1. 1 3D Exact Analysis of Functionally Graded and Laminated Piezoelectric Plates and Shells G.M. Kulikov and S.V. Plotnikova Speaker: Svetlana Plotnikova Department of Applied Mathematics & Mechanics

2. 2 (n)i(n)i n (n)i(n)i n n n 33 3 n (1) (2) (3) Figure 1. Geometry of laminated shell Base Vectors of Midsurface and SaS Indices: n = 1, 2, …, N; i n = 1, 2, …, I n ; m n = 2, 3, …, I n -1 N - number of layers; I n - number of SaS of the nth layer r(  1,  2 ) - position vector of midsurface  ; R (n)i - position vectors of SaS of the nth layer e i - orthonormal vectors; A , k  - Lamé coefficients and principal curvatures of midsurface c  = 1+k   3 - components of shifter tensor at SaS  (n)1,  (n)2, …,  (n)I - sampling surfaces (SaS)  (n)i - transverse coordinates of SaS  [n-1],  [n] - transverse coordinates of interfaces Kinematic Description of Undeformed Shell

3. 3 (n)i(n)i n (n)i(n)i n ( (4) (5) (6) Figure 2. Initial and current configurations of shell Base Vectors of Deformed SaS Position Vectors of Deformed SaS u (  1,  2 ) - displacement vectors of SaS  (  1,  2 ) - derivatives of 3D displacement vector at SaS Kinematic Description of Deformed Shell

4. 4 Green-Lagrange Strain Tensor at SaS Linearized Strain-Displacement Relationships Presentation of Displacement Vectors of SaS (7) (8) (9)

5. 5 Presentation of Derivatives of Displacement Vectors of SaS Strain Parameters Component Form of Strains of SaS Remark. Strains (12) exactly represent all rigid-body shell motions in any convected curvilinear coordinate system. It can be proved through Kulikov and Carrera (2008) (10) (11) (12)

6. 6 Description of Electric Field Electric Field Vector at SaS  – electric potential  1    – electric potentials of SaS (n)i(n)i n (13) (14) (15)

7. 7 Displacement Distribution in Thickness Direction Distribution of Derivatives of 3D Displacement Vector Strain Distribution in Thickness Direction Higher-Order Layer-Wise Shell Formulation (16) (17) (18) (19) L (  3 ) - Lagrange polynomials of degree I n - 1 (n)i(n)i n

8. 8 Electric Potential Distribution in Thickness Direction Distribution of Electric Field Vector Distribution of Derivative of Electric Potential (20) (21) (22)

9. 9 Variational Equation Stress Resultants Electric Displacement Resultants W – work done by external electromechanical loads (23) (24) (25) (26)

10. 10 Material Constants in Thickness Direction (27) (28) (29) C ijkl, e kij and  ik – values of elastic, piezoelectric and dielectric constants on SaS of the nth layer (n)i(n)i n (n)i(n)i n (n)i(n)i n

11. 11 Constitutive Equations Presentations for Stress and Electric Displacement Resultants (n) (30) (31) (32) (33) (34) C ijk, e kij,  ik – elastic, piezoelectric and dielectric constants of the nth layer (n)

12. 12 Numerical Examples 1. Simply Supported Three-Layer Plate under Mechanical Loading Analytical solution Figure 3. PVDF [0/90/0] square plate (h = 0.01 m, p 0 = 3 Pa) (r=s=1) Table 1. Results for a piezoelectric three-ply plate with a /h = 4 under mechanical loading (Lage at al.),, VariableExactI n =3I n =5I n =7I n =9I n =11  u 1 (0, a/2, 0.005)  10 12, m 1.7191.68791.7188 u 3 (a/2, a/2,0.005)  10 11, m 1.5291.51701.5285  11 (a/2, a/2, 0.005)  10  1, Pa 3.3713.31583.37153.3714  12 (0, 0, 0.005), Pa 2.6392.60302.6391  13 (0, a/2, 0.0023), Pa 3.0813.17223.06973.07903.0789  23 (a/2,0, 0), Pa 2.6142.23962.62162.61392.6140  (a/2, a/2, 0)  10 3, V 1.2801.27071.2798  D 1 (0, a/2, 0)  10 11, C/m 2 2.4142.38882.41392.4138  D 3 (a/2, a/2, 0.005)  10 11, C/m 2 4.9705.44554.97704.96974.9696

13. 13 Figure 4. Distributions of transverse shear stresses, electric displacement and electric potential through the thickness of the three-ply plate subjected to mechanical loading for I 1 = I 2 = I 3 = 7: present analysis ( ) and Heyliger (  ), where z = x 3 /h.

