Skip to main content
SpringerLink
Log in

Dynamic analysis of interfacial multiple cracks in piezoelectric thin film/substrate

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

The dynamic fracture characteristics of mode-III cracks are investigated. This is a crucial problem in piezoelectric devices. The theoretical solution to this problem is described using the integral-transform method (Laplace and Fourier transforms) and the Chebyshev point method. A crack-propagation model is provided to obtain the stress and electrical-displacement fields near the crack tips. The results show that crack propagation is related to the electromechanical coupling coefficient and film thickness. The effect of film thickness has not been considered in previous literature. In the case of multiple cracks, according to their mutual effects, the nondimensional DSIF inside the crack tip is more significant than that outside the crack tip, regardless of the number of cracks. When the film thickness is small, the change in the DSIF is significant, indicating appropriate circumstances, and a thinner film thickness is more conducive to safe design. Negative electrical-displacement loads always prevent crack propagation, whereas positive electric-displacement loads can promote or prevent crack propagation. Numerical examples are provided to highlight the result.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Data availability

All the numerical calculated data used to support the findings of this study can be obtained by calculating the equations in the paper, and piezoelectric material parameters are taken from references [14, 15] and [34,35,36].

References

  1. Shindo, Y., Tanaka, K., Narita, F.: Singular stress and electric fields of a piezoelectric ceramic strip with a finite crack under longitudinal shear. Acta Mech. 120, 31–45 (1997)

    Article  MATH  Google Scholar 

  2. Deeg W.F.J.: The analysis of dislocation, crack, and inclusion problems in piezoelectric solids. Ph. D. Thesis. Stanford University (1980).

  3. Sosa, H.A., Pak, Y.E.: Three-dimensional eigenfunction analysis of a crack in a piezoelectric material. Int. J. Solids Struct. 26(1), 1–15 (1990)

    Article  MATH  Google Scholar 

  4. Zhang, T.Y.: Effects of static electric field on the fracture behavior of piezoelectric ceramics. Acta Mech. Sin. 18, 537–550 (2002)

    Article  Google Scholar 

  5. Pak, Y.E.: Crack extension force in a piezoelectric material. J. Appl. Mech. 57(3), 647–653 (1990)

    Article  MATH  Google Scholar 

  6. Suo, Z., Kuo, C.-M., Barnett, D.M., Willis, J.R.: Fracture mechanics for piezoelectric ceramics. J. Mech. Phys. Solids 40(4), 739–765 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kwon, S.M., Lee, K.Y.: Eccentric crack in a rectangular piezoelectric medium under electromechanical loadings. Acta Mech. 148, 239–248 (2001)

    Article  MATH  Google Scholar 

  8. Sosa, H.: On the fracture mechanics of piezoelectric solids. Int. J. Solids Struct. 29(21), 2613–2622 (1992)

    Article  MATH  Google Scholar 

  9. Liu, L.L., Feng, W.J., Ma, P., et al.: Fracture analysis of a penny-shaped dielectric crack in a piezoelectric cylinder. Acta Mech. 226, 3045–3057 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Pak, Y.E.: Linear electro-elastic fracture mechanics of piezoelectric materials. Int. J. Fract. 112(1), 79–100 (1992)

    Article  Google Scholar 

  11. Yang, P.S., Liou, J.Y., Sung, J.C.: Subinterface crack in an anisotropic piezoelectric bimaterial. Int. J. Solids Struct. 45, 4990–5014 (2008)

    Article  MATH  Google Scholar 

  12. Ou, Z.C., Chen, Y.H.: Near-tip stress fields and intensity factors for an interface crack in metal/piezoelectric bimaterials. Int. J. Eng. Sci. 42, 1407–1438 (2004)

    Article  Google Scholar 

  13. Govorukha, V., Kamlah, M., Sheveleva, A.: Influence of concentrated loading on opening of an interface crack between piezoelectric materials in a compressive field. Acta Mech. 226, 2379–2391 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, S.F., Mataga, P.A.: Dynamic crack propagation in piezoelectric materials-part I. Electrode solution. J. Mech. Phys. Solids 44(11), 1799–1830 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, S.F., Mataga, P.A.: Dynamic crack propagation in piezoelectric materials-part II. Vacuum solution. J. Mech. Phys. Solids 44(11), 1831–1866 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  16. Narita, F., Shindo, Y.: Dynamic anti-plane shear of a cracked piezoelectric ceramic. Theor. Appl. Fract. Mech. 29(3), 169–180 (1998)

