Moving contact problems involving a rigid punch and a functionally graded coating

Frictional contact mechanics analysis for a rigid moving punch of an arbitrary profile and a functionally graded coating/homogeneous substrate system is carried out. The rigid punch slides over the coating at a constant subsonic speed. Smooth variation of the shear modulus of the graded coating is defined by an exponential function and the variation of the Poisson's ratio is assumed negligible. Coulomb's friction law is adopted. Hence, tangential force is proportional to the normal applied force through the coefficient of friction. An analytical method is developed utilizing the singular integral equation approach. Governing partial differential equations are derived in accordance with the theory of elastodynamics. The mixed boundary value problem is reduced to a singular integral equation of the second kind, which is solved numerically by an expansion-collocation technique. Presented results illustrate the effects of punch speed, coefficient of friction, material inhomogeneity and coating thickness on contact stress distributions and stress intensity factors. Comparisons indicate that the difference between elastodynamic and elastostatic solutions tends to be quite larger especially at higher punch speeds. It is shown that use of the elastodynamic theory provides more realistic results in contact problems involving a moving punch.     

Electro-mechanical sliding frictional contact of a piezoelectric half-plane under a rigid conducting punch

This paper investigates the two-dimensional sliding frictional contact of a piezoelectric half-plane in the plane strain state under the action of a rigid flat or a triangular punch. It is assumed that the punch is a perfect electrical conductor with a constant electric potential. By using the Fourier integral transform technique and the superposition theorem, the problem is reduced to a pair of coupled Cauchy singular integral equations and then is numerically solved to determine the unknown contact pressure and surface electric charge distribution. The effects of the friction coefficient and electro-mechanical loads on the normal contact stress, normal electric displacement, in-plane stress and in-plane electric displacement are discussed in detail. It is found that the friction coefficient has a significant effect on the electro-mechanical sliding frictional contact behaviors of the piezoelectric materials.     

Harmonic oscillations of a rigid punch on a porous elastic layer

The plane dynamic contact problem of the harmonic oscillations of a rigid punch on the free surface of an elastic layer of porous isotropic material with linear properties is considered. The Fourier transformation of the problem is reduced to a Fredholm integral equation of the first kind in the contact pressure. The properties of the kernel of the fundamental integral equation are investigated and a numerical method of solving it is constructed. Numerical results are compared with existing results in classical limiting cases.     

An equivalent mechanical model for fluid sloshing in a rigid cylindrical tank equipped with a rigid annular baffle

A mass-spring mechanical model for linear sloshing of fluid in a rigid cylindrical tank with a rigid annual baffle is developed. By means of the subdomain method, the complicated fluid domain is divided into several subdomains with pure boundary conditions and interface conditions. Combined with the continuity conditions of velocity and pressure on interfaces, the velocity potential of fluid in each subdomain is analytically solved. The mass-spring model for fluid sloshing is established by generating the same hydrodynamic shear force and overturning moment as those from the exact solutions when the tank is subjected to horizontal excitations. The present model gives all the mechanical parameters including the convective mass-spring oscillators and the impulsive mass as well as their positions. Comparative studies between the present solutions and the available results verify the correctness of the mechanical model. Based on the normalized equivalent masses and spring stiffnesses, the dynamic responses of fluid in tanks are discussed with respect to the fluid height, the baffle position and the inner radius of the baffle, respectively. The present model is especially suitable for dealing with complicated liquid-structure systems. The motivation and novelty of this study are confirmed by an application of the present model to a multi-tank system.     

The indentation of a transversely isotropic half-space by a rigid punch

Zusammenfassung Der Verfasser gibt eine allgemeine Lösung für die Verteilung des Druckes zwischen einem axialsymmetrischen Stempel und einem transversal-isotropen Halbraum. Es wird gezeigt, dass die Verteilung des Druckes für den flachen Stempel mit allgemeiner Belastung unabhängig ist von den elastischen Eigenschaften des Halbraums und auch genau dieselbe, als ob der Halbraum isotrop wäre.     

