Wavelet Spectral Finite Elements for Wave Propagation in Composite Plates with Damages

Ratneshwar Jha, Clarkson UniversityS. Gopalakrishnan, Indian Institute of Science

Grant Number FA9550-09-1-0275Period: April 2009 – Nov 2012

Program Manager: Victor Giurgiutiu/David Stargel

AFOSR Annual Structural Mechanics Grantees/Contractors Meeting

28 July 2009 Fairborn, OH

Project Objectives

Develop Methodology for Prediction of Damage Type and Severity in Composite Plates Using Lamb Waves

Task 1

Develop Spectral Finite Element (SFE) models for damages in composite plates (delamination, arbitrarily oriented crack, transverse crack)

Task 2

Perform extensive experimental studies with various type and level of damages; Obtain characteristic signals for damages using ‘time reversal’ technique; Implement efficient damage detection methods

Outline

• Background• Time Reversal • Modified Time Reversal• Experimental Setup• Results and Discussions• Conclusions

Spectral Finite Element

Spectral finite element method highly suitable for wave propagation analysis due to its frequency domain approach; Yields models that are many orders smaller than conventional FEM

Wavelet spectral finite element can accurately model 2-D plate structures of finite dimensions, unlike the existing Fourier transform based SFE which cannot handle waveguides of short lengths

WSFE – 1 Element

2-D FE - 6432 four-noded plane stress quad elements

Axial velocity at midpoint of edge AB in [04/904] cantilever plate with LX =0.5 m and LY=0.25 m due to tip impulse load applied in axial directions along AB

Introduction

• In the case of detection of small delaminations, the spectral finite element analysis is more sensitive to small stiffness changes and has been used extensively .

• Such spectral finite element models are available only for few regular geometries, which is the motivation for the development of an approximate model.

• In this presentation an approximate spectral finite element model is developed, where the wave number is approximated using a suitable approximation function of frequency , the coefficients of which are obtained from the experimentally measured dispersion relation

3/)1()3/3(2

)3/2(1

nnaaak

Wave number is approximated using a suitable approximation function of frequency

where ω is the frequency

dk

dωCg

(1)

Therefore from equation (1),

gn

n Caaad

dk/11)3/)1((1)3/3(

21)3/2(

1

(2)

WAVE NUMBER APPROXIMATION

This approximated wave number is now applied to the equation for finding the spectral amplitudes of the displacement fields and the stiffness matrix is calculated in the same way as an exact model

0 1 2 3 4

x 10-4

0

0.2

0.4

0.6

0.8

1

Time (Sec)

Load

(N)

0 2 4 6 8

x 104

0

0.5

1

1.5

2

2.5

3x 10

-5

Frequency (kHz)

Spe

ctra

l am

plitu

de

0 0.5 1 1.5

x 10-3

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Time (sec)

Load

(N) 0 5 10 15

x 104

0

2

4

6

x 10-5

frequency(Hz)

spec

tral a

mpl

itude

Single frequency tone-burst sinusoidal pulse modulated at 20kHz.The spectral amplitude over the frequency axis is shown in inset

Broad band pulse used to study the response of the beam. The spectral amplitude over the frequency axis is shown in the inset

Signals used for SFE Simulations

0 2000 4000 6000 8000 10000 120000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Wavenum

ber,

kb

Frequency (Hz)

Theoryn=4n=5n=6n=7

. Spectrum relation for the beam (An aluminum beam of 0.8 m length and cross sectional area 10 mm x 10 mm ) using Timoshenko beam theory and approximate model

0 0.2 0.4 0.6 0.8 1 1.2 1.4

x 10-3

-6

-4

-2

0

2

4

6 T

ransvers

e v

elo

city a

t im

pact

node(m

m/s

ec)

Time(sec)

theory

approx(n=7)

Comparison of transverse velocity history at the cantilever tip due to a modulated sine pulse applied at the same location using approximate model (n=7)

and using theory

0 0.2 0.4 0.6 0.8 1 1.2

x 10-3

0

1

2

3

4

5

6

7

8

Tra

nsvers

e v

elo

city

at im

pact node(m

m/s

ec)

Time(sec)

theory

n=7

n=6

n=5

n=4

Comaprison of transverse velocity history at the cantilever tip due to the broad band pulse applied at the same location using approximate model

and using theory

0 0.2 0.4 0.6 0.8 1 1.2

x 10-3

-1

-0.5

0

0.5

1

Tra

nsve

rse

ve

locity a

t im

pa

ct n

od

e(m

m/s

ec)

Time(sec)

Theory

approx (n=7)

Comparison of the transverse velocity history calculated at the tip of the cantilever using approximate spectral model and using theoretical SFEM results for the delaminated beam (L=0.8 m, L1=0.2 m and delamination

length=15 mm)

0 0.5 1 1.5

x 10-3

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (sec)

Tra

nsvers

e v

elo

city

at im

pact node (

mm

/sec)

theory

approx (n=7)

Comparison of the transverse velocity history at the tip of the cantilever beam using approximate model (n=7) and theory for the delaminated beam (L=1 m, L1=0.75 m, delamination length=20 mm)

• Numerical results shows that the approximate model using only seven points (n=7) itself shows excellent match with the theory, in both healthy and delaminated beam.

• This implies that the model is suitable for the detection of damage location in a delaminated beam

Time Reversal Method

• Ing and Fink (1998)– Increase Lamb wave resolution using Time Reversal Mirrors– Detect flaws using a pulse-echo technique

• Sohn and Park (2004 and 2007)– Baseline free damage detection– Detect presence and location of flaws using pitch-catch technique

(using A0 Lamb wave mode)

– Analyze effects of temperature on TRM

• Butenasand and Kazys (2006)– Numerical validation of TRM

• Giurgiutiu (2008)– Experimental and analytical validation of TRM using A0, S0 and

A0+S0 modes

Modified Time Reversal Method

Significant reduction in actuation hardware

Damage Index

Quantifies difference between actuated and time reversed signals

Sohn-Park et al.

Current

2

2 2

( )N

L

N

I VDI

I

Modified Damage Index

• Sohn-Park normalize input and time reversed signals with their peak amplitudes– DI quantifies changes in frequency and signal

shape– DI does not capture amplitude change

• Current procedure normalizes both input and time reversed signals with a common value

- DI quantifies amplitude change as well

Experimental Setup

Actuation signal Tone burst

• 20 kHz• 9.5 cycles

Damage Index Values (Baseline)

Damage Index Values (Path 1-2)

Damage Index Values (Path 1-3)

Damage Index Values (Path 2-3)

Results (Signal Path 1-2)

Results (Signal Path 1-3)

Results (Signal Path 2-3)

Conclusions

• Results show that modified time reversal method identifies damage – Presence– Location– Severity

• Baseline (healthy) data for test specimen not needed