Defect Detection In Plates Using Guided Waves.pdf

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Doctoral Thesis

Defect detection in plates using guided waves Author(s): Fromme, Paul Publication Date: 2001 Permanent Link: https://doi.org/10.3929/ethz-a-004304781

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ETH Library

Diss ETHNo

14397

Defect detection in

plates using guided

A thesis submitted to the SWISS FEDERAL INSTITUTE OF TECHNOLOGY

for the

degree

of

Doctor of Technical Sciences

presented by Paul Fromme Dipl -Ing University

of Karlsruhe

(TH)

born November 28, 1971 citizen of

Accepted

on

Germany

the recommendation of

Prof Dr M B Prof Dr P

Sayir, examiner Cawley, coexammer

Prof Dr S K Datta,

Zurich,

coexammer

2001

waves

Acknowledgements This work

was

of Mechanical contributed

path

or

such

Cawley

my time

another to this

an

as a

research assistant at the Institute

I would like to thank all the

my supervisor, for

Sayir,

providing

Prof Dr P

form

during

ETH Zurich

Systems,

in one

Prof Dr M B and

carried out

thesis,

in

guiding

people

who

particular

me

this interesting research

on

excellent research environment

and Prof Dr

S K

coexammers, for

Datta, my

thoroughly

reviewing this thesis and the interesting and fruitful discussions

Prof Dr J

Andreas

Dual, for his support during

Allenspach,

Leutenegger,

Tunaboylu,

Paolo Buiatti,

Bernard

Masserey,

for writing their

scope of my research

my time

as a

lecturer

Georgios Kotsalis,

Dieter

Profunser,

Semesterarbeiten

or

project and thus contributing

Joachim

Britta

Lackner, Tobias

Schmid, and

Diplomarbeiten

to this thesis

The

fatigue engineering group at RUAG Aerospace, Emmen, for giving possibility to make 'real-life' experiments, especially Markus Wyss, Zehnder, Mirco Figholmo, and Gustav Bolfmg My

office companions, Christian Lmder and Frank

Daniel

May,

for

having

a

me

the

Simon

good time,

Gsell, Tobias Leutenegger, Bernard Masserey, and Jaquelme Vollmann,

for their Markus

Yaman

withm the

help with Pfaffmger,

wave

propagation problems and reviewing part of my work,

Andreas

Hochuh, and last but

not

least, Traude Junker

Paul Fromme

Zurich, November

2001

Table of contents

Abstract

iii

Zusammenfassung

v

1

Introduction

1

1 1

Motivation

1 2

Selection of

1 3

Previous work

7

13 1

Theoretical work

7

13 2

Experimental

14

Content of the thesis

9

2

Measurements

11

1

mode

wave

5

work

8

remarks

11

2 3 1

Introductory Experimental setup Specimen geometry Aluminum plate

2 3 2

Tensile specimen

16

2 4

Excitation

2 5

Excitation transducer

20

2 5 1

Piezoelectric transducer

20

2 1 2 2 2 3

12 14 14

signal

18

acoustical transducer

2 5 2

Electromagnetic

2 6

Measurement and data

3

Scattering

at

a

handling

circular hole

of the scattering

3 1

Geometry

3 2

Lamb

3 3

Solution using classical

3 3 1

Wave

3 3 2

Scattering

3 4

Solution using Mmdlm's

3 4 1

Wave

3 42

Scattering

3 5

Solution using

3 5 1

Wave

3 5 2

Scattering

3 6

Comparison

wave

(EMAT)

problem

plate theory

circular hole

theory

propagation at

a

circular hole an

asymptotic expansion

propagation at

a

29 29

31 31

propagation a

26

30

propagation

at

22

circular hole

with measurements

32 34 34 37 39 39 41 42

Table of contents

//

hole with

defect

4

Scattering

4 1

Description

4 2

Possible

4 2 1

Modification of the scattering at

4 2 2

Conformai mapping

4 2 3

Superposition

4 3

Finite difference

4 3 1

FDM

4 3 2

4 5

plate Scattering implementation Comparison Numerical study of defect detectabihty

5

Application

4 3 3 4 4

Wave

at

a

a

of the geometry

45

solutions

analytical

46

algorithm

circular hole

a

problems,

crack and circular hole

modeling

propagation

46 48

of two separate for

45

a

51

Mindhn type

plate

51

and tensile specimen

in a

49

54 56 57 63

69

to NDT

5 1

Outline

5 2

Measurements at

52 1

Influence of

5 2 2

Measurements at

5 2 3

Broadband excitation

73

5 3

Measurements at tensile specimens

75

5 3 1

Description

5 3 2

Intermediate measurements

5 3 3

Description

5 3 4

Influence of

5 3 5

On-line monitoring of crack

69

plates

notch

a

a

69

the scattered field

69

complicated geometry

72

on

of the first measurement in

the

series

laboratory

of the second measurement a

crack

on

series

the scattered field

growth

6

Conclusions and Outlook

6 1

Measurements

6 2

Theoretical calculations

6 3

Application

6 4

Outlook

to NDT

76 82 87 91 96

99 99 101 102 103

Bibliography

105

Curriculum vitae

111

Abstract

The scattering of the first antisymmetric Lamb wave mode A0 at obstacles in plate-like structures is studied in this dissertation The propagation in an isotro¬ pic, homogeneous plate, the scattering at a circular hole, and the scattering at a hole with a defect are investigated experimentally and theoretically Guided flexural waves have the advantage of propagating over large distances in plates, thus allowing the fast and efficient detection of defects in large structures This method holds promise for the nondestructive testing of aircraft Airplane fuselage and wings often consist of aluminum face sheets, connected with fasten¬ ers or

containing holes, which

mation at their boundaries

typical

scattered

indicates the can

As

are sources

When the field

of stress concentration and crack for¬

guided

wave

obtained A

displacement development of a fatigue crack, is

hits such

a

discontinuity,

and thus the

a

the scattered field

change growth in

of such cracks

be monitored a

model system to gam

flexural

wave

notch at

an

with

an

a

well-founded

obstacle

in

the

understanding of the interaction of the plate, the case of a through hole with a

arbitrary angle is studied In the experiments, the A0 mode is excited selectively by means of a piezoelectric transducer with a well-defined time sig¬ nal The used frequency range is below the cut-off frequencies of the higher wave modes in the plate The scattered field is measured on a grid around the hole with a heterodyne laser-interferometer Using fast Fourier transformation, the ampli¬ tude and phase values of the scattered field are extracted from the measured time The introduction of a small imperfection, like a notch, at the boundary of series the cavity changes the measured scattered field significantly The first antisymmetric Lamb wave mode A0 physically represents a flexural wave propagating along the structure It can be described well using approximate theories Therefore no three-dimensional theory needs to be implemented, and a fast calculation is achieved Different approximate analytical approaches to cal¬ culate the wave propagation and the scattering at a circular hole, employing clas¬ sical plate theory, Mmdlm's theory, and an asymptotic expansion of the threedimensional theory are compared Good agreement between the experimental data and the analytical solutions is found for the extent of validity of the different models The influence of tered field

is

a

defect like

modelled

a

crack

cretizmg Mmdlm's equations of transient

wave

or a

notch at the hole

numerically implementing

propagation

is

boundary

on

the scat¬

finite difference scheme Dis-

on a staggered, Cartesian grid, the by explicit time integration The stress-

motion

calculated

a

Abstract

IV

free

boundary

conditions at the hole and

a

notch

approximation of the boundaries This way

a

are

implemented

on a

Cartesian

stable and fast numerical calcula¬

tion of the scattered field around the hole and notch

is

achieved Good agreement

with the

analytical calculation and the measurements for the propagation and the scattering at an undamaged hole is found The numerical calculations agree well

with the measurements for

a

notch

or

a

crack at the hole boundaries

Accurate

descriptions of the influence of a defect on the scattered field can be made The detectabihty of a defect is studied numerically for a parameter variation, and the predictions are compared to the experiments The method is applied experimentally to a variety of specimen, proving its use¬ fulness for nondestructive testing purposes In aluminum plates well-defined geometries like a notch at different angles relative to the propagation direction of the incident wave, and a line of holes symbolizing the multiple scattering at a line of rivets are studied Broadband excitation and measurements at only a few points are investigated to achieve a fast defect detection Fatigue cracks at holes in

tensile specimens

realistic

problem

are

studied

The cracks

in

collaboration with

initiated and

an

industrial partner

as

a

tensile load¬

propagated by cyclic servo-hydraulic material testing machine An on-line monitoring of the crack length during the crack propagation is implemented and found to give repeatable results The minimum detectable crack length is evalu¬ ated and problems like crack closure are studied Thorough theoretical and experimental know-how on the interaction of flexural Accurate predictions on waves with obstacles in plate-like structures is gained the detectabihty of fatigue cracks at fastener holes, an important problem in aero¬ space industry, can be made The practical applicability of the method is shown ing of the test specimen

in a

are

Zusammenfassung In dieser Arbeit wird die

Streuung des ersten anti-symmetrischen Modes A0 der Unstetigkelten in Platten untersucht Die Wellenausbreitung in einer isotropen, homogenen Platte, die Streuung an einer kreisrunden Bohrung und die Streuung an einer Bohrung mit einem Riss oder einer Kerbe wird experi¬ Lambwellen

an

mentell und theoretisch untersucht Strukturwellen haben den ausbreiten und daher fur

Vorteil, dass

eine

sie

sich über grosse Distanzen

schnelle und effiziente Fehlerdetektion

in in

Platten grossen

Strukturen geeignet sind Eine

mögliche Anwendung dieser Methode ist die zer¬ störungsfreie Prüfung von Flugzeugen Der Rumpf und die Flügel von Flugzeu¬ gen bestehen oft aus Aluminiumplatten, die Aussparungen enthalten und durch Nieten verbunden sind An diesen Bohrungen gibt es eine Spannungsuberhohung und daher eine erhöhte Gefahr der Bildung von Ermudungsnssen Strukturwel¬ len, die sich in der Platte ausbreiten, werden an diesen Bohrungen gestreut, und Das Auftreten eines es ergibt sich ein typisches Streuungsfeld um die Bohrung bewirkt dieses eine Ermudungsrisses Veränderung Streuungsfeldes Die Messung dieser Änderung erlaubt die Detektion von Rissen, und das Risswachstum in Pro¬ ben kann überwacht werden Als einfaches

Modellsystem,

ständnis fur die Interaktion runde

um

von

die Machbarkeit nachzuweisen und

Welle und Defekt

zu

erhalten, wird

ein

eine

Ver¬

kreis¬

Bohrung in einer isotropen, homogenen Platte untersucht Die selektive Anregung des A0-Modes erfolgt durch einen piezoelektrischen Transducer mit einem vorgegebenen Zeitsignal Der betrachtete Frequenzbereich hegt unterhalb der Cut-off-Frequenzen der höheren Wellenmodes in der Platte Die Messung des Streuungsfeldes erfolgt punktweise auf einem Messgitter um die Bohrung mit Die Amplituden- und Phaseninforma¬ einem heterodynen Laserinterferometer tion des Streuungsfeldes wird mittels Founertransformation bestimmt Das Ein¬ bringen einer kleinen Fehlstelle, beispielsweise durch Sagen einer Kerbe an der Bohrung, hat einen signifikanten Emfluss auf das gemessene Streuungsfeld Physikalisch betrachtet ist der erste anti-symmetrische Mode A0 der Lambwellen eine Biegewelle Mit der Approximation der Mmdlm'schen Theorie, die den Em¬ fluss der Biegung, des Schubes und der Rotationstragheit berücksichtigt, kann die Ausbreitung dieses Modes im untersuchten Frequenzbereich gut beschrieben werden Dies erlaubt eine schnellere Berechnung als bei Berücksichtigung der vollen dreidimensionalen Theorie Um die Ausbreitung der Welle in der Platte und die Streuung an einem kreisrunden Loch zu beschreiben, werden noch zwei weitere Naherungslosungen verwendet, namhch die klassische Theorie fur Bie-

Zusammenfassung

VI

gewellen

in

Platten und

eine

asymptotisch hergeleitete Theorie,

welche die

chen

physikalischen Effekte wie die Mmdlm'sche Theorie berücksichtigt den Geltungsbereich der Näherung der verschiedenen Theorien ergibt sich gute Übereinstimmung mit den Messergebnissen Eine

Fehlstelle,

wie

em

Riss oder

eine

Kerbe

am

glei¬ Fur eine

Lochrand, erzeugt zusatzliche

berücksichtigende Randbedingungen Zur Berechnung des kombinierten Streuungsfeldes wird eine numerische Modellierung mit der Fimten-DifferenzenMethode implementiert Die Bewegungsgleichungen gemäss Mmdlm werden auf einem gestaffelten kartesischen Gitter diskretisiert und die transiente Wellenaus¬ breitung durch explizite Zeitintegration berechnet Die spannungsfreien Randbe¬ dingungen an der Bohrung und der Kerbe werden auf einer kartesischen Approximation der Rander implementiert Dies erlaubt eine schnelle und nume¬ risch stabile Berechnung des kombinierten Streuungsfeldes um Bohrung und Kerbe Fur den Fall einer kreisrunden Bohrung ergibt sich eine gute Übereinstim¬ mung der numerischen Ergebnisse mit der analytischen Berechnung und den Messergebnissen Fur eine Fehlstelle an der Bohrung kann mit der numerischen Berechnung der gemessene Emfluss einer Kerbe oder eines Risses gut vorherge¬ zu

sagt werden Die Detektierbarkeit

einer

Kerbe wird

an

die numerisch evaluiert und die minimal detektierbare

Hand

einer

Parameterstu¬

bestimmt

Risslange zerstörungsfreie Prüfung von Strukturen wird experimentell an verschiedenen Proben gezeigt An Aluminiumplatten mit einer Bohrung wird der Emfluss des Winkels zwischen Kerbe und Ausbreitungs¬ richtung der Welle untersucht Zur Simulation einer Nietreihe, wie sie typischer¬ weise in Flugzeugen vorkommt, wird das kombinierte Streuungsfeld um Bohrungen auf einer Linie experimentell und theoretisch untersucht Auch fur diese kompliziertere Geometrie hat eine Kerbe an einer der Bohrungen einen gut messbaren Emfluss auf das Streuungsfeld Zur Minimierung der Messdauer und des Messaufwandes wird die Möglichkeit der Anregung mit breitbandigem Fre¬ quenzinhalt und die Messung an nur wenigen Punkten der Struktur untersucht Die Detektierbarkeit von Ermudungsnssen an Bohrungen in Zugproben wird in Zusammenarbeit mit einem Industriepartner evaluiert Die Risse werden durch zyklische Ermüdung der Proben in einer Zugmaschine erzeugt Die Implementie¬ rung einer on-line Überwachung der Risslange wahrend der Ermudungsversuche ergibt wiederholbare Resultate Die kleinste detektierbare Risslange und Pro¬ Die Anwendbarkeit der Methode fur die

bleme

wie

das Schhessen des Risses ohne Last werden untersucht

Im Rahmen dieser Studie konnten

grundlegende Erkenntnisse über die Streuung Biegewellen an Unstetigkelten in Platten gewonnen werden Die Detektier¬ barkeit von Ermudungsnssen an Nietlochern kann vorhergesagt werden von

1

Introduction

1.1

Motivation

Fig.

Fatigue

1.1

crack at

Technical

a

hole in

an

aluminum tensile

specimen.

machinery, systems, and components, e.g., airplanes, cars, pumps, and are subject to varying or cyclic service loads and environmen¬ tal influences. Such operation conditions can lead to wear, corrosion, and damag¬ ing of the components. The problem is relevant in aircraft industry, where a common maintenance problem is the development of fatigue and corrosion cracks in aircraft fuselage and wings. Due to stress concentration and the contact of different materials, fastener and rivet holes are frequently sources of crack growth. The longer service life span of aircraft increases the need to periodically check the structure for damage. A variety of nondestructive methods for the detection of flaws has been devel¬ oped and used successfully [7]. Over the years the research focus has shifted from simpler methods like liquid penetration and visual/optical testing to more sophisticated techniques, mostly employing electromagnetic or elastic waves of varying frequency. Many electromagnetic methods, e.g. eddy current testing [61], radiography [27], and thermography ([25], [18]), have been employed in industry for a long time and been proven to be very efficient. Mechanical waves as used in ultrasonic testing (UT) have a well established perpipes

in

refineries,

2

Introduction

formance for the detection of defects However, UT the

wavelength

size one aims

of the mechanical

waves

used

is

is

rather time-consuming

usually

in

as

the order of the defect

to detect and thus small

compared to the thickness of the structure high frequencies dampen out quickly and good signal transmitted or reflected wave can usually only be achieved

Waves at such rather

strength of either working through the

thickness of the structure

manual scanning around the the

area

of

suspected

Therefore classical UT involves defects

Bar-Cohen

[8]

proposes

of robotic devices to automate the scanning

procedure An alternate, more elegant and promising approach is the use of guided waves, resulting in a propagation direction along the structure and reducing the need for scanning Guided waves result from multiple reflections of the pressure and shear waves over the specimen cross section, such that a standing wave mode through the thickness is obtained (see eg [1]) From the measurement of the guided wave at a few points on the surface of the structure it is possible to detect defects in a large area with a fast and cost-effective method [36] The method has been used successfully for defect detection in rods and beams Beams can be approximated as one-dimensional structures, the guided wave propagating with constant ampli¬ tude only along the beam Dual et al [16] used bending waves to detect small use

notches

in

aluminum beams

Another important

application is the detection of corrosion and cracks in tubes widely used in oil and chemical industries and for water and distribution Such pipes are often buried in the ground or surrounded supply materials and therefore not readily accessible for classical pointby insulating It is advantageous having to access the pipe at only a few points and wise UT [6] generating a guided wave mode that travels along the pipe for distances of sev¬ and pipes, which

eral meters This underneath ing

a

are

allows,

e

g

,

the remote inspection of the parts of

street, where excavation would be very

pipework use

one-dimensional

waves

along

with

a

costly

given circumferential mode

the pipe,

so

a

pipe buried

Most studies involv¬

shape

that travel

that the energy per cross-section

is

only

constant

Recently guided circumferential or nonaxisymmetnc waves were studied for hol¬ low cylinders in order to access the remote side of a tube that can only be reached from one side ([33], [64]) This leads to a two-dimensional propagation problem along the curved surface of the tube and shows similarities to the propagation of guided waves in plates, where beamspreading and circular wavefronts have to be considered For

guided

modes metric

can

waves

exist

in

plates not many studies exist Two types of Lamb wave isotropic, homogeneous plates, either symmetric or antisym¬

in

The first symmetric mode describes

a

longitudinal

wave, while the first

Introduction

3

antisymmetric mode contrast to the bulk

can

be

physically

seen

used

waves

as a

bending

motion of the

UT, the propagation of these modes

plate

In

disper¬ frequency This

in

is

wavelength and propagation velocity depend on the signal poses some experimental challenges, but can be overcome by using either narrowband excitation or advanced signal processing [24] Due to the two-dimensional propagation of waves in plates the amplitude of the wave decreases with distance from the source This decrease in amplitude can lead to problems with the signal to noise ratio, as the wave travels from the excitation sive,

l e

,

distortion of the

transducer to the defect and then further measurement methods that

etry

are

are

Guided

advantageous

on

to the measurement

spot Therefore

sensitive to small excitations like laser mterferom-

bending

waves

were

successfully employed for plates [65] The method

the measurement of the material properties of composite has also been proven

testing of

in

the

large structures,

as

laboratory to be promising for compared to classical UT [59]

the nondestructive

Transducer

O^

Fig

o

Qn

Schematic

1 2

showing

from the rivet

a

line of rivets connecting two

O

plates,

with cracks originating

holes, possible transducer localization, and direction of incident

wave

Applications

that

oil tanks

and

planes

[14]

consist of

and fasteners

readily to mmd are the detection of corrosion patches in fatigue problems in aeroplanes [50] The fuselage and wmgs of large plate-like parts, which are connected with lines of rivets come

Manual scanning around each rivet hole

and therefore cost-intensive,

might

be

gained by being

as

it

able to

increases

perform

is

very

time-consuming

the downtime of the

such checks

plane Much automatically over large

4

Introduction

parts of the

structure with the

use

of

guided

waves

Placing

transducer

a

on

the

guided wave can be excited that interrogates the whole line of rivets (Fig 1 2) Alternatively transducer arrays may be fabricated that allow control over the propagation direction of the excited wave, checking different parts of the structure consecutively [32] Due to the rather low cost of piezoelectric transduc¬ specimen,

ers one a

way

a

could also think about integrating the transducers into the structure smart structure may be

the structure eliminated can

is in service

or

at least the

This

fabricated, performing the damage checks while

Part of the

period,

mandatory

checks for

an

aircraft

with which such checks have to be

might be performed,

be increased

Aircraft

are

not the

only possible application,

for nondestructive testing

is

but

are

a

prime

target

as

the need

well established and the increased need for cost-effi¬

ciency leads to Due to the

places

longer service life of the aircraft and demands less down-time optimized but complicated design of aircraft, many difficult to access

exist like the interior of the fuel tanks

measurement

sonnel tems

However, if

can

be

in

the wmgs

Automated built-m

systems will reduce the laborious and hazardous

friendly, proven, and new designed, application markets

the testing of small scale electronic

devices,

checking by

per¬

cost-efficient measurement sys¬

user

like the automotive

as

used

in

industry and industry,

the computer

might be developed The problem arising from the use of guided waves is the fact that the wavelength is usually of the order of magnitude or larger than the thickness of the structure, and hence large compared to the typical flaw size one aims to detect The large ratio between wavelength and defect length reduces the sensitivity and makes an accurate study of the scattering characteristics necessary, to determine experi¬ mental constraints and gam a well-founded theoretical understanding of the inter¬ action between

where from

wave

change

and flaw This opinion the measured time

is in

contrast to other studies

the

of

[23],

defect

is development the influ¬ to signal changes any attempt In Chapter 5 3 1 it can be ence of the optically visible defect on the scattering seen that without an understanding of the physics, wrong calls may be made as other changes in the setup can also influence the measured signal The model system investigated in this thesis is a circular through hole in an alu¬ minum alloy plate with a notch at an arbitrary angle Specimens with these traits can be fabricated easily and the stress-free boundaries at the hole and notch can be modelled exactly, allowing for adequate repetitions and verification of the measurements and calculations This interesting theoretical problem has only partially been discussed in literature Analytical solutions can be found for the a

deduced, without

in

at

linking

series

the observed

a

Introduction

5

geometrically simpler problem

of the

cavity. Only few, mostly numerical,

scattering

studies exist

of on

guided

waves

at

a

circular

the

scattering characteristics development of a concise

of Lamb

waves at a notch or crack in the plate. The analytical model of the combined scattered field would allow an accurate and fast prediction of the detectabihty of such a defect. Furthermore, the inverse problem might be solved and the defect size evaluated directly.

