Two problems with mixed boundary conditions for an elastic orthotropic strip

Two problems are considered for an elastic orthotropic strip: the contact problem and the crack problem. Both problems are reduced to integral equations of the first kind with different kernels, containing a singularity: logarithmic for the first problem and singular for the second problem. Regular and singular asymptotic methods are employed to construct approximate solutions of these integral equations. Numerical results are presented.     

On a mixed boundary value problem for an infinite elastic cone

Zusammenfassung Die Verfasser haben ein gemischtes Randwertproblem für einen unendlichen elastischen Kegel mit Spitzenwinkel2 auf die Lösung einer endlichen Wiener-Hopf-Gleichung zurückgeführt. Der wesentliche Schritt besteht darin, eine FunktionK(s), die in einem Streifen regulär ist, als ein Produkt von Faktoren darzustellen, die auf übereinanderliegenden Halbebenen regulär sind. In dem hier behandelten Fall für willkürliches ist diese Methode besonders schwierig. In dem Sonderfalle des elastischen Halbraumes =/2 können jedoch bekannte Resultate erlangt werden. Das deutet darauf hin, dass die Methode auf das Kegelproblem anwendbar ist, vorausgesetzt dass die erforderliche Faktorzerlegung durchgeführt werden kann.

---

---

This work was supported in part by the Office of Ordnance Research under contract No. DA-11-022-ORD-2195.     

On an integral equation of the problem for an elastic strip with a slit

A new singular integral equation is obtained that describes the elastic equilibrium of a strip with both an inner and an edge slit (crack) and has a considerable advantage over existing equations /1–9/, etc.) from the viewpoint of a numerical realization and clarification of the analytical relationship with an analogous equation for a half-plane. Numerical results are given of a computation of the stress intensity coefficients at the tips of the inner and edge cracks that refine data in the literature.     

The contact of a solid with an elastic half-space through a thin coating

More accurate equations of the deformation of thin plates, which are more convenient for solving contact problems for bodies with coatings and containing, as a special case, the equations of all known applied theories, are derived by an asymptotic analysis of the first fundamental problem of the theory of elasticity. The equations of the deformation of thin-walled elastic bodies are classified, their qualitative correspondence to the equations of the theory of elasticity is clarified, and the forms of the features that arise along the shift lines of the boundary conditions in the corresponding contact problems are established. A criterion for selecting approximate models to describe the properties of the coatings depending on the geometrical and mechanical characteristics of the coating and the substrate and also on their degree of adhesion is given.     

On the bending of plates with transverse shear deformation and mixed periodic boundary conditions

The method of matched asymptotic expansions is used to find a homogenized problem whose solution is an approximation to the solution of a mixed periodic boundary value problem in the theory of bending of thin elastic plates. A critical size for the fixed parts of the boundary is found such that the boundary condition of the homogenized problem is an intermediate case between that for the clamped edge plate and that for the free boundary plate.     

Existence and uniqueness of weak solutions for a thin plate with elastic boundary conditions

Robin-type problems are studied for thin elastic plates with transverse shear deformation. These problems are reduced to analogous ones for the corresponding homogeneous equilibrium equation, whose solutions are then represented as single and double layer potentials. The unique solvability of the systems of boundary integral equations yielded by this procedure is discussed in Sobolev spaces.     

On the rotating Navier-Stokes equations with mixed boundary conditions

The stationary and nonstationary rotating Navier-Stokes equations with mixed boundary conditions are investigated in this paper. The existence and uniqueness of the solutions are obtained by the Galerkin approximation method. Next, θ-scheme of operator splitting algorithm is applied to rotating Navier-Stokes equations and two subproblems are derived. Finally, the computational algorithms for these subproblems are provided.     

On multilayer films on the boundary of a half-space

Generalized boundary conditions on multilayer films bounding a half-space and consisting of alternating infinitely thin strongly and weakly permeable layers are derived. The solution of the problem for the Laplace equation in a half-plane D bounded by a three-layer film is expressed in simple quadratures in terms of the solution of the classical Dirichlet problem in D without a film.     

Oscillations of an elastic rod with nonlinear boundary conditions

We study the dynamics of a building with a nonlinear seismic insulation system whose motion is described by the equation of transverse oscillations of a rod. We obtain the connection between the displacement of the lower section of the rod and the forces and moments in the section. We propose a numerical procedure for solving a nonlinear Volterra integral equation of second kind with dry friction damping. We determine the region of variation of the parameters of the building and the seismic insulation system in which it is possible to use the rigidbody model of the building. Two figures.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 76–80.     

On low eigenvalues of the Laplacian with mixed boundary conditions

By means of the so-called α-symmetrization we study the eigenvalue problem for the Laplace operator with mixed boundary conditions. We obtain various bounds for combinations of the low eigenvalues and some sharp comparison results for the first eigenfunction in terms of Bessel functions.     

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.     

On an integral representation of the solution of the Laplace equation with mixed boundary conditions

We obtain an integral representation of the solution of the Laplace equation with three distinct boundary conditions. Depending on the statement of the problem, the homogeneous boundary value problem may have nontrivial solutions; in other cases, the solution of the homogeneous problem is zero. Note that the inhomogeneous problem is always solvable.     

Boundary value problems of electroelasticity for a plate with an inclusion and a half-space with a slit

Problems of determining the mechanical and electrical fields in a piezoelectric plate reinforced with an inclusion or in a half-space weakened by a cut are considered. Using the methods of the theory of analytic functions these problems are reduced to a system of singular integro-differential equations (for a plate) or to a singular integral equation with a fixed singularity (for a half-space). Approximate and exact solutions of the problems are obtained by the method of orthogonal polynomials and integral transforms.     

A minimum principle for an integral equation with mixed boundary conditions

We have considered an integral equation, which describes the dynamic problem of a viscoelastic body and submitted it to a boundary condition with fading memory. It has come out that the solution to this mixed initial-boundary value problem is a strict minimum of a proper class of functional with a weight function and vice versa.

---

Sunto è preso in considerazione un problema misto con una condizione al contorno di tipo viscoelastico per un'equazione integrale, che descrive il problema dinamico di un corpo viscoelastico. Si mostra che la soluzione a tale problema rappresenta un minimo per un funzionale bilineare con una funzione peso e viceversa.

     

The deformation of a half-space with a gradient elastic coating under arbitrary axisymmetric loading

An analytical-numerical method of solving the Neumann boundary-value problem for an elastic half-space with a gradient elastic coating is proposed. The problem is formulated and the construction of the fundamental solution (Green’s function) is described. The method enables a solution of the problem to be obtained for a fairly wide class of types of non-uniformity of the medium, and effects related to the non-uniformity are investigated analytically. A procedure for calculating the displacement, stress and strain fields is described. Particular attention is devoted to analysing the mechanical characteristics in the transition region from the coating to the elastic substrate.     

On a boundary value problem in the half-space

In this paper, we study the existence of positive solutions and sign-changing solutions for the following boundary value problem in the half-space

     

Time-periodic Stokes equations with inhomogeneous Dirichlet boundary conditions in a half-space

The time-periodic Stokes problem in a half-space with fully inhomogeneous right-hand side is investigated. Maximal regularity in a time-periodic L^p setting is established. A method based on Fourier multipliers is employed that leads to a decomposition of the solution into a steady-state and a purely oscillatory part in order to identify the suitable function spaces.     

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.