Asymptotic solutions of coupled equations of supercritically axially moving beam

In supercritical regime, the coupled model equations for the axially moving beam with simple support boundary conditions are considered. The critical speed is determined by linear bifurcation analysis, which is in agreement with the results in the literature. For the corresponding static equilibrium state, the second-order asymptotic nontrivial solutions are obtained through the multiple scales method. Meantime, the numerical solutions are also obtained based on the finite difference method. Comparisons among the analytical solutions, numerical solutions and solutions of integro-partial-differential equation of transverse which is deduced from coupled model equations are made. We find that the second-order asymptotic analytical solutions can well capture the nontrivial equilibrium state regardless of the amplitude of transverse displacement. However, the integro-partial-differential equation is only valid for the weak small-amplitude vibration axially moving slender beams.     

Analysis of non-Fourier heat conduction in a solid sphere under arbitrary surface temperature change

In this paper, the non-Fourier heat conduction in a solid sphere under arbitrary surface thermal disturbances is solved analytically. Four cases including sudden, simple harmonic periodic, triangular and pulse surface temperature changes are investigated step-by-step. The analytical solutions are obtained using the separation of variables method and Duhamel’s principle along with the Fourier series representation of an arbitrary periodic function and the Fourier integral representation of an arbitrary non-periodic function. Using these analytical solutions, the temperature profiles of the solid sphere are analyzed, and the differences in the temperature response between the “hyperbolic” and “parabolic” are discussed. These solutions can be applicable to all kinds of non-Fourier heat conduction analyses for arbitrary boundary conditions occurred in technology. And as application examples, particular attention is devoted to the cases of triangular surface temperature change and pulse surface temperature change. The examples presented in this paper can be used as benchmark problems for future numerical method validations.     

Free vibration analyses of axially loaded laminated composite beams based on higher-order shear deformation theory

The dynamic stiffness matrix method is introduced to solve exactly the free vibration and buckling problems of axially loaded laminated composite beams with arbitrary lay-ups. The Poisson effect, axial force, extensional deformation, shear deformation and rotary inertia are included in the mathematical formulation. The exact dynamic stiffness matrix is derived from the analytical solutions of the governing differential equations of the composite beams based on third-order shear deformation beam theory. The application of the present method is illustrated by two numerical examples, in which the effects of axial force and boundary condition on the natural frequencies, mode shapes and buckling loads are examined. Comparison of the current results to the existing solutions in the literature demonstrates the accuracy and effectiveness of the present method.     

Propagation crack problems in interfaces between dissimilar orthotropic media

By the theory of complex functions, the general representation of the solution is derived for crack problems of arbitrary index of self-similarity expanding along the interfaces between dissimilar orthotropic media. The problems of Riemann and Hilbert can be formulated immediately and the analytical solution was obtained with this method. Some problems were analyzed as examples. The analytical solution for the cracked body subjected to an arbitrary load can be obtained with the summary method based on the present solution.     

Approximate solution of the problem of the three-dimensional boundary layer in an incompressible fluid

A method of successive approximations is proposed for the solution of the equations of the three-dimensional incompressible boundary layer on bodies of arbitrary shape. A coordinate system connected with the streamlines of the external nonviscous flow is used. It is assumed that the velocity across the external streamlines is small. When the intensity of secondary flow is low the equations describing the boundary layer in an incompressible fluid are reduced to a form analogous to the equations for the boundary layer on axially symmetrical bodies. An approximate analytical solution is obtained for the velocity and for the friction in the form of equations which can be used for any problems of a three-dimensional incompressible boundary layer. The method developed was applied to the problem of the three-dimensional boundary layer at a plate with a cylindrical obstacle in the presence of a slip angle.     

正交异性复合材料界面上反平面动态自相似扩展裂纹问题的解

采用复变函数论的方法，对复合材料界面上的裂纹扩展问题进行研究。并根据任意的自相似指数的断裂动力学问题，进行自相似求解，导出解析解的一般表示。应用该法可以迅速地将所论问题转化为Riemann-Hil-bert问题，并可以相当简单地得到问题的闭合解。文中分别对裂纹中心受阶跃载荷，裂纹面受到瞬时脉冲载荷作用下的界面裂纹扩展问题进行求解。得到了裂纹的位移。尖端的应力和动态应力强度因子的解析解。应用该解并通过叠加原理。就可以很容易的求得任意复杂问题的解。     

Ⅲ型界面裂纹面受变载荷Pxmtn作用下的自相似解

通过复变函数论的方法，对Ⅲ型界面裂纹表面受变载荷$Px^mt^n$作用下的动态扩 展问题进行了研究. 采用自相似函数的方法可以获得解析解的一般表达式. 应用 该法可以很容易地将所讨论的问题转化为Riemann-Hilbert问题, 然后应 用Muskhelishvili方法就可以较简单地得到问题的闭合解. 利用这些解 并采用叠加原理，就可以求得任意复杂问题的解.     

