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


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.


Transformation method Elastic wave cloaking Nonlocal continuum theories Peridynamics 

Mathematics Subject Classification (2010)

45A05 74A05 74A20 



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.


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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

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