粘弹性岩体中支护圆形洞室施工过程的解析分析

地下洞室的开挖与支护是逐步的连续过程。对具有流变效应的粘弹性岩体,流变时效与施工效应发生耦合,变形与时间相关。针对深埋圆形洞室的施工,用半径时变函数模拟断面开挖过程。当岩体模拟为任一粘弹性材料时,将方程进行拉普拉斯变换求得位移通解,逆变换后代入边界条件确定待定函数,最终得到用洞周面力表达的围岩应力、位移统一解。区分开挖与支护时段,将半径时变函数、洞周面力不同表达式代入,利用支护后围岩与弹性支护接触条件建立关于支护力的Volterra积分方程。当岩石模拟为Boltzmann粘弹模型时,代入材料参数可求解积分方程得到支护力的确切表达,并进一步求得开挖过程及任意时刻支护后应力、位移分段解析表达式。表达式和算例分析表明:加支护后的径向位移增长呈指数形式变化且最终稳定于某一数值。最终洞型相同时,采用不同断面开挖速度且挖完立即支护时,开挖较快的情况位移变化较剧烈,而支护后最终稳定位移较小;但是,相应支护阶段产生的位移较大,支护力也较大。文中给出的方法可用于计算圆形洞室半径任意开挖并加支护后的应力、位移,适用于任一粘弹模型岩体的施工分析。     

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。     

弹性力学空间轴对称问题的一种复变函数法

本文引用了复变量广义解析函数概念,证明了空间轴对称问题的拉甫应力函数可以用两个适当选择的广义解析函数表示,据此即可导出应力分量、位移分量及边界条件的复变函数表示式,进而就可利用复变函数法求解空间轴对称问题。本文用这种方法求解了含有一个球形空腔的圆轴在两端受拉时的解答,表明了以此法分析求解空间轴对称问题的可能性。     

无限粘弹介质中圆孔时变问题的解析分析

本文推导粘弹介质中圆孔孔径时变时的应力和位移.由粘弹解与弹性解的对应关系得到粘弹时变应力解.用直接解方程法求径向位移,最终归结为求解关于待定函数的l阶非齐次微分方程.将半径时变函数泰勒展开,用幂级数解法得到一般情况下的解.在寻找定解条件时,采用了对待定函数的光滑化处理,认为在t=0的微小邻域内函数仍满足微分方程,通过积分得到与待定系数数目相同的定解条件,从而获得本问题径向位移解析解.对Maxwell粘弹模型的求解证明了该法的可靠性.文中解适用于任意粘弹模型和孔径任意时变的情况.     

应用边界积分法求圆形夹杂问题的解析解

边界元方法作为一种数值方法,在各种科学工程问题中得到了广泛的应用.本文参考了边界元法的求解思路,从Somigliana等式出发,利用格林函数性质,得到了一种边界积分法,使之可以用来寻求弹性问题的解析解.此边界积分法也可以从Betti互易定理得到.应用此新方法,求解了圆形夹杂问题.首先设定夹杂与基体之间完美连接,将界面处的位移与应力按照傅里叶级数展开,根据问题的对称性与三角函数的正交性来简化假设,减少待定系数的个数.其次选择合适的试函数(试函数满足位移单值条件以及无体力的线弹性力学问题的控制方程),应用边界积分法,求得界面处的位移与应力的值.然后再求解域内位移与应力.得到了问题的精确解析解,当夹杂弹性模量为零或趋向于无穷大时,退化为圆孔或刚性夹杂问题的解析解.求解过程表明,若问题的求解区域包含无穷远处时,所取的试函数应满足无穷远处的边界条件.若求解区域包含坐标原点,试函数在原点处位移与应力应是有限的.结果表明了此方法的有效性.     

应用边界积分法求圆形夹杂问题的解析解

边界元方法作为一种数值方法, 在各种科学工程问题中得到了广泛的应用.本文参考了边界元法的求解思路, 从Somigliana等式出发, 利用格林函数性质,得到了一种边界积分法, 使之可以用来寻求弹性问题的解析解.此边界积分法也可以从Betti互易定理得到. 应用此新方法, 求解了圆形夹杂问题.首先设定夹杂与基体之间完美连接, 将界面处的位移与应力按照傅里叶级数展开,根据问题的对称性与三角函数的正交性来简化假设, 减少待定系数的个数.其次选择合适的试函数(试函数满足位移单值条件以及无体力的线弹性力学问题的控制方程),应用边界积分法, 求得界面处的位移与应力的值. 然后再求解域内位移与应力.得到了问题的精确解析解, 当夹杂弹性模量为零或趋向于无穷大时,退化为圆孔或刚性夹杂问题的解析解. 求解过程表明,若问题的求解区域包含无穷远处时, 所取的试函数应满足无穷远处的边界条件.若求解区域包含坐标原点, 试函数在原点处位移与应力应是有限的.结果表明了此方法的有效性.      