1.2

Selection of wave mode

3

4

5

6

7

Frequency-Wckriess [Ml-fe !wn|

Fig.

1.3

Typical dispersion relation for symmetric and antisymmetric aluminum alloy plate; frequency-thickness region of an

Lamb

wave

current

modes in

study

below

1 MHz-mm marked.

The

guided

travelling along the structure has a wave mode through the guided waves in homogeneous, isotropic plates, of the Lamb [31] wave modes, either symmetric or antisymmet-

wave

thickness of the structure. For the mode is

one

6

ne

Introduction

The

higher

various

modes

modes have different

only

exist above

wave

speeds

and

wavelengths,

and the

given cutoff

frequency For a simple excita¬ tion like an impact, a multimode signal is induced in the plate, for which the eval¬ uation of the measured signal can be quite complicated Therefore one usually aims at exciting only a single mode in the plate Below the cutoff frequencies of the higher wave modes only three modes can exist the first antisymmetric mode A0, the first symmetric mode S0, and the shear mode SH The dispersion dia¬ grams for the symmetric and antisymmetric modes are shown in Fig 1 3 In a number of studies the symmetric mode S0 was chosen, as there is basically no dispersion for frequency-thickness relations up to 1 MHz-mm As the displace¬ ment is constant over the thickness, a flaw at each depth has the same influence The problem on the wave and can thus be detected with the same resolution associated with this mode is the experimental realization of a single-mode signal [10], as the displacement is m-plane, and quite complicated transducer setups have been employed can

For the scope of this

mode

thesis, the

This mode

a

wave

mode

was

be excited rather

selected

A0

can

transducer and measured using

a

easily by

as

the first antisymmet¬ of

piezoelectric (see Chapter 2) As heterodyne this mode is highly dispersive in the frequency range of interest (Fig 1 3), it is usually avoided However, applying Fourier transform for the data evaluation and studying amplitude and phase variations instead of time of flight measure¬ ments, this poses no problem The dispersive nature of the excited pulse can be further used to measure material properties Applying the known dispersion properties, a desired signal shape in the measurement area can be achieved, e g a contraction of the signal in the time domain The displacement of the antisymmetric mode A0 is a transversal movement of the plate For low frequencies this out-of-plane displacement is a pure bending mode Going to higher frequencies, the effects of shear and rotatory inertia have to be taken into account Different approximative theories exist, describing the propagation and scattering of the flexural waves (see Chapter 3) Defects at or close to the surface have a larger influence on the bending stiffness of the plate than anomalies in the center, and can therefore be more easily detected As fatigue cracks usually start to grow at the surface corner of the holes, the anti¬ ric

means

a

interferometer

,

symmetric mode

is

very sensitive to this kind of defect

Introduction

7

1.3

Previous work

1.3.1

Theoretical work

The literature reviewed here

ject of this concerning

scattering The

is

limited to isotropic,

homogeneous plates, the sub¬ grouped into three distinct areas, guided, flexural waves in a plate, the

The theoretical aspects

study respectively

at

a

can

be

the propagation of hole, and the influence of a crack be described

or

notch

on

the scattered field

theory as the first antisymmet¬ The displacement of the wave is primarily a ric mode A0 of Lamb waves [31] bending of the plate For low frequencies, l e when the wavelength is large com¬ pared to the plate thickness, the propagation can be described approximately using classical plate theory (CPT), taking only bending stiffness and inertia into account For higher frequencies, shear and rotatory inertia have to be considered according to the theory of Mindhn [40] Alternatively, approximate theories can be derived from an asymptotic expansion of the full three-dimensional theory for different levels of accuracy, as done by Niordson [42] A review of theories, describing the motion of waves in plates, are for example given by Achenbach waves can

in

three-dimensional

,

[1],

Graff

[22],

and Viktorov

The scattering of flexural

lytically by

Pao and Chao

using Mmdlm's

hole,

[67] at

waves

an

[45] They

theory

obstacle

in a

solved the

plate

case

of

has been a

analyzed

circular cavity,

of plates, and derived three scattered

waves

ana¬ l e

a

to fulfill

the

boundary conditions at the hole A similar analysis employing Kirchhoff type boundary conditions was done by Staudenmann [57], who also studied the scat¬ tering

study

at different

types of circular inclusions Vemula and Noms did

for thin

Further

and Mindlin type

plates [43] analytical work

exists

on

plates [66],

using

an

optical

a

similar

theorem

the scattering of the first symmetric Lamb

mode

S0 at a circular hole by Pao [44], and McKeon and Hinders [39], who also give a good review of other papers Sih [55] analytically studied the case of a flexural wave incident on a crack The study makes strict assumptions on the ori¬ entation of the crack relative to

pler boundary

Finite element methods

used ural be a

(FEM)

by Paskaramoorthy,

wave

at

expanded

a

propagation direction of the

wave

to achieve

sim¬

conditions

circular cavity

to

the scattered field

[47]

and at

a

a wave

function expansion

crack

a

were

flex¬

Their method

can plate [46] cavities or irregularly shaped the difference in [12] investigated in a

geometries, like

complicated boundary Chang and of a longitudinal wave

more

crack at the hole

combined with

Shah and Datta to calculate the scattered field of

Mai at

a

hole due to two cracks at opposite

8

Introduction

sides of the method mode

hole, also

was

inclusions

Harker

Ymg [68]

ing the Lamb

[26]

waves

posed

the

stable

algorithm

use

of

a

hybrid

FEM

modeling

Cho and Rose

[13]

A

hybrid boundary

study

to

the

element

reflection and

edge

of the lower Lamb modes above the lowest cutoff

conversion

Cylindrical

using

employed by

parallel

to the

for FDM

in

frequency analyzed by Wang and

were

studied the scattering of Lamb

with finite difference methods

staggered grid

a

surface

plate

at

waves

a

crack, simulat¬

(FDM) Madanaga [34]

seismology, resulting

in

a

pro¬

more

Experimental work

1.3.2

experimental papers on the scattering of Lamb waves at an be found Chang and Mai [12] compared their numerical plate experimental results, using a wedge transducer to excite the first sym¬

However, only obstacle

a

few

in a

data with

can

metric mode

S0 Measurements were made at two points with contact type trans¬ They achieved good qualitative agreement between measured and

ducers

calculated time

might nitude

series

and power spectra, but did not get

be due to the

large wavelength

size

an

used, which has the

exact match

same

This

order of mag¬

at the applied frequencies of about 0 5 MHz Malyarenko and Hinders [35] experimentally studied the scattering of the S0 mode at a through hole in a plate in the frequency range between 1 MHz and 2 MHz, below the cutoff frequencies of the higher modes, to reconstruct flaws using fan beam tomography They used longitudinal contact transducers at vari¬ ous positions around the hole and analyzed the arrival times Though both the S0 and A0 modes were excited, only the arrival times of the S0 mode were used, as this mode propagates faster and is nondispersive Chan and Cawley [11] investigated the propagation of higher Lamb modes in an attenuative plate By selecting the angle of incidence of a water coupled broad¬ band transducer, desired higher Lamb modes in a polyethylene plate were excited and their group velocity and attenuation were measured Alleyne and Cawley [4] as

the

transducer

studied the reflection and transmission of the steel

plate experimentally,

S0

and

Aj

modes at

a

notch

in a

using similar transducers and comparing their results

to numerical calculations using FEM

The

[57],

experimental part

of this thesis builds

who studied the scattering of

through

hole

in an

on

the PhD thesis of Staudenmann

structural, low-frequency bending

waves

isotropic plate experimentally and compared his results

culations using CPT His

study

focused

on

holes with

a

radius

at

a

to cal¬

approximately

as

Introduction

large

as

the

9

wavelength

and did not incorporate cracks

or

other asymmetries He

used

frequencies in a very narrow spread between 9 kHz and 15 kHz, resulting in This is large compared to the plate thickness, a wavelength of about 30 mm avoiding higher order effects such as rotatory inertia and shear In contrast to his work, a number of components of the experimental setup were replaced, allowing a far better accuracy of the measurements and the use of a much wider frequency range for the excitation

1.4

Content of the thesis

Chapter 2 Building on In

setup for the

improved,

the method of measurement used

in

this

previous research at the Institute of measurement of

guided

waves

is

study

further

automated,

and the measurement range îextended Flexural

excited and measured with excellent

large frequency

range the scattered field

with and without

description

repeatability

a

defect

is

measured

of the influence of

a

defect

on

a

described

is

and

waves in

signal

to

detail

measurement

the scattering

its precision

plates

noise

grid

can

be

For

ratio

around

This allows the accurate on

in

Mechanics, the experimental

a

a

hole

geometrical

Different excitation

transducers, like electromagnetic acoustical transducers and line

excitation by piezoelectric ceramic plates are investigated Broadband excitation measurements at single points are introduced to achieve a faster measure¬

custom cut

and

ment

Chapter

3 gives

an

overview

propagation of flexural the

physical

effects

of the approximate theories used to describe the

waves in

they

homogeneous, isotropic plates

describe

The scattering at

and links them to

circular hole

in a plate is plate theory, Mmdlm's theory, and an asymptotic expansion are compared to experimental results Good agreement between the measurements and the analytical calculations is obtained The validity of the different approximations is studied In Chapter 4 analytical attempts at solving the scattering problem with a crack at A numerical solution employing an arbitrary angle of the hole are presented finite difference methods is implemented Mmdlm's equations of motion are discretized on a staggered Cartesian grid and the stress-free boundary conditions at the hole and notch are introduced The wave propagation is calculated by explicit

calculated, and the different approaches

time

a

using classical

integration Consistent agreement with the experiments

tion of all parameters

defect

is

studied and

the

relations

describing geometrical accurate predictions can be made

is

The

found for

a varia¬

detectabihty

of

a

10

In

Introduction

Chapter

5 the

application

of the method to nondestructive testing

is

described

Experimental more realistic cases are studied, involving complicated geometries and fatigue grown cracks Multiple scattering at a line of holes, symbolizing a line of rivets in an aircraft fuselage is measured Fast measurements at single points

are

detection

made using broadband excitation, minimizing the time for defect In

cooperation with the fatigue engineering

space, Emmen the

fatigue holes

cracks

in

applicability

aluminum specimens

generated by cyclic

are

center of RUAG Aero¬

of the measurement method for the detection of

tensile

machine The influence of the cracks to be well described

by

is investigated Fatigue cracks at circular loading in a servo-hydraulic material testing on

the scattered field

is

measured and found

the numerical model using finite difference methods An

on-line monitoring of the crack

length during the cyclic tensile loading is imple¬ experimentally Good correlation between measured and calculated change in signal and the optically measured crack length is found, allowing an

mented

evaluation of the defect

Chapter

6

and gives

sums an

experimental

size

up the close agreement between measurements and calculations

outlook method

on

possible

further improvements and

applications

of the

2

Measurements

2.1

Introductory

For the scope of this

flexural wave,

remarks

thesis, the first antisymmetric Lamb This

employed

was

wave

mode has

a

wave

mode

number of

A0, 1 e a advantages,

pointed out by Sayir [53] The excitation by piezoelectric transducers is well repeatable, allowing an averaging of the measured signal to increase the signal to noise ratio Defined time functions can be prescribed, giving control over the fre¬ quency content of the excitation signal The dispersive nature of the excited pulse can be further used to measure material properties [65] Applying the known dis¬ persion properties, a desired signal shape in the measurement area can be achieved, e g a contraction of the signal in the time domain The employed measurement method and the experimental setup have been devel¬ oped over the years at the Institute of Mechanics Initial measurements were made by Goodbread [21], who studied the mechanical properties of spongy bones using low frequency vibrations and guided waves Basic methods devel¬ oped by him, namely excitation using piezoelectric transducers glued to the spec¬ imen

and measurement with

used for the

of

study

agation of flexural

a

a

laser

interferometer, have been further refined and

variety of problems Kreis and

waves

in

Sayir [29]

studied the prop¬

thm

transversely isotropic plates Veidt and Sayir plates from the measure¬ Dual [17],[15] studied a range of frequencies

determined the material parameters of composite

[65]

ment of the

phase velocity

over

in anisotropic tubes, exciting desired modes guided axisymmetnc and measuring the phase velocity for a determination of material parameters Dual et al [16] applied the measurement method to the detection of defects in beams A flexural wave was excited at the end of an aluminum bar by a piezo¬ wave

modes

electric transducer and the reflection of the flexural sion

to

a

longitudinal

Staudenmann rods and

large ness

[57]

=

plates 1

worked

at on

a

notch of varying

mm)

wave

depth

and the mode

was

scattering of

(hole

radius r0

for the

narrow

=

15

mm)

low-frequency

in a

flexural

thm aluminum

frequency spread

conver¬

studied

the propagation and scattering of flexural

He studied the

circular hole 2h

wave

waves in

waves

plate (plate

from 9 kHz to 15 kHz

at

a

thick¬

(wave¬

length X from 33 mm to 25 mm) In comparison to his work, several components were replaced, allowing a far better accuracy of the measurements and the use of Measure¬ a much wider frequency range for the excitation and measurement ments were made for different combinations of hole radius, plate thickness, and wavelength The influence of flaws on the scattered field was studied

12

Measurements

2.2

Experimental setup

The setup consists of modular components controlled mterface from

a

central computer, shown

following sub-chapters two different

The

aim

detection,

a

Fig

2

via

calculations,

experimental parameters a fatigue in a

tensile specimen with

a

were

large

crack

a

GPIBin

the

twofold and therefore

employed

To compare the

aluminum

generated

an

Demodulator

Laser-

is

wide range For the

The geometry of the specimen resembles that of

Interferometer

Lab View and

1, and further described

of the measurements

types of specimens (Chapter 2 3)

measurements with theoretical

choice of the

in

aircraft

plate allows the study of the crack

at the hole

was

used

fuselage

Bandpass

Oscilloscope

Filter

Aluminum Plate

Fig

Schematic of the

2 1

In the

laboratory,

experimental setup

the setup

out vibrations transmitted

was

by

placed on an optical table (Fig 2 2) to dampen building However, due to the selected fre¬

the

(higher than most external noise and vibration sources) and averag¬ signal, this precaution is not necessary For the experiments performed at the fatigue engineering center of RUAG Aerospace, Emmen, Switzerland (Chapter 5) no optical table was available Good qualitative measurements were made in the rather rough environment of an airplane hangar and only a slightly higher variation of the measurements was found quency range

ing of the

Measurements

Fig.

Experimental setup

2.2

The

13

experimental

with aluminum

plate

on

optical

sequence goes from excitation

over

table in the

laboratory

at ETH.

measurement to data anal¬

signal (Chapter 2.4) is generated as a voltage signal, ampli¬ fied and applied to the transducer (Chapter 2.5), where it is converted into a flexural wave in the plate. The wave propagates along the structure and is scat¬ tered at the obstacle. The scattered field is measured using a heterodyne laserinterferometer (Chapter 2.6). The voltage signal is bandpass filtered, stored and averaged in an oscilloscope. The function generator triggers the oscilloscope, so

ysis.

The excitation

that excitation and measurement start at the

same

is transferred to the PC and evaluated there

time. The measured time series

Using fast Fourier (FFT), amplitude phase frequency f0 of the excitation signal are extracted for each measurement point. They can either be displayed on a circle around the hole or as a pseudo-colored surface around the hole, shown in Fig. 2.3. At the free boundary of the hole the incident wave is scattered and a high amplitude directly at the boundary results. A flaw like a transform

crack

peak

or a

in

and

notch introduces additional free and

using

Matlab.

values at the center

boundaries, from which result

of the scattered field.

these

a

local

amplitude change Through good understanding of the geometry of the scattered wave and the influ¬ ence of a notch or a crack can be gained. The characteristics of the scattered field around a hole are further described in Chapter 3.6.

ments

a

a

measure¬

14

Measurements

a

E

2

<0, *

30

2ft

Peak duc

m

to

amplitude

the notch

y-ass

Fig.

Measured

2.3

2h

=

amplitude

1 mm, r0

of scattered field at

10 mm,

=

f0

=

2.3

Specimen Geometry

2.3.1

Aluminum

100

kHz, X

a =

hole with 10 mm, 2

a

notch in

mm

a

plate

notch at 45°.

plate

plate with a size of 1000 mm by 1000 mm, shown in Fig. 2.4 experiments to ascertain the accuracy of the measurement method and for the comparison of the experimental results with the theoretical calculations. Variations in the experimental parameters, like different ratios between hole radius, plate thickness, and wavelength, and different positions of the flaw relative to the propagation direction of the incident wave can be studied. The large size permits a time separation between the wave scattered at the hole and the part scattered at the plate boundaries. A piezoelectric disc with a diameter of 10 mm and a thickness of 1 mm acts as a point source for the wave, which propagates radially outwards. The distance y2 between the transducer and the hole is selected much larger than the hole radius. Therefore the wavefront can be assumed to be a straight line when the wave reaches the hole. This allows for a simpler comparison to the theoretical calcula¬ A

large

was

aluminum

used for the

tions.

