The Hertz contact problem, coupled Volterra integral equations and a linear complementarity problem

A. Gauthier, P.A. Knight, S. McKee

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

This paper is concerned with the indentation of an elastic half-space by an axisymmetric punch under a monotonically applied normal force and under the assumption of Coulomb friction with coefficient μ in the region of contact. Within an inner (unknown) circle the contact is adhesive, while in the surrounding annulus the surface moves inwards with increasing load. In this paper it is shown how this problem is equivalent to two coupled Abel's equations with an unknown free point, the inner circumference of the annulus. It is further shown that a product integration finite difference approximation of those integral equations leads to a mixed linear complementarity problem (mixed LCP). A method based on Newton's method for solving non-smooth nonlinear equations is demonstrated to converge under restrictive assumptions on the physical parameters defining the system; and numerical experimentation verifies that it has much wider applicability. The method is also validated against the approach of Spence. The advantage of the mixed LCP formulation is that it provides the radius of the inner adhesive circle directly using the physical parameters of the problem.
LanguageEnglish
Pages322-340
Number of pages18
JournalJournal of Computational and Applied Mathematics
Volume206
Issue number1
DOIs
Publication statusPublished - Sep 2007

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Mixed Complementarity Problem
Linear Complementarity Problem
Volterra Integral Equations
Adhesives
Ring or annulus
Contact Problem
Integral equations
Circle
Contact
Product Integration
Abel Equation
Nonsmooth Equations
Unknown
Coulomb Friction
Circumference
Finite Difference Approximation
Indentation
Newton-Raphson method
Nonlinear equations
Newton Methods

Keywords

  • Hertz contact problem
  • Abel's integral equations
  • Linear complementarity problem

Cite this

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title = "The Hertz contact problem, coupled Volterra integral equations and a linear complementarity problem",
abstract = "This paper is concerned with the indentation of an elastic half-space by an axisymmetric punch under a monotonically applied normal force and under the assumption of Coulomb friction with coefficient μ in the region of contact. Within an inner (unknown) circle the contact is adhesive, while in the surrounding annulus the surface moves inwards with increasing load. In this paper it is shown how this problem is equivalent to two coupled Abel's equations with an unknown free point, the inner circumference of the annulus. It is further shown that a product integration finite difference approximation of those integral equations leads to a mixed linear complementarity problem (mixed LCP). A method based on Newton's method for solving non-smooth nonlinear equations is demonstrated to converge under restrictive assumptions on the physical parameters defining the system; and numerical experimentation verifies that it has much wider applicability. The method is also validated against the approach of Spence. The advantage of the mixed LCP formulation is that it provides the radius of the inner adhesive circle directly using the physical parameters of the problem.",
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AU - Gauthier, A.

AU - Knight, P.A.

AU - McKee, S.

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N2 - This paper is concerned with the indentation of an elastic half-space by an axisymmetric punch under a monotonically applied normal force and under the assumption of Coulomb friction with coefficient μ in the region of contact. Within an inner (unknown) circle the contact is adhesive, while in the surrounding annulus the surface moves inwards with increasing load. In this paper it is shown how this problem is equivalent to two coupled Abel's equations with an unknown free point, the inner circumference of the annulus. It is further shown that a product integration finite difference approximation of those integral equations leads to a mixed linear complementarity problem (mixed LCP). A method based on Newton's method for solving non-smooth nonlinear equations is demonstrated to converge under restrictive assumptions on the physical parameters defining the system; and numerical experimentation verifies that it has much wider applicability. The method is also validated against the approach of Spence. The advantage of the mixed LCP formulation is that it provides the radius of the inner adhesive circle directly using the physical parameters of the problem.

AB - This paper is concerned with the indentation of an elastic half-space by an axisymmetric punch under a monotonically applied normal force and under the assumption of Coulomb friction with coefficient μ in the region of contact. Within an inner (unknown) circle the contact is adhesive, while in the surrounding annulus the surface moves inwards with increasing load. In this paper it is shown how this problem is equivalent to two coupled Abel's equations with an unknown free point, the inner circumference of the annulus. It is further shown that a product integration finite difference approximation of those integral equations leads to a mixed linear complementarity problem (mixed LCP). A method based on Newton's method for solving non-smooth nonlinear equations is demonstrated to converge under restrictive assumptions on the physical parameters defining the system; and numerical experimentation verifies that it has much wider applicability. The method is also validated against the approach of Spence. The advantage of the mixed LCP formulation is that it provides the radius of the inner adhesive circle directly using the physical parameters of the problem.

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KW - Abel's integral equations

KW - Linear complementarity problem

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