一维六方准晶双材料中圆孔边共线界面裂纹的反平面问题

研究了一维六方准晶双材料中圆孔边不对称共线界面裂纹的反平面问题。利用Stroh公式和复变函数方法得到了声子场和相位子场耦合作用下的复势函数,给出了裂纹尖端应力强度因子和能量释放率的解析表达式。通过数值算例,讨论了圆孔半径和裂纹长度对应力强度因子的影响,以及耦合系数、声子场应力和相位子场应力对能量释放率的影响。结果表明:当圆孔半径不变时,应力强度因子随右裂纹长度的增大趋向稳定值。当相位子场应力取一定值时,能量释放率达到最小值,说明特定的相位子场应力可以抑制裂纹的扩展。     

Analysis of arbitrarily shaped planar cracks in two-dimensional hexagonal quasicrystals with thermal effects. Part I: Theoretical solutions

An extended displacement discontinuity (EDD) boundary integral equation method is proposed for analysis of arbitrarily shaped planar cracks in two-dimensional (2D) hexagonal quasicrystals (QCs) with thermal effects. The EDDs include the phonon and phason displacement discontinuities and the temperature discontinuity on the crack surface. Green's functions for unit point EDDs in an infinite three-dimensional medium of 2D hexagonal QC are derived using the Hankel transform method. Based on the Green's functions and the superposition theorem, the EDD boundary integral equations for an arbitrarily shaped planar crack in an infinite 2D hexagonal QC body are established. Using the EDD boundary integral equation method, the asymptotic behavior along the crack front is studied and the classical singular index of 1/2 is obtained at the crack edge. The extended stress intensity factors are expressed in terms of the EDDs across crack surfaces. Finally, the energy release rate is obtained using the definitions of the stress intensity factors.     

The interaction between a screw dislocation and a rigid line in a confocal elliptical inhomogeneity

The interaction between a screw dislocation and an elastic elliptical inhomogeneity which contains a confocal rigid line is investigated. The screw dislocation is located inside either the elliptical inhomogeneity or the infinite matrix. By using the complex potential method, explicit series solutions of complex potentials are obtained. The image force acting on the screw dislocation and the stress intensity factor at the tip of the rigid line are derived. As a result, the analysis and discussion show that the influence of the rigid line on the interaction effects between a screw dislocation and an elliptical inhomogeneity is significant. The rigid line enhances the repulsive force exerted on the dislocation produced by the stiff inhomogeneity and abates the attractive force produced by the soft inhomogeneity. For the soft inhomogeneity, there is an unstable equilibrium position when the dislocation is inside the matrix and there is a stable equilibrium position when the dislocation is inside the inhomogeneity. The stress intensity factor contour around the rigid line tip shows that when a dislocation with positive burgers vector is in the upper half-plane, stress intensity factor will be positive; while in the lower half-plane, stress intensity factor will be negative; and in the x-axis, it will be zero. The absolute value of the stress intensity factor will increase when the dislocation approaches the tip of the rigid line. The stress intensity factor at the rigid line tip is enhanced by a harder matrix and abated by a softer matrix.     

On a partially debonded rigid line inclusion penetrating a circular inhomogeneity

We derive closed-form solutions to the mixed boundary value problem of a partially debonded rigid line inclusion penetrating a circular elastic inhomogeneity under antiplane shear deformation. The two tips of the rigid line inclusion are just mutual mirror images with respect to the inhomogeneity/matrix interface, and the upper part of the rigid line inclusion is debonded from the surrounding materials. By using conformal mapping and the method of image, closed-form solutions are derived for three loading cases: (i) the matrix is subjected to remote uniform stresses; (ii) the matrix is subjected to a line force and a screw dislocation; and (iii) the inhomogeneity is subjected to a line force and a screw dislocation. In the mapped ξ-plane, the solutions for all the three loading cases are interpreted in terms of image singularities. For the remote loading case, explicit full-field expressions of all the field variables such as displacement, stress function and stresses are obtained. Also derived is the near tip asymptotic elastic field governed by two generalized stress intensity factors. The generalized stress intensity factors for all the three loading cases are derived.     

