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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. 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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. 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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. Fritzen
VIth International Workshop NDT
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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. Fritzen
VIth International Workshop NDT
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)
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. Fritzen
VIth International Workshop NDT
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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. 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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. Fritzen
VIth International Workshop NDT
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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. Fritzen
VIth International Workshop NDT
in Progress 2011
Third Order Plate Theory for Wave Propagation Modelling
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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. Fritzen
VIth International Workshop NDT
in Progress 2011
Third Order Plate Theory for Wave Propagation Modelling
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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. 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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. Fritzen
VIth International Workshop NDT
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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. Fritzen
VIth International Workshop NDT
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)?
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. Fritzen
VIth International Workshop NDT
in Progress 2011
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:
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. 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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. Fritzen
VIth International Workshop NDT
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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. Fritzen
VIth International Workshop NDT
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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. Fritzen
VIth International Workshop NDT
in Progress 2011
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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. 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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. Fritzen
VIth International Workshop NDT
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)
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. Fritzen
VIth International Workshop NDT
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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. Fritzen
VIth International Workshop NDT
in Progress 2011
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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. 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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. Fritzen
VIth International Workshop NDT
in Progress 2011
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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. Fritzen
VIth International Workshop NDT
in Progress 2011
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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. Fritzen
VIth International Workshop NDT
in Progress 2011
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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. Fritzen
VIth International Workshop NDT
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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. Fritzen
VIth International Workshop NDT
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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. Fritzen
VIth International Workshop NDT
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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. 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
Institute of Mechanics and Control
Engineering – Mechatronics
Prof. Dr.-Ing. C.-P. Fritzen
VIth International Workshop NDT
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