非线性粘弹性平面问题的数学模型及其失记效应

A Mathematical Model for Nonlinear Viscoelastic Plane Problems and Its Fugue Effects

本文以位移为基本未知量,利用非线性粘弹性力学中的Leaderman本构关系和线性几何假设,建立了非线性粘弹性平面问题的数学模型;在粘弹性泊松比为常数的情况下,利用Titchmarsh定理和Laplace变换法证明了非线性粘弹性平面问题与非线性弹性平面问题之间存在着某些对应关系。对应关系为粘弹性问题的求解提供了一种新的思路,利用这些关系可直接从相应弹性问题获得粘弹性平面问题的部分响应,与传统的时域有限差分法相比,计算时间明显缩短。另外,对应关系也揭示了粘弹性结构的失记效应,即结构的部分响应仅与外部输入的现时值有关,而与其历史无关。

In this paper, based on the Leaderman consitutive relations in nonlinear viscoelasticity and the linear geometrical assumption, a mathematical model of nonlinear viscoelastic plane problems for the displacements is established. In the case of constant viscoelastic Poisson ratio, the Titchmarsh theorem and the Laplace transformation method are used to prove that some relations exist between solutions for plane problems of visco and elasticity. The corresponding relations provide a new method for viscoelastic problems. Part of responses of viscoelastic plane problems could be obtained directly from the corresponding elastic ones by these relations. Compared with the traditional finite difference method in time domain, the computational time is gready reduced. In addition, the corresponding relations reveal the fugue effects of viscoelastic structures; mat is to say, the part of responses of viscoelastic structures is dependent on the present value of the external input rather than its history.