Continuation of some solutions in potential theory and their applications to elastic contact and crack problems

The following mixed boundary value problem is considered: arbitrary tangential displacements are prescribed inside a circle, while the tangential and normal stresses outside the circle are zero. In this case, a direct and simple formula is derived for the tangential displacements outside the circle in terms of prescribed displacements inside, thus making the tangential displacement known all over the boundary. The original problem is no longer mixed, and the complete solution becomes readily available. Another solution for the case, when arbitrary tangential displacements are prescribed outside a circle, is derived in a similar manner. The reciprocal theorem is used to derive the continuation formulae for the tangential stresses inside and outside a circle. Application of these results to contact and crack problems is demonstrated.     

Stress field around an arbitrary thin inclusion in a transversely isotropic elastic half-space

We consider a thin flat inclusion of arbitrary shape located inside a transversely isotropic elastic half-space in the plane parallel to its boundary z = 0. An arbitrary tangential displacement is prescribed on the inclusion. The boundary of the half-space is stress-free. We need to find the complete field of stresses and displacements in this half-space. A governing integral equation is derived by the generalized method of images, introduced by the author. The case of circular inclusion is considered as an example. Two methods of solution of the governing integral equation are derived. A detailed solution is presented for the particular cases of radial expansion, torsion and lateral displacement of the inclusion. The solution is also valid for the case of isotropy. The governing integral equation for the case of isotropy is derived.     

New closed-form Green function and integral formula for a thermoelastic quadrant

In this study a new Green’s function and a new Green-type integral formula for a boundary value problem (BVP) in thermoelastostatics for a quadrant are derived in closed form. On the boundary semi-straight-lines twice mixed homogeneous mechanical boundary conditions (one boundary semi-straight-line is free of loadings and normal displacements and tangential stresses are prescribed on the other one) are prescribed. The thermoelastic displacements are subject by a heat source applied in the inner points of the quadrant and by mixed non-homogeneous boundary heat conditions (on one boundary semi-straight-line the temperature is prescribed and the heat flux is given on the other one). When thermoelastic Green’s function is derived the thermoelastic displacements are generated by an inner unit point heat source, described by δ Dirac’s function. All results are obtained in elementary functions that are formulated in a special theorem. A closed-form solution for a particular BVP of thermoelastostatics for a quadrant also is included. Using the proposed approach it is possible to extend the obtained for quadrant results to any other canonical Cartesian domain.     

A normal crack in an elastic half-space with stress-free surface

The problem of an elastic half-space with stress-free surface and a crack of arbitrary shape with prescribed displacements or tractions is reduced to an equivalent system of integral equations on the crack. For a pressurized crack in a plane perpendicular to the free surface, a scalar integral equation is derived. In properly chosen function spaces, unique solvability of the integral equation and regularity of solutions for regular data are proven.     

Liouville Property for Solutions of the Linearized Degenerate Thin Film Equation of Fourth Order in a Halfspace

We consider a boundary value problem in a half-space for a linear parabolic equation of fourth order with a degeneration on the boundary of the half-space. The equation under consideration is substantially a linearized thin film equation. We prove that, if the right hand side of the equation and the boundary condition are polynomials in the tangential variables and time, the same property has any solution of a power growth. It is shown also that the specified property does not apply to the normal variable. As an application, we present a theorem of uniqueness for the problem in the class of functions of power growth.     

An energy approach to the proof of the existence of Rayleigh waves in an anisotropic elastic half-space

An approach based on investigating the energy functional is applied for the first time to the classical problem of Rayleigh waves in an anisotropic half-space with a free boundary. The main object of the investigation is an ordinary differential operator in a variable characterizing the depth. An investigation of the spectrum by variational methods enables a new proof to be given of the existence of a Rayleigh wave in a linear elastic half-space with arbitrary anisotropy, which does not rest on the Stroh formalism.     

Variational statement of deformation problems for a composite latticed plate with various types of lattices

The variational statement of various boundary value problems for tangential displacements and forces in a latticed plate with an arbitrary piecewise smooth contour is investigated. The lattice consists of several families of bars made of a homogeneous composite material with a matrix of relatively low shear stiffness. The energy method reduces the problem to the variational problem of minimizing the energy functional. The conditions on the plate contour are established under which the functional is minimal and positive definite, which ensures that the problem is well posed.     

On the problem of determining the structure of a layered medium and the shape of an impulse source

For the equation of wave propagation in the half-space ? ₊ ² + = {(x, y) ∈ ?² | y > 0} we consider the problem of determining the speed of wave propagation that depends only on the variable y and the shape of a point impulse source on the boundary of the half-space. We show that, under some assumptions on the shape of the source and the structure of the medium, both unknown functions of one variable are uniquely determined by the displacements of boundary points of the medium. We estimate stability of a solution to the problem.     

