FINITE ELEMENT MODELS WITH NODE-DEPENDENT KINEMATICS ADOPTING LEGENDRE POLYNOMIAL EXPANSIONS FOR THE ANALYSIS OF LAMINATED PLATES

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1 FINITE ELEMENT MODELS WITH NODE-DEPENDENT KINEMATICS ADOPTING LEGENDRE POLYNOMIAL EXPANSIONS FOR THE ANALYSIS OF LAMINATED PLATES G. Li, A.G. de Miguel, E. Zappino, A. Pagani and E. Carrera Department of Mechanical and Aerospace Engineering Politecnico di Torino Full 21 st International Conference on Composite Materials ICCM21 25 August 2017, Xi an

2 Introduction CUF 2D NDK HLE Case 1 Case 2 Case 3 Conclusion Acknowledgment MUL2 - MULtilayered structures & MULtifield effects

3 Overview 1 Carrera Unified Formulation (CUF) for refined 2D models; 2 Node-dependent kinematic 2D FEM models; 3 Hierarchical Legendre Expansions as shape functions for 2D elements; 4 Numerical cases; 5 Conclusions.

4 { { An Example: A Higher-order Deformation Theory for Plate Written in CUF Displacement description u = u 0 (x, y) +z u 1 (x, y) + +z N u N (x, y) v = v 0 (x, y) +z v 1 (x, y) + +z N v N (x, y) w = w 0 (x, y) +z w 1 (x, y) + +z N w N (x, y) u T = { u v w } = { F τu τ F τv τ F τw τ } = Fτu T τ F τ(z) = z τ 1 τ = 1, 2,, N + 1 FEM discretization u(x, y, z) = N i (x, y)u i (z) = N i (x, y)f τ (z)u iτ K ijτs = N j F s b T CbF τ N i dv V K ijτs : Fundamental Nucleus 3 x 3 Nucleus K xx K xy K xz K yx K yy K zx K zy K yz K zz 3 X M X N =1 PVD u(x, y, z) = N i (x, y)f τ(z) U iτ δu(x, y, z) = N j (x, y)f s(z) δu js δl int = δε T σdv = δu T b T CbudV V V = δu T js F sn j b T CbN i F τdv U iτ = δu T js K ijτs U iτ V δl ext = δu T pdv = δu T js N j F spdv = δu T js P js V V } } } } } } s=1 s=2,n-1 s=n =2, M-1 =M

CUF for Two Major Frameworks of Refined 2D Models Equivalent Single-Layer models (ESL) Layer-Wise models (LW) u = F 0 u 0 + F 1 u 1 + + F N u N Note: F τ defined on the whole through-thickness

u k = F t u k t + F b u k b + F ru k r Note: F τ defined on the thickness domain of layer k; Continuity constraints at layer interfaces: u k t = u k+1 b ; Can adopt interpolation polynomials;

5 CUF for Two Major Frameworks of Refined 2D Models Equivalent Single-Layer models (ESL) Layer-Wise models (LW) u = F 0 u 0 + F 1 u F N u N Note: F τ defined on the whole through-thickness domain; Applicable to series expansion; Taylor, Fourier, Exponential, Hyperbolic, et al.; Taylor Expansions (TE) are most commonly used. u k = F t u k t + F b u k b + F ru k r Note: F τ defined on the thickness domain of layer k; Continuity constraints at layer interfaces: u k t = u k+1 b ; Can adopt interpolation polynomials; Lagrange, Legendre, Chebyshev, et al.; Lagrange Expansions (LE) are the most widely used. Carrera, E., Cinefra, M., Li, G. and Kulikov, G.M. MITC9 shell finite elements with miscellaneous through-the-thickness functions for the analysis of laminated structures. Composite Structures, 154(2016), pp

6 3D Constitutive Equations Stress-strain relations for an orthotropic material: σ k p = C k pp ɛ k p + C k pn ɛ k n σ k n = C k np ɛ k p + C k nn ɛ k n where C k C k 11 C k 12 C k C k pp = C k 12 C k 22 C k 26 C k 13 C k 16 C k 26 C k pn = 0 0 C k C k C k C k np = C k 55 C k 45 0 C k 13 C k 23 C k nn = C k 45 C k C k 33

Plate Elements with Node-Dependent Kinematics (NDK) Introduction of the dependency of kinematics on FEM nodes u = N i F τ u iτ u = N i F i τu iτ Variable LW/ESL nodal capabilities; Modeling of

7 Plate Elements with Node-Dependent Kinematics (NDK) Introduction of the dependency of kinematics on FEM nodes u = N i F τ u iτ u = N i F i τu iτ Variable LW/ESL nodal capabilities; Modeling of patches; Global-local analysis. Zappino, E., Li, G., Pagani, A. and Carrera, E. Global-local analysis of laminated plates by node-dependent kinematic finite elements with variable ESL/LW capabilities. Composite Structures, 172(2017), pp.1 14.

8 Plate Elements with Node-Dependent Kinematics (NDK) Assembly technique Node d Node c Node a Node b Assembly technique of stiffness matrix for 2D elements adopting NDK.

