The integral equations of plane contact problems for high values of the parameter

Typical integral equations of the first kind, that arise when investigating various plane static contact problems, are considered. These are, for example, problems for a layer of finite thickness, problems for an infinite circular cylinder and a space with an infinite cylindrical cavity of finite inner radius, and problems for an infinite wedge-shaped region. To solve these, it is proposed to use a regular asymptotic method of high values of the characteristic parameter λ.     

The Equation of Motion of a Vortex Layer of Small Thickness

The equation governing the evolution of a vortex layer whose thickness is small compared to its radius of curvature and in which the vorticity is everywhere the same is obtained, viscous and compressibility effects being neglected. The method of matched asymptotic expansions is applied and results in an extension, which is unexpectedly simple, of Birkhoff's integro-differential equation for vortex sheets. The equation is applied to long waves on a straight vortex layer of uniform thickness.     

The solution of the Hertz axisymmetric contact problem

The main terms of the asymptotic form of the solution of the contact problem of the compression without friction of an elastic body and a punch initially in point contact are constructed by the method of matched asymptotic expansions using an improved matching procedure. The condition of unilateral contact is formulated taking account of tangential displacements on the contact surface. An asymptotic solution of the problem for the boundary layer is constructed by the complex potential method. An asymptotic model is constructed, extending the Hertz theory to the case where the surfaces of the punch and elastic body in the vicinity of the contact area are approximated by paraboloids of revolution. The problem of determining the convergence of the contacting bodies from the magnitude of the compressive force is reduced to the problem of calculating the so-called coefficient of local compliance, which is an integral characteristic of the geometry of the elastic body and its fixing conditions.     

The indentation of a punch in the form of an elliptic paraboloid into the plane boundary of an elastic body

The three-dimensional contact problem for an elastic body of arbitrary geometry with a single plane face, into which a punch in the shape of an elliptic paraboloid is indented, is considered. The curvilinear boundary of the body is partially clamped, and the remaining boundary (outside the contact region) is stress-free. It is assumed that the dimensions of the contact area are small compared with the characteristic dimension of the body. Using the method of matched asymptotic expansions a model problem of unilateral contact without friction is derived for the boundary layer, which is solved using the apparatus of Hertz's theory. Asymptotic models of the contact interaction of different degrees of accuracy are constructed, including corrections to the geometry and clamping conditions of the elastic body. The sensitivity of the parameters of the elliptic region of the contact to these factors is investigated.     

On the contact interaction between a strip-shaped punch and a linearly-deformable base through a covering of variable thickness

The contact problem of the frictionless penetration of a punch with strip-shaped section into the surface of a linearly-deformable base protected by a thin elastic layer (covering) of variable thickness, the stiffness of which is comparable to or smaller than that of the supporting elastic body, is investigated. A Fredholm integral equation of the second kind is obtained for the unknown contact pressure with a coefficient in front of the leading term that is a fairly arbitrary function of the longitudinal coordinate. To solve it the Bubnov-Galerkin projection method is used in which the coordinate elements are chosen to be a system of orthogonal polynomials and delta-shaped functions [1, 2] (variational-difference method), together with an algorithm for the required asymptotic expansions [3] when the above-mentioned coefficient is small. In the special case of an elastic half-space protected by a covering of constant thickness, the results obtained are compared with the corresponding characteristics given in [4].     

Mathematical Justification of an Obstacle Problem in the Case of a Plate

In this paper the modeling of a thin plate in unilateral contact with a rigid plane is properly justified.Starting from the three-dimensional nonlinear Signorini problem,by an asymptotic approach the convergence of the displacement field as the thickness of the plate goes to zero is studied.It is shown that the transverse mechanical displacement field decouples from the in-plane components and solves an obstacle problem.     

The contact problem for a two-layer spherical base

Analytical methods for solving problems of the interaction of punches with two-layer bases are described using in the example of the axisymmetric contact problem of the theory of elasticity of the interaction of an absolutely rigid sphere (a punch) with the inner surface of a two-layer spherical base. It is assumed that the outer surface of the spherical base is fixed, that the layers have different elastic constants and are rigidly joined to one anther, and that there are no friction forces in the contact area. Several properties of the integral equation of this problem are investigated, and schemes for solving them using the asymptotic method and the direct collocation method are devised. The asymptotic method can be used to investigate the problem for relatively small layer thicknesses, and the proposed algorithm for solving the problem by the collocation method is applicable for practically any values of the initial parameters. A calculation of the contact stress distribution, the parameters of the contact area, and the relation between the displacement of the punch and the force acting on it is given. The results obtained by these methods are compared, and a comparison with results obtained using Hertz, method is made for the case in which the relative thickness of the layers is large.     

