Quadruple Trigonometrical Series Equations and Their Application to an Inclusion Problem in the Theory of Elasticity

Closed form solution of quadruple series equations involving cosine kernels has been obtained by reducing the series equations into triple Abel's type integral equations which in turn are reduced to a single integral equation. Making use of finite Hilbert transforms the solution of the single integral equation is obtained in closed form. This solution is used to solve an electrostatic problem. The results of this paper have also been used in a two-dimensional elastostatic problem under anti-plane shear and the effect of rigid line inclusions with thickness on the Griffith cracks has been examined. The expressions for shear stress and stress intensity factor at the tip of the crack are obtained. Finally, some numerical results for the stress intensity factor and shear stress distribution are obtained.     

The breaking of a non-homogeneous fiber embedded in an infinite non-homogeneous medium

The torsion of an infinite non-homogeneous elastic cylindrical fiber, containing a penny-shaped crack embedded in an infinite non-homogeneous elastic material is considered. The cylinder and elastic medium have different shear moduli. Using integral transformation techniques the solution of the problem is reduced to the solution of dual integral equations. Later on the solution of the dual integral equations is transformed into the solution of a Fredholm integral equation of the second kind, which is solved numerically. Closed form expressions are obtained for the stress intensity factor and numerical values for the stress intensity factors are graphed to demonstrate the effect of non-homogeneity of the fiber and infinite medium. In the end the stress singularity is obtained when the crack touches the infinite non-homogeneous medium (matrix).     

含圆形孔口和水平直裂纹的平面问题的应力分析

利用复变函数方法和积分方程理论研究了既含有圆形孔口又含有水平裂纹的无限大平面的平面弹性问题,将复杂的解析函数的边值问题化成了求解只在裂纹上的奇异积分方程的问题.此外,还给出了裂纹尖端附近的应力场和应力强度因子的公式.     

A family of analytic extremals in problems of the optimal control of the rotation of a body

The problem of the optimal control of the rotation of an absolutely rigid body about the centre of mass is investigated. The main purpose of the control is to vary the angular velocity vector from its initial value to the required terminal value in a finite time so that the manoeuvre would require the smallest power consumption, which is characterized by an integral quadratic functional. The principal torque produced by the external forces applied to the body serves as the control. The change in orientation is not taken into account, i.e., the problem of the overspeed–braking control of the body, is studied. A new class of analytic extremals based on the use of space-time deformations of the solutions of the dynamical Euler equations for the free rotation of a rigid body is described. Sufficient conditions for the existence of such extremals for all types of symmetries are presented.     

Method of Decreasing the Order of a Partial Differential Equation by Reducing to Two Ordinary Differential Equations

Using additional unknown functions and additional boundary conditions in the integral method of heat balance, we obtain approximate analytic solutions to the non-stationary thermal conductivity problem for an infinite solid cylinder that allow to estimate the temperature state practically in the whole time range of the non-stationary process. The thermal conducting process is divided into two stages with respect to time. The initial problem for the partial differential equation is represented in the form of two problems, in which the integration is performed over ordinary differential equations with respect to corresponding additional unknown functions. This method allows to simplify substantially the solving process of the initial problem by reducing it to the sequential solution of two problems, in each of them additional boundary conditions are used.     

Energy release rate along a kinked path

Considering anti‐plane elasticity we provide an existence result for the energy release rate along a piecewise C^{1, 1} path that admits a kink. We provide two representations: an asymptotic one in terms of the stress intensity factor and an integral one in terms of the Eshelby tensor. Both the formulas make use of an implicit coefficient, depending on the kink angle and obtained by a minimum problem. Copyright © 2010 John Wiley & Sons, Ltd.     

