PROGRESSIVE DAMAGE ANALYSIS OF COMPOSITE STRUCTURES VIA ONE-DIMENSIONAL CARRERA UNIFIED FORMULATIONS

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*Department of Mechanical and Aerospace Engineering, Politecnico di Torino Full

1 PROGRESSIVE DAMAGE ANALYSIS OF COMPOSITE STRUCTURES VIA ONE-DIMENSIONAL CARRERA UNIFIED FORMULATIONS Erasmo Carrera*, Ibrahim Kaleel* and Marco Petrolo* *Department of Mechanical and Aerospace Engineering, Politecnico di Torino Full International Conference on Composite Structures, ICCS September 2016, Porto (Portugal)

2 Introduction CUF Taylor Models Component-Wise Approach Progressive Damage Model Numerical Results Conclusion MUL2 - Our Research Group Marie Curie Project on Composites Full

3 The FULLCOMP project FULLy integrated analysis, design, manufacturing and health-monitoring of COMPosite structures The FULLCOMP project is funded by the European Commission under a Marie Sklodowska-Curie Innovative Training Networks grant for European Training Networks (ETN). The FULLCOMP partners are: 1 Politecnico di Torino (Italy) - Coordinator 2 University of Bristol (UK) 3 Ecole Nationale Superieure d arts et Metiers (Bordeaux, France) 4 Leibniz Universitaet Hannover (Germany) 5 University of Porto (Portugal) 6 University of Washington (USA) 7 RMIT (Australia) 8 Luxembourg Institute of Technology 9 Elan-Ausy, Hamburg, (Germany) FULLCOMP has recruited 12 PhD students who will work in an international framework to develop integrated analysis tools to improve the design of composite structures. 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. The FULLCOMP research activity is aimed at many engineering fields, e.g. aeronautics, automotive, mechanical, wind energy, and space.

4 Overview 1 Description of the Carrera Unified Formulation for refined models (CUF). 2 Main 1D CUF capabilities overview (1D Taylor- and Lagrange-based models). 3 Numerical examples dealing with different applications (aerospace structures, composites, Bio Structures). 4 Introduction to the Component-Wise approach (CW). 5 Progressive Damage Model 6 Numerical Results

5 Brief Overview of Beam Refinement Methods and Contributors 1 Shear correction factors (Timoshenko, Sokolnikoff, Cowper, Gruttmann, etc.). 2 Warping functions and Saint-Venant solutions (El Fatmi, Ladéveze, etc.). 3 Variational asymptotic method (Berdichevsky, Hodges, Yu, etc.). 4 Generalized beam theory (Schardt, Camotim, Silvestre, etc.). 5 Higher-order models (Washizu, Reddy, Kapania, Carrera, etc.).

6 Introduction CUF Taylor Models Component-Wise Approach Progressive Damage Model Numerical Results Conclusion 1D Advanced Structural Models Actual Wing Our Model 1D Carrera Unified Formulation, CUF - FEM Version Ni(y) Classical 1D FE + Fτ(x,z) Cross-Section Functions

7 { Fundamental Nucleus Equations by CUF Finite Element Formulation (FEM) i, j: Shape function indexes u = F τ (x, z)n i (y) u τi (1D) (depend on the FE δl int = δq T τi Kijτs q discretization). sj δl ine = δq T Mijτs q τ, s: Expansion function τi sj indexes (depend on the model δl ext = Pδu T order). 3 x M x M τs-block Assembly Technique 3 x 3 Nucleus{ K xx K xy K xz K yx K yy K yz τ = 1 K zx K zy K zz Fundamental Nucleus } } s = 1 s = 2, M-1 s = M } } } τ } τ = 2, M-1 = M K ijτs xx = C 22 Ω F τ,x F s,x dω N i N j dy +... l

Introduction CUF Taylor Models Component-Wise Approach Progressive Damage Model Numerical Results Conclusion 1D CUF Applications Overview - Books Thin-Walled and Reinforced

Diagrams Aeroelasticity Load Factors and Non-Structural Masses Rotors and Rotating Blades Biomechanics Multifield Analysis Nanostructures Analysis of Aerospace Structures via

