The stressed state of an elastic space with a rectangular crack

By using the method of mechanical quadratures we reduce the resolvent integral equation for the problem of an infinite elastic space with a rectangular crack to a system of linear algebraic equations. We give the results of numerical experiments in varying the stress intensity factor on one side of the crack in the case of tension in the direction perpendicular to the plane of the crack. One figure. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 24–28.     

Shape sensitivity of a plane crack front

The 3D‐elasticity model of a solid with a plane crack under the stress‐free boundary conditions at the crack is considered. We investigate variations of a solution and of energy functionals with respect to perturbations of the crack front in the plane. The corresponding expansions at least up to the second‐order terms are obtained. The strong derivatives of the solution are constructed as an iterative solution of the same elasticity problem with specified right‐hand sides. Using the expansion of the potential and surface energy, we consider an approximate quadratic form for local shape optimization of the crack front defined by the Griffith criterion. To specify its properties, a procedure of discrete optimization is proposed, which reduces to a matrix variational inequality. At least for a small load we prove its solvability and find a quasi‐static model of the crack growth depending on the loading parameter. Copyright © 2003 John Wiley & Sons, Ltd.     

The method of integral equation for scattering problem with a mixed crack

We consider the scattering of an electromagnetic time‐harmonic plane wave by an infinite cylinder having a mixed open crack (or arc) in R² as the cross section. The crack is made up of two parts, and one of the two parts is (possibly) coated by a material with surface impedance λ. We transform the scattering problem into a system of boundary integral equations by adopting a potential approach, and establish the existence and uniqueness of a weak solution to the system by the Fredholm theory. Copyright © 2011 John Wiley & Sons, Ltd.     

Isoperimetric estimates of the solutions of a class of pseudodifferential equations and their application to crack problems

Isoperimetric estimates are obtained of solutions of boundary value problems for a class of pseudodifferential equations. This class of equations includes the equation of problems on plane normal discontinuity cracks located in a homogeneous linearly elastic space and an inhomogeneous space whose Young's modulus has a power-law dependence on the distance to the plane of the crack. As it applies to crack problems, the established inequalities yield, in particular, isoperimetric estimates of the maximum opening of the crack and its volume under arbitrary loads.     

CR Structures on the 3‐Sphere and the Blow‐Up of the Projective Plane

Motivated by a problem of characterizing CR‐structures on the 3‐sphere, we give a geometric construction of formal deformations of a complex surface, which is the complement of a ball in the projective plane. They are described by cohomology groups of the blow‐up X of the projective plane. Moreover it will be shown that the space of these formal deformations is an infinite dimensional space with a natural stratification by finite dimensional subspaces. This stratification re ects algebro‐geometric properties of X. It is expected that our construction will clarify the complex geometric nature of the space of non‐embeddable CR‐structures on the 3‐sphere.     

Driving force acting on a running crack

The asymptotic analysis of the dynamic crack problem for the anti‐plane shear mode is provided. The field near the crack tip is studied in detail for a nonlinear elastic incompressible material whose stored energy behaves asymptotically as a power of the first invariant of the strain tensor at large strains. It is shown that the hardening parameter characterizes fully the singularity degree of the near‐crack‐tip field. Based on the latter knowledge the driving force acting on the crack tip is calculated. Possible scenarios of the crack propagation are discussed.     

Numerical simulation of the non‐linear crack problem with non‐penetration

Here the numerical simulation of some plane Lamé problem with a rectilinear crack under non‐penetration condition is presented. The corresponding solids are assumed to be isotropic and homogeneous as well as bonded. The non‐linear crack problem is formulated as a variational inequality. We use penalty iteration and the finite‐element method to calculate numerically its approximate solution. Applying analytic formulas obtained from shape sensitivity analysis, we calculate then energetic and stress characteristics of the solution, and describe the quasistatic propagation of the crack under linear loading. The results are presented in comparison with the classical, linear crack problem, when interpenetration between the crack faces may occur. Copyright © 2004 John Wiley & Sons, Ltd.     