Furthermore,

even

if

more

to transverse contraction of the

than

one

mode is excited

piezoelectric disc),

(e.g.

the modes

the can

S0-mode due be separated

Measurements

15

due to their different propagation

speeds

and thus arrival times at the hole

distance from the transducer and hole to the

plate

selected

are

achieved area in

Any

so

wave

that

time

a

plate boundary

separation between the different pulses

reflected at the

plate

has

is

boundaries reaches the measurement

the vicinity of the hole after the incident

piezoelectric transducer,

The

and the width of the

pulse,

directly from the pulse duration minimum plate size is

arriving

From the calculation of the

passed

and the propagation time for the reflected waves, the derived

min(2y1>2y3>x)>cgN/f0, with group quency

velocity

c„

and N

(2 1) of the excitation

cycles

pulse

with

a

center fre¬

f0 x

=

1000

mm

iL yj

=

350

mm

1 -

i

Y2

=

300

£

Piezo

mm

Hole(s) 4f

r^A v^

"kS^P \^/

r*--

dH y3

=

350

=

100

mm

mm

1 '

Fig

2 4

Geometry of plate

specimens

The plate material is an aluminum alloy (Alusuisse Anticorodal 110, EN AWAlSilMgMn T6), having a Young's modulus E of 6 9-1010 N/m2 and a density p of 2700 kg/m Poisson's ratio V is assumed as 0 31 Plates with a thickness 2h of 0

5, 1,2, and 4

was

drilled

radius and

mm were

used One hole with

a

radius r0 between 0 5 and 40

mm

through the plate Different combinations of plate thickness, hole wavelength were studied to ascertain the validity of the approximate

16

Measurements

theoretical calculations hole radius

of 10

horizontal line

The standard For these

mm

(distance

case

was

parameters

between hole centers

a

plate

dH

thickness of 1

plate

a

=

100

mm

with three holes

mm)

was

and

a

on

a

manufactured

study the influence of multiple scattering plate is suspended vertically to avoid static bending To simulate a defect at the hole boundary, a fine saw blade is used to cut a notch through the thickness of the plate The notch has a width of about 0 2 mm, a blunt tip, and lies at an angle (Po to the vertical (see Fig 2 1) Notches with a length of up to 4 mm at 0°, 45°, 60°, 90° and 180° were investigated Initially the surface of the plate around the hole was prepared to allow for a good reflection of the laser beam As the measurements were run automatically and over night, it had to be ascertained that the laser beam was reflected well at every measurement position Goodbread [21] and Staudenmann [57] applied retroreflective tape, as the laser interferometer used by them was very sensitive to changes in the reflection of the laser beam One of their problems was the inaccu¬ rate amplitude measurement of the laser interferometer The measurement of the amplitude was very dependent on the laser beam reflection, leading them to rely more on the phase measurement However, in order to describe the influence of a notch or crack on the scattered field, it is best to use the complex description (amplitude and phase), as shown in Chapter 4 5 After some studies it was found that the use of the retro-reflective tape was not necessary with the Polytec vibrometer used in this study With this interferometer it is possible to achieve a suffi¬ cient reflection directly off the aluminum plate Even a polishing of the plate to

The

surface, tried for

some

tion of the laser beam

of the measurements,

is

achieved off the

points of the measurement

2.3.2

Tensile

Fatigue

of

grid

a

not necessary

Maximum reflec¬ at

a

few

scratch

specimen

cracks

sharp edge

with

is

untreated, dull surface, except

a

were

generated

in

crack and crack

The geometry of the specimen

tensile specimens to

closure, which was

can

study

the influence of the

not be achieved with

selected to resemble

a

a

fastener hole

notch in

the

fuselage of a fighter jet used by the Swiss Air Force Cyclic tensile loading in a servo-hydraulic testing machine (MTS 810) with a maximum tensile stress of 135N/mm2 was applied to the specimen After 20'000 to 300'000 cycles, depending on the stress level, a fatigue crack developed at the hole boundary The crack always started with a quarter-elliptical form at the front or back surface on

Measurements

17

the side of the hole 2

(Fig

trated face a

5)

through the

are

The crack

monitored using

was

about the

same

center of RUAG

the front surface and withm the hole

45°)

fatigue testing

Aerospace, Emmen,

(short)

same

length rate

made to

was

When the crack has pene¬

microscope

at the

increase

2 mm,

Tensile specimen

2 5

Fig

(0

and

The tensile

develops

on

optical

thickness of the specimen, the

small starter notch

crack

length

an

was

the front and back

on

For

specify

some

sur¬

of the specimens

the location where the

made at the

fatigue

engineering

Switzerland

with piezoceramic

plate, hole,

and crack

in

polished

surface

Two measurement

thick,

40

hole with

mm a

series

were

wide and 250

radius of 3 25

made

mm

long,

first, standard

specimens 3 17

made from Al-2024 PL-T3,

drilled

mm was

Due to the rather short

In the

through

were

mm

used A

the middle of the specimen

no time separation between the length reflected the at pulse pulse clamping jaw could be achieved The specimen moves slightly in the clamping jaw during the first few thousand loading cycles This had a measurable influence on the crack length monitoring measurements, described in Chapter 5 3 1 For the second measurement series, specimens 500 mm long, made from Al-7075 PL-T6, were used Al-7075 PL-T6 has a Young's Modulus E of 7 1-1010 N/m2, a density p of 2796 kg/m3, and a Poisson's ratio V of 0 33 Two of the specimens had slightly smaller holes (radius

first incident

r0

=

3 12

and the

mm)

specimens,

a

for measurements with

line transducer

the specimen and avoids 40

mm

wide,

the hole

of the specimens,

8

mm

long

was

multiple and 1

a

fastener Due to the limited width of the

used to generate

mm

thick

an

that propagates A

along piezoelectric plate

glued to the specimen 50 mm from plate with the plate strip and the amplitude modulation over the width of the was

The interaction of the piezoceramic

free surfaces at the sides leads to

a wave

reflections at the sides

18

Measurements

The

specimen.

amplitude

of the excitation

pulse,

at the sides is

additional

higher and, depending

amplitude

maxima

Before and after the measurement series in Emmen,

over

on

the

the width

experiments

frequency

can

develop. speci¬

with the

were performed in the laboratory at ETH Zürich. A mechanical tensile loading apparatus (Fig. 2.6) with defined mounting and tensile loading of the specimens up to 20 kN was rebuilt in the workshop of the institute. The applied force is measured using a strain gauge, calibrated with normed lead weights.

mens

Fig.

Mechanical

2.6

2.4

Excitation

A sinusoid

signal

on

a

with

short tensile

a

Hanning

window

was

usually

chosen

short-time, narrowband signal with the

around the center

specimen

and

laser

Signal

multiplied by

to achieve

loading apparatus optical table.

tensile

interferometer

as

the excitation

energy concentrated

frequency f0 (Fig. 2.7a, b). The signal has to be short in time to (Chapter 2.6) cutting out the interference of incident wave and wave scattered at the hole and the notch from the reflection at the plate boundaries, arriving some time later at the measurement spot. The arrival times of the different pulses are calculated from the theoretical group velocities. A nar¬ row bandwidth of the excitation pulse avoids extensive signal distortion due to the dispersive character of the A0 mode. The energy of the pulse is concentrated around the center frequency f0 and a good signal to noise ratio is achieved. Most allow

a

time window

Measurements

19

of the experiments

were

In order to improve the

carried out with the

efficiency

narrow

sary time, well-controlled broadband excitation

the

prescribed

ments,

a

time

signal

bandwidth excitation

of the measurements and to reduce the

of the excitation

linear sweep from 20 to 100 kHz

was

2

neces¬

implemented by adapting

In contrast to previous

(Fig

pulse

7c, d)

was

used

as

measure¬

the excita¬

signal to measure the amplitude of the scattered field at different frequencies simultaneously [20] The linear sweep started with the low frequencies to achieve a contraction of the wave pulse due to the dispersion, resulting in lower group velocities at the lower frequencies The signal is generated in a programmable function generator (Stanford Research Systems DRS 345) and then amplified to 200 V peak to peak (Krohn-Hite KH 7500) tion

0 05

0

c)

Fig

2 7

01

Time

tmsj

015

0 2

50

Frequenoy [kHz]

Typical excitation signals a) sinusoid in a Hanning window, center frequency fo 50 kHz, 5 cycles, b) amplitude spectrum of sinusoid in a Hanning window, c) linear sweep from 20 to 100 kHz, d) amplitude spectrum of linear sweep =

20

Measurements

2.5

Excitation transducer

2.5.1

Piezoelectric transducer

The material of the ceramics a

piezoelectric

source

transducer

27, polarized for thickness

piezoceramic disc with

point a

Pz

for the

wave

a

selected

was

extension mode

diameter of 10

mm

and

a

Ferroperm

as

For the

plate

thickness of 1

piezo-

specimens

mm

acts

as a

propagating radially outwards For the tensile specimen

plate 40 mm by 8 mm, 1 mm thick was selected as a line trans¬ plate strip was cut from a larger plate (50 mm by 25 mm), using a Different sizes of the piezoceramic plate were evaluated experimen¬

piezoceramic

ducer The wafer

tally,

saw

but did not result

The transducer

in a more

uniform

wave in

the tensile specimen

glued to the plate using a two-component fast cure epoxy adhesive (PermaBond Double Bubble) Initially, some transducers got loose dur¬ ing the cyclic tensile testing, due to the high shearing stresses in the adhesive layer Care had to be taken to achieve a uniform and thick enough adhesive layer On the other hand, a thick layer results in a more complex transfer function of the transducer, tem

No

as

was

the piezoceramic disc and the adhesive act

backing

transfer function The

wires were

mass

was

used,

as

it further

Sufficient excitation

affixed with

a

increases

amplitude up to fast-drying bond (HBM

a

X

as

the

a

mass-spring sys¬

complexity

few |im

60)

or

was

of the

achieved

soldered to the

piezoceramic

When

voltage is applied to the piezoelectric transducer, the disc contracts and expands This generates a vertical force to the plate surface and excites primarily the first antisymmetric mode A0, as the resulting normal stress in the plate is anti¬ symmetric For the frequencies used in this study, the energy transferred to the longitudinal mode S0 and the shear-horizontal mode SH is negligible Since we operate well below the cutoff frequencies for the higher wave modes, only the desired mode A0 is excited The approach shown here is feasible for frequencies below the cutoff frequencies, as no selection between different modes is neces¬ sary, avoiding the need for prescribing the wavelength at a given frequency by an angle of incidence as in classical UT The excited frequencies of up to 200 kHz are well below the eigenfrequencies of the piezoelectric disc, so that a linear transfer curve is achieved [53] Only in one case the symmetric mode S0 was also excited, but even at the rather high frequency of 200 kHz the amplitude of the S0 pulse was only about 10% of the amplitude of the An. pulse The transfer function of the circular piezoelectric disc was studied in a term project (Semesterarbeit) with D Profunser [48], supervised by MB Sayir and

Measurements

P Fromme

21

For

a

simple

theoretical

model, the piezoelectric disc

the thickness direction due to the

rigid, contracting only

in

adhesive

viscoelastic and the force

area

is

assumed

underneath

is

as

discretized

Assuming

a wave

applied by

is

assumed

applied voltage

as

The

the transducer to the

propagating radially outwards,

the

applied stresses and displacements are calculated Close agreement with an experimentally measured transfer curve up to about 100 kHz can be seen in Fig 2 8 For a better agreement at high frequencies, a more accurate modeling according to [15] would be necessary Measured transfer curves of different piezoelectric discs showed close agreement as long as care was taken that the adhesive layer had about the same thickness

4Ù an

S

î»

Fig

2 8

Transfer function of 0 5

mm

a

piezoelectnc transducer (Pz 27, d 10 mm, h plate measured (dashed), calculated (solid),

thick aluminum

=

=

1

mm) on [48]

from

a

22

Measurements

Electromagnetic

2.5.2

acoustical transducer

(EMAT)

magnetic field circular

permanent magnet

eddy

Fig

current

Principle

2 9

density

J Lorentz force

for the excitation of transversal

acoustical transducer

Electromagnetic contact

the

means

eddy

wire

current

=

acoustical transducers

density (J),

JxB

a

waves in

were

in

(dF)

plates

broadband point

investigated

the

(B)

using

an

electromagnetic

source

of

a

as an alternative, nonpermanent magnet and

plate by an alternating (dV) of the plate

current

is

in a

per volume

dV

The Lorentz force

most

inducted

Lorentz force

(2 2) vertical to the

wave, if the direction of the

(Fig

as a

of excitation The magnetic field

coil, generate dF

(EMAT)

plate

surface and generates

current and

magnetic field lines

a

transversal

eddy m-plane design ideas for EMATs have been studied in literature [62], prescribing the wavelength by a meander coil Here a different approach is

2

9)

are

Different

studied, where

a

broadband, non-contacting point

source

is

built, that

can

be

positioned on one side of a plate to generate a transversal wave propagating radi¬ ally outwards Circular wire coils are glued to one side of a circular permanent magnet (Fig 2 10 left) The magnet is radially polarized, resulting in primarily radial magnetic field lines The eddy currents inducted by the wire coil in the plate are circular and independent of the angle

Measurements

According in a

[60],

to

circular

23

the

J0(r,z)

to the

to current I

°°

2

|Vkb/a

-L-R-l

=

current density in the plate due plate surface, is calculated as

tangential eddy

coil, parallel

wire

a

J1(k)J1(kr/a)x

„

0

,

(k|ir

(2 3) q(d-z')/a

s

.

+

q)

,,

e^

(k|ir

+

q)

qd/a

eH

-q(d-z')/a

.

e

-(k|ir-q)

,2

.

,,

,2

,,

-qd/a

eH

-(k|ir-q)

Jj is the Bessel function of the first kind, [i^ the relative permeability of the plate material, d the plate thickness, and b the distance of the wire coil to the plate sur¬ face The origin of the coordinate system

is

z' defines the relative

plate

z'

=

z

z

coordinate

in

the

selected

in

the center of the

b

-

wire

coil

(2 4)

With

p

as a

=

Vl^Cûoa

function of total

plate material, integrand k as

q

=

The skm

permeability |i, angular frequency

and radius of the

wire

coil a, q

V(k2 ip2) depth

of the

—

eddy

is

CO,

defined

conductivity G dependence

in

of the of the

(2 6)

+

C0|IG

is

(2 5)

current

density

(2 7)

inversely proportional to the frequency of the alternating current and for the frequencies varies from a quarter of the plate thickness to the plate thick-

studied

24

Measurements

The

of the inducted

phase

relative coordinate z' from

Eq (2 3)

eddy

The total

current

eddy

also

density

current to

varies

depth

with

z' at radius

frequency r is

and

calculated

as

z'

Jtot(r,z')

Assuming in

the

dV

for

a

fj(r,ç)dç

(2 8)

constant magnetic field B in the plate and neglecting the inductance coils, the transfer function was calculated in the term project of Dieter

a

wire

Profunser

=

dV

[48] Substituting rd0

=

right angle

in

cylindrical

coordinates

dz'

dr

(2 9)

between magnetic field lines and

resulting eddy

current gives the

vertical force per volume

dF(r, z')

=

J(r, z')

and the normal stress

G

B

rd0

dr

by integration

dz',

over

the

(2 10) thickness

plate

d

f

d

=

° =

Calculating in a

dz'

=

°(0

dF(r,z')

_

rAt* rd0

R B

Ar dr

the summed

eddy

current

f J(r>z') density

dz'

for all

loops

rotational symmetric stress distribution and thus

pagating

wave as in

Chapter

2 5 1 for the

(2 H)

a

piezoelectric

of the

wire

coil results

rotational symmetric pro¬ transducer

Measurements

25

The transfer function to the is

in the center underneath the transducer

displacement

given by wn

2%\

1

8coVphD

'EMAT ((D)

(2.12)

JA(r)[J0(Kr)-iY0(Kr) For

a

single

A(r)

wire coil

=

B

•

f

J0(iKr)

+

e~kb/a J^kJJ^kr/a)

x

is calculated

A(r)

-!-§- f

+

iY0(iKr)]rdr

as

(2.13) .,

(k|ir

Fig.

2.10

,2

.

(knr

resulting

-q(d-z')/a -(k|ir-q)e^ .,

q)e^

.,

The

q(d-z')/a

,

.

+

+

qd/a

q)

eH

,

transfer function is calculated

Left: Broadband

plate, from

point

source

[54];

Right: waves

in

Broadband a

tensile

e

using

a

kdk

Matlab program

dz

[48].

EMAT for the excitation of transversal

circular coils from thin wire

transversal

-qd/a

.2

.,

_(knr-q)

glued

line

on

transducer

specimen.

waves

in

a

radially polarized permanent magnet, EMAT

for

the

excitation

of

26

Measurements

In the term

sured

Good

were built (Fig 2 10 left) and their transfer function mea¬ qualitative agreement with the theoretical model can be seen in

To achieve

2 11

Fig

project of B Schmid [54], supervised by M B Sayir and P Fromme,

transducers

matching

a

broadband line

source in

the tensile specimens,

an

alter¬

design, shown in Fig 2 10 right was used [63] The amplitude of the excited wave is directly proportional to the strength of the magnetic field and the alter¬ nating current in the wire loop A power amplifier (ENI 1140 LA) was used to drive the EMATs The resulting displacement in the plate is rather small and above 100 kHz no useful wave pulses could be excited Therefore and to avoid positioning inaccuracies of the transducers, piezoelectric transducers were used nate

for the measurements of the scattered fields

I

}

I

Companson of measured (left) and calculated (right) transfer functions for source EMAT at different radii, from [54]

2 11

Fig

2.6

Measurement and data

The wavefront

facilitating

can

be assumed to be

consists of two

a

straight

line when it reaches the

hole, and

a

scattered

boundary layer

wave is

wave

is

hole,

scattered at the stress-

generated

The scattered

close to the hole and

a part propa¬ gating radially outwards from the hole In the vicinity of the hole, incident and

wave

scattered

wave

surements

velocity

overlap

This

of the

parts

point

handling

the theoretical simulation The incident

free boundaries of the

a

is

A0

in

a

time,

due to the mode

so

that

length

only

a

single pulse is visible in the mea¬ signal and the low group selected is large enough that the

of the excitation

The specimen

size

Measurements

27

reflections from the

ary

reflections,

as

boundaries reach the measurement

plate

This way

[Eq (2 1)]

time

a

shown

The scattered field

on

a

in

2

Fig

12,

is

time later

achieved

grid

measurement

available

area some

separation between the scattered field and the bound¬

0 1

mm

This allows

field,

as no

ducer

is

implicit

made

a

laser

around the hole

recorded using

is

a

(Polytec OFV 303 / OFV 3001) The demodulator output is a voltage signal proportional to the velo¬ city of the out-of-plane component of the displacement of the plate surface The measurement spot is defined by the laser beam diameter, which is well below commercially

heterodyne

measurement of variations

point-wise

average

interferometer

over a

rather

The laser interferometer

is

moved

tioning system (Aerotech Unidex 12), allowing the defect without disturbance

largest

cause

tive to the hole

0 1

a

parallel

an

inaccurate

center, which could

to the

measurement

The measurements

of variation due to

are

well

the scattered

plate

in

on a

posi¬

the vicinity of

repeatable,

with the

positioning of the laser beam rela¬

be achieved with

only

in

surface of the measuring trans¬

large

an

accuracy of about

mm

10

sspd

whwî

0

pfetll ImuikImw

seailwed El hole

05

Time

Fig

Measured time

2 12

wave

Two

types of

the hole

a

signal

15

1

with time window to cut out

scattered at hole from reflections at

measurement

radial

grid

was

cles around the hole and

grids

used,

were

2

Ims]

plate

overlap

of incident

wave

and

boundaries

used For measurements

moving the measurement

spot

in

on

the vicinity of concentric

cir¬

recording a time signal every A(p degree on radii Ar apart For the experiments on the plate with three holes and propagation charac¬ teristics in the tensile specimens, a Cartesian grid with step size Ax in the hon-

28

Measurements

zontal and

Ay

the vertical direction

in

used

was

The

voltage signal is bandpass filtered (Krohn-Hite KH 3988) around the center frequency f0 and averaged in a digital storage oscilloscope (LeCroy 9304A) The function generator triggers the oscilloscope, so that excitation and measurement start at the

The measured time

time

same

puter for evaluation using code series

with

written

10 000 values

usually

off the reflections caused

by

the

is

in

then transferred to the

series are

stored A time

plate boundaries,

windowing

grid a time applied to cut

is

which contain

no

information

about the scattering at the hole The arrival times of the different

pulses

lated from the theoretical group

(FFT)

and the

soid

is

around

and

amplitude

These values

are

the

phase values equivalent of the

assumed for the incident a

velocity

hole with

a

The measurements

notch

is

wave

shown

in

com¬

Matlab At each point of the

Fast Fourier transform

at the center

theoretical

frequency f0

results, where

The

amplitude

Fig

2 3

of

a

are

an

are is

calcu¬

applied

extracted

infinite

sinu¬

scattered field

typical

automatically, using a program written in Labgrid, pulses and frequencies, and further measure¬ ment parameters can be selected by the user A measurement on a narrowly spaced grid with several excitation frequencies was usually run over night From the measurement of the whole scattered field a good notion of the geometry of View The

were

different excitation

the scattered

wave was

scattered

field,

From the

complex

However, such

obtained To characterize the influence of

measurements

were

wave

scattered field due to

a

are

too

on

was

the cut

field, the geometry and propagation

time-consuming

testing purposes notch

or

Such measurements

broadband excitation and

notch

scattered at the notch could be obtained

measurements

tion for nondestructive

a

made before and after the notch

difference of the scattered

characteristics of the

sufficient

made

are

crack,

were

shown

a

For

a

measurement at

made for the in

in

Chapter

the industrial

fast detection of

plate

5 2 3

one

applica¬ changes in the

point

or on a

line

is

with the three holes using

3

Scattering

3.1

Geometry

at

of the

circular hole

a

scattering problem

The

simplest case of the scattering problem is a circular cavity through the thick¬ of the plate This represents an undamaged rivet or fastener hole and can be studied theoretically as well as experimentally with great accuracy and repeat¬ ability The measured scattered field of a flexural wave around the hole can be compared with analytical calculations The scattering at a hole with some dam¬ age at the boundary is studied numerically in Chapter 4

ness

Fig

Geometry of the scattering

3 1

For the

analytical investigation,

sinusoid, propagating

w,

=

in

the

at

the hole

the incident flexural

direction, and

x

is

given

is

Cylindrical

orr

a

=

as

shown

the stress-free

or(p

scattered

=

orz

wave

=

in

Fig

boundary

0,

must

3 1

r

occur

=

can

selected

infinite

by

be fulfilled with

theory

chosen at the center of the circular cavity coordinates

(r,(p)

are

introduced

conditions at the hole

(pe [-Jt,

r0

Jt],

The introduced scattered

(3 2)

waves propagate radially angular dependence The boundary condi¬ different degrees of accuracy, depending on the

outwards from the hole and have tions

as an

(3 1)

The origin of the coordinate system

satisfy

taken

U,e

with radius r0, To

wave is

to describe the

an

wave

propagation

Scattenng

30

3.2

Lamb

at

a

circular hole

propagation

wave

The

dispersion relation for guided waves in plates was derived by Lamb [31] Following the description of the work by Graff [22], the possible modes in a plate are either shear-horizontal, symmetric or antisymmetric Their dispersion relation can be deduced either from considering multiple reflections through the thickness of the plate or by the formulation of a standing wave mode [1] For the symmetric or longitudinal modes, the dispersion relation is given by

tan

(k2--u22)2

pu, h) =

4k2D,u,

tan(D2h)

and for the antisymmetric

transversal modes

or

by

(k2-o)22)2

tan(u2h)

(3 4)

=

4k21)ll)2

tanCt^h)

Lamé constants X

=

-Ev/(2v

ulus E and Poisson's ratio V,

plate

thickness 2h

wave

velocity

sion

wave

u1

k1

=

-

The cutoff For the

fc

For the

fc

k, k

,

c2 =

x>2

are

=

used

J\i/p,

co/c,, =

k2

frequencies

-

2 + v

1

) and \i

Compression

wave

are

for the

CO,

wave

number k number

E/(2(1

=

angular frequency

wave

k

-

=

of

phase velocity

velocity

co/c, the

v)), with

+

c1

wave

Jjp,

p

=

shear

modes

are

wave

determined

andfc

=

^q,

antisymmetric modes the cutoff frequencies

=

4p;P.

P

=

0,2,4,...

