Solution Of The Elastic Wedge Paradox

Paradox problem is one of fundamental but classical research project in the theory of elasticity. But the research achievements obtained by now, such as the stress function methods and complex variable methods solved the problem under the Lagrange system, all the methods belong to the tradition semi-inverse solution method, that is, the 'distribution of the stress must be assumed in some expression in advance, which based on the special problems and lake of the generation. Looking from the analogy theory between the computational structural Mechanics and the Optimal control, the Hamilton system theory can be introduced into the theory of Elasticity. So the powerful mathematical physics methods such as the separation of variables and eigenfunction expansion can be applied directly into the elasticity. In this paper some typical paradoxes on the elastic wedge is studied in symplectic space which consist of original variables and their dual variables. It shows further that the special paradox in Euclidean space under the Lagrange system is just Jordan form solutions in symplectic space under Hamiltonian system. So the paradox solutions can be obtained by solving the Jordan form solution. The main content of this paper includes:1) The paradox problem of the two-material elastic wedge subjected to a concentrated couple on the vertex.2) The paradox problem of the cylindrical orthogonal anisotropic elastic wedge subjected to a concentrated couple on the vertex.3) The paradox problem of the cylindrical orthogonal anisotropic elastic wedge subjected to uniform tractions on the surfaces.