A BEM for transient thermoelastic analysis of a functionally graded layer on a homogeneous substrate under thermal shock

Transient thermoelastic analysis of isotropic and linear thermoelastic bimaterials, which are constituted by a functionally graded (FG) layer attached to a homogeneous substrate, subjected to thermal shock is presented in this paper. For this purpose, a boundary element method for transient linear coupled thermoelasticity is developed. The material properties of the FG layer are assumed to be continuous functions of the spatial coordinates. The boundary-domain integral equations are derived by using the fundamental solutions of linear coupled thermoelasticity for the corresponding isotropic, homogeneous and linear thermoelastic solids in the Laplace-transformed domain. For the numerical solution, a collocation method with piecewise quadratic approximation is implemented. Numerical results for the dynamic stress intensity factors are presented and discussed. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interaction of a rigid punch with a base-secured elastic rectangle with stress-free sides

The plane contact problem of the indentation of a rigid punch into a base-sucured elastic rectangle with stress-free sides is considered. The problem is solved by a method tested earlier and reduces to a system of two integral equations in functions describing the displacement of the surface of the rectangle outside the punch and the normal or shear stress on its base. These functions are sought in the form of the sum of trigonometric series and an exponential function with a root singularity. The ill-posed infinite systems of algebraic equations obtained as a result of this are regularized by introducing small positive parameters. Because the matrix elements of the systems, and also the contact stresses, are defined by poorly converging numerical and functional series, the previously developed method of summation of these series is used. The contact pressure distribution and the dimensionless indenting force are found. Examples of a plane punch calculation are given.     

Static frictional indentation of an elastic half-plane by a rigid unsymmetrical punch

Static rigid 2-D indentation of a linearly elastic half-plane in the presence of Coulomb friction which reverses its sign along the contact length is studied. The solution approach lies within the context of the mathematical theory of elastic contact mechanics. A rigid punch, having an unsymmetrical profile with respect to its apex and no concave regions, both slides over and indents slowly the surface of the deformable body. Both a normal and a tangential force may, therefore, be exerted on the punch. In such a situation, depending upon the punch profile and the relative magnitudes of the two external forces, a point in the contact zone may exist at which the surface friction changes direction. Moreover, this point of sign reversal may not coincide, in general, with the indentor's apex. This position and the positions of the contact zone edges can be determined only by first constructing a solution form containing the three problem's unspecified lengths, and then solving numerically a system of non-linear equations containing integrals not available in closed form.The mathematical procedure used to construct the solution deals with the Navier-Cauchy partial differential equations (plane-strain elastostatic field equations) supplied with boundary conditions of a mixed type. We succeed in formulating a second-kind Cauchy singular integral equation and solving it exactly by analytic-function theory methods.Representative numerical results are presented for two indentor profiles of practical interest—the parabola and the wedge.     

Hertz problem for a rigid punch moving across the surface of a semi-infinite elastic solid

The elastodynamic problem of a rigid punch moving at a constant sub-Rayleigh speed across the surface of an elastic half-space is investigated in the present paper. The unknown contact region is determined as part of solution from the unilateral or Signorini conditions. Numerical results are plotted showing how the eccentricity of the contact ellipse changes with the punch speed. Some asymptotic properties of the solution for the case where the punch speed is comparable with the Rayleigh wave speed are explored in details.     

Stability of localized solutions under rigid loading in a heuristic buckling model

The stability of static localized buckling solutions which bifurcatefrom the critical load for a strut on an elastic foundationis analysed under conditions of rigid loading (i.e. with prescribedend shortening) using a non-periodic modal analysis. The characteristicof such deflection patterns is in marked contrast with stabilityunder the more straightforward case of dead loading (behaviourunder prescribed load). The model presented incorporates initialdestabilization and subsequent restabilization of the responseat the lowest order.     