Hierarchical Legendre Expansions as shape

1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 9 10 11 12

33 34 Nodal modes Edge modes Surface modes

9 Hierarchical Legendre Expansions as shape functions of 2D elements Polynomial degree Nodal modes Edge modes Surface modes HLE as shape functions for 2D p version elements. Szabó, B. and Babuška, I. Finite Element Analysis. John Wiley & Sons, Szabó, B., Düster, A. and Rank, E. The p-version of the finite element method. Encyclopedia of computational mechanics (2004). Pagani, A., De Miguel, A.G., Petrolo, M. and Carrera, E. Analysis of laminated beams via Unified Formulation and Legendre polynomial expansions. Composite Structures, 156(2016), pp

10 Hierarchical Legendre Expansions as shape functions of 2D elements Nodal modes: N 4 E 3 N 3 N i (ξ, η) = 1 4 (1 ξ iξ)(1 η i η) i = 1, 2, 3, 4 Edge modes: N i (ξ, η) = 1 (1 η)φp(ξ) i = 5, 9, 13, 18, 2 E 4 s E 2 N i (ξ, η) = 1 (1 + ξ)φp(η) i = 6, 10, 14, 19, 2 N i (ξ, η) = 1 (1 + η)φp(ξ) i = 7, 11, 15, 20, 2 N i (ξ, η) = 1 (1 ξ)φp(η) i = 8, 14, 16, 21, 2 Surface modes: *Basis functions: N i (ξ, η) = φ p(ξ)φ q(η) p, q 4 2p 1 ξ φ p(ξ) = L 2 p 1 (x)dx = Lp(ξ) L p 2(ξ) p = 2, 3, 1 4p 2 N 1 N 2 E 1 Nodal modes Edge modes Surface modes Szabó, B., Düster, A. and Rank, E. The p-version of the finite element method. Encyclopedia of computational mechanics (2004).

11 Three-layered cross-ply plate, [0 /90 /0 ], a/h = 2 and 100, loaded on top surface B Y Z X C D Pagano, N.J. 2D FEM 1/4 model with symmetry for the composite plate. Exact solutions for rectangular bidirectional composites and sandwich plates. Journal of Composite Materials, Vol 4, pp (January 1970). Composites 1.4(1970):

12 Three-layered cross-ply plate, [0 /90 /0 ], a/h = 2 Thick plate ( a h = 2), with LE4 as thickness functions. w σ yy 10 σ yz σ zz DOFs Element Mesh ( a 2, b 2, 0) ( a 2, b 2, h 6 ) ( a 2, 0, 0) ( a 2, b 2, h 2 ) Q Q Q Q HLE HLE HLE HLE HLE HLE Pagano Kulikov and Plotnikova Carrera et al Relative error of yz (%) HLE2 Q9 1X1 Q9 2X2 Q9 Mesh-nxn HLEp-Mesh 1x1 Carrera, E., Cinefra, M., Li, G. and Kulikov, G.M. MITC9 shell finite elements with miscellaneous through-the-thickness functions for the analysis of laminated structures. Composite Structures, 154(2016), pp Kulikov, GM and Plotnikova, SV Exact 3D stress analysis of laminated composite plates by sampling surfaces method. Composite Structures, 94(12), pp HLE3 Q9 4X4 HLE4 HLE7 Q9 5X DOFs DOFs vs Accuracy

13 Three-layered cross-ply plate, [0 /90 /0 ], a/h = 100 Thin plate ( h a = 100), with LE4 as thickness kinematics. w σ yy 10 σ yz σ zz DOFs Element Mesh ( a 2, b 2, 0) ( a 2, b 2, h 6 ) ( a 2, 0, 0) ( a 2, b 2, h 2 ) Q Q Q MITC MITC MITC HLE HLE HLE HLE HLE HLE HLE HLE HLE HLE HLE HLE Pagano Kulikov and Plotnikova Carrera et al Relative error of yz (%) HLE6 HLE8 HLE5 Q9 5X5 Q9 Mesh-nxn MITC9 Mesh-mxm HLEp Mesh-1x1 HLEp Mesh-2x2 MITC9 5X5 HLE DOFs DOFs vs Accuracy Q9 10X10 MITC9 8X8 Q9 15X15 MITC9 10X10

Three-layered cross-ply plate, [0 /90 /0 ], σ xz and σ yz 7 6 5 a/h=2 HLE7, Mesh 1x1 Sigma_XZ 10 yz 4 3 2 1 a/h=2: Q9 Mesh-5x5 HLE7 Mesh-1x1 a/h=100: MITC9 Mesh-10x10 HLE8 Mesh-1x1 0-0.5-0.