8 Introduction CUF Taylor Models Component-Wise Approach Progressive Damage Model Numerical Results Conclusion 1D CUF Applications Overview - Books Thin-Walled and Reinforced Structures Buckling, Free Vibration and Dynamic Response Analysis Composite Structures FGM Structures Variable Kinematics Models Axiomatic/Asymptotic Analyses and Best Theory Diagrams Aeroelasticity Load Factors and Non-Structural Masses Rotors and Rotating Blades Biomechanics Multifield Analysis Nanostructures Analysis of Aerospace Structures via the Component-Wise Approach Analysis of Civil Structures via the Component-Wise Approach Component-Wise Approach for the Multiscale Analyses of Composites FEM 2D and Smart Structures 1D

9 { The Taylor CUF 1D Models, TE u x = u x1 +x u x2 + z u x3 +x 2 u x4 + xz u x5 + z 2 u x u y = u y1 +x u y2 + z u y3 +x 2 u y4 + xz u y5 + z 2 u y u z = u z1 }{{} + x u z2 + z u z3 } {{ } + x 2 u z4 + xz u z5 + z 2 u z6 } {{ } +... N = 0 N = 1 N = 2 τ = 1 τ = 2, τ = 3 τ = 4, τ = 5, τ = 6 3 DOFs 9 DOFS 18 DOFs Classical models, such as Timoshenko, can be obtained as particular cases of the linear models. Assembly Technique 3 x M x M τs-block 3 x 3 Nucleus{ K xx K xy K xz K yx K yy K yz τ = 1 K zx K zy K zz τ = 2, M-1 } } } τ = M Fundamental Nucleus K ijτs xx = C 22 Ω F τ,x F s,x dω N i N j dy +... l } } s = 1 s = 2, M-1 s = M }

10 Wave propagation in a Thin-Walled Cylinder z x t E. Carrera, A. Varello, Dynamic Response of Thin-Walled Structures by Variable Kinematic One-Dimensional Models Journal of Sound and Vibration, 331(24), pp , N = 8 beam model, 6000 DOFs

11 Plastic beam under bending load Analytical P 6 CUF ABAQUS L w h Force [N] Material model : Isotropic von-mises plasticity Ideal elastic-perfect plastic stress-strain relationship Plastic strength factor is estimated Emphasizes the importance of refined higher order models Displacement [m] Type NDOF Uz at yield load Uz at limit load Plastic Strength Factor Value Error Value Error [%] Value Error [%] [m] [%] [m] [%] [%] Analytical ABQ - 3D 22, Taylor EBBM TBM N = N = N =

12 Lagrange-Based 1D Models, LE u x = L τ u xτ u y = L τ u yτ u z = L τ u zτ L9 polynomials - Isoparametric L τ = 1 4 (r 2 + r r τ )(s Cross-Section Elements 3D Geometry from CAD LE Modeling Beam element Beam node Lagrange node above the first beam node cross-section Lagrange node above the second beam node cross-section DOFs: pure displacements of each Lagrange node (3 DOFs per Lagrange node) Lagrange nodes can be placed above the physical surface of the structure Computational Model Cross-Section nodes Disconnected nodes

13 The Component-Wise Approach Only displacements as unknowns. Each component of the structure is modeled via beams only. No need of reference surfaces. This might be useful in a CAD-FEM interface scenario. No need of homogenization techniques.

Aircraft Wing Shell-like modal shapes by means of a BEAM MODEL Mode 54-246.7 Hz Z X c E. Carrera, A. Pagani and M.

14 Aircraft Wing Shell-like modal shapes by means of a BEAM MODEL Mode Hz Z X c E. Carrera, A. Pagani and M. Petrolo, Component-wise Method Applied to Vibration of Wing Structures, Journal of Applied Mechanics, 80(4), doi: / Natural frequencies [Hz] Mode 1D-LE SOLID Computational costs CUF LE 20,000 DOFs SOLID 190,000 DOFs

15 λ N/mm λ N/mm Thin-walled lipped channel beam F =101λ N F Material model : Isotropic von-mises plasticity Ideal elastic-perfect plastic stress-strain relationship Type NDOF Displacement λ = λ = GBT 3, ABAQUS - Shell 69, ABAQUS - 3D Brick 1,342, L9 24 L9 44 L9 Lagrange node CUF - CW - (30-B4 elements) 14L9 23, L9 40, L9 72,

