Y. A. Antipov

Eigenvalue problems for doubly periodic elastic structures and phononic band gaps

We consider the problem of elastic waves propagating in a two–dimensional array of circular cavities, taking rigorous account of coupling between shear and dilational waves. A technique, originally due to Rayleigh, is derived that involves an elegant identity between the singular and non–singular components of the stress fields in the array. This leads to an infinite linear system which can be truncated and solved in order to determine the complete structure of the propagating modes. Of particular interest is the possibility of exhibiting phononic band gaps, i.e. domains of frequency for which all propagating vibration in the material is suppressed.

Mathematical model of delamination cracks on imperfect interfaces

Abstract A mathematical model of a crack along a thin and soft interface layer is studied in this paper. This type of interface could arise in a ceramic support that has been coated with a layer of high surface area material which contains the dispersed catalyst. Asymptotic analysis is applied to replace the interface layer with a set of effective contact conditions. We use the words “imperfect interface” to emphasise that the solution (the temperature or displacement field) is allowed to have a non-zero jump across the interface. Compared to classical formulations for cracks in dissimilar media (where ideal contact conditions are specified outside the crack), in our case the gradient field for the temperature (or displacement) is characterised by a weak logarithmic singularity. The scalar case for the Laplacian operator as well as the vector elasticity problem are considered. Numerical results are presented for a two-phase elastic strip containing a finite crack on an imperfect interface.

DIFFRACTION OF A PLANE WAVE BY A CIRCULAR CONE WITH AN IMPEDANCE BOUNDARY CONDITION

We consider the boundary-value problem for the Helmholtz equation in a circular cone with an impedance boundary condition on its face. A new approach for its solution is proposed. The scheme of solution includes applying the Kontorovich--Lebedev transform, derivation of a second-order difference equation in a strip of a complex variable, and reduction of the latter to an integral equation of the convolution type with variable coefficients. It is also shown that the equation is equivalent to a 2 × 2 matrix Riemann--Hilbert problem with a discontinuous coefficient. We analyze the behavior of the solution of the integral equation at the ends of the contour and construct an approximate solution using a collocation method. The diffraction coefficient is found in terms of the solution of the integral equation. Numerical results for the diffraction coefficient and comparative analysis of the results for the impedance cone with the limiting cases of the acoustically soft and hard cones are reported. A full high f...

An exact solution of the 3-D-problem of an interface semi-infinite plane crack

Abstract The problem of a semi-infinite plane interface crack between three-dimensional (3-D) isotropic half-spaces is considered. Mathematically the problem is reduced to the analysis of the 3×3 matrix Wiener–Hopf problem. Explicit expressions for the factors of the 3×3 matrix are determined in quadratures. Exact-closed formulae for the stresses, discontinuities of the displacements, the stress intensity factors and the weight functions are found.

The modified Helmholtz equation in a semi-strip

We study the modified Helmholtz equation in a semi-strip with Poincar´ e type boundary conditions. On each side of the semi-strip the boundary conditions involve two parameters and one real-valued function. Using a new transform method recently introduced in the literature we show that the above boundary-value problem is equivalent to a 2 × 2-matrix Riemann–Hilbert (RH) problem. If the six parameters specified by the boundary conditions satisfy certain algebraic relations this RH problem can be solved in closed form. For certain values of the parameters the solution is not unique, furthermore in some cases the solution exists only under certain restrictions on the functions specifying the boundary conditions. The asymptotics of the solution at the corners of the semi-strip is investigated. In the case that the 2 × 2 RH problem cannot be solved in closed form, the Carleman–Vekua method for regularising it is illustrated by analysing in detail a particular case.