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.     

Diffraction of a monochromatic plane shear wave on a reinforced cylindrical cavity in an elastic half-space

This article examines a boundary-value problem concerning the diffraction of a monochromatic plane shear wave on a reinforced cylindrical cavity in an elastic half-space. It is assumed that longitudinal shear stresses are absent and that the normal displacements over the entire boundary are specified. Through the use of a special form of the Lamé representation in cylindrical coordinates, the problem is reduced to the determination of scalar functions which satisfy the Helmholtz equation. The coefficients of the Fourier expansions of these functions in the angular coordinate are written as the sum of Fourier and Weber integrals. The densities of these integrals are determined exactly. A specific example is examined.Translated from Dinamicheskie Sistemy, No. 5 pp. 42–49, 1986.     

Smoothness of solutions of a special one-sided problem

We consider a planar domain, namely a curvilinear quadrilateral. We study a variational inequality of special form on the set of functions that are monotonically increasing on part of the boundary. This problem corresponds to a one-sided problem for an elliptic equation. A boundary condition of first kind is prescribed on part of the boundary, while on the other part of the boundary the tangential derivative is nonnegative and the product of the tangential and oblique derivatives is zero. We establish that the first derivatives of the solution satisfy a Hölder condition. Bibliography: 5 titles.Translated fromProblemy Matematicheskogo Analiza, No. 12, 1992, pp. 173–186.     

The determination of the thermal properties of a heat conductor in a nonlinear heat conduction problem

This paper considers the inverse determination of the positive unknown thermal properties K(T), C(T) and the unknown temperature T(_{x, t}) in the nonlinear transient heat conduction equation. In addition to prescribed initial and/or boundary values, specified continuously differentiable temperature data T(x₀, t) with non-zero derivative at a single sensor location x = x₀ is given. When K(T) and C(T) obey a certain relationship which enables one to linearise exactly the nonlinear heat equation then their dependence upon T is obtained explicitly, whilst the unknown temperature T(x, t) is obtained implicitly and is then calculated numerically. Results are presented and discussed for infinite, semi-infinite and finite slabs.     

External circular crack under arbitrary shear loading

An exact closed form solution in terms of elementary functions has been obtained to the governing integral equation of an external circular crack in a transversely isotropic elastic body. The crack is subjected to arbitrary tangential loading applied antisymmetrically to its faces. The recently discovered method of continuity solutions was used here. The solution to the governing integral equation gives the direct relationship between the tangential displacements of the crack faces and the applied loading. Now a complete solution to the problem, with formulae for the field of all stresses and displacements, is possible.     

Vorticity-velocity-pressure formulation for Stokes problem

We propose a three-field formulation for efficiently solving a two-dimensional Stokes problem in the case of nonstandard boundary conditions. More specifically, we consider the case where the pressure and either normal or tangential components of the velocity are prescribed at some given parts of the boundary. The proposed computational methodology consists in reformulating the considered boundary value problem via a mixed-type formulation where the pressure and the vorticity are the principal unknowns while the velocity is the Lagrange multiplier. The obtained formulation is then discretized and a convergence analysis is performed. A priori error estimates are established, and some numerical results are presented to highlight the perfomance of the proposed computational methodology.

     

Outer solution for elastic torsion by the method of boundary layer residual states

By the method of boundary layer residual state (BLRS), it is possible to specify the unknown parameters in the general form of the outer asymptotic solution of the governing differential equations for linear boundary value problems (BVP) without any reference to the inner asymptotic solutions of the same problem and the matching procedure. The method accomplishes this task by rationally assigning a portion of the prescribed boundary data to the outer solution. Specifically, the method requires certain weighted averages of the outer solution to be equal to the same averages of the data over the (localized) boundary where the data is prescribed. These weighted averages are consequences of a reciprocity relation inherent in the BVP and the stipulation that the difference between the outer solution and the exact solution (called the residual solution) of the BVP be a boundary layer phenomenon.¶The weighted average requirements are only necessary conditions for the residual state to be a boundary layer. Unfortunately, there are generally countably infinite number of (2) states, many more than the available degrees of freedom in the outer solution to satisfy them. We must show that there is no over-determination or non-uniqueness of the outer asymptotic solution, the abundance of necessary conditions notwithstanding. The present note describes an approach to assuring a well-specified outer solution (up to the expected accuracy) by way of the problem of Saint-Venant torsion. The same approach also also applies to other linear BVP, deducing the appropriate outer solution whenever the determination of the relevant inner solutions is not practical.     

