1. A New Concept of Sampling Surfaces and its Implementation for Layered and Functionally Graded Doubly-Curved Shells G.M. Kulikov, S.V. Plotnikova and S.A. Mamontov Speaker: Gennady M. Kulikov Department of Applied Mathematics & Mechanics

2. Kinematic Description of Undeformed Shell
n = 1, 2, …, N; in = 1, 2, …, In; mn = 2, 3, …, In – 1
N - number of layers; In - number of SaS of the nth layer
Kinematic Description of Undeformed Shell
(n)1,  (n)2, …, (n)I - sampling surfaces (SaS)
q(n)i - transverse coordinates of SaS
q[n-1], q[n] - transverse coordinates of interfaces 3
(1)
(2)
(3)
Position Vectors and Base Vectors of SaS
components of shifter tensor at SaS; A(1, 2), k(1, 2) – Lamé coefficients and principal curvatures
r(1, 2) – position vector of midsurface  ; ei(1, 2) – orthonormal base vectors; c = 1+k3 – n n
(n)i n
(n)i n

3. Kinematic Description of Deformed Shell
u (1, 2) – displacement vectors of SaS
(n)i n
(4)
(5)
(6)
Base Vectors of Deformed SaS
Position Vectors of Deformed SaS
u (1, 2, 3) – displacement vector
(1, 2) – values of derivative of displacement vector at SaS

4. Green-Lagrange Strain Tensor at SaS Linearized Strain Tensor at SaS Displacement Vectors of SaS in Orthonormal Basis ei :
(7)
(8)
(9)

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

6. Displacement Distribution in Thickness Direction
(13)
(14)
(15)
Displacement Distribution in Thickness Direction
Strain Distribution in Thickness Direction
L (3) – Lagrange polynomials of degree In - 1
(n)i n

7. Presentation for Derivative of Displacements at SaS
(16)
(17)
Derivatives of Lagrange Polynomials at SaS

8. Variational Equation for Analytical Development
 – total potential energy, W – work done by external electromechanical loads
(18)
(19)
(20)
(21)
Stress Resultants
Constitutive Equations
Cijkl – elastic constants of the nth layer
(n)

9. Distribution of Elastic Constants in Thickness Direction
(22)
(23)
(24)
Cijkl – values of elastic constants of the nth layer on SaS
(n)i n
Presentations for Stress Resultants
c= 1+k3 – components of shifter tensor

10. Navier-Type Solutions
1. FG Rectangular Plate under Sinusoidal Loading
Analytical solution
Table 1. Results for a square plate with a = b =1 m , a/h = 3 and  = 0
Dimensionless variables in crucial points
Exact results have been obtained by authors using Vlasov's closed-form solution (1957)  I1
-u1(0.5)
u3(0)
s11(0.5)
s13(0)
s33(0) 3 7 11 15 19
Exact

11. 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 I1 = 11 and Vlasov's closed-form solution (), and Kashtalyan's closed-form solution ()
Exact results have been obtained by Kashtalyan (2004)  I1
-u1(0.5)
u3(0)
s11(0.5)
s13(0)
s33(0) 3 7 11 15 19
Exact

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

13. Table 3. Results for a Metal/Ceramic square plate with a/h = 5 I1
u3(0.5)
11(0.5)
12(0.5)
13(0)
33(0.25) 3 7 11 15 19 23
Batra
2.5559
2.7562
2.3100
0.8100
Table 4. Results for a Metal/Ceramic square plate with a/h = 10  I1
u3(0.5)
11(0.5)
12(0.5)
13(0)
33(0.25) 3 7 11 15 19 23
Batra
2.2148
2.6424
2.3239
0.8123

14. Dimensionless variables in crucial points
3. Two-Phase FG Cylindrical Shell under Sinusoidal Loading
Analytical solution
Dimensionless variables in crucial points
Mori-Tanaka method is used for evaluating material properties of the Metal/Ceramic shell with
where Vc , Vc – volume fractions of the ceramic phase - +
on bottom and top surfaces