J(k

+

and

2|i)/p, shear

k2

=

co/c2, and

defined

wave

1,3,5,...

density p,

c,

number of the compres¬

symmetric modes this gives the cutoff frequencies

=

=

mod¬

Young's

andfc

=

^q,

q

=

by considering

k

—>

as

0,2,4,...

(3 5)

1,3,5,...

(3 6)

are

q

=

0

Scattering

at

circular hole

a

The fundamental modes

S0

and

A0

have

Eq (3 5)

Eq (3 6)

The other

a

cutoff

frequency

wave

modes

of zero,

as

can

be

only above a cer¬ the Aj-mode has the lowest cutoff frequency at 1 56 MHz tain frequency, e g for a 1 mm thick aluminum plate A typical dispersion diagram for an aluminum plate is shown in Fig 1 3 Experimentally it is advantageous to work with a sin¬ gle mode signal [5], and therefore often one of the fundamental modes below the cutoff frequencies of the higher wave modes is employed In this thesis, the first antisymmetric mode A0 was used, as pointed out in Chapter 1 2 Different approximations can be used to simplify Eq (3 4) for the description of the first antisymmetric mode A0, a flexural wave Usually the development starts with the physical effects taken into consideration in classical plate theory (CPT) (Chapter 3 3) and Mmdlm's theory [40] (Chapter 3 4) One can show that CPT is seen in

and

31

can

exist

,

the first approximation of the equations governing the propagation of flexural

physical effects, namely shear and approach development of the full three-dimen¬ rotatory sional equations in terms of a dimensionless parameter e, describing the relation of wavelength to plate thickness [51], [52] Similar to an approach set out by Sun et al [58], Niordson [42] studied flexural waves in a homogeneous, isotropic plate This asymptotic approach was further investigated and applied to the scat¬ tering at a circular cavity in the diploma thesis of G Kotsahs [28] (see Chapter 3 5) In this thesis, mostly the theory of Mmdlm has been implemented and used Initially, also CPT and the asymptotic expansion were used waves

Mmdlm's inertia

theory

considers additional

A different

3.3

Solution

3.3.1

Wave

using

the

is

classical

plate theory

propagation

The

simplest approach to describe flexural waves in plates is using classical plate theory (CPT), taking only inertia and bending stiffness into account according to 2phw+DAAw with

=

out-of-plane displacement

4Gh3

(3 7)

0,

2Eh3 =

3(1-^)~3(1-a)2)'

w

of the

plate

The

plate

modulus D

is

given

by

(3 8)

Scattenng

32

with shear modulus G This

approach

is

valid

only

at low

at

a

circular hole

frequencies

when the

wavelength is large compared to the plate thickness The wavelength X for co/(2ji) can be calculated from the dispersion relation as given frequency f

a

=

*

f

=

(3 9)

Inserting Eq (3 1) wave

number k

k

solving

for non-trivial solutions gives the

(3 10)

Eh

Scattering

This scattering of

and

J3(1_Z!)P^

=

*V

3.3.2

Eq (3 7)

into

as

at

a

plate theory can be seen as a simpler version separately by Noms and Vemula [43] and [57] Following their approach, the scattered wave is assumed in problem

3 4 2 and

Chapter

Staudenmann

circular hole for classical

was

studied

the form of

ws

£ (a1nHn(2)(kr) a2nHn(1)(ikr))cos(n(p)elrat +

=

n

=

The first part describes ond

(3 11)

0

a wave

and the second part

propagating outwards (Hankel function of the

sec¬

boundary layer around the hole (Hankel function of the first kind) The angular dependence is given by the sum over the cosine functions With this approach, the boundary conditions can only be fulfilled in kind)

a

the Kirchhoff approximation

Mrr

0,

=

satisfying

Qr+±Mr(p(p 'o

a

=

0,

r

=

r0,

(3 12)

combination of vertical force and derivative of the twisting moment

Scattering

at

a

The incident

circular hole

expressed

wave is

e-ikrcoscp

£

=

n

33

in a

Fourier Bessel

series

n

7n(H)nJn(kr)cos(n(p),

yn

=0

The coefficients of the scattered incident and scattered

wave are

=

0

n>1

(3 13)

calculated from the substitution of the

boundary conditions and the projection in [cos(n(p)] Following the diploma thesis of G Kotsahs [28] wave

into the

tangential direction and introducing non-dimensional

for

n

Cao(ro) Cbo(lro)

a10

Cao(r0) Cbo(ir0)

a20

=

radius

r

=

kr, the equations

are

[(1+v)J0(r0)-(1-v)J2(r0)]

(3 14)

-Ji(r0)

0 and

Can(ro) Cbn(lro)

Can(r0) Cbn(ir0)

^[2(1

a2n

(3 15) +v)Jn(r0)-(1 -v)[Jn_2(r0) +Jn 2(r0)]] +

-2(-i)"[(1 -v)^Jn(r0)-l(l +(1 -v)^)(Jn_1(r0)-Jn 1(r0))] +

for

n >

1, using

Can(r) Cbn(ir)

(1-v)H(n2)(r),rr-vH(2)(r)

=

=

(3 16)

(1-v)H(n1)(ir)rr-vH(1>(ir)

Can(r)=(1-v)23H(n2)(r)

1+(1-v)^

-

H(2)(r)r (3 17)

Cbn(ir)=(1

-v)^Hln'V)

+

(J

-(1

-v)^JH(1)(ir)r

Scattenng

34

For the numerical

tered

evaluation, usually only the first

a

circular hole

30 coefficients of the scat¬

numerically that the higher coefficients have a negligible influence on the scattered field, as they only describe very local oscillations and the coefficients converge quickly to zero From the asymptotic expansion in Chapter 3 5 it is shown that this solution is thm

ments are

are

valid for X

only in a

wave

calculated It

at

plate

(Chapter

made

~

even

3

6)

Solution

3.4.1

Wave

l e

it

the scattering of low

,

frequency

waves

comparison to results using Mmdlm's

a

at low

3.4

For

h,

r »

From

found

was

shown that for small holes

is

(X»

r=

at

theory

a

large

hole

and experi¬

h) significant

errors

frequencies [19]

Mindlin's

using

theory

propagation to shorter

higher frequencies, corresponding

and rotatory inertia have to be considered without normal pressure q

on

the

plate

wavelengths,

the effects of shear

Therefore, the theory of Mmdlm [40]

faces

is

used

Starting

from the

integrated

equations of motion

^xx ^yx_Q +

dMyx

9Myy

dx

dy

: +

3

q„

=

K

and the relations between

M

*y

(3 18)

2ph*w

dy

9¥*

92¥

£T£L "_n 3 dt2

y

+

at2

2ph3

C_x ^ dx

2ph3 92¥x

_

x

=

dy

=

dx

3t2

plate-stress

dWï)

M....

and

=

plate-displacement components

D(^ v^' +

=1^Ä ^1 2

\dx

3y.

Qx^2K2Ghf^ ¥xl +

the equations of motion

(3 19)

+

are

Q¥

=

obtained

2K2Ghf^

+

Vy

Scattering

at

circular hole

a

35

v)g] K2Gh(^¥x

+

§[(1-v)A¥y (1+v)|]-K2Gh(¥y

+

[(1

-

v)A¥x

(1

+

+

-

+

K2Gh(Aw + 0)

ph

=

ph3 92¥x

dw\

dx)

"

dw\

12

at2

ph3

92¥y

12

at2

(3 20)

^)^^^

ay J

"

^ at2

wlthO^ ^

(3 21)

+

Substituting,

2phw+DAAw

The term

k

=

^-(l 3

denotes

value of this factor with the

for the

single equation

a

that

Rayleigh velocity

4^(1 -<xk2)(1 -k2)

of

=

is

to

=

adjust

can

choose the

high frequencies (co —> oo) propagate surface waves According to this condition, the obtained as a function of v, given by Mmdlm as

waves

at very

(2-k2)2,

For most values of V this gives

K2

(3 22)

3K2G9t4

K2(1-vy

value of the correction factor

with the exact solution

Vw-^L^w

^-^

+

v

obtained

w is

non-dimensional correction factor One

a

so

displacement

a

0
close

Alternatively

correspondence

a

=

of the

the correction factor

can

Jr72V,

dispersion

relation

be set to

Jt2/12

the cutoff

(3 23)

(3 24)

frequency

of the thickness-shear motion

in

the low

frequency

range

Three types of number wave

kj),

(wave

a

waves

flexural

number

plate, a propagating flexural wave (real boundary layer (imaginary wave number k2) and a

exist

k3)

in

the

wave

shear

Scattenng

36

The

k-,

=

1

=

/„u,f

±— c

k2

numbers

wave

P

can

be derived

Jhco(2

+

v

vV2h(1 -v)V

±^j2T7T^^hm(2

K2(1-v)) +

n

k_

=

-v)2 + h2co2(2-K2(1 -v))2

J12cPK4(1-v)2 h2m2(2-K2(1^ +

(3 25)

cnh,v1 -va/2

plate

wave

P

velocity

(3 26)

p(1-v2) In

circular hole

±2^ Œ |3c2K2(1_v)_h2m2

3

with

a

as

K2(1-v)) +Jl2c2K'4(1 v " V P

+

at

coordinates the equations of motion

polar

&»

+

-T"»

+

rr

and the

7(v¥r

=

Mcpcp

=

Mr(p

Q(p

rr

r

(32?)

=

r3
"

r»

-Qr

r

=

by

^^¥r 3

at2

r

stress-displacement relations by

Mrr

Qr

r

given

7&**-°* 2j¥w*'

>rr+lMrr-lM0 +lA^r 9r

are

=

=

=

+

r¥rr

+

V¥(p(p)

7(¥r ¥cpcp+rv¥rr) +

-^r(v-1)(-¥(p 2K2Gh(\|/r+

+

w,r )

2K2Ghf¥(p+ lwJ(p

¥r(p+r¥(pr)

(3 28)

Scattering

circular hole

a

Scattering

3.4.2

Using

dary

at

Mmdlm

0,

=

0,

as an

Qr

formulated

wave is

=

average

over

the

plate

[45],

the boun¬

thickness with

(3 29)

0

in

terms of three

potentials

and consists of the

£(a1nHn(2)(k1r) a2nHn(1)(ik2r))cos(n(p)elrat

(3 30)

+

=

n

a

=

the work of Pao and Chao

following

wave

ws

and

and

be fulfilled

Mr(p

The scattered flexural

can

circular hole

a

theory

conditions

Mrr

at

37

shear

=

0

boundary layer

with the two components

^(a1n(G1-1)Hn(2)(k1r) a2n(G2-1)Hn(1)(k2r) a3nHn(1)(k3r))x +

¥r= n

=

+

°

-cos(n(p)e

¥cp

X

=

n

(am(1

-°-i)Hn(2)(kir)

+

a2n(1

-G2)Hn(1)(k2r)-a3nHn(1)(k3r))x

=0

1I

,

l

x

cot

-sin(n(p)e

Evaluating scattered

the

and

ftf

and

2

^^T^J

°2

boundary

waves are

conditions

calculated for

Can ^bn ^cn

an

Cdn Cen Cfn

b„

Cgn Chn Cm

cn

=

A©'

°2^lkT)

in

n >

polar coordinates,

<332>

the coefficients of the

1 from

Bm =

B2n B3n

(3 33)

Scattenng

38

The coefficients

B1n

B2n

=

"2

are

(-|)"(01

-2(-')"-1

=

a

circular hole

by

given

-1

at

)[n2vJn(k1

rQ)

-

r0v|r(Jn(k1 r0)) r^cyk, r0))] -

^Wo»

(3 34)

2n,

B3 n^H^-I^Ck^-r^J^r,,))) o

Cbn

4^-1f2vHn1)(k2^o)-vr0^H(1V2r0))-r2^(H(1V2r0))

=

^

o

J-lv-1^W-r0^W.)))

=

H

Cdn C.„ °fn

°lKHn2)("l^)

=

"2£ftVo))

=

=

(335)

r^1)(V0) o

Cgn

=

4n(CT1-1)(Hn2)(k1ro)-ro|r(Hn2)(k1r0))) o

Chn

=

-J"(('2-1)(Hn1)(k2ro)-ro|r(Hn1)(k2ro))) o

The calculation involves shear

much

an

inversion

of the matrix

than the other

larger k3 problems arise As the shear wave does (measured in the experiments), we do not late

wave

only

inated

is

wave

not give

The

wave

numbers, an

number of the

therefore numerical

out-of-plane displacement explicitly To calcu¬

need to calculate it

the coefficients of the flexural wave, the third line of

Eq (3 33)

is

elim¬

Scattering

at

circular hole

a

Uli

f\

f\

°an-c-°.gn

Cfnr

r

udn

For

n

=

~

q-ugn

0 the

C

~

^bn

39

—2HC ^hn q

Bun 1n~

Cfnr

r

~

uen

boundary

q

Q-B3n

(3 36)

B2n-Q-B3n

q-uhn

condition of the twisting moment

vanishes The remaining coefficients

can

trivial,

is

as

Ca0 Cb0

right

J20

^-l^vJ^k^ r^J^r,,)) +

=

B20

(3 38)

k^J^k^)

=

CPT, the first 30 coefficients of the scattered

solution

is

valid for all relations of

The scattering of studied

(p)

hand side coefficients

Bio

As for

(0

(3 37)

Cd0 Ce0 with

sin

be calculated from

an

incident flexural

For

[37],[38]

typical

wave

are

wavelength, plate thickness, wave

with

calculated

curved wavefront

a

transducer positions

used

(Chapter 2 3), the difference due to the approximation a plane wavefront was found to be negligible

in

This

and hole radius

the

was

also

experiments

of the curved wavefront

with

3.5

Solution

using

an

asymptotic expansion

The work

presented in this section was mostly developed in the framework of the diploma (Diplomarbeit) of G Kotsahs [28], supervised by M B Sayir and P Fromme, and in the following cooperation thesis

3.5.1

The

Wave

nine

ticity

in

propagation

partial a

differential equations

solid

(eg

Graff

describing

[22], App A)

are

the

case

of

linear, isotropic elas¬

non-dimensionalized

A small

Scattenng

40

at

circular hole

a

parameter

e

2

=

£

Jt

=

k

(3 39)

h,

X

giving the relation of ment of the middle

u3

=

U3

e2 V3

+

+

e4 W3

and the other unknowns

considered,

as

thickness to

plate plane

are

+

wavelength,

is

introduced The

0(e6)

(3 40)

developed

the odd powers

displace¬

in

terms of e

the

trivially

give

Only same

even

powers of

e are

The

first

equations

approximation

U3

+

—.r

AAU3

=

0,

(3 41)

^-^ AÜ3,

(3 42)

3(1-v2) second approximation

V3

+

—^-T

AAV3

=

15(1-v)

3(1-v) and third approximation

VV3

+

AAW3

—.r

=

3(1-v2) 17-7v

15(1-v)

+

_(33v2

derived

gle

dispersion

relation has the

same

right

and

good agreement

9

<

U3

according

as

K

For

of the

Eq (3 22), a

a sin¬

obtained

plate according is

derived

according typical value of v

to

to Poisson's ratio

the difference

Eq (3 23), dispersion relations

)

9t4

hand side vary

to

between the two

displacement

form

and the choice of the correction factor

correction factor selected

v)

equations and neglecting higher order terms,

Mmdlm The coefficients of the V

+

525(1-v)

differential equation for the transversal

The

424v-422)(1

+

are

Summing

up all

Ay

is

is

found

=

0 3 and the

less than 3%

Scattering

U3

at

circular hole

a

+

41

5-

3(1 -v2)

(3 44)

4

17-7v

2 e

Scattering

3.5.2

The

scattered

diploma

5-=-: 15(1

at

wave

a

+

~"°

-v)

e "

the

up to

second approximation The

[28]

medium sized hole

a

9t4

525(1 -v)

of

case

asymptotic development of the equations, or

^3

circular hole

thesis of G Kotsahs

(X»r=h)

d u3

(33v +424v-422)(1+v)

.-4 Au3

r

=

a

X

large

»

h

is

calculated

was

hole

is

studied,

assumed For

(X»r»h), separate asymptotic

would have to be carried out As the solution described

a

1 e

,

in

the

for the

small hole expansions

Chapter 3 4 2 is more (within 2%) for the case in

general (arbitrary hole radii) and gives the same results of a large hole, only the outline of the asymptotic solution for the scattering at a large circular hole is given here The boundary conditions are solved in terms of e In the first approximation (e ), the stress-free boundary conditions at the hole are satisfied An integration over the thickness leads to the Kirchhoff approximation and Eq (3 14)andEq (3 15), showing that the solution shown in Chapter 3 3 2 is indeed the first approxima¬ tion for

coupling £

r

=

X

h

»

remain

Residual stresses and

are

taken

as

size

driving

smaller due to shear and curvature terms for two

The left side of the matrix equations has the

Eq (3 15),

while the

side

right

approximation Again residual ing to similar

for which dent

an

were

equations

has

a

wave a

solved

given

by

separate problems

form

as

Eq (3 14)

wave

has

a

further component

the

plate

in

thickness

scattered field

was

coupling stays

about constant with

and rotatory inertia

in £

,

inci¬

the thickness direction and the assumed

distribution, the boundary conditions The

can only be fulfilled eight separate scattering problems

Mathematica, and the contribution of the different effects

in

in

and

the residual stresses from the first

solved As the normal radial stress G^ of the

is

cubic distribution linear

over

average

is

same

stresses due to shear and curvature remain, lead¬

The incident

in £

separate system

a

wave

scattered as

one

the

studied

rises

[28]

with

£

to the

While the relative contribution of the curvature

decreasing wavelength, the is significant for e > 0 2

and

influence of shear

Scattering

42

3.6

Fig.

x

-

axis

b)

[mm]

Amplitude (normalized: Uj

3.2

10 mm,

=

r0

f0

circular hole

a

with measurements

Comparison

a)

at

=

=

kHz, X

100

x

-

axis

[mm]

1 mm, 1) of the scattered field around a hole; 2h 10 mm: a) measured, b) calculated using Mindlin's =

=

theory. A

typical

measured scattered field around

excitation with

a

surement is made

5°

on

radii 0.5

center on a

mm

a

hole in

of 100 kHz

frequency grid

circular

can

around the

apart. The incident

an

with

aluminum

a

in

plate

for

3.2a. The

an

Fig. signal recorded every nearly straight wavefront

seen

hole, with

wave

propagates from the direction of the

be

mea¬

a

y axis and is scattered at the hole

positive (low amplitude) can be seen, the so called shadow area, where only little energy arrives. Directly at the hole a high ampli¬ tude (light) results from the scattering at the free surface. Further outwards a characteristic hill and valley pattern develops due to the constructive and destruc¬ tive interference of incident and scattered waves. Rather strong amplitude varia¬ boundary.

tions

over

Behind the

hole,

short distances

wavelength, In Fig. 3.2b

a

high

and low

the scattered

Pao and Chao

[45],

a

dark

are

area

evident. In the backscattered

amplitude

every half

field,

appears due to interference.

field, calculated using Mindlin's theory and following

is shown. Measurement and

analytical description

show

good

agreement. To verify the validity of the various approximations, the scattered field around

a

hole is measured for different ratios between

thickness and hole radius. The measurements

on a

hole

using

are

compared

with

analytical

calculations

wavelength, plate

concentric circle around the CPT and Mindlin

theory.

Scattering

at

a

circular hole

43

18

o

«

«s

se

f«

m

a»

»s

ms

Angle I"!

Fig.

3.3

2h 1 mm, 10 mm, Amplitude (normalized: Uj 1) at r 13 mm; r0 fo 20 kHz, X 22 mm: measured (dots), Mindlin's theory (solid), CPT (dashed). =

=

=

=

=

=

Good agreement between both

approximate theories and experiments is found, as Fig. frequency f0 20 kHz. This corresponds to a of much 22 mm, wavelength larger than the plate thickness of 1 mm and about as the hole diameter of 20 mm. The experimental points, measured every as large and the for Mindlin and CPT are normalized with the amplitude of the curves 5°, incident wave. This case is similar to the geometry studied by Staudenmann [57]. Compared to his experimental results, better accuracy of the measurements and an almost perfect match with the analytical calculations is achieved. shown in

3.3, for

a

center

=

t ia

ie

o

48

0

m

1»

1»

Î2S

Ï70

MB

315

ftn|îê i"J

Fig.

3.4

Amplitude

fg

=

100

(normalized: Uj

kHz, X

=

10

mm:

=

1)

measured

at

r

(dots),

=

13 mm;

Mindlin's

2h

=

1 mm,

theory (solid),

r0 CPT

=

10 mm,

(dashed).