    Article  Google Scholar 

  17. Gu, B., Wang, X.Y., Yu, S.W., Gross, D.: Transient response of a Griffith crack between dissimilar piezoelectric layers under anti-plane mechanical and in-plane electrical impacts. Eng. Fract. Mech. 69, 565–576 (2002)

    Article  Google Scholar 

  18. To, A.C., Li, S., Glaser, S.D.: On scattering in dissimilar piezoelectric materials by a semi-infinite interfacial crack. Q. J. Mech. Appl. Math. 58(2), 309–331 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gao, C.F., Wang, M.Z.: Collinear permeable cracks between dissimilar piezoelectric materials. Int J Solids Struct. 37, 4969–4986 (2000)

    Article  MATH  Google Scholar 

  20. Chi, S., Chung, Y.L.: Cracking in coating-substrate composites with multi-layered and FGM coating. Eng. Fract. Mech. 70, 1227–1243 (2003)

    Article  Google Scholar 

  21. Sedaghati, R., Dargahi, J., Singh, H.: Design and modeling of an endoscopic piezoelectric tactile sensor. Int. J. Solids Struct. 42, 5872–5886 (2005)

    Article  MATH  Google Scholar 

  22. Wang, Y.C., Chen, Y.W.: Application of piezoelectric PVDF film to the measurement of impulsive forces generated by cavitation bubble collapse near a solid boundary. Exp. Therm. Fluid Sci. 32, 403–414 (2007)

    Article  Google Scholar 

  23. Ding, S.H., Li, X.: Periodic cracks in a functionally graded piezoelectric layer bonded to a piezoelectric half-plane. Theor. Appl. Fract. Mech. 49, 313–320 (2008)

    Article  Google Scholar 

  24. Peng, X.L., Li, X.F.: Transient response of the crack tip field in a magnetoelectroelastic half-space with a functionally graded coating under impacts. Arch. Appl. Mech. 79, 1099–1113 (2009)

    Article  MATH  Google Scholar 

  25. Asadi, E., Fariborz, S.J., Fotuhi, A.R.: Anti-plane analysis of orthotropic strips with defects and imperfect FGM coating. Eur. J. Mech. A/Solid. 34, 12–20 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ding, S.H., Li, X.: The collinear crack problem for an orthotropic functionally graded coating-substrate structure. Arch. Appl. Mech. 84, 291–307 (2014)

    Article  MATH  Google Scholar 

  27. Bayat, J., Ayatollahi, M., Bagheri, R.: Fracture analysis of an orthotropic strip with imperfect piezoelectric coating containing multiple defects. Theor. Appl. Fract. Mech. 77, 41–49 (2015)

    Article  Google Scholar 

  28. Zhang, Y.N., Li, J.L., Xie, X.F.: Dynamic propagation characteristics of a mode-III interfacial crack in piezoelectric bi-materials. Adv. Mater. Sci. Eng. 1, 1–21 (2022)

    Google Scholar 

  29. Hu, S.S., Liu, J.S., Li, J.L.: Fracture analysis of griffth interface crack in fine-grained piezoelectric coating/substrate under thermal loading. Adv. Math. Phys. 1, 1–15 (2020)

    MATH  Google Scholar 

  30. Chen, H.S., Wei, W.Y., Liu, J.X., Fang, D.N.: Propagation of a semi-infinite conducting crack in piezoelectric materials: Mode-I problem. J. Mech. Phys. Solids 68(1), 77–92 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Li, X.F., Tang, G.J.: Transient response of a piezoelectric ceramic strip with an eccentric crack under electromechanical impacts. Int. J. Solids Struct. 40, 3571–3588 (2003)

    Article  MATH  Google Scholar 

  32. Wang, X.Y., Yu, S.W.: Transient response of a crack in piezoelectric strip subjected to the mechanical and electrical impacts:mode-III problem. Int. J. Solids Struct. 37, 5795–5808 (2000)

