Wave propagation analysis in transversely isotropic piezoelastic medium based on nonlocal strain gradient theory

Trinh Thi Thanh Hue, Do Xuan Tung
Author affiliations


  • Trinh Thi Thanh Hue Faculty of Building and Industrial Construction, Hanoi University of Civil Engineering, 55 Giai Phong Street, Hanoi, Vietnam
  • Do Xuan Tung Faculty of Civil Engineering, Hanoi Architectural University, Km 10 Nguyen Trai Street, Thanh Xuan, Hanoi, Vietnam https://orcid.org/0000-0001-6731-4109




dispersion equation, nonlocal, gradient, transversely isotropic, piezoelectric


The purpose of this research is to study the propagation of surface waves in transversely isotropic piezoelastic medium based on nonlocal strain gradient theory. A characteristics equation for the existence of surface waves is discussed. This equation could be easily reduced to the ones of the gradient strain theory, nonlocal theory, and classical theory. It has also been concluded that there exist cut-off frequency for the wave propagating in size-dependent materials based on the nonlocal strain gradient theory. The dispersion equation which surface wave speed satisfies is derived from the free traction condition on the surface of half-space with consideration of electrically open circuit conditions. The effect of the nonlocal parameter, the strain gradient parameter on the existence of surface waves as well as the Rayleigh wave propagation is illustrated through some numerical examples.


Download data is not yet available.


Metrics Loading ...


A. C. Eringen and D. G. B. Edelen. On nonlocal elasticity. International Journal of Engineering Science, 10, (1972), pp. 233–248. DOI: https://doi.org/10.1016/0020-7225(72)90039-0

A. C. Eringen. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54, (1983), pp. 4703–4710. DOI: https://doi.org/10.1063/1.332803

F. Yang, A. C. M. Chong, D. C. C. Lam, and P. Tong. Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures, 39, (2002), pp. 2731–2743. DOI: https://doi.org/10.1016/S0020-7683(02)00152-X

A. C. Eringen. Theory of micropolar plates. Zeitschrift für angewandte Mathematik und Physik ZAMP, 18, (1967), pp. 12–30. DOI: https://doi.org/10.1007/BF01593891

E. C. Aifantis. Strain gradient interpretation of size effects. International Journal of Fracture, (1999), pp. 299–314. DOI: https://doi.org/10.1007/978-94-011-4659-3_16

D. X. Tung. Wave propagation in nonlocal orthotropic micropolar elastic solids. Archives of Mechanics, 73, (3), (2021).

D. X. Tung. Surface waves in nonlocal transversely isotropic liquid-saturated porous solid. Archive of Applied Mechanics, 91, (2021), pp. 2881–2892. DOI: https://doi.org/10.1007/s00419-021-01940-2

C. W. Lim, G. Zhang, and J. N. Reddy. A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. Journal of the Mechanics and Physics of Solids, 78, (2015), pp. 298–313. DOI: https://doi.org/10.1016/j.jmps.2015.02.001

F. Ebrahimi, M. R. Barati, and A. Dabbagh. A nonlocal strain gradient theory for wave propagation analysis in temperature-dependent inhomogeneous nanoplates. International Journal of Engineering Science, 107, (2016), pp. 169–182. DOI: https://doi.org/10.1016/j.ijengsci.2016.07.008

M. Arefi. Analysis of wave in a functionally graded magneto-electro-elastic nano-rod using nonlocal elasticity model subjected to electric and magnetic potentials. Acta Mechanica, 227, (2016), pp. 2529–2542. DOI: https://doi.org/10.1007/s00707-016-1584-7

L. H. Ma, L. L. Ke, J. N. Reddy, J. Yang, S. Kitipornchai, and Y. S. Wang. Wave propagation characteristics in magneto-electro-elastic nanoshells using nonlocal strain gradient theory. Composite Structures, 199, (2018), pp. 10–23. DOI: https://doi.org/10.1016/j.compstruct.2018.05.061

D.-J. Yan, A.-L. Chen, Y.-S. Wang, C. Zhang, and M. Golub. Propagation of guided elastic waves in nanoscale layered periodic piezoelectric composites. European Journal of Mechanics - A/Solids, 66, (2017), pp. 158–167. DOI: https://doi.org/10.1016/j.euromechsol.2017.07.003

D.-J. Yan, A.-L. Chen, Y.-S. Wang, C. Zhang, and M. Golub. In-plane elastic wave propagation in nanoscale periodic layered piezoelectric structures. International Journal of Mechanical Sciences, 142–143, (2018), pp. 276–288. DOI: https://doi.org/10.1016/j.ijmecsci.2018.04.054

H. Askes and E. C. Aifantis. Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results. International Journal of Solids and Structures, 48, (2011), pp. 1962–1990. DOI: https://doi.org/10.1016/j.ijsolstr.2011.03.006

Y. Huang, P. Wei, Y. Xu, and Y. Li. Modelling flexural wave propagation by the nonlocal strain gradient elasticity with fractional derivatives. Mathematics and Mechanics of Solids, 26, (2021), pp. 1538–1562. DOI: https://doi.org/10.1177/1081286521991206

D. X. Tung. Dispersion equation of Rayleigh waves in transversely isotropic nonlocal piezoelastic solids half-space. Vietnam Journal of Mechanics, 41, (2019), pp. 363–371. DOI: https://doi.org/10.15625/0866-7136/14621

J. Achenbach. Wave propagation in elastic solids. Elsevier, (2012).

A. Chakraborty. Wave propagation in anisotropic media with non-local elasticity. International Journal of Solids and Structures, 44, (2007), pp. 5723–5741. DOI: https://doi.org/10.1016/j.ijsolstr.2007.01.024

S. Gopalakrishnan and S. Narendar. Wave propagation in nanostructures: Nonlocal continuum mechanics formulations. Springer International Publishing, (2013). DOI: https://doi.org/10.1007/978-3-319-01032-8

J. N. Sharma, M. Pal, and D. Chand. Propagation characteristics of Rayleigh waves in transversely isotropic piezothermoelastic materials. Journal of Sound and Vibration, 284, (2005), pp. 227–248. DOI: https://doi.org/10.1016/j.jsv.2004.06.036

J. N. Sharma and V. Walia. Further investigations on Rayleigh waves in piezothermoelastic materials. Journal of Sound and Vibration, 301, (2007), pp. 189–206. DOI: https://doi.org/10.1016/j.jsv.2006.09.018




How to Cite

T. T. T. Hue and D. X. Tung, Wave propagation analysis in transversely isotropic piezoelastic medium based on nonlocal strain gradient theory, Vietnam J. Mech. 45 (2023) 358–375. DOI: https://doi.org/10.15625/0866-7136/19604.



Research Article


Funding data