Propagation of a Stress Pulse in a Heterogeneous Elastic Bar


The propagation of a wave pulse due to low-speed impact on a one-dimensional, heterogeneous bar is studied. Due to the dispersive character of the medium, the pulse attenuates as it propagates. This attenuation is studied over propagation distances that are much longer than the size of the microstructure. A homogenized peridynamic material model can be calibrated to reproduce the attenuation and spreading of the wave. The calibration consists of matching the dispersion curve for the heterogeneous material near the limit of long wavelengths. It is demonstrated that the peridynamic method reproduces the attenuation of wave pulses predicted by an exact microstructural model over large propagation distances.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15


  1. 1.

    Askes H, Metrikine AV, Pichugin AV, Bennett T (2008) Four simplified gradient elasticity models for the simulation of dispersive wave propagation. Philos Mag 88:3415–3443

    Article  Google Scholar 

  2. 2.

    Barker L (1971) A model for stress wave propagation in composite materials. J Compos Mater 5:140–162

    Article  Google Scholar 

  3. 3.

    Bedford A, Drumheller D (1994) Elastic wave propagation. Wiley, Hoboken, pp 121–122

    Google Scholar 

  4. 4.

    Butt SN, Timothy JJ, Meschke G (2017) Wave dispersion and propagation in state-based peridynamics. Computational Mechanics 60:725–738

    MathSciNet  Article  Google Scholar 

  5. 5.

    Capdeville Y, Guillot L, Marigo J-J (2010) 1-D non-periodic homogenization for the seismic wave equation. Geophys J Int 181:897–910

    Google Scholar 

  6. 6.

    Caruso A, Bertotti B, Giupponi P (1966) Ionization and heating of solid material by means of a laser pulse. Il Nuovo Cimento B (1965-1970) 45:176–189

    Article  Google Scholar 

  7. 7.

    Chen Z, Bakenhus D, Bobaru F (2016) A constructive peridynamic kernel for elasticity. Comput Methods Appl Mech Eng 311:356–373

    MathSciNet  Article  Google Scholar 

  8. 8.

    Chiu S (1970) Difference method for multiple reflection of elastic stress waves. J Comput Phys 6:17–28

    Article  Google Scholar 

  9. 9.

    Dayal K (2017) Leading-order nonlocal kinetic energy in peridynamics for consistent energetics and wave dispersion. J Mech Phys Solids 105:235–253

    MathSciNet  Article  Google Scholar 

  10. 10.

    Fish J, Chen W, Nagai G (2002) Non-local dispersive model for wave propagation in heterogeneous media: one-dimensional case. Int J Numer Methods Eng 54:331–346

    MathSciNet  Article  Google Scholar 

  11. 11.

    Grady D (1997) Physics and modeling of shock-wave dispersion in heterogeneous composites. Le Journal de Physique IV 7(C3):C3–669

    Google Scholar 

  12. 12.

    Grady D (1998) Scattering as a mechanism for structured shock waves in metals. J Mech Phys Solids 46:2017–2032

    Article  Google Scholar 

  13. 13.

    Gu X, Zhang Q, Huang D, Yv Y (2016) Wave dispersion analysis and simulation method for concrete shpb test in peridynamics. Eng Fract Mech 160:124–137

    Article  Google Scholar 

  14. 14.

    Hu R, Oskay C (2017) Nonlocal homogenization model for wave dispersion and attenuation in elastic and viscoelastic periodic layered media. J Appl Mech 84:031003

    Article  Google Scholar 

  15. 15.

    Mal AK, Bar-Cohen Y, Lih S-S (1992) Wave attenuation in fiber-reinforced composites. In: STP1169: Mechanics and Mechanisms of Material Damping. ASTM International

  16. 16.

    Mikata Y (2012) Analytical solutions of peristatic and peridynamic problems for a 1d infinite rod. Int J Solids Struct 49:2887–2897

    Article  Google Scholar 

  17. 17.

    Mutnuri VS, Gopalakrishnan S (2017) Elastic wave propagation in in-homogenous peridynamic bar. In: Nanosensors, biosensors, info-tech sensors and 3D systems 2017. International Society for Optics and Photonics, vol 10167, p 101671L

  18. 18.