    Article  MATH  Google Scholar 

  33. Wang, B.L., Han, J.C., Du, S.Y.: Dynamic response for non-homogeneous piezoelectric medium with multiple cracks. Eng. Fract. Mech. 61, 607–617 (1998)

    Article  Google Scholar 

  34. Li, X.F.: Transient response of a piezoelectric material with a semi-infinite mode-III crack under impact loads. Int. J. Fract. 111(2), 119–130 (2001)

    Article  Google Scholar 

  35. Noble, B.: Methods based on the Wiener-Hopf technique. Pergamon Press, New York (1958)

    MATH  Google Scholar 

  36. Freund, L.B.: Dynamic fracture mechanics. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

This study was supported by the National Natural Science Foundation of China (11972019), Shanxi Provincial Key Research and Development Project (202102090301027), and Shanxi Postgraduate Innovation Project (2022Y668).

Author information

Authors and Affiliations

  1. School of Materials Science and Engineering, Taiyuan University of Science and Technology, Taiyuan, 030024, China

    Yani Zhang

  2. School of Applied Science, Taiyuan University of Science and Technology, Taiyuan, 030024, China

    Junlin Li & Xiufeng Xie

Authors
  1. Yani Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Junlin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiufeng Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Junlin Li.

Ethics declarations

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

$$\begin{aligned} R_{1} & = \left[ { - e_{15}^{(1)} \frac{{\varepsilon_{11}^{(2)} }}{{\varepsilon_{11}^{(1)} }}(1 - e^{{ - 2\left| \zeta \right|d_{2} }} ) + e_{15}^{(2)} \frac{{\varepsilon_{11}^{(2)} }}{{\varepsilon_{11}^{(1)} }}(1 - e^{{ - 2\left| \zeta \right|d_{2} }} )} \right]\left[ { - \frac{{e_{15}^{(2)} }}{{\varepsilon_{11}^{(2)} }}(1 + e^{{ - 2\alpha_{2} d_{2} }} )} \right]e^{{ - 2\alpha_{1} d_{1} }} \\ & \quad - \frac{{\alpha_{2} }}{\left| \zeta \right|}\left\{ {c_{44}^{(2)} \left( {1 + \frac{{e_{15}^{(2)} e_{15}^{(2)} }}{{\varepsilon_{11}^{(2)} }}} \right)(1 - e^{{ - 2\alpha_{2} d_{2} }} )\left[ { - \frac{{\varepsilon_{11}^{(2)} (1 - e^{{ - 2\left| \zeta \right|d_{2} }} )(1 + e^{{ - 2\left| \zeta \right|d_{1} }} )}}{{\varepsilon_{11}^{(1)} (1 - e^{{ - 2\left| \zeta \right|d_{1} }} )}} - (1 + e^{{ - 2\left| \zeta \right|d_{2} }} )} \right]} \right\}e^{{ - 2\alpha_{1} d_{1} }} \\ \end{aligned}$$
