Elastodynamic response of a layered anisotropic plate : comparative approaches. Towards the development of hybrid methods
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1 Journées Ondes du Sud-Ouest 10 mars 2016 Elastodynamic response of a layered anisotropic plate : comparative approaches. Towards the development of hybrid methods PhD thesis of Pierric MORA, successfully defended on 17 th December 2015 Direction : Marc DESCHAMPS & Éric DUCASSE Financial support: CEA & REGION AQUITAINE Institut de Mécanique et d'ingénierie (I 2 M) - CNRS UMR 5295 University of Bordeau, FRANCE Response of layered anisotropic plates 1 / 30 Physical Acoustics Dpt.
2 Physical Acoustics Department (Apy) 1. Ultrasons Matériau (UM) Michel CASTAINGS, Christophe BACON, Stéphane BASTE, Marc DESCHAMPS, Anissa MEZIANE, Mathieu RÉNIER, Éric DUCASSE, Samuel RODRIGUEZ, Christine BIATEAU NDE/NDT/SHM in composite structures (high interest and needs of the Industry) High activity on modeling (numerical, analytical, hybrid) of complete processus (generation, propagation, diffraction, detection, non-linear acoustics) : from the understanding of phenomena to numerical imaging 2. Optoacoustics (OA) Bertrand AUDOIN, Clément ROSSIGNOL, Yannick GUILLET Development of eperimental devices for picosecond acoustics (ultrasound-light interaction) High resolution cell imaging (with possible higher performances than optical imaging) Vibrations of nano-objects 3. Functional Materials for Acoustics (FAMAS) Christophe ARISTÉGUI, Aleander SHUVALOV, Olivier PONCELET, Thomas BRUNET Design and manufacturing of metamaterials for acoustics (booming activity) Theory of propagation in periodical media (including interfaces, waveguides, defects ) Dynamic homogenization of heterogeneous materials (new constitutive laws, effective media for waves) Fundamental studies/numerical methods for field calculation Response of layered anisotropic plates 2 / 30 Physical Acoustics Dpt.
3 Motivation Submarine acoustics Seismology Composite materials (aircraft structures) CEA: development of the software I2M : guided wave topological imaging Response of layered anisotropic plates 3 / 30 Physical Acoustics Dpt.
4 Motivation I2M: guided wave topological imaging Non-dispersive regime TRANSDUCER [ S. Rodriguez, M. Deschamps, M. Castaings & E. Ducasse (2014). Guided wave topological imaging of isotropic plates. Ultrasonics, 54(7), ] Response of layered anisotropic plates 4 / 30 Physical Acoustics Dpt.
5 Motivation I2M: guided wave topological imaging dispersive regime TRANSDUCER [ S. Rodriguez, M. Deschamps, M. Castaings & E. Ducasse (2014). Guided wave topological imaging of isotropic plates. Ultrasonics, 54(7), ] Response of layered anisotropic plates 5 / 30 Physical Acoustics Dpt.
6 Motivation CEA: development of the software z Today at CEA: modal-k n (ω,θ) method... z θ Given phase direction θ + Harmonic regime e i ω t + Thickness discretized modal wave numbers k n (ω,θ) & mode shapes Semi Analytical Finite Elements (S.A.F.E.) method [ PhD thesis by K. Jezzine, 2006 ] [ PhD thesis by V. Baronian, 2009 ] [ PhD thesis by L. Taupin, 2012] Response of layered anisotropic plates 6 / 30 Physical Acoustics Dpt.
7 Motivation modal-k n (ω,θ) good for: Cylinders & 2D plates in vacuum, in far field modal-k n (ω,θ) bad for: 3D-anisotropic plates, embedded structures, in near field Alternative methods to efficiently deal with: 3D-anisotropy near field embedded plates? Response of layered anisotropic plates 7 / 30 Physical Acoustics Dpt.
8 Outline 1 Healthy plate 1.1 Modal-ω n (k,k y ) 1.2 Fourier-Fourier-Laplace domain 2 Plate + Defect 2.1 Boundary Element Method for a crack 2.2 Towards hybrid methods? Stiffener Crack Response of layered anisotropic plates 8 / 30 Physical Acoustics Dpt.
9 Time-domain modal approach [ω n (k,k y )] Former uses of modes ω n (k) : - (few) theoretical uses J.H. Rosenbaum (1960). The long time response of a layered elastic medium to eplosive sound. Journal of Geophysical Research, 65(5), S.B. Dong, & R.B. Nelson (1972). On natural vibrations and waves in laminated orthotropic plates. Journal of applied mechanics, 39(3), K. Aki, & PG. Richards (1980). Quantitative seismology. - almost no use for field calculation in plates Reason: 3D-anisotropy? E. Kausel (1994). Thin layer method: Formulation in the time domain. International journal for numerical methods in engineering, 37(6), E. Ducasse & M. Deschamps (2014). Time-domain computation of the response of composite layered anisotropic plates to a localized source. Wave Motion 51(8) Response of layered anisotropic plates 9 / 30 Physical Acoustics Dpt.
