a study of attenuation and acoustic energy anisotropy of ... · institute of mechanics and control...

Institute of Mechanics and Control

Engineering – Mechatronics

Prof. Dr.-Ing. C.-P. Fritzen

VIth International Workshop NDT in Progress 2011, October 10-12, 2011

Prague, Czech Republic

A Study of Attenuation and Acoustic Energy Anisotropy of Lamb Waves

in Multilayered Anisotropic Media for NDT and SHM applications

Miguel Angel Torres A., Henning Jung and Claus-Peter Fritzen




VIth International Workshop NDT

in Progress 2011

Contents

Outline

(1) Motivation

(2) Modelling of Guided Waves

Plate Theory

Spectral Element Method

(3) Dispersion Characteristics

(4) Importance of Dispersion Knowledge

(5) Examples

(6) Conclusions





in Progress 2011

Motivation: Fields of Application

Impact Damage in Aeronautic and Aerospace Structures

Courtesy of NASA

Delamination in Composite Structures

Courtesy of IMA

Crack Propagation Detection

Courtesy of BAM

Pipeline Inspection

Courtesy of NatGeo





in Progress 2011

Motivation: Current Techniques

• Many SHM/NDT approaches based on

propagation of elastic waves

• Data-based, model-free (Machine Learning)

• Setup: time consuming, costly, many pre-tests

• Optimization: trial and error

Active Sensing (UT)

Passive Sensing (AE)





in Progress 2011

Motivation: Requirements

Detailed analysis of wave propagation phenomena

Use of predictive modeling tools

Analyzing guided wave interaction in structures and damages

Technique suitable to achieve fast/real-time damage detection

Cost effective techniques

Improvement of sensor networks and reliable NDT/SHM systems





in Progress 2011

Contents

Outline

(1) Motivation


Plate Theory




(5) Examples

(6) Conclusions





in Progress 2011

Third Order Plate Theory for Wave Propagation Modelling

The displacements fields in terms of the thickness z are expanded to a

third order. The stress resultants in the three dimensional system for a

given propagation direction θ and fiber orientation φ are shown below.

ѱ: Rotation function

ϕ: Dilatation function

u0 ,v0 ,w0: Particle displacements

2 30 , , , , , , , ,x x xu u x y t z x y t z x y t z x y t

2 30 , , , , , , , ,y y yv v x y t z x y t z x y t z x y t

20 , , , , , ,z zw w x y t z x y t z x y t





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The strain energy of each layer can be represented as:

Additionally,

The plate constitutive equations:

From the strain energy density in the 3-D elasticity theory

The equations of motion:

From the dynamic version of the principle of virtual displacement

2 211 12 13 16 22 23 26

2 2 2 233 36 66 44 45 55

1( 2 2 2 2 2

2

2 2 ) .

x x y x z x xy y y z y xy

V

z z xy xy yz yz xz xz

U C C C C C C C

C C C C C C dV

,..., .x xyx xy

U U





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The system can be expressed in a matrix form D, and by imposing

boundary conditions and setting its determinant to zero, a characteristic

function relating the angular frequency to the wavenumber is obtained.

For a given ω, a resulting complex wavenumber k = kRe + ikIm is obtained.

The real part kRe is used to describe the phase velocity of waves

The imaginary part kIm describes the amplitude decay

det , , , , 0ijk CD





in Progress 2011

Contents

Outline

(1) Motivation


Plate Theory




(5) Examples

(6) Conclusions





in Progress 2011

What is the Spectral Element Method (SEM)?

• Finite elements with high degree of polynomial interpolation

• Carefully chosen nodal base and numerical integration rule

• Combines advantages of both methods

global pseudospectral-

method „classical“ FEM

Numerically

efficient,

high accuracy

Geometric flexibility

spectral element method,

SEM





in Progress 2011

• Excellent interpolation properties less nodes per wavelength

• Discrete orthogonality ijji δξψ )(

Nodal base and shape functions on a 1D reference element: 11

Shape functions : Lagrange interpolation polynomials i

)()1()()1( 1

22 ξLoξξLξ NNLocal Nodes: roots of the polynomial

derivate of the Legendre polynom Lobatto polynom =

GLL: Gauss-Lobatto-Legendre

What is the Spectral Element Method (SEM)?