Closed-form solutions for two collinear dielectric cracks in a magnetoelectroelastic solid

The problem of two collinear electromagnetically dielectric cracks in a magnetoelectroelastic material is investigated under in-plane electro-magneto-mechanical loadings. The semi-permeable crack-face boundary conditions are adopted to simulate the case of two collinear cracks full of a dielectric interior. Utilizing the Fourier transform technique, the boundary-value problem is reduced to solving singular integral equations with Cauchy kernel, which then are solved explicitly. The intensity factors of stress, electric displacement, magnetic induction, crack opening displacement (COD) and the energy release rates near the inner and outer crack tips are determined in closed forms for two cases of possible far-field electro-magneto-mechanical loadings respectively. Numerical results for a BaTiO₃–CoFe₂O₄ composite are carried out to show the effects of applied mechanical loadings on the crack-face electric displacement and magnetic induction, the stress intensity factor and the COD intensity factor, respectively. The obtained results reveal that when the applied mechanical loading is stress, applied electromagnetic loadings have no influences on the stress intensity factor. When the applied mechanical loadings is train, the applied positive electromagnetic loadings decrease the intensity factors of stress and COD, and the applied negative ones increase the intensity factors of stress and COD. The variations of energy release rates are also given to show the effects of the geometry of two collinear dielectric cracks.     

The periodic problem for a system of coplanar thin ribbons under longitudinal shear

We give the general outline of the construction of systems of singular integral equations for linearly and circularly periodic two-dimensional problems of the theory of cracks and thinwalled inclusions and the solution of these problems using the method of collocations taking account of the limited number of intervals of integration. As an example we study the dependence of the generalized stress intensity factors on identical systems of thin elastic ribbons forming one, two, and three columnsin an infinite isotropic medium using the conditions of longitudinal shear under the influence of a uniform stress field at infinity. In a particular case we obtain results for the corresponding systems of cracks or absolutely rigid inclusions and films. The method makes it possible to study the interaction of rows of closely placed defects. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 91–99.     

Stressed state of an anisotropic body with elliptical holes, elastic inclusions, and cracks

A method is proposed for studying the two-dimensional stressed state of a multiply connected anisotropic body with cavities and elastic and rigid inclusions, as well as planar cracks and rigid laminar inclusions. Generalized complex potentials, conformal mapping, and the method of least squares are used. The problem is reduced to solving a system of linear algebraic equations. Formulas are given for finding the stress intensity factors in the case of cracks and laminar inclusions. For an anisotropic plate with a single elliptical hole or a crack and an elastic (rigid) inclusion, some numerical results are presented from a study of the effect of the rigidity of the inclusion and the closeness of the contours to one another on the distribution of stresses and the stress intensity factor. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 175–187, 1999.     

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.     

A Yoffe-type moving crack in one-dimensional hexagonal piezoelectric quasicrystals

A Yoffe-type moving crack in one-dimensional hexagonal piezoelectric quasicrystals is considered. The Fourier transform technique is used to solve a moving crack problem under the action of antiplane shear and inplane electric field. Full elastic stresses of phonon and phason fields and electric fields are derived for a crack running with constant speed in the periodic plane. Obtained results show that the coupled elastic fields inside piezoelectric quasicrystals depend on the speed of crack propagation, and exhibit the usual square-root singularity at the moving crack tip. Electric field and phason stresses do not have singularity and electric displacement and phonon stresses have the inverse square-root singularity at the crack tip for a permeable crack. The field intensity factors and energy release rates are obtained in closed form. The crack velocity does not affect the field intensity factors, but alters the dynamic energy release rate. Bifurcation angle of a moving crack in a 1D hexagonal piezoelectric quasicrystal is evaluated from the viewpoint of energy balance. Obtained results are helpful to better understanding crack advance in piezoelectric quasicrystals.     

