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

2. 2 Description of Temperature Field (1) (2) Temperature Gradient  in Orthonormal Basis N - number of layers; I n - number of SaS of the nth layer n = 1, 2, …, N; i n = 1, 2, …, I n ; m n = 2, 3, …, I n - 1  (n)1,  (n)2, …,  (n)I - sampling surfaces (SaS)  (n)i - transverse coordinates of SaS  [n-1],  [n] - transverse coordinates of interfaces n n 33 3 T(  1,  2,  3 ) – temperature; c  = 1+k   3 – components of shifter tensor at SaS A  (  1,  2 ), k  (  1,  2 ) – Lamé coefficients and principal curvatures of midsurface 

3. 3 Temperature Distribution in Thickness Direction Temperature Gradient Distribution in Thickness Direction T (n)i (  1,  2 ) – temperatures of SaS; L (n)i (  3 ) – Lagrange polynomials of degree I n - 1 nn (3) (4) (5)  (n)i (  1,  2 ) – temperature gradient at SaS n i

4. 4 Relations between Temperature Gradient and Temperature c (n)i (  1,  2 ) – components of shifter tensor at SaS n  (6) (7) (8) Derivatives of Lagrange Polynomials at SaS

5. 5 Description of Electric Field Electric Field E in Orthonormal Basis Electric Potential Distribution in Thickness Direction Electric Field Distribution in Thickness Direction Relations between Electric Field and Electric Potential (9) (10) (11) (12)  (n)i (  1,  2 ) – electric potentials of SaS n E (n)i (  1,  2 ) – electric field at SaS n i

6. 6 n 33 3 n (13) (14) (15) Position Vectors and Base Vectors of 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 N - number of layers; I n - number of SaS of the nth layer n = 1, 2, …, N; i n = 1, 2, …, I n ; m n = 2, 3, …, I n - 1 r(  1,  2 ) – position vector of midsurface  ; e i (  1,  2 ) – orthonormal base vectors of midsurface 

7. 7 u (  1,  2 ) – displacement vectors of SaS (n)i(n)i n ( (16) (17) (18) Base Vectors of Deformed SaS Position Vectors of Deformed SaS (n)i(n)i n  (  1,  2 ) – values of derivative of displacement vector at SaS Kinematic Description of Deformed Shell u (  1,  2,  3 ) – displacement vector

8. 8 Green-Lagrange Strain Tensor at SaS Linearized Strain Tensor at SaS Displacement Vectors of SaS in Orthonormal Basis (19) (20) (21)

9. 9 Derivatives of Displacement Vectors in Orthonormal Basis Strain Parameters of SaS Linearized Strains of SaS Remark. Strains (24) exactly represent all rigid-body shell motions in any convected curvilinear coordinate system. It can be proved through results of Kulikov and Carrera (2008) (22) (23) (24)

10. 10 Displacement Distribution in Thickness Direction Strain Distribution in Thickness Direction Presentation for Derivative of Displacement Vector (25) (26) (27)

11. 11 Variational Formulation of Heat Conduction Problem Variational Equation q (n) – heat flux of the nth layer; Q n – specified heat flux i Heat Flux Resultants (28) (29) (30)

12. 12 (31) (32) (33) Fourier Heat Conduction Equations Basic Functional of Heat Conduction Theory k (n) – thermal conductivity tensor of the nth layer ij Heat Flux Resultants

13. 13  (n) = T (n) - T 0 – temperature rise; W – work done by external loads (34) (35) (36) (37) (38) Stress Resultants Electric Displacement Resultants Entropy Resultants Formulation of Thermopiezoelectric Problem

14. 14 Constitutive Equations (39) (40) (41) (42) (43) (44) Stress, Electric Displacement and Entropy Resultants (n) C ijk, e kij,  ij,  ik, r i,  (n) – physical properties of the nth layer (n) (n)

15. 15 Numerical Examples 1. Elastic Rectangular Plate under Mechanical Loading Analytical solution Table 1. Results for a single-layer square plate with E = 10 7 Pa, = 0.3, a = b = 1 m and a/h = 3, Dimensionless variables in crucial points InIn -u 1 (0.5)u 3 (0)  11 (0.5)  13 (0)  33 (0) 30.39863724945046801.2788782433895911.9993021468548370.49774361511680030.4752837277628003 70.43589339371319481.3425549534665132.1240324282884130.70227626660600940.4943950643281928 110.43589339426031201.3425546895430952.1240184103140480.70230220832235380.4944039935419638 150.43589339426031211.3425546895424912.1240184101937820.70230220847675800.4944039936052152 190.43589339426031201.3425546895424912.1240184101937800.70230220847675780.4944039936052150 Exact0.43589339426031201.3425546895424912.1240184101937810.70230220847675780.4944039936052149    Exact results have been obtained by authors using Vlasov's closed-form solution (1957)

