The Symplectic Method In Viscoelasticity

The Hamiltonian system is a direct method by which the order of differential governing equations can be reduced. Since the difficulty of solving high-order differential equations in the traditional methods, such as the semi-inverse method, is overcome, the Hamiltonian system gained much attention in recent years and has been applied successfully into elasticity. However this method can not be applied directly into viscoelasticity for the energy non-conservation. Based on the investigation of the character of viscoelastic material, the Hamiltonian system is introduced into viscoelasticity in this dissertation and the fundamental problems are discussed in the symplectic space.With the aid of the symplectic character and the integral transformation, the plane problem is transformed into problem of solving zero-eigenvalue eigensolutions and non zero-eigenvalue eigensolutions, which are solutions of Saint-Venant problems and local effect problems respectively. Meanwhile, the adjoint relationships of the symplectic orthogonality in the Laplace domain are generalized to the time domain. Therefore the problem can be discussed directly in the eigensolution space of the time domain and the iterative application of Laplace transformation is not needed. In addition, an effective method of solving non-homogeneous equations and boundary conditions is given with the help of the expansion of the symplectic eigenvalue eigensolutions. Based on this method, the Saint-Venant problems are studied in the zero-eigenvalue solution space and the whole character of creep and relaxation for viscoelasticity is revealed. On the other hand, the local effect near the boundary is thoroughly discussed in the whole symplectic eigenvalue solution space and the stress and strain fields of some typical problems are obtained, in which the stress concentrations caused by the boundary condition restraints are well exhibited.As everyone knows, temperature has an important effect on materials, especially on viscoelastic material. The variable substitution method is applied in this dissertation to transform the non-homogeneous lateral boundary condition and temperature effect into homogeneous problem. Thus the key point of thermo-viscoelasticity problem is to find a special solution of the non-homogeneous equations. Based on the research of viscoelasticity, the symplectic system is applied into thermo-viscoelasticity. Numerical results show the thermal effect on the whole character of viscoelasticity in the simple tensional problems and bending problems, and exhibit the local effects caused by the boundary condition restraints and uneven distribution of thermal condition.Based on the research of the plain problem, the Hamiltonian system is generalized to three-dimensional cylinder problem. By studying the canonical equations of the original problem and employing the adjoint relationships of the symplectic orthogonality and the symplectic eigensolution expansion technology, an effective method of solving three-dimensional problems is given. As a particular case, the axisymmetric problem is thoroughly discussed. Since the lateral boundary condition and end condition can also be transformed into problem of non-homogenous equation, the analytical solution of the axisymmetric problem can be given. Moreover the non-homogeneous problems caused by the boundary conditions and typical temperature conditions are discussed in the numerical results, in which the creep and relaxation character and thermal effect of viscoelasticity are well described.The thick walled cylinder problem is also investigated in this dissertation under Hamiltonian system. The dual equations and corresponding boundary conditions are described in the symplectic space. Besides the sub-symplectic system is constructed so that the partial differential equations can be transformed into the ordinary differential equations. Based on this method, solutions of tension, torsion, bending and local effect problems are obtained. These solutions exhibit the characters of the creep and relaxation of viscoelasticity in the thick walled cylinder problems.