Stress singularities for three-dimensional corners using the boundary integral equation method

The boundary integral equation method is developed to study three-dimensional asymptotic singular stress fields at vertices of a pyramidal notch or inclusion in an isotropic elastic space. Two-dimensional boundary integral equations are used for the infinite body with pyramidal notches and inclusions when either stresses or displacements are specified on its surface. Applying the Mellin integral transformation reduces the problem to one-dimensional singular integral equations over a closed, piece-wise smooth line. Using quadrature formulas for regular and singular integrals with Hilbert and logarithmic kernels, these integral equations are reduced to a homogeneous system of linear algebraic equations. Setting its determinant to zero provides a characteristic equation for the determination of the stress singularity power. Numerical results are obtained and compared against known eigenvalues from the literature for an infinite region with a conical notch or inclusion, for a Fichera vertex, and for a half-space with a wedge-shaped notch or inclusion.     

Aleksandrov–Bakelman–Pucci Type Estimates for Integro-Differential Equations

In this work we provide an Aleksandrov–Bakelman–Pucci type estimate for a certain class of fully nonlinear elliptic integro-differential equations, the proof of which relies on an appropriate generalization of the convex envelope to a nonlocal, fractional-order setting and on the use of Riesz potentials to interpret second derivatives as fractional order operators. This result applies to a family of equations involving some nondegenerate kernels and, as a consequence, provides some new regularity results for previously untreated equations. Furthermore, this result also gives a new comparison theorem for viscosity solutions of such equations which depends only on the L ^∞ and L ⁿ norms of the right-hand side, in contrast to previous comparison results which utilize the continuity of the right-hand side for their conclusions. These results appear to be new, even for the linear case of the relevant equations.     

Solution of three-dimensional problems of the dynamic theory of elasticity for bodies with cracks using hypersingular integrals

Expressions are derived for the singular kernels of a system of integral equations that describe a three-dimensional dynamic problem for a cracked elastic body under harmonic loading. The order of singularity of the kernels of the integral equations is analyzed. Relations that allow reducing hypersingular surface integrals to ordinary contour ones are obtained. Test calculations are conducted. Translated from Prikladnaya Mekhanika, Vol. 36, No. 1, pp. 88–94, January, 2000.     

Numerical solution of singular integral equations in stress concentration problems under longitudinal shear loading

Summary The paper deals with numerical solutions of singular integral equations in stress concentration problems for longitudinal shear loading. The body force method is used to formulate the problem as a system of singular integral equations with Cauchy-type singularities, where unknown functions are densities of body forces distributed in the longitudinal direction of an infinite body. First, four kinds of fundamental density functions are introduced to satisfy completely the boundary conditions for an elliptical boundary in the range 0≤φ _k ≤2π. To explain the idea of the fundamental densities, four kinds of equivalent auxiliary body force densities are defined in the range 0≤φ _k ≤π/2, and necessary conditions that the densities must satisfy are described. Then, four kinds of fundamental density functions are explained as sample functions to satisfy the necessary conditions. Next, the unknown functions of the body force densities are approximated by a linear combination of the fundamental density functions and weight functions, which are unknown. Calculations are carried out for several arrangements of elliptical holes. It is found that the present method yields rapidly converging numerical results. The body force densities and stress distributions along the boundaries are shown in figures to demonstrate the accuracy of the present solutions. Received 26 May 1998; accepted for publication 27 November 1998     

An internal crack problem for an infinite elastic layer

The elastostatic plane problem of an infinite elastic layer with an internal crack is considered. The elastic layer is subjected to two different loadings, (a) the elastic layer is loaded by a symmetric transverse pair of compressive concentrated forces P/2, (b) it is loaded by a symmetric transverse pair of tensile concentrated forces P/2. The crack is opened by an uniform internal pressure p ₀ along its surface and located halfway between and parallel to the surfaces of the elastic layer. It is assumed that the effect of the gravity force is neglected. Using an appropriate integral transform technique, the mixed boundary value problem is reduced to a singular integral equation. The singular integral equation is solved numerically by making use of an appropriate Gauss–Chebyshev integration formula and the stress-intensity factors and the crack opening displacements are determined according to two different loading cases for various dimensionless quantities.     

Applications of the dual integral formulation in conjunction with fast multipole method to the oblique incident wave problem

In this paper, the dual integral formulation is derived for the modified Helmholtz equation in the propagation of oblique incident wave passing a thin barrier (zero thickness) by employing the concept of fast multipole method (FMM) to accelerate the construction of an influence matrix. By adopting the addition theorem, the four kernels in the dual formulation are expanded into degenerate kernels that separate the field point and the source point. The source point matrices decomposed in the four influence matrices are similar to each other or only to some combinations. There are many zeros or the same influence coefficients in the field point matrices decomposed in the four influence matrices, which can avoid calculating the same terms repeatedly. The separable technique reduces the number of floating‐point operations from O((N)²) to O(N log^a(N)), where N is the number of elements and a is a small constant independent of N. Finally, the FMM is shown to reduce the CPU time and memory requirement, thus enabling us to apply boundary element method (BEM) to solve water scattering problems efficiently. Two‐moment FMM formulation was found to be sufficient for convergence in the singular equation. The results are compared well with those of conventional BEM and analytical solutions and show the accuracy and efficiency of the FMM. Copyright © 2008 John Wiley & Sons, Ltd.     