16. 16 InIn -u 1 (0.5)u 3 (0)  11 (0.5)  13 (0)  33 (0) 316.69442044348862112.442017807340519.916901245964341.7062021316580780.4978329738513218 7 16.81014395019634113.172094102493720.043946355190682.3833731392144060.4999497585886086 11 16.81014395019644113.172094100491420.043946331897512.3833737911768810.4999497657893389 15 16.81014395019644113.172094100491320.043946331897502.3833737911769090.4999497657893389 19 16.81014395019645113.172094100491420.043946331897512.3833737911769100.4999497657893400 Exact 16.81014395019644113.172094100491420.043946331897512.3833737911769110.4999497657893391 Table 2. Results for a single-layer square plate with E = 10 7 Pa, = 0.3, a = b = 1 m and a/h = 10 Exact results have been obtained by authors using Vlasov's closed-form solution (1957) Figure 1. Through-thickness distributions of transverse stresses of the plate with a/h = 1: present analysis ( ─ ) for I 1 = 11 and Vlasov's closed-form solution (  )    

17. 17 2. Piezoelectric Plate in Cylindrical Bending for Heat Flux Boundary Conditions Analytical solution Dimensionless variables in crucial points

18. 18 Table 4. Results for a single-layer Cadmium-Selenide plate in case of heat flux boundary conditions with a/h = 10 InIn  (0.5)  (0.5) u 1 (0.5)u 3 (0) 31.035337086088057-1.329936076735578-4.7500695155952884.830450207594048 71.035337187913652-1.329961621213969-4.7499491493302884.829776282323792 111.035337187913652-1.329961621213969-4.7499491493302884.829776282323684 151.035337187913653-1.329961621213969-4.7499491493302884.829776282323684 191.035337187913652-1.329961621213969-4.7499491493302884.829776282323683    Table 3. Results for a single-layer Cadmium-Selenide plate in case of heat flux boundary conditions with a/h = 2 InIn  (0.5)  (0.5) u 1 (0.5)u 3 (0) 31.514830974804894-1.276585132031578-6.9920902137774794.039437427219185 71.516041850892555-1.287176996356866-6.9362283600927254.045924923554766 111.516041850893569-1.287176996676169-6.9362283593885624.045924921944983 151.516041850893570-1.287176996676170-6.9362283593885624.045924921942870 191.516041850893570-1.287176996676169-6.9362283593885634.045924921942869 231.516041850893570-1.287176996676170-6.9362283593885634.045924921942869   

19. 19 3. Piezoelectric Plate in Cylindrical Bending for Convective Boundary Conditions Dimensional variables in crucial points Analytical solution

20. 20 Table 5. Results for a single-layer Cadmium-Selenide plate in case of convective boundary conditions with a/h = 2 InIn  (-0.5)  (0.5)  (0.5) u 1 (0.5)u 3 (0.5)  11 (0.5)  13 (-0.25)  33 (0) D 3 (0)q 3 (0.5)  (0.5) 50.288810.63570-5.2859-2.90889.9580-15.6823.1848-2.8412-1.3398-10.9203.9685 70.288810.63570-5.2859-2.90879.9580-13.0983.69683.9052-1.3255-10.9293.9685 90.288810.63570-5.2859-2.90879.9580-13.0893.69383.8899-1.3256-10.9293.9685 110.288810.63570-5.2859-2.90879.9580-13.0893.69383.8898-1.3256-10.9293.9685 130.288810.63570-5.2859-2.90879.9580-13.0893.69383.8898-1.3256-10.9293.9685 Exact0.28880.6357-5.286-2.9099.958-13.09 Table 6. Results for a single-layer Cadmium-Selenide plate in case of convective boundary conditions with a/h = 10 InIn  (-0.5)  (0.5)  (0.5) u 1 (0.5)u 3 (0.5)  11 (0.5)  13 (-0.25)  33 (0) D 3 (0)q 3 (0.5)  (0.5) 50.728940.90045-10.953-4.13112.8608-6.91371.7532-9.9104-3.3167-2.98655.6212 70.728940.90045-10.953-4.13112.8608-6.71821.96352.0573-3.3154-2.98655.6212 90.728940.90045-10.953-4.13112.8608-6.71821.96342.0579-3.3154-2.98655.6212 110.728940.90045-10.953-4.13112.8608-6.71821.96342.0579-3.3154-2.98655.6212 Exact0.72890.9004-10.95-4.1312.861-6.718                 Exact results have been obtained by Dube, Kapuria, Dumir (1996)

