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n = 1, 2, …, N ; i n = 1, 2, …, I n ; m n = 2, 3, …, I n – 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. Kinematic Description of Undeformed Shell.

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n = 1, 2, …, N ; i n = 1, 2, …, I n ; m n = 2, 3, …, I n – 1

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  1. A New Concept of Sampling Surfacesand its Implementation for Layeredand Functionally GradedDoubly-Curved ShellsG.M. Kulikov, S.V. Plotnikova and S.A. MamontovSpeaker: Gennady M. KulikovDepartment of Applied Mathematics & Mechanics

  2. 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 3 3 (1) (2) (3) (n)i (n)i n n 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 Position Vectors and Base Vectors of SaS r(1, 2) – position vector of midsurface  ; ei(1, 2) – orthonormal base vectors; c= 1+k3 – components of shifter tensor at SaS; A(1, 2), k(1, 2)–Lamé coefficients and principal curvatures n n

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

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

  5. (10) (11) (12) 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) Derivatives of Displacement Vectors in Orthonormal Basis ei: Strain Parameters of SaSLinearized Strains of SaS

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

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

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

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

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

  11. Table 2. Results for a FG square plate with a = b = 1m , a/h = 3 and = 0.1      Exact results have been obtained by Kashtalyan (2004) 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 ()

  12. 2. Two-Phase FG Rectangular Plate under SinusoidalLoading Analytical solution 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 - + Dimensionless variables in crucial points

  13.          Table 3. Results for a Metal/Ceramic square plate witha/h = 5 Table 4. Results for a Metal/Ceramic square plate witha/h = 10

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

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

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

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

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

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

  20. Dimensionless variables in crucial points Lamination scheme: Table 5. Results for a three-layer rectangular plate with a/h = 4 using a regular 6464 mesh           3D Finite Element Solutions 1. Three-Layer Rectangular Plate under Sinusoidal Loading

  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 Spherical shell with R = 10,  = 89.99, E = 107 and  = 0.3 is modeled by 641 mesh Dimensionless variables Table 6 Results for a thick spherical shell with R / h = 2     

  23.          Figure 4. Distribution of stresses 11 and 33 through the shell thickness: SaS formulation for I1 = 7 and Lamé’s solution () Table 7. Results for thick and thin spherical shells   

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

  25. 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!

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