1. 1 3D Thermopiezoelectric Analysis of Laminated and Functionally Graded Plates and Shells by Sampling Surfaces Method G.M. Kulikov and S.V. Plotnikova Speaker: Gennady M. Kulikov Department of Applied Mathematics & Mechanics

2. 2 Description of Temperature Field n n 33 3 (1) (2) Temperature Gradient  in Orthonormal Basis 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  (n)1,  (n)2, …,  (n)I - sampling surfaces (SaS)  (n)i - transverse coordinates of SaS  [n-1],  [n] - transverse coordinates of interfaces T(  1,  2,  3 ) – temperature; c  = 1+k   3 – components of shifter tensor 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 Kinematic Description of Undeformed Shell (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 r(  1,  2 ) – position vector of midsurface  ; e i (  1,  2 ) – orthonormal base vectors of midsurface  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

7. 7 u (  1,  2 ) – displacement vectors of SaS (n)i(n)i n Kinematic Description of Deformed Shell ( (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 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 e i : (19) (20) (21)

9. 9 Derivatives of Displacement Vectors in Orthonormal Basis e i : 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 Thermal Problem Heat Flux Resultants (28) (29) (30) (31) Basic Functional of Heat Conduction Theory q (n) – heat flux of the nth layer; Q n – specified heat flux i

12. 12 Fourier Heat Conduction Equations k (n) – thermal conductivity tensor of the nth layer ij (32) (33) (34) Distribution of Thermal Conductivities in Thickness Direction Heat Flux Resultants

13. 13 Formulation of Thermopiezoelectric Problem  (n) = T (n) - T 0 – temperature rise; W – work done by external loads (35) (36) (37) (38) (39) Stress Resultants Electric Displacement Resultants Entropy Resultants

14. 14 Constitutive Equations (40) (41) (42) (43) C ijkI, e kij,  ij,  ik, r i,  (n) – material constants of the nth layer (n) Distribution of Material Constants in Thickness Direction

15. 15 – values of Stress, Electric Displacement and Entropy Resultants (44) (45) (46) (47) material constants on SaS of the nth layer

16. 16 Numerical Examples I1I1 -u 1 (0.5)u 3 (0)  11 (0.5)  13 (0)  33 (0) 30.39863724945046801.2788782433895911.9993021468548370.49774361511680030.4752837277628003 7 0.43589339371319481.3425549534665132.1240324282884130.70227626660600940.4943950643281928 11 0.43589339426031201.3425546895430952.1240184103140480.70230220832235380.4944039935419638 15 0.43589339426031211.3425546895424912.1240184101937820.70230220847675800.4944039936052152 19 0.43589339426031201.3425546895424912.1240184101937800.70230220847675780.4944039936052150 Exact0.43589339426031201.3425546895424912.1240184101937810.70230220847675780.4944039936052149 Dimensionless variables in crucial points Exact results have been obtained by authors using Vlasov's closed-form solution (1957)    1. FG Rectangular Plate under Mechanical Loading Analytical solution Table 1. Results for a square plate with a = b =1 m, a/h = 3 and  = 0

17. 17 I1I1 -u 1 (0.5)u 3 (0)  11 (0.5)  13 (0)  33 (0) 30.41583365026526521.3479772412572942.0732399068074750.49832267990213840.4596565434470269 7 0.45369779791337921.4146360436827282.1932658589898440.70207003397206540.4877173446168515 11 0.45369779841425761.4146357713109622.1932702584590390.70209576763559460.4877129579452970 15 0.45369779841425761.4146357713103682.1932702586500210.70209576779048560.4877129578319663 19 0.45369779841425751.4146357713103682.1932702586500210.70209576779048540.4877129578319657 Exact 1.41464 Table 2. Results for a FG square plate with a = b =1 m, a/h = 3 and  = 0.1 Figure 1. Through-thickness distributions of transverse stresses of FG plate with a/h = 1: present analysis (─) for I 1 = 11 and Vlasov's closed-form solution (  ), and Kashtalyan's closed-form solution (  ) Exact results have been obtained by Kashtalyan (2004)   

18. 18 2. Two-Phase FG Rectangular Plate under Temperature Loading Analytical solution Dimensionless variables in crucial points Mori-Tanaka method is used for evaluating material properties of the Metal/Ceramic plate with where V c, V c – volume fractions of the ceramic phase on bottom and top surfaces 