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Nodal Base:

5 GLL nodes

Plate Theory

Resulting ODE:

)(tFKqqCqM Central Difference Scheme

- For diagonal mass matrix

Solution:

Kinematics: Shape functions:

A Spectral Element

Workflow:





in Progress 2011

Contents

Outline

(1) Motivation


Plate Theory




(5) Examples

(6) Conclusions





in Progress 2011

Exact Theory vs. Plate Theory

For a frequency-thickness product of 1.2 MHzmm, the error in group

velocity is below 3%

Comparison between exact theory (solid) and approximated (dashed) at α=30

Phase Velocity Dispersion Curve Group Velocity Dispersion Curve





in Progress 2011

Models of Attenuation

Hysteretic model: Imaginary part of the complex stiffness matrix does

not depend on the frequency. The stiffness matrix is expressed as:

Kelvin-Voigt model: Assumes a linear dependence of the viscoelastic

coefficients with frequency. The complex stiffness matrix is expressed as:

.C C i

ω: Angular frequency

ῶ: Characterization Frequency

η: Viscoelastic Constants

.C C i





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Focusing of Lamb Waves

Deviation of the group velocity (Cg(ϑ)) from the wave vector (k(θ)) also

results in acoustic energy anisotropy described by the phonon focusing

factor which measures the acoustic ray anisotropy. The factor is given

by:

The knowledge of this effect is important in SHM applications for:

Avoidance of dead zones in sensor networks

Better understanding of the detection of these waves

Analysis of several wave arrivals required for source localization

2

21

2

1

2

1

1

ph

ph

ph

ph

dCC

ddA

d d CC

d

Cph: Phase velocity

θ: Angle of propagation

ϑ: Group velocity angle





in Progress 2011

Contents

Outline

(1) Motivation


Plate Theory




(5) Examples

(6) Conclusions





in Progress 2011

Dispersion Knowledge Importance: In General

Modal wave analysis provides an improved understanding of the

propagation and interaction of guided waves. It offers potential for:

Sensor location

Sensor reduction

Increased source location accuracy

Improve probability of detection (POD)

Insight into the origin of the source (in case of passive techniques)





in Progress 2011

Mode Behaviour Importance: In Active Techniques

A number of modes of particle vibration are possible, but the two most

common are symmetrical and asymmetrical (shear horizontal modes

also exist!). Mode shapes for the fundamental modes of propagation

are depicted below.

-2 -1 0 1 2

x 10-8

-1

-0.5

0

0.5

1

Displacement [m]

Pla

te N

orm

al [m

m]

Ao Mode at 100kHz

-2 -1 0 1 2

x 10-9

-1

-0.5

0

0.5

1Ao Mode at 1MHz

Plate Normal [m]

Pla

te N

orm

al [m

m]

-1 0 1

x 10-9

-1

-0.5

0

0.5

1Ao Mode at 2MHz

Plate Normal [m]

Pla

te N

orm

al [m

m]

-1 0 1

x 10-8

-1

-0.5

0

0.5

1So Mode at 100kHz

Displacement [m]

Pla

te N

orm

al [m

m]

-2 -1 0 1 2

x 10-9

-1

-0.5

0

0.5

1So Mode at 1MHz

Displacement [m]

Pla

te N

orm

al [m

m]

-2 -1 0 1 2

x 10-9

-1

-0.5

0

0.5

1So Mode at 2MHz

Displacement [m]

Pla

te N

orm

al [m

m]

In-plane Motion Out of Plane Motion

Modes of Propagation

Courtesy of ndt-ed.org





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Dispersion Knowledge Importance: In Passive Techniques

The asymmetric wave mode will interact most strongly with damage

lying normal to the plane of the wave propagation such as:

Delamination

Skin/Core debonding

Impact Damage

The symmetric wave mode will interact most strongly with damage

lying perpendicular to the plane of wave propagation such as:

Matrix cracking

Matrix splitting

Core crushing

Rotor Blade Delamination

Courtesy of Helicopter

Magazine

Matrix Cracking

Courtesy of IKB





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Contents

Outline

(1) Motivation


Plate Theory




(5) Examples

(6) Conclusions





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Example 1: 1.5mm Thick Glass Fibre Reinforced Plastic Plate

