An Effective Model of a Fractured Medium

A homogeneous isotropic elastic medium intersected by three systems of fractures on which the jumps of stresses are proportional to displacements is considered. An effective model of this medium is described by equations differing from the respective equations of the elastic medium by additional terms. On the basis of the equations of the effective model, the wave field excited by a point source is established. An investigation of the integral representation of the wave field shows that the velocities of the longitudinal and transversal waves and of the Rayleigh wave are functions of the frequency and the wave numbers. Formulas for the phase and group velocities of these waves are derived. Bibliography: 3 titles.     

The scattering of an harmonic elastic wave by a volume inclusion with a thin interlayer

The boundary element method is used to investigate the propagation of harmonic elastic waves in an infinite matrix with a volume inclusion with a thin interlayer between the inclusion and the matrix. A boundary-integral formulation of the problem is based on a consideration of a two-phase medium, consisting of the matrix and the inclusion, on the contact surface of which conditions of proportional dependence between the forces and jumps in the displacements, which model the interlayer, are satisfied. These conditions are taken into account implicitly in the boundary integral equations obtained, which are subsequently regularized and discretized on the grid of boundary elements introduced. The numerical results obtained demonstrate the effect of the interlayer on the dynamic contact stresses for a spherical inclusion in the field of a plane longitudinal wave.     

Dynamic contact between a spherical inclusion and a matrix upon incidence of an elastic wave

Using the boundary element method, we have studied the dynamic displacements and stresses in an infinite elastic matrix with a spherical elastic inclusion, caused by the propagation of an elastic wave. The original problem has been reduced to a system of boundary integral equations for the contact displacements and tractions on the interface between the inclusion and matrix. Based on the numerical solution of these equations, we have analyzed the influence of the direction of wave propagation and frequency on the important physical parameters, depending on the elastic characteristics of composite constituents.     

压电压磁复合材料中一对平行裂纹对弹性波的散射

利用Schmidt方法对压电压磁复合材料中一对平行对称裂纹对反平面简谐波的散射问题进行了分析,借助富里叶变换得到了以裂纹面上的间断位移为未知变量的对偶积分方程．在求解对偶积分方程的过程中,裂纹面上的间断位移被展开成雅可比多项式的形式,最终获得了应力强度因子、电位移强度因子、磁通量强度因子三者之间的关系．结果表明,压电压磁复合材料中平行裂纹动态反平面断裂问题的应力奇异性与一般弹性材料中的动态反平面断裂问题的应力奇异性相同,同时讨论了裂纹间的屏蔽效应．     

Dual boundary element method for problems of the theory of thin inclusions

We have proposed to apply the dual boundary element method in problems of the theory of thin inclusions. The contact conditions on the boundary of a thin inclusion are considered as jumps of displacements and stresses in the body on the median surface of this defect. Thus, the relations between the unknown discontinuities and average values of the displacements and stresses are a model of inclusions. For rectilinear boundary elements, we have constructed models of inclusions, taking into account the tension, shear, and bending of a thin inclusion. Examples for rectilinear and curved inclusions have been considered. Comparison of the results obtained by the proposed technique with data based on the direct approach shows the efficiency of the proposed method.     

基于降阶解法的三维分层地基状态空间解

从以位移形式表达的三维弹性力学控制方程出发,经双重Fourier变换,并运用基于Cayley-Hamilton定理的降阶解法,推导出位移变量及其1阶导数的解,再利用物理方程求得单层地基的传递矩阵;结合边界条件和层间连续条件,进一步得到多层地基的状态空间解;编制相应程序进行数值分析,对多层地基中有软弱下卧层和坚硬下卧层时的沉降情况进行了比较和讨论．     

The diffraction of plane elastic waves by a delaminated rigid inclusion when there is smooth contact in the delamination region

A solution of the problem of the diffraction of harmonic elastic waves by a thin rigid strip-like delaminated inclusion in an unbounded elastic medium, in which the conditions for plane deformation are satisfied, is proposed. We mean by a delaminated inclusion an inclusion, one side of which is completely bonded to the elastic medium, while the second does not interact in any way with it, or this interaction is partial. It is assumed that the conditions for smooth contact are satisfied in the delamination region. The method of solution is based on the use of previously constructed discontinuous solutions of the equations describing the vibrations of an elastic medium under plane deformation conditions. The problem therefore reduces to solving a system of three singular integral equations in the unknown stress and strain jumps at the inclusion. An approximate solution of the latter enabled formulae to be obtained that are convenient for numerical realization when investigating the stressed state in the region of the inclusion and its displacements when acted upon by incident waves.     