Exact solution of the plane problem for a composite plane with a cut across the boundary between two media

Abstract An exact closed solution is obtained for the problem of stress concentration in a composite elastic plane near a straight cut orthogonal to the dividing line between two media and which cuts it in half. The solution is constructed using the scheme of /1/ for the factorization of a special matrix coefficient of a Riemann problem. This Riemann problem is obtained by reducing the system of singular integral equations with a fixed singularity, which corresponds to the given problem of elasticity theory. The matrix coefficient of the Riemann problem does not satisfy the restrictions of /2/, and therefore the method described in /2/ produces an essential singularity at infinity for the factorizing matrices. The application of the scheme of /1/, based on the apparatus of boundary-value Riemann problems, on Riemann surfaces of algebraic functions /3/ enabled the essential singularity at infinity to be neutralized (by inversion of the corresponding Abelian integral). The solution of the problem is obtained in quadratures in a form suitable for numerical realization. Working formulas are given for the stress intensity factors. A numerical example is examined.

Transient loading of a rapidly advancing Mode-II crack in a viscoelastic medium

The problem that is studied concerns a semi-infinite crack in an infinite viscoelastic medium, propagating at uniform speed but subject to time-dependent loading. A particular loading, consisting of a pair of concentrated shear forces which come into existence as the crack passes their location, is studied in detail. Both sub-Rayleigh and transonic speed ranges are considered. The solution displays a richer range of possible behaviour than the corresponding solution for an elastic medium, which is likely to reflect in practice on the range of propagation speeds that may be achievable.

Partially stiffened elastic half-plane with an edge crack

In the present paper two contact problems referring to the partially stiffened elastic half-plane along its edge with a reinforcement (or stringer) either in the form of a rigid plate or a smooth rigid stamp, are considered. The half-plane contains an edge crack which is arbitrarily pressurized, and is perpendicular to the bounding edge of the half-space. It is assumed that the mid-point of the stringer is located in the axis of the crack. Each of the above two half-plane contact problems is first reduced to a system of two singular integral equations with fixed singularities. Then by employing the generalized method of integral transforms, this system is further reduced to a system of Wiener–Hopf equations that is equivalent to the Riemann matrix boundary value problem. Exact analytical solutions of the two problems are presented in series form. Asymptotic approximations for the stress intensity factor and the energy release rate at the crack tip are also given. Finally, numerical results for the contact stresses, crack opening displacements, stress intensity factor and crack energy are displayed.

An efficient solution of Prandtl-type integrodifferential equations in a section and its application to contact problems for a strip☆

Abstract Two Prandtl-type integrodifferential equations are solved exactly, one equation arising from the antiplane problem for an elastic layer, one of whose boundaries is rigidly attached, the other boundary being rigidly attached everywhere except along a section where it is elasticity attached, the other equation arising from the plane problem of a strip-shaped membrane uniformly extending at infinity and strengthened by elastic inclusions. In both cases the integral equation leads, with the help of a Fourier transformation, to a vector Riemann problem, which reduces by a method similar to one presented earlier [1]to an infinite Poincare-Koch algebraic system. Explicit formulae are found for the system unknowns together with recurrence relations that are convenient for numerical implementation. Computational formulae are found for the axial forces at the ends of the stringer, together with tangential contact stresses and their intensity factors. In the neighbourhood of the ends of the stringer an asymptotic expansion for the contact stresses is constructed, which, besides powers of radicals, contains products of radicals in integer powers of logarithms. Numerical results are presented.

An interface crack between elastic materials when there is dry friction

Abstract An analytic solution of the plane problem of a crack (finite or semi-infinite) along the interface between two elastic half-planes is given. Under tensile and shear forces, the crack opens over an interval (unknown in advance). In the vicinity of the crack tips the edges join smoothly and Coulombs law of dry friction applies. The materials are perfectly bonded everywhere except along the crack. A closed exact solution is found in the case of a semi-infinite crack. The slip direction, the slip zone length, and formulae for the contact stress and displacement jumps are determined. The problem of a finite crack is reduced to the vector (third-order) Riemann problem in the theory of analytic functions, for which an effective solution is constructed by the method proposed in [1]. An explicit relationship between the smaller and larger slip zone lengths is found by asymptotic analysis. A numerical analysis is carried out. Situations are determined in which the coefficient of friction has practically no effect on the length of the slip zone (to within 5%) and when the effect is substantial (20% or more). An effective analytic solution is found for Comninous equation [2], which corresponds to the problem of an interface crack ignoring the friction between its edges.