Viscoelastic Cylinder Problem, The Dual System

The research background is based on creep and stress relaxation phenomenon in engineering structures. The problem is focused on viscoelastic cylinders in this paper. The work is supported by the National Natural Science Foundation of China (19972014). A Hamiltonian system, the duality system, of the viscoelastic theory is established in cylinders via the investigation. In the systematic system, a direct method solution for symplectic eigenvalue problems is put forward. The symplectic method updated the solving system of the viscoelasticity to a new platform.By means of the theory and the computational methods and via the Laplacian transform, the viscoelastic problem can be come down to fundamental problems under the phase space. Then Hamiltonian system is introduced into the space. The problems are changed into a series of symplectic eigenvalue problems under the duality system. It can be solved by the modern canonical mathematic tools, for example, adjoint symplectic orthonormality and the expansion theorem etc. The problem of viscoelastic cylinders can be solved by using the analytical and semi-analytical method. For this aim, the first step is to discuss constitutive relations of viscoelasticity and then bring forward a differential form for three dimensions viscoelastic constitutive relations (Kelvin model, Maxwell model and three-level structure model). By introducing dual variables, the dual governing equations and boundary conditions, which are composed by mixed variables under whole state space, are obtained. Then, the solving method is studied besides the transform relationship of the problem between the time space and the phase space. The fundamental problems are boiled to zero eigenvalue solutions and all their Jordan normal form and non-zero eigenvalue solutions. Using elastic-viscoelastic correspondence, the viscoelastic solutions are obtained by inversion of Laplacian transform. Thus a new systematic solving method for viscoelasticity is established.Based on the zero eigenvalue solutions and non-zero eigenvalue solutions of the problem of viscoelastic cylinders, as particular example, Kelvin model's creep and Maxwell model's stress relaxation are studied for the simple extension. The numerical results show that the model is reasonable, which is in agreement with viscoelastic characteristics. And the method is efficient.The symplectic system of the viscoelastic cylinders is discussed in this paper only. For whole viscoelastic problems, the Hamiltonian system should be researched further. However the method provides a new way for other viscoelastic problems.