Interaction of an anti-plane singularity with interfacial anti-cracks in cylindrically anisotropic composites

Anti-plane problem for a singularity interacting with interfacial anti-cracks (rigid lines) under uniform shear stress at infinity in cylindrically anisotropic composites is investigated by utilizing a complex potential technique in this paper. After obtaining the general solution for this problem, the closed solution for the interface containing one anti-crack is presented analytically. In addition, the complex potentials for a screw dislocation dipole inside matrix are obtained by the superimposing method. Expressions of stress singularities around the anti-crack tips, image forces and torques acting on the dislocation or the center of dipole are given explicitly. The results indicate that the anisotropy properties of materials may weaken the stress singularity near the anti-crack tip for the singularity being a concentrated force but enhance the one for the singularity being a screw dislocation and change the equilibrium position of screw dislocation. The presented solutions are valid for anisotropic, orthotropic or isotropic composites and can be reduced to some new or previously known results.     

静止裂纹尖端实验的HRR奇异场

用近代光学试验方法(面内云纹和投影云纹),测量了不同应变硬化指数材料(n=3.350～9.180)、平面应力Ⅰ型双边裂纹试件、裂纹尖端附近位移场和应变场。由试验结果分析了裂纹尖端位移奇异性,得到J主导区和围绕裂纹尖端附近HRR场分布。分析了HRR分布随载荷、材料不同的变化规律。     

压电材料平面应力状态的直线裂纹问题一般解

研究了含直线裂纹系的压电材料平面应力问题单个裂纹和双裂纹问题的封闭解答表明，在裂纹尖端，应力、电场强度和电位移有１／２阶的奇异性并与前人结果比较了产生电场奇异性的物理因素     

On the solution singularity in the plane elasticity problem for a cantilever strip

The plane elasticity problem of bending of a cantilever strip whose material is assumed to be incompressible in the transverse direction is solved. It is shown that, in the classical statement of of the boundary condition for the fixed edge of the strip, the solution has a singularity at the corner points of the edge. Several cases of the strip fixation and loading characterized by the presence or absence of the solution singularity are considered. The strength of glass beams of three types, for which the theory of elasticity predicts whether the normal stress has a singularity, is studied experimentally. It is shown that the limit stresses for the beams of the types under study are practically the same, which testifies that the solution singularity does not have any physical nature.     

一种处理弹性动力边界元法中奇异积分的方法

利用边界元法求解瞬态弹性动力学问题时，时域基本解函数的分段连续性和奇异性为该问题的求解带来很大的困难。为了解决时域基本解中的奇异性问题，本文依据柯西主值的定义，对经过时间解析积分之后的时域基本解进行奇异值分解，将其分成奇异和正则积分两部分；其中正则部分可通过采用常规高斯积分方法来计算，而奇异部分具有简单的形式，可以利用解析积分计算。经过上述操作之后，就可以达到直接消除时域基本解中奇异积分的目的。和传统方法相比，本文方法并不依赖静力学基本解来消除奇异性，是一种直接求解方法。最后给定两个数值算例来验证本文提出方法的正确性和可行性，结果表明使用本文算法可以解决弹性动力学边界积分方程中的奇异性问题。     

Mixed Convection on a Horizontal Surface Embedded in a Porous Medium: the Structure of a Singularity

The mixed convection boundary-layer flow on a horizontal impermeable surface embedded in a saturated porous medium and driven by a local heat source is considered. Similarity solutions are obtained for specific outer flow variations and these are shown to have a solution only for parameter values greater than some critical value. When this is not the case the solution develops a singularity at a finite distance from the leading edge. The nature of this singularity is also discussed.     

Singular stress fields near contact boundaries in a composite bolted joint

Singular stresses arising in the neighborhood of contact surfaces introduced in laminated orthotropic plates by mechanical joining with clamp-up were investigated by using local asymptotic solutions and full-field numerical analysis. Three-dimensional B-spline approximation of the displacements and a penalty function-based contact solution was used in the numerical analysis. Recent work has shown that fracture in bolted composite joints may initiate near the outer edge of the bolt head or washer away from the hole edge, particularly if the joint is preloaded. Material and geometric discontinuities exist in these regions, resulting in singular stress behavior. Asymptotic stress analysis was performed to obtain the power of singularity in these regions as a function of the bolt-head (washer) stiffness. Frictionless contact conditions were assumed. It was found that the characteristics of the stress singularity for such practically important combinations as titanium bolt-head and carbon fiber composite plate are similar to a crack in terms of the power of singularity and uniqueness of the singular term. Coefficients of the singular terms of the asymptotic expansion were determined by comparison with the numerical solution in the close vicinity of the singular contour. Good agreement between the asymptotic and numerical solution in the transition regions was observed.     