    Nicely C, Tang S, Qian D (2018) Nonlocal matching boundary conditions for non-ordinary peridynamics with correspondence material model. Comput Methods Appl Mech Eng 338:463–490

    MathSciNet  Article  Google Scholar 

  19. 19.

    Seleson P, Parks M (2011) On the role of the influence function in the peridynamic theory. Int J Multiscale Comput Eng 9:689–706

    Article  Google Scholar 

  20. 20.

    Silling S (2019) Attenuation of waves in a viscoelastic peridynamic medium. Mathematics and Mechanics of Solids 24:3597–3613

    MathSciNet  Article  Google Scholar 

  21. 21.

    Silling SA (2000) Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 48:175–209

    MathSciNet  Article  Google Scholar 

  22. 22.

    Silling SA, Askari E (2005) A meshfree method based on the peridynamic model of solid mechanics. Comput Struct 83:1526–1535

    Article  Google Scholar 

  23. 23.

    Silling SA, Zimmermann M, Abeyaratne R (2003) Deformation of a peridynamic bar. J Elast 73:173–190

    MathSciNet  Article  Google Scholar 

  24. 24.

    Swift DC, Niemczura JG, Paisley DL, Johnson RP, Luo S-N, Tierney IVTE (2005) Laser-launched flyer plates for shock physics experiments. Rev Sci Instrum 76:093907

    Article  Google Scholar 

  25. 25.

    Turner JA, Anugonda P (2001) Scattering of elastic waves in heterogeneous media with local isotropy. J Acoust Soc Am 109:1787–1795

    Article  Google Scholar 

  26. 26.

    Van Pamel A, Sha G, Rokhlin SI, Lowe MJ (2017) Finite-element modelling of elastic wave propagation and scattering within heterogeneous media. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473:20160738

    Article  Google Scholar 

  27. 27.

    Vogler T, Borg J, Grady D (2012) On the scaling of steady structured waves in heterogeneous materials. J Appl Phys 112:123507

    Article  Google Scholar 

  28. 28.

    Wang L, Xu J, Wang J (2017) Static and dynamic Green’s functions in peridynamics. J Elast 126:95–125

    MathSciNet  Article  Google Scholar 

  29. 29.

    Weckner O, Abeyaratne R (2005) The effect of long-range forces on the dynamics of a bar. J Mech Phys Solids 53:705–728

    MathSciNet  Article  Google Scholar 

  30. 30.

    Weckner O, Brunk G, Epton MA, Silling SA, Askari E (2009) Green’s functions in non-local three-dimensional linear elasticity. Proceedings of the Royal Society A 465:3463–3487

    MathSciNet  Article  Google Scholar 

  31. 31.

    Weckner O, Silling SA (2011) Determination of the constitutive model in peridynamics from experimental dispersion data. Int J Multiscale Comput Eng 9:623–634

    Article  Google Scholar 

  32. 32.

    Wildman RA, Gazonas GA (2014) A finite difference-augmented peridynamics method for reducing wave dispersion. Int J Fract 190:39–52

    Article  Google Scholar 

  33. 33.

    Xu X, Foster JT. Deriving peridynamic influence functions for one-dimensional elastic materials with periodic microstructure. Journal of Peridynamics and Nonlocal Modeling, to appear

  34. 34.

    Youssef HM, El-Bary AA (2014) Thermoelastic material response due to laser pulse heating in context of four theorems of thermoelasticity. J Therm Stresses 37:1379–1389

    Article  Google Scholar 

Download references


Helpful discussions with Dr. Marta D’Elia, Prof. Yue Yu, and Mr. Huaiqian You are gratefully acknowledged. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of Honeywell International Inc. for the US Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

Author information



Corresponding author

Correspondence to Stewart A. Silling.

Ethics declarations

The author is an Editor-in-Chief of the Journal of Peridynamics and Nonlocal Modeling. He played no part in the assignment of this manuscript to Associate Editors or peer reviewers and was separated and blinded from the editorial system from submission inception to decision.


This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the US Department of Energy or the US Government.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Silling, S.A. Propagation of a Stress Pulse in a Heterogeneous Elastic Bar. J Peridyn Nonlocal Model (2021).

Download citation


  • Peridynamic
  • Material stability
  • Fracture
  • Elastic
  • Waves
  • Damage