E1 (GPa) E2 (GPa) E3 (GPa) G12 (GPa) G13 (GPa) G23 (GPa) ѵ12= ѵ13= ѵ23 ρ (kg/m3)

30.7 15.2 10 4 3.1 2.75 0.3 1700 Material Properties in GPa, α=90°

f=200 kHz

Time [ms]

Experiment

Simulation

Experiment

Simulation

Experiment

Simulation





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Example 1: 1.5mm Thick Glass Fibre Reinforced Plastic Plate

Simulated Displacement Fields

In-Plane Out-of-Plane

Caustics of the SH0 mode indicate the energy focusing in these

directions for this wave mode

The energy of the S0 and A0 modes is highly concentrated in the fibre

direction

Fib

re D

ire

ctio

n

Videos/GFRP/GFRP_S0_SH0.avi

Videos/GFRP/GFRP_A0.avi





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Example 2: 5.1mm Thick Unidirectional CFRP(at 95kHz)

Structure

In-Plane Motion

Out-of-Plane Motion

0 0.11 0.22

x[m]

α

0

0.1

4

0

.28

y[m

]

S2

S1

C11 C12 C13 C22 C23 C33 C44 C55 C66

125 6.3 5.4 13.9 7.1 14.5 3.7 5.4 5.4

η11 η12 η13 η22 η23 η33 η44 η55 η66

3 0.9 0.4 0.6 0.23 0.6 0.12 0.3 0.5

Material Properties in GPa

Layup: [0°]18 (Castaings07)

Fibre Direction

Videos/DFG-II-01/DFG-II-01_S0_SH0.avi

Videos/DFG-II-01/DFG-II-01_A0.avi





in Progress 2011

Example 2: 5.1mm Thick Unidirectional CFRP (at 95kHz)

Wave Curve Attenuation Curve Focusing Curve

Attenuation is affected a great deal by the anisotropy of the material

The phonon focusing factor precisely tracks the angular dependent

energy concentration effect

The cuspidal regions of the SH0 in the focusing curve mode explain the

energy patterns containing caustics





in Progress 2011

Example 3: 4.7mm Thick Multilayered CFRP(at 95kHz)

Out-of-Plane Motion

0 0.11 0.22

x[m]

0

0

.14

0

.28

y[m

] S2

S1

α

C11 C12 C13 C22 C23 C33 C44 C55 C66

70 23.9 6.2 33 6.8 14.7 4.2 4.7 21.9

η11 η12 η13 η22 η23 η33 η44 η55 η66

1.8 0.9 0.3 1.4 0.2 0.5 0.17 0.2 0.5

Material Properties in GPa

Layup: [-45° 0° 45° 45° 0° -45°

-45° 0° 45°]s

In-Plane Motion

(Castaings07)

Fibre Direction

Videos/DFG-II-03/DFG-II-03_A0.avi

Videos/CFRP/CFRP_Blank_Plate_CFRP_S0_SH0.avi





in Progress 2011

Example 3: 4.7mm Thick Multilayered CFRP(at 95kHz)

Wave Curve Attenuation Curve

Wave velocities and attenuation are not strongly related to the frequency

and orientation of propagation

The multilayered and multi-oriented composition of the structure

mitigates the anisotropic impact of each layer

Cuspidal regions of the SH0 in the focusing curve indicate the energy

concentration at approximately α = 48°, 138°, 228° and 318°

Focusing Curve





in Progress 2011

Contents

Outline

(1) Motivation


Plate Theory




(5) Examples

(6) Conclusions





in Progress 2011

Conclusions

Comparisons to experimental data have been presented in order to

validate the models

Accurate estimates of velocity and attenuation in anisotropic

laminates in the frequency range of Lamb wave applications has

been presented

The effects of energy focusing and importance of guided wave modal

analysis has been introduced

The proposed methodology can help to the improvement of the

understanding of the source and its localization

Optimization of sensor networks in terms of sensor placement and

number of sensors can be accomplished by understanding the wave

propagation phenomena and can be used in pre-design-phase to

reduce cost





in Progress 2011

THANK YOU FOR YOUR ATTENTION!!!

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