Griffith crack moving in a piezoelectric strip

Summary  The dynamic problem of an impermeable crack of constant length 2a propagating along a piezoelectric ceramic strip is considered under the action of uniform anti-plane shear stress and uniform electric field. The integral transform technique is employed to reduce the mixed-boundary-value problem to a singular integral equation. For the case of a crack moving in the mid-plane, explicit analytic expressions for the electroelastic field and the field intensity factors are obtained, while for an eccentric crack moving along a piezoelectric strip, numerical results are determined via the Lobatto–Chebyshev collocation method for solving a resulting singular integral equation. The results reveal that the electric-displacement intensity factor is independent of the crack velocity, while other field intensity factors depend on the crack velocity when referred to the moving coordinate system. If the crack velocity vanishes, the present results reduce to those for a stationary crack in a piezoelectric strip. In contrast to the results for a stationary crack, applied stress gives rise to a singular electric field and applied electric field results in a singular stress for a moving crack in a piezoelectric strip. Received 14 August 2001; accepted for publication 24 September 2002 The author is indebted to the AAM Reviewers for their helpful suggestions for improving this paper. The work was supported by the National Natural Science Foundation of China under Grant 70272043.     

Anti-plane shear of kinked interface crack in bonded dissimilar anisotropic solid

Complex potentials are derived to describe the anti-plane singular shear stress fields around a kinked crack, the main portion of which is embedded along the interface of two dissimilar anisotropic elastic media. This is accomplished by formulating the problem as singular integral equations with generalized Cauchy kernels. The shear stress singularity at the kink differs from the familiar inverse square root of the local distance; it is found to influence the magnitude of the Mode III crack tip stress intensity factor, K₃. Numerical results of K₃ are obtained and displayed in graphical forms for different degree of material anisotropy and crack dimensions.     

A Further Note on the Onset of Convection in a Layer of a Porous Medium Saturated by a Non-Newtonian Fluid of Power-Law Type

The singular behavior of the Horton–Rogers–Lapwood problem when a Newtonian fluid is replaced by a standard power-law fluid is investigated further. Using weakly nonlinear stability theory, an estimate is made of the amplitude of convection at which the convection is initiated (in the case of a fluid with index n > 1) or levels off (in the case of a fluid with n < 1).     

The integral representation of fractionally exponential functions and their application to dynamic problems of linear visco-elasticity

Fractionally exponential functions are written in the integral form and distribution functions with an Abelian singularity are obtained for the corresponding relaxation and retardation spectra. A principle is stated, defining the dynamic problems for which weakly singular functions can be used as the kernels of the integral operators. A one-dimensional sound wave traveling in a semiinfinite visco-elastic medium is considered. The generalized exponential functions of fractional order, proposed by Yu. N. Rabotnov [1, 2] as the kernels of Boltzmann-Volterra integral relations, have found wide applications in theory of linear visco-elasticity. This is explained partly by the great mathematical flexibility of the F-operators when applying the Volterra principle to the solution of elastically hereditary problems and partly by the fact that almost all weakly singular kernels possessing an Abelian singularity are connected in some way or other with the F-functions. For example, the resolvent of the elementary weakly singular Abelian kernel is an F-function. The product of an exponential function with an Abelian kernel represents a particular case of the product of two F-functions with different fractional parameters, while the resolvent of such a kernel is the product of an exponential function with an F-function [3, 4]. Since the e-functions are defined by slowly convergent series, their various asymptotic forms [2, 5–8] are commonly used in practical calculations. The theory of F-functions can be developed further in the context of their integral representations, which enables a more exact physical interpretation to be given to their parameters and on occasion simplifies computational operations.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 1, pp. 103–110, January–February, 1970.In conclusion, the author thanks Yu. N. Rabotnov for discussion.     

The equations of an inverted pendulum with an arbitrary number of links and an asymmetric follower force

This paper formulates a problem for and gives the differential equations of plane-parallel motion of an inverted n-link pendulum subject to an asymmetric follower force applied at the upper end via a spring. The physical nonlinearities of springs are taken into account. The possible mechanisms of energy dissipation are described __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 106–114, May 2007.     

On the Principal Boundary Value Problem in the Two-Dimensional Anisotropic Theory of Elasticity

By definition, the principal problem of the two-dimensional theory of elasticity consists in solving the equation for the Airy’s stress function in a region with its first order derivatives assigned at a boundary. In this paper, an indirect formulation of this problem based on integral equations with weakly singular kernels is proposed. In a bounded region with a Lyapunov boundary it is reduced to the solution of weakly singular integral equations. Differential properties of its solution are investigated.     

