Viscoelastic cylinder reinforced with thin elastic casing in a heterogeneous temperature field

The problem of a viscoelastic cylinder reinforced with a thin elastic casing with consideration given to a heterogeneous temperature field is presented. The method of numerical solution of the problem is discussed. The results of computation are presented in the form of curves.     

Flow of thin liquid film over a rough rotating disk in the presence of a transverse magnetic field

The development of a flow of a viscous conducting fluid over a rough spinning disk in the presence of a transverse magnetic field has been analysed for different patterns of surface roughness of the disk and different initial distributions of the height of the liquid lubricant. The numerical solution of the governing equation of motion subject to initial and boundary conditions has been obtained by a finite-difference method. The temporal evolution of the free surface of the fluid and the rate of retention of the liquid lubricant on the spinning disk have been obtained for different values of the two parameters M , the Hartmann number and N_ratio, the ratio of the surface tension effect to the centrifugation effect. In the absence of the magnetic field, the results have been observed to agree with those of [6]. It has been observed that the effect of surface roughness is to enhance the relative volume of the fluid retained on the spinning disk and this is further enhanced by the presence of the magnetic field.     

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 a fast method for computing the field of waves reflected by a thin elastic layer

A method for computing the field of waves reflected by a thin elastic layer is proposed. The method takes into account all kinds of exchange type reflections and is based on the summation of kinematic and dynamic analogues.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 152–162, 1989.     

Time-marching numerical schemes for the electric field integral equation on a straight thin wire

We derive and analyse four algorithms for computing the current induced on a thin straight wire by a transient electric field. They all involve solving the thin wire electric field integral equations (EFIEs) and consist of a very accurate differential equations solver together with various schemes to approximate the vector potential integral equation. We carry out a rigorous numerical stability analysis of each of these methods. This has not previously been done for solution schemes for the thin wire EFIEs. Each scheme is shown to be stable and convergent provided the radius of the wire is small enough for the thin wire equations to be a valid model.     

The interaction of a system of weakened zones at the interface of elastic media in a tensile stress field

The interaction of a system of crack-like defects with distributed cohesive forces over the whole surface of the edges, located at the interface of two elastic half-planes and which open under the action of forces at infinity, is considered. A dislocation approach is used to describe the model of each defect: the discontinuity in the asymmetric shifts is specified in the form of a basis function with free parameters that satisfies a number of physical constraints. The free parameters of the model are determined when finding an analytical solution of the problem. The key questions are: what is the minimum load at which just one of these weakened zones is converted into the nucleus of a crack or when one of the connecting bridges separating these zones is fractured and, also, under what conditions can the interaction of the defects be neglected ? The model is extended with a relation which enables an explicit opening - bonding force dependence to be obtained.     

Impact of a cone on a thin elastic membrane

The problem of computation of parameters of motion of a thin elastic membrane under the Impact of a rigid body, was considered by various authors (see [1] together with bibliography) without, however, yielding a rational solution. This paper presents a full qualitative analysis of solution of this problem for the case of normal impact of a circular cone moving with constant velocity on an infinite elastic membrane of constant thickness. Although this is the simplest case, it is important, insofar as it brings to light the characteristic features of the problem. In the “membrane” approximation the thickness of the layer is found to be an unessential parameter, therefore the problem, as postulated by us, is self-similar and its solution is reducible to the problem for ordinary differential equations.     

A numerical study of singular stress field of 3D cracks

We investigate the stress distribution and the variation of the mode I stress intensity factor along a straight three-dimensional (3D) crack by the finite element method. The results are checked against plane strain theory near the mid-crack and against the 3D theory of Zhu at the free surface. Although Zhu's formulation is not perfect and has some typographical errors. The surface stress distribution of his results are in line with the present study by the finite element method. The stress intensity factors at the free surface are found to be much lower than that at the mid-crack.     

On the determination of higher order terms of singular elastic stress fields near corners

Summary In two-dimensional elasticity stresses at reentrant corners exhibit singular behavior. The stress field is of the form , where (r, ) are polar coordinates centered at the tip of the corner, andf _i (; _i are smooth functions. For practical use of this series the eigenvalues _i (which are generally complex numbers) are required in order of ascending real part. The problem then is to find the roots of a transcendental equation (eigenequation) in the complex plane and arranged in order of ascending real part.A theorem is proved on the number, location and nature of the roots of this equation with the real part in fixed intervals of length . Excellent initial estimates of the real part of the complex roots become available, and so are bounds, within which single real roots exist. This enables the determination of any number of roots in ascending order of real part. The critical angles at which the eigenvalues change nature are also determined. It is shown that for certain cases and for the symmetric mode of deformation, the eigenvalue =1 does not represent a rigid body rotation, therefore it has to be included in the series representation of the stresses. The coefficientsK _i can be determined by recently developed extraction techniques, thus allowing complete determination of the elastic field and enabling its correlation with experimental data on brittle fracture, crack initiation, plastic zone estimation etc.Dedicated to Professor Ivo Babuka on the occasion of his 60th birthdayPresented at the Conference: The Impact of Mathematical Analysis on the Solution of Engineering Problems, 17–19 September 1986, University of Maryland, College Park, Maryland, USA     

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.     