$$\begin{aligned} R_{2} & = \left[ {e_{15}^{(1)} \frac{{\varepsilon_{11}^{(2)} }}{{\varepsilon_{11}^{(1)} }}(1 - e^{{ - 2\left| \zeta \right|d_{2} }} ) - e_{15}^{(2)} (1 - e^{{ - 2\left| \zeta \right|d_{2} }} )} \right]\left( { - 1 - e^{{ - 2\alpha_{2} d_{2} }} } \right)e^{{ - 2\alpha_{1} d_{1} }} \\ & \quad = (1 - e^{{ - 2\left| \zeta \right|d_{2} }} )\left( {1 + e^{{ - 2\alpha_{2} d_{2} }} } \right)\left( {\frac{{e_{15}^{(2)} \varepsilon_{11}^{(1)} - e_{15}^{(1)} \varepsilon_{11}^{(2)} }}{{\varepsilon_{11}^{(1)} }}} \right)e^{{ - 2\alpha_{1} d_{1} }} \\ \end{aligned}$$
$$\begin{aligned} R_{3} & = \left[ {(1 - e^{{ - 2\left| \zeta \right|d_{2} }} )\frac{{e_{15}^{(2)} \varepsilon_{11}^{(1)} - e_{15}^{(1)} \varepsilon_{11}^{(2)} }}{{\varepsilon_{11}^{(1)} }}} \right]\left[ {\frac{{e_{15}^{(2)} }}{{\varepsilon_{11}^{(2)} }}(1 + e^{{ - 2\alpha_{2} d_{2} }} )} \right] \\ & \quad - \frac{{a_{2} }}{\left| \zeta \right|}\left\{ {c_{44}^{(2)} \left( {1 + \frac{{e_{15}^{(2)} e_{15}^{(2)} }}{{\varepsilon_{11}^{(2)} }}} \right)(1 - e^{{ - 2\alpha_{2} d_{2} }} )\left[ { - \frac{{\varepsilon_{11}^{(2)} (1 - e^{{ - 2\left| \zeta \right|d_{2} }} )(1 + e^{{ - 2\left| \zeta \right|d_{1} }} )}}{{\varepsilon_{11}^{(1)} (1 - e^{{ - 2\left| \zeta \right|d_{1} }} )}} - (1 + e^{{ - 2\left| \zeta \right|d_{2} }} )} \right]} \right\} \\ \end{aligned}$$
$$R_{4} = (1 - e^{{ - 2\left| \zeta \right|d_{2} }} )\left( {1 + e^{{ - 2\alpha_{2} d_{2} }} } \right)\left( {\frac{{e_{15}^{(2)} \varepsilon_{11}^{(1)} - e_{15}^{(1)} \varepsilon_{11}^{(2)} }}{{\varepsilon_{11}^{(1)} }}} \right)$$
$$\begin{aligned} R_{5} & = - \frac{{\alpha_{1} }}{\left| \zeta \right|}\left[ {(1 + e^{{ - 2\left| \zeta \right|d_{2} }} ) + \frac{{\varepsilon_{11}^{(2)} (1 - e^{{ - 2\left| \zeta \right|d_{2} }} )(1 + e^{{ - 2\left| \zeta \right|d_{1} }} )}}{{\varepsilon_{11}^{(1)} (1 - e^{{ - 2\left| \zeta \right|d_{1} }} )}}} \right]c_{44}^{(1)} \left( {1 + \frac{{e_{15}^{(1)} e_{15}^{(1)} }}{{\varepsilon_{11}^{(1)} }}} \right)(1 - e^{{ - 2\alpha_{1} d_{1} }} ) \\ & \quad + \frac{{e_{15}^{(1)} }}{{\varepsilon_{11}^{(1)} }}(1 + e^{{ - 2\alpha_{1} d_{1} }} )]\left[ {e_{15}^{(1)} \frac{{\varepsilon_{11}^{(2)} }}{{\varepsilon_{11}^{(1)} }}(1 - e^{{ - 2\left| \zeta \right|d_{2} }} ) - e_{15}^{(2)} (1 - e^{{ - 2\left| \zeta \right|d_{2} }} )} \right] \\ \end{aligned}$$
$$R_{6} = (1 + e^{{ - 2\alpha_{1} d_{1} }} )(1 - e^{{ - 2\left| \zeta \right|d_{2} }} )\left( {e_{15}^{(2)} - \frac{{e_{15}^{(1)} \varepsilon_{11}^{(2)} }}{{\varepsilon_{11}^{(1)} }}} \right)$$
