A Symplectic Method Of EigenSolutions In The Viscoelasticity

The background of research is based on creep and stress relaxation phenomenon in engineering structures. The symplectic eigenvalue solutions of viscoelastic problems(viscoelastic plane problems, viscoelastic cylinders and viscoelastic hollow circular cylinder)are discussed in this article.The work is supported by the National Natural Science Foundation of China(19902014 and 10202024). Hamiltonian systems and the duality systems are established in the viscoelastic problems. A direct method solution for symplectic eigenvalue problems is put forward under the sysplectic system. The symplectic method updated the solving system of the viscoelasticity to a new platform.The article discusses constitutive relations of the viscoelastic problems at the first step, then the differential forms of constitutive equation are put forward(Maxwell, Kelvin and three-level Structure model). After Laplace transformation of the above equations, the viscoelastic problems become elastic problems under the phase space. By introducing dual variables, the dual governing equations and boundary conditions, which are composed by mixed variables under whole state space, are obtained. It can be solved by modern canonical mathematic tools, for example, adjoint symplectic orthonormality and the expansion theorem etc. Under the phase space, the zero eigenvalue solutions and all their Jordan normal forms and non-zero eigenvalue solutions are obtained. Using elastic-viscoelastic correspondence, the viscoelastic solutions are obtained by inversion of Laplace transform.Based on the zero eigenvalue solutions and non-zero eigenvalue solutions of the three viscoelastic problems, discusses the zero and non-zero eigen value solutions of plane problems, draw the real parts and imaginary parts of the solutions for three kinds of models (Maxwell, Kelvin and three-level Structure model) when the eigenvalues change; discusses the creep and stress relaxation phenomenon of the viscoelastic cylinder and viscoelastic hollow circular cylinder. The results show that the model is reasonable, which is in agreement with viscoelastic characteristics and the method is efficient.