ANALYTICAL SOLUTIONS FOR THE LAYERED GEO-MATERIALS SUBJECTED TO AN ARBITRARY POINT LOAD IN THE CARTESIAN COORDINATE

This paper presents analytical solutions for the stress and displacement field in elastic layered geo-materials induced by an arbitrary point load in the Cartesian coordinate system. The point load solutions can be obtained by referring to the integral transform and the transfer matrix technique. However, former solutions usually exist in the cylindrical coordinate system subjected to axisymmetric loading. Based on the proposed solutions in the Cartesian coordinate, it is very easy to solve asymmetric problems and consider the condition with internal loads in multi-layered geo-materials. Moreover, point load solutions can be used to construct solutions for analytical examination of elastic problems and incorporated into numerical schemes such as boundary element methods. The results discussed in this paper indicate that there is no problem in the evaluation of the point load solutions with high accuracy and efficiency, and that the material non-homogeneity has a significant effect on the elastic field due to adjacent loading.     

On Solutions of Some Non-Linear Differential Equations Arising in Newtonian and Non-Newtonian Fluids

Some steady as well as unsteady solutions of the equations of motion for an incompressible Newtonian and non-Newtonian (second-grade) fluids are obtained by applying different methods including the Lie symmetry group method. The flows considered are axially symmetric with the swirling motion, and the governing equations for second-grade fluid flow have been modeled. Expressions for streamlines, velocity and vorticity components are constructed explicitly in each case. Exact analytical solutions in second-grade fluid are obtained and compared with the corresponding viscous solutions.     

变边界粘弹性轴对称问题的复变函数法

对边界几何形状、位置随时间变化的变边界结构,给出了用复变函数求解粘弹问题的解析方法。文中用拉普拉斯变换结合平面弹性复变方法,对内外边界变化时粘弹性轴对称问题进行求解。引入两个与时间、空间相关的解析函数,给出了变边界情况下应力、位移以及边界条件与解析函数的关系。当解析函数形式部分确定,则可用边界条件求解其中与时间相关的待定函数。求解待定函数的方程一般情况下为一系列积分方程,特殊情况可求得解析解。对轴对称问题中应力边值问题、位移边值问题以及混合边值问题,分别利用边界条件求得相关系数,从而得到了应力与位移的解析表达。当取Boltzmann粘弹模型时,进行不同边值问题的分析。分析显示,应力、位移的形态与大小均与边界变化过程相关,与固定边界粘弹性问题有较大不同。本文解答可用于粘弹性轴对称问题内外边界任意变化及各种边值问题的力学分析。此外,该法可进一步进行荷载非对称、复杂孔型变边界问题的求解。     

An Analytical Solution to the One-Dimensional Solute Advection-Dispersion Equation in Multi-Layer Porous Media

An analytical solution to the one-dimensional solute advection-dispersion equation in multi-layer porous media is derived using a generalized integral transform method. The solution was derived under conditions of steady-state flow and arbitrary initial and inlet boundary conditions. The results obtained by this solution agree well with the results obtained by numerically inverting Laplace transform-generated solutions previously published in the literature. The analytical solution presented in this paper provides more flexibility with regard to the inlet conditions. The numerical evaluation of eigenvalues and matrix exponentials required in this solution technique can be accurately and efficiently computed using the sign-count method and eigenvalue evaluation methods commonly available. The illustrative calculations presented herein have shown how an analytical solution can provide insight into contaminant distribution and breakthrough in transport through well defined layered column systems. We also note that the method described here is readily adaptable to two and three-dimensional transport problems.     

On XFEM applications to dislocations and interfaces

A method for modelling dislocations in systems with arbitrary materials interfaces is described. The method is based on the extended finite element method (XFEM) where dislocations are modelled in the manner of the Volterra dislocation model. A method for calculating the Peach–Koehler force by J-integrals in this framework is studied. The method is compared to closed form solutions for interface problems and excellent accuracy is obtained. The convergence and accuracy of the method is studied in two problems where analytical solutions are available: an edge dislocation interacting with a free-surface and an edge dislocation interacting with a bimaterial interface. The applicability of the method to more complicated problems is illustrated by the modelling of slip misorientation of an edge dislocation with a glide plane intersecting a material interface and dislocations in a multi-material domain with non-parallel interfaces.     

基于混合变量条形传递函数方法的圆柱壳屈曲分析

提出了一种分析交各向异性圆柱壳和阶梯圆柱壳稳定性问题的混合变量条形传递函数方法。首先基于Fluegge薄壳理论，通过定义广义位移变量和对应的广义力变量，建立了圆柱壳混合变量能量泛函；然后通过引入条形单元，定义混合状态变量和采用传递函数方法对超级壳单元求解，得到具有多种边界条件圆柱壳屈曲问题的半解析解；最后通过位移连续和力平衡条件，可以得到阶梯圆柱壳屈曲问题的解。理论解推导过程表明此方法在引入边界条件和进行阶梯圆柱壳求解时非常方便。算例分析的结果验证了本方法的正确性。     

