The symplectic system method in the stress analysis of 2D elasto-viscoelastic fiber reinforced composites

Abstract

With the aid of the variational method and Laplace transformation, the symplectic system method is employed into two-dimensional elastic-viscoelastic fiber reinforced composites. The fundamental eigenfunctions of the governing equations are generalized to the time domain. Therefore the problem can be discussed directly in the time domain, and the iterative application of Laplace transformation is not needed. Using this method, all the stress components of the inner fiber and outer matrix, and hence the stress transfer in the interface between the fiber and matrix, are expressed analytically. The results obtained by the approach are accurate, because all the boundary conditions prescribed on the surfaces and ends of the composites can be satisfied. Numerical example demonstrate that both the shear stress and the normal stress decrease with time due to the viscoelastic property of the matrix, and that stress concentration occurs near the end.

An Eigenexpansion Method in 2D Viscoelastic Materials

“…The solutions have been obtained by the former viscoelastic researches [43]. Thus the final solution of the problem can be obtained by the linear combinations of the eigensolutions…”

“…The symplectic method can not be applied directly into viscoelasticity since the energy consumption exists. However in the Laplace domain, the problem can be transformed into problem of conservative system in which the Hamiltonian formulation is applicable [40][41][42][43][44].…”

The Quasi-Static Analysis for Two-Dimensional Thin Plates

“…respectively [4]. The coefficients are For the case of 0 κ = , the zero eigenfunctions can be easily to be found.…”

“…Zhong [2] developed the eigenfuction expansion method for elastic problems on the basis of the mathematical theory of symplectic geometry. In contrast to the well known semi-inverse method, this approach takes original variables and their dual variables as the basic variables, and hence the difficulty of solving high order differential equations in the traditional methods is overcome [3,4]. This approach can explain the approximation of Saint-Venant principle theoretically: the exact solution consists of two parts, the Saint-Venant solution and the local solution.…”

An Eigenfunction Expansion Method in the Stress Concentration Analysis

“…Zhong [2] introduced the Hamiltonian formulation into the theory of elasticity and put forward a direct method. Since the constitutive equations and the geometrical equations of viscoelasticity in the Laplace domain are similar to the elastic counterparts in the time domain, the Hamiltonian formulation can be generalized [3][4][5][6]. This paper discusses firstly the Hamiltonian formulation for a viscoelastic media of the plane-strip in Laplace domain.…”