From Boltzmann’s Equation to the Incompressible Navier–Stokes–Fourier System with Long-Range Interactions

We establish a rigorous demonstration of the hydrodynamic convergence of the Boltzmann equation towards a Navier–Stokes–Fourier system under the presence of long-range interactions. This convergence is obtained by letting the Knudsen number tend to zero and has been known to hold, at least formally, for decades. It is only more recently that a fully rigorous mathematical derivation of this hydrodynamic limit was discovered. However, these results failed to encompass almost all physically relevant collision kernels due to a cutoff assumption, which requires that the cross sections be integrable. Indeed, as soon as long-range intermolecular forces are present, non-integrable collision kernels have to be considered because of the enormous number of grazing collisions in the gas. In this long-range setting, the Boltzmann operator becomes a singular integral operator and the known rigorous proofs of hydrodynamic convergence simply do not carry over to that case. In fact, the DiPerna–Lions renormalized solutions do not even make sense in this situation and the relevant global solutions to the Boltzmann equation are the so-called renormalized solutions with a defect measure developed by Alexandre and Villani. Our work overcomes the new mathematical difficulties coming from the consideration of long-range interactions by proving the hydrodynamic convergence of the Alexandre–Villani solutions towards the Leray solutions.     

Boundary integral equations for 2D elasticity and its application in discrete dislocation dynamics simulation in finite body: 1. General theory

A unified approach, originating from Cauchy integral theorem, is presented to derive boundary integral equations for two dimensional elasticity problems. Several sets of boundary integral equations are derived and their relations are revealed. Explicit expressions for materials with different symmetry planes are listed. Special attention is given to the formulation that is based on the tractions and the tangential derivatives of displacements along solid boundary, since its integral kernels have the weakest singularities. The formulation is further extended to include singular points, such as dislocations and line forces, in a finite body, so that the singular stress field can be directly obtained from solving the integral equations on the external boundary, without involving the linear superposition technique that was often used in the literature. Its application in simulating discrete dislocation motion in a finite solid body is discussed.     

Eigenvalues of Self-Similar Solutions of the Dafermos Regularization of a System of Conservation Laws via Geometric Singular Perturbation Theory

The Dafermos regularization of a system of n conservation laws in one space dimension admits smooth self-similar solutions of the form u=u(X/T). In particular, there are such solutions near a Riemann solution consisting of n possibly large Lax shocks. In Lin and Schecter (2004, SIAM. J. Math. Anal. 35, 884–921), eigenvalues and eigenfunctions of the linearized Dafermos operator at such a solution were studied using asymptotic expansions. Here we show that the asymptotic expansions correspond to true eigenvalue–eigenfunction pairs. The proofs use geometric singular perturbation theory, in particular an extension of the Exchange Lemma.     

Stochastic functional expansion in elasticity of heterogeneous solids

The Volterra-Wiener functional expansion is employed to the analysis of statistic properties for random heterogeneous solids. For simplicity, the technique is displayed on an elastic suspension of spheres. The basis function in the expansion is chosen as that corresponding to the so-called “perfect disorder” of spheres (PDS), recently introduced by the authors. An infinite hierarchy of equations for the kernels in the expansion is derived whose truncating after the nth equation is shown to yield results for the averaged statistical characteristics which are valid to order cⁿ_f, where c_f is the volume fraction of the spheres. The kernels for the first and the second approximations, n = 1, 2, are found and related to the displacement fields in an infinite elastic body containing, respectively, one and two spherical inhomogeneities. Within the frame of the so-called singular approximation the overall tensor of elastic moduli for a suspension of perfectly disordered spheres is shown to coincide to the order c²_f with a formula, earlier obtained by means of the method of the effective field.     

Method for determining power-type complex singularities in solutions of singular integral equations with generalized kernels and complex conjugate unknowns

We develop a method for determining power-type complex singularities of solutions for a class of one-dimensional singular integral equations with generalized kernels and complex conjugate unknown functions. By analyzing the characteristic part of a singular integral equation, we reduce the problem of determining the solution singularity exponents at the ends of the integration interval to two independent transcendental equations for these exponents. We show that the distribution of admissible singularity exponents is of continuous character. We present numerical results for a two-dimensional elasticity problem whose mathematical statement leads to a singular integral equation of the class under study. We also reveal the drawbacks of one classical approach to the determination of stress field singularities.     

Bending integral expressions of a cylinder with cracks

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Some bounded results of θ(t)-type singular integral operators

Let B be a Banach space in UMD with an unconditional basis. The boundedness of the θ(t)-type singular integral operators in L _B ^p (R ⁿ), (1≤p<+∞) and H _B ¹ (R ⁿ) spaces are discussed. Foundation item: the Education Commission of Shandong Province (J98P51) Biography: Zhao Kai (1960-)     

Interaction of a die with a layered elastic foundation

Plane and axisymmetric contact problems for a three-layer elastic half-space are considered. The plane problem is reduced to a singular integral equation of the first kind whose approximate solution is obtained by a modified Multhopp-Kalandiya method of collocation. The axisymmetric problem is reduced to an integral Fredholm equation of the second kind whose approximate solution is obtained by a specially developed method of collocation over the nodes of the Legendre polynomial. An axisymmetric contact problem for an transversely isotropic layer completely adherent to an elastic isotropic half-space is also considered. Examples of calculating the characteristic integral quantities are given. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 165–175, May–June, 2006.