Solution for a problem of linear plane elasticity with mixed boundary conditions on an ellipse by the method of boundary integrals

A numerical boundary integral scheme is proposed for the solution of the system of field equations of plane, linear elasticity in stresses for homogeneous, isotropic media in the domain bounded by an ellipse under mixed boundary conditions. The stresses are prescribed on one half of the ellipse, while the displacements are given on the other half. The method relies on previous analytical work within the Boundary Integral Method [1], [2].The considered problem with mixed boundary conditions is replaced by two subproblems with homogeneous boundary conditions, one of each type, having a common solution. The equations are reduced to a system of boundary integral equations, which is then discretized in the usual way and the problem at this stage is reduced to the solution of a rectangular linear system of algebraic equations. The unknowns in this system of equations are the boundary values of four harmonic functions which define the full elastic solution inside the domain, and the unknown boundary values of stresses or displacements on proper parts of the boundary.On the basis of the obtained results, it is inferred that the tangential stress component on the fixed part of the boundary has a singularity at each of the two separation points, thought to be of logarithmic type. A tentative form for the singular solution is proposed to calculate the full solution in bulk directly from the given boundary conditions using the well-known Boundary Collocation Method. It is shown that this addition substantially decreases the error in satisfying the boundary conditions on some interval not containing the singular points.The obtained results are discussed and boundary curves for unknown functions are provided, as well as three-dimensional plots for quantities of practical interest. The efficiency of the used numerical schemes is discussed, in what concerns the number of boundary nodes needed to calculate the approximate solution.     

Plane strain deformation of an initially unstressed elastic medium

Selim and Ahmed [1] used the eigenvalue approach by assuming distinct eigenvalues to calculate the elastic deformation due to an inclined load at any point as a result of an inclined line load of initially stressed orthotropic elastic medium. They studied the plane strain problem and obtained the corresponding results for an unstressed orthotropic medium as a particular case. In the present paper, it is shown that all the eigenvalues do not remain distinct, but become repeated when the elastic medium is free from the initial compressive stresses. Further, the displacements and stresses for an unstressed elastic medium have been independently obtained. The variation of the displacements and stresses due to normal and tangential line load are also shown graphically.     

Exact solutions of some mixed problems of uncoupled thermoelasticity for a truncated hollow circular cone with a groove along the generatrix

An elastic body of finite dimensions in the form of a truncated hollow circular cone with a groove along the generatrix is considered. The uncoupled problem of thermoelasticity is formulated for this body for different types of boundary conditions on all the surfaces. These are the conditions for specifying the displacements or sliding clamping on surfaces with fixed angular coordinates and the conditions for specifying the stresses on surfaces with a fixed radial coordinate (shear stresses are assumed to be zero). It is assumed that the temperature is a specified function of all the spherical coordinates. Some auxiliary functions, related to the displacements, are introduced first, and equations for these functions are then derived using Lamé's equations. A finite integral Fourier transformation with respect to one of the angular variables is then employed. After this, by solving certain Sturm-Liouville problems, a new integral transformation is constructed and is applied to the equations with respect to the other angular variable. As a result a one-dimensional system of differential equations is obtained, to solve which an integral Mellin transformation is employed in a special way. Finally, exact solutions of some problems of thermoelasticity are constructed in series for this body.     

Modeling nonlinear deformation of a plate with an elliptic inclusion by John’s harmonic material

The exact analytical solution of a nonlinear plane-strain problem has been obtained for a plate with an elastic elliptic inclusion with constant stresses given at infinity. The mechanical properties of the plate and inclusion are described with the model of John’s harmonic material. In this model, stresses and displacements are expressed in terms of two analytical functions of a complex variable that are determined from nonlinear boundary-value problems. Assuming the tensor of nominal stresses to be constant inside the inclusion has made it possible to reduce the problem to solving two simpler problems for a plate with an elliptic hole. The validity of the adopted hypothesis has been justified by the fact that the derived solution exactly satisfies all the equations and boundary conditions of the problem. The existence of critical plate-compression loads that lead to the loss of stability of the material has been established. Two special nonlinear problems for a plate with a free elliptic hole and a plate with a rigid inclusion have been solved.     

Plane problem of interaction between an elastic longitudinal wave and a cylindrical shell with axial cut

We have proposed a procedure for investigating the stressed state of a thin-walled round cylindrical shell with a cut along its generatrix, which is placed into an elastic space. A longitudinal elastic wave is incident on the shell. The procedure is based on the use of the Rayleigh expansion in partial waves. We have obtained relations for determining the displacement jumps and angle of rotation at the lips of the shell cut, the radial component of the displacement vector and the normal force in the cylindrical shell, as well as the stresses and displacements in the elastic medium. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 131–137, January–March, 2008.     

Non-local theory solution of two collinear mode-I cracks in piezoelectric materials

In this paper, the basic solution of two collinear cracks in a piezoelectric material plane subjected to a uniform tension loading is investigated by means of the non-local theory. Through the Fourier transform, the problem is solved with the help of two pairs of integral equations, in which the unknown variables are the jumps of displacements across the crack surfaces. To solve the integral equations, the jumps of displacements across the crack surfaces are directly expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the interaction of two cracks, the materials constants and the lattice parameter on the stress field and the electric displacement field near crack tips. Unlike the classical elasticity solution, it is found that no stress and electric displacement singularities are present at crack tips. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to using the maximum stress as a fracture criterion in piezoelectric materials.     