Asymptotic analysis of a thin‐layer device with Tresca's contact law in elasticity

In this paper, we consider a thin elastic layer between a rigid body and an elastic one. A Tresca law is assumed between the two elastic bodies. The Lamé coefficients of the thin layer are assumed to vary with respect to its height ϵ. This dependence is shown to be of primary importance in the asymptotic behaviour of the device, a critical case leading to a non‐classical contact law when deleting the bond. Copyright © 1999 John Wiley & Sons, Ltd.     

Torsion of a nonhomogeneous layer

A rotation of a nonhomogeneous, with respect to thickness, layer is considered. Pliability functions are studied that connect the transformations of stresses and displacements. Some estimates, asymptotic formulas, distribution and character of zeros and poles in the complex plane are given. A solution to the problem of rotation of a nonhomogeneous layer with a ring die is constructed. Triple integral equations are reduced to successively solvable Fredholm equations of the second kind and an infinite system of linear algebraic equations that is normal in the sense of Poincaré—Koch.Translated from Dinamicheskie Sistemy No. 8, pp. 22–30, 1989.     

Shape and topology optimization of contact problems by level set methods

This paper deals with the numerical solution of a topology and shape optimization problems of an elastic body in unilateral contact with a rigid foundation. The contact problem with the prescribed friction is considered. The structural optimization problem consists in finding such shape of the boundary of the domain occupied by the body that the normal contact stress along the contact boundary of the body is minimized. In the paper shape as well as topological derivatives formulae of the cost functional are provided using material derivative and asymptotic expansion methods, respectively. These derivatives are employed to formulate necessary optimality condition for simultaneous shape and topology optimization. Level set based numerical algorithm for the solution of the shape optimization problem is proposed. Level set method is used to describe the position of the boundary of the body and its evolution on a fixed mesh. This evolution is governed by Hamilton – Jacobi equation. The speed vector field driving the propagation of the boundary of the body is given by the shape derivative of a cost functional with respect to the free boundary. Numerical examples are provided. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Ventcel-Type Transmission Conditions for a Poisson Problem at High–Low Diffusion

This work looks at the asymptotic behavior of the solution to the Poisson equation defined in a bounded domain of \(\mathbb {R}^{N}\) (\(N=2,3)\) with a thin layer of thickness \(\delta \) which tends to 0. We use a method based on multi-scale expansion to derive and justify an asymptotic expansion of the solution with respect to the thickness \(\delta \) up to any order. We then provide Ventcel-type transmission conditions modeling the effect of the thin layer with accuracy up to \(O(\delta ^{2})\).     

Asymptotic expansions of stress tensor for linearly elastic shell

In this paper, the asymptotic expansions of stress tensor for linearly elastic shell have been proposed by new asymptotic analysis method, which is different from the classical asymptotic analysis. The new asymptotic analysis method has two distinguishing features: one is that the displacement is expanded with respect to the thickness variable of the middle surface not to the thickness; another is that the first order term and the second order term of the displacement variable can be algebraically expressed by the leading term. To decompose stress tensor totally into 2-D variable and thickness variable, we have three steps: operator splitting, variables separation and dimension splitting. In the end, a numerical experiment of special hemispherical shell by FEM (finite element method) is provided. We derive the distribution of displacements and stress fields in the middle surface.     

Two-layer film flow over a rotating disk

Unsteady two-layer liquid film flow on a horizontal rotating disk is analyzed using asymptotic method for small values of Reynolds number. This analysis of non-linear evolution equation elucidates how a two-layer film of uniform thickness thins when the disk is set in uniform rotation. It is observed that the final film thickness attains an asymptotic value at large time. It is also established that viscous force dominates over centrifugal force and upper layer thins faster than lower layer at large time.     

