Abstract
With the use of the Laplace integral transformation and state space formalism, the classical axial symmetric quasistatic problem of viscoelastic solids is discussed. By employing the method of separation of variables, the governing equations under Hamiltonian system are established, and hence, general solutions including the zero eigensolutions and nonzero eigensolutions are obtained analytically. Due to the completeness property of the general solutions, their linear combinations can describe various boundary conditions. Simply by applying the adjoint relationships of the symplectic orthogonality, the eigensolution expansion method for boundary condition problems is given. In the numerical examples, stress distributions of a circular cylinder under the end and lateral boundary conditions are obtained. The results exhibit that stress concentrations appear due to the displacement constraints, and that the effects are seriously confined near the constraints, decreasing rapidly with the distance from the boundary.
Highlights
In the modern engineering designs, the knowledge in material behavior is necessary to obtain predictive numerical simulations
Zhang et al discussed crack problems in linear viscoelastic materials by generalizing the Heaviside function to represent the displacement discontinuity across the crack surface 6
In the Hamiltonian system, the boundary conditions just correspond to the fundamental variables displacements and stresses
Summary
In the modern engineering designs, the knowledge in material behavior is necessary to obtain predictive numerical simulations. Based on differential constitutive relations, Mesquita and Coda provided the important algebraic equations and presented a method for the treatment of two dimensional coupling problems between the finite element method and the boundary element method by discussing Kelvin and Boltzmann models 7. Zhong developed the Hamiltonian system method for deriving exact analytical solutions to some basic problems in elastic mechanics 8, 9. This method is developed on the basis of the mathematical theory on Hamiltonian geometry, by which the method of separation of variables can be applied by introducing dual variables. Based on the adjoint symplectic relationships of the general solutions in the time domain, the eigensolution expansion method is introduced to satisfy the boundary conditions. In the region far from the boundary, the effect vanished and usually can be neglected
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