Deformation of characteristic curves of the plane ideal plasticity equations by point symmetries

Conservation Laws, Hodograph Transformation and Boundary Value Problems of Plane Plasticity

“…This system was investigated, using the group of admitted symmetries: for its invariant solutions see [1], all its conservation laws and highest symmetries were described in [22] and for the reproduction of solutions by point transformations see [24,25,29]. Being semi-inverse method, group analysis provides analytical solutions and then one can determine the boundary conditions for obtained solutions.…”

Some symmetry group aspects of a perfect plane plasticity system

“…Taking into account solutions (19), (21), (22), we can express operator Y 1 in terms of r, ϕ, θ c , σ. Namely, for the higher sing solution tan θ p = η/ξ we have:…”

“…Finally, the complete Lie algebra of all admissible point symmetries of (1) was determined in [19] and all conservation laws as well Lie-Backlund symmetries were constructed. Moreover, in the series of papers [20], [22], [24] point transformation groups were used to deform some known solutions and in [21] conservation laws were applied to solve the main boundary problems.…”

“…admit transformations ϕ → −ϕ, σ → −σ, θ p,c → −θ p,c ;(33) r → −r respectively. The transformation group, corresponding to operator X 4 has been calculated and called "quasi-scale" transformation in[22].…”

Solution of Boundary Value Problems of Plasticity with the Use of Conservation Laws

“…Its main point is that we get symmetries to act on the given solution and we obtain new solutions of the same differential equation system. It was used in the papers [2][3][4][5]. There may be an obstruction for realisation of this method which is "poorness" of the admitted group of symmetries or difficulty to interpret the found "multiplied" solutions.…”