Scattenng

44

For

a

higher frequency

tion between CPT

side, especially This

is

in

shorter

a

wavelength

a

circular hole

of 10 mm,

a

devia¬

side and Mmdlm and the

experimental data on the other (around 180°), is evident in Fig 3 4

the backscattered region

due to the effects of shear and rotatory inertia,

at shorter to

of 100 kHz with

on one

at

wavelengths Describing

good agreement

still rather

small,

the

having a stronger influence plate according to Mmdlm's theory leads

with the measured values However, the difference of CPT

so

that it

can

be used

first approximation for the

as a

is

case con¬

sidered here, where both wavelength and hole radius are large compared to the plate thickness A geometrically different case is shown in Fig 3 5, with a hole radius of 5 mm and a plate thickness of 2 mm In contrast to the two previous cases, the hole radius is not large compared to the plate thickness Even for a low frequency of 20 kHz, a significant, systematic difference between measurement and Mmdlm's theory on one side and CPT on the other side is evident This results from an additional boundary layer close to the hole due to the effect of shear and rotatory inertia, which are neglected in CPT, but are well described using Mmdlm's the¬ ory Here, the wavelength of 31 mm is large compared to both hole radius and plate thickness The solution using CPT can be shown by an asymptotic expan¬ sion to be the first approximation, when both wavelength and hole radius are large compared to the plate thickness

a 18

1«

0

4S

m

135

2»

180

2?0

315

360

Angte [•]

Fig

3 5

Amplitude (normalized Uj X 31 mm measured (dots),

=

=

1

)

at

r

=

6 mm, 2h

Mindlin's

=

1 mm, rg 5 mm, CPT (dashed)

theory (solid),

=

fg

=

20

kHz,

4

Scattering

4.1

Description

Fig

The influence of notch

stress-free

Sep At the

or

a

boundary

=


crack-tip,

crack, while the

=

a

on

at

a

hole with

a

defect

notch /crack

the scattered field

length

at

a

angle (p0

is

to the

conditions must be satisfied


a

of the geometry

defect

crack of

hole with

a

Geometry of the scattering

4 1

ness

at

°>

=

singularity

corners


in

=



investigated x-axis

by

A

arises

+

through-thick¬

assumed Additional

the scattered

re[r0, r0

the stress-field

is

wave

a]

due to the

(4 1)

sharp edge

of the crack with the hole must be stress-free

of the

Except

for

0°, 180°), the symmetry of the scattered field for the special cases ((p0 hole disturbed This may be used for the detection of a defect is undamaged Several analytical approaches were tried to model the defect, but did not give accurate results for arbitrary crack length and position Some of the approaches two

show

=

promise

Chapter

4 2

reasonable

with

further

In order to gam

theoretical a

effort, the scattering

work

and

are

therefore

functional model for the hole with

stated

in

notch with

numerically The employed finite (FDM) explained Chapter 4 3 Results from the numeri¬ cal modeling are compared to measured changes in the scattered field for the model system of a hole with a notch in a large plate Good agreement is found in Chapter 4 4 for all examined parameter variations, such as notch position, notch length, plate thickness, and excitation frequency Therefore it can be concluded that the FDM model accurately describes the influence of the defect The defect detectabihty is studied in Chapter 4 5, allowing predictions of the minimum detectable crack length and the optimum excitation frequency for a given geome¬ difference method

try

is

was

modelled

a

in

Scattering

46

4.2

Possible

4.2.1

Modification of the

analytical

The scattered field around tional stress-free This work

at

hole with

a

defect

circular hole

a

circular hole

was

conditions at the

boundary

modified to

corner

implement

the addi¬

of the crack with the hole

[30] and the diploma thesis supervised by MB Sayir and P Fromme From the experiments it is known that the out-of-plane displacement has a peak directly in front of the crack and a low amplitude behind the crack (shadow area) An addi¬ of A

done

a

solutions

scattering a

at

was

in

tional scattered

the term project of J Lackner

both

Allenspach [3],

and

wave

boundary layer

were

introduced As the

driving

term

a

disturbance function

C lei CK

%

y

j«:

with <J)

having

an

=

U

\

(®)

=

(p

-

(4 3)

(p0,

exponential

U

form

Assumed

4 2

Solving

the

(Fig

4

2),

was

assumed

-

I

-v\Fig

(4 2)

e

exponential

form of the

driving

boundary conditions at the hole edge of the crack is satisfied

term

and

adjusting

the

amplitude

U

so

that

the stress-free

°
=

0


=


(4 4)

Scattering

at

a

hole with

a

defect

47

the combined scattered field is calculated. Different forms for the disturbance function and choice of the parameters

were

tried, all showing similar effects.

mS^'

x

Fig.

4.3

*

fflca

fmiti]

-

â^e

|fsmj

Comparison between calculated (left) and measured (right) change 1 ) for a 2 mm notch at 90°, 2h 1 mm, (normalized: Uj f0 100kHz, X=10mm. =

amplitude

in

=

r0

=

10 mm,

=

For short notches at certain

angles

and excitation

frequencies,

a

agreement of the calculations with experiments could be found,

Fig.

good

shown in

4.3.

n

Fig.

rather as

4.4

Measured

complex

field for

2

a

mm

-

sms

difference in

notch at 90°, 2h

[ram]

magnitude (normalized: Uj =

1 mm, r0

=

10 mm,

f0

=

=

100

1) of the scattered 10 mm. kHz, X =

Scattering

48

However,

generally

no

a

agreement could be achieved This

is

direction at

hole,

its

Conformai

4.2.2

Building [49]

on

angle

the work of Muskhehshvili

The idea

good

it

can

be

as a

is

modification of

implicitly that the

seen

taken

as

complex

the form of two lobes with the

more

(Fig

4

4)

[41]

with

a

was

to

use

crack to

a

and Bowie a

[9],

Roberts and Rich

hole with cracks

in a

plate

sub¬

their conformai mapping to transform the circular hole

Applying

the transform also

the incident wave, the scattered field could be calculated and re-transformed

to the real coordinate

Z

=

i

x +

system Introducing

a

complex

coordinate

(4 5)

y

the real space, and

Ç in

no

mapping

jected bending boundary of the hole

in

it

to the orientation of the notch

calculated the stress intensity factors for to

on

Assuming

wave

propagation direction

However, from experiments an

defect

a

due to the wrong choice of the propagation

difference between the measurements has mam

hole with

variety of notch lengths and angles

characteristics of the additional scattered the scattering at the circular outwards

a

valid choice of the disturbance function and the free

parameters could be found, and for

radially

at

=

b

-

i

(4 6)

c

the transformed space, the transform from

bc-space

to xy-space

is

given

by

1

z

=

{i4ß

tcK+rK+i+ß+(i+rK)

K defines the number of

Vç2K+2pçK+i]|K

symmetrically positioned

cracks at the hole

(4 7)

boundary

The parameter

ffrl

(4^

Scattering

is r0

at

a

hole with

defect

49

function of the relation between crack

a

1. For

=

lated

only

one

(- 1

(4-4ß)Z

(K=l),

length

and normalized hole radius

the inverse transform

can

be

explicitly

calcu¬

Mapping

4.5

Applying

+

2ß

+

ß2 + 2Z 2ß2Z Z2 + 2ßZ2 ß2Z2

(1 -2ß

-

+

of the contour of

this method to the

occur.

-

-

+

-

.(4.9)

I- 4(-2+2ß)2Z2

lems

crack

as

1

Fig.

a

ß2-(2-2ß2)Z + (1

a

hole with

a

crack

on a

scattering calculation,

it

-2ß

+

ß2)Z2)2

circular contour

was

(ß

=

-0.995).

found that several

It would have to be checked that the conformai

mapping

prob¬

is also valid

for the three-dimensional stress distribution in the

larity

at the crack

tip

in the real system is

plate and that the stress singu¬ accurately mapped. Furthermore, the

transform maps the interior of the hole to its exterior and vice versa, thus

negating a simple implementation of the incident wave. No good agreement the experimental influence of a crack / notch could be found.

4.2.3

Superposition

A further

possibility

circular hole and

a

of two separate

problems,

with

crack and circular hole

would be the separate calculation of the scattered fields for

crack, superimposing the

a

two scattered fields. As the scatter-

Scattering

50

mg at

a

circular hole

tered field around

a

is

given

in

Chapter

3 4 for

a

a

hole with

Mmdlm type

crack would have to be calculated for

wave

the orientation of the crack

One

use

at an arbitrary angle to possibility might be the

at

an

plate,

a

defect

the scat¬

incident flexural

of the Joukowski mapping

2 w

transforming tered field at lems

(4 10)

z + —,

=

as in

a a

circle to

Chapter

of the incident

an

ellipse

or

line

4 2 2 concerning

In

4 6

same

prob¬

of the mapping and transform

wave occur

w-plane

Joukowski mapping of a circle to

[56], chapter 7,

and Mmdlm's

to model the scat¬

circular hole The

a

admissibility

z-plane

Fig

(degenerate ellipse),

crack from the known solution for

the scattering of

a

a

degenerate ellipse

flexural

wave

at

a

crack

is

studied for CPT

theory The numerical solution for the coefficients of the scattered wave is quite lengthy, and severe restrictions on the incident wave are made to achieve simple boundary conditions Two incident waves are superimposed, so that they are symmetrical to the crack and the only non-vamshmg boundary con¬ dition is the bending moment The only single propagating wave that can be studied with this theory has a propagation direction along the crack For the experiments, only a crack at 0° and 180° could be studied with this theory, while for the tensile specimen experiments (Chapter 5 3), the crack is at 90° to the inci¬ dent wave It might be interesting to study the possibility of extending the theory presented in [56], to incorporate incident waves at an arbitrary angle This leads to significantly more complex boundary conditions, as the twisting moment and vertical force do not vanish a priori No further literature on the subject could be found, and it was deemed beyond the scope of this thesis to achieve such a com¬ plicated analytical model

Scattering

4.3

at

a

hole with

defect

a

Finite difference

Therefore

51

modeling

numerical simulation

investigated and found to give good results, complicated geometry. Especially the experiments with the tensile specimen would be difficult to model analytically, as the wave propaga¬ tion in a plate strip, the scattering at a hole with a crack, and the reflection of this scattered wave at the specimen boundaries have to be considered. Finite differ¬ ence methods (FDM) are used to calculate the propagation and scattering charac¬ teristics of a flexural wave in a plate according to Mindlin's theory. a

even

4.3.1

FDM

for

was

a

algorithm

1,N„-1

1,N„-1

ff,N.-l

,N„-1

for

a

Mindlin type

2,NV-1

2,NV-1

plate

NX-1,NV-1NX-1,NV-1

2,NV-1

2,2

-1,2

1,2 w

MT,M„

Ay

,i

&*-

1,1

Nx-l,l

Nx-l,l

Nx-l,l

X-l.l

Qyfy 2,1

Nx,l

Ax

Fig.

4.7

Staggered grid used for finite difference calculation. One cell marked and numbering scheme shown. Displacement, rotation angles, moments, and forces calculated at the four points of the cell Ax/2, Ay/2 apart.

Scattering

52

at

hole with

a

a

defect

Mmdlm's equation of motions

[Eq (3 18), Eq (3 19)] are discretized on a Carte¬ proposed by [34], shown in Fig 4 7 The bending and twisting moments Mx, M M per unit of length, the transverse shear forces Qx, Qy per unit of length, the displacement w, and the rotation angles *PX, *Py are cal¬ culated at different grid points, half a grid step (Ax/2, Ay/2) apart They are arranged such that the respective centered first derivatives m x and y can be cal¬ culated from neighboring points (half grid step) Calculating moments and forces besides the displacement uses more memory space, but allows an easy imple¬ mentation of stress-free boundary conditions, compared to e g [26] The result¬ ing equations for the inner region of the plate are given by Eq (4 11) and Eq (4 12)

staggered grid,

sian,

n+

1

n

W

n +

n

w

1

-

Vx

"

¥y

n

Qx

ph

Qx

-

ix+i,iy

n

n

Qy

Ax

ix, iy

ix,

"

iy+i

Qy ix,iy

Ay

+

n

Mx

+

M,

-

Mxy

Ax V

-

Mxy

'x-'y+1

'x, 'y

Ay

(4 11)

J

n-1

n

2

Vx

Qx

Phc

1

n

AT

n

12At'

n +

-1 W

n-1

n

2

Vx

as

¥v

"

Vv

+

/

12At'

Phc

n

n

Qy

M„

+

-y

y

'y

n

Mv

-

'x> 'y

direction The

grid

size is

Ax,

Ay

-

v

y

The superscripts denote the time step resp

Ay,

MXy v

Chapter

Mxy

'x+1-'y

(At), subscripts see

n "

'x, 'y

the indices

3 4 for other

Ay j

m

x-

symbols

and yused

Scattering

at

a

hole with

defect

a

53

f

\ n

=D

Mx

n

Vx V

'x+Vy

D

=

Ax

lx.ly+1

V

lx.ly+1

lx.'y

vD

+

—

Ay j

((

\

1 -vr

¥y

=

K2Gh

Qy

K2Gh

W

vv

is

is

" ,

^V

Ay+

Ix. ly j

j

boundary conditions,

as

the

only boundary

at the free side boundaries of the tensile

grid is selected large enough to achieve a pulse as described in Chapter 2 3 1 for so that the boundary conditions can be ful¬

in

zero

grid

the

corners

of the

plate

0,

Therefore

Mxy

and

Qx

resp

is

Qy

calculated, can

are

directly

be

Eq (4 11)

ing moment

is

calculated from the equation for

at the boundaries

can

(4 13)

resp

The missing condition be

seen

to be

(ix

zero

=

1,NX, iy

due to the

=

l,Ny),

boundary

*PX

resp

*Py

Evaluat¬

the shear force and twist¬ conditions However, the

Mx resp My would have to be calculated half a grid step out¬ plate boundary Interpolating linearly to get a boundary value of zero,

moment

side of the

selected

at the boundaries

xy

bending

is

The points, where the twisting moment M

easily

M

ing

study

1

separation of incident and reflected

selected set to

this

in

In the other directions the

the experiments The filled

lx.'y-1

modelled with stress-free

reflection of interest specimen

1

W

-

-

\

n

'x.'y

lx, ly

(4 12)

Ay

'x- 'y y

\

time

Vx

"

Vx

Ax

'x-i,'yy

n =

\

1 WZ+

W

-

'-'y

w

plate

\

¥x

Ax

n

W

y

n

'x. 'y

n

The

lx.'y

(

i», i„

Qx

Ax

n

Vy

"

Ay

J_

Vx

"

'x+1-'y

n

n

'xy

lx.'y J

n

Vx v

J_

¥y

"

n

¥y

"

n

¥y

n

¥y v

vD

+

—

Ix.'y }

n

My

n

Vx

"

Scattering

54

the value outside the

plate

must be the

at

a

hole with

negative of the bending

a

defect

moment half

a

grid step inside the plate This leads to the following conditions in Eq (4 14) at the left, right, upper and lower boundary, respectively With these conditions, a stress-free right-angled plate is modelled accurately, including the corners

n+1

2

Vx

1,ly

n +

-

1,l„

1

Vx

n-1

n =

Vx

, 2

Nx.'y

Vx

-

Nx,iy

"

Mx

24At2

n-1

Vx

24At —3—

Ph AX 1,lv

1,lv

n =

+

Vx

-

Nx,iy

—3— Ph Ax

"

Mx Nx_1ly

(4 14) n +

1

¥y

=

2

'x.Ny

¥y

¥y

2

Wave

n-1

¥y

"

i_, 1

+

¥y

To test the accuracy of the

in

a

24At —3—

pn Ay

i_, 1

propagation

"

My —3— pn Ay ,x;N

"

ix,Ny

n =

ix, 1

4.3.2

¥y

"

ix,Ny

n+1

24At2

n-1

n

plate

algorithm

,

"

My |

1

and tensile described

model of the tensile specimen described

specimen

above, the

wave

propagation

in a

Chapter 2 3 2 and the plate described in Chapter 2 3 1 is studied Wavelength, phase velocity, and group velocity are calculated from numerical data and compared to analytically derived values The tensile specimen is modelled as a strip of a plate with 40 mm width, 1000 mm length, and a thickness of 3 17 mm Free boundaries at all edges are assumed The material parameters are selected according to Chapter 2 3 2 The excitation is achieved by prescribing the displacement w at a line over the width of the specimen Excitation frequencies of 40 and 160 kHz and a grid size of 0

25, 0 5, and 1 0

0 025 (is to allow

the

mm a

are

used

in

in

the simulation

stable simulation

even

The time step

at the smallest

grid

stability criteria in Eq (4 15) For a grid size of 1 0 mm a run with a larger time step of 0 1 (is, inside the stability limits

is

size,

selected

according

simulation

is

as

to

also

Scattering

at

a

hole with

a

defect

55

max(Ax, Ay)

CPJ

JL Ax

+

^2cp

JL

(4 15)

Ay

Vp(1 -v2) A time shot of the

plate displacement w is recorded to measure the wavelength wavelength matches the analytically calculated value withm the accuracy of the grid size The time signals at a measurement grid with 25 points along the length and 5 points over the width of the specimen are recorded From these time series the phase and group velocity are calculated For the phase velocity, the val¬ ues calculated from the simulation typically he withm 2% of the analytical value A slightly higher deviation of about 4% can be found for the time series recorded at the side boundaries of the plate This is due to a boundary effect at the stressfree surfaces, which also results in an amplitude modulation over the specimen width with higher amplitudes at the sides This effect was also observed in the experimental data in Chapter 5 3 2 No significant effect of the grid size or time increment could be found The group velocity of the simulations was found to typically he withm 1% of the analytical values for the grid size of 0 25 and 0 5 mm For a grid size of 1 0 mm and an excitation frequency of 160 kHz the calcu¬ lated values show a slightly stronger variation with a median value about 5% lower than the theoretical group velocity This could be the result of the grid size not being small enough compared to the wavelength of 12 mm The model of the plate has a size of 1000 mm by 1000 mm and a thickness of 1 mm Free boundaries at all edges are assumed The material parameters are selected according to Chapter 2 3 1 The excitation is achieved by prescribing the displacement w at one point, resulting in a wave propagating radially outward Excitation frequencies of 50 and 100 kHz and a grid size of 1 0 mm are used in The

the simulation The time step 100 kHz are

The

according

calculated

on

to the

selected

stability

as

criteria

rays from the excitation

phase velocity

in

0 2 (is to allow

in in

Eq (4 15)

a

stable simulation at

Phase and group

analytical velocity has an error of less slightly higher deviation of less than

The calculated group

For the simulation at 100

kHz,

a

velocity

three directions at 0°, 45°, and 90°

the simulation at 50 kHz lies withm 1% of the

value for all three directions than 3%

is

Scattering

56

2%

is

found for the

direction The

error

cal value

stability

4 8

Staggered Grid with hole, notch, boundary selected, so that twisting

conditions

are

4

Mxy

is

radius The

scattering

at

implemented

(see Fig

moment

an

The calculated

phase velocity

in

a

defect

0° and 90°

a

criterion

To calculate the

tour

in

the 45° direction

hole with

Scattering implementation

4.3.3

Fig

phase velocity

a

slightly smaller error is found for the group velocity is higher and lies only withm 10% of the analyti¬ This is due to the large grid size used, which only narrowly fulfils the

is

the same, while

at

grid

a

and

grid approximation

moment

Mxy

of hole contour Grid

calculated at the

corners

circular hole

The hole

is

in the plate, the stress-free boundary approximated with a right-angled con¬

8) The corners are selected as the points at which the twisting calculated, lying either on the radius or the closest outside of the size is

selected with Ax

integer multiple of Ax If the grid

equal

size is

to

Ay,

so

that the hole radius r0

selected small

compared

is

to the hole

wavelength (eg Ax 0 25 mm, r0 10 mm, X 10 mm), The no noticeable disadvantage due to this Cartesian approximation was found boundary conditions, as given in Eq (4 14), are applied to the grid points lying on the Cartesian hole contour Care has to be taken with the numbering scheme, as not to mix up indices and apply the different boundary conditions at different grid points radius and the smallest

=

=

=

Scattering

at

a

hole with

a

an

expressed

4.4

in

use

terms of radius

of

57

grid, Mmdlm's equations of motion, angle (p (Eq (3 27), Eq (3 28)), could also be discretized on a radial grid This turns out to be more complicated as additional coupling terms due to the curvature exist These terms pose a problem in the cal¬ culation of boundary conditions, which can not be expressed as conveniently as for a Cartesian grid Furthermore, the cell size gets larger with increasing radius (r A(p), imposing a restriction due to the stability criteria A straight boundary, as in the tensile specimen, can only be approximated with a ragged contour Due to these difficulties and the good results achieved with a Cartesian grid, it was decided to use only the Cartesian grid A possible modification might be the use of a radial grid in the vicinity of the hole and a change to a Cartesian grid further out However, a rather complicated mesh would be necessary to implement this A crack or notch at the hole boundary, through the thickness of the plate, is implemented in a similar way The boundary conditions (Eq (4 14)) are applied on two parallel lines and one point at the tip of the notch This simulates a notch with a width of Ax and a blunt tip The sharp edge of a crack and effects like crack closure are not considered No numerical problems or instabilities were found To simulate a quarter-elliptical crack that does not go through the thick¬ ness of the plate, as encountered in the measurements at the tensile specimen, elements with a reduced height, and thus bending stiffness, might be used, with¬ out considering mode conversion to the symmetric S0-mode As

alternative to the

defect

r

Cartesian

a

and

Comparison

Il4

VvVV A

4%

Fig

4 9

W

13§

1B0

35

Zm

315

35B

m

135

%m

225

270

315

3S0

Comparison of analytically calculated (green), FDM calculated (red) and measured (blue) amplitude (normalized U1 1), 2h 1 mm, r0 10 mm, f0 50 kFlz, X 14 mm left r 14 mm, right r 20 mm =

=

=

=

=

=

=

Scattering

58

The

scattering

at

an

undamaged

hole and at

a

hole with

at

a

a

hole with

notch

or

a

defect

crack

was

modelled for the

experimentally measured cases. Shown here is the comparison for the plate specimens, as it was possible to measure a wider parameter range, specifically different angles between the propagation direction of the incident wave and the notch, and variations in the ratio of wavelength, hole diameter, plate thickness, and notch length. The scattering in the tensile specimen can also be modelled quite well, as shown in Chapter 5.3.4. The comparison with the analytical model and experimental data for the scatter¬ ing at an undamaged circular hole in a large plate is shown in Fig. 4.9. Measure¬ ments were made on circles around the hole, using a radial grid. The calculation using FDM shows a good qualitative and quantitative agreement with the mea¬ sured and analytically calculated values. Only around 0° (front) and 180° (back) a slight divergence of about 10% is visible. This is probably due to the Cartesian approximation of the circular hole boundary in the modeling. A longer straight scattering surface is presented to the incident wave.

q)

Fig.