19. 19 I1I1 u 1 (0.5)u 3 (0.5)  11 (0.5)  12 (0.5)  13 (0.25)  33 (0)  (0) q 1 (0)q 3 (-0.5)  (0) 3-1.20964.4213-3.1907-6.47754.3279-266.580.397800.249940.5911986.305 5-1.21174.4198-4.1612-6.48905.0419-6.31220.393790.247420.7239285.605 7-1.21014.4111-4.1765-6.48044.2085-8.78940.393750.247400.7312585.596 9-1.21014.4111-4.1764-6.48044.2259-8.68030.393750.247400.7315885.596 11-1.21014.4111-4.1764-6.48044.2265-8.68300.393750.247400.7316085.596 13-1.21014.4111-4.1763-6.48044.2264-8.68290.393750.247400.7316085.596 15-1.21014.4111-4.1763-6.48044.2264-8.68290.393750.247400.7316085.596 Batra-1.21014.4111-4.1764-6.48044.2264-8.68290.39380.7316 Table 3. Results for a Metal/Ceramic square plate with a/h = 5 Table 4. Results for a Metal/Ceramic square plate with a/h = 10 I1I1 u 1 (0.5)u 3 (0.5)  11 (0.5)  12 (0.5)  13 (0.25)  33 (0)  (0) q 1 (0)q 3 (-0.5)  (0) 3-1.20953.6353-3.0497-6.47734.7296-1259.00.42763 0.7105192.759 5-1.21403.6416-4.1605-6.50155.3713-13.7900.42404 0.8021692.184 7-1.21243.6337-4.1555-6.49284.4563-9.76550.42401 0.8072392.176 9-1.21243.6337-4.1555-6.49284.4699-9.15310.42401 0.8074792.176 11-1.21243.6337-4.1555-6.49284.4704-9.16270.42401 0.8074892.176 13-1.21243.6337-4.1555-6.49284.4703-9.16220.42401 0.8074892.176 15-1.21243.6337-4.1555-6.49284.4703-9.16220.42401 0.8074892.176 Batra-1.21243.6337-4.1555-6.49284.4703-9.16220.42400.8075            

20. 20 Figure 2. Through-thickness distributions of temperature, heat flux, transverse displacement and stresses of the Metal/Ceramic plate for I 1 = 13

21. 21 where V c, V c – volume fractions of the ceramic phase  3. Two-Phase FG Cylindrical Panel under Temperature Loading on bottom and top surfaces Analytical solution Dimensionless variables in crucial points Mori-Tanaka method is used for evaluating material properties of the Metal/Ceramic shell with

22. 22 Figure 3. Through-thickness distributions of temperature, heat flux, transverse displacement and stresses of the Metal/Ceramic cylindrical panel for I 1 = 13

23. 23 4. FG Piezoelectric Three-Layer Cylindrical Shell Distribution of Material Constants in Thickness Direction C ijkl, e ikl,  ik – material constants of PZT4,  – material gradient index, z =  3 / h (2) Analytical solution Lamination scheme

24. 24 Table 5. Results for a FG piezoelectric three-layer cylindrical shell with R/h = 5 and  = -1.5 InIn u 1 (-0.5)u 2 (-0.5)u 3 (-0.5)  (1/6)  11 (-0.5)  22 (-0.5)  12 (-0.5)  13 (1/6)  23 (1/6)  33 (1/6) D 3 (1/6) 5-19.21611.205-7.5907-12.9621.56482.24861.4933-5.1543 -4.8146 3.6287 3.3692 -6.8473 -7.0749 0.79383 0.78867 7-19.21611.205-7.5908-12.9621.55812.24181.4933-5.1015 -5.1017 3.5600 3.5404 -6.9305 -6.9487 0.79379 0.79384 9-19.21611.205-7.5908-12.9621.55852.24221.4933-5.1073 -5.1079 3.5674 3.5656 -6.9230 -6.9248 0.79379 0.79380 11-19.21611.205-7.5908-12.9621.55852.24231.4933-5.1078 -5.1080 3.5681 3.5676 -6.9223 -6.9227 0.79379 0.79380 13-19.21611.205-7.5908-12.9621.55852.24231.4933-5.1080 3.5682 3.5680 -6.9221 -6.9223 0.79379 Table 6. Results for a FG piezoelectric three-layer cylindrical shell with R/h = 5 and  = 1.5 InIn u 1 (-0.5)u 2 (-0.5)u 3 (-0.5)  (1/6)  11 (-0.5)  22 (-0.5)  12 (-0.5)  13 (1/6)  23 (1/6)  33 (1/6) D 3 (1/6) 5-6.87823.5063-2.3484-4.747910.53414.6239.6493-64.090 -63.982 73.114 72.895 52.928 52.712 0.62399 0.62307 7-6.87813.5062-2.3484-4.747910.57714.6669.6491-64.075 -64.084 73.094 73.117 52.903 52.926 0.62397 0.62392 9-6.87813.5062-2.3484-4.747910.57614.6659.6491-64.077 -64.079 73.097 73.101 52.906 52.909 0.62397 11-6.87813.5062-2.3484-4.747910.57614.6649.6491-64.077 -64.078 73.097 73.098 52.906 52.907 0.62397 13-6.87813.5062-2.3484-4.747910.57614.6649.6491-64.077 -64.078 73.097 73.098 52.906 0.62397               