Theory of a reinforced layered medium with random initial irregularities

The author examines an elastic medium reinforced with slightly distorted elastic layers. The basic equations are obtained by the method proposed in [1,2]. It is assumed that the functions describing the initial distortions of the reinforcing layers form a random field. With the help of the method of canonical expansions [3], expressions are derived for the statistical characteristics of the stresses, strains and displacements in the reinforced medium. The theory is used to account for the known experimental fact of the reduction in the moduli of elasticity of layered glass-reinforced plastics as compared with the values calculated for an ideal reinforced medium. In particular, it is shown that this reduction may be considerable even when the initial irregularities are relatively small.Mekhanika Polimerov, Vol. 2, No. 1, pp. 11–19, 1966     

Joint deformation of a circular inclusion and a matrix

An elastic infinite plane containing a circular inclusion with given jumps of tractions and displacements along the interface and nonzero conditions at infinity is considered. Explicit expressions are derived for the Goursat-Kolosov complex potentials of this problem. The solution constructed can be used to examine various circular interfacial defects, including interfacial cracks and rigid parts of the interface. The problem under consideration is fundamental for the superposition method, which solves many problems in which a circular region is an element of a polyphase elastic medium. In such cases, the well-posedness of the problem, which depends on the interrelation between the jumps of tractions and displacements, follows from the very superposition method. The application techniques of this method are demonstrated for singular problems on the action of a point force and an edge dislocation located inside an inclusion or in the matrix. Computational results for the tractions arising at the interface under the action of a point force concentrated in the inclusion are given.     

2D numerical simulation of submerged hydraulic jumps

Hydraulic jumps are usually used to dissipate energy in hydraulic engineering. In this paper, the turbulent submerged hydraulic jumps are simulated by solving the unsteady Reynolds averaged Navier–Stokes equations along with the continuity equation and the standard k–? equations for turbulence modeling. The Lagrangian moving grid method is employed for the simulation of the free surface. In the developed model, kinematic free-surface boundary condition is solved simultaneously with the momentum and continuity equations, so that the water elevation can be obtained along with velocity and pressure fields as part of the solution. Computational results are presented for Froude numbers ranging from 3.2 to 8.2 and submergence factors ranging from 0.24 to 0.85. Comparisons with experimental measurements show that numerical model can simulate the velocity field, variation of free surface, maximum velocity, Reynolds shear and normal stresses at various stations with reasonable accuracy.     

Radiation problem of normal stress loading of bore surface

The radiation of elastic waves at normal harmonic stress loading of a ring strip of the cylindrical bore-surface in the infinite elastic medium is considered. The wave field is represented by contour integrals. By using the saddle-point method stress and displacement of asymptotic expansions in the far field were obtained. It is shown that in the far field P-waves bring radial displacements and stresses, and perpendicular to these, displacements and stresses are caused by S-waves. In the paper orientation diagrams of radial and circular displacements and frequency dependencies of relative distribution of radiation power source on types of waves are presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

General Solution for the Two-Dimensional System of Static Lame’s Equations with an Asymmetric Elasticity Matrix

We study a two-dimensional system of equations of linear elasticity theory in the case when the symmetric stress and strain tensors are related by an asymmetric matrix of elasticity moduli or elastic compliances. The linear relation between stresses and strains is written in an invariant form which contains three positive eigenmodules in the two-dimensional case. Using a special eigenbasis in the strain space, it is possible to write the constitutive equations with a symmetric matrix, i.e., in the same way as in the case of hyperelasticity. We obtain a representation of the general solution of two-dimensional equations in displacements as a linear combination of the first derivatives of two functions which satisfy two independent harmonic equations. The obtained representation directly implies a generalization of the Kolosov–Muskhelishvili representation of displacements and stresses in terms of two analytic functions of complex variable. We consider all admissible values of elastic parameters, including the case when the system of differential equations may become singular. We provide an example of solving the problem for a plane with a circular hole loaded by constant stresses.     

Nonshallow spherical shells

In this paper, spherical bodies of shell type are discussed, where the displacement vector is independent of the thickness coordinate. Their internal geometry is alterable towards the thickness (nonshallow spherical shells). A planar deformation model for spherical bodies of shell type is obtained. General representations of the system of equilibrium equations are expressed with the help of three holomorphic functions for the spherical shells. The components of the stresses and displacements and the boundary conditions for the components of the stresses and displacements are also expressed with the help of three holomorphic functions. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 51, Differential Equations and Their Applications, 2008.     

压电压磁复合材料中界面裂纹对弹性波的散射

利用Schmidt方法分析了压电压磁复合材料中可导通界面裂纹对反平面简谐波的散射问题．经过富里叶变换得到了以裂纹面上的间断位移为未知变量的对偶积分方程A·D2在求解对偶积分方程的过程中,裂纹面上的间断位移被展开成雅可比多项式的形式．数值模拟分析了裂纹长度、波速和入射波频率对应力强度因子、电位移强度因子、磁通量强度因子的影响A·D2从结果中可以看出,压电压磁复合材料中可导通界面裂纹的反平面问题的应力奇异性形式与一般弹性材料中的反平面问题应力奇异性形式相同．