The contact problem when there are friction forces in the contact area for a three-component cylindrical base

The plane contact problem of elasticity theory on the interaction when there are friction forces in the contact area of an absolutely rigid cylinder (punch) with an internal surface of a cylindrical base, consisting of two circular cylindrical layers rigidly connected to one another and with an elastic space, is considered. The layers and space have different elastic constants. A vertical force and a counterclockwise torque, act on the punch, and the punch – base system is in a state of limiting equilibrium,. An exact integral equation of the first kind with a kernel represented in an explicit analytical form, is obtained for the first time for this problem using analytical calculation programs. The main properties of the kernel of the integral equation are investigated, and it is shown that the numerator and denominator of the kernel symbols can be represented in the form of polynomials in products of the powers of the moduli of the displacement of the layers and the half-space. A solution of the integral equation is constructed by the direct collocation method, which enables the solution of the problem to be obtained for practically any values of the initial parameters. The contact stress distributions, the dimensions of the contact area, the interconnection between the punch displacement and the forces and torques acting on it are calculated as a function of the geometrical and mechanical parameters of the layers and the space. The results of the calculations in special cases are compared with previously known results.     

Impact response of a cracked layer embedded in a composite

The impact response of a laminate composite with a crack or flaw normal to the interface is studied in terms of the intensification of the dynamic stresses around the crack border. Analytically, the laminate is modeled by a single layer with the crack sandwiched between two other layers of dissimilar material. Fourier and Laplace transforms are employed such that the problem reduces to the solution of a system of dual integral equations. Numerical results for the dynamic stress intensity factor are obtained by solving a Fredholm integral equation. The dynamic stress intensity factors are shown to fluctuate as a function of time, reaching a maximum that depends on the elastic properties of the composite and the flaw size.Institute of Fracture and Solid Mechanics, Lehigh University, Bethlehem, Pennysylvania 18015. Published in Mekhanika Polimerov, No. 5, pp. 835–840, September–October, 1978.     

A family of trigonometric extremals in the problem of reorienting a spherically symmetrical body with minimum energy consumption

The problem of the optimal control of the reorientation of an absolutely rigid, spherically symmetric body is investigated. An integral quadratic functional, which characterizes the total energy consumption, is chosen as the criterion of the efficiency of the manoeuvre. The resultant torque of the applied external forces serves as the control. Application of the formalism of the Pontryagin's maximum principle leads to an analysis of a third-order non-linear vector differential equation, whose general solution is still unknown at the present time. It is shown that this equation has a particular solution described by trigonometric functions of time, which can be used to completely reconstruct the explicit solution for the corresponding extremal rotation. An analogy with the free rotation of a certain axisymmetric body is proposed.     

Dynamic stresses in a compound body with circular crack under sliding contact on an interface

We have considered the three-dimensional problem of harmonic loading of a circular crack in an elastic composite consisting of two dissimilar half-spaces under sliding contact on the surface of their bonding. The defect is situated in one of the half-spaces perpendicularly to the interface of materials. Using the representations of solutions in the form of Helmholtz potentials, we have reduced the problem to a boundary integral equation for the function of dynamic defect opening. Based on the numerical solution of this equation, we have obtained the frequency dependences of mode I stress intensity factor near the crack for different relations between the elastic moduli of components of the composite and the depths of crack location with respect to the interface.     

Singular integral equation method for contact problem for rigidly connected punches on elastic half-plane

In the contact problem of a rigid flat-ended punch on an elastic half-plane, the contact stress under punch is studied. The angle distribution for the stress components in the elastic medium under punch is achieved in an explicit form. From obtained singular stress distribution, the punch singular stress factor (abbreviated as PSSF) is defined. A fundamental solution for the multiple flat punch problems on the elastic half-plane is investigated where the punches are disconnected and the forces applied on the punches are arbitrary. The singular integral equation method is suggested to obtain the fundamental solution. Further, the contact problem for rigidly connected punches on an elastic half-plane is considered. The solution for this problem can be considered as a superposition of many particular fundamental solutions. The resultant forces on punches are the undetermined unknowns in the problem, which can be evaluated by the condition of relative descent between punches. Finally, the resultant forces on punches can be determined, and the PSSFs at the corner points can be evaluated. Numerical examples are given.     

