Algorithm for the solution of elastoplastic half-space impact: force-indentation linearisation method

Akuro Big-Alabo, Philip Harrison, Matthew P Cartmell

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6 Citations (Scopus)

Abstract

The governing equation of a half-space impact is generally nonlinear and it is normally solved using numerical techniques that are mostly conditionally stable and require many iteration steps for convergence of the solution. In this paper, we present the force-indentation linearisation method (FILM), an approximate technique that produces closed-form solutions of piecewise linearisation of the governing nonlinear differential equation and is capable of producing accurate impact response for an elastoplastic half-space impact. In contrast to the existing numerical techniques, which discretise the impact force or variable of interest in the time-domain, the present technique discretises the impact force with respect to the indentation using successive piecewise linear approximations. Generalised closed-form solutions were derived for each piecewise approximation, and this was used to develop an iterative algorithm for updating the solutions from one piecewise approximation to the next. The results of the present technique matched with results obtained by direct numerical integration of the governing nonlinear differential equation for a half-space impact, and the FILM was found to converge to the results of the numerical solution after a few iterations, typically between five and 10 iterations. The FILM is simple, is inherently stable, converges quickly, gives accurate results and can be implemented manually; these features make it potentially more attractive than the comparable numerical methods.
LanguageEnglish
Pages850-858
Number of pages9
JournalProceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science
Volume229
Issue number5
Early online date27 Jun 2014
DOIs
Publication statusPublished - 30 Apr 2015

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Indentation
Linearization
Differential equations
Numerical methods

Keywords

  • half-space
  • impact
  • elastoplastic
  • contact law
  • force-indentation linearisation method

Cite this

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title = "Algorithm for the solution of elastoplastic half-space impact: force-indentation linearisation method",
abstract = "The governing equation of a half-space impact is generally nonlinear and it is normally solved using numerical techniques that are mostly conditionally stable and require many iteration steps for convergence of the solution. In this paper, we present the force-indentation linearisation method (FILM), an approximate technique that produces closed-form solutions of piecewise linearisation of the governing nonlinear differential equation and is capable of producing accurate impact response for an elastoplastic half-space impact. In contrast to the existing numerical techniques, which discretise the impact force or variable of interest in the time-domain, the present technique discretises the impact force with respect to the indentation using successive piecewise linear approximations. Generalised closed-form solutions were derived for each piecewise approximation, and this was used to develop an iterative algorithm for updating the solutions from one piecewise approximation to the next. The results of the present technique matched with results obtained by direct numerical integration of the governing nonlinear differential equation for a half-space impact, and the FILM was found to converge to the results of the numerical solution after a few iterations, typically between five and 10 iterations. The FILM is simple, is inherently stable, converges quickly, gives accurate results and can be implemented manually; these features make it potentially more attractive than the comparable numerical methods.",
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N2 - The governing equation of a half-space impact is generally nonlinear and it is normally solved using numerical techniques that are mostly conditionally stable and require many iteration steps for convergence of the solution. In this paper, we present the force-indentation linearisation method (FILM), an approximate technique that produces closed-form solutions of piecewise linearisation of the governing nonlinear differential equation and is capable of producing accurate impact response for an elastoplastic half-space impact. In contrast to the existing numerical techniques, which discretise the impact force or variable of interest in the time-domain, the present technique discretises the impact force with respect to the indentation using successive piecewise linear approximations. Generalised closed-form solutions were derived for each piecewise approximation, and this was used to develop an iterative algorithm for updating the solutions from one piecewise approximation to the next. The results of the present technique matched with results obtained by direct numerical integration of the governing nonlinear differential equation for a half-space impact, and the FILM was found to converge to the results of the numerical solution after a few iterations, typically between five and 10 iterations. The FILM is simple, is inherently stable, converges quickly, gives accurate results and can be implemented manually; these features make it potentially more attractive than the comparable numerical methods.

AB - The governing equation of a half-space impact is generally nonlinear and it is normally solved using numerical techniques that are mostly conditionally stable and require many iteration steps for convergence of the solution. In this paper, we present the force-indentation linearisation method (FILM), an approximate technique that produces closed-form solutions of piecewise linearisation of the governing nonlinear differential equation and is capable of producing accurate impact response for an elastoplastic half-space impact. In contrast to the existing numerical techniques, which discretise the impact force or variable of interest in the time-domain, the present technique discretises the impact force with respect to the indentation using successive piecewise linear approximations. Generalised closed-form solutions were derived for each piecewise approximation, and this was used to develop an iterative algorithm for updating the solutions from one piecewise approximation to the next. The results of the present technique matched with results obtained by direct numerical integration of the governing nonlinear differential equation for a half-space impact, and the FILM was found to converge to the results of the numerical solution after a few iterations, typically between five and 10 iterations. The FILM is simple, is inherently stable, converges quickly, gives accurate results and can be implemented manually; these features make it potentially more attractive than the comparable numerical methods.

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