The effect of heat flow on the contact area between a continuous rigid punch and a frictionless elastic half-space

A general proof is given of the theorem that, if a continuous rigid punch indenting an elastic half space is heated, the separation of the solids will increase and part of the contact area will be lost. It is also shown that if the punch is convex, the contact area cannot be multiply-connected.

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Résumé On donne une preuve générale du théorème que, si un poinçon continu rigid, pénétrant un demi-espace élastique, est chauffé, la séparation des solides grandira et une partie de la région de contact sera perdue. On montre également que la région de contact ne peut pas être multiplement connexe si le poinçon est convexe.

     

Linear and nonlinear dynamics of a circular cylindrical shell connected to a rigid disk

The dynamics of a circular cylindrical shell carrying a rigid disk on the top and clamped at the base is investigated. The Sanders–Koiter theory is considered to develop a nonlinear analytical model for moderately large shell vibration. A reduced order dynamical system is obtained using Lagrange equations: radial and in-plane displacement fields are expanded by using trial functions that respect the geometric boundary conditions.The theoretical model is compared with experiments and with a finite element model developed with commercial software: comparisons are carried out on linear dynamics.The dynamic stability of the system is studied, when a periodic vertical motion of the base is imposed. Both a perturbation approach and a direct numerical technique are used. The perturbation method allows to obtain instability boundaries by means of elementary formulae; the numerical approach allows to perform a complete analysis of the linear and nonlinear response.     

Factors in the adhesion strength of polymers toward a solid substrate

The true adhesion is looked upon as the sum of the experimentally determined strength of the glued area, internal stresses, and unaccountable technological losses (unrealized contacts). It has been established that the internal stresses do not depend on the value of the adhesion strength.Translated from Mekhanika Polimerov, No. 2, pp. 356–359, March–April, 1976.     

Buckling and vibration analysis of nonlocal axially functionally graded nanobeams based on Hencky-bar chain model

Buckling and free vibration analyses of nonlocal axially functionally graded Euler nanobeams is the main objective of this paper. Due to its simplicity, the Eringen's differential constitutive model is adopted for describing the nonlocal size dependency of nanostructure beam. The nonlocal equilibrium equation is derived using the principle of the minimum potential energy principle, and discretized by using the link-spring model known in literature as Hencky bar-chain model. The general applicability of the proposed approach allows analyses of functional graded microbeams without any restriction on variability, boundary and loading conditions. A comparison with results available in the literature shows the reliability of the method.     

Wave operators in a three-particle model of a rigid kernel

For the three-particle scattering problem in the model of a rigid kernel one establishes the existence, the isometry, and the completeness of the wave operators. One proves the absolute continuity of the Hamiltonian system.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 152, pp. 45–49, 1986.     

Interaction of a hollow cylinder of finite length and a plate with a cylindrical cavity with a rigid insert

Two problems of the interaction of a hollow circular cylinder with load-free ends and an unbounded plate with a cylindrical cavity and a symmetrically imbedded rigid insert are considered. Homogeneous solutions are found and the generalized orthogonality of these solutions is used when the modified boundary conditions are satisfied. As a result, we have a system of two integral equations in functions of the displacements of the outer and inner surfaces of the hollow cylinder. These functions are sought in the form of sums of a trigonometric series and a power function with a root singularity. The ill-posed infinite systems of linear algebraic equations obtained are regularized by the introduction of small positive parameters. Since the elements of the matrices of the systems as well as the contact stresses are defined by poorly converging numerical and functional series, an efficient method for calculating of the remainders of the above-mentioned series is developed. Formulae are found for the contact pressure distribution function and the integral characteristic. Examples of the calculation of the interaction of the cylinder and the plate with an insert are given.The method of solving contact problems described here has been used earlier1, 2 and the generalized orthogonality of the solutions found for bodies of finite dimensions, that is, for a rectangle and cylinders of finite length, is its basis. Problems for hollow cylinders with a band ² and an insert reduce to a system of two integral equations, and the problem for a rectangle¹ reduces to one integral equation. Solving these integral equations, ill-posed systems of linear algebraic equations are obtained which are subject to regularization³.     

