Elastodynamics of Linearized Isotropic State-Based Peridynamic Media

Abstract

The peridynamic theory has been used to model and simulate numerically various kinds of mechanical behavior of solids. This work is devoted to analytical solutions of the elastodynamic behavior of linearized isotropic state-based peridynamic materials. First, we present the solutions of the dispersion relations, group velocities, and phase velocities of longitudinal and transverse waves, and examine in detail the effects of the Poisson’s ratio on these properties. It is shown that the elastodynamic behavior of the state-based peridynamic material with a negative Poisson’s ratio is remarkably different from that of the material with a positive Poisson’s ratio. We then derive the general solutions of initial-value problems, and obtain the Green’s function in a closed form. Finally, we study the evolution of a displacement discontinuity in the state-based peridynamic medium, and find that each component of the discontinuity in the three-dimensional theory varies independently according to the same vibrational mode. The results may have implications in investigations of wave propagations, including discontinuities such as phase transitions and kink propagations.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Notes

  1. 1.

    Le and Bobaru [26] classify the linearized state-based peridynamic theories into two types: One is objective, and the other is not. We use the objective type.

References

  1. 1.

    Silling, S.A.: Reformation of elasticity theory for discontinuities and longrange force. J. Mech. Phys. Solids 48, 175–209 (2000)

    ADS  MathSciNet  Article  Google Scholar 

  2. 2.

    Silling, S.A., Epton, M., Weckner, O., Xu, J., Askari, E.: Peridynamic states and constitutive modeling. J. Elast. 88, 151–184 (2007)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Silling, S.A.: Linearized theory of peridynamic states. J. Elast. 99, 85–111 (2010)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Mindlin, R.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Kröner, E.: Elasticity theory of materials with long range cohesive forces. Int. J. Solids Struct. 3, 731–742 (1967)

    Article  Google Scholar 

  6. 6.

    Willis, J.: Variational and related methods for the overall properties of composites. Adv. Appl. Mech. 21, 1–78 (1981)

    ADS  MathSciNet  Article  Google Scholar 

  7. 7.

    Askes, H., Aifantis, E.: Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results. Int. J. Solids Struct. 48, 1962–1990 (2011)

    Article  Google Scholar 

  8. 8.

    Maugin, G.: Continuum Mechanics Through the Twentieth Century—A Concise Historical Perspective. Springer, Dordrecht (2013)

    Google Scholar 

  9. 9.

    S̆ilhavý, M.: Higher gradient expansion for linear isotropic peridynamic materials. Math. Mech. Solids 22, 1483–1493 (2017)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Silling, S.A., Lehoucq, R.B.: Peridynamic theory of solid mechanics. Adv. Appl. Mech. 44, 73–168 (2010)

    Article  Google Scholar 

  11. 11.

    Oterkus, S., Madenci, E., Agwai, A.: Peridynamic thermal diffusion. J. Comput. Phys. 265, 71–96 (2014)

    ADS  MathSciNet  Article  Google Scholar 

  12. 12.

    Wang, L.J., Xu, J., Wang, J.: A peridynamic framework and simulation of non-Fourier and nonlocal heat conduction. Int. J. Heat Mass Transf. 118, 1284–1292 (2018)

    Article  Google Scholar 

  13. 13.

    Gerstle, W., Silling, S.A., Read, D., Tewary, V., Lehoucq, R.B.: Peridynamic simulation of electromigration. Comput. Mater. Continua 8, 75–92 (2008)

    Google Scholar 

  14. 14.

    Dayal, K., Bhattacharya, K.: Kinetics of phase transformations in the peridynamic formulation of continuum mechanics. J. Mech. Phys. Solids 54, 1811–1842 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  15. 15.

    Aguiar, A.R., Royer-Carfagni, G.F., Seitenfuss, A.B.: Wiggly strain localizations in peridynamic bars with non-convex potential. Int. J. Solids Struct. 138, 1–12 (2018)

    Article  Google Scholar 

  16. 16.

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

    ADS  MathSciNet  Article  Google Scholar 

  17. 17.

    Zimmermann, M.: A continuum theory with long-range forces for solids. Ph.D. Thesis, Massachusetts Institute of Technology (2005)

  18. 18.

    Weckner, O., Brunk, G., Epton, M.A., Silling, S.A., Askari, E.: Green’s functions in non-local three-dimensional linear elasticity. Proc. R. Soc. A 465, 3463–3487 (2009)

    ADS  MathSciNet  Article  Google Scholar 

  19. 19.

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

    MathSciNet  Article  Google Scholar 

  20. 20.

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

    Article  Google Scholar 

  21. 21.

    Seleson, P., Parks, M.L., Gunzburger, M., Lehoucq, R.B.: Peridynamics as an upscaling of molecular dynamics. Multiscale Model. Simul. 8, 204–227 (2009)

    MathSciNet  Article  Google Scholar 

  22. 22.

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

    Article  Google Scholar 

  23. 23.

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

    Article  Google Scholar 

  24. 24.

    Bažant, Z.P., Luo, W., Chau, V.T., Bessa, M.A.: Wave dispersion and basic concepts of peridynamics compared to classical nonlocal damage models. J. Appl. Mech. 83, 111004 (2016)

    ADS  Article  Google Scholar 

  25. 25.

    Butt, S.N., Timothy, J.J., Meschke, G.: Wave dispersion and propagation in state-based peridynamics. Comput. Mech. 60, 1–14 (2017)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Le, Q.V., Bobaru, F.: Objectivity of state-based peridynamic models for elasticity. J. Elast. 131, 1–17 (2017)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Sarego, G., Le, Q.V., Bobaru, F., Zaccariotto, M., Galvanetto, U.: Linearized state-based peridynamics for 2-D problems. Int. J. Numer. Methods Eng. 108, 1174–1197 (2016)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Madenci, E.: Peridynamic integrals for strain invariants of homogeneous deformation. Z. Angew. Math. Mech. 97, 1236–1251 (2017)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (2011)

    Google Scholar 

  30. 30.

    Eringen, A.C.: Continuum Physics II: Continuum Mechanics of Single-Substance Bodies. Academic Press, New York (1975)

    Google Scholar 

  31. 31.

    Huang, Z.P.: Fundamentals of Continuum Mechanics, 2nd edn. Higher Education Press, Beijing (2012), in Chinese

    Google Scholar 

Download references

Acknowledgements

The work is supported by the National Natural Science Foundation of China under Grant 11521202. Part of the work was completed when Linjuan Wang was visiting the Department of Mechanical Engineering at Massachusetts Institute of Technology under support of the Chinese Scholarship Council, and Professor Rohan Abeyaratne.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jianxiang Wang.

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

Wang, L., Xu, J. & Wang, J. Elastodynamics of Linearized Isotropic State-Based Peridynamic Media. J Elast 137, 157–176 (2019). https://doi.org/10.1007/s10659-018-09723-7

Download citation

Keywords

  • Elastodynamics
  • Nonlocal continuum theory
  • Dispersion relation
  • Wave propagation
  • Peridynamics
  • Green’s function

Mathematics Subject Classification

  • 45A05
  • 74A05
  • 74B99
  • 74J05