25. 25 Figure 4. Distribution of shear stresses  13 = 10  6  13 (0, 0, z) and electric displacement D 3 = D 3 (L/2, 0, z) through the shell thickness: SaS formulation for I 1 = I 2 = I 3 = 9 and 3D solution of Wu and Tsai (  ) _ _

26. 26 ______ 5. FG Piezoelectric Rectangular Plate under Temperature Loading Analytical solution Distribution of Material Constants in Thickness Direction C ijkI, e ijk,  ij,  ij, r i,  – material constants of cadmium celenide

27. 27 I1I1  (-0.25) q 3 (-0.25)  (-0.25) D 3 (-0.25)u 1 (-0.25) 30.2701193906411858–133.71860866506474.790575240600712–9.749312104894471–2.723401339623277 70.2779956120573660–132.54436142526754.794299928460303–2.509769305877469–2.772079585945968 110.2779955945354090–132.54444581015874.794300922222834–2.509667562498987–2.772080267312558 150.2779955945353866–132.54444581006214.794300922230537–2.509667562748993–2.772080267314651 190.2779955945353849–132.54444581006084.794300922230507–2.509667562748973–2.772080267314632 Zhong 4.79 –2.51–2.77 I1I1 u 3 (-0.25)  11 (-0.25)  12 (-0.25)  13 (-0.25)  33 (-0.25) 3 2.469329978124763–4.466281473757366–2.847320399542269–1.598363296467524–142.5296650859701 7 2.427520269386693–3.391022140268101–3.230854772325724–1.068475784605092–1.166648209403661 11 2.427520284332587–3.391013719175138–3.230865616130692–1.068338926873448–1.171449910294010 15 2.427520284332663–3.391013719313675–3.230865616087885–1.068338926549097–1.171449913966712 19 2.427520284332651–3.391013719313694–3.230865616087861–1.068338926549103–1.171449913972714 Zhong2.43–3.39–3.23–1.07–1.17 Table 7. Results for FG square plate at point P(a/4, b/4) with a = b = 1 m, h = 0.1 m,  = 1 and  0 = 1 K        

28. 28 Figure 5. Distribution of temperature, electric potential, electric displacement, transverse displacement and stresses at point P(a/4, b/4) through the plate thickness for I 1 = 11

29. 29 Thanks for your attention! Conclusions  An implementation of the method of sampling surfaces for FG piezoelectric layered shells has been proposed. This method allows the use of 3D constitutive equations and can be applied efficiently to analytical solutions for FG piezoelectric plates and shells, which asymptotically approach the 3D exact solutions of electroelasticity and thermoelectroelasticity as well.  A refined layer-wise theory of FG piezoelectric shells has been developed in which strains exactly represent rigid-body motions of the shell in any convected curvilinear coordinate system.  An implementation of the method of sampling surfaces for FG piezoelectric layered shells has been proposed. This method allows the use of 3D constitutive equations and can be applied efficiently to analytical solutions for FG piezoelectric plates and shells, which asymptotically approach the 3D exact solutions of electroelasticity and thermoelectroelasticity as well.  A refined layer-wise theory of FG piezoelectric shells has been developed in which strains exactly represent rigid-body motions of the shell in any convected curvilinear coordinate system.