The perturbed stress state of a closed cylindrical shell with longitudinal cracks on its inner surface

Using the Rice-Levy line spring model we find an approximate solution of the problem of elastic balance of a closed infinite circular cylindrical shell weakened by a periodic system of longitudinal interior cracks of identical length. The stress intensity factors are found at the center of a crack for various numbers of cracks with various parameters. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

Strip-saturation model for piezoelectric plane weakened by two collinear cracks with coalesced interior zones

A strip-saturation model is proposed for a transversely isotropic piezoelectric plane weakened by two collinear equal cracks, when developed saturation zones at the interior tips of the cracks get coalesced. The plane is subjected to unidirectional, normal (to the crack length) in-plane tension and electric displacement. The developed saturation zones are arrested by distributing over their rims the normal, cohesive, unidirectional saturation-limit electrical displacement. The solution is obtained using Stroh formulation and complex variable technique. Closed form expressions are derived for crack opening displacement (COD), crack potential drop (COP), field intensity factors, length of saturation zone, energy release rate. Case study carried out for PZT-4 to show the effects of inter-crack distance on the stress intensity factor. The variations of energy release rates are plotted for PZT-4, PZT-5H and BaTiO₃ to study the effects of the geometry of the two cracks.     

Resonance spectrum for one-dimensional layered media

We consider the “weighted” operator P_k=????_x a(x)? _x on the real line with a step-like coefficient which appears when propagation of waves through a finite slab of a periodic medium is studied. The medium is transparent at certain resonant frequencies which are related to the complex resonance spectrum of P_k. If the coefficient is periodic on a finite interval (locally periodic) with k identical cells, then the resonance spectrum of P_k has band structure. In the article, we study a transition to semi-infinite medium by taking the limit k→?∞?. The bands of resonances in the complex lower half plane are localized below the band spectrum of the corresponding periodic problem (k=∞) with k???1 or k resonances in each band. We prove that as k→?∞?, the resonance spectrum converges to the real axis.     

Properties of the solution set of nonlinear evolution inclusions

In this paper we examine nonlinear, nonautonomous evolution inclusions defined on a Gelfand triple of spaces. First we show that the problem with a convex-valued,h*-usc inx orientor fieldF(t, x) has a solution set which is anR _δ-set inC(T, H). Then for the problem with a nonconvex-valuedF(t, x) which ish-Lipschitz inx, we show that the solution set is path-connected inC(T, H). Subsequently we prove a strong invariance result and a continuity result for the solution multifunction. Combining these two results we establish the existence of periodic solutions. Some examples of parabolic partial differential equations with multivalued terms are also included. This work was done while the authors were visiting the Florida Institute of Technology.     

Interactions in an infinite medium of planar cracks with completely rigid planar inclusions

The problem of determining the interactions in an infinite medium of planar cracks with absolutely rigid inclusions leads to a system of integral equations, the regular kernals of which represent the interaction. The system of integral equations is completely determined under boundary conditions for the equilibrium of the inclusions as a rigid body. An approximate solution for the system of integral equations is used. The dependence of the magnitude of the external load on parameters characterizing the distribution in the medium of disc-shaped cracks and inclusions is graphically presented.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 29, pp. 63–68, 1989.     

On small perturbations of the Schrödinger operator with a periodic potential

Consideration is given to small perturbations of a potential, periodic with respect to the variables x_j, j=1, 2, 3, by a function periodic in x₁ and x₂ that decays exponentially as |x₃| → ∞. It is shown that in the neighborhood of energies corresponding to the extrema in the third quasimomentum component of nondegenerate eigenvalues of the Schrödinger operator, with the periodic potential considered in a cell, there exists a unique solution (up to within a numerical factor) to the integral equation describing both the eigenvalues and resonance levels. The asymptotic behavior of the eigenvalues and resonance levels is investigated.     

