Impression of a semiinfinite stamp in an elastic strip with regard for friction and adhesion

We consider the problem of contact interaction between a semiinfinite stamp with rectilinear base and an elastic strip with one rigid side. Friction forces in the contact region are taken into account. These forces lead to the division of the contact region into slipping and adhesion zones. With the use of the Wiener–Hopf method, a system of integral equations is reduced to an infinite system of algebraic equations. The computational results of stresses and strains at the boundary and at inner points of the elastic strip are presented. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 138–149, January–March, 2008.     

A dynamic contact problem which reduces to a singular integral equation with two fixed singularities

The problem of the harmonic sheat oscillations of an elastic strip, coupled to an elastic half-space is considered. Using the method of integral transformations, the problem is reduced to a singular integral equation in the contact stresses in the region where the strip and the half-space are coupled when there are two fixed singularities at points bounding the integration intervals. One of the main results of this paper is the method of solving this equation numerically, taking into account the true singularity of the solution and based on the use of special quadrature formulae for singular integrals. The approximate solution obtained provides the possibility of numerically investigating the effect of the oscillation frequency and the ratio of the elastic constants of the strip and the half-space on the stress distribution in the contact area.     

The contact problem for an elastic strip and a wavy punch when there is friction and wear

The plane problem of the mutual wear of a wavy punch and an elastic strip, bonded to an undeformable foundation under the condition of complete contact between the punch and the strip is considered. An analytical expression for the contact pressure is constructed using the general Papkovich–Neuber solution, the two harmonic functions in which are represented in the form of Fourier integrals after which the problem reduces to a non-linear system of differential equations. In the case of a small degree of wear of the strip, this system becomes linear and admits of a solution in explicit form. The harmonics, constituting the profile of the punch and the contact pressure, move along the strip with respect to one another and are shifted in time. Conditions are obtained that ensure the hermetic nature of the contact between the wavy punch and the strip when there is friction and wear.     

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.     

The rolling of a wheel with a reinforced tyre along a plane without slip

A model of a wheel with a reinforced tyre, whose surface is simulated by a flexible strip (tread) attached to parts of two tori (the sidewalls of the tyre) is considered. The disk of the wheel (a rigid body) has six degrees of freedom and is in contact with the plane along part of the tread. Based on several assumptions, the potential energy functional of the deformed wheel is found as a function of the deformations of the centre line of the tread. On the assumption that the wheel is rolling without slip in the region of contact of the tread with the plane along a previously unknown section of the tread, the complete system of equations of motion is obtained. The equilibrium of the wheel and the steady state of rolling in a straight line with given swivel and tilt are investigated, and all characteristics of the motion are found (the contact region, the tyre deformation, and the forces and torques applied to the wheel disk).     

Determination of the stress state of a circular disk with a crack

With the use of complex potentials from the solution of a contact problem for slits in a multiply-connected region, a solution is found for a problem of the theory of elasticity for an isotropic circular disk with an arbitrary radial crack. The case of an edge crack is among the cases for which a solution is found. The types of loading examined are uniform tension on an outside edge, internal pressure on the edges of cracks, and concentrated forces at arbitrary points of a disk. The unknown coefficients in the complex potentials are found from the boundary conditions on the outside edge of the disk by the series method, the colocation method, or the least squares method. Detailed numerical studies are conducted to determine the effect of the geometric characteristics and the points of application of concentrated forces on the character of the stress distribution and the stress intensity factor. A comparative characteristic of the methods used to find the coefficients is presented.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19 pp. 50–61, 1988.     

Contact Problem for Second Order Parabolic Systems in a Strip with a Nonsmooth Interface Curve

Differential Equations - We prove the existence of a classical solution of the contact problem for Petrovskii parabolic systems of the second order with Dini continuous coefficients in a strip...     

The stress state of a plastic layer with a variable yield strength under a flat deformation

We study the stress state of a plastic layer with a variable yield strength in a strip under a flat deformation with a tensile load. We approximately calculate the first integrals of the system of plastic equilibrium equations, obtain an analog of the first Hencky theorem, and solve the conjugation problem for stresses on the contact boundary.     

