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

Akuro Big-Alabo, Philip Harrison, Matthew P Cartmell

Research output: Contribution to journalArticle

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.
Language English 850-858 9 Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 229 5 27 Jun 2014 10.1177/0954406214541431 Published - 30 Apr 2015

### Fingerprint

Indentation
Linearization
Differential equations
Numerical methods

### Keywords

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

### Cite this

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.",
keywords = "half-space, impact, elastoplastic, contact law, force-indentation linearisation method",
author = "Akuro Big-Alabo and Philip Harrison and Cartmell, {Matthew P}",
year = "2015",
month = "4",
day = "30",
doi = "10.1177/0954406214541431",
language = "English",
volume = "229",
pages = "850--858",
journal = "Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science",
issn = "0954-4062",
number = "5",

}

TY - JOUR

T1 - Algorithm for the solution of elastoplastic half-space impact

T2 - Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science

AU - Big-Alabo, Akuro

AU - Harrison, Philip

AU - Cartmell, Matthew P

PY - 2015/4/30

Y1 - 2015/4/30

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.

KW - half-space

KW - impact

KW - elastoplastic

KW - contact law

KW - force-indentation linearisation method

U2 - 10.1177/0954406214541431

DO - 10.1177/0954406214541431

M3 - Article

VL - 229

SP - 850

EP - 858

JO - Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science

JF - Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science

SN - 0954-4062

IS - 5

ER -