15. Figure 2. Through-thickness distributions of stresses of Metal/Ceramic cylindrical shell for I1 = 11

16. Exact Geometry Solid-Shell Element Formulation
Biunit square in (1, 2)-space mapped into
the exact geometry four-node shell element
in (x1, x2, x3)-space
Nr (1, 2) - bilinear shape functions
 = ( - c)/ - normalized surface coordinates
Displacement Interpolation
(25)
(26)
Assumed Strain Interpolation

17. Assumed Stress Resultant Interpolation
(27)
(28)
Assumed Stress Resultant Interpolation
Assumed Displacement-Independent Strain Interpolation

18. Hu-Washizu Mixed Variational Equation
W – work done by external loads applied to edge surface 
(29)
(30)

19. Finite Element Equations
(31)
(32)
– element displacement vector, Ur – nodal displacement vector (r = 1, 2, 3, 4)
K – element stiffness matrix of order 12NSaS12NSaS, where

20. 3D Finite Element Solutions
1. Three-Layer Rectangular Plate under Sinusoidal Loading
Table 5. Results for a three-layer rectangular plate with a/h = 4 using a regular 6464 mesh
Dimensionless variables in crucial points
Lamination scheme: 
Formulation
u3(0)
s11(-0.5)
s11(0.5)
s22(-1/6)
s22(1/6)
10s12(-0.5)
10s12(0.5)
s13(0)
10s23(0)
s33(0.5)
In = 3
In = 4
In = 5
In = 6
In = 7
Exact SaS
2.8211
1.1443
1.0878
2.8065
1.0000
Pagano
2.82
-1.10
1.14
-1.19
1.09
2.81
-2.69
-3.51
-3.34
1.00

21. Figure 3. Through-thickness distributions of displacements and stresses for three-layer rectangular plate for In = 7 using a regular 6464 mesh; exact SaS solution () and Pagano's closed form solution ()

22. 2. Spherical Shell under Inner Pressure p3 = -p0
Table 6 Results for a thick spherical shell with R / h = 2
Spherical shell with R = 10,  = 89.99, E = 107 and  = 0.3 is modeled by 641 mesh
2. Spherical Shell under Inner Pressure p3 = -p0
Dimensionless variables I1
u3(0)
11(–0.5)
11(0.5)
33(–0.5)
33(0) 3
2.281
5.345
2.438
–0.5882
–0.3759 5
2.300
4.616
2.087
–0.9770
–0.2576 7
4.572
2.066
–0.9978
–0.2626
Lamé’s solution
4.566
–1.0000

23. Table 7. Results for thick and thin spherical shells
R / h
SaS formulation for I1 = 7
Exact Lamé’s solution
u3(0)
11(–0.5)
11(0.5)
33(0) 4
2.945
4.583
3.332
–0.3766
4.582 10
3.291
4.784
4.284
–0.4501
4.783
4.282
100
3.480
4.976
4.926
–0.4950
4.975
4.925    
Figure 4. Distribution of stresses 11 and 33 through the shell thickness:
SaS formulation for I1 = 7 and Lamé’s solution () 

24. 3. Hyperbolic Composite Shell under Inner Pressure p3 =–p0cos4q2 –
Figure 5. Distribution of stresses through the shell thickness for I1 = 7
Single-layer graphite/epoxy hyperbolic shell with

25. Thanks for your attention!
Conclusions
An efficient concept of sampling surfaces inside the shell body has been proposed. This concept permits the use of 3D constitutive equations and leads for the large number of sampling surfaces to numerical solutions for layered composite shells which asymptotically approach the 3D solutions of elasticity
A robust exact geometry four-node solid-shell element has been developed which allows the solution of 3D elasticity problems for thick and thin shells of arbitrary geometry using only displacement degrees of freedom
Thanks for your attention!