14 Three-layered cross-ply plate, [0 /90 /0 ], σ xz and σ yz a/h=2 HLE7, Mesh 1x1 Sigma_XZ 10 yz a/h=2: Q9 Mesh-5x5 HLE7 Mesh-1x1 a/h=100: MITC9 Mesh-10x10 HLE8 Mesh-1x z Variation of 10 σ yz along z e Sigma_XZ e e+02 a/h=100 HLE8, Mesh 1x1 Contour plot of σ xz 4.393e+04

15 Cross-ply plate subjected to point load on the top surface Z P z Feature positions: X Y ESL-FSDT Nodal modes Edge modes Surface modes LW-LE4 Geometry and loading. Assignment of nodal kinematics w DOFs: HLE3-LE HLE4-LE HLE5-LE HLE6-LE HLE7-LE HLE8-LE HLE8-LE4/FSDT 5751 HLE8-LE4 HLE8-LE4/FSDT z Through-the-thickness variation of w HLE7-LE4 HLE6-LE4 HLE5-LE4 HLE4-LE4 HLE3-LE4 w at ( a 2, b 2, 0) Element Kinematics w HLE8 LE HLE8 LE4/FSDT ABAQUS 3D Carrera (closed-form) Carrera, E. and Ciuffreda, A. Bending of composites and sandwich plates subjected to localized lateral loadings: a comparison of various theories. Composite Structures, 68(2), pp

16 Three-layered cross-ply composite plate under local pressure, a/h = 10 Z X Y Geometry and loading, a/h = 10, (0 /90 /0 ) Biscani, F., Giunta, G., Belouettar, S., Carrera, E. and Hu, H. Variable kinematic plate elements coupled via Arlequin method. International Journal for Numerical Methods in Engineering, 91(12), pp Zappino, E., Li, G., Pagani, A. and Carrera, E. Global-local analysis of laminated plates by node-dependent kinematic finite elements with variable ESL/LW capabilities. Composite Structures, 172(2017), pp Assignement of nodal kinematics.

17 Three-layered cross-ply composite plate under local pressure, a h = 10 Displacement and stress evaluation on the three-layered plate under local pressure Element Mesh Kinematics w[10 5 m] σxx[mpa] σyy[mpa] -10σxz[MPa] -10σyz[MPa] σzz[mpa] DOFs ( a 2, b 2, h 2 ) ( a 2, b 2, h 2 ) ( a 2, b 2, h 2 ) ( 5a 12, b 2, 0) ( a 2, 5b 12, 0) ( a 2, b 2, h 2 ) HLE3 5x5 LE HLE4 5x5 LE HLE5 5x5 LE HLE6 5x5 LE HLE6 5x5 FSDT HLE6 5x5 LE4/FSDT Zappino et al Biscani et al xz 3-10 yz HLE6-LE4 HLE6-LE4/FSDT 2 1 HLE6-LE4 HLE6-LE4/FSDT z Variation of 10σ xz through z at B( 5a 12, b 2 ) z Variation of 10σ yz through z at C( a 2, 5b 12 )

18 Conclusions With node-dependent kinematic plate elements adopting HLE as shape functions: 1 FEM models with variable LW/ESL nodal capabilities can be conveniently constructed; 2 Local kinematic refinement can be carried out without modifying the mesh neither using any ad hoc coupling method; 3 With sufficiently high polynomial degree p, 2D elements adopting HLE can achieve comparable accuracy with fewer computational resources compared with pure h-version refinement; 4 Node-dependent kinematics can help to further reduce the computational consumption when the structure undergoes strong local effects; 5 Such an approach is ideal for global-local analysis.

Acknowledgment FULLComp: FULLy integrated analysis, design, manufacturing and health-monitoring of COMPosite structures Funded by the European Commission under a Marie Sklodowska-Curie Innovative

19 Acknowledgment FULLComp: FULLy integrated analysis, design, manufacturing and health-monitoring of COMPosite structures Funded by the European Commission under a Marie Sklodowska-Curie Innovative Training Networks grant for European Training Networks (grant agreement No ). Full The full spectrum of the design of composite structures will be dealt with, such as manufacturing, health-monitoring, failure, modeling, multiscale approaches, testing, prognosis, and prognostic. Research activities are aimed at engineering fields such as aeronautics, automotive, mechanical, wind energy and space.

20 FINITE ELEMENT MODELS WITH NODE-DEPENDENT KINEMATICS ADOPTING LEGENDRE POLYNOMIAL EXPANSIONS FOR THE ANALYSIS OF LAMINATED PLATES G. Li, A.G. de Miguel, E. Zappino, A. Pagani and E. Carrera Department of Mechanical and Aerospace Engineering Politecnico di Torino Full 21 st International Conference on Composite Materials ICCM21 25 August 2017, Xi an

FINITE ELEMENT MODELS WITH NODE-DEPENDENT KINEMATICS ADOPTING LEGENDRE POLYNOMIAL EXPANSIONS FOR THE ANALYSIS OF LAMINATED PLATES

2 st International Conference on Composite Materials Xi an, 20-25 th August 207 FINITE ELEMENT MODELS WITH NODE-DEPENDENT KINEMATICS ADOPTING LEGENDRE POLYNOMIAL EXPANSIONS FOR THE ANALYSIS OF LAMINATED