Archive of Applied Mechanics

, Volume 88, Issue 8, pp 1275–1304 | Cite as

Longitudinal impact into viscoelastic media

  • George A. Gazonas
  • Raymond A. Wildman
  • David A. Hopkins
  • Michael J. Scheidler


We consider several one-dimensional impact problems involving finite or semi-infinite, linear elastic flyers that collide with and adhere to a finite stationary linear viscoelastic target backed by a semi-infinite linear elastic half-space. The impact generates a shock wave in the target which undergoes multiple reflections from the target boundaries. Laplace transforms with respect to time, together with impact boundary conditions derived in our previous work, are used to derive explicit closed-form solutions for the stress and particle velocity in the Laplace transform domain at any point in the target. For several stress relaxation functions of the Wiechert (Prony series) type, a modified Dubner–Abate–Crump algorithm is used to numerically invert those solutions to the time domain. These solutions are compared with numerical solutions obtained using both a finite-difference method and the commercial finite element code, COMSOL Multiphysics. The final value theorem for Laplace transforms is used to derive new explicit analytical expressions for the long-time asymptotes of the stress and velocity in viscoelastic targets; these results are useful for the verification of viscoelastic impact simulations taken to long observation times.


1-D viscoelastodynamics Numerical inverse Laplace transform Modified Dubner–Abate–Crump Asymptotic impact behavior Mathematica MATLAB COMSOL 



We would like to thank our colleague, Dr. Rob Jensen (ARL), for providing the torsional DMA PC data plotted in Fig. 8a.