Creep of a thermoplastic cylindrical shell under axisymmetric loading

The solution of the problem of creep of an axisymmetrically loaded thermoplastic cylindrical shell is considered. The strains and stresses for the zero-moment zone and with allowance for the edge effect are predicted on the basis of Kachanov's variational method using a computer. An algorithm is constructed for polyethylene and PVC shells at various values of the load. The analysis of the results obtained is illustrated by the calculation data for individual variants of the program.Leningrad Mozhaiskii Military Engineering Academy. Translated from Mekhanika Polimerov, No. 3, pp. 512–518, May–June, 1969.     

Two-dimensional nonlinear advection-diffusion in a model of surfactant spreading on a thin liquid film

The spreading of a localized monolayer of dilute, insoluble surfactant, discharged from a point source that moves at constant speed over a thin liquid film coating a planar substrate, is described according to lubrication theory by a pair of coupled nonlinear evolution equations for the monolayer concentration and the film depth h. Numerical and asymptotic techniquesare here used to show that the extent and structure of sucha spreading asymmetric monolayer can be well approximated bya single nonlinear advection–diffusion equation involving alone. At large times the solution is composed of three, spatiallydistinct, asymptotic regions: (i) a quasi-steady ‘nose’region (containing the source), in which there is a dominantbalance between two-dimensional nonlinear diffusion and advection;(ii) an ‘advective’ region, in which longitudinaladvection balances transverse diffusion; and (iii) a ‘tail’region, in which unsteady diffusion is dominant. In each region,local similarity solutions are obtained either exactly (inthe advective region) or approximately (elsewhere) by rescalingnumerical solutions of the initial-value problem. If the sourceconcentration decreases with time, it is demonstrated that the monolayer’s width is greatest in the tail region, whereasfor a source of increasing concentration the monolayer is widestin the advective region. For the simpler one-dimensional problemof a monolayer spreading from a line source, the same balanceshold but with transverse diffusion eliminated; here self-similarsolutions are found in all three regions that agree closelywith numerical solutions of the initial-value problem. Received 7 October, 1998. Revised 11 April, 2000. ⁺ antoine@mip.ups-tlse.fr Present address: Division of Theoretical Mechanics, Schoolof Mathematical Sciences, University of Nottingham , UniversityPark, Nottingham NG7 2RD, UK. Oliver.Jensen@nottingham.ac.uk.     

Axisymmetric contact of a punch of polynomial profile with an elastic half-space when there is friction and adhesion

The axisymmetric problem of the contact interaction of a punch of polynomial profile and an elastic half-space when there is friction and partial adhesion in the contact area is considered. Using the Wiener–Hopf method the problem is reduced to an infinite system of algebraic Poincare–Koch equations, the solution of which is obtained in series. The radii of the contact area and of the adhesion zone, the distribution of the contact pressures and the indentation of the punch are obtained.     

Two-dimensional model of a strip detector with two tunnel junctions

The article describes a two-dimensional model of a strip X-ray detector with a superconducting absorbing strip and two tunnel junctions at its ends, The motion of nonequilibrium current carriers (quasiparticles) is considered in the framework of a diffusion equation with boundary conditions of third kind. The detector signals Q _l and Q _r and the spectral line shape are computed as a function of the strip size, quasiparticle capture efficiency in the tunnel junction, the quasiparticle loss parameter on the absorber lateral boundaries, and the quasiparticle recombination parameter R. Analytical solutions for the detector signals Q _l and Q _r are derived for the case R = 0 . Numerical calculations are generally used. When boundary and recombination losses are allowed for, the detector signals turn out to depend on the photon absorption coordinate in the transverse direction. This broadens and distorts the spectral line shape.