$$\begin{aligned} R_{7} & = - \frac{{\alpha_{1} }}{\left| \zeta \right|}(1 - e^{{ - 2\alpha_{1} d_{1} }} )e^{{ - 2\alpha_{2} d_{2} }} c_{44}^{(1)} \left( {1 + \frac{{e_{15}^{(1)} e_{15}^{(1)} }}{{\varepsilon_{11}^{(1)} }}} \right)\left[ {(1 + e^{{ - 2\left| \zeta \right|d_{2} }} ) + \frac{{\varepsilon_{11}^{(2)} (1 - e^{{ - 2\left| \zeta \right|d_{2} }} )(1 + e^{{ - 2\left| \zeta \right|d_{1} }} )}}{{\varepsilon_{11}^{(1)} (1 - e^{{ - 2\left| \zeta \right|d_{1} }} )}}} \right] \\ & \quad + (1 - e^{{ - 2\left| \zeta \right|d_{2} }} )(1 + e^{{ - 2\alpha_{1} d_{1} }} )e^{{ - 2\alpha_{2} d_{2} }} \frac{{e_{15}^{(1)} (e_{15}^{(1)} \varepsilon_{11}^{(2)} - e_{15}^{(2)} \varepsilon_{11}^{(1)} )}}{{\varepsilon_{11}^{(1)} \varepsilon_{11}^{(1)} }} \\ \end{aligned}$$
$$R_{8} = (1 + e^{{ - 2\alpha_{1} d_{1} }} )(1 - e^{{ - 2\left| \zeta \right|d_{2} }} )e^{{ - 2\alpha_{2} d_{2} }} \frac{{e_{15}^{(2)} \varepsilon_{11}^{(1)} - e_{15}^{(1)} \varepsilon_{11}^{(2)} }}{{\varepsilon_{11}^{(1)} }}$$
$$\begin{aligned} R_{9} & = - \frac{{\varepsilon_{11}^{(2)} }}{{\varepsilon_{11}^{(1)} }}e^{{ - 2\left| \zeta \right|d_{1} }} \frac{{1 - e^{{ - 2\left| \zeta \right|d_{2} }} }}{{1 - e^{{ - 2\left| \zeta \right|d_{1} }} }}\left[ {\frac{{\alpha_{2} }}{\left| \zeta \right|}c_{44}^{(2)} \left( {1 + \frac{{e_{15}^{(2)} e_{15}^{(2)} }}{{\varepsilon_{11}^{(2)} }}} \right)\frac{{e_{15}^{(1)} }}{{\varepsilon_{11}^{(1)} }}(1 - e^{{ - 2\alpha_{2} d_{2} }} )(1 + e^{{ - 2\alpha_{1} d_{1} }} )} \right. \\ & \quad \left. { + \frac{{\alpha_{1} }}{\left| \zeta \right|}c_{44}^{(1)} \left( {1 + \frac{{e_{15}^{(1)} e_{15}^{(1)} }}{{\varepsilon_{11}^{(1)} }}} \right)\frac{{e_{15}^{(2)} }}{{\varepsilon_{11}^{(2)} }}(1 - e^{{ - 2\alpha_{1} d_{1} }} )(1 + e^{{ - 2\alpha_{2} d_{2} }} )} \right] \\ \end{aligned}$$
$$\begin{aligned} R_{10} & = - \left[ {\frac{{\alpha_{2} }}{\left| \zeta \right|}c_{44}^{(2)} \left( {1 + \frac{{e_{15}^{(2)} e_{15}^{(2)} }}{{\varepsilon_{11}^{(2)} }}} \right)(1 - e^{{ - 2\alpha_{2} d_{2} }} )(1 + e^{{ - 2\alpha_{1} d_{1} }} )} \right. \\ & \quad \left. { + \frac{{\alpha_{1} }}{\left| \zeta \right|}c_{44}^{(1)} \left( {1 + \frac{{e_{15}^{(1)} e_{15}^{(1)} }}{{\varepsilon_{11}^{(1)} }}} \right)(1 - e^{{ - 2\alpha_{1} d_{1} }} )(1 + e^{{ - 2\alpha_{2} d_{2} }} )} \right]\frac{{\varepsilon_{11}^{(2)} }}{{\varepsilon_{11}^{(1)} }}e^{{ - 2\left| \zeta \right|d_{1} }} \frac{{1 - e^{{ - 2\left| \zeta \right|d_{2} }} }}{{1 - e^{{ - 2\left| \zeta \right|d_{1} }} }} \\ \end{aligned}$$