10 Time-domain modal approach [ω n (k,k y )] Wave equation: Source + continuity + null stress at top & bottom Anisotropic stiffness tensor C (a b) im = C ijkm a j b k 3D Response of layered anisotropic plates 10 / 30 Physical Acoustics Dpt.
11 Time-domain modal approach [ω n (k,k y )] Modal wave equation:? + continuity + null stress at top & bottom 3D z y Integral transforms: k t (none) Eigenvalue problem... z Response of layered anisotropic plates 11 / 30 Physical Acoustics Dpt.
![Time-domain modal approach [ω n (k,k y )].](/docs-images/84/89167980/images/12-1.jpg ".. Infinite number of modes Re(k n ) Propagative modes ω n Dispersion curves ω Short times?")
12 Time-domain modal approach [ω n (k,k y )]... Infinite number of modes Re(k n ) Propagative modes ω n Dispersion curves ω Short times? Real frequencies Orthogonal complete basis No evanescent modes! k This is a broad band modal response! Im(k n ) Evanescent modes Short distances? Infinite number of modes Heaviside unitstep function Response of layered anisotropic plates 12 / 30 Physical Acoustics Dpt.
13 Time-domain modal approach [ω n (k,k y )]... Infinite number of modes [ω n (k,k y )] [k n (θ,ω)] Re(k n ) No need to sort modes! Good for 3D anisotropy ω n Dispersion curves Propagative modes Requires sorting modes according to direction of energy propagation Bad for 3D anisotropy ω Ok for computing the field in many points Real frequencies Orthogonal complete basis No evanescent modes! Im(k n ) Ok for computing the field in few points Evanescent modes Short times? k This is a broad band modal response! Short distances? Infinite number of modes Heaviside unitstep function Response of layered anisotropic plates 12 / 30 Physical Acoustics Dpt.
14 Time-domain modal approach [ω n (k,k y )] What about viscoelasticity? becomes Anisotropic viscoelastic tensor C'' (a b) im = C ijkm a j b k Time derivative? We prefer to define it in ω arbitrary Op t (ω) is problematic for the ω n (k,k y ) eigenvalue statement Is it a limitation of SAFE-ω n (k,k y )? answer: perturbation theory (but modal basis is not orthogonal) ω n = ω n + iω n damped oscillation Response of layered anisotropic plates 13 / 30 Physical Acoustics Dpt.
15 Time-domain modal approach [ω n (k,k y )] What about embedded waveguides? Water Plate Incomplete modal basis Orthogonality relationships? Eigenvalue statement? Concrete, Substratum, etc. is the substructure method a solution? Hayashi, T., Inoue, D. (2014) Calculation of leaky Lamb waves with a semi-analytical finiteelement method, Ultrasonics 54(6) Park, J., & Kausel, E. (2006). Response of layered half-space obtained directly in the time domain, part I: SH sources. Bulletin of the Seismological Society of America, 96(5), Response of layered anisotropic plates 14 / 30 Physical Acoustics Dpt.
16 Partial-waves method in the Laplace Domain Wave equation: + continuity + null stress at top & bottom 3D y Integral transforms: k (Fourier) t s (Laplace) of ODE z z... Response of layered anisotropic plates 15 / 30 Physical Acoustics Dpt.
17 Partial-waves method in the Laplace Domain Cooley J. W., Tukey J. W. (1965) An algorithm for the machine calculation of comple Fourier series, Math. Comp. 19, Phinney R. A. (1965) Theoretical calculation of the spectrum of first arrivals in layered elastic mediums, J. Geophys. Res. 70(20) Radiation continuum Guided mode Transient leaky mode Bromwich Mellin formula Response of layered anisotropic plates 16 / 30 Physical Acoustics Dpt.
18 Partial-waves method in the Laplace Domain Total field = incident field z U Σ z = U inc Σ z,inc Upgoing waves: ξ e κ, α=1,2,3, Im(κ α )<0 Downgoing waves: ξ e κ, α=4,5,6, Im(κ α )>0 Polarizations and vertical wavenumbers obtained from Christoffel's equation Response of layered anisotropic plates 17 / 30 Physical Acoustics Dpt.
19 Partial-waves method in the Laplace Domain Total field = incident field + refracted field z U U inc U ref = + Σ z Σ z,inc Σ z,ref... Polarizations and vertical wavenumbers obtained from Christoffel's equation Unknown coefficients: resolution of a global linear system Response of layered anisotropic plates 17 / 30 Physical Acoustics Dpt.
