‘Homotopy’ of Prandtl and Nadai solutions

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.…”

“…In the theory of group analysis such kind of systems is called automorphic: if one particular solution is known, then the orbit of this solution under the group of admitted symmetries forms locally a general solution of the given system. In such a way, for example, some new exact solutions for the plane plasticity system were constructed in [29]. Using conservation laws permits to find out both nonsingular and singular solutions.…”

Some symmetry group aspects of a perfect plane plasticity system

“…Two straight lines ϕ = ±α are boundaries of the channel. For more details about this solution see [24]. In the case c 2 < 1, there is not any envelope for slip line families.…”

“…The boundary condition for the above solution is τ rϕ | r=R = k cos 2θ p = k cos 2g(R) = −k, σ| r=R = −p. The homotopy of the above solution with Prandtl solution (50) was analyzed in [24]. The generalization of Nadai solution was obtained by Mikhlin [11] for τ rϕ = q, |q| < k, when r = R. For the corresponding velocity solution see [23].…”

“…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.…”