4.10

Change r0

=

in

%

[mmj

ä)

amplitude (normalized: Uj

10 mm,

f0

=

50

kHz, X

=

14

=

1) due

t

to

{mm)

a

2

mm

notch at 90°, 2h

=

1 mm,

mm:

a) measured amplitude with notch; b) measured change in amplitude due to notch; c) calculated amplitude with notch; d) calculated change in amplitude due to notch The

modeling of the influence of a notch experimental results in Fig. 4.10. After a

on

the scattered field is

first measurement for

compared to undamaged

an

Scattering

hole,

a

using

at

hole with

a

notch of 2 fine

a

the difference

surface of the notch

tion of such

an

notch from

a

4

cut at

an

due to the notch

is

of 90°

on

the hole

measured again shown

in

Fig

(Fig

4 10b

4

boundary 10a)

and

At the free

generated, which changes the amplitude and allows the detec¬

wave is

up to about 30%) an

angle

was

in

amplitude measurement The calculation of the change in amplitude (Fig 4 lOd) show good

and the

10c)

agreement with the

was

additional scattered

significantly

amplitude (Fig

59

The scattered field

amplitude

in

scattered field

length

mm

blade

saw

defect

a

measurements

When inspecting

large structures, the crack orientation (vertical to the direction of tensile stress) is not necessarily known a prion, or it is not possible to place the transducers at the required positions For the inspection of a line of rivets one will place the transducer at some distance from the line, resulting in different incident angles for each rivet hole (see Fig 1 2) Therefore it is important to quantify the measurement method for defects at arbitrary angles to the propagation direction of the incident wave In Fig 4 11 and Fig 4 12 the measured and calculated changes in the amplitude of the scattered field are shown for a 2 mm notch at four different angles and a center frequency of 50 kHz The case of the notch at 90° (at the side of the hole) was already shown and discussed in Fig 4 10 For all cases good qualitative and quantitative agreement between experiment and FDM cal¬ culation

is

evident

The difference

icantly larger dent

amplitude

in

than for

for

a

defect at the side of the hole

defect oriented

a

in

(45°,

90°

is

signif¬

the propagation direction of the

aspect of the notch

inci¬

'visible' to the incident

larger amplitude of about 30%> of the amplitude of the inci¬ dent wave is for a notch at a 90° angle (Fig 4 lie) A notch at 45° results in about 20%o change in amplitude (Fig 4 lib) The notch at 0° has a much smaller effect of less than 10% (Fig 4 11a), as only the cross contraction is obstructed A slight asymmetry is visible in the experimental data, as the notch was cut at a slight angle and not perfectly centered When the notch lies in the shadow area of the hole (180°, Fig 4 lid) an even smaller change of only about 5%> occurs, as wave

wave

The

(0°, 180°),

as

largest change

a

in

very little of the energy of the incident

fore

in

is

practical applications

wave

reaches the defect position There¬

for NDT purposes,

one

should

aim

positioning

is

more

vertical to the propagation direction

is

then

cantly larger

and allows

a

detection of smaller defects The

ther parameter and best

case scenario

mens

(Chapter

5

expected crack position The change in amplitude

at

the excitation transducer such that the

mam

or

less

signifi¬

focus of the fur¬

detectabihty study is on a defect at 90°, as this represents the and is the angle at which all fatigue cracks in the tensile speci¬

3)

are

located

Scattering

60

at

defect

a

i:

f*

"V

hole with

a

61

.

!

I

-20

"»2

30

Fig.

4.11

Measured

f0

=

50

change

in

amplitude (normalized Uj=l),

kHz, X=14mm,

propagating

a

2mm, notch

=

bottom): a)

from top to

a

at

0°; b)

=

a

2h

=

1 mm,

r0

=

10mm,

angles (incident wave 180°. 45°; c) a 90°; d) a

different =

=

=

«

i 0

20

turn)

I -20' -30

»[mm!

Fig.

4.12

calculation

FDM

r0

=

10 mm,

wave

a)

a

f0

=

50

of

change

kHz, X

=

in

amplitude (normalized Uj

14 mm,

a

propagating from top to bottom): 0°; b) a 45°; c) a 90°; d) a

=

=

=

=

=

=

2 mm, notch at different

180°.

l), 2h 1 mm, angles (incident =

Scattering

at

a

hole with

defect

a

61

-9k ï 1

15

if* 6

20

20

61

i: » 20 3«

Fig

4 13

Measurement and FDM calculation of

2h

1 mm, r0 measured a) a

Fig

4 14

=

=

1 mm,

f0 b) a

=

100 =

change

kHz, X

=

in

10 mm,

amplitude (normalized Uj a

=

2 mm, FDM calculation

=

1

),

90°,

c)

a

=

1 mm,

d)

a

=

2

mm

FDM calculation of complex difference in magnitude 1 mm, r0 10 mm, a 90°, a 2 mm, 1), 2h 50 kHz, X 14 mm, b) f0 100 kHz, X 10 mm, calculation c) f0 50 kHz, X 14 mm, d) f0 100 kHz, X 10 mm

Measurement

(normalized Uj measured a) f0 FDM

10 mm,

=

and =

=

=

=

=

=

=

=

=

=

=

=

=

Scattering

62

The influence of the defect 1 and 2

mm

100 kHz

The

The calculated nomenon

increase

which

amplitude,

is

length

the scattered field

on

notches at the side of the hole

is

of the notch

changes

observed

leads to

length

about 20%> for the 1

is

and

(90°)

an

at

increased

At the shorter notch

(1 mm)

a

an

Fig is

4 13 and

mostly

4 14 also

Fig

area

Fig 4 13 for frequency of change of the mm

notch

interesting phe¬

forward scattered

suggests that the

the ratio between defect

governed by length and

notch

behind the notch

defect

in

center

notch and 50%> for the 2

mm

a

wave

longer notch (2 mm) has a qualitatively similar influence on the area behind (with a larger rel¬ ative change), but additionally generates a backscattered wave that significantly changes the amplitude in the area above the notch This characteristic can be seen more clearly from the difference in complex mag¬ nitude, taking also the phase information into account Fig 4 14 shows the mea¬ sured and calculated change of the complex magnitude for a 2 mm notch at the two excitation frequencies of 50 kHz and 100 kHz For the smaller ratio between defect length and wavelength a second, backscattered lobe is generated, that can not be observed for the larger ratio This suggests a nonlinear character of the scattering mechanism, making it difficult to model analytically The evaluation of develops,

the

hole with

shown a

agree well with the measurements and

that influences

a

an increase

mam

influence

and

on

wavelength, wavelength show similar

of the

size

The

the scattered field

as a

reduction of the

influences

on

the

scattered field This effect tudes

is

on one

also evident

in

Fig

4

15, where the measured and calculated ampli¬

radius around the hole hole and

hole with

are

displayed

Shown

notch at

are

the

amplitudes

for

for the excitation

undamaged angle frequencies of 20, 50, and 100 kHz, corresponding to wavelengths of 22, 14, and 10 mm respectively The FDM calculation matches the measured curves very well for all frequencies and accurately reflects the change due to the defect It can be clearly seen that the change in amplitude increases with rising frequency, from about 10%o (relative to the amplitude of the incident wave) at 20 kHz to 30%> at 50 kHz and 100%> at 100 kHz The 10% change in amplitude (seen here at 20 kHz in Fig 4 15a) was taken as the approximate detection limit of this method Smaller changes below 5%> can be detected in the controlled laboratory environment, but allowing for minor deviations in the setup or the rougher mea¬ surement conditions in an aircraft hangar (e g Chapter 5 3), this was deemed to an

be

on

a

a

2

mm

a

45°

the safe side

From the comparison between the numerical calculations and the measurements

for

a

variation of all

accurately

experimental parameters

model the scattered field around

a

it

can

be concluded that

hole with

a

we

can

notch using the FDM

Scattering

at

a

hole with

approach. Especially

a

defect

63

the amount of

change

in

amplitude

can

be well

predicted,

allowing a numerical study of the defect detectabihty in Chapter 4.5 instead of the time-consuming experimental realization of all possible parameter variations.

Fig.

4.15

Measurement and FDM calculation of

amplitude (normalized Uj

=

1),

r

=

11 mm;

1 mm, r0 10 mm, a 315°, measured: no defect (black dashed), a 2 mm notch (blue, solid); FDM calculation: no defect (red, dotted), a 2 mm notch

2h

=

=

=

=

=

(magenta, dash-dotted); a) f0

c)f0=100kHz,

4.5 A

Numerical

variety

=

20

kHz, X

=

22 mm;

50

kHz, X

=

14 mm;

X=10mm.

study of defect detectabihty

of parameters define the geometry of the

the defect and the incident detectable crack

b) f0

=

wave

and

can

have

hole, the position and size of

an

influence

on

the minimum

length. Namely the plate thickness, hole radius, notch length, and the wavelength (dependent on the excitation frequency) can vary relative to each other. Furthermore different positions of the notch, given by the angle between notch and the propagation direction of the incident wave, have to be

Scattering

64

considered in can

be

principle.

One of the four

shown that

easily

only

length parameters

selected

as a

thickness

hole radius of 10 mm,

wavelength detectabihty of

to

plate

a

of

ca.

from the evaluation of

4 5-

was

hole with

be

center

a

an

easy

comparison

frequency

of 100

defect

as

it

4.11 and

angles was already Fig. 4.12.

—

to the

case

was

kHz, corresponding

10 mm, and the notch at the side of the hole

Fig.

a

eliminated,

set to 1 mm, and the standard

defect at different

a

can

a

the relation between the sizes and not the absolute

size matter for the theoretical calculation. To allow

measurements, the

at

(a

studied in

=

»

-i

90°).

The

Chapter

4.4

1

m

'/•

"0

01

02

03

04

§8

06

01

§•

OS

a/A.

Fig.

4.16

complex change in magnitude due to notch at length a to wavelength X; 2h 1 mm, r0 10 mm, f [20 1000] kHz, X [22 2.3] mm: a 2 mm (blue, 1 mm (red, dash-dotted, squares), a 0.5 mm (black solid, dotted, circles), a diamonds); 2h 1 mm, r0 1 mm, f= 1 MHz, X 23 mm, a [0.25 2] mm (magenta, dash-dotted, squares). FDM calculation of maximum a

=

90°

versus

=

relation from notch

=

=

=

=

=

=

The main influence

=

=

=

=

the change in the scattered field due to the notch was seen Fig. 4.13, Fig. 4.14, and Fig. 4.15 in Chapter 4.4 to be the ratio between notch length and the wavelength of the incident wave. In Fig. 4.16 the maximum change of the complex magnitude (including phase information) on

from the evaluation of

Scattering

at

due to the

a

hole with

65

notch, normalized with the amplitude of the incident

for three notch

of

lengths

that for all three notch

lengths the in amplitude

further

significant increase wavelength. The

fifth of the

totally

same

different case, where for

between quarter and double the

a

Fig.

4.17

It

for

can

be

similar form and that

no

plate

notch

a

(r

=

2h)

thickness. The

plate

above is

curve

^

M

15

a

thickness. It

larger

seen

one

geometrically length is varied

linear increase up to

in

*

§

than about

a

the notch

same

Fig.

a

4.16.

,

10

I

IS

»)

Radius

[nrnj

bj

Riftuslitinsj

c)

Radius

[rrtffll

a]

Radius

20

ffflm}

detectabihty of a a) complex change in magnitude, maximum change (black, solid, circles), change 1 mm behind notch (blue, dotted, squares), change 1 mm before notch (red, dashed, diamonds); b) maximum amplitude at hole (red, dash-dotted, squares), amplitude at notch (black, solid, diamonds); c) change in amplitude; d) complex change in magnitude, f 1 MHz, X 2.3 mm, a 0.25 mm. FDM calculation for the influence of the hole radius a

=

90°, 2h

=

=

1 mm,

f

=

=

The two other parameters to be checked In

occurs

1

5

notch at

ness.

show

small hole

>

0

curves

characteristic also appears for

ratio of a/X =1/5 and rather constant

p

wave, is shown

half, full and double plate thickness, stretching the

of the relation between defect size and

interesting part seen

defect

a

Fig.

100 kHz and

4.17a the a

1

mm

complex change

100

kHz, X

=

r

on

10 mm,

a

the =

1 mm;

=

are

in

notch is shown for

the hole radius and the

magnitude a

for

a

center

plate thick¬ frequency of

variation of the hole radius. An

Scattering

66

increase of the maximum

change

is

visible, but

as can

be

at

seen

a

hole with

in

a

defect

Fig. 4.17b,

this

increase correlates

mostly with the higher amplitude of the scattered field at a Analyzing the change in amplitude (without phase information) in

larger hole. Fig. 4.17c, no

influence of the hole radius

on

this parameter

ever, the direction of the main lobes of the scattered

can

be found. How¬

changes up to about 15° for varying hole radii, due to the increased secondary scattering at a larger hole. For a smaller wavelength (higher frequency) and smaller notch length also no significant dependence of the detectabihty on the hole size is evident in Fig. 4.17d. Varying the ratio of notch length to plate thickness, while keeping the ratio between notch length and wavelength constant, no systematic dependence is recognizable in Fig. 4.18. While slight variations can be observed, overall the amount of change is rather constant.

Fig.

4.18

wave

length a and plate detectabihty of a notch at a 90°, 2h 1 mm, rg 5 mm, f [24 2120] kHz, X [20 1.25] mm, a/X 1/10: a) maximum amplitude at hole (red, dash-dotted, diamonds), amplitude at notch (black, solid, squares); b) change in amplitude, maximum change (black, solid, squares), change 1 mm behind notch (blue, dotted, circles), change 1 mm before notch (red, dashed, diamonds); c) complex change in magnitude. FDM calculation for the influence of the relation between notch

thickness 2h =

on

the

=

=

=

=

=

Scattering

at

Therefore it

a

hole with

can

be

a

defect

safely

67

concluded that the

mam

influence

the defect detect¬

on

abihty is the ratio between defect size and wavelength A maximum change of the complex magnitude of about the amplitude of the incident wave on average results in a change in amplitude of ca 10% at a distance of 1 mm from the notch and hole, where a measurement using the heterodyne laser interferometer can be conveniently conducted, and is therefore taken as the detection limit As can be 1/10 From a systematic seen from Fig 4 16 this corresponds to a ratio a/A, =

evaluation of the calculated and measured scattered fields around without

a

(described

defect, in

the scattered

it

Chapter wave

tion transducer and

gant NDT in

the

is

4

found that the second lobe of

4)

only generated

is

propagates backwards

might be

fuselage

of

an

the

in

measured with the

measurement with

The selection of

for

a

single

a

ratio of

general

same

transducer

a

a

hole with and

backscattered

a/A,

>

1/8

wave

This part of

direction of the excita¬

transducer, allowing

positioned

above

a row

an

ele¬

of rivets

aircraft

wavelength smaller than five times the notch length does not larger change in amplitude and thus a better detection of the defect With the current experimental setup higher frequencies are more difficult to han¬ dle, especially concerning the generation of a sufficiently high excitation ampli¬

result

a

in a

tude and the relative positioning of excitation transducer and measurement spot Due to these

experimental

difficulties the

frequencies that are higher than wavelength of the exci¬ the times eight length of the defect

use

of

necessary should be avoided The optimum range for the tation

signal

one aims

therefore lies between five and

to detect

68

Scattering

at

a

hole with

a

defect

5

Application

5.1

Outline

Measurements

Chapter

2 3

were

large

On

scattered field

to NDT

two

kinds

and thm aluminum

made

plates,

using

of

described

specimens,

the influence of

a

notch

in

the

on

studied For a single hole in the plate, notches were intro¬ angles and the change in amplitude was measured for different relations of wavelength (excitation frequency), plate thickness, hole radius, and notch length The comparison of these measurements to the numerical calcula¬ tions is described in Chapter 4 4 and the numerical study of the defect detectabillty in Chapter 4 5 To simulate the multiple scattering at a line of fasteners in an airplane fuselage, the scattered field around three holes and the detectabihty of a notch at one of the holes is investigated Broadband excitation and measurements at only one line or a single point are studied for fast and efficient monitoring was

duced at different

measurements

The influence of real

grown cracks

fatigue

on

the scattered field

tensile specimens The ratio of hole radius to specimen thickness

is

is

measured

the

in

same as in

fighter planes of the Swiss Air Force Two measurement series were made at the fatigue engineering center of RUAG Aerospace, Emmen The scattered field for different crack lengths is measured At optimized positions of the specimens the amplitude is monitored during the cyclic tensile loading in a servo-hydraulic test¬ ing machine to achieve an on-line measurement of the crack length

5.2

Measurements at

5.2.1

Influence of

The

dependence

From

an

Chapter

a

that the

between defect thick aluminum

on

the scattered field

of the scattered field

evaluation

4 5

notch

plates

size

of the mam

and

influence

wavelength

made

on

a

notch

amplitude change on

was

the defect

Shown here

analyzed

introduced

are

circles around

it

in

Chapter

was

detectabihty

4 4

found

is

in

the ratio

measurements at

a

1

mm

hole for several excitation

on a single plate, frequencies, corresponding to different wavelengths After a first measurement 14 mm), a notch of 2 mm (solid line) for the undamaged hole at f0 50 kHz (A, the thickness of the the hole boundary and the cut at was length through plate scattered field was measured again (dashed line) as shown in Fig 5 2 The notch =

was in

introduced at 315° to the

amplitude

x axis

of about 20%> of the

In the

=

vicinity of the notch,

amplitude

of the incident

a

marked

wave

can

change

be seen,

Application

70

while there

on a

the other side of the hole

second notch at 90°,

again

no

noticeable

change

a

of the hole where the second notch

was

change

occurs.

to NDT

By introducing

of about 20%> is observed

the side

on

introduced for the third measurement

(dotted line).

45

Fig.

§0

im

180

amplitude (normalized: Uj 1) 14 mm: no notch (solid), 315° and 90° (dotted).

Measured

5.1

fo

=

50

=

kHz, X

notch at

22S

=

2m

at

2

r

315

=

Wù

13 mm, 2h

mm

=

1 mm, r0

notch at 315°

Notch 2

Notch 2

[mm]

10 mm,

=

(dashed),

2

mm

[mm] ^

\ Notch 2

45

90

13&

180

226

270

[mm]

315

360

0

6

S

»

)S5

S

17Q

31S

M

Angle H

Fig. 5.2

amplitude (normalized: Uj 1) at r 13 mm, 2h 1 mm, r0 (solid), 2 mm notch at 315° (dashed), 2 mm notch at 315° (dotted); left: f0 20 kHz, X 22 mm; right: f0 100 kHz, X 10 mm. Measured no

=

=

Repeating the measurement resulting in larger (22 mm) about 10%o

=

=

notch

can

be

seen

=

at different center

10 mm;

and 90°

=

frequencies of 20 and 100 kHz, (10 mm) wavelengths, an influence of larger wavelength at 20 kHz on the left side of

and shorter

for the

=

=

Application

to NDT

Fig. 5.2,

while the

the

side.

71

for the shorter

change

wavelength

at 100 kHz is about 40%o

sufficiently high frequency has to be used for of small cracks, as evaluated in Chapter 4.5. 2 mm) with a Repeating the measurement at a thicker plate (2h right

Thus,

a

=

on

the detection

smaller hole

frequency of 20 kHz (A, 31 mm) the maximum change in complex amplitude is 30%o directly behind the notch (Fig. 5.3a). Measuring the amplitude again on a circle around the hole (Fig. 5.3c), a change in amplitude of 10% is visible, with a clear loss of symme¬ try in the measurement line. In contrast to the thinner plate with the larger hole, the change in amplitude is also noticeably on the opposite side of the hole, as the wavelength is about as large as the hole circumference. For a higher frequency 13 mm), the change in complex amplitude is about 100%o directly (100 kHz, A, at the notch (Fig. 5.3b). The influence on the amplitude measured on a circle is ca. 40%o in the vicinity of the notch. (r0

=

5

mm),

similar effects

are seen.

For

a

low

=

=

Notch 2

0

ß

B

1351»22S2?031S3ffi

o)

Fig.