An Invariant Measure and the Probability of a Fall in the Problem of an Inhomogeneous Disk Rolling on a Plane

This paper addresses the problem of an inhomogeneous disk rolling on a horizontal plane. This problem is considered within the framework of a nonholonomic model in which there is no slipping and no spinning at the point of contact (the projection of the angular velocity of the disk onto the normal to the plane is zero). The configuration space of the system of interest contains singular submanifolds which correspond to the fall of the disk and in which the equations of motion have a singularity. Using the theory of normal hyperbolic manifolds, it is proved that the measure of trajectories leading to the fall of the disk is zero.     

The problem of the bending of a beam lying on an elastic base

The plane contact problem of the transmission of a normal force of specified strength onto an elastic anisotropic, wedge-shaped plate by an elastic beam of variable flexural stiffness is considered. The beam is coupled to one of the edges of the plate and its other edge is stress-free. The solution of the problem is obtained in closed form by reducing it to a Karleman boundary-value problem with shear for a strip. A conclusion is reached concerning the nature of the discontinuity of the normal contact stress at the vertex of the wedge.     

A Contact Problem of the Interaction of a Semi-Finite Inclusion with a Plate

A piece wise-homogeneous plane made up of twodifferent materials and reinforced by an elastic unclusion is considered on a semi-finite section where the different materials join. Vertical and horizontal forces are applied to the inclusion which haz a variable thichness and a variable elasticity modulus.Under certain conditions the problem is reduced to integrodifferential equations of third order. The solution is constructed effectively by applying the methods of theory of analytic functions to a boundary value problem of the Carleman type for a strip. Asymptotic estimates of normal contact stress are obtained.     

State of stress of an anisotropic plate with a closed rod pressed into a curvilinear hole in it

A method is proposed for solving the problem of elastic equilibrium in the case of an anisotropic plate with a closed rod pressed into a curvilinear hole in it (or stretched over an anisotropic disk). This method is based on representing the boundary conditions in the form of contour integrals of an arbitrary function holomorphic within the region of that plate. The normal magnitude of the jump of the displacement vector at the contact line is given as the function of the arc. Friction at the contact line is assumed negligible. The stress-strain state of the rod (ring) is described by the equations in the theory of thin curvilinear beams.I. V. Franko L'vov State University. Translated from Mekhanika Polimerov, No. 2, pp. 304–309, March–April, 1976.     

Stokes waves with constant vorticity: I. Numerical computation

Periodic traveling waves are numerically computed in a constant vorticity flow subject to the force of gravity. The Stokes wave problem is formulated via a conformal mapping as a nonlinear pseudodifferential equation, involving a periodic Hilbert transform for a strip, and solved by the Newton‐GMRES method. For strong positive vorticity, in the finite or infinite depth, overhanging profiles are found as the amplitude increases and tend to a touching wave, whose surface contacts itself at the trough line, enclosing an air bubble; numerical solutions become unphysical as the amplitude increases further and make a gap in the wave speed versus amplitude plane; another touching wave takes over and physical solutions follow along the fold in the wave speed versus amplitude plane until they ultimately tend to an extreme wave, which exhibits a corner at the crest. Touching waves connected to zero amplitude are found to approach the limiting Crapper wave as the strength of positive vorticity increases unboundedly, while touching waves connected to the extreme waves approach the rigid body rotation of a fluid disk.     

Principal Eigenvalue Minimization for an Elliptic Problem with Indefinite Weight and Robin Boundary Conditions

This paper focuses on the study of a linear eigenvalue problem with indefinite weight and Robin type boundary conditions. We investigate the minimization of the positive principal eigenvalue under the constraint that the absolute value of the weight is bounded and the total weight is a fixed negative constant. Biologically, this minimization problem is motivated by the question of determining the optimal spatial arrangement of favorable and unfavorable regions for a species to survive. For rectangular domains with Neumann boundary condition, it is known that there exists a threshold value such that if the total weight is below this threshold value then the optimal favorable region is like a section of a disk at one of the four corners; otherwise, the optimal favorable region is a strip attached to the shorter side of the rectangle. Here, we investigate the same problem with mixed Robin-Neumann type boundary conditions and study how this boundary condition affects the optimal spatial arrangement.     