Asymptotic analysis of the Stokes flow in a thin cylindrical elastic tube

The purpose of this article is to perform an asymptotic analysis for an interaction problem between a viscous fluid and an elastic structure when the flow domain is a three-dimensional cylindrical tube. We consider a periodic, non-steady, axisymmetric, creeping flow of a viscous incompressible fluid through a long and narrow cylindrical elastic tube. The creeping flow is described by the Stokes equations and for the wall displacement we consider the Koiter's equation. The well posedness of the problem is proved by means of its variational formulation. We construct an asymptotic approximation of the problem for two different cases. In the first case, the stress term in Koiter's equation contains a great parameter as a coefficient and dominates with respect to the inertial term while in the second case both the terms are of the same order and contain the great parameter. An asymptotic analysis is developed with respect to two small parameters. Analysing the leading terms obtained in the second case, we note that the wave phenomena takes place. The small error between the exact solution and the asymptotic one justifies the below constructed asymptotic expansions.     

A numerical study of the dynamics of elastic articulated systems in a fluid

We propose a method and an algorithm for computing the dynamics of elastic structures of articulated form in a fluid flow taking account of the weakening in certain structural elements. In describing the motion we use two sets of radius-vectors, which are approximated in the computations by parametric local splines of first degree. The possibilities of the proposed method are illustrated using the example of the study of the dynamics of transition processes in an articulated anchor-buoy structure, which arise when there is an abrupt change in the direction of the fluid flow velocity. We determine the kinematic and force characteristics of the structure under various changes in the direction of the flow velocity. We determine the structural elements in which the weakening occurs. Three figures. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 128–134.     

Unilateral contact of a plate with a thin elastic obstacle

We study the problem of contact of an elastic body with a beam. The most attention is paid to describing boundary conditions on the possible contact set. Moreover, we study asymptotic properties of solutions and the energy functional as the rigidity parameters tend to infinity or the length of the beam (or the zone of possible contact) changes.     

Contact conditions of elastic media with a thin interlayer

Contact conditions are proposed for elastic media approximately describing the influence of thin interlayers with linear and nonlinear rheological properties. The boundary conditions are constructed on the basis of assumptions on the quasistatic loading of thin interlayers.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 173, pp. 163–166, 1988.     

Asymptotics for a thin elastic fiber in contact with a rigid foundation

In this work a 3-D contact elasticity problem for a thin fiber and a rigid foundation is studied. We describe the contact condition by a linear Robin-boundary-condition (by meaning of the penalized and linearized non-penetration and friction conditions). The Robin parameters are scaled differently in the longitudinal and cross-sectional directions. The dimension of the problem is reduced by a standard ([3], [4]) asymptotic approach with an additional expansion suggested to fulfil the contact conditions. The 3-D contact conditions result into 1-D Robin-boundary-conditions for corresponding ODEs. The Robin-coefficients of the 1-D problem depend on the ones from the 3-D statement and on the cross-section of the fiber. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Direct numerical simulation of particles in a fluid interacting with a wall

We develop a code to be applied in the context of the cleaning of wafer surfaces by hydrodynamic forces. Our goal is to study the detachment of (submicron) particles, exposed to a shear flow, from a wall by means of direct numerical simulation. The particles are treated as rigid bodies fully interacting with the fluid. To simulate moving particles in the fluid we implement an immersed boundary method with direct forcing into OpenFOAM. The particle-wall interaction is treated with a soft contact model. As first simple examples we study the elastic normal impact of a cylinder onto a wall as well as the onset of sliding of a cylinder pressed to a horizontal wall by gravity under a time-depended drag force. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Random vibrations of thin elastic plates

Zusammenfassung Das Verfahren des mittleren Fehlerquadrates für Zufallsfunktionen zweiter Ordnung wird auf das Problem der Querschwingungen dünner elastischer Platten unter Zufallsbelastung angewandt. Es werden Ausdrücke gefunden für die Kovarianz und die Spektraldichte der seitlichen Verschiebung und der Spannung in Abhängigkeit der Kovarianz und Spektraldichte der Zufallsfunktion für die Belastung.