20 Partial-waves method in the Laplace Domain What about viscoelasticity? Hysteretic model (non causal): Usually: C C + i sign(ω) C Adopted here: C C + i sign(im(s)) C This is not a holomorphic function. Consequences? Prospects: replace by a (similar) causal dissipation law Response of layered anisotropic plates 18 / 30 Physical Acoustics Dpt.
21 Partial-waves method in the Laplace Domain Numerical eample 1: Monolayer carbon-epoy plate Parameters measured by G. Neau: G. Neau (2003), Lamb waves in anisotropic viscoelastic plates. Study of the wave fronts and attenuation, PhD Thesis, Univ. Bordeau & Imperial College. elastic C' accurately estimated (by Lamb modes) viscoelastic C'' difficult to measure (100% error) and geometric diffraction neglected (100% additional error) Ecitation Laser measurement Response of layered anisotropic plates 19 / 30 Physical Acoustics Dpt.
22 Partial-waves method in the Laplace Domain Numerical eample 2: Zero Group Velocity modal region of an aluminum plate in water water Input: transducer 20 Dispersion curves in vacuum aluminum water S 1 -S -2 Group velocity (km/s) Propagative ZGV Backward-propagative Frequency (MHz) Response of layered anisotropic plates 20 / 30 Physical Acoustics Dpt.
23 Partial-waves method in the Laplace Domain Numerical eample 2: Zero Group Velocity modal region of an aluminum plate in water Response of layered anisotropic plates 21 / 30 Physical Acoustics Dpt.
24 Healthy Plate: Conclusion Modal-ω n (k,k y ) Fourier-Fourier-Laplace 3D anisotropy Embedded plate Non established regime Potential for C ijkm measurements by 3D Lamb waves Additional information needed for optimal sampling P. Mora, E. Ducasse, M. Deschamps, Transient elastodynamic field in a 3D embedded multilayered anisotropic plate (in revision). Response of layered anisotropic plates 22 / 30 Physical Acoustics Dpt.
25 Integral equation along the crack: Σ z (total) = 0 Source Secondary source + F s, F d Incident field Green tensor Diffracted field The unknown 23 Null for a Neumann defect Response of layered anisotropic plates 23 / 30 Physical Acoustics Dpt.
26 Integral equation : Discretized boundary integral equation : F 1 F 2 F N δ λ / 50 m = 1.. N Linear 24 system Response of layered anisotropic plates 24 / 30 Physical Acoustics Dpt.
27 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
28 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
29 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
30 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
31 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
32 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
33 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
34 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
35 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
36 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
37 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
38 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
39 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
40 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
41 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
42 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
43 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
44 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
45 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
46 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
47 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
48 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
49 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
50 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
51 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
52 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
53 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
54 Validation case: Aluminum in vacuum, bulk wave regime z With crack Vertical displacement z crack Without crack Normal stress at z crack Response of layered anisotropic plates 25 / 30 Physical Acoustics Dpt.
55 Validation case: Aluminum in vacuum, bulk wave regime B C A Computation times: A B 26 C Validation with Finite Elements 8 s This work 10 min Finite Element Response of layered anisotropic plates 26 / 30 Physical Acoustics Dpt.
56 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
57 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
58 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
59 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
60 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
61 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
62 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
63 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
64 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
65 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
66 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
67 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
68 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
69 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
70 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
71 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
72 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
73 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
74 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
75 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
76 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
77 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
78 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
79 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
80 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
81 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
82 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
83 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
84 Realistic case: Composite plate in water, guided wave regime f = 150 khz Water λ So 3 cm λ SHo 2 cm λ Ao 1 cm 0.4 cm 1 cm Carbon-epoy 8 layers, anisotropic + viscoelastic Response of layered anisotropic plates 27 / 30 Physical Acoustics Dpt.
85 Prospects for general defects Hybrid methods Previously at CEA: Finite Elements & modal-k n (ω) coupling ω n (k,k y )? FFL? [ PhD thesis by V. Baronian, 2009 ] [ PhD thesis by L. Taupin, 2012 ] [ PhD thesis by A. Tonnoir, 2015 ] Response of layered anisotropic plates 28 / 30 Physical Acoustics Dpt.
86 Prospects for general defects Hybrid Methods Favorable configuration (not yet implemented) Semi-analytical Green tensor Response of layered anisotropic plates 29 / 30 Physical Acoustics Dpt.
87 Prospects Thesis in progress : Hamza Hafidi Alaoui (2015- ) Topological imaging in layered and comple media, with INRIA, Magique 3D Aditya Krishna (2016- ) Elastodynamic response of layered anisotropic tubular structures, Region Aquitaine & CEA Tech Response of layered anisotropic plates 30 / 30 Physical Acoustics Dpt.
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