16 1 Response based on Continuum Damage Mechanics and Hashin s Initiation criterion 2 Three damage variables are introduced : d f (fiber), d m (matrix) and d s (shear) - Tension and Compression handled separately d I = ˆɛ I,f (ˆɛ I ˆɛ I,0 ) ˆɛ I (ˆɛ I,f ˆɛ I,0 ) I [ft, fc, mc, mt] 3 Linear damage evolution law σ = C(d I ) : ɛ 4 Mesh objectivity obtained by introducing characteristic length scale ˆɛ I,f = 2 G I,c L c X I I [ft, fc, mc, mt] 5 Viscous regularization scheme introduced to alleviate convergence issues ḋ v I = 1 η I ( di d v I ) I [ft, fc, mc, mt]

17 Failure Envelope Tensile Fiber Mode Compressive Fiber Mode Tensile Matrix Mode (σ 11 + ˆσ 33 > 0) ˆσ 22 = X T ˆσ 22 = X C 1 Y 2 T 2 (ˆσ 11 + ) 1 ˆσ (ˆσ S 2 13 ˆσ ) 1 11 ˆσ (ˆσ S ) ˆσ2 = 1 T L Compressive Matrix Mode (ˆσ 11 + ˆσ 33 < 0) 1 Y C ( YC 2S T ) 2 1 (ˆσ 11 + ˆσ 33 )+ 1 (ˆσ 4S ˆσ 33 ) S 2 T T 2 (ˆσ 13 + ˆσ ) ˆσ 33 (ˆσ S ) ˆσ2 = 1 L Nomenclature : 2 - Fiber direction, 1 & 3 - Transverse direction ˆσ - effective stress

18 Failure Strength Estimation w 1 Composite Laminate subjected to uni-axial tension CUF 1D BEAM 2 Laminate : HTA/6376-C 3 Each ply is modeled as a component (5 L9 elements) L t t Each ply is a component 90 0 ply 0 0 ply Geometry of the beam Material properties of HTA /6376-C E 11 (GPa) E 22 = E 33 (GPa) G 12 = G 13 (GPa) G 23 (GPa) ν 12 = ν 13 ν Orthotropic damage initiation properties for HTA /6376-C X T (MPa) X C (MPa) Y T (MPa) Y C (MPa) S L (MPa) S T (MPa)

19 Stacking Sequence : [90/0] 2s Various stages of damage propagation at fixed end ABAQUS 2D Shell CUF - CW 3 Force [N] Fully virgin composite Displacement [mm] 90 0 lamina damaged Experimental DDM 1 CUF-CW ABAQUS-2D shell Failure Strength DOF - 900* 17,391 40,000 Fully damaged composite 1. Moure et al., Damage evolution in open-hole laminated composite plates subjected to in-plane loads, 2015

20 Stacking Sequence : [90/0] 4s ABAQUS 2D shell CUF-CW 7 6 Force [N] 5 4 σ yy Displacement [m] Experimental DDM 1 CUF-CW ABAQUS-2D shell Failure Strength (MPa) DOF - 900* 27,621 40,000 Tensile Matrix Damage (d mt ) 1. Moure et al., Damage evolution in open-hole laminated composite plates subjected to in-plane loads, 2015

21 Main Conclusions and Perspectives Accurate 3D stress fields obtained through CUF is successfully employed for damage propagation analysis Framework is able to predict the failure strength of composites (un-notched) within acceptable limits Computationally efficient higher-order non-linear finite element framework is developed Future extensions Failure strength estimation of notched composites Component wise (fiber and matrix) damage propagation Extension of damage analysis for explicit scheme Inclusion of other damage initiation criteria

22 PROGRESSIVE DAMAGE ANALYSIS OF COMPOSITE STRUCTURES VIA ONE-DIMENSIONAL CARRERA UNIFIED FORMULATIONS Erasmo Carrera*, Ibrahim Kaleel* and Marco Petrolo* *Department of Mechanical and Aerospace Engineering, Politecnico di Torino Full International Conference on Composite Structures, ICCS September 2016, Porto (Portugal)

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