, Volume 54, Issue 15, pp 2411–2420 | Cite as

Dynamic stress concentration of a cylindrical cavity in vertical exponentially inhomogeneous half space under SH wave

  • Guanxixi Jiang
  • Zailin YangEmail author
  • Cheng Sun
  • Xinzhu Li
  • Yong Yang


Dynamic stress concentration factor around a cylindrical cavity which is in vertically inhomogeneous half space is investigated by applying complex function method and multi-polar coordinates system. The mass density of the half space is inhomogeneous while the shear modulus is a constant. Utilizing conformal mapping method, the governing equation with variable coefficients is transformed to be a normalized Helmholtz equation. Then, incident wave, reflected wave and scattering wave in the half space are obtained. With the help of the boundary condition at the cylindrical cavity, the undetermined coefficients in scattering wave are solved. Then, dynamic stress concentration factor with different influencing parameters around the cavity is calculated and discussed.


SH wave scattering Complex function method Multi-polar coordinates system Vertical exponentially inhomogeneous Dynamic stress concentration factor (DSCF) 



This work is supported by the National Natural Science Foundation of China (Grant No. 11872156), National Key Research and Development Program of China (Grant No. 2017YFC1500801) and the program for Innovative Research Team in China Earthquake Administration.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Pao Y, Mao C (1973) Diffraction of elastic waves and dynamic stress concentration. Crane and Russak, New YorkCrossRefGoogle Scholar
  2. 2.
    Trifunac M (1973) Scattering of plane SH waves by a semi-cylindrical canyon. Earthq Eng Struct Dyn 1:267–281CrossRefGoogle Scholar
  3. 3.
    Wong H, Trifunac M (1974) Scattering of plane SH waves by a semi-elliptical canyon. Earthq Eng Struct Dyn 3:157–169CrossRefGoogle Scholar
  4. 4.
    Liu D, Gai B, Tao G (1982) Applications of the method of complex to dynamic stress concentrations. Wave Motion 4:293–304MathSciNetCrossRefGoogle Scholar
  5. 5.
    Liu D, Han F (1991) Scattering of plane SH-wave by cylindrical canyon of arbitrary shape. Soil Dyn Earthq Eng 10:249–255CrossRefGoogle Scholar
  6. 6.
    Liu G, Ji B, Chen H et al (2009) Antiplane harmonic elastodynamic stress analysis of an infinite wedge with a circular cavity. J Appl Mech 76:061008–1CrossRefGoogle Scholar
  7. 7.
    Qi H, Yang J (2012) Dynamic analysis for circular inclusions of arbitrary positions near interfacial crack impacted by SH-wave in half-space. Eur J Mech A/Solids 36:18–24MathSciNetCrossRefGoogle Scholar
  8. 8.
    Xu H, Yang Z, Wang S (2016) Dynamics response of complex defects near bimaterials interface by incident out-plane waves. Acta Mech 227:1251–1264MathSciNetCrossRefGoogle Scholar
  9. 9.
    Liu Q, Zhang C, Todorovska M (2016) Scattering of SH waves by a shallow rectangular cavity in an elastic half space. Soil Dyn Earthq Eng 90:147–157CrossRefGoogle Scholar
  10. 10.
    Kara H (2016) A note on response of tunnels to incident SH-waves near hillsides. Soil Dyn Earthq Eng 90:138–146CrossRefGoogle Scholar
  11. 11.
    Le T, Lee V, Trifunac M (2017) SH waves in a moon-shaped valley. Soil Dyn Earthq Eng 101:162–175CrossRefGoogle Scholar
  12. 12.
    Dravinski M, Sheikhhassani R (2013) Scattering of a plane harmonic SH wave by a rough multilayered inclusion of arbitrary shape. Wave Motion 50:836–851MathSciNetCrossRefGoogle Scholar
  13. 13.
    Sheikhhassani R, Dravinski M (2014) Scattering of a plane harmonic SH wave by multiple layered inclusions. Wave Motion 51:517–532MathSciNetCrossRefGoogle Scholar
  14. 14.
    Sheikhhassani R, Dravinski M (2016) Dynamic stress concentration for multiple multilayered inclusions embedded in an elastic half-space subjected to SH-waves. Wave Motion 62:20–40MathSciNetCrossRefGoogle Scholar
  15. 15.
    Liu Z, Ju X, Wu C et al (2017) Scattering of plane \(\text{ P }_{1}\) waves and dynamic stress concentration by a lined tunnel in a fluid-saturated poroelastic half-space. Tunn Undergr Space Technol 67:71–84CrossRefGoogle Scholar
  16. 16.
    Panji M, Ansari B (2017) Transient SH-wave scattering by the lined tunnels embedded in an elastic half-plane. Eng Anal Bound Elem 84:220–230MathSciNetCrossRefGoogle Scholar
  17. 17.
    Shyu W, Teng T (2014) Hybrid method combines transfinite interpolation with series expansion to simulate the anti-plane response of a surface irregularity. J Mech 30:349–360CrossRefGoogle Scholar
  18. 18.
    Shyu W, Teng T, Chou C (2017) Anti-plane response caused by interactions between a dike and the surrounding soil. Soil Dyn Earthq Eng 92:408–418CrossRefGoogle Scholar
  19. 19.
    Daros C (2013) Green’s function for SH-waves in inhomogeneous anisotropic elastic solid with power-function velocity variation. Wave Motion 50:101–110MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kowalczyk S, Matysiak S, Perkowski D (2016) On some problems of SH wave propagation in inhomogeneous elastic bodies. J Theor Appl Mech 54:1125–1135CrossRefGoogle Scholar
  21. 21.
    Zhang N, Gao Y, Pak R (2017) Soil and topographic effects on ground motion of a surficially inhomogeneous semi-cylindrical canyon under oblique incident SH waves. Soil Dyn Earthq Eng 95:17–28CrossRefGoogle Scholar
  22. 22.
    Kara H, Aydogdu M (2018) Dynamic response of a functionally graded tube embedded in an elastic medium due to SH-Waves. Compos Struct 206:22–32CrossRefGoogle Scholar
  23. 23.
    Martin P (2009) Scattering by a cavity in an exponentially graded half-space. J Appl Mech 76:031009–1CrossRefGoogle Scholar
  24. 24.
    Liu Q, Zhao M, Zhang C (2014) Antiplane scattering of SH waves by a circular cavity in an exponentially graded half space. Int J Eng Sci 78:61–72MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ghafarollahi A, Shodja H (2018) Scattering of SH-waves by an elliptic cavity/crack beneath the interface between functionally graded and homogeneous half-spaces via multipole expansion method. J Sound Vib 435:372–389ADSCrossRefGoogle Scholar
  26. 26.
    Hei B, Yang Z, Sun B et al (2015) Modelling and analysis of the dynamic behavior of inhomogeneous continuum containing a circular inclusion. Appl Math Model 39:7364–7374MathSciNetCrossRefGoogle Scholar
  27. 27.
    Hei B, Yang Z, Wang Y et al (2016) Dynamic analysis of elastic waves by an arbitrary cavity in an inhomogeneous medium with density variation. Math Mech Solids 21:931–940MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.College of Aerospace and Civil EngineeringHarbin Engineering UniversityHarbinChina
  2. 2.Key Laboratory of Advanced Material of Ship and Mechanics, Ministry of Industry and Information TechnologyHarbin Engineering UniversityHarbinChina

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