The coupled thermoelastic problem for a plane with a rectilinear crack

We solve the thermoelastic problem for a plane with a rectilinear heat-conducting crack whose conductivity depends on its opening. By modeling the crack as a thin inclusion of variable thickness we reduce the problem to a system of singular integrodifferential equations for the potential densities of the temperature field. We study the behavior of the unknown functions at the ends of the contour of integration and, using a numerical-iteration method, we also determine the solution of the problem. We find an approximate asymptotic solution in the case of a weakly conducting crack.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 31, 1990, pp. 54–58.     

On the Neumann problem for the Helmholtz equation in a plane angle

We prove that the solution of the Neumann problem for the Helmholtz equation in a plane angle Ω with boundary conditions from the space H_−1/2(Γ), where Γ is the boundary of Ω, which is provided by the well‐known Sommerfeld integral, belongs to the Sobolev space H₁(Ω) and depends continuously on the boundary values. To this end, we use another representation of the solution given by the inverse two‐dimensional Fourier transform of an analytic function depending on the Cauchy data of the solution. Copyright © 2000 John Wiley & Sons, Ltd.     

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.     

On stresses in a strip with a crack

The solution of the problem of a loaded crack in an infinite strip is given using the method of superposition of three problems (a loaded crack in the infinite plane; an infinite homogeneous strip with normal and tangent stresses that are given on nonhomogeneous boundaries; an infinite strip with longitudinal generators which are free from load and an arbitrary load at the end), which makes it possible to satisfy the boundary conditions exactly.Translated from Dinamicheskie Sistemy, No. 9, pp. 65–71, 1990.     

In-plane loading of a cracked elastic solid by a disc inclusion with a Mindlin-type constraint

The paper examines the problem related to the axisymmetric interaction between an external circular crack and a centrally placed penny-shaped rigid inclusion located in the plane of the crack. The interface between the inclusion and the elastic medium exhibits a Mindlin-type imperfect bi-lateral contact. Analytical results presented in the paper illustrate the manner in which the lateral translational stiffness of the inclusion and the stress intensity factor at the boundary of the external circular crack are influenced by the inclusion/crack radii ratio.     

The hp‐version of the BEM with quasi‐uniform meshes for a three‐dimensional crack problem: The case of a smooth crack having smooth boundary curve

The article considers a three‐dimensional crack problem in linear elasticity with Dirichlet boundary conditions. The crack in this model problem is assumed to be a smooth open surface with smooth boundary curve. The hp‐version of the boundary element method with weakly singular operator is applied to approximate the unknown jump of the traction which is not L₂‐regular due to strong edge singularities. Assuming quasi‐uniform meshes and uniform distributions of polynomial degrees, we prove an a priori error estimate in the energy norm. The estimate gives an upper bound for the error in terms of the mesh size h and the polynomial degree p. It is optimal in h for any given data and quasi‐optimal in p for sufficiently smooth data. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Closed-Loop Control of the Motion of a Sphere Rolling on a Moving Horizontal Plane

A uniform sphere is rolling without slipping on a horizontal plane. The motion of the sphere is controlled via the control of the acceleration of the plane. At the time t=0, the sphere and the plane are stationary and the center of the sphere is located at a point A in the plane. Given a time interval [0, t _f], the problem dealt with here is: Find a closed-loop strategy for the acceleration of the moving plane such that, at the time t=t _f, the plane and the sphere will be nearly at rest and the center of the sphere will be in a given neighborhood of the origin. By introducing the concept of path controllability, a closed-loop strategy for the solution of the above-mentioned problem is proposed and its efficiency is demonstrated by solving numerically some examples.     

A study of the perturbed stress state of a closed cylindrical shell with a system of part-through transverse cracks

We study the elastic equilibrium of a closed infinite circular cylindrical shell with a system of surface cracks of identical length and depth. We use the method of singular integral equations together with the modeling of solid matter in the plane of a part-through crack by irregularly distributed “line springs”. We conduct a numerical analysis of the variation of the relative stress intensity factor at the center of a crack as a function of the parameters of a crack and the number of cracks. We study cracks located on both the interior and exterior surface of the shell. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 63–65.     