Le problème de Signorini dans la théorie des plaques minces de Kirchhoff–LoveSignorini's problem in the Kirchhoff–Love theory of plates

In the framework of the Kirchhoff–Love asymptotic theory of elastic thin plates we consider the unilateral contact problem with friction for a plate on a rigid foundation (Signorini problem with friction). First, we notice, when the thickness vanishes, that the order of the friction force must be lower than that of the contact pressure. These two measures are connected by Coulomb law. Consequently, at least formally, the friction force must be vanishing when the thickness goes to zero. We actually prove that any sequence of solution of the sequence of three-dimensional scaled Signorini problems with friction strongly converges to the unique solution of a two-dimensional Signorini plate problem without friction. To cite this article: J.-C. Paumier, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 567–570.     

Exact solutions of functionally graded piezoelectric material sandwich cylinders by a modified Pagano method

The three-dimensional (3D) coupled analysis of simply-supported, functionally graded piezoelectric material (FGPM) circular hollow sandwich cylinders under electro-mechanical loads is presented. The material properties of each FGPM layer are regarded as heterogeneous through the thickness coordinate, and obey an exponent-law dependent on this. The Pagano method is modified to be feasible for the study of FGPM sandwich cylinders. The modifications are as follows: a displacement-based formulation is replaced by a mixed formulation; a set of the complex-valued solutions of the system equations is transferred to the corresponding set of real-valued solutions; a successive approximation method is adopted to approximately transform each FGPM layer into a multilayered piezoelectric one with an equal and small thickness for each layer in comparison with the mid-surface radius, and with the homogeneous material properties determined in an average thickness sense; and a transfer matrix method is developed, so that the general solutions of the system equations can be obtained layer-by-layer, which is significantly less time-consuming than the usual approach. A parametric study is undertaken of the influence of the aspect ratio, open- and closed-circuit surface conditions, and material-property gradient index on the assorted field variables induced in the FGPM sandwich cylinders.     

Approximate models for wave propagation across thin periodic interfaces

This work deals with the scattering of acoustic waves by a thin ring that contains regularly spaced inhomogeneities. We first explicit and study the asymptotic of the solution with respect to the period and thickness of the inhomogeneities using so-called matched asymptotic expansions. We then build simplified models replacing the thin ring with Approximate Transmission Conditions that are accurate up to third order with respect to the layer width. We pay particular attention to the study of these approximate models and the quantification of their accuracy.     

On Upper Bounds for Minimum Distance and Covering Radius of Non-binary Codes

We consider upper bounds on two fundamental parameters of a code; minimum distance and covering radius. New upper bounds on the covering radius of non-binary linear codes are derived by generalizing a method due to S. Litsyn and A. Tietäväinen lt:newu and combining it with a new upper bound on the asymptotic information rate of non-binary codes. The upper bound on the information rate is an application of a shortening method of a code and is an analogue of the Shannon-Gallager-Berlekamp straight line bound on error probability. These results improve on the best presently known asymptotic upper bounds on minimum distance and covering radius of non-binary codes in certain intervals.     

Averaging a transport equation with small diffusion and oscillating velocity

A complete asymptotic expansion is constructed for the transport equation with diffusion term small with respect to the convection. Error estimates are obtained by using matched asymptotic expansion technique and building all the boundary layer terms in time and in space, necessary for obtaining an accurate error estimate. Copyright © 2003 John Wiley & Sons, Ltd.     

Frictionless elliptical contact of thin viscoelastic layers bonded to rigid substrates

A three-dimensional unilateral contact problem for thin viscoelastic layers bonded to rigid substrates shaped like elliptic paraboloids is considered. Two cases are studied: (a) Poisson’s ratios of the layer materials are not very close to 0.5 and (b) the layer materials are incompressible with Poisson’s ratio of 0.5. Poisson’s ratios are assumed to be time independent. In the present paper we derive the general solutions to the problems of elliptical contact between thin compressible or incompressible layers of arbitrary viscoelastic materials. The approach is based on the analytical method developed by the authors for the elliptical contact of thin biphasic cartilage layers. The obtained analytical solution is valid for monotonically increasing loading conditions.     

Passage of a wave through a thin layer

The problem is considered of the passage of an acoustic wave given by its ray expansion through a thin layer. The method of boundary-layer asymptotic expansions is applied to obtain ray representations of the reflected and transmitted waves (the small parameters are the thickness of the layer and the wave length).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklov AN SSSR, Vol. 173, pp. 20–29, 1988.