Numerical Approximation of Basic Boundary-Contact Problems

2017 ◽  
Author(s):  
Manana Chumburidze ◽  
David Lekveishvili
Keyword(s):  
Numerical Approximation ◽  
Singular Integral Equations ◽  
Contact Problems ◽  
Potential Method ◽  
Static System ◽  
Elastic Materials ◽  
Generalized Fourier Series ◽  
Boundary Contact ◽  
Basic Boundary ◽  
Fourier Series Analysis

This paper is devoted to the development of approximation method for numerical solution of basic boundary-contact problems of coupled thermo-elasticity in the Green-Lindsay formulation. In particular, we consider a static system of partial differential equations for two-dimensional isotropic inhomogeneous elastic materials in assumptions that surfaces are sufficiently smooth. The tools applied in this development are based on singular integral equations, the potential method and the generalized Fourier series analysis.

Asymptotic Distribution of Eigenelements of the Basic Two-Dimensional Boundary-Contact Problems of Oscillation in Classical and Couple-Stress Theories of Elasticity

2000 ◽  
Vol 7 (1) ◽  
pp. 11-32
Author(s):  
T. Burchuladze ◽  
R. Rukhadze
Keyword(s):  
Couple Stress ◽  
Correlation Method ◽  
Contact Problems ◽  
Asymptotic Formulas ◽  
Two Dimensional ◽  
Closed Curves ◽  
Dimensional Boundary ◽  
Isotropic Elastic Medium ◽  
Boundary Contact ◽  
Basic Boundary

Abstract The basic boundary-contact problems of oscillation are considered for a two-dimensional piecewise-homogeneous isotropic elastic medium bounded by several closed curves. Asymptotic formulas for the distribution of eigenfunctions and eigenvalues of the considered problems are derived using the correlation method.

scholarly journals CONTACT PROBLEMS FOR NONLINEARLY ELASTIC MATERIALS: WEAK SOLVABILITY INVOLVING DUAL LAGRANGE MULTIPLIERS

2010 ◽  
Vol 52 (2) ◽  
pp. 160-178 ◽  
Author(s):  
A. MATEI ◽  
R. CIURCEA
Keyword(s):  
Lagrange Multipliers ◽  
Variational Equation ◽  
Small Deformation ◽  
Contact Problems ◽  
Point Theory ◽  
The Body ◽  
Elastic Materials ◽  
Weak Formulation ◽  
Nonlinearly Elastic ◽  
Weak Solvability

AbstractA class of problems modelling the contact between nonlinearly elastic materials and rigid foundations is analysed for static processes under the small deformation hypothesis. In the present paper, the contact between the body and the foundation can be frictional bilateral or frictionless unilateral. For every mechanical problem in the class considered, we derive a weak formulation consisting of a nonlinear variational equation and a variational inequality involving dual Lagrange multipliers. The weak solvability of the models is established by using saddle-point theory and a fixed-point technique. This approach is useful for the development of efficient algorithms for approximating weak solutions.

An Accelerated Retarded Potential Method for Time Dependent Acoustic Wave Scattering

1995 ◽  
Author(s):  
Toufic S. Abboud ◽  
Joseph M. Gharib ◽  
Jean Claude Nédélec ◽  
Toni G. Sayah
Keyword(s):  
Finite Element Method ◽  
Integral Equation ◽  
Wave Scattering ◽  
Numerical Approximation ◽  
Mesh Size ◽  
Integral Equation Method ◽  
Potential Method ◽  
Boundary Integral ◽  
Time Dependent ◽  
Acoustic Wave Scattering

Abstract We are interested in the numerical approximation of the problem of the scattering of a transient acoustic plane wave by a bounded obstacle in IR2 or IR3, using the boundary integral equation method. In the frequency domain it has been recently developed a boundary finite element method where the mesh size is like O(λ1/3) instead of O(λ) (λ is the wavelength) and where the obstacle is convex. This paper presents the implementation of the idea on the retarted potential representation.

Load History Effects on Fretting Contacts of Isotropic Materials

2004 ◽  
Vol 126 (2) ◽  
pp. 385-390 ◽  
Author(s):  
P. T. Rajeev ◽  
H. Murthy ◽  
T. N. Farris
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Singular Integral Equations ◽  
Contact Problems ◽  
Two Dimensional ◽  
Load History ◽  
Jet Engine ◽  
Isotropic Materials ◽  
History Effects ◽  
Cauchy Singular Integral

The load history that blade/disk contacts in jet engine attachment hardware are subject to can be very complex. Using finite element method (FEM) to track changes in the contact tractions due to changing loads can be computationally very expensive. For two-dimensional plane-strain contact problems with friction involving similar/dissimilar isotropic materials, the contact tractions can be related to the initial gap function and the slip function using coupled Cauchy singular integral equations (SIEs). The effect of load history on the contact tractions is illustrated by presenting results for an example fretting “mission.” For the case of dissimilar isotropic materials the mission results show the effect of the coupling between the shear traction and the contact pressure.