An asymptotic form of the energy functional for an elastic body with a crack and a rigid inclusion. The plane problem

The plane problem in the linear theory of elasticity for a body with a rigid inclusion located within it is investigated. It is assumed that there is a crack on part of the boundary joining the inclusion and the matrix and complete bonding on the remaining part of the boundary. Zero displacements are specified on the outer boundary of the body. The crack surface is free from forces and the stress state in the body is determined by the bulk forces acting on it. The variation in the energy functional in the case of a variation in the rigid inclusion and the crack is investigated. The deviation of the solution of the perturbed problem from the solution of the initial problem is estimated. An expression is obtained for the derivative of the energy functional with respect to a zone perturbation parameter that depends on the solution of the initial problem and the form of the vector function defining the perturbation. Examples of the application of the results obtained are studied.     

Critical behavior of nonintersecting Brownian motions at a tacnode

We study a model of n one‐dimensional, nonintersecting Brownian motions with two prescribed starting points at time t = 0 and two prescribed ending points at time t = 1 in a critical regime where the paths fill two tangent ellipses in the time‐space plane as n → ∞. The limiting mean density for the positions of the Brownian paths at the time of tangency consists of two touching semicircles, possibly of different sizes. We show that in an appropriate double scaling limit, there is a new family of limiting determinantal point processes with integrable correlation kernels that are expressed in terms of a new Riemann‐Hilbert problem of size 4 × 4. We prove solvability of the Riemann‐Hilbert problem and establish a remarkable connection with the Hastings‐McLeod solution of the Painlevé II equation. We show that this Painlevé II transcendent also appears in the critical limits of the recurrence coefficients of the multiple Hermite polynomials that are associated with the nonintersecting Brownian motions. Universality suggests that the new limiting kernels apply to more general situations whenever a limiting mean density vanishes according to two touching square roots, which represents a new universality class. © 2011 Wiley Periodicals, Inc     

On the peculiarities of a crack growth path in anisotropic solid

One of the possibilities to increase the resistance of a structure to catastrophic fracture is to force a main line crack to deviate from its path. In this connection the influence of the elastic moduli of an anisotropic material on the possibilities of crack rotation are studied. In particular a linear elastic problem for a straight Mode I crack, located on a symmetry axis of an orthotropic plane is considered. The strength properties of the material are supposed to be isotropic. For studying a direction of a crack growth path several crack models are considered. It is shown that a thin elongated elliptical hole as a crack model leads to more plausible results concerning crack rotation conditions than an ideal cut model. The maximal tensile stresses are taken as a crack growth criterion. It is shown that for some class of orthotropic materials a crack deviates from the straight path just after it starts to grow even in the conditions of uniaxial normal tension. The problem of the stability of a straight crack path under Mode I loading is also considered. This problem is reduced to the problem of the fracture direction determination for thin elongated elliptical cavity slightly inclined to the initial direction. In the frame of the proposed approach the conditions of instability are obtained. It is shown that for some class of orthotropic materials a straight crack path is unstable in the conditions of uniaxial normal tension. This class of materials is wider than one for which a crack deviates from the straight crack path just after its start. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The embedding formula for an electromagnetic diffraction problem

Embedding formulas are powerful tools that enable one to reduce the dimension of the space of variables for a diffraction problem. Let a scatterer be finite, planar, and perfectly conducting. The idea of the method is to replace the initial problem of diffraction of a plane wave by construction of an edge Green’s function, i.e., to solve a problem with a source located near the edge of a scatterer. An embedding formula is an integral relation connecting the solution of the initial plane wave incidence problem with the edge Green’s function. Earlier, embedding formulas have been derived for acoustic and elasticity problems. Here we derive an embedding formula for an electromagnetic problem. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 324, 2005, pp. 247–261.     

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.