Skip to main content
Log in

Experimental study and numerical simulation of the vertical bounce of a polymer ball over a wide temperature range

  • Published:
Journal of Materials Science Aims and scope Submit manuscript


The dependence to temperature of the rebound of a solid polymer ball on a rigid slab is investigated. An acrylate polymer ball is brought to a wide range of temperatures, covering its glass to rubbery transition, and let fall on a granite slab while the coefficient of restitution, duration of contact, and force history are measured experimentally. The ball fabrication is controlled in the lab, allowing the mechanical characterization of the material by classic dynamic mechanical analysis. Finite element simulations of the rebound at various temperatures are run, considering the material as viscoelastic and as satisfying a WLF equation for its time–temperature superposition property. A comparison between the experiments and the simulations shows the strong link between viscoelasticity and time–temperature superposition properties of the material and the bounce characteristics of the ball.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others


  1. Cross R (1999) The bounce of a ball. Am J Phys 67:222–227

    Article  Google Scholar 

  2. Lewis GJ, Arnold JC, Griffiths IW (2011) The dynamic behavior of squash balls. Am J Phys 79:291–296

    Article  Google Scholar 

  3. Collins F, Brabazon D, Moran K (2011) Viscoelastic impact characterisation of solid sports balls used in the Irish sport of hurling. Sports Eng 14:15–25

    Article  Google Scholar 

  4. Akay A, Hodgson TH (1978) Acoustic radiation from the elastic impact of a sphere with a slab. Appl Acoust 11:285–304

    Article  Google Scholar 

  5. Dintwa E, Van Zeebroeck M, Ramon H, Tijskens E (2008) Finite element analysis of the dynamic collision of apple fruit. Postharvest Biol Technol 49:260–276

    Article  Google Scholar 

  6. Hertzsch JM, Spahn F, Brilliantov V (1995) On low-velocity collisions of viscoelastic particles. J Phys II France 5:1725–1738

    Article  Google Scholar 

  7. Hertz H (1881) Ueber di Berührung fester elastischer Körper. J Reine Angew Math 92:156–171

    Google Scholar 

  8. Kuwabara G, Kono K (1987) Restitution coefficient in a collision between two spheres. Jpn J Appl Phys 26:1230–1233

    Article  Google Scholar 

  9. Ismail KA, Stronge WJ (2008) Impact of viscoplastic bodies: dissipation and restitution. J Appl Mech 75:061011.1–061011.5

    Article  Google Scholar 

  10. Tabor D (1948) A simple theory of static and dynamic hardness. Proc R Soc Lond A 192:247–274

    Article  Google Scholar 

  11. Jenckel E, Klein E (1952) Die Bestimmung von Relaxationszeiten aus der Rückprallelastizität. Z Naturf A 7:619–630

    Google Scholar 

  12. Tillett JPA (1954) A study of the impact of spheres on plates. Proc Phys Soc B 67:677–688

    Article  Google Scholar 

  13. Hunter SC (1960) The Hertz problem for a rigid spherical indenter and a viscoelastic half-space. J Mech Phys Solids 8:219–234

    Article  Google Scholar 

  14. Lifshitz JM, Kolsky H (1964) Some experiments on anelastic rebound. J Mech Phys Solids 12:35–43

    Article  Google Scholar 

  15. Calvit HH (1967) Experiments on rebound of steel balls from blocks of polymers. J Mech Phys Solids 15:141–150

    Article  Google Scholar 

  16. Raphael T, Armeniades CD (1967) Correlation of rebound tester and torsion pendulum data on polymer samples. Polym Eng Sci 7:21–24

    Article  Google Scholar 

  17. Briggs LJ (1945) Methods for measuring the coefficient of restitution and the spin of a ball. Research Paper RP1624, National Bureau of Standards

  18. Robbins RF, Weitzel DH (1969) An automated resilience apparatus for polymer studies from −196 to +180 °C. Rev Sci Inst 40:1014–1017

    Article  Google Scholar 

  19. Drane PJ, Sherwood JA (2004) Characterization of the effect of temperature on baseball COR performance. In: Hubbard M et al (eds) The engineering of sports 5, vol 2. pp 59–65

  20. Nathan AM, Smith LV, Faber WL, Russell DA (2011) Corked bats, juiced balls, and humidors: the physics and cheating in baseball. Am J Phys 79:575–580

    Article  Google Scholar 

  21. Nagurka M, Huang S (2006) A mass-spring-damper model of a bouncing ball. Int J Eng Ed 22:393–401

    Google Scholar 

  22. Safranski D, Gall K (2008) Effect of chemical structure and crosslinking density on the thermo-mechanical properties and toughness of (meth)acrylate shape memory polymer networks. Polymer 49:4446–4555

    Article  Google Scholar 

  23. Weese J (1993) A regularization method for nonlinear ill-posed problems. Comput Phys Commun 77:429–440

    Article  Google Scholar 

  24. Williams ML, Landel RF, Ferry JD (1955) The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids. J Am Chem Soc 77:3701–3707

    Article  Google Scholar 

  25. Matlab (2011) The MathWorks Inc., Natick, MA, USA

  26. Aguiar CE, Laudares F (2003) Listening to the coefficient of restitution and the gravitational acceleration of a bouncing ball. Am J Phys 71:499–501

    Article  Google Scholar 

  27. Abaqus (2010) Dassault Systems Simulia Corp., Providence, RI, USA

  28. Simo JC (1987) On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects. Comput Meth Appl Mech Eng 60:153–173

    Article  Google Scholar 

  29. Johnson KL (1985) Contact mechanics. Cambridge University Press, Cambridge, pp 351–355

    Book  Google Scholar 

  30. Carslaw HS, Jaeger HS (1959) Conduction of heat in solids, 2nd edn. Clarendon Press, Oxford, pp 233–234

    Google Scholar 

Download references


The authors are grateful to several colleagues from PIMM laboratory: J. Lédion for lending his thermal chamber, M. Schneider for making his digital oscilloscope available, and F. Coste for his advice on signal acquisition. Moreover, M. Brieu from LML is acknowledged for giving access to his mechanical testing machine for sensor calibration.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Julie Diani.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Diani, J., Gilormini, P. & Agbobada, G. Experimental study and numerical simulation of the vertical bounce of a polymer ball over a wide temperature range. J Mater Sci 49, 2154–2163 (2014).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: