1. Identification of material properties using full field measurements on vibrating plates Mr Baoqiao GUO, Dr Alain GIRAUDEAU, Prof. Fabrice PIERRON LMPF research group École Nationale Supérieure d’Arts et Métiers (ENSAM) Châlons en Champagne - France

2. Introduction Theory Experimental Results Conclusion 2 / 19 Presentation of the procedure  Thin plates point clamped  Sine driving movement  Inertial excitation  Full field measurements  Data processing using Virtual Fields Method (V.F.M.) Introduction

3. Introduction Theory Experimental Results Conclusion 3 / 19 Virtual Fields Method Thin plates Sine vibrating response Measured Chosen Unknown Principle of Virtual Work u *,  *=  ( u *) Virtual fields Theory Constitutive law Thin plates  k

4. Introduction Theory Experimental Results Conclusion 4 / 19 Actual out of plane deflection:  (x,y,t) = d.cos  t + Re [ (w r (x,y) +j.w i (x,y)). exp(j  t) ] Virtual fields:  (x,y,t) = d.cos  t + Re [ (w r *(x,y) +j.w i *(x,y)). exp(j  t) ] Two fields to measure Theory + w(x,y,t) Plate deformation (bending) Driving mouvement = d.coswt Two fields to select

5. Introduction Theory Experimental Results Conclusion 5 / 19 V.W.E.F. = 0 Measured ( w(x,y), k(x,y)) Selected k*(x,y), w*(x,y) ( ,w(x,y),w*(x,y)) Two selected virtuals fields : VF1, VF2 Theory

6. Introduction Theory Experimental Results Conclusion 6 / 19 Measurements Deflectometry CCD O M P Q Deflection fields Curvature fields Slope fields In phase  /2 lag  dd . d  Experimental

7. Introduction Theory Experimental Results Conclusion 7 / 19 Image Processing Grid Images At restDeformed Phases xx y y - Slopes x - y Spatial phase shifting Experimental

8. Introduction Theory Experimental Results Conclusion 8 / 19 Experimental set up Experimental

9. Introduction Theory Experimental Results Conclusion 9 / 19 Slope fields Out of resonance 80 Hz Near resonance 100 Hz Experimental Plate : PMMA 200 x 160 x 3 mm 3

10. Introduction Theory Experimental Results Conclusion 10 / 19 Noise filtering: polynomial fitting Deflection field: integration Curvature fields: differentiation Experimental No data (hole) High gradients: uncertain measurements Remove data before fitting

11. Introduction Theory Experimental Results Conclusion 11 / 19 Experimental Use of piecewise virtual fields 3x3 5x5 Zero contribution from the clamping area ! Use of optimal special virtual fields Avril S., Grédiac M., Pierron F. Sensitivity of the virtual fields method to noisy data, Computational Mechanics, vol. 34, n° 6, pp. 439-452, 2004. Toussaint E., Grédiac M., Pierron F., The virtual fields method with piecewise virtual fields, International Journal of Mechanical Sciences, vol. 48, n° 3, pp. 256-264, 2006.

12. Introduction Theory Experimental Results Conclusion 12 / 19 Influence of the degree of the polynomial fitting (80 Hz) Results 0 4 2 8 6 8 10 12 14 16 18 Choice: degree 10 Polynomial degree

13. Introduction Theory Experimental Results Conclusion 13 / 19 Reference values Results Coupons: PMMA beams h = 4mm, l = 10mm, L= 107-114mm Clamped-free conditions Free vibrations, first bending mode ~80 Hz 3%2.3%C. Var. 1.08 10 -4 s 4.90 GPa Mean  E

14. Introduction Theory Experimental Results Conclusion 14 / 19 Reference values Results Assumption: constant = 0.3 (manufacturer datasheet) (???)

15. Introduction Theory Experimental Results Conclusion 15 / 19 Influence of number of virtual elements (80 Hz) Results reference 3 x 3 5 x 5 GPa CV # 0.4 % GPa CV # 0.6 % 10 -4 GPa.s CV # 4 % 10 -4 GPa.s CV # 23 %

16. Introduction Theory Experimental Results Conclusion 16 / 19 Influence of the frequency Results GPa 10 -4 GPa.s reference 80 Hz 100 Hz

17. Introduction Theory Experimental Results Conclusion 17 / 19 Influence of the frequency Results Out of resonance 80 Hz Near resonance 100 Hz Poor SNR: take pictures at other times

18. Introduction Theory Experimental Results Conclusion 18 / 19 Influence of the frequency Results reference 80 Hz 100 Hz constant ??? Material model ???

19. Introduction Theory Experimental Results Conclusion 19 / 19 Conclusion Novel procedure for damping measurements At or out-of resonance Based on full-field slope measurements Main assets –Direct method (no updating) –Insensitive to clamping dissipation –Poisson’s damping Future work –Explore wider range of frequencies –Apply to anisotropic plates (composites)