An asymptotic method of solving transient dynamic contact problems of the theory of elasticity for a strip

An asymptotic method is proposed for solving transient dynamic contact problems of the theory of elasticity for a thin strip. The solution of problems by means of the integral Laplace transformation (with respect to time) and the Fourier transformation (with respect to the longitudinal coordinate) reduces to an integral equation in the form of a convolution of the first kind in the unknown Laplace transform of contact stresses under the punch. The zeroth term of the asymptotic form of the solution of the integral equation for large values of the Laplace parameter is constructed in the form of the superposition of solutions of the corresponding Wiener-Hopf integral equations minus the solution of the corresponding integral equation on the entire axis. In solving the Wiener-Hopf integral equations, the symbols of the kernel of the integral equation in the complex plane is presented in special form — in the form of uniform expansion in terms of exponential functions. The latter enables integral equations of the second kind to be obtained for determining the Laplace-Fourier transform of the required contact stresses, which, in turn, is effectively solved by the method of successive approximations. After Laplace inversion of the zeroth term of the asymptotic form of the solution of the integral equations, the asymptotic solution of the transient dynamic contact problem is determined. By way of example, the asymptotic solution of the problem of the penetration of a plane punch into an elastic strip lying without friction on a rigid base is given. Formulae are derived for the active elastic resistance force on the punch of a medium preventing the penetration of the punch, and the law of penetration of the punch into the elastic strip is obtained, taking into account the elastic stress wave reflected from the strip face opposite the punch and passing underneath it.     

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.     

On the optimization of direct methods for solving fredholm integral equations of the second kind with infinitely smooth kernels

We give a direct method, optimal inL ₂, for solving the Fredholm integral equation of the second kind with operators acting into the space of functions harmonic in a disk or into the space of functions that can be analytically extended to an infinite strip. The exact order of the error of this method is determined.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 12, pp. 1695–1701, December, 1993.     

Conformal mappings onto prescribed regions via optimization techniques

Summary All well known extremal principles for conformal mappings of simply connected regionsR yield mappings onto disksD. It is shown here that given an arbitrary star shaped regionD as range a corresponding extremal principle is valid just by replacing the ordinary modulus in by a suitable positively homogeneous functional. If the star shaped regionD (bounded or not) is a convex polygon the extremal principle is equivalent to a linear (but infinite) programming problem, which can be solved approximately by passing to an ordinary (i.e. finite) linear programming problem. A numerical example whereR is a disk and the rangeD is an infinite strip is given.     

On a method of obtaining spectral relationships for integral operators of mixed problems of mechanics of continuous media

The method of orthogonal polynomials, and its generalization, the method of orthogonal functions /1,2/ applied for the investigation of complex mixed problems of the mechanics of continuous media, are based on the utilization of spectral relationships that invert the main (singular) part of the kernel of the integral equation of the problem under consideration. A sufficiently general approach to the derivation of spectral relationships that is based on potential theory is proposed. Eigenfunctions are obtained in the problem of impressing a strip stamp in an elastic halfspace as are also the odd eigenfunctions of a logarithmic series in the case of two symmetric intervals. An an application of the results obtained, the solution is constructed for any value of a certain dimensionless parameter, for the plane contact problem of the impression of a rigid stamp into the surface of an elastic strip which is under an interlayer of the type of a covering resting on an undeformable foundation.     

The calculation of the energy losses in a sliding elastic contact with friction and wear (the plane problem)

The plane problem of the sliding contact of a punch with an elastic foundation when there is friction and wear is considered. Assuming the existence of a steady solution in a moving system of coordinates, relations are derived between the sliding velocity, the wear, the contact stresses and the displacements for an arbitrary dependence of the wear rate on the contact pressure. Taking into account the presence of a deformation component of the friction force, an equation is written for the balance of the mechanical energy for the punch - elastic base system considered. It is shown that the equality of the work of the external force in displacing the punch to the losses due to friction and the change in the shape of the foundation due to wear is satisfied when the work done by the contact stresses on the increments of the boundary displacements is equal to zero, and the frictional losses must be determined taking into account the non-uniformity of the distributions of the shear contact stresses and the sliding velocity in the contact area. Two special cases of the foundation in the form of a wide and narrow strip are considered, for which the total coefficient of friction is calculated, taking into account the deformation component of the friction force.