5.3

Notch 2

[mm] 0

43

B0

13513Û22S2703153BÛ

d)

Angle [*)

[mm]

Angle fl

change in amplitude (normalized: Uj 1), 2h 2 mm, r0 5 mm; complex change in magnitude of the scattered field due to a 2 mm notch at 270°: 100 kHz, X 13 mm; a) f0 20 kHz, X 31 mm; b) f0 measured amplitude at r 7 mm, no notch (solid), 2 mm notch at 270° (dashed): 100 kHz, X 13 mm. c) f0 20 kHz, X 31 mm; d) f0 Measured

=

=

=

=

=

=

=

=

=

=

=

A defect

can

previous

measurement at

=

be detected from either an

change undamaged hole a

in the measured is

available,

amplitude or, if no a comparison to

from

Application

72

theoretical ment

on a

5.2.2

calculations,

or

from

a

disturbance

in

the symmetry for

a

to NDT

measure¬

circle around the hole

Measurements at

a

complicated geometry

40

| S

1 ,

f"

20

0 «

105

20

150

100

50

0

50

100

150

o

^3 0 x

-

50 axis

100

2 15 1

05 0

150

[mm] 01 005 0 -0 05 01

0 x

Fig

5 4

Amplitude (normalized 2h Ax

=

=

Uj

-

=

50 axis

=

=

150

[mm]

1) of the scattered field around three holes, 100 mm, measurement grid resolution

1 mm, r0 10 mm, hole distance d0 Ay 2 mm, fg 20 kHz, X 22 mm =

100

=

=

a) measured, b) calculated using classical plate theory, c) difference amplitude due to a 2 mm notch at 300° (center hole) The

measured

a single hole in an infinite plate is the simplest case to test the accu¬ validity of the measurements and theoretical descriptions However, more complicated geometries are present in engineering applications, e g a line of rivet holes in an aircraft fuselage In order to be able to detect defects in such samples, the combined scattered field has to be described and measured very accurately A line of three holes is drilled into a plate and measurements on a case

racy and

of

in

Application

Cartesian

to NDT

grid

are

73

before and after

performed

The measured combined scattered field incident

wave

has

a

different

can

be

amplitude, phase,

a

notch

seen in

and

cut at the center hole

is

5 4a At each

Fig

angle

of

agates radially outwards from the transducer, located 300 hole

incidence,

mm

hole, the

as

it prop¬

above the center

The chessboard-like pattern results from the interference of the different

scattered

This

be well described

theoretically by a superposition according to CPT, without considering secondary scattering (Fig 5 4b) After the first measurement (Fig 5 4a), a notch of 2 mm length is cut through the thickness of the plate at the center hole using a fine saw blade The notch simulates a defect, like a fatigue or corrosion crack at a fastener in an airplane component The scattered field is measured again and the difference in amplitude due to the notch is shown in Fig 5 4c At the free sur¬ faces of the notch, an additional scattered wave is generated, which changes the scattered field significantly up to about 10% in amplitude waves

case can

of the three scattered fields at each hole

5.2.3

Broadband excitation

As this measurement

grid is too time-consuming for industrial applications, sought [20] It can be seen in Fig 5 4 that even at some points close to the notch, the amplitude changes only slightly Thus, A broadband a measurement at a single frequency might not give enough data excitation was used and the amplitude spectrum on two lines in front of and behind the hole(s) was measured For the single, undamaged hole in the 2 mm thick plate, good symmetry of the measured amplitude spectra at y ±10 mm can be seen in Fig 5 5 The amplitude at each frequency is normalized, so that the a

on

a

Cartesian

faster measurement

was

=

average

over

the measured line

ment after the 2

mm

notch

in

was

x

direction

is one

Repeating

hole,

a

the

measure¬

clear asymmetry

is

higher frequencies and behind the hole Making similar ±40 mm) for the plate with three holes, in Fig 5 6 measurements at two lines (y the symmetry of the amplitude spectra for the undamaged plate and the signifi¬ cant change in amplitude due to the 2 mm notch are visible The change in ampli¬ tude again has the largest value of about 30%o for the higher frequencies It can be clearly distinguished from changes due to a slight movement of the measurement noted, especially

at the

the

introduced at the

=

position

Application

74

|«

I'!

ff

to NDT

Ȥi %{Tmn\

2% Fig.

5.5

Measured 2h

=

amplitude spectra (normalized)

2 mm, rg

=

at

lines in front of and behind

5 mm, excitation: linear sweep,

fg

=

kHz, X

20-100

=

one

hole,

31-13

mm:

10 mm (in front of hole); a) no notch, y -10 mm (behind hole); b) no notch, y -10 mm (behind hole); d) 2 mm notch at 270°, c)2mm notch at 270°, y 10 mm (in front of hole). y =

=

=

=

Fig.

5.6

Measured 2h

amplitude spectra (normalized)

1 mm, r0 notch: a) y

=

=

at

lines in front of and behind three

10 mm, excitation: linear sweep,

f0

=

20-100

kHz, X

=

hole,

22-10 mm;

40 mm (in front of hole); b) y -40 mm (behind hole); change 40 mm (in front amplitude due to a 2 mm notch at 300° (center hole): c) y hole); d) y -40 mm (behind hole). no

=

=

=

=

in of

Application

to NDT

75

In order to further reduce the time needed for the measurements, the non-normal¬

ized

amplitude spectra at two points, ca 30 mm away from the hole, are ana¬ lyzed On the side closest to the notch a marked change in amplitude can be seen for the higher frequencies in Fig 5 7 (left), while the change for lower frequen¬ On the far side of the hole (Fig 5 7, right) no change can be cies is again small seen up to about 70 kHz, and only a smaller change at the higher frequencies The measurements for the undamaged case are symmetric except for small devia¬ tions around 80 kHz From a change in this symmetry the notch can be detected with

fast measurement at

a

notch, either further sured

some

distance from the hole However, to pinpoint the

measurements close to the hole have to be made

amplitude spectra

have to be

compared

to calculations

the

or

mea¬

incorporating the

effect of the notch

%

20

30

«

SO

60

70

66

00

%

«0

20

30

«

Repawn IhHzj

Fig

Measured

5 7

r0

=

y

5.3

=

amplitude spectra

10 mm, excitation

(solid),

2

mm

40 mm,

x

=

points

at two

linear sweep,

notch at 300°

right

10

60

f mmmy

f0

=

(center hole),

20 mm, y

=

40

Measurements at tensile

in

70

front of three

20-100

kHz, X

measurement

iO

90

100

imt}

=

holes, 2h

22-10 mm,

position, left

x

=

no

=

1 mm,

notch

-20 mm,

mm

specimens

The measurement method is applied to fatigue cracks in tensile specimens Com¬ pared to the previously described measurements at a hole with a notch in a large, thm plate, several experimental constraints have to be considered Tensile speci¬ mens

with

a

reduced

cross

generate the fatigue cracks lic testing machine thickness

(Fig

5

8)

section around the hole must be

at the hole

Therefore or

a

width for

used,

in

order to

by cyclic tensile loading in a servo-hydrau¬ plate-strip like geometry with either increased the clamping at the ends is necessary For the

Application

76

two measurement series conducted at the

fatigue engineering

to NDT

center of RUAG

specimens with varying width were cut from sheet material used as planking in the fuselage of fighter jets. The hole radius (3.25 mm) is about the same size as the specimen thickness (3.17 mm). Aerospace, Emmen,

Fig.

5.8

Tensile

tensile

specimen

C9809 with

varying thickness, growth

two

fatigue

cracks at the

hole, foil

for resistance measurement of crack

Before the first measurement series in Emmen, made with old

some

experiments

in the labora¬

from RUAG

specimens Aerospace containing a newly manufactured specimens, to adjust the experimental para¬ meters and gain some understanding of the specific constraints in tensile speci¬ mens. The first series was run on short, standard specimens made from Al-2024 PL-T3. A significant influence of the crack on the scattered field was observed. However, due to the lacking experimental experience, not all results could be achieved with sufficient accuracy. The measurement parameters were optimized by further measurements on the damaged specimens in the laboratory at ETH. For the second measurement series, longer specimens made from Al-7075 PL-T6 were used, to avoid the influence of reflections of the wave at the clamping jaws. tory

at ETH

were

crack and the

5.3.1

Description

of the first measurement series

For the measurement series

the

fatigue laboratory

of RUAG

Aerospace, servo-hydraulic testing machine. Two types of laser interferometer were used. The single-point interferometer, giving a measurement of the velocity at one point of the structure, was mounted on the positioning system and placed on a table in front of the test¬ ing machine (Fig. 5.9 left). This allows as well the measurement of the amplitude Emmen the

at

experimental setup

was

built around their 100 kN

Application

at

one

to NDT

point during

77

the

cyclic

tensile

loading ('Monitoring'),

of the scattered field around the hole when the

as

the measurement

was interrupted two-point interferometer was mounted on the backside directly at the testing machine (Fig. 5.9 right). It allows the measurement of the difference between the movement of two points of the structure. The laser beams were adjusted to measure at points above and below the presumed position of the crack, and thus directly get the difference signal ('Difference'). The mounting on the vibrating machine and the focusing of the laser beams proved to be rather problematic. Retro-reflective tape had to be used to gain sufficient reflection of the beams on the specimen surface.

('Scan').

Fig.

5.9

cyclic loading

A

hydraulic clamping jaws with mount fixture of two-point laser power feed; single-point laser interferometer mounted on positioning system; right: specimen in hydraulic clamping jaws with two-point Left:

specimen

in

interferometer and

laser interferometer mounted in fixture

optical microscope could be placed on a bracket in the front to measure the length on the front surface optically (labelled cFR (front right) / cFL (front left)). Inserting a mirror into the hole, the crack length within the hole was mea¬ sured. It was labelled aFR (front right), aFL (front left), aBR (back right), aBL (back left), see also Fig. 5.10. The crack length on the back of the specimen could not be measured during the tests. The nomenclature, as seen when standing in front of the testing machine, was not changed, even when the specimen was turned around to measure a crack on the backside optically. Additionally a rotat¬ ing eddy current probe was used to gain a signal for the crack initiation, given by An

crack

Application

78

to NDT

in the measured resistance Measuring with the microscope or the eddy probe, it could happen that one touched and shifted the interferometers slightly, as they were only fixed provisionally This moved the measurement spot and lead to a jump in the measured amplitude during the monitoring measure¬ ments For the second series, the single point interferometer was affixed securely a

change

current

at the

Fig

testing machine

5 10

Geometry of tensile

Six tensile specimens

specimen

were

(long)

manufactured

face of the specimen around the hole the laser beam In four of the fine

saw

blade,

to

ensure

side

cracks at the

same

six

was

fatigue

specimens

blow-up

of hole and crack nomenclature

by RUAG Aerospace The front sur¬ polished to gam a better reflection of

specimens,

that the

(cFL/aFL) The last two cracks grew simultaneously

with

a

small starter notch

was

cut using

a

crack started to grow at the front left

were

left without

a

notch

so

that several

For specimen Z0004O05 the two

side of the hole

quarter-elliptical overlapped (aBL+aFL>3 17mm), while in

specimen Z0004O06 the cracks BL and FR grew

The first specimen

was

subjected

to

cyclic

tensile

loading

with

a maximum

ten-

Application

to NDT

79

sile stress of 150 N/mm was

according (R

For the next five specimens the

lowered to 135 N/mm

=

0

1),

to

to achieve

sinusoid with

a

a

slower crack

maximum

growth

tensile stress

The force varied

amplitude of 7 5 kN around a median of 9 2 kN The was constantly subjected to tensile stress 15 Hz The monitoring measurement was triggered a measurement was always taken at the maximum

an

that the specimen

so

loading frequency was set to by the loading force, so that force, when the crack

was

open

Monitoring

measurements

were

made for all

six

specimens

typical amplitude, difference, and crack length measurement can be seen in 5 11 The crack growth began at ca 30 000 cycles from the starter notch At 60 000 cycles the crack had grown through the thickness of the specimen and at 66 700 cycles reached a critical length, so that the measurement was stopped Two excitation frequencies of 20 kHz and 40 kHz were used, resulting in wave¬ lengths of 38 mm and 26 mm, respectively The wavelength at 20 kHz was selected as approximately the same as the width of the specimen to achieve a standing wave mode across the width As the maximum amplitude of this mode is at the free side boundaries of the plate strip and the wavelength is large com¬ pared to the thickness and crack size, this frequency proved not to be very sensi¬ tive to small defects at the hole, and usually only a significant change in the measured amplitude could be seen when the crack had grown through the thick¬ ness of the specimen The measurement of the phase of the signal showed a much larger variance than the amplitude, and while showing similar effects was less reliable The further evaluation concentrated on the changes in the measured amplitude at 40 kHz For the first 15 000 cycles a variation of both monitored signals can be seen, without any crack growth As the specimen is rather short and the wavelength rather long, reflections of the flexural wave at the clamping jaw can not be sepa¬ rated in time from the first incident wave The specimen moved slightly during the initial cycles in the clamping jaws and this caused the observed change in amplitude For the second measurement series, longer specimens and higher fre¬ A

Fig

quencies

were

used, and this initial setting

was

not observed any

sured values then stay rather constant till about 55 000 crack

length

of about 2

mm

A noticeable

increase

more

The

mea¬

cycles, corresponding

to

a

marks the detection of the

crack, and at 59 000 cycles especially the difference measurement shows a sig¬ nificantly stronger increase The two jumps in the difference measurement are due to focusing problems of the interferometer As the crack was optically mea¬ sured to have grown through the thickness between 57 500 and 60 000 cycles, the strong

increase

in

the difference measurement marks this point

The scattering

Application

80

characteristics at this

point change,

not

as

only

the

bending

to NDT

stiffness is reduced

due to the

quarter-elliptical crack, but a scattering and therefore phase difference between the two points in front of and behind the crack occurs. The increase in the amplitude of the difference measurement shown is too small after 62 000 cycles as the measuring range was exceeded.

tooo

i soi

ma =

4m

fc

^ Ü

Fig.

5.11

!~

Measured r0

=

amplitude during

3.25mm, f0

=

growth, specimen Z0004K03, 2h 3.17mm, 26mm; top: measured amplitude (single-point

crack

40kHz, X

=

=

interferometer) at x -4 mm, y 0.5 mm; middle: measured difference amplitude (two-point interferometer) between x -4 mm, y ± 0.75 mm; bottom: optically measured crack length (diamonds: apL, squares: Cj?l). =

=

=

The

=

specimens also showed the beginning amplitude for a crack of about the 1 2 For Z0004O05 to mm mm. length specimen eddy current mea¬ surement erroneously indicated a crack at the back side of the specimen after 20 000 cycles and therefore the specimen was turned around. This results in the jump in amplitude as the measurement position changes (Fig. 5.12). No crack could be found optically and the first cracks developed after 80 000 cycles. Sev¬ eral cracks developed simultaneously as the position was not given by a starter monitoring

measurements

strong variations in the

notch.

on

the other short

and

a

marked increase in

Application

to NDT

i

Fig.

5.12

too

Measured ro

=

81

amplitude during

3.25mm, fo

=

growth, specimen Z0004O05, 2h 3.17mm, 26mm; top: measured amplitude (single-point

crack

40kHz, X

=

=

interferometer) at x 4 mm, y 1.5 mm; middle: (two-point interferometer) between x -4 mm, y measured crack length. =

=

=

measured difference =

±

0.75 mm; bottom:

amplitude optically

The

amplitude was now measured on the side indicated L. A rise in amplitude change in phase value is observed after ca. 100 000 cycles, when the longest crack in the specimen is about 2.5 mm long, and the longest crack on the L side has a length of 1.5 mm. The difference was measured on the other side of the hole (R). From about 80 000 cycles a slow increase in amplitude and at 107 000 cycles a strong increase is visible, marking that crack FR has grown through the thickness. The jump at 85 000 cycles in the amplitude measurement is due to the shifting of the interferometer at an interruption for a scan measurement. At 90 000 cycles the maximum tensile stress level was lowered to 100 N/mm to achieve a slower crack growth. This again results in a jump in the amplitude mea¬ and

surement.

The

monitoring scans showed a significant change in amplitude for a crack length of about 2 mm. When the crack had grown through the thickness of the specimen, the amplitude increase got stronger, especially the increase of the dif¬ ference measurement. Problems arose from the interruptions of the monitoring measurements to either make

a

scan

measurement of the scattered field

or

to

Application

82

measure

current

the crack

probe

A

length optically

slight shifting

with the microscope

the two-point interferometer

focus

As the difference measurement

small crack

For

some

use

eddy

Espe¬

sensitive and often lost

significantly more sensitive to through the plate thickness), it was not

was

this measurement method for the second measurement

of the specimens the monitoring measurement

the scattered field at certain crack

sure

to be very

proved

the crack grew

lengths (before

with the rotating

of the interferometers could not be avoided

cially

decided not to

or

to NDT

lengths ('Scan')

was

interrupted

The

scan

series

to

mea¬

measurements

done without tensile loading of the specimen to save on time the machine hydraulic pump were running The measured differences in amplitude even large crack lengths were very small compared to the expected changes This

were

and for is

due to the effect of crack

for the small excitation

closure, where the

amplitudes

two faces of the crack touch and

of the flexural

wave

appear to be almost

intact

5.3.2

Intermediate measurements in the

laboratory

The minimal tensile force to open the cracks surements

in

a

mechanical tensile

specimen Z0004K04 with

evaluated

laboratory

in

apparatus, shown

loading

crack about 7 5

a

was

mm

long,

no

in

Fig

mea¬

2 6

For

strong asymmetries

in

amplitude or phase between the two sides of the hole are visible without tensile loading in Fig 5 13 For a tensile force of 10 kN the peak in amplitude and jump To evaluate the necessary force even for in phase at the crack can be seen smaller cracks, specimen Z0004O05 with three cracks was subjected to different tensile forces and the scattered field measured For the scattered field does not sile

loading

force for the

change

scan

any

more

measurements

a

tensile force above 7

with the force was

set to 10

(Fig kN,

5

14)

to be

on

kN,

The ten¬ the safe

side

Higher excitation frequencies of 75 kHz and 160 kHz corresponding to shorter 12 mm), well wavelengths were tried Good results were found for 160 kHz (A, below the cutoff frequencies of the higher wave modes In Fig 5 15a, the sharp increase in amplitude before the through-crack (below the hole) and also the high peak at the two part-through cracks on the other side of the hole is visible =

Application

to NDT

83

Hi o

w

Fig

5 13

amplitude

=

=

10

0

10

phase with and without tensile loading, specimen (cj?l=7 42 mm, after 85 000 cycles), 2h 3 17mm, 3 25 mm, f0 40 kHz, X 26 mm a) amplitude, F 0 kN, b) amplitude, 10 kN, c) phase, F 0 kN, d) phase, F 10 kN

Measured

Z0004K04, r0 F

to

and

crack

one

=

=

=

=

=

=

32) ! «1

Ktmml

«[«ml

I

&

m s

Fig

5 14

«

amplitude with different tensile loading, specimen Z0004O05, three 2 13 mm, cFL =4 42 mm, aFL 2 00 mm, (cBL=4 11 mm, aBL 6 16 mm, after 115 000 cycles), 2h 3 17 mm, rg 3 25 mm, fg 40 kHz, CFR X 26 mm a) F 0 kN, b) F 5 kN, c) F 7 kN, d) F 10 kN Measured cracks

=

=

=

=

=

=

=

=

=

=

=

Application

84

Fig.

5.15

Measured

amplitude

and

phase

with tensile

loading (F

Z0004O05, three cracks (cgL=4.11mm, aBL apL

r0

Fig.

5.16

=

2.00 mm,

3.25 mm,

=

Measured

=

Plotting

the

dent wave,

lines

=

=

mm:

parallel

three

=

12

kN), specimen

10

=

4.42

=

Cj?l

mm,

cycles), a) amplitude, b) phase.

after

mm,

2.13

2h

115 000

=

mm,

3.17mm,

10 kN), length with tensile loading (F 2.13 mm, (cBL 4.11mm, aBL 6.16 mm, after 115 000 cycles), mm, cFR f0 160 kHz, X 12 mm: a) left side (L), b) right to

=

cracks

=

=

=

=

=

(R).

amplitude one

position

from the hole

on

2.00 aFL 3.25 mm,

can

on

lines

parallel to the propagation direction of the inci¬ approximate crack length on both sides of the through crack (Fig. 5.16b) a strong drop in amplitude

discern the

hole. On the side with the at the

kHz, X

Z0004O05,

4.42 mm,

3.17 mm, r0

=

side

=

amplitude

specimen CpL 2h

f0

Cj?r 160

6.16

=

=

to NDT

of the crack

boundary).

(x

=

-0.75

mm)

is visible out to y

One measurement line further out

=

(y

-9 =

mm

-9.5

(6

mm

mm)

no

Application

such

sharp

to NDT

85

decrease is

visible, corresponding well with the optically measured

crack

length of 6.16 mm. For the two quarter-elliptical cracks the increase in amplitude before and decrease at the crack is not as sharp, but clearly visible till 7 mm compared to the measurement at y 8 mm (Fig. 5.16a). This again cor¬ y relates well with the crack lengths between 4 mm and 4.5 mm. From such mea¬ surements where the laser interferometer is moved on lines parallel to the length of the specimen, an on-line measurement algorithm of the crack length during the cyclic tensile loading might be derived, without the need for theoretically solving the inverse problem. The wave propagation characteristics in the new, longer tensile specimen were measured as a preparation of the second measurement series. Good results were achieved at 40 kHz and 160 kHz, with a sufficient time separation between the incident pulse from the piezoelectric excitation transducer and the reflections at the clamping of the specimen ends. =

Fig.

=

5.17

CO

-40

-20

0

20

«o

«0

«

^0

-ÎQ

0

20

«

00

amplitude

Measured

cracks, 2h

=

and

phase

3.17 mm, rg

=

with tensile

3.25 mm,

fg

=

loading (F 160

=

kHz, X

=

10

kN), long specimen, no mm: a) amplitude, b)

12

phase. At the

-50 mm) in Fig. 5.17, a quite complicated position of the excitation (x and distribution underneath the transducer is visible. Multiple amplitude phase reflections over the width of the specimen result in a standing mode across the width with amplitude maxima approximately a wavelength (12 mm) apart. The =

wave

propagates along the specimen with

from the lines of 271

a

rather

Application

to NDT

straight wavefront,

visible

phase measurement. An amplitude variation over the width is visible. Around the undamaged hole a more or less symmetric scattered field with the desired high amplitudes at the sides of the hole (places of crack growth) develops. The visible asymmetries result from an off-center hole (up to 0.4 mm) and slightly inclined bonding of the piezoceramic plate. Different sizes for the excitation transducer were tested, namely a narrower (4 mm instead of 8 mm) and shorter (not across the width of the specimen) plate. For the result¬ ing incident wave no straighter wavefront or more uniform amplitude distribution jumps

in the

could be obtained.