A transversely isotropic medium containing a penny-shaped crack subjected to a non-uniform axisymmetric loading via an anchored smooth rigid disk

A smooth rigid circular anchor disk encapsulated by a penny-shaped crack is embedded in and unbounded transversely isotropic medium. The lamellar rigid disk exerts a nonuniform axisymmetric loading to the upper face of the crack. With the aid of an appropriate stress function and Hankel transform, the governing equations are converted to a set of triple integral equations which in turn are reduced to a Fredholm integral equation of the second kind. For some transversely isotropic materials the normalized stiffness of the system falls well outside of the envelope pertinent to isotropic media. It is shown that mode I stress intensity factor is independent of the material properties and solely depends on the ratio of the radius of the rigid disk to that of the crack; moreover, for the cases where this ratio is less than about 0.9 a simple explicit approximate expression for the mode I stress intensity factor is derived. In contrast, the normalized mode II stress intensity factor is independent of the mentioned geometrical parameters but depends on the elastic properties of the material; depending on the material properties, the normalized mode II stress intensity factor can vary between 0 to ∞ for transversely isotropic materials and between 0 to π/4 for isotropic materials.     

功能梯度条硬币型裂纹扭转冲击响应

研究非均匀条中硬币型裂纹的扭转冲击问题．材料的剪切模量假定按特定的梯度变化．采用Laplace 和Hankel 变换将问题化为求解Fredholm积分方程,通过将Bessel函数渐进展开获得裂纹尖端动态应力场．考查非均匀参数和功能梯度条高度对裂尖动态断裂行为的影响．动应力强度因子和能量密度因子的清晰表达式表明,作为裂纹扩展力,对于这里所研究的问题,二者是等价的．动应力强度因子的数值结果显示,增加剪切模量的非均匀参数可以抑制动应力强度因子的幅度,而条形域的高度对动态断裂特性的影响较小．     

两非均匀半平面粘结的非均匀平面的裂纹问题*

本文对两个非均匀半平面粘结的非均匀平面的裂纹问题作了分析,文中假定两种材料的泊松比v相同,但杨氏模量随坐标x按不同形式的指数函数变化.本文使用非均匀平面问题的单裂纹解及富氏变换方法, 使问题归为一个柯西型奇异积分方程,最后对应力强度因子的计算给出了若干数值例子.     

Free convection from a point heat source in a stratified fluid

A non-stationary problem of free convection from a point heat source in a stratified fluid is considered. The system of equations is reduced to a single equation for a special scalar function which determinos the velocity field, and the temperature and salinity distribution. Relations are found connecting the spatial and temporal scales of the phenomenon with the parameters of the medium and the intensity of the heat source. The magnitude of the critical source intensity at which the fluid begins to move in a jet-flow mode is established.The structure of convective flows above the heat sources depends, in the stratified media, essentially on the nature of the stratification /1/ which may be caused by a change in the temperature of the medium /2, 3/ or its salinity /4–7/, and by the form of the heat source. When a temperature gradient exists within the medium, an ascending jet forms above the point source, mushrooming outwards near the horizon of the hydrostatic equilibrium. In the case of a fluid with salinity gradient, the jet is surrounded by a sheet of descending salty fluid, and a regular system of annular convective cells is formed around it /1/.The height of the stationary jet computed in /2, 3/ on the basis of conservative laws agrees with experiment. However, this approach does not enable the temperature and velocity distribution over the whole space to be found and does not enable the problem of determining the flow to be investigated. A stationary solution of the linearized convection equations /8/ does not correspond to detail to the observed flow pattern /1, 5–7/. In this connection the study of the non-linear, non-stationary convection equations is of interest.The purpose of this paper is to construct a non-linear, non-stationary free convection equation above a point heat source, and to analyse the scales of the resulting structure and the critical conditions under which the flow pattern changes.     