粘弹性双材料界面裂缝的缝端位移场及动态断裂分析

本文研究的是经常在实际工程中遇到的粘弹性双材料界面裂缝的动断裂问题.由于粘弹性自身的复杂性,使得粘弹性双材料界面裂缝缝端应力的奇异性较弹性呈现出更为复杂的形式,从而使动断裂问题的分析变得更为困难.根据此情况,本文采用复阻尼理论反映粘弹性体的运动规律,用复势理论和平面问题复变函数解答的科洛索夫公式推导了粘弹性双材料界面裂缝缝端位移场及动态应力强度因子的求解公式,利用特解边界元进行了粘弹性双域耦合动力响应计算,按求得的公式用位移外推法计算了单边裂纹板在动荷载作用下的动态应力强度因子.分析了粘性,弹模比和缝长对动态应力强度因子的影响,得出了一些有益的结论.     

粘弹性轴对称平面问题的动态响应

1.引言关于粘弹性轴对称平面问题的一些特殊情况已有若干论述,然而多是讨论材料不可压的情形;有关可压缩粘弹体的求解,往往只是分析准静态问题或讨论一些特例.在文[7]中作组合筒应力分析时给出的动态响应一般解,亦限于不可压材料.Huang等讨论了圆筒高速旋转对材料可压性的影响.Valanis和Sachman分析过弹性波问题,讨论了若干情况下的求解方法. 本文讨论可压粘弹材料在轴对称平面应变状态下的动态响应,从基本方程出发,导出位移应满足的方程,用Laplace变换方法求得在象空间中的一般解.具体分析了厚壁     

简支圆板的非轴对称热弯曲

由于热弹性耦合问题的复杂性, 能得到解析解的主要是轴对称问题和比较简单的问题.利用Green函数, 根据双调和方程边值问题的边界积分公式和自然边界积分方程.在简支板的非轴对称问题的基础上,利用傅立叶级数及卷积的几个公式,求得了非轴对称变温边界条件下圆板的弯曲解,有较好的收敛速度和计算精度,计算过程相对简单.算例表明了方法的有效性.     

含有功能梯度界面相压电纤维复合材料的平面问题

基于复变函数理论和线弹性压电材料本构方程,研究了含有功能梯度界面相压电纤维复合材料平面问题的电-弹场.首先基于复变函数理论,对功能梯度界面相分层均匀化处理,然后将基体、分层均匀化后的界面相以及压电纤维的复势设定为含有待定系数的级数解形式,再通过边界条件建立相应的方程组求解复势中的待定系数,最后求得在各种载荷下压电复合材料的电-弹场.研究结果表明功能梯度界面相的材料属性对压电复合材料的电-弹场有重要的影响.     

Analytical solutions of dynamic mode I crack under the conditions of displacement boundary

By the approaches of the theory of complex variable functions, the problems of dynamic mode I crack under the condition of displacement boundary are investigated. For this kind of dynamic crack extension problems with arbitrary index of self-similarity, the universal representations of analytical solutions are facilely deduced by the methods of self-similar functions. Analytical solutions of the stresses, displacements and stress intensity factors are readily acquired using the methods of self-similar functions. The problems studied can be very easily translated into Riemann–Hilbert problems and their closed solutions are gained rather straightforward in terms of this technique. According to corresponding material properties, the mutative rule of stress intensity factor was illustrated very well. Using those solutions and superposition theorem, the solutions of arbitrarily complex problems can be attained.     

Stresses in rotating heterogeneous viscoelastic composite cylinders with variable thickness

An analytical solution is presented for the rotation problem of a two-layer composite elastic cylinder under a plane strain assumption. The external cylinder has variable-thickness formulation, and is made of a heterogeneous orthotropic material. It contains a fiber-reinforced viscoelastic homogeneous isotropic solid core of uniform thickness. The thickness and elastic properties of the external cylinder are taken as power functions of the radial direction. By the boundary and continuity conditions, the radial displacement and stresses for the rotating composite cylinder are determined. The effective moduli and Illyushin’s approximation methods are used to obtain the viscoelastic solution to the problem. The effects of heterogeneity, thickness variation, constitutive, time parameters on the radial displacement, and stresses are investigated.     

