A free-vibration thermo-elastic analysis of laminated structures by variable ESL/LW plate finite element

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1 A free-vibration thermo-elastic analysis of laminated structures by variable ESL/LW plate finite element Authors: Prof. Erasmo Carrera Dr. Stefano Valvano Bologna, 4-7 July 2017

2 Research group at Politecnico di Torino MULtilayered structures MULtifield interaction FEM for mulilayered structures in multifield analysis

3 Unified Formulation 2D approximation of mechanical displacements and temperature using the thickness functions u k (x, y, z) = F 0 (z) u k (x, y) + 0 F 1(z) u k (x, y) F N(z) u k N (x, y) v k (x, y, z) = F 0 (z) v k (x, y) + 0 F 1(z) v k (x, y) F N(z) v k N (x, y) w k (x, y, z) = F 0 (z) w k (x, y) + 0 F 1(z) w k (x, y) F N(z) w k N (x, y) Θ k (x, y, z) = F 0 (z) Θ k (x, y) + 0 F 1(z) Θ k (x, y) F N(z) Θ k N (x, y) in compact form: u k (x, y, z) = F τ (z)u k τ (x, y) ; δu k (x, y, z) = F s (z)δu k s (x, y) ; τ, s = 0, 1,..., N Θ k (x, y, z) = F τ (z)θ k τ (x, y) ; δθ k (x, y, z) = F s (z)δθ k s (x, y) ; τ, s = 0, 1,..., N

Taylor Polynomials Legendre Polynomials u k = F 0 u k + 0 F 1u k +... + 1 F Nu k = N F τu k τ Θ k = F 0 Θ k + 0 F 1Θ k +... + 1 F NΘ k = N F τθ k τ τ = 0, 1,..., N F 0 = (z) 0 = 1 ; F 1 = (z) 1 = z ;.

4 Taylor Polynomials Legendre Polynomials u k = F 0 u k + 0 F 1u k F Nu k = N F τu k τ Θ k = F 0 Θ k + 0 F 1Θ k F NΘ k = N F τθ k τ τ = 0, 1,..., N F 0 = (z) 0 = 1 ; F 1 = (z) 1 = z ;... ; F N = (z) N u k = F t u k t + F b u k b + F ru k r = F τ u k τ Θ k = F t Θ k t + F b Θ k b + F rθ k r = F τ Θ k τ τ = t, b, r ; r = 2,..., N F t = P 0+P 1 2 ; F b = P 0 P 1 2 ; F r = P r P r 2 Equivalent Single Layer Approach

Layer Wise Legendre polynomials expansion: Variable-Kinematic Legendre polynomials expansion: u k = F t u k t + F b u k b + F ru k r = F τ u k τ Θ k = F t Θ k t + F b Θ k b + F rθ k r = F τ Θ k τ τ =

5 Layer Wise Legendre polynomials expansion: Variable-Kinematic Legendre polynomials expansion: u k = F t u k t + F b u k b + F ru k r = F τ u k τ Θ k = F t Θ k t + F b Θ k b + F rθ k r = F τ Θ k τ τ = t, b, r ; r = 2,..., N F t = P 0+P 1 2 ; F b = P 0 P 1 2 ; F r = P r P r 2 u = F t u t + F b u b + F r u r = F τ u τ Θ = F t Θ t + F b Θ b + F r Θ r = F τ Θ τ τ = t, b, r ; r = 2,..., N F t = P 0+P 1 2 ; F b = P 0 P 1 2 ; F r = P r P r 2 Interlaminar continuity condition: u k t = u k+1 b ; k = 1, n l 1 Interlaminar continuity condition is guaranteed in specified zones

6 Finite Element Method Approximation of variables in the reference midplane surface using the Langrangian shape functions: MITC To overcome the problem of the membrane and shear locking, the strain components are calculated using a specific interpolation strategy: u τ = N i (ξ, η) u τi For example: ɛ xx γ xz ɛ yy γ yz ɛ xy ɛ xx = N A1 ɛ xxa1 + N B1 ɛ xxb1 + N C1 ɛ xxc1 + N D1 ɛ xxd1 + N E1 ɛ xxe1 + N F1 ɛ xxf1

