Paradox Solution on Elastic Wedge Dissimilar Materials

According to the Hellinger Reissner variational principle and introducing proper transformation of variables,the problem on elastic wedge dissimilar materials can be led to Hamiltonian system,so the solution of the problem can be got by employing the separation of variables method and symplectic eigenfunction expansion under symplectic space,which consists of original variables and their dual variables.The eigenvalue -1 is a special one of all symplectic eigenvalue for Hamiltonian system in polar coordinate.In general,the eigenvalue 1 is a single eigenvalue,and the classical solution ofan elastic wedge dissimilar materials subjected to a unit concentrated couple at the vertex is got directly by solving the eigenfunction vector for eigenvalue 1.But the eigenvalue 1 becomes a double eigenvalue when the vertex angles and modulus of the materials satisfy certain definite relationships and the classical solution for the stress distribution becomes infinite at this moment,that is,the paradox should occur.Here the Jordan form eigenfunction vector for eigenvalue 1 exists,and solution of the paradox on elastic wedge dissimilar materials subjected to a unit concentrated couple at the vertex is obtained directly by solving this special Jordan form eigenfunction.The result shows again that the solutions of the special paradox on elastic wedge in the classical theory of elasticity are just Jordan form solutions in symplectic space under Hamiltonian system.