An application of the symplectic system in two-dimensional viscoelasticity

This paper redescribes fundamental problem of the two-dimensional viscoelasticity in symplectic system. With the aid of the symplectic character and integral transformation, solutions of duality equations are obtained, or Saint-Venant solutions of extension and bend and local solutions of boundary effects. Thus the original problem is reduced to finding zero eigenvalue eigensolutions and non-zero eigenvalue eigensolutions. Meanwhile, adjoint relationships of the symplectic orthogonality in the Laplace domain are generalized to in the time domain. After obtaining fundamental eigensolutions, the problem can be discussed in the eigensolution space of the time domain without the need of the Laplace transformation and inverse one. As its application, a direct method is shown and some examples are discussed, which reveal relations between the creep or relaxation and eigensolutions. The symplectic method and numerical method provide an idea for other researching as well.