Advertisement

Journal of Elasticity

, Volume 137, Issue 2, pp 237–246 | Cite as

On the Invariance of Governing Equations of Current Nonlocal Theories of Elasticity Under Coordinate Transformation and Displacement Gauge Change

  • Linjuan Wang
  • Jianxiang WangEmail author
Research Note

Abstract

The invariance of elastodynamic wave equations under coordinate transformations provides a way to achieve elastic wave cloaking. Under general coordinate transformations, it has been proved that the conventional elastodynamic wave equation (the Navier equation) changes its form. In addition to the conventional Navier equation, various nonlocal theories of elasticity have been developed to encompass more general descriptions of behaviour of solids. Whether the forms of the governing equations of these nonlocal theories of elasticity remain invariant under coordinate transformations has not been investigated. In this note, we examine the form-invariance of current nonlocal theories of elasticity, including the Mindlin 1964 theory, the strain gradient theory, the stress gradient theory, the peridynamic theory, the Kunin 1982 theory, the Eringen 1983 theory, and the Kröner 1967 theory under coordinate transformation and displacement gauge change. These theories are classified into three types and their invariance is examined in terms of three criteria. It is found that only the peridynamic theory, the Kunin 1982 theory, and the Eringen 1983 theory satisfy form-invariance. We further show that the operations of degeneration and a general coordinate transformation on nonlocal elasticity are not communicative. The results shed new light on the properties of the nonlocal theories of elasticity.

Keywords

Transformation method Elastic wave cloaking Nonlocal continuum theories Peridynamics 

Mathematics Subject Classification (2010)

45A05 74A05 74A20 

Notes

Acknowledgements

The work is supported by the National Natural Science Foundation of China under Grant 11521202. The authors thank the anonymous reviewers whose incisive comments have helped to improve the technical content of this work.

References

  1. 1.
    Pendry, J.B., Schurig, D., Smith, D.R.: Controlling electromagnetic fields. Science 312, 1780–1782 (2006) MathSciNetCrossRefADSGoogle Scholar
  2. 2.
    Chen, T., Weng, C.N., Chen, J.S.: Cloak for curvilinearly anisotropic media in conduction. Appl. Phys. Lett. 93, 114103 (2008) CrossRefADSGoogle Scholar
  3. 3.
    Norris, A.N.: Acoustic cloaking theory. Proc. R. Soc. A 464, 2411–2434 (2008) MathSciNetCrossRefADSGoogle Scholar
  4. 4.
    Milton, G.W., Briane, M., Willis, J.R.: On cloaking for elasticity and physical equations with a transformation invariant form. New J. Phys. 8, 248 (2006) CrossRefADSGoogle Scholar
  5. 5.
    Willis, J.R.: Dynamics of Composites. Continuum Micromechanics CISM Courses and Lectures, pp. 265–290. Springer, Berlin (1997) CrossRefGoogle Scholar
  6. 6.
    Brun, M., Guenneau, S., Movchan, A.B.: Achieving control of in-plane elastic waves. Appl. Phys. Lett. 94, 061903 (2009) CrossRefADSGoogle Scholar
  7. 7.
    Norris, A.N., Shuvalov, A.L.: Elastic cloaking theory. Wave Motion 48, 525–538 (2011) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Liu, Y., Liu, W., Su, X.: Precise method to control elastic waves by conformal mapping. Theor. Appl. Mech. Lett. 3, 021012 (2013) CrossRefGoogle Scholar
  9. 9.
    Hu, J., Chang, Z., Hu, G.: Approximate method for controlling solid elastic waves by transformation media. Phys. Rev. B 84, 201101 (2011) CrossRefADSGoogle Scholar
  10. 10.
    Chang, Z., Hu, J., Hu, G., Tao, R., Wang, Y.: Controlling elastic waves with isotropic materials. Appl. Phys. Lett. 98, 121904 (2011) CrossRefADSGoogle Scholar
  11. 11.
    Chang, Z., Hu, G.: Elastic wave omnidirectional absorbers designed by transformation method. Appl. Phys. Lett. 101, 054102 (2012) CrossRefADSGoogle Scholar
  12. 12.
    Norris, A.N., Parnell, W.J.: Hyperelastic cloaking theory: transformation elasticity with pre-stressed solids. Proc. R. Soc. A 468, 2881–2903 (2012) MathSciNetCrossRefADSGoogle Scholar
  13. 13.
    Xiang, Z.H., Yao, R.W.: Realizing the Willis equations with pre-stresses. J. Mech. Phys. Solids 87, 1–6 (2016) MathSciNetCrossRefADSGoogle Scholar
  14. 14.
    Askes, H., Aifantis, E.C.: 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) CrossRefGoogle Scholar
  15. 15.
    Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Aifantis, E.C.: On the role of gradients in the localization of deformation and fracture. Int. J. Eng. Sci. 30, 1279–1299 (1992) CrossRefGoogle Scholar
  17. 17.
    Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 9, 4703–4710 (1983) CrossRefADSGoogle Scholar
  18. 18.
    Kunin, I.: Elastic Media with Microstructure I: One-Dimensional Models. Springer, Berlin (1982) CrossRefGoogle Scholar
  19. 19.
    Silling, S.A.: Reformation of elasticity theory for discontinuities and longrange force. J. Mech. Phys. Solids 48, 175–209 (2000) MathSciNetCrossRefADSGoogle Scholar
  20. 20.
    Silling, S.A.: Linearized theory of peridynamic states. J. Elast. 99, 85–111 (2010) MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kröner, E.: Elasticity theory of materials with long range cohesive forces. Int. J. Solids Struct. 3, 731–742 (1967) CrossRefGoogle Scholar
  22. 22.
    Milton, G.W., Willis, J.R.: On modifications of Newton’s second law and linear continuum elastodynamics. Proc. R. Soc. A 463, 2881–2903 (2007) MathSciNetCrossRefGoogle Scholar
  23. 23.
    Silling, S.A.: Origin and effect of nonlocality in a composite. J. Mech. Mater. Struct. 9, 245–258 (2014) CrossRefGoogle Scholar
  24. 24.
    Seleson, P., Parks, M.L., Gunzburger, M., Lehoucq, R.B.: Peridynamics as an upscaling of molecular dynamics. Multiscale Model. Simul. 8, 204–227 (2009) MathSciNetCrossRefGoogle Scholar
  25. 25.
    Wang, L.J., Abeyaratne, R.: A one-dimensional peridynamic model of defect propagation and its relation to certain other continuum models. J. Mech. Phys. Solids 116, 334–349 (2018) MathSciNetCrossRefADSGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of EngineeringPeking UniversityBeijingP.R. China
  2. 2.CAPT, HEDPS and IFSA Collaborative Innovation Center of MoEPeking UniversityBeijingP.R. China

Personalised recommendations