  1. 1.
    Gazonas, G.A., Scheidler, M.J., Velo, A.P.: Exact analytical solutions for elastodynamic impact. Int. J. Solids Struct. 75–76, 172–187 (2015)CrossRefGoogle Scholar
  2. 2.
    Gazonas, G.A., Velo, A.P., Wildman, R.A.: Asymptotic impact behavior of Goupillaud-type layered elastic media. Int. J. Solids Struct. 96, 38–47 (2016)CrossRefGoogle Scholar
  3. 3.
    Gazonas, G.A., Wildman, R.A., Hopkins, D.A.: Elastodynamic impact into piezoelectric media. U.S. Army Research Laboratory, ARL-TR-7056, Sept (2014)Google Scholar
  4. 4.
    Gazonas, G.A., Wildman, R.A., Hopkins, D.A., Scheidler, M.J.: Longitudinal impact of piezoelectric media. Arch. Appl. Mech. 86, 497–515 (2016)CrossRefzbMATHGoogle Scholar
  5. 5.
    Lee, E.H., Kanter, I.: Wave propagation in finite rods of viscoelastic material. J. Appl. Phys. 24(9), 1115–1122 (1953)CrossRefzbMATHGoogle Scholar
  6. 6.
    Berry, D.S., Hunter, S.C.: The propagation of dynamic stresses in visco-elastic rods. J. Mech. Phys. Solids 4, 72–95 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Behring, A.G., Oetting, R.B.: On the longitudinal impact of two, thin, viscoelastic rods. J. Appl. Mech. 37(2), 310–314 (1970)CrossRefGoogle Scholar
  8. 8.
    Musa, A.B.: Numerical solution of wave propagation in viscoelastic rods (standard linear solid model). In: 4th International Conference on Energy and Environment 2013 (ICEE 2013), IOP Conference Series: Earth and Environmental Science, vol. 16; Putrajaya, Malaysia, pp. 1–6 (2013)Google Scholar
  9. 9.
    COMSOL Multiphysics Reference Manual, Version 4.4. COMSOL, Inc., Burlington (2013)Google Scholar
  10. 10.
    Gurtin, M.E., Sternberg, E.: On the linear theory of viscoelasticity. Arch. Ration. Mech. Anal 11, 20–356 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Christensen, R.M.: Theory of Viscoelasticity: An Introduction. Academic Press, New York (1982)Google Scholar
  12. 12.
    Wineman, A.S., Rajagopal, K.R.: Mechanical Response of Polymers: An Introduction. Cambridge University Press, New York (2000)Google Scholar
  13. 13.
    Walton, J.R.: On the steady-state propagation of an anti-plane shear crack in an infinite general linearly viscoelastic solid. Q. Appl. Math. 40(1), 37–52 (1982)CrossRefzbMATHGoogle Scholar
  14. 14.
    Willis, J.R.: Crack propagation in viscoelastic media. J. Mech. Phys. Solids 15, 229–240 (1967)CrossRefzbMATHGoogle Scholar
  15. 15.
    Fodor, G.: Laplace Transforms in Engineering. Akadémiai Kiadó, Budapest (1965)zbMATHGoogle Scholar
  16. 16.
    Schapery, R.A.: Viscoelastic behavior and analysis of composite materials. In: Sendeckyj, G.P. (ed.) Mechanics of Composite Materials, vol. 2, pp. 85–168. Academic Press, New York (1974)Google Scholar
  17. 17.
    Nunziato, J.W., Schuler, K.W.: Shock pulse attenuation in a nonlinear viscoelastic solid. J. Mech. Phys. Solids 21, 447–457 (1973)CrossRefGoogle Scholar
  18. 18.
    Nunziato, J.W., Walsh, E.K., Schuler, K.W., Barker, L.M.: Wave propagation in nonlinear viscoelastic solids. In: Truesdell, C. (ed.) Handbuch Der Physik; Volume VIa/4 Mechanics of Solids IV, pp. 1–108. Springer, New York (1974)Google Scholar
  19. 19.
    Coleman, B.D., Gurtin, M.E., Herrera, R.I.: Waves in materials with memory, I. The velocity of one-dimensional shock and acceleration waves. Arch. Ration. Mech. Anal. 19, 1–19 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Fisher, G.M.C., Gurtin, M.E.: Wave propagation in the linear theory of viscoelasticity. Q. Appl. Math. 23(3), 257–263 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Widder, D.V.: The Laplace Transform. Princeton University Press, Princeton (1946)Google Scholar
  22. 22.
    Kaplan, W.: Operational Methods for Linear Systems. Addison-Wesley, Reading (1962)zbMATHGoogle Scholar
  23. 23.
    LePage, W.R.: Complex Variables and the Laplace Transform for Engineers. Dover Publications, New York (1980)zbMATHGoogle Scholar
  24. 24.
    Wylie, C.R.: Advanced Engineering Mathematics, 3rd edn. McGraw-Hill, New York (1966)zbMATHGoogle Scholar
  25. 25.
    Doetsch, G.: Introduction to the Theory and Application of the Laplace Transform. Springer, New York (1974)zbMATHGoogle Scholar
  26. 26.
    Mathematica Edition: Version 10.3. Wolfram Research, Champaign (2015)Google Scholar
  27. 27.
    Laverty, R., Gazonas, G.A.: An improvement to the Fourier series method for inversion of Laplace transforms applied to elastic and viscoelastic waves. Int. J. Comput. Meth. 3(1), 57–69 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    MATLAB 9.2.0. The MathWorks, Inc., Natick, Massachusetts, United States (2017)Google Scholar
  29. 29.
    Berenger, J.P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114(2), 185–200 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Knauss, W.G., Zhao, J.: Improved relaxation time coverage in ramp-strain histories. Mech. Time Depend. Mater. 11, 199–216 (2007)CrossRefGoogle Scholar
  31. 31.
    Chadwick, P., Powdrill, B.: Application of the Laplace transform method to wave motions involving strong discontinuities. Proc. Camb. Philos. Soc. 60, 313–324 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Martin, P.A.: The pulsating orb: solving the wave equation outside a ball. Proc. R. Soc. A 472, 20160037 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Faciŭ, C., Molinari, A.: On the longitudinal impact of two phase transforming bars. Elastic versus a rate-type approach. Part II: the rate-type case. Int. J. Solids Struct. 43, 523–550 (2006)CrossRefzbMATHGoogle Scholar
  34. 34.
    Critescu, N., Suliciu, I.: Viscoplasticity. Martinus Nijhoff Publishers, The Hague-Boston-London (1982)Google Scholar

Copyright information

© This is a U.S. government work and its text is not subject to copyright protection in the United States; however, its text may be subject to foreign copyright protection 2018

Authors and Affiliations

  • George A. Gazonas
    • 1
  • Raymond A. Wildman
    • 1
  • David A. Hopkins
    • 1
  • Michael J. Scheidler
    • 1
  1. 1.U.S. Army Research LaboratoryAberdeen Proving GroundUSA

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