无限粘弹性平面中椭圆孔口变边界过程的复变函数解答

工程中存在一类几何边界随时间变化的变边界结构，例如土木工程中处于施工阶段的结构。本文以粘弹性岩体中隧道开挖为背景，尝试用变边界问题对应关系和平面弹性复变方法求取无限平面中椭圆孔口自相似变边界情况下的解析解答。首先建立了复变函数法求解变边界粘弹性问题的基本步骤和公式。然后通过建立逆映射函数将已知?平面复位势转至z平面，从而解耦参与拉普拉斯变换的时间与孔口映射函数所带来的时间，从而导出了粘弹性类材料的应力与位移的统一表达。作为一个例子，本文选择Boltzmann粘弹性模型，代入模型参数后得到积分形式的位移、应力解析解，通过与数值解的比较验证了该解答的可靠性，并通过一个算例分析了变边界过程对位移、应力的影响。分析结果显示，采用不同变边界过程的位移、应力变化形态和数值均有差别。本文解答可用于进行地下椭圆孔型隧道在开挖过程中的力学分析，为实际工程提供初步设计的手段。此外，本文给出的方法可用于推导任意形状孔型变边界问题的解答。

Complex variable solutions for elliptical hole involving time-dependent boundary in viscoelastic infinite plane

The structures involving time-dependent boundary regions are commonly encountered in engineering, i.e. structures under construction in civil engineering. For tunnel excavations in rheological rock mass, this paper presents the analytical solutions of problem with elliptical hole in viscoelastic infinite plane by using the complex variable method and corresponding principle of time-dependent boundary problem in combination. First, basic formulations of complex variable method are established for general viscoelastic problems involving time-dependent boundary regions. Then, based on the derived potentials with respect to ? in the reference, the potentials in z-plane are obtained by introducing the inverse mapping function, and therefore the variable t used in Laplace transformation is decoupled with the t which comes from mapping function. At last, the expressions of displacement and stress are derived for the general cases of viscoelasticity. In addition, the Boltzmann viscoelastic model is chosen as an example to obtain the exact solutions of stress and displacement in integral form by substituting the material parameters into the general expressions. The comparison between the specific analytical and FEM solutions is made to validate the correctness of the derivation and further analyses are performed to illustrate the influence of boundary varying process on the relationship between stress (displacement) and time. The results show that the variations of displacement and stress are correlated with boundary varying speeds. The solutions can be used in mechanical analysis and preliminary design of underground elliptical tunnel excavation. Furthermore, the method in this paper is also suitable for the analysis of the underground excavation problems in arbitrary sharp.