$$\begin{aligned} R_{11} & = - \frac{{\varepsilon_{11}^{(2)} }}{{\varepsilon_{11}^{(1)} }}\frac{{1 - e^{{ - 2\left| \zeta \right|d_{2} }} }}{{1 - e^{{ - 2\left| \zeta \right|d_{1} }} }}\left[ {\frac{{\alpha_{2} }}{\left| \zeta \right|}c_{44}^{(2)} \left( {1 + \frac{{e_{15}^{(2)} e_{15}^{(2)} }}{{\varepsilon_{11}^{(2)} }}} \right)\frac{{e_{15}^{(1)} }}{{\varepsilon_{11}^{(1)} }}(1 - e^{{ - 2\alpha_{2} d_{2} }} )(1 + e^{{ - 2\alpha_{1} d_{1} }} )} \right. \\ & \quad \left. { + \frac{{\alpha_{1} }}{\left| \zeta \right|}c_{44}^{(1)} \left( {1 + \frac{{e_{15}^{(1)} e_{15}^{(1)} }}{{\varepsilon_{11}^{(1)} }}} \right)\frac{{e_{15}^{(2)} }}{{\varepsilon_{11}^{(2)} }}(1 - e^{{ - 2\alpha_{1} d_{1} }} )(1 + e^{{ - 2\alpha_{2} d_{2} }} )} \right] \\ \end{aligned}$$
$$\begin{aligned} R_{12} & = - \frac{{\varepsilon_{11}^{(2)} }}{{\varepsilon_{11}^{(1)} }}\frac{{1 - e^{{ - 2\left| \zeta \right|d_{2} }} }}{{1 - e^{{ - 2\left| \zeta \right|d_{1} }} }}\left[ {\frac{{\alpha_{2} }}{\left| \zeta \right|}c_{44}^{(2)} \left( {1 + \frac{{e_{15}^{(2)} e_{15}^{(2)} }}{{\varepsilon_{11}^{(2)} }}} \right)(1 - e^{{ - 2\alpha_{2} d_{2} }} )(1 + e^{{ - 2\alpha_{1} d_{1} }} )} \right. \\ & \quad \left. { + \frac{{a_{1} }}{\left| \zeta \right|}c_{44}^{(1)} \left( {1 + \frac{{e_{15}^{(1)} e_{15}^{(1)} }}{{\varepsilon_{11}^{(1)} }}} \right)(1 - e^{{ - 2\alpha_{1} d_{1} }} )(1 + e^{{ - 2\alpha_{2} d_{2} }} )} \right] \\ \end{aligned}$$
$$\begin{aligned} R_{13} & = \frac{{\alpha_{2} }}{\left| \zeta \right|}c_{44}^{(2)} \left( {1 + \frac{{e_{15}^{(2)} e_{15}^{(2)} }}{{\varepsilon_{11}^{(2)} }}} \right)\frac{{e_{15}^{(1)} }}{{\varepsilon_{11}^{(1)} }}(1 - e^{{ - 2\alpha_{2} d_{2} }} )(1 + e^{{ - 2\alpha_{1} d_{1} }} ) \\ & \quad + \frac{{\alpha_{1} }}{\left| \zeta \right|}c_{44}^{(1)} \left( {1 + \frac{{e_{15}^{(1)} e_{15}^{(1)} }}{{\varepsilon_{11}^{(1)} }}} \right)\frac{{e_{15}^{(2)} }}{{\varepsilon_{11}^{(2)} }}(1 - e^{{ - 2\alpha_{1} d_{1} }} )(1 + e^{{ - 2\alpha_{2} d_{2} }} ) \\ \end{aligned}$$
$$\begin{aligned} R_{14} & = \frac{{\alpha_{2} }}{\left| \zeta \right|}c_{44}^{(2)} \left( {1 + \frac{{e_{15}^{(2)} e_{15}^{(2)} }}{{\varepsilon_{11}^{(2)} }}} \right)(1 - e^{{ - 2\alpha_{2} d_{2} }} )(1 + e^{{ - 2\alpha_{1} d_{1} }} ) \\ & \quad + \frac{{\alpha_{1} }}{\left| \zeta \right|}c_{44}^{(1)} \left( {1 + \frac{{e_{15}^{(1)} e_{15}^{(1)} }}{{\varepsilon_{11}^{(1)} }}} \right)(1 - e^{{ - 2\alpha_{1} d_{1} }} )(1 + e^{{ - 2\alpha_{2} d_{2} }} ) \\ \end{aligned}$$
