∂ N 12 A 2
ð
Þ
∂α 1
þ
∂ N 22 A 1
ð
Þ
∂α 2
þ N 21
∂A 2
∂α 1
À N 11
∂A 1
∂α 2
þ Q 23 A 1 A 2
1
R 2
þ F
0
ð Þ
2
¼ 2ρhA 1 A 2 € u
0
ð Þ
2 ,
ð5:153Þ
∂
∂α 1
Q 13 A 2
ð
Þþ
∂
∂α 2
Q 23 A 1
ð
ÞÀN 11
A 1 A 2
R 1
À N 22
A 1 A 2
R 2
þ F
0
ð Þ
3
¼ 2ρhA 1 A 2 € u
0
ð Þ
3 :
ð5:154Þ
A thin shell can sometimes be treated as a membrane that does not resist bending,
without bending moments and transverse shear forces. In this case Eqs. (5.152),
(5.123), and (5.154) reduce the following membrane theory:
∂ N 11 A 2
ð
Þ
∂α 1
þ
∂ N 21 A 1
ð
Þ
∂α 2
þ N 12
∂A 1
∂α 2
À N 22
∂A 2
∂α 1
þ F
0
ð Þ
1 ¼ 2ρhA 1 A 2 € u
0
ð Þ
1 ,
ð5:155Þ
∂ N 12 A 2
ð
Þ
∂α 1
þ
∂ N 22 A 1
ð
Þ
∂α 2
þ N 21
∂A 2
∂α 1
À N 11
∂A 1
∂α 2
þ F
0
ð Þ
2 ¼ 2ρhA 1 A 2 € u
0
ð Þ
2 ,
ð5:156Þ
ÀN 11
1
R 1
À N 22
1
R 2
þ F
0
ð Þ
3 ¼ 2ρh€ u
0
ð Þ
3 :
ð5:157Þ
References
1. J.S. Yang, H.G. Zhou, Amplification of acoustic waves in piezoelectric semiconductor plates.
Int. J. Solids Struct. 42, 3171–3183 (2005)
2. Y.X. Luo, C.L. Zhang, W.Q. Chen, J.S. Yang, Piezotronic effect of a thin film with elastic and
piezoelectric semiconductor layers under a static flexural loading. ASME J. Appl. Mech. 86,
051003 (2019)
3. R.D. Mindlin, in An Introduction to the Mathematical Theory of Vibrations of Elastic Plates,
ed. by J.S. Yang, (World Scientific, Singapore, 2006)
4. R.D. Mindlin, High frequency vibrations of crystal plates. Q. Appl. Math. 19, 51–61 (1961)
5. R.D. Mindlin, High frequency vibrations of piezoelectric crystal plates. Int. J. Solids Struct. 8,
895–906 (1972)
6. J.S. Yang, The Mechanics of Piezoelectric Structures (World Scientific, Singapore, 2006)
7. J.L. Bleustein, H.F. Tiersten, Forced thickness-shear vibrations of discontinuously plated
piezoelectric plates. J. Acoust. Soc. Am. 43, 1311–1318 (1968)
8. J.S. Yang, X.M. Yang, J.A. Turner, Amplification of acoustic waves in piezoelectric semiconductor shells. J. Intell. Mater. Syst. Struct. 16, 613–621 (2005)
9. H. Kraus, Thin Elastic Shells (Wiley, New York, 1967)
10. H.S. Tzou, Piezoelectric Shells (Kluwer, Dordrecht, 1993)
140
5 Extension and Bending of Plates
ð
Þ
∂α 1
þ
∂ N 22 A 1
ð
Þ
∂α 2
þ N 21
∂A 2
∂α 1
À N 11
∂A 1
∂α 2
þ Q 23 A 1 A 2
1
R 2
þ F
0
ð Þ
2
¼ 2ρhA 1 A 2 € u
0
ð Þ
2 ,
ð5:153Þ
∂
∂α 1
Q 13 A 2
ð
Þþ
∂
∂α 2
Q 23 A 1
ð
ÞÀN 11
A 1 A 2
R 1
À N 22
A 1 A 2
R 2
þ F
0
ð Þ
3
¼ 2ρhA 1 A 2 € u
0
ð Þ
3 :
ð5:154Þ
A thin shell can sometimes be treated as a membrane that does not resist bending,
without bending moments and transverse shear forces. In this case Eqs. (5.152),
(5.123), and (5.154) reduce the following membrane theory:
∂ N 11 A 2
ð
Þ
∂α 1
þ
∂ N 21 A 1
ð
Þ
∂α 2
þ N 12
∂A 1
∂α 2
À N 22
∂A 2
∂α 1
þ F
0
ð Þ
1 ¼ 2ρhA 1 A 2 € u
0
ð Þ
1 ,
ð5:155Þ
∂ N 12 A 2
ð
Þ
∂α 1
þ
∂ N 22 A 1
ð
Þ
∂α 2
þ N 21
∂A 2
∂α 1
À N 11
∂A 1
∂α 2
þ F
0
ð Þ
2 ¼ 2ρhA 1 A 2 € u
0
ð Þ
2 ,
ð5:156Þ
ÀN 11
1
R 1
À N 22
1
R 2
þ F
0
ð Þ
3 ¼ 2ρh€ u
0
ð Þ
3 :
ð5:157Þ
References
1. J.S. Yang, H.G. Zhou, Amplification of acoustic waves in piezoelectric semiconductor plates.
Int. J. Solids Struct. 42, 3171–3183 (2005)
2. Y.X. Luo, C.L. Zhang, W.Q. Chen, J.S. Yang, Piezotronic effect of a thin film with elastic and
piezoelectric semiconductor layers under a static flexural loading. ASME J. Appl. Mech. 86,
051003 (2019)
3. R.D. Mindlin, in An Introduction to the Mathematical Theory of Vibrations of Elastic Plates,
ed. by J.S. Yang, (World Scientific, Singapore, 2006)
4. R.D. Mindlin, High frequency vibrations of crystal plates. Q. Appl. Math. 19, 51–61 (1961)
5. R.D. Mindlin, High frequency vibrations of piezoelectric crystal plates. Int. J. Solids Struct. 8,
895–906 (1972)
6. J.S. Yang, The Mechanics of Piezoelectric Structures (World Scientific, Singapore, 2006)
7. J.L. Bleustein, H.F. Tiersten, Forced thickness-shear vibrations of discontinuously plated
piezoelectric plates. J. Acoust. Soc. Am. 43, 1311–1318 (1968)
8. J.S. Yang, X.M. Yang, J.A. Turner, Amplification of acoustic waves in piezoelectric semiconductor shells. J. Intell. Mater. Syst. Struct. 16, 613–621 (2005)
9. H. Kraus, Thin Elastic Shells (Wiley, New York, 1967)
10. H.S. Tzou, Piezoelectric Shells (Kluwer, Dordrecht, 1993)
140
5 Extension and Bending of Plates