14. 14 2. Simply Supported Three-Layer Plate under Electric Loading Analytical solution Figure 5. PVDF [0/90/0] square plate (h = 0.01 m,  0 = 200 V) ( r=s=1 ) Table 2. Results for a piezoelectric three-ply plate with a /h = 4 under electric loading (Lage at al.),, VariableExactI n =3I n =5I n =7I n =9I n =11  u 1 (0,a/2, 0.005)  10 10, m 3.2233.19223.2226 u 3 (a/2, a/2,0.005)  10 9, m 3.3133.30893.3131  22 (a/2, a/2, 0.01/6)  10  3, Pa 2.8412.84402.84082.8407  12 (0, 0, 0.005)  10  2, Pa 5.5435.51745.5427  13 (0, a/2, 0.003)  10  2, Pa 2.9252.46602.93152.9246  23 (a/2,0, 0.01/6)  10  2, Pa 2.328 1.7841 2.0834 2.3174 2.3252 2.3283 2.3282 2.3284  33 (a/2, a/2, 0)  10  1, Pa 3.6293.97403.60343.62923.6290  D 1 (0, a/2, 0.005)  10 6, C/m 2 1.7391.73931.7389  D 3 (a/2, a/2, 0.005)  10 6, C/m 2 3.1003.07053.10023.1003

15. 15 Figure 6. Distributions of transverse shear stresses, electric displacement and electric potential through the thickness of the three-ply plate subjected to electric loading for I 1 = I 2 = I 3 = 7: present analysis ( ) and Heyliger (  ), where z = x 3 /h.

16. 16 3. FG Piezoelectric Square Plate under Mechanical Loading Figure 7. PZT-4 FG square plate with grounded interfaces under mechanical loading (r=s=1) Analytical solution Material constants

17. 17 InIn u 1 (-0.5)u 3 (0)  (0)  11 (0.5)  12 (0.5)  13 (0)  33 (0) D 1 (-0.5)D 3 (0) 5 4.2738-2.4856-12.138-7.86093.0219-1.1787-0.287380.913270.20612 7 4.2738-2.4856-12.138-7.85443.0229-1.1761-0.281480.945030.21328 9 4.2738-2.4856-12.138-7.85433.0229-1.1762-0.281500.945160.21323 11 4.2738-2.4856-12.138-7.85433.0229-1.1762-0.281500.945160.21323 InIn u 1 (-0.5)u 3 (0)  (0)  11 (0.5)  12 (0.5)  13 (0)  33 (0) D 1 (-0.5)D 3 (0) 5 1.1493-0.92158-4.4469-15.3366.0057-1.1786-0.212384.8146-0.23971 7 1.1493-0.92158-4.4469-15.3606.0068-1.1760-0.218324.8517-0.24667 9 1.1493-0.92158-4.4469-15.3606.0068-1.1760-0.218294.8519-0.24662 11 1.1493-0.92158-4.4469-15.3606.0068-1.1760-0.218294.8519-0.24662 Table 3. Results for FG piezoelectric plate with a/h = 10 and  =  1 under mechanical loading Table 4. Results for FG piezoelectric plate with a/h = 10 and  = 1 under mechanical loading            

18. 18 Figure 8. Mechanical loading of the FG piezoelectric square plate: distributions of transverse shear stress, electric potential and electric displacement through the thickness of the plate for I 1 = 9, present analysis ( ) and Zhong and Shang (  ).

