Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-29T02:15:02.640Z Has data issue: false hasContentIssue false
  • English
  • Français

Investigation on Transient Responses of a Piezoelectric Crack by using Durbin and Zhao Methods for Numerical Inversion of Laplace Transforms

Published online by Cambridge University Press:  07 August 2013

Y.-S. Ing  and
H.-F. Liao
Y.-S. Ing*
Affiliation:
Department of Aerospace Engineering, Tamkang University, Taipei, Taiwan 25137, R.O.C.
H.-F. Liao
Affiliation:
Department of Aerospace Engineering, Tamkang University, Taipei, Taiwan 25137, R.O.C.
*
*[email protected]
Get access

Abstract

This study applies the numerical inversion of Laplace transform methods to study the piezoelectric dynamic fracture problem, recalculating Chen and Karihaloo's [1] analysis on the transient response of a impermeable crack subjected to anti-plane mechanical and in-plane electric impacts. Three numerical methods were adopted for calculating the dynamic stress intensity factor: Durbin method, Zhao method 1, and Zhao method 2. The results obtained were more accurate than the results in Chen and Karihaloo's [1] study. Through the calculation, this study presents a better range of parameters for the above three methods, and compares the advantages and disadvantages of each method in detail.

Keywords

PiezoelectricCrackTransientInversionLaplace transform
Type
Research Article
Information
Journal of Mechanics , Volume 30 , Issue 1 , February 2014 , pp. 3 - 10
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1.Chen, Z. T. and Karihaloo, B. L., “Dynamic Response of a Cracked Piezoelectric Ceramic Under Arbitrary Electro-Mechanical Impact,” International Journal of Solids and Structures, 36, pp. 5125–5133 (1999).Google Scholar
2.Schmittroth, L. A., “Numerical Inversion of Laplace Transforms,” Communications of the ACM, 3, pp. 171–173 (1960).Google Scholar
3.Bellman, R. E., Kalaba, R. E. and Lockett, J., Numerical Inversion of Laplace Transform, American Elsevier, New York (1966).Google Scholar
4.Miller, M. K. and Guy, W. T., “Numerical Inversion of the Laplace Transform by Use of Jacobi Polynomials,” SIAM Journal on Numerical Analysis, 3, pp. 624–635 (1966).Google Scholar
5.Week, W. T., “Numerical Inversion of Laplace Transforms Using Laguerre Functions,” Journal of the Association for Computing Machinery, 13, pp. 419–429 (1966).Google Scholar
6.Dubner, H. and Abate, J., “Numerical Inversion of Laplace Transforms by Relating Them to the Finite Fourier Cosine Transform,” Journal of the Association for Computing Machinery, 15, pp. 115–123 (1968).Google Scholar
7.Durbin, F., “Numerical Inversion of Laplace Transforms: An Efficient Improvement to Dubner and Abate's Method,” The Computer Journal, 17, pp. 371–376 (1974).CrossRefGoogle Scholar
8.Crump, K. S., “Numerical Inversion of Laplace Transforms Using a Fourier Series Approximation,” Journal of the Association for Computing Machinery, 23, pp. 89–96 (1976).CrossRefGoogle Scholar
9.Albrecht, P. and Honig, G., “Numerical Inversion of Laplace Transforms: [Numerische Inversion Der Laplace-Transformierten],” Angewandte Informatik, 8, pp. 336–345 (1977).Google Scholar
10.Honig, G. and Hirdes, U., “A Method for the Numerical Inversion of Laplace Transforms,” Journal of Computational and Applied Mathematics, 10, pp. 113–132 (1984).Google Scholar
11.Zakian, V. and Coleman, R., “Numerical Inversion of Rational Laplace Transforms,” Electronics Letters, 7, pp. 777–778 (1971).CrossRefGoogle Scholar
12.Singhal, K., Vlach, J. and Vlach, M., “Numerical Inversion of Multidimensional Laplace Transform,” Proceedings of the IEEE, 63, pp. 1627–1628 (1975).Google Scholar
13.Hosono, T., “Numerical Inversion of Laplace Transform and Some Applications to Wave Optics,” Radio Science, 16, pp. 1015–1019 (1981).Google Scholar
14.Therapos, C. P. and Diamessis, J. E., “Numerical Inversion of a Class of Laplace Transforms,” Electronics Letters, 18, pp. 620–622 (1982).Google Scholar
15.Shih, D. H., Shen, R. C. and Shiau, T. C., “Numerical Inversion of Multidimensional Laplace Transforms,” International Journal of Systems Science, 18, pp. 739–742 (1987).Google Scholar
16.Kwok, Y. K. and Barthez, D., “An Algorithm for the Numerical Inversion of Laplace Transforms,” Inverse Problems, 5, pp. 1089–1095 (1989).Google Scholar
17.Evans, G. A., “Numerical Inversion of Laplace Transforms Using Contour Methods,” International Journal of Computer Mathematics, 49, pp. 93–105 (1993).Google Scholar
18.Wu, J. L., Chen, C. H. and Chen, C. F., “Numerical Inversion of Laplace Transform Using Haar Wavelet Operational Matrices,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 48, pp. 120–122 (2001).Google Scholar
19.Abate, J. and Valkó, P. P., “Multi-Precision Laplace Transform Inversion,” International Journal for Numerical Methods in Engineering, 60, pp. 979–993 (2004).Google Scholar
20.Zhao, X., “An Efficient Approach for the Numerical Inversion of Laplace Transform and its Application in Dynamic Fracture Analysis of a Piezoelectric Laminate,” International Journal of Solids and Structures, 41, pp. 3653–3674 (2004).CrossRefGoogle Scholar
21.Milovanović, G. V. and Cvetković, A. S., “Numerical Inversion of the Laplace Transform,” Electronics and Energetics, 18, pp. 515–530 (2005).Google Scholar
22.Narayanan, G. V. and Beskos, D. E., “Numerical Operational Methods for Time-Dependent Linear Problems,” International Journal for Numerical Methods in Engineering, 18, pp. 1829–1854 (1982).Google Scholar
23.Cheney, W. and Kincaid, D., Numerical Mathematics and Computing, 2nd Edition, Brooks/Cole Publishing Company, Monterey, (1985).Google Scholar
24.Loeber, J. F. and Sih, G. C., “Diffraction of Antiplane Shear Waves by a Finite Crack,” Journal of the Acoustical Society of America, 44, pp. 90–98 (1968).Google Scholar
25.Ing, Y. S. and Ma, C. C., “Dynamic Fracture Analysis of a Finite Crack Subjected to an Incident Horizontally Polarized Shear Wave,” International Journal Solids Structures, 34, pp. 895–910 (1997).Google Scholar