Three-dimensional Green's functions in anisotropic trimaterials

Three-dimensional elastostatic Green's functions in anisotropic trimaterials are derived, for the first time, by applying the generalized Stroh's formalism and Fourier transforms. The Green's functions are expressed as a series summation with the first term corresponding to the full-space solution and other terms to the image solutions due to the interfaces. The most remarkable feature of the present solution is that the image solutions can be expressed by a simple line integral over a finite interval [0,2π]. By partitioning the trimaterial Green's function into a full-space solution and a complementary part, the line integral involves only regular functions if the singularity is within one of the three materials, being treated analytically owning to the explicit expression of the full-space solution. When the singularity is on the interface, which occurs if the field and source points are both on the same interface, the involved singularity is handled with the interfacial Green's functions.A numerical example is presented for a trimaterial system made of two anisotropic half spaces bonded perfectly by an isotropic adhesive layer, showing clearly the effect of material layering on the Green's displacements and stresses. Furthermore, by comparing the present Green's solution to the direct (two-dimensional) 2D integral expression which is also derived in this paper, it is shown that, the computational time for the calculation of the Green's function can be substantially reduced using the present solution, instead of the direct 2D integral method.     

Entropy layer in the problem of hypersonic flow over thin blunt bodies which are close to two-dimensional

In the problem of the hypersonic flow of a nonviscous thermally nonconducting gas over thin blunt bodies which are close to two-dimensional the solution is constructed in the entropy layer. The construction is achieved by a generalization of the method developed ia [1] in application to bodies close to two-dimensional. The use of an approximate model identifying the effect of the blunting on the gas with the effect of a concentrated force distributed over the edge is important in the construction. The solution is represented in the form of asymptotic expansions. The equations of the hypersonic theory of small perturbations, which is the null approximation in the process of construction of the solution in the form of a series in powers of a small parameter determined as the square of the relative thickness of the body or the relative width of the perturbed region, are obtained in the null approximation in this case. The surface of the blunt body proves to be singular for the null approximation, since the entropy function p/?^x grows without limit as the surface is approached. The attempt to construct the succeeding approximations leads to strengthening of the singularity. This necessitates the use of the method of deformed coordinates (the PLG method). Basic to the latter is the removal of the singularity, which is not inherent to the exact solution of the problem, through asymptotic expansions with respect to a small parameter not only of the unknown variables, but also of the independent variables, with the subsequent determination of the deformation of the independient variables on the basis of the “quenching” of the singularity. Use of the PLG method allows one to construct a solution which is uniformly applicable in the entire stream, including the entropy layer. In practice, the construction of such a solution leads to the determination of the displacement of the streamlines near the surface of the body, as a result of which the singularity is “absorbed” by the body and the solution outside the body proves to be freed of the singularity. In the null approximation this displacement of the streamlines can be determined in closed form.     

Critical value sets and dynamical systems

Elementary applications of singularity theory to dynamical systems are outlined. The main idea is to consider the geometry of the wrinkled surface generated by the time development of a one variable solution as a function of initial conditions or system parameters. Applications of singularity theory to vibro-impacting systems are discussed in some detail and further applications to dynamics are briefly reviewed.     

BASIC SOLUTION FOR THE THREE-PHASE COMPOSITE CONSTITUTIVE MODEL IN ANTIPLANE PIEZOELECTRICITY

A basic solution in series form for the three-phase composite cyclindrical model in antiplane piezoelectricity subjected to the action of a singularity in the intermediate matrix region is presented. The solution is obtained through the complex potential approach in conjuction with the techniques of analytical continuation, singularity analysis, Laurent series expansion in an annular region and Cauchy integral formulae, etc. Based on the complex potentials obtained, explicit expressions for the distribution of stress and electric displacement in the three regions are also derived.     

反平面压电中三相复合本构模型的基本解

给出了三相圆柱压电复合本构模型在随奇点作用时的基本解答,该解答是通过复势技术结合解析开拓、奇点分析、圆环域上的Laurent级数展开及Cauchy积分公式的运用等而获得的,在所获得的复势的基础上,文中得到在基体上的奇点作用时各区域上的应力及电位移分布的表达式。     

A note on faxen laws for nonspherical particles

The advantages of new forms of Faxen laws with derivatives of finite order are discussed and specifically illustrated for prolate spheroids. Earlier ideas on the connection between the Faxen laws and certain “associated solutions” (single-particle solutions of the Stokes flow where the ambient velocity is a uniform stream or linear field) have been refined for the case where the associated solution has a relatively simple singularity solution.     

Axisymmetric nuclei of strain and their applications for multilayered solids

In this paper, the effect of several axisymmetric elastic singularities (i.e., point forces, double forces, sum of two double forces and centers of dilatation) on the elastic response of a multilayered solid is investigated. The boundary conditions in an infinite solid at the plane passing through the singularity are derived first using Papkovich–Neuber harmonic functions. Then, a Green’s function solution for multilayered solids is obtained by solving a set of simultaneous linear algebraic equations using both the boundary conditions for the singularity and the layer interfaces. Finally, the elastic solutions in a single layer on an infinite substrate due to point defects and infinitesimal prismatic dislocation loops are presented to illustrate the application of these Green’s function solutions.     