7 Governing Equations and Fundamental Nucleus Principle of Virtual Displacements (PVD) for mechanical problems δɛ k T σ k dv = δl e V Governing equations in compact form: δu kτi : K kτsij u ksj = P kτi K xx K xy K xz where K kτsij = K yx K yy K yz K zx K zy K zz kτsij

8 Assembling Approaches

9 Partially coupled thermo-mechanical problems Static Analysis Principle of Virtual Displacements ( mxπ ) ( nyπ ) Θ(x, y, z) = Θ(z)sin sin a b Θ(z) is assumed linear V δɛ k T σ k dv = δl e σ k = σ k u σ k Θ = Ck ɛ k λ k Θ k V λ k = C k α k δɛ k T σ k u dv = δɛ k T σ k Θ dv V Cinefra M., Valvano S., and Carrera E., Thermal stress analysis of laminated structures by a variable kinematic MITC9 shell element, Journal of Thermal Stresses, 39(2), ,

10 Partially coupled thermo-mechanical problems Static Analysis Principle of Virtual Displacements δɛ k T σ k u dv = δɛ k T σ k Θ dv V V ( mxπ ) ( nyπ ) Θ(x, y, z) = Θ(z)sin sin a b Θ(z) is calculated via the Fourier Heat conduction equations Linear Profile a/h=100 a/h=10 a/h=4 a/h=2 Temperature z Cinefra M., Valvano S., and Carrera E., Heat conduction and Thermal Stress Analysis of laminated composites by a variable kinematic MITC9 shell element, Curved and Layered Structures, 2, ,

11 Fully coupled thermo-mechanical problems Static Analysis Principle of Virtual Displacements V {δɛ k T σ k δθ k η k δϑ k T h k } dv = δl e In compact form: (4 4) δu k kτsij ksj kτi : τi K uu K uθ u P u = δθ k : K τi Θu K ΘΘ Θ P Θ σ k = C k ɛ k λ k Θ k η k = λ k ɛ k + χ k Θ k h k = κ k ϑ k ɛ mn = u mn χ = ρc v Θ 0 ϑ m = Θ m K xx K xy K xz K xθ K kτsij K = yx K yy K yz K yθ K zx K zy K zz K zθ K Θx K Θy K Θz K ΘΘ kτsij

T dv δ k σk δθk ηk δϑk h k dv = ρ δut u V V In compact form: ksj δukτi : K k τsij uksj = M k τsij u

12 Intro CUF FEM & MITC Gov. Eq. PVD Results Conclusions Fully coupled thermo-mechanical problems Free-Vibrations Analysis o R R n T T dv δ k σk δθk ηk δϑk h k dv = ρ δut u V V In compact form: ksj δukτi : K k τsij uksj = M k τsij u (4 4 ) K k τsij = " K uu K Θu K uθ KΘΘ (4 4 ) #k τsij M k τsij = ksj + K k τsij uksj = 0 M k τsij u " M uu 0 #k τsij 0 0 Harmonic Solution K k τsij ω2n M k τsij uksj = 0

13 Mixed ESL/LW Preliminaries Static Analysis Results

14 Composite Square Plate [0 /90 /0 ] Mechanical Analysis p(x, y, z top ) = ˆp z sin ˆp z = 1, 0 ( πx ) ( πy ) sin a b B.C.= Simply-Supported Material Properties: E L /E T = 25 G LT /E T = 0, 5 G TT /E T = 0, 2 ν LT = ν TT = 0, 25 Pagani A., Valvano S., and Carrera E., Analysis of laminated composites and sandwich structures by variable-kinematic MITC9 plate elements, Journal of Sandwich Structures and Materials,

15 ESL Approach, Taylor vs Legendre Polynomials a/h = 4 a/h = 100 DOFs ŵ ˆσxx ˆσxz ˆσyz ŵ ˆσxx ˆσxz ˆσyz top bottom top bottom 3D [Pagano 1970] LW4a [Petrolo et al. 2015] ET4a [Petrolo et al. 2015] LW ET ET ET ET ET EL EL EL EL thickness locking correction no correction

16 Thermal loads ( πx ) ( πy ) T(x, y, z) = ˆT(z) sin sin a b Carrera E., Valvano S., A variable kinematic shell formulation applied to thermal stress of laminated structures, Journal of Thermal Stresses, 40(7): , ˆT(z = top) = +1.0, ˆT(z = bottom) = 1.0, a = b = 1, h = 0.1 Mechanical properties: E 1 /E 2 = 25, E 2 = E 3 G 12 /E 2 = 0.5, G 23 /E 2 = 0.2 GPa, G 12 = G 13 ν 12 = ν 13 = ν 23 = 0.25 Thermal properties: α 2 /α 1 = 3, α 1 = α 3 K 1 /K 2 = 36.42/0.96, K 2 = K 3