Non - Classical Mixed Interface Problems for Anisotropic Bodies

The paper deals with the three-dimensional mathematical problems of the elasticity theory of anisotropic piece-wise homogeneous bodies. Non - classical mixed type boundary value problems are studied when on one part (S₁) of the interface the rigid contact conditions (jumps of displacement and stress vectors) are given, while conditions of the non - classical interface Problem H or Problem G are imposed on the remaining part (S₂) of the interface, i. e., in both cases jumps of the normal components of displacement and stress vectors are known on S₂ and, in addition, in the first one (Problem H) the tangent components of the displacement vector and in the second one (Problem G) the tangent components of the stress vector are given from the both sides on S₂. The investigation is carried out by the potential method and the theory of pseudodifferential equations on manifolds with boundary.     

Rational and Semirational Solutions of the Nonlocal Davey–Stewartson Equations

In this paper, the partially party‐time () symmetric nonlocal Davey–Stewartson (DS) equations with respect to x is called x‐nonlocal DS equations, while a fully symmetric nonlocal DSII equation is called nonlocal DSII equation. Three kinds of solutions, namely, breather, rational, and semirational solutions for these nonlocal DS equations are derived by employing the bilinear method. For the x‐nonlocal DS equations, the usual (2 + 1)‐dimensional breathers are periodic in x direction and localized in y direction. Nonsingular rational solutions are lumps, and semirational solutions are composed of lumps, breathers, and periodic line waves. For the nonlocal DSII equation, line breathers are periodic in both x and y directions with parallels in profile, but localized in time. Nonsingular rational solutions are (2 + 1)‐dimensional line rogue waves, which arise from a constant background and disappear into the same constant background, and this process only lasts for a short period of time. Semirational solutions describe interactions of line rogue waves and periodic line waves.     

An Averaging Method for Singularly Perturbed Systems of Semilinear Differential Inclusions with C0-Semigroups

We consider a system of two semilinear differential inclusions with infinitesimal generators of C ₀-semigroups. The nonlinear terms are of high frequency with respect to time and periodic with a specified period. Moreover, they are condensing in the state variables (x,y) with respect to a suitable measure of noncompactness. The goal of the paper is to give sufficient conditions to guarantee, for >0 sufficiently small, the existence of periodic solutions and to study their behaviour as 0. The main tool to achieve this is the topological degree theory for uppersemicontinuous, condensing vector fields.     

On the structure of cluster point sets of iteratively generated sequences

LetX be a closed subset of a topological spaceF; leta(·) be a continuous map fromX intoX; let {x _i} be a sequence generated iteratively bya(·) fromx ₀ inX, i.e.,x _i+1 =a(x _i),i=0, 1, 2, ...; and letQ(x ₀) be the cluster point set of {x _i}. In this paper, we prove that, if there exists a pointz inQ(x ₀) such that (i)z is isolated with respect toQ(x ₀), (ii)z is a periodic point ofa(·) of periodp, and (iii)z possesses a sequentially compact neighborhood, then (iv)Q(x ₀) containsp points, (v) the sequence {x _i} is contained in a sequentially compact set, and (vi) every point inQ(x ₀) possesses properties (i) and (ii). The application of the preceding results to the caseF=E ⁿ leads to the following: (vii) ifQ(x ₀) contains one and only one point, then {x _i} converges; (viii) ifQ(x ₀) contains a finite number of points, then {x _i} is bounded; and (ix) ifQ(x ₀) containsp points, then every point inQ(x ₀) is a periodic point ofa(·) of periodp.     

Periodicity of the Spectrum of a Finite Union of Intervals

A set Ω, of Lebesgue measure 1, in the real line is called spectral if there is a set Λ of real numbers such that the exponential functions e _λ (x)=exp (2πiλx), λ∈Λ, form a complete orthonormal system on L ²(Ω). Such a set Λ is called a spectrum of Ω. In this note we present a simplified proof of the fact that any spectrum Λ of a set Ω which is finite union of intervals must be periodic. The original proof is due to Bose and Madan.