$$\begin{aligned} R_{15} & = e^{{ - 2\left| \zeta \right|d_{2} }} \left[ {\frac{{\alpha_{2} }}{\left| \zeta \right|}c_{44}^{(2)} \left( {1 + \frac{{e_{15}^{(2)} e_{15}^{(2)} }}{{\varepsilon_{11}^{(2)} }}} \right)\frac{{e_{15}^{(1)} }}{{\varepsilon_{11}^{(1)} }}(1 - e^{{ - 2\alpha_{2} d_{2} }} )(1 + e^{{ - 2\alpha_{1} d_{1} }} )} \right. \\ & \quad \left. { + \frac{{\alpha_{1} }}{\left| \zeta \right|}c_{44}^{(1)} \left( {1 + \frac{{e_{15}^{(1)} e_{15}^{(1)} }}{{\varepsilon_{11}^{(1)} }}} \right)\frac{{e_{15}^{(2)} }}{{\varepsilon_{11}^{(2)} }}(1 - e^{{ - 2\alpha_{1} d_{1} }} )(1 + e^{{ - 2\alpha_{2} d_{2} }} )} \right] \\ \end{aligned}$$
$$\begin{aligned} R_{16} & = e^{{ - 2\left| \zeta \right|d_{2} }} \left[ {\frac{{\alpha_{2} }}{\left| \zeta \right|}c_{44}^{(2)} \left( {1 + \frac{{e_{15}^{(2)} e_{15}^{(2)} }}{{\varepsilon_{11}^{(2)} }}} \right)(1 - e^{{ - 2\alpha_{2} d_{2} }} )(1 + e^{{ - 2\alpha_{1} d_{1} }} )} \right. \\ & \quad \left. { + \frac{{\alpha_{1} }}{\left| \zeta \right|}c_{44}^{(1)} \left( {1 + \frac{{e_{15}^{(1)} e_{15}^{(1)} }}{{\varepsilon_{11}^{(1)} }}} \right)(1 - e^{{ - 2\alpha_{1} d_{1} }} )(1 + e^{{ - 2\alpha_{2} d_{2} }} )} \right] \\ \end{aligned}$$
$$\begin{aligned} R & = \left[ {e_{15}^{(2)} (1 - e^{{ - 2\left| \zeta \right|d_{2} }} ) - e_{15}^{(1)} \frac{{\varepsilon_{11}^{(2)} }}{{\varepsilon_{11}^{(1)} }}(1 - e^{{ - 2\left| \zeta \right|d_{2} }} )} \right]\left( {\frac{{e_{15}^{(1)} }}{{\varepsilon_{11}^{(1)} }} - \frac{{e_{15}^{(2)} }}{{\varepsilon_{11}^{(2)} }}} \right)(1 + e^{{ - 2\alpha_{1} d_{1} }} )(1 + e^{{ - 2\alpha_{2} d_{2} }} ) \\ & \quad + \left[ {(1 + e^{{ - 2\left| \zeta \right|d_{2} }} ) + \frac{{\varepsilon_{11}^{(2)} (1 - e^{{ - 2\left| \zeta \right|d_{2} }} )(1 + e^{{ - 2\left| \zeta \right|d_{1} }} )}}{{\varepsilon_{11}^{(1)} (1 - e^{{ - 2\left| \zeta \right|d_{1} }} )}}} \right]\left[ {\frac{{\alpha_{2} }}{\left| \zeta \right|}c_{44}^{(2)} \left( {1 + \frac{{e_{15}^{(2)} e_{15}^{(2)} }}{{\varepsilon_{11}^{(2)} }}} \right)(1 - e^{{ - 2\alpha_{2} d_{2} }} )(1 + e^{{ - 2\alpha_{1} d_{1} }} )} \right. \\ & \quad \left. { + \frac{{\alpha_{1} }}{\left| \zeta \right|}c_{44}^{(1)} \left( {1 + \frac{{e_{15}^{(1)} e_{15}^{(1)} }}{{\varepsilon_{11}^{(1)} }}} \right)(1 - e^{{ - 2\alpha_{1} d_{1} }} )(1 + e^{{ - 2\alpha_{2} d_{2} }} )} \right] \\ \end{aligned}$$

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, Y., Li, J. & Xie, X. Dynamic analysis of interfacial multiple cracks in piezoelectric thin film/substrate. Acta Mech 234, 705–727 (2023). https://doi.org/10.1007/s00707-022-03390-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-022-03390-5

Search

Navigation