17 Mixed ESL-LW Variable-Kinematics Transverse mechanical displacement w

18 Mixed ESL-LW Variable-Kinematics Transverse mechanical displacement w

19 Mixed ESL-LW Variable-Kinematics In-plane mechanical stress σ xx, (a/h = 2)

20 Mixed ESL-LW Variable-Kinematics In-plane mechanical stress σ xx, (a/h = 2)

21 Mixed ESL/LW Free-Vibration Analysis Results

22 Isotropic and Composite Simply-Supported Square Plates Aluminum Properties: E = 73 GPa, ν = 0.3 ρ = 2800 Kg m 3, c V = 897 J Kg K α = 25 E 6 1 K, κ = 130 W m K Composite Properties: E L /E T = /6.909 (GPa) G LT /G TT = 3.45/1.38 (GPa) ν LT = ν TT = 0, 25 ρ = 1940 Kg m 3, c V = 846 J Kg K α L /α T = 0.57 E 6/35.6 E 6 ( 1 K ) κ L /κ T = 36.42/0.96 ( W m K )

23 1 Layered Isotropic Plates a/h = 2 a/h = 100 EL4 M EL4 T EL1 M EL1 T EL4 M EL4 T EL1 M EL1 T Ref Analytical [1] Frequencies DOFs [1] Brischetto S., Carrera E., "Coupled thermo-mechanical analysis of one-layered and multilayered plates", Composite Structures (2010) 92,

24 2 Layered Composite Plates a/h = 2 a/h = 100 LW4M LW4T LW1M LW1T EL4M EL4T EL1M EL1T LW4M LW4T LW1M LW1T EL4M EL4T EL1M EL1T Ref Analytical [1] Frequencies DOFs [1] Brischetto S., Carrera E., "Coupled thermo-mechanical analysis of one-layered and multilayered plates", Composite Structures (2010) 92,

25 3 Layered Composite Plates

26 3 Layered Composite Plates a/h = 100 LW4 M LW4 T EL4 M EL4 T EL4 Case 1 M EL4 Case 1 T EL4 Case 2 M EL4 Case 2 T Frequencies DOFs

27 3 Layered Composite Plates a/h = 2 LW4 M LW4 T EL4 M EL4 T EL4 Case 1 M EL4 Case 1 T EL4 Case 2 M EL4 Case 2 T Frequencies DOFs

28 Intro CUF FEM & MITC Gov. Eq. PVD Results Conclusions Three-dimensional view of the Temperature θ

29 5 Layered Composite Sandwich Plates

30 5 Layered Composite Sandwich Plates a/h = 100 LW4 M LW4 T EL4 M EL4 T EL4 Case 1 M EL4 Case 1 T EL4 Case 2 M EL4 Case 2 T Frequencies DOFs DOFs %

31 5 Layered Composite Sandwich Plates a/h = 2 LW4 M LW4 T EL4 M EL4 T EL4 Case 1 M EL4 Case 1 T EL4 Case 2 M EL4 Case 2 T Frequencies DOFs DOFs %

32 Intro CUF FEM & MITC Gov. Eq. PVD Results Conclusions Three-dimensional view of the Temperature θ

33 Conclusions Unified Formulation is the ideal tool for the implementation of variable kinematic theories. In fact, the theory approximation order and the modelling technique (ESL, LW) are free parameters of the FEM arrays, which are written in a compact and very general form. The present variable kinematic models, in general for static and free-vibration analysis, allow to locally improve the solution with a reduction of computational costs with respect to Layer-Wise solutions. The Mixed ESL/LW variable kinematic is effective for the free-vibration analysis of sandwich structures. Strong computational cost reductions can be obtained with high solution accuracy, respect to the full Layer-Wise model. The results show confidence for future extension of the present variable-kinematic methodology to thermography investigations analysis, and to the free-vibration analysis of multilayered piezoelectric components.

34 Intro CUF FEM & MITC Gov. Eq. PVD Results Conclusions Thanks for the attention

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

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