On solutions of the problem in stresses with the use of Maxwell stress functions

The spatial problems of elasticity are mainly solved in displacements [1, 2], i.e., the Lamé equations are taken as the initial equations. This is related to the lack of general solutions for the system of basic equations of elasticity expressed in stresses. In this connection, a new variational statement of the problem in stresses was developed in [3, 4]; this statement consists in solving six generalized equations of compatibility for six independent components of the stress tensor, while the three equilibrium equations are transferred to the set of boundary conditions. This method is more convenient for the numerical solution of problems in stresses and has been tested when solving various boundary value problems. In the present paper, analyzing the completeness of the Saint-Venant identities and using the Maxwell stress functions, we obtain a new resolving system of three differential equations of strain compatibility for the three desired stress functions φ, ξ, and ψ. This system is an alternative to the three Lamé equilibrium equations for three desired displacement components u, v, w and is simpler in structure. Moreover, both of these systems of resolving equations can be solved by the new recursive-operator method [5, 6]. In contrast to well-known methods for constructing general solutions of linear differential equations and their systems, the solutions obtained by the recursive-operator method are constructed as operator-power series acting on arbitrary analytic functions of real variables (not necessarily harmonic), and the series coefficients are determined from recursive relations (matrix in the case of systems of equations). The arbitrary functions contained in the general solution can be determined directly either from the boundary conditions (the obtained system of inhomogeneous equations with a right-hand side can also be solved by the recursive-operator method [6]) or by choosing them from various classes of analytic functions (elementary, special); a complete set of particular solutions can be obtained in the same function classes, and the coefficients of linear combinations of particular solutions can be determined by the Trefftz method, the least-squares method, and the collocation method.     

横观各向同性圆柱体端部问题的三维弹性理论解

从位移的通解出发，用分离变量法得到横观各向同性圆柱体的位移和应力的特征函数展开式，并把位移势函数的解用付里叶积分的形式表示。利用留数运算，该积分解可以转换成类似于特征函数的展开式。通过混合端部边界问题，得到与特征函数解成双正交关系的另一组函数。利用这种双正交关系，可以处理不同的端部边界问题。     

Static analysis of anisotropic functionally graded magneto-electro-elastic beams subjected to arbitrary loading

This paper considers the analytical and semi-analytical solutions for anisotropic functionally graded magneto-electro-elastic beams subjected to an arbitrary load, which can be expanded in terms of sinusoidal series. For the generalized plane stress problem, the stress function, electric displacement function and magnetic induction function are assumed to consist of two parts, respectively. One is a product of a trigonometric function of the longitudinal coordinate (x) and an undetermined function of the thickness coordinate (z), and the other a linear polynomial of x with unknown coefficients depending on z. The governing equations satisfied by these z-dependent functions are derived. The analytical expressions of stresses, electric displacements, magnetic induction, axial force, bending moment, shear force, average electric displacement, average magnetic induction, displacements, electric potential and magnetic potential are then deduced, with integral constants determinable from the boundary conditions. The analytical solution is derived for beam with material coefficients varying exponentially along the thickness, while the semi-analytical solution is sought by making use of the sub-layer approximation for beam with an arbitrary variation of material parameters along the thickness. The present analysis is applicable to beams with various boundary conditions at the two ends. Two numerical examples are presented for validation of the theory and illustration of the effects of certain parameters.     

粘弹性饱和土体中半封闭圆形隧洞的动力响应分析

基于Biot波动方程,研究分析了粘弹性饱和土体中半封闭圆形隧洞的动力响应问题．假定衬砌材料为多孔介质,引入了更符合工程实际的半封闭边界条件．通过引入势函数,在Laplace变换域中得到隧洞边界上作用轴对称荷载和流体压力条件下应力、位移和超孔隙水压力的解答．利用Laplace数值逆变换得到时域中的解,分析了隧洞边界的半透水特性对隧洞动力响应问题的影响,结果表明：隧洞边界的半透水特性对应力、位移场的变化和超孔隙水压力的消散有很大的影响,透水和不透水下条件的解仅是本文的两个特例。     

An analytical solution of an axisymmetric problem of deforming an isotropic half-space with an elastic fixed boundary under the action of a distributed load

An analytical solution to the axisymmetric problem on the action of a distributed load on an isotropic half-space when the load is given by a function dependent on the radial coordinate is obtained. The surface of the half-space is elastically fixed outside the circular domain of load application, the shear stresses are absent along the entire boundary, and the stresses vanish at infinity. At the boundary and inside the elastic half-space, the solutions are represented by the formulas for the stress tensor components and for the displacement vector components.