one-parameter, and 20 individual infinitesimal generators (total 32 elements). The symmetry algebra of the axisymmetric problem equations has dimension 5; the optimal system of one-dimensional subalgebras consists of 1 one-parameter and 22 individual infinitesimal generators (total 23 elements).