19. 19 4. FG Piezoelectric Square Plate under Electric Loading InIn u 1 (0.5)u 3 (0)  (0)  11 (0.5)  12 (0.5)  13 (0)  33 (0) D 1 (0.5)D 3 (0) 5208.76-41.091-200.62182.37147.650.18798-0.22143783.824854.2 7208.76-41.090-200.68182.80147.660.14102-0.25627783.284856.0 9208.76-41.090-200.68182.80147.660.14140-0.25555783.274856.0 11208.76-41.090-200.68182.80147.660.14140-0.25556783.274856.0 13 208.76-41.090-200.68182.80147.660.14140-0.25556783.274856.0 InIn u 1 (0.5)u 3 (0)  (0)  11 (0.5)  12 (0.5)  13 (0)  33 (0) D 1 (0.5)D 3 (0) 527.45715.11673.803192.64143.51-0.18798-0.221431086.04854.2 727.45715.11673.827193.26143.50-0.14102-0.256271086.74856.0 927.45715.11673.827193.26143.50-0.14140-0.255551086.74856.0 1127.45715.11673.827193.26143.50-0.14140-0.255561086.74856.0 13 27.45715.11673.827193.26143.50-0.14140-0.255561086.74856.0 Table 6. Results for FG piezoelectric plate with a/h = 10 and  = 1 under electric loading       Table 5. Results for FG piezoelectric plate with a/h = 10 and  =  1 under electric loading      

20. 20 Figure 9. Electric loading of the FG piezoelectric square plate: distributions of transverse shear stresses and electric potential through the thickness of the plate for I 1 = 9, present analysis ( ) and Zhong and Shang (  ).

21. 21 InIn u 1 (-0.5)u 2 (-0.5)u 3 (-0.5)  (0)  11 (-0.5)  22 (-0.5)  12 (-0.5)  13 (0)  23 (0)  33 (0) D 3 (0) 3251.59-539.53740.232.8488-14.325-136.36-82.66156.821-40.71340.49415.226 5254.32-543.71742.092.8509-13.457-127.94-83.39757.592-41.21442.25115.986 7254.30-543.65742.072.8511-13.433-127.70-83.38957.589-41.21042.24715.975 9254.30-543.65742.072.8511-13.431-127.67-83.38957.589-41.21042.24715.975 11254.30-543.65742.072.8511-13.430-127.67-83.38957.589-41.21042.24715.975 5. Piezoelectric Laminated Orthotropic Cylindrical Shell Table 7. Results for a piezoelectric three-layer shell with S = 2 under mechanical loading Figure 10. Three-layer [PZT4/PZT4F/PZT4] cylindrical shell under mechanical loading (r=s=1) Analytical solution   

22. 22 Figure 11. Distributions of transverse shear stresses, electric potential and electric displacement through the thickness of the three-layer shell under mechanical loading for I 1 = I 2 = I 3 = 7: present analysis ( ) and Heyliger (  )

23. 23 InIn u 1 (-0.5)u 2 (-0.5)u 3 (-0.5)  (0)  11 (-0.5)  22 (-0.5)  12 (-0.5)  13 (0)  23 (0)  33 (0) D 3 (0) 3 -8.047116.158-0.644212.6894-1.5080-12.7512.53837.5733-6.57420.8436-37.197 5 -7.978016.144-0.595562.6878-1.6051-13.7062.52858.0537-6.93891.1103-36.807 7 -7.977816.143-0.593842.6879-1.6021-13.6762.52838.0504-6.93651.1176-36.808 9 -7.977816.143-0.593842.6879-1.6026-13.6812.52838.0503-6.93651.1175-36.808 11 -7.977816.143-0.593842.6879-1.6026-13.6812.52838.0503-6.93651.1175-36.808 6. Piezoelectric Laminated Orthotropic Cylindrical Shell Table 8. Results for a piezoelectric three-layer shell with S = 2 under electric loading Analytical solution Figure 12. Three-layer [PZT4/PZT4F/PZT4] cylindrical shell under electric loading ( r=s=1)     

24. 24 Figure 13. Distributions of transverse shear stresses, electric potential and electric displacement through the thickness of the three-layer shell under electric loading for I 1 = I 2 = I 3 = 7

25. 25 7. FG Piezoelectric Anisotropic Cylindrical Shell Figure 14. Four-layer FG [PZT/45/-45/PZT] cylindrical shell under mechanical loading (R/h=4) (r=1) Analytical solution Material constants of PZT Figure 15. Through-thickness distribution of elastic constants of the top FG piezoelectric layer