21. 21 Figure 2. Through-thickness distributions of temperature, heat flux, electric displacement, transverse stresses and entropy of the single-layer Cadmium-Selenide plate in case of convective boundary conditions for I n = 11

22. 22 4. Laminated Piezoelectric Rectangular Plate under Temperature Loading Analytical solution Dimensionless variables in crucial points

23. 23 Table 7. Results for a two-layer PZT5A/Cadmium-Selenide plate with a/h = 2 InIn  (0)  (0) u 1 (0.5)u 3 (0.5)  11 (0.5)  12 (0.5)  13 (0)  33 (0) D 3 (0)q 3 (0)  (0.5) 30.605973.3660-2.392410.317-4.8826-5.07681.6475 1.4076 -2.3110 -3.8790 10.220 -0.38923 -3.0893 -2.6675 6.2427 50.605833.3689-2.380110.215-4.6437-5.05061.3421 1.3515 -1.7144 -1.6464 -0.17554 -0.31134 -3.3443 -3.3422 6.2428 70.605833.3689-2.380110.215-4.6431-5.05061.3549 1.3500 -1.6483 -1.6475 -0.30981 -0.31035 -3.3465 6.2428 90.605833.3689-2.380110.215-4.6432-5.05061.3501 -1.6477 -0.31034 -0.31035 -3.3465 6.2428 110.605833.3689-2.380110.215-4.6432-5.05061.3501 -1.6477 -0.31035 -3.3465 6.2428 Table 8. Results for a two-layer PZT5A/Cadmium-Selenide plate with a/h = 10 InIn  (0)  (0) u 1 (0.5)u 3 (0.5)  11 (0.5)  12 (0.5)  13 (0)  33 (0) D 3 (0)q 3 (0)  (0.5) 30.867486.6728-2.84606.2762-1.7075-6.03941.6735 3.3784 -6.5647 -8.2434 1.8313 -1.8259 -3.5125 -3.5162 6.2434 50.867486.6726-2.84556.2730-1.6988-6.03842.7117 2.7139 -6.3918 -6.3860 -1.8001 -1.8020 -3.5268 6.2434 70.867486.6726-2.84556.2730-1.6989-6.03842.7132 -6.3876 -1.8020 -3.5268 6.2434 90.867486.6726-2.84556.2730-1.6989-6.03842.7132 -6.3876 -1.8020 -3.5268 6.2434            

24. 24 Figure 3. Through-thickness distributions of temperature, heat flux, electric potential, transverse stresses and entropy of the two-layer PZT5A/Cadmium-Selenide plate for I 1 = I 2 = 9

25. 25 InIn u 1 (-0.5)u 3 (-0.5)  11 (-0.5)  12 (-0.5)  13 (0.125)  23 (0.125)  (-0.5) D 3 (0.25) 31.13186.4450-4.82411.95467.32678.54982.5402-1.5100-4.1738 51.13286.4493-4.81741.95677.48208.81932.5433-1.5156-1.8940 71.13286.4493-4.81741.95677.48058.81682.5433-1.5156-1.8911 91.13286.4493-4.81741.95677.48058.81682.5433-1.5156-1.8911 5. Antisymmetric Angle-Ply Plate Covered with PZT Layers Figure 4. [PZT/60/  60/PZT] square plate with grounded interfaces under mechanical loading for r = s = 1 Table 9. Results for the unsymmetric angle-ply plate with a/h = 4 under mechanical loading  ̃    Analytical solution

26. 26 Figure 5. Mechanical loading of the unsymmetric angle-ply plate: distributions of transverse shear stresses and electric displacement through the thickness of the plate for I 1 = I 2 = I 3 = I 4 =7

27. 27 InIn u 1 (-0.5)u 3 (-0.5)  11 (-0.5)  12 (-0.5)  13 (0.125)  23 (0.125) D 3 (0.25) 5 35.26890.07825.65752.2851.8363-2.694818.563-7.2148-17212 7 35.26890.07825.65752.2851.7915-2.650718.487-7.1405-17213 9 35.26890.07825.65752.2851.7917-2.650818.488-7.1407-17213 6. Antisymmetric Angle-Ply Plate Covered with PZT Layers Table 10. Results for the unsymmetric angle-ply plate with a/h = 4 under electric loading ̃    ̃ Analytical solution Figure 6. [PZT/60/  60/PZT] square plate with grounded interfaces under electric loading for r = s = 1

28. 28 Figure 7. Electric loading of the unsymmetric angle-ply plate: distributions of transverse stresses through the thickness of the plate for I 1 = I 2 = I 3 = I 4 =7

29. 29 Thanks for your attention!