WP2 / D4 : Damped Composites Modelling

1. WP2 / D4 : Damped Composites Modelling The main objective of this work package is to investigate, upgrade and propose new models for linear and non linear vibration of damped (viscoelastic) sandwich and multilayered composite structures. • Task 2.1: Modeling of multilayered composites with viscoelastic layers • Task 2.2: Non linear vibrations of damped sandwich and laminated composites • Task 2.3: Models validations

2. T2.1- Modeling of multilayered composites with viscoelastic layers Efficient analytical models and computational methods (finite element techniques) for damped composites with viscoelastic layer. • Review and assessments of various kinematic models for sandwich structures • Non Linear multiscale modelling of sandwich structures modeling. • Finite element frequency-dependent dynamic analysis of viscoelastic composite structures. • A shell finite element of three layered sandwich structures and laminated composites with viscoelastic layers.

3. ESL models Equivalent Single-Layer (ESL) models Review and assessment

4. ID-ZZT models IC-ZZT models Zigzag models Review and assessment

5. F • P1: Deflection • P2: Shear stress • P3: Normal stress • Global criterion • Core criterion It depends mainly on the global stiffness It depends mainly on the core’s stiffness • P4: First natural frequency • P5: First loss factor ratio Validations tests Review and assessment

6. Results Review and assessment Misfit: Maximum relative error is more than 10%. Good: Maximum relative error is less than 10%. Very good: Maximum relative error is less than 5%. Excellent: Maximum relative error is less than 1%.

7. Element 2D (Q8) K1 Element 1D (Hermite) 0 0 C1T K2 C2T C1 C2 0 Arlequin multiscale modelling multi-scale modelling Arlequin method (H. Ben Dhia 1998)

8. 2D-1D coupling in sandwich structures (3) 1D (zig-zag)-2D coupling 2D element Displacement 1D zigzag element multi-scale modelling Deformation Shear stress

9. Deformation (Incompatible mesh) Deformation (Compatible mesh) Shear stress 2D-2D coupling 2D coarse elements 2D refine elements multi-scale modelling

10. 2D coarse elements 2D refine elements 2D-2D coupling multi-scale modelling

11. 1D sandwich elements 2D elements 1D (sandwich)-2D coupling multi-scale modelling

12. 1D sandwich elements 2D elements Reason of the inaccuracy in 2D-1D non linear coupling Details multi-scale modelling thickness variation

13. Viscoelastic frequency-dependent damping model Damping material LD-400 FE frequency-dependent dynamic analysis

14. The used Sandwich finite element model FE frequency-dependent dynamic analysis

15. Finite element dynamic analysis • Free vibration analyses • Method of complex eigenvalues • Energy method • Frequency response analysis • Transient response analysis FE frequency-dependent dynamic analysis

16. Experimental dynamic analysis ISI-SYS laser vibrograph FE frequency-dependent dynamic analysis POLYTEC laser vibrometer

17. Experimental dynamic analysis FE frequency-dependent dynamic analysis

18. z he1 elastic hv viscoelastic x elastic he2 Shell finite element and Numerical Algorithm for vibrations of viscoelastic structures A triangular sandwich finite element 8 d.o.f / node Longitudinal displacements of faces, rotations and deflection A shell finite element of three layered sandwich structures Classical laminate, Kirchhoff assumptions in the faces, Reissner/Mindlin theory in the core, No slips occurs at the interfaces, All points of the elastic layers on a normal have the same rotations

19. ([K (w)] - w2 [M]) [U] = 0 U :complex eigenmode w2 :complex eigenvalue - Constant complex modulus - Low damping -QR method -Asymptotic approach (Ma et He 1992) -Iterative algorithm (Chen et al. 1999) Algorithms for complex eigenvalue problem Numerical Algorithm for vibrations of viscoelastic structures The new proposed Algorithm: Continuation algorithmis based on homototy technique and ANM method.

20. Validation (with a simple viscoelasticity model) Numerical Algorithm for vibrations of viscoelastic structures Free vibrations 4 first bending modes Damped vibrations, h=1 4 first bending modes • Abaqus simulation uses volume elements and MSEC. • Eve simulation using our shell element + ANM.

21. Task 2.2- Non linear vibrations of damped sandwich and laminated composites Development of analytical and numerical methods to investigate non-linear vibration analysis of damped sandwich structures, particularly for complex geometries (i.e. arch, ring…) and to the dynamic analyses (free vibration, frequency and time response analyses) of structures made from frequency and temperature dependent viscoelastic materials. • … Simplified approach for the non linear vibration of sandwich structures • … Non-linear vibrations of sandwich ring • … Analysis with Abaqus FE code for damped sandwich beams • …

22. Non linear geometrical effect Harmonic balance method Frequency amplitude equation Galerkin method Determined by solving simple problems Piezoelectric Elastic Elastic layer Viscoelastic Piezoelectric Elastic Simplified approach for the non linear vibration of sandwich structure Non linear vibration Non linear vibrations of damped sandwich - non linear geometrical effect - piezoelectric effect - non linear geometrical effect - viscoelastic effect

23. Assumptions and limits • Assumptions • The approximate deflection is harmonic in time • The approximate deflection is parralel to single mode in space • The approximate solution has a complex amplitude to be defined • We neglect the axial inertia term • Limits • Periodic responses • Transverse harmonic excitations and Free vibrations • The frequencies are near the resonance ones Non linear vibrations of damped sandwich

24. y u1 u2 R1 q R2 W/h x R3 Wnl/Wn W/h Wnl/Wn Non linear vibrations sandwich ring Non linear vibrations of damped sandwich Non linear response for various load amplitude Variation of the non linear response with the material loss factor

25. Non-linear terms Analysis with Abaqus FE code A new sandwich beam element has been developed to compute non-linear forced vibrations of viscoelastically damped sandwich beams Non linear vibrations of damped sandwich • A UEL subroutine have been written • The Riks method is used to compute unstable paths

26. Analysis with Abaqus FE code - Comparison with results of fully non-linear direct-integration dynamic analyses (2D model and use of the Hilbert-Hughes-Taylor integration scheme) Non linear vibrations of damped sandwich Theses results allow to validate both the beam kinematics assumptions and the harmonic balance approximations

27. z x Uxs,Uys,Uxa,Uya W, Rx, Ry, Rz 0.8 mm 0.254 mm 50 mm 1.2 mm 300 mm T2.3 Models Validations

28. Experimental validations Geometrical parameters BEAM1 BEAM2 h1=0.0012 mh1=0.0012 m h2=0.0001016 mh2=0.000254 m h3=0.0008 mh3=0.0008 m b=0.05 mb=0.05 m L=0.3 mL=0.3 m • Materials: • aluminium 2024 T6 • E=64 GPa, υ=0.32, ρ=2695·Ns2/m4 • 3M viscoelastic damping polymer • ISD-112, υ=0.49,ρ=1300·Ns2/m4 EADS sandwich beam

29. Experimental validations Modal frequencies, Damping ratios, FRF EADS sandwich beam

30. Experimental validations: Dynamic characteristics: EADS sandwich beam BEAM1, BC1 BEAM2, BC2

31. Dynamic characteristics: EADS sandwich beam BEAM2, BC1

32. Dissemination • 7 Papers submitted to International dedicated journals • 10 participation to national and International conferences • Details are in the annual reports Thanks …!

33. Conclusion ….