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.
References
Pendry, J.B., Schurig, D., Smith, D.R.: Controlling electromagnetic fields. Science 312, 1780–1782 (2006)
Chen, T., Weng, C.N., Chen, J.S.: Cloak for curvilinearly anisotropic media in conduction. Appl. Phys. Lett. 93, 114103 (2008)
Norris, A.N.: Acoustic cloaking theory. Proc. R. Soc. A 464, 2411–2434 (2008)
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)
Willis, J.R.: Dynamics of Composites. Continuum Micromechanics CISM Courses and Lectures, pp. 265–290. Springer, Berlin (1997)
Brun, M., Guenneau, S., Movchan, A.B.: Achieving control of in-plane elastic waves. Appl. Phys. Lett. 94, 061903 (2009)
Norris, A.N., Shuvalov, A.L.: Elastic cloaking theory. Wave Motion 48, 525–538 (2011)
Liu, Y., Liu, W., Su, X.: Precise method to control elastic waves by conformal mapping. Theor. Appl. Mech. Lett. 3, 021012 (2013)
Hu, J., Chang, Z., Hu, G.: Approximate method for controlling solid elastic waves by transformation media. Phys. Rev. B 84, 201101 (2011)
Chang, Z., Hu, J., Hu, G., Tao, R., Wang, Y.: Controlling elastic waves with isotropic materials. Appl. Phys. Lett. 98, 121904 (2011)
Chang, Z., Hu, G.: Elastic wave omnidirectional absorbers designed by transformation method. Appl. Phys. Lett. 101, 054102 (2012)
Norris, A.N., Parnell, W.J.: Hyperelastic cloaking theory: transformation elasticity with pre-stressed solids. Proc. R. Soc. A 468, 2881–2903 (2012)
Xiang, Z.H., Yao, R.W.: Realizing the Willis equations with pre-stresses. J. Mech. Phys. Solids 87, 1–6 (2016)
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)
Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964)
Aifantis, E.C.: On the role of gradients in the localization of deformation and fracture. Int. J. Eng. Sci. 30, 1279–1299 (1992)
Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 9, 4703–4710 (1983)
Kunin, I.: Elastic Media with Microstructure I: One-Dimensional Models. Springer, Berlin (1982)
Silling, S.A.: Reformation of elasticity theory for discontinuities and longrange force. J. Mech. Phys. Solids 48, 175–209 (2000)
Silling, S.A.: Linearized theory of peridynamic states. J. Elast. 99, 85–111 (2010)
Kröner, E.: Elasticity theory of materials with long range cohesive forces. Int. J. Solids Struct. 3, 731–742 (1967)
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)
Silling, S.A.: Origin and effect of nonlocality in a composite. J. Mech. Mater. Struct. 9, 245–258 (2014)
Seleson, P., Parks, M.L., Gunzburger, M., Lehoucq, R.B.: Peridynamics as an upscaling of molecular dynamics. Multiscale Model. Simul. 8, 204–227 (2009)
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)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Wang, L., Wang, J. On the Invariance of Governing Equations of Current Nonlocal Theories of Elasticity Under Coordinate Transformation and Displacement Gauge Change. J Elast 137, 237–246 (2019). https://doi.org/10.1007/s10659-018-09715-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10659-018-09715-7