Forward displacement analysis of a quadratic 4-DOF 3T1R parallel manipulator

A quadratic parallel manipulator refers to a parallel manipulator with a quadratic characteristic polynomial. This paper revisits the forward displacement analysis (FDA) and the Type II singularity analysis of a quadratic 4-DOF 3T1R (SCARA) parallel manipulator: the Quadrupteron. It will be proved that there exists a one-to-one correspondence between the two formulas, each producing one solution to the FDA, and the two singularity-free regions. Therefore, a unique solution to the FDA can be obtained in a straightforward way for such a parallel manipulator if the singularity-free region in which it works is specified. The Type II singularity analysis in the joint space will also be investigated in order to identify the conditions on the inputs to keep the Quadrupteron working in the same singularity-free region in its Cartesian workspace.     

压电材料平面应力状态的直线裂纹问题一般解

研究了含直线裂纹系的压电材料平面应力问题单个裂纹和双裂纹问题的封闭解答表明，在裂纹尖端，应力、电场强度和电位移有１／２阶的奇异性并与前人结果比较了产生电场奇异性的物理因素     

Analysis of a three-dimensional dissimilar material joint with one real singularity using a conservative integral

In the present study, a conservative integral based on the Betti reciprocal principle is formulated to determine the intensity of singularity at a vertex of the interface in three-dimensional dissimilar material joints with one real singularity. Eigenanalysis formulated using a three-dimensional finite element method (FEM) is used to calculate the order of stress singularity, angular functions of displacements and stresses. Models with various element sizes and various integral areas are used to investigate the effect of the integration area on the accuracy of the results. The results are compared with those obtained from the boundary element method (BEM) using a curve-fitting technique to calculate the intensity of singularity. In addition, models of various lengths and various material combinations are used to investigate the stress singularity characteristics in three-dimensional dissimilar material joints. The results of the present study indicate that the conservative integral can be used to determine the intensity of singularity in three-dimensional bi-material joints. The accuracy of the results can be improved by mesh refinement. Finally, the relationships among the intensity of singularity, the order of stress singularity and the model geometry are discussed.     

Steady penetration of a rigid cone into pressure-dependent plastic material

The objective of the present paper is to find a semi-analytical axisymmetric solution for steady penetration of a rigid cone into pressure-dependent plastic material obeying the double-shearing model. As expected, the solution is singular near the maximum friction surface. It is important to mention that the singularity is not due to the geometry of the problem but the friction law. The type of the singularity is the same as in plane-strain solutions based on the double-shearing model and in classical plasticity. This allows for calculating the strain rate intensity factor. The solution is illustrated by a numerical example.     

Numerical analysis of singular solutions of two-dimensional problems of asymmetric elasticity

An algorithm for the numerical analysis of singular solutions of two-dimensional problems of asymmetric elasticity is considered. The algorithm is based on separation of a power-law dependence from the finite-element solution in a neighborhood of singular points in the domain under study, where singular solutions are possible. The obtained power-law dependencies allow one to conclude whether the stresses have singularities and what the character of these singularities is. The algorithm was tested for problems of classical elasticity by comparing the stress singularity exponents obtained by the proposed method and from known analytic solutions. Problems with various cases of singular points, namely, body surface points at which either the smoothness of the surface is violated, or the type of boundary conditions is changed, or distinct materials are in contact, are considered as applications. The stress singularity exponents obtained by using the models of classical and asymmetric elasticity are compared. It is shown that, in the case of cracks, the stress singularity exponents are the same for the elasticity models under study, but for other cases of singular points, the stress singularity exponents obtained on the basis of asymmetric elasticity have insignificant quantitative distinctions from the solutions of the classical elasticity.     

On the nature of singularities inherent,under a given analytic distribution of the external pressure,in solutions of the prandtl equations near the point of separation

A comprehensive analysis of a classical two-dimensional boundary layer is developed with the aim of revealing possible types of singularities related to separation. According to the basic assumption, the limit flow regime with a singular point of vanishing skin friction obeys an analytic solution where the frictional intensity approaches zero according to quadratic law rather than varing linearly with distance. Small deviations from the limit regime are described in terms of eigenfunctions, three of which involve singularities. The flow field for all supercritical regimes is continued into a small region centered about the singular characteristic springing from the point of vanishing skin friction in the undisturbed limit solution. The solvability condition for a key boundary-value problem posed in this region as a result of matching the global-scaled solution upstream gives three different types of singularities of the Prandtl equation. According to the weakest-type singularity the skin friction varies as the square root of the cube of the local distance. The next type includes a sudden change in the wall shear stress derivative with respect to the coordinate along the solid surface, it was ruled out of the basic limit solution. Then the famous Landau-Goldstein singularity with the skin friction being proportional to the square root of the local distance evolves. A still more complicated flow pattern may be composed of the singularity with a sudden change in the wall shear stress derivative superseded by the Landau-Goldstein singularity at some small distance downstream. A qualitative comparison between theoretical predictions and experimental data for a circular cylinder in an incompressible stream is made with the emphasis on conceivable explanations for the nonmonotonic behavior of frictional intensities in the transitional range of Reynolds numbers.The final stage of this study was carried out with the support to O.S. Ryzhov of the Alfred P. Sloan Foundation and the U.S. Air Force Office of Scientific Research under Grant AFOSR 88-0037.