Problems of the concentration of elastic stresses near defects in spherically multilayered media

A spherically multilayered medium, whose elastic parameters change abruptly on the spherical surfaces, with defects in the form of cracks or thin rigid inclusions, is considered. The method of solving problems of the stress concentration near such defects is based on the introduction of linear combinations of the displacements and stresses as the fundamental unknowns. This enables the difficulties related to the presence of an arbitrary number of layers to be effectively overcome. The method is described initially for an unbounded elastic medium and defects of spherical form, situated on the surfaces where the elastic parameters change (interphase defects) and a way of extending this to the case of an elastic medium of finite dimensions, defects of other forms and not situated on these surfaces, is indicated. The method is described in detail as it applies to the case of a two-layer medium with an interphase crack when a torsion centre at the origin of coordinates acts on the medium. The problem is reduced to an integral equation, an effective method of solving it is given, and a formula is obtained for the stress intensity factor.     

Integral representation of a solution of the Neumann problem for the Stokes system

The Neumann problem for the Stokes system is studied on a domain in R ³ with Ljapunov bounded boundary. We construct a solution of this problem in the form of appropriate potentials and determine unknown source densities via integral equation systems on the boundary of the domain. The solution is given explicitly in the form of a series.     

Quadratic integrals and the reducibility of the equations of motion of a complex mechanical system in a central field

A mechanical system, consisting of a non-variable rigid body (a carrier) and a subsystem, the configuration and composition of which may vary with time (the motion of its elements with respect to the carrier is specified), is considered. The system moves in a central force field at a distance from its centre which considerably exceeds the dimensions of the system. The effect of the system motion about the centre of mass on the motion of the centre of mass, which is assumed to be known, is ignored (the analogue of the limited problem [1] for a rigid body). The necessary and sufficient conditions for a quadratic integral of the motion around the centre of mass to exist are obtained in the case when there is no dynamic symmetry. It is shown that, for a quadratic integral to exist, it is necessary that the trajectory of the motion of the centre of mass should be on the surface of a certain circular cone, fixed in inertial space, with its vertex at the centre of the force field. If the trajectory does not lie on the generatrix of the cone, only one non-trivial quadratic integral can exist and the initial system, in the presence of this quadratic integral, reduces to autonomous form. For the motion of the centre of mass along the generatrix or the motion of the system around a fixed centre of mass, the necessary and sufficient conditions for a non-trivial quadratic integral to exist are obtained, which are generalizations of the energy integral, the de Brun integral [2] and the integral of the projection of the kinetic moment. When three non-trivial quadratic integrals exist, the condition for reduction to an autonomous system describing the rotation of the rigid body around the centre of mass and integrable in quadratures are indicated [3, 4].     

Smooth boundary integration method in a linear two-dimensional problem of incompressible fluid and solid

This paper presents a new application of a theoretical and computational method of smooth boundary integration which belongs to the methods of boundary integral equations. Smooth integration is not a method of approximation. In its final analytical form, a smooth-kernel integral equation is computerized easily and accurately.

Smooth integration is associated with a “pressure-vorticity” formulation which covers linear problems in elasticity and fluid mechanics. The solution presented herein is essentially the same as that reported in an earlier paper for regular elasticity. The constraint of incompressibility does not cause difficulties in the pressure-vorticity formulation.

The linear fluid mechanics problem formulated and solved in this paper covers Stokes' problem of a slow viscous flow, and has a wider interpretation. The translational inertia forces are incorporated in the linear problem, as in Euler's dynamic theory of inviscid flow. The centrifugal inertia forces are left for the non-linear problem. The linear problem is perceived as a step in solution of the non-linear problems.     

Effect of shearing forces on the stressed state of a half space with crack

Using the method of boundary integral equations, we study the stressed state in the neighborhood of a plane crack perpendicular to the boundary of a half space. The crack surfaces are subjected to the action of shearing forces. The problem is reduced to two-dimensional hypersingular integral equations, and their regular kernels, taking into account interaction between the crack and boundary of the half space, are written in explicit form. The dependences of stress intensity factors on the angular coordinate are presented for different loads of the crack. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 112–120, January–March, 2008.