弹性地基上各向异性板的静力分析

根据弹性地基上各向异性矩形板弯曲挠度的微分方程精确的求得了适用于各种载荷的非齐次解和各类齐次解。其中由三角函数和双曲线函数组成的齐次解能满足四个边为任意边界条件的问题;由代数多项式和双正弦级数组成的齐次解能满足四个角为任意边界条件的问题。通过适当选取建立了满足任意边界条件和任意载荷作用的一般解。解中的积分常数完全由边界条件来决定。以四边简支承受均布载荷和局部分布载荷的对称迭层复合材料方板为例进行了计算和分析。其结果与已有文献结果是一致的。由于集中载荷不能求得作用点的弯矩,故在例题中改用局部分布载荷因而求得了最大弯矩。     

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。     

压电复合材料中的Eshelby夹杂问题

通过采用解析延拓和共形映射技术，获得了压电复合材料中有关Eshelby夹杂几个典型问题的精确弹性解答，即横观各向同性压电介质中任意形状的Eshelby夹杂与圆柱异相夹杂间相互作用；一般各向异性压电介质中任意形状的Eshelby夹杂与双压电材料所形成界面的相互作用．成功求解这些问题的关健在于构造一个辅助函数．与Ru所采用的方法不同，所引入的辅助函数在无穷远点不存在极点，从而使得所展开的分析更加自然合理．分析结果清楚地揭示出Eshelby夹杂的存在对压电复合材料机电耦合响应将产生不容被忽视的影响．很典型的一个例于是当一个Eshelby椭圆夹杂与圆柱异相夹杂相互作用时，每个夹杂体内部的应力场和电场都将是不均匀的；另一个例于是位于界面附近的Eshelby夹杂有可能是界面发生损伤的一个重要原因．     

High-order derivatives of Green’s functions in magneto-electro-elastic materials

Three-dimensional Green’s functions and their arbitrary order derivatives in general anisotropic magneto-electro-elastic materials are derived by using Fourier transform. They are analytical solutions expressed in line integral forms, and can be evaluated by a standard numerical integration method. With this method, we can obtain results with high accuracy. Besides, a numerical finite difference method is also given to evaluate the second-order derivatives quickly. When setting the appropriate material coefficients to zero, the piezoelectric, piezomagnetic, and purely anisotropic elastic Green’s functions and their derivatives can all be obtained from the current solutions.     

动态扩展裂纹的若干反平面问题的研究

采用复变函数论,对反平面条件下的动态裂纹扩展问题进行研究。通过自相似函数的方法可以获得解析解的一般表达式。应用该法可以很容易地将所讨论的问题转化为Riemann—Hilbert问题,并可以相当简单地得到问题的闭合解。文中分别对裂纹面受均布载荷、坐标原点受集中增加载荷、坐标原点受瞬时冲击载荷以及裂纹面受运动集中载荷Px／t作用下的动态裂纹扩展问题进行求解,得到了裂纹扩展位移、裂纹尖端的应力和动态应力强度因子的解析解。应用该解并通过叠加原理,就可以求得任意复杂问题的解。     

Exact Electromagnetothermoelastic Solution for a Transversely Isotropic Piezoelectric Hollow Sphere Subjected to Arbitrary Thermal Shock

This paper presents analytical study for electromagnetothermoelastic transient behavior of a transversely isotropic hollow sphere, placed in a uniform magnetic field, subjected to arbitrary thermal shock. Exact solutions for the transient responses of stresses, perturbation of magnetic field vector, electric displacement and electric potential in the transversely isotropic piezoelectric hollow sphere are obtained by means of the Hankel transform, the Laplace transform and their inverse transforms. An interpolation method is used to solve the Volterra integral equation of the second kind caused by interactions among electric, magnetic, thermal and elastic fields. From the sample numerical calculations, it is seen that the present method is suitable for the transversely isotropic hollow sphere, placed in a uniform magnetic field, subjected to arbitrary thermal shock. Finally, the result can be used as a reference to solve other transient coupling problems of electromagnetothermoelasticity.     

Functional and generalized separable solutions to unsteady Navier–Stokes equations

The paper studies unsteady Navier–Stokes equations with two space variables. It shows that the non-linear fourth-order equation for the stream function with three independent variables admits functional separable solutions described by a system of three partial differential equations with two independent variables. The system is found to have a number of exact solutions, which generate new classes of exact solutions to the Navier–Stokes equations. All these solutions involve two or more arbitrary functions of a single argument as well as a few free parameters. Many of the solutions are expressed in terms of elementary functions, provided that the arbitrary functions are also elementary; such solutions, having relatively simple form and presenting significant arbitrariness, can be especially useful for solving certain model problems and testing numerical and approximate analytical hydrodynamic methods. The paper uses the obtained results to describe some model unsteady flows of viscous incompressible fluids, including flows through a strip with permeable walls, flows through a strip with extrusion at the boundaries, flows onto a shrinking plane, and others. Some blow-up modes, which correspond to singular solutions, are discussed.