Asymptotic Stability of a Stationary Flowing Regime of an Ideal Incompressible Fluid

We give sufficient conditions for asymptotic stability of a stationary solution to a flowing problem of a homogeneous incompressible fluid through a given planar domain. We consider a planar problem for the Euler equation and boundary conditions for the curl and the normal component of the velocity; moreover, the latter is given on the whole boundary of the flow domain and the curl is given only on the inlet part of the boundary. We establish asymptotic stability of a stationary flow (in linear approximation), assuming it to have no rest points and to satisfy some smallness condition which means that the perturbations leave the flow domain before they become to affect the main flow. In particular, we prove asymptotic stability for an arbitrary stationary flow in a rectangular canal close to the Couette flow without rest points. Moreover, we show that stability of the main flow in the L ₂-norm under curl perturbations implies its stability in higher-order norms depending, for example, on the derivatives of the curl.     

Steady solutions of the Navier–Stokes equations with threshold slip boundary conditions

We establish the wellposedness of the time‐independent Navier–Stokes equations with threshold slip boundary conditions in bounded domains. The boundary condition is a generalization of Navier's slip condition and a restricted Coulomb‐type friction condition: for wall slip to occur the magnitude of the tangential traction must exceed a prescribed threshold, independent of the normal stress, and where slip occurs the tangential traction is equal to a prescribed, possibly nonlinear, function of the slip velocity. In addition, a Dirichlet condition is imposed on a component of the boundary if the domain is rotationally symmetric. We formulate the boundary‐value problem as a variational inequality and then use the Galerkin method and fixed point arguments to prove the existence of a weak solution under suitable regularity assumptions and restrictions on the size of the data. We also prove the uniqueness of the solution and its continuous dependence on the data. Copyright © 2006 John Wiley & Sons, Ltd.     

Temperature distribution in a parallelopiped under the influence of heat source with boundary conditions of second kind

In the present paper, the general expressions for temperature distribution in an orthotropic plate of finite dimensions have been derived under the influence of an arbitrary volume heat source and arbitrary initial temperature distribution. The boundary conditions prescribed on the faces are of second kind. The solution is obtained by constructing an eigen value problem. The expression for unsteady temperature distribution consists of two parts—steady and pseudo-steady. The pseudo-steady parts of the solution have been obtained with the help of finite integral transform technique. The general expressions are further studied under different cases. The whole analysis is presented in the dimensionless form with the help of various dimensionless parameters,viz., F _o , P _o , K _i , etc.     

Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic matrix functions

Classical interpolation problems are concerned with the problem of finding an analytic function on the unit disk bounded by one which takes on prescribed values at certain prescribed points inside the disk (Pick-Nevanlinna) or on the boundary of the disk (Loewner). We consider the problem for matrix-valued functions having as many as poles ( a nonnegative integer) inside the disk but still uniformly bounded by one on the boundary of the disk. Our technique is an adaptation of that of Sz.-Nagy and Koranyi to spaces with an indefinite inner product. The problem arises in the broadband matching problem for electrical circuits and certain multichannel scattering problems in physics.Research supported by US National Science Foundation Grant MCS-8101678.     

Vibration of a stamp partially adhering to an elastic medium : PMM, vol. 42, no. 6, 1978, pp. 1085–1092

There is examined the problem of vibration of a stamp of arbitrary planform occupying a space Ω and vibrating harmonically in an elastic medium with plane boundaries. It is assumed that the elastic medium is a packet of layers with parallel boundaries, at rest in the stiff or elastic half-space. Contact of three kinds is realized under the stamp: rigid adhesion in the domain Ω₁, friction-free contact in domain Ω₂, there are no tangential contact stresses, and “film” contact without normal force in domain Ω₃ (there are no normal contact stresses, only tangential stresses are present.). It is assumed that the boundaries of all the domains have twice continuously differentiable curvature and Ω = Ω₁ Ω₂ Ω₃.

The problem under consideration assumes the presence of a static load pressing the stamp to the layer and hindering the formation of a separation zone. Moreover, a dynamic load, harmonic in time, acts on the stamp causing dynamical stresses which are of the greatest interest since the solution of the static problem is obtained as a particular case of the dynamic problem for ω = 0 (ω is the frequency of vibration). The general solution is constructed in the form of a sum of static and dynamic solutions.

A uniqueness theorem is established for the integral equation of the problem mentioned and for the case of axisymmetric vibration of a circular stamp partially coupled rigidly to the layer, partially making friction-free contact, the problem is reduced to an effectively solvable system of integral equations of the second kind, which reduce easily to a Fredholm system.

These results are an extension of the method elucidated in [1], where by the approach in [1] must be altered qualitatively to obtain them.