26. 26 Table 9. Results for a FG piezoelectric angle-ply shell with  =  1 under mechanical loading    InIn u 1 (-0.5)u 2 (-0.5)u 3 (0)  (-0.5)  11 (-0.5)  22 (-0.5)  12 (-0.5)  13 (-0.125)  23 (0.125)  33 (0.125) D 3 (0.25) 5 3.32890.932637.4317-2.6981-3.70682.0244-4.482862.895-9.988462.693-5.9486 7 3.32890.932647.4316-2.6984-3.70702.0244-4.482862.891-9.985562.693-5.9535 9 3.32890.932647.4316-2.6984-3.70692.0242-4.482962.891-9.985562.693-5.9536 11 3.32890.932647.4316-2.6984-3.70692.0242-4.482962.891-9.985562.693-5.9536 InIn u 1 (-0.5)u 2 (-0.5)u 3 (0)  (-0.5)  11 (-0.5)  22 (-0.5)  12 (-0.5)  13 (-0.125)  23 (0.125)  33 (0.125) D 3 (0.25) 5 1.55281.26984.15583.9143-6.30305.2026-2.441452.534-5.341760.071-6.4591 7 1.55281.26984.15583.9143-6.30155.2042-2.441452.533-5.340460.070-6.4810 9 1.55281.26984.15583.9143-6.30155.2042-2.441452.533-5.340460.070-6.4818 11 1.55281.26984.15583.9143-6.30155.2042-2.441452.533-5.340460.070-6.4819 Table 10. Results for a FG piezoelectric angle-ply shell with  = 1 under mechanical loading   

27. 27 Figure 16. Distributions of stresses and electric displacement through the thickness direction of the FG piezoelectric angle-ply cylindrical shell under mechanical loading for I 1 = I 2 = I 3 = I 4 = 9: present analysis ( ) and authors’ 3D exact solution (  )

28. 28 InIn u 1 (-0.5)u 2 (-0.5)u 3 (0)  (-0.5)  11 (-0.5)  22 (-0.5)  12 (-0.5)  13 (-0.125)  23 (0.125)  33 (0.125) D 3 (0.25) 5 2.4020-0.921667.411518.401-2.09892.79890.44301-2.8587-24.12963.358-165.68 7 2.4019-0.921557.410918.399-2.09902.79620.44295-2.8635-24.12163.353-165.81 9 2.4019-0.921557.410918.399-2.09902.79620.44295-2.8635-24.12163.353-165.82 11 2.4019-0.921557.410918.399-2.09902.79620.44295-2.8635-24.12163.353-165.82 InIn u 1 (-0.5)u 2 (-0.5)u 3 (0)  (-0.5)  11 (-0.5)  22 (-0.5)  12 (-0.5)  13 (-0.125)  23 (0.125)  33 (0.125) D 3 (0.25) 5 2.12440.587527.872426.175 -6.210312.716-1.1296-6.0595-17.931108.42-363.48 7 2.12440.587527.872426.175-6.208412.718-1.1296-6.0650-17.925108.42-364.66 9 2.12440.587527.872426.175-6.208412.718-1.1296-6.0650-17.925108.42-364.70 11 2.12440.587527.872426.175-6.208412.718-1.1296-6.0650-17.925108.42-364.71 8. FG Piezoelectric Anisotropic Cylindrical Shell under Electric Loading Table 11. Results for a FG piezoelectric angle-ply shell with  =  1 under electric loading      Table 12. Results for a FG piezoelectric angle-ply shell with  = 1 under electric loading     

29. 29 Figure 17. Distributions of stresses and electric displacement through the thickness direction of the FG piezoelectric angle-ply cylindrical shell under electric loading for I 1 = I 2 = I 3 = I 4 = 9: present analysis ( ) and authors’ 3D exact solution (  )

30. 30 Thanks for your attention! Conclusions 1.SaS method gives the possibility to obtain exact 3D solutions of electroelasticity for thick and thin FG piezoelectric plates and shells with a prescribed accuracy 1.SaS method gives the possibility to obtain exact 3D solutions of electroelasticity for thick and thin FG piezoelectric plates and shells with a prescribed accuracy. 2.New higher-order layer-wise theory of FG piezoelectric shells has been developed by using of only displacements of SaS. This is straightforward for finite element developments.Conclusions 1.SaS method gives the possibility to obtain exact 3D solutions of electroelasticity for thick and thin FG piezoelectric plates and shells with a prescribed accuracy 1.SaS method gives the possibility to obtain exact 3D solutions of electroelasticity for thick and thin FG piezoelectric plates and shells with a prescribed accuracy. 2.New higher-order layer-wise theory of FG piezoelectric shells has been developed by using of only displacements of SaS. This is straightforward for finite element developments.