scholarly journals Some Problems of Thermoelastic Equilibrium of a Rectangular Parallelepiped in Terms of Asymmetric Elasticity

2001 ◽  
Vol 8 (4) ◽  
pp. 767-784
Author(s):  
N. Khomasuridze
Keyword(s):  
Thermal Field ◽  
Separation Of Variables ◽  
Contact Problems ◽  
Rectangular Parallelepiped ◽  
Effective Solution ◽  
Contact Conditions ◽  
Thermoelastic Equilibrium ◽  
Boundary Contact ◽  
Asymmetric Elasticity ◽  
Surface Disturbances

Abstract An effective solution of a number of boundary value and boundary contact problems of thermoelastic equilibrium is constructed for a homogeneous isotropic rectangular parallelepiped in terms of asymmetric and pseudo-asymmetric elasticity (Cosserat's continuum and pseudo- continuum). Two opposite faces of a parallelepiped are affected by arbitrary surface disturbances and a stationary thermal field, while for the four remaining faces symmetry or anti-symmetry conditions (for a multilayer rectangular parallelepiped nonhomogeneous contact conditions are also defined) are given. The solutions are constructed in trigonometric series using the method of separation of variables.

Some Criteria for Coating Effectiveness in Heavily Loaded Line Elastohydrodynamically Lubricated Contacts—Part II: Lubricated Contacts

2015 ◽  
Vol 138 (2) ◽  
Author(s):  
Ilya I. Kudish ◽  
Sergey S. Volkov ◽  
Andrey S. Vasiliev ◽  
Sergey M. Aizikovich
Keyword(s):  
Analytical Solution ◽  
Contact Problem ◽  
Half Space ◽  
Contact Problems ◽  
Functionally Graded ◽  
Elastic Solids ◽  
Elastic Materials ◽  
Lubricated Contacts ◽  
Dry Contact ◽  
Appl Math

Contacts of indentors with functionally graded elastic solids may produce pressures significantly different from the results obtained for homogeneous elastic materials (Hertzian results). It is even more so for heavily loaded line elastohydrodynamically lubricated (EHL) contacts. The goal of the paper is to indicate two distinct ways the functionally graded elastic materials may alter the classic results for the heavily loaded line EHL contacts. Namely, besides pressure, the other two main characteristics which are influenced by the nonuniformity of the elastic properties of the contact materials are lubrication film thickness and frictional stress/friction force produced by lubricant flow. The approach used for analyzing the influence of functionally graded elastic materials on parameters of heavily loaded line EHL contacts is based on the asymptotic methods developed earlier by the authors such as Kudish (2013, Elastohydrodynamic Lubrication for Line and Point Contacts: Asymptotic and Numerical Approaches, Chapman & Hall/CRC Press, Boca Raton, FL), Kudish and Covitch (2010, Modeling and Analytical Methods in Tribology, Chapman & Hall/CRC Press, Boca Raton, FL), Aizikovich et al. (2002, “Analytical Solution of the Spherical Indentation Problem for a Half-Space With Gradients With the Depth Elastic Properties,” Int. J. Solids Struct., 39(10), pp. 2745–2772), Aizikovich et al. (2009, “Bilateral Asymptotic Solution of One Class of Dual Integral Equations of the Static Contact Problems for the Foundations Inhomogeneous in Depth,” Operator Theory: Advances and Applications, Birkhauser Verlag, Basel, p. 317), Aizikovich and Vasiliev (2013, “A Bilateral Asymptotic Method of Solving the Integral Equation of the Contact Problem for the Torsion of an Elastic Halfspace Inhomogeneous in Depth,” J. Appl. Math. Mech., 77(1), pp. 91–97), Volkov et al. (2013, “Analytical Solution of Axisymmetric Contact Problem About Indentation of a Circular Indenter Into a Soft Functionally Graded Elastic Layer,” Acta Mech. Sin., 29(2), pp. 196–201), Vasiliev et al. (2014, “Axisymmetric Contact Problems of the Theory of Elasticity for Inhomogeneous Layers,” Z. Angew. Math. Mech., 94(9), pp. 705–712), Aizikovich et al. (2008, “The Deformation of a Half-Space With a Gradient Elastic Coating Under Arbitrary Axisymmetric Loading,” J. Appl. Math. Mech., 72(4), pp. 461–467), and Aizikovich et al. (2010, “Inverse Analysis for Evaluation of the Shear Modulus of Inhomogeneous Media in Torsion Experiments,” Int. J. Eng. Sci., 48(10), pp. 936–942). More specifically, it is based on the analysis of contact problems for dry contacts of functionally graded elastic solids and the lubrication mechanisms in the inlet and exit zones as well as in the central region of heavily loaded lubricated contacts. The way the solution of the EHL problem for coated/functionally graded materials is obtained provides a very clear structure of the solution. The solution of the EHL problem in the Hertzian region is very close to the solution of the dry contact problem while in the inlet and exit zones the solutions of the EHL problem with the right asymptotes coming from the solution of the dry contact problem can be related to the solutions of the classic EHL problem for homogeneous materials.

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