Fig.

5.18

Measured time traces,

kHz, X

long specimen,

no

cracks, 2h

=

3.17mm, r0

=

3.25mm,

a) time 0 kN, blue (amplified 40 dB): black F F 5 kN, red F 10 kN; b) time signal of laser interferometer: black F 0 kN, blue F 5 kN, red F 10 kN; c) comparison shape of time signals (different amplitudes), F 10 kN: black laser, red piezoelectric transducer.

Îq

=

160

signal

of

=

12 mm, measurement at

piezoelectric

=

transducer

=

=

x

=

-3.5 mm, y

=

-4.5

mm:

=

=

=

=

For further

monitoring measurements, small piezoelectric plates (1 mm by 1 mm, were applied as measurement transducers in the vicinity of the hole, about 3 mm in front of the expected crack position, i.e., at the position where a large increase in amplitude is expected. The wiring on the upper electrode was soldered at 350°C as the surface is too small to use regular glue. The maximum 1

mm

thick)

Application

measured of 400

Fig.

to NDT

voltage

volt

87

of

a

few millivolts is small

peak-to-peak

and

a

certain

5.18a before the main measured

pulse.

compared

amount

to the excitation

of cross-talk is

The measurement

fied 40 dB with the built-in

loading

5.3.3

Fig.

of the

Description

5.19

mm

Servo-hydraulic testing machine with long specimen point laser interferometer mounted on machine

were

measured time the laser inter¬

of the second measurement series

For the second measurement 500

in

voltage is ampli¬ sensitivity to ten¬

amplifier of the KH 3988 filter. The specimen is small and good agreement of the traces from the piezoelectric transducer and a measurement with ferometer at the same position can be seen in Fig. 5.18c. sile

voltage

visible

manufactured

in

clamping jaws

series, eight tensile specimens with

by

RUAG

Aerospace

a

and

single-

length

of

from Al-7075 PL-T6 sheets

Application

88

with as

a

the

thickness of 3.17

specimens

mm.

Six

specimens

for the first series. Two

had the

specimens

same were

to NDT

hole radius of 3.25 made with

a

mm

hole radius

of 3.12

mm for measurements with fasteners in the hole during the cyclic tensile loading. For the new material, experience with the fatigue loading had to be gained, and for the first two specimens the level of the maximum stress was var¬ ied during the experiments, leading to jumps in the measured amplitude, as seen in Fig. 5.20. The maximum stress was then set to 100 N/mm .

äOO

300*

toad sum a Igvsl m 100 Nar

Fig.

amplitude using single-point laser interferometer during crack growth, specimen Z0004OK11, 2h 3.17mm, r0 3.25mm, f0=160kHz, X=12mm; top: measured amplitude with two changes in stress level; bottom: optically measured crack length. Measured

5.20

=

a problem occurred with the glue layer used to piezoelectric plates to the specimens. Quite a few of the plates became loose during the cyclic loading. No such problem had occurred during the first measurement series, but the failure is probably due to a too thin layer of glue. The plates were reapplied with a thicker glue layer. This was a problem espe¬ cially for the measurement transducers, as they are rather small. They could not be reapplied at exactly the same position without taking the specimen out of the testing machine and were therefore left off. Another problem that occurred with the piezoceramic discs as measurement transducers was the significant cross-talk from the excitation signal. That pulse had to be eliminated with the time window, causing larger uncertainties. For further applications of this measurement tech¬ nique, the wiring should be better shielded.

During

this measurement series

=

bond the

Application

to NDT

89

f §»

1

12

12

Cyaass

41

Fig.

5.21

toe*

amplitude during crack growth, specimen Z0004O13, 2h 3.17mm, a) single-point laser interferometer, f0=160kHz, X=12mm; r0 b) piezoelectric transducer, fo 160kHz, X 12 mm; c) single-point laser interferometer, f0 40 kHz, X 26 mm; d) optically measured crack length. Measured =

=

3.25mm:

=

=

For four of the

=

=

specimens comparable monitoring measurements during crack growth amplitude was monitored using the single-point laser interferometer attached securely to the servo-hydraulic testing machine and the piezoelectric discs. Typical measured signals can be seen in Fig. 5.21, where the crack started to grow at 140 000 cycles and went through the thickness of the specimen at 153 000 cycles. The measurement with the laser interferometer at an excitation frequency of 160 kHz shows an increase around 140 000 cycles, which gets more significant around 153 000 cycles, when the crack penetrates through the thickness. However, the variation of the measured values is large, compared to the measurement at 40 kHz and using the piezoelectric transducer at 160 kHz. Problems with the trigger on the maximum force occurred, which might have caused the measurement to start at slightly different positions of the specimen, as the measurement spot moves with the tensile loading of the specimen. The ampli¬ tude measured at 40 kHz shows the increase slightly later, but the variation in the measured signals is also significantly smaller. In Chapter 5.3.5, the measured amplitudes vs. the crack length are compared for the different measurements. The measured signal from the piezoelectric transducer (Fig. 5.21b) shows a significould be made. The

Application

90

cant variation before

smaller This

crack

a

is

present and the relative change

in

amplitude

is

being positioned farther away from the crack, but also due to the measurement problems described above Therefore the signals measured from the piezoelectric transducers are not as sensitive to the crack growth as the signals from the laser interferometer However, this type of transducer shows promise when its handling is improved in further measurement series and is a lot cheaper than a laser interferometer is on

the

to NDT

side due to it

one

For the measurement with

around 300 000

cycles,

tener The fastener

had

was

a

very

fastener

in

the

long compared

hole, the crack

initiation time

to the measurements without

removed at the interruptions to

measure

whether

a

a

was

fas¬

crack

eddy current probe and the microscope This caused amplitude and lead to a rather strong variation of the mea¬ sured signal No significant change in the measured amplitude could be seen, until the crack had grown to about 3 mm length and was visible next to the head emerged,

jumps

in

using the

the measured

of the fastener Due to time restriction

on

the machine use,

the second specimen with the fastener could be made

on

mass

of the fastener leads to smaller

no

measurement for

Further studies should be

measurabihty of small cracks The amplitudes of the flexural motion of the around the hole and the fastener plate clamps the free flanks of the crack together, of of the flexural wave It is possible that higher transmission a means providing wave modes with the largest displacement around the middle of the thickness might be better suited

Fig

5 22

the influence of the fastener

run

Detail of fastener wiring

in

on

the

hole with two measurement

piezoelectnc

transducers and

Application

to NDT

91

Influence of

5.3.4

a

crack

on

the scattered field

The

monitoring measurement at specimen Z0004O17 (without a starter notch) was interrupted five times to make scan measurements of the scattered field at different crack lengths. The first measurement was taken after 50 000 cycles for the undamaged hole, the second after 115 000 cycles for a small quarter-elliptical crack on the back side (aBR 1.08 mm), the third for a larger quarter-elliptical crack on the back side (122 000 cycles, aBR 2.65 mm), the fourth when the crack had entirely penetrated through the thickness of the specimen (126 000 cycles, cFR 2.53mm), and the fifth for a longer crack (130 000 cycles, =

=

=

cFR

=

4.76

mm). 50

—2

5

Fig.

5.23

Measured 2h

a)

=

no

10

Hi

*i«flflfffl§i*&

ill S

,»J|^B^^1,l|li_

—2

i

=

crack; b) aBR

3.25 mm, =

f0

2.65 mm;

=

160

c) cFR

kHz, X =

the measured

tion

=

12

mm)

specimen Z0004O17, =

12

2.53 mm;

For the

quarter-elliptical cracks, frequency of 160 kHz (A,

II! S

,jâË^m^^'r%

amplitude (normalized Uj=l),

3.17 mm, r0

ill

H^aîillfllufas

one

crack,

mm:

d)

change

is rather

cFR

in

=

4.76

mm.

amplitude

small,

as

at

can

an

be

excita¬ seen

in

Fig. 5.24a, b. For the smaller crack, the change in amplitude is only about 10%o of the amplitude of the incident wave and not significantly larger than the variation of the measured values. With increasing crack length, the difference in amplitude is about 20%o, but still not clearly visible from an asymmetry of the scattered field in Fig. 5.23b. At the part-through crack, a mode conversion can take place, and for the detection of very small cracks the

gated.

use

of other modes should be investi¬

Application

92

Fig.

5.24

Measured

crack, 2h

a)

aBR

=

change

in

=

a

shadow

area

=

=

When the crack has grown

crack,

amplitude (normalized Uj

3.17 mm, r0 3.25 mm, f0 160 1.08 mm; b) aBR 2.65 mm; c) Cj?r =

through

the entire

behind the crack and

tered field is visible in

Fig.

5.23. The

l), specimen Z0004O17, 12 mm: kHz, X =

one

=

=

2.53 mm;

thickness,

d) a

Cj?r

=

peak

4.76

mm.

in front of the

clear loss of symmetry of the scat¬

increase in

directly change in amplitude is about 100%o, but the amplitude close to the hole can again decrease with increasing crack length. Further away from the hole and crack, the change in amplitude can be up to 50%o, allowing the detection of such a through-thickness crack from a measurement at some distance from the hole. Analyzing the phase measurements, no change is visible for the quarter-elliptical cracks in Fig. 5.25, as only very locally the bending stiffness is reduced. Once the crack is through the thickness of the specimen, it presents an obstacle for the flexural wave and the measured phase changes noticeably. in front of the crack. Even for

largest

a

to NDT

small crack

a

Measurements of the scattered field

were

quency of 40

a

(cFR

=

amplitude

2.53

mm),

occurs

the

also made at the lower excitation fre¬

of 26 mm. As the wavelength large compared to the hole radius, the increase in amplitude of the scat¬ tered field at the undamaged hole (Fig. 5.26) is noticeably smaller than for 160 kHz. For the part-through cracks the change in amplitude is small and not visible in a direct comparison of measured amplitudes (Fig. 5.27). It can only be is rather

kHz, corresponding

to

wavelength

Application

seen

from

to NDT

a

93

comparison

information. Here for the ible in where

of the

Fig. 5.28, but still much smaller the complex difference is more

thickness similar conclusions

Fig.

complex

5.25

Measured

f0 a)

=

measured

larger part-through

can

crack

a

values, including the phase

change

of about 15% is vis¬

than for the measurement at 160 than 50%o. For the cracks

kHz,

through

the

be drawn.

phase, specimen Z0004O17,

one

crack 2h

=

3.17mm, r0

=

3.25mm,

160kHz, X=12mm:

no

crack; b) aBR

FDM calculations

=

2.65 mm;

c)

Cj?r

=

2.53 mm;

d)

=

4.76

mm.

a through-thickness notch Comparing the measured and calculated changes in Fig. 5.29 for the two excitation frequencies of 40 kHz and 160 kHz, a qualitative agreement can be seen. The important changes in the scattered field, like the peak in front of the crack and the places of increase and decrease in amplitude are accurately found. However, the quantitative agreement is not as good as for the case of a notch in a thin plate, shown in Fig. 4.10. This is on one side due to the modeling of the crack as a notch, disregarding the sharp tip with the stress con¬ centration. On the other side, experimental deviations like the amplitude modula¬ tion over the width of the specimen and the off-center position and direction of the crack are not considered for the FDM calculations. To incorporate these effects, a much finer grid for the finite difference calculations around the notch on

the

are

used to calculate the influence of

Cj?r

scattered field.

would be necessary.

Application

94

I«

I:

-10

%{mm]

-s

I:

û

to)

x

to NDT

[mm]

^' 10

il:

-5

d)

Fig.

5.26

amplitude (normalized Uj=l), 3.25 mm, Îq 40 kHz, X crack; b) aBR 2.65 mm; c) cFR 2.53

«[»m]

specimen Z0004O17,

Measured 2h

a)

3.17 mm, rg

=

no

=

=

=

=

=

26

one

crack,

mm:

d)

mm;

cFR

=

4.76

mm.

P02 f.'

I»; • .

s

»

IL x[mm]

ml

l<

L; 41. K[miH

Fig.

5.27

Measured

crack, 2h

a) aBR

=

change

in

d)

x

amplitude (normalized Uj

[mm]

=

l), specimen Z0004O17,

3.17 mm, rg 3.25 mm, Îq 40 kHz, X 26 mm: 1.08 mm; b) aBR 2.65 mm; c) cFR 2.53 mm; d) cFR =

=

=

=

=

=

=

4.76

mm.

one

Application

to NDT

95

10

Fig.

5.28

complex

Measured

Z0004O17,

2h

2.65mm; aBR X 12 mm, aBR

=

b)f0

=

=

change =

=

r0

X

40kHz,

2.65 mm;

=

d) fg

0

S

10

magniitude

in

3.17mm,

-5

=

(normalized Uj=l), specimen X 26mm, a) f0 40 kHz, 26mm, cFR 4.76 mm; c) f0=160kHz,

3.25mm:

=

=

=

=

160

kHz, X

=

12 mm, Cj?r

=

4.76

mm.

15

I

I

1

nie«

^m

Moi

noOS

<%»_,

-0.5

15 0

10

0

10

I

I

10

10

5

s

°

"-6

0

1

-10

-10

5.29

-0.5

-10,0

0 m

Fig.

0

-s

0

fmmj

n{mml

change in amplitude (normalized Uj 1), specimen Z0004O17, one crack (cFR 4.76 mm), 2h 3.17 mm, r0 3.25 mm; measured: a) f0 40 kHz, X 26 mm; b) f0 12 mm; FDM 160kHz, X calculation: c) f0 40 kHz, X 26 mm; d) f0 160 kHz, X 12 mm. Measurement and FDM calculation of

=

=

=

=

=

=

=

=

=

=

=

=

Application

96

5.3.5

On-line

monitoring

of crack

to NDT

growth

05 §i23ase?i»io

Crack

Fig

5 30

tengtli [mml

changes in amplitude using single-point laser interferometer against optically measured crack length, amplitudes normalized as one at zero crack length, 2h 3 17 mm, r0 3 25 mm, f0 160 kHz, X 12 mm specimen Z0004O13 (crack FR, black), specimen Z0004O14 (crack BR, red), specimen Z0004K15 (crack FR, magenta), specimen Z0004O17 (crack BR, blue) Measured

=

The sensitivity and

=

repeatability

=

=

of the measurement method

can

be best

tained when comparing the results of the four available monitoring ments for the different specimens

each specimen with the

The measured

amplitudes

are

normalized for

length, and plotted the measured crack for excitation an optically lengths frequency of against 160 kHz m Fig 5 30 A significant increase m amplitude, larger than the varia¬ tion at zero crack length, can be seen m all measurements for a crack length of 2 mm, and therefore a crack of this length can be certainly detected The ampli¬ tude rises sharply when the crack grows through the thickness of the plate, and then decreases for longer cracks, as is also seen m the scattered field m Fig 5 23 The variation between the different monitoring curves is rather large This is due as

zero

crack

well to the fact that the cracks start to grow at different

back of the specimen, at

amplitude

measured at

ascer¬

measure¬

slightly

as

to

locations,

e

g front

or

the setup and thus measuring

slight misalignments For such a high frequency

different locations

m

it

can

be found from

Application

to NDT

97

FDM calculations that

of

half

only

a

even

small

millimeter have

an

changes

in

influence

the crack

on

or

measurement

the measured

amplitudes

position of up to

20%

Comparing

one

of the measured monitoring

thro ugh-thickness

notches

curves

to FDM calculations

for

length, good agreement is found in Fig 5 31 The increase in amplitude for small crack lengths is over-estimated as the FDM calculation assumes through-thickness notches, while the cracks in the specimen are still quarter-elliptical

01

of varying

23*S06?»f10

Gracfe

Fig

changes in amplitude using single-point laser interferometer against optically measured crack length, amplitudes normalized as one at zero crack length, 2h 3 17mm, r0 3 25 mm, f0 160 kHz, X 12 mm specimen Z0004O13 (crack FR, black diamond), FDM calculation (through notch, blue, squares) Measured

5 31

=

Doing Fig

length fmml

a

similar

analysis

=

=

for the excitation

=

frequency of 40 kHz, it is found in amplitude curves is significantly

5 32 that the variation among the measured

smaller and again agrees well with the FDM calculation The smaller variation due to the

larger wavelength (26

ment less sensitive to

length above

compared to

12

mm), making

the

is

measure¬

geometric variations, but also to small changes in the crack significant change in amplitude for all measurements can be seen only crack length of 2 5 mm, close to the thickness of the specimen

A a

mm

Application

98

to NDT

Craok tenglh fmml

Fig

changes in amplitude using single-point laser interferometer against optically measured crack length, amplitudes normalized as one at zero crack length, 2h 3 17 mm, rg 3 25 mm, fg 40 kHz, X 26 mm specimen Z0004O13 (crack FR, black), specimen Z0004O14 (crack BR, red), specimen Z0004K15 (crack FR, magenta), specimen Z0004017 (crack BR, black), FDM calculation (brown, squares) Measured

5 32

=

=

=

=

The on-line monitoring measurements allow 2

sured

a

For further measurement series,

length signals should

mm in

be eliminated and

certain detection of cracks sources

comparable

curves

1

mm

and

same

might

position This will allow

a

is

predict

the

changes

the

given

by

a

ca

mea¬ new

small starter

better comparison of the measured

reduce the minimal detectable crack

In order to better

in

measurements with the

specimen geometry run, where the position of the crack

notch at the

of variations

in

length

to the order of

the scattered field due to small

cracks, the finite difference model should be improved

to incorporate a finer grid part-through cracks For the detection of small cracks in the tensile specimen, possibly with a fastener, a further study might investigate the experimental suitability of higher guided wave modes or

around the crack and

Rayleigh

waves

tures like

an

a

The

aircraft

challenging

task

possibly

model

application of the fuselage with rows

described method to of

fasteners,

large

real-life struc¬

rivets and holes will

provide

Conclusions and Outlook

6

Starting in

from

specific problem, the detection of fatigue cracks at fastener holes general experimental and theoretical study of the underlying prob¬ performed, employing guided waves The model system investigated

aircraft,

lems

was

a

a

wave mode A0, a flexural arbitrary angle With the insight gained from this fundamental research, the developed measurement method was re¬ applied to the specific problem In collaboration with the fatigue engineering center of RUAG Aerospace, Emmen a monitoring system for the crack length in tensile specimens was implemented and found to allow the reliable detection of was

the scattering of the first antisymmetric Lamb

wave, at

a

circular hole with

a

notch at

an

defects However, the nondestructive testing method range of problems,

of which

are

mentioned

is

usable for

a

much broader

Chapter 1 1 Almost all tech¬ nical systems contain thm-walled structures like plates and shells, connected with joints, at which stress concentration and damage development can occur Guided waves allow a fast measurement of large areas of the structures, and therefore a significant cost reduction compared to conventional testing methods can be some

in

achieved

6.1

Measurements

With the chosen measurement ural

Improving

tering

in

plates

method, the

excitation and measurement of flex¬

with

was

performed

a

accuracy

the precision and further automating the measurement

measurements

Therefore piezoceramic transducers in

good

and

repeatability procedures, the measurement range of the setup was significantly extended compared to previous studies at the institute New types of excitation transducers were investigated Electromagnetic acoustical transducers were built as point and line sources and the achieved excitation could be well modelled Though EMATs have the large advantage of giving a non-contact wave excitation, the frequency range of the prototypes built was limited and further improvements, especially concerning the electronics aspects will be necessary until they can be employed for actual scat¬ waves

broad

frequency spectrum

were

with

a

used, allowing the

excitation of

waves

linear transfer function and sufficient

amplitude Line excitation was achieved by using custom cut piezoelectric ceramic plates, resulting in waves with a nearly plane wavefront in the strip-like tensile specimen A simplified model of the excitation was implemented for the low frequency range, and could be extended by including more physical effects

100

Conclusions and Outlook

The part that

was

significant

a

ing

ducer

is

influence

an

ratio and

cancelling an

types of

the

coupling,

,

the

glue layer,

of the measurements

repeatability

is

several measurements, improving the

hangar signals

can

occur

The cost of the piezoceramics

into the structure for

excitation time

i e

a

hav¬

Since the excitation trans¬

out spurious influences that

aircraft

permitting the integration

was

the transfer function

over

averaging

environment, like Two

to model

on

fixed to the structure, the

This allows noise

problematic

in is

excellent

signal-toa

harsher

rather

low,

permanent on-line monitoring

used, either narrowband signals with

were

the energy concentrated around the center

frequency, or broadband signals, allowing the extraction of information over a range of frequencies Due to the good signal-to-noise ratio, good results could be obtained with both approaches While the measurement at a single frequency is more straight-forward and allows an

easy

comparison

to

theoretical

achieved with the evaluation of

calculations, faster

broadband

measurements

were

measured at

a single point signal, The scattered field was measured using a heterodyne laser interferometer, mounted on a positioning system and moved parallel to the plate This allows an automated, non-contact, pomtwise measurement of local variations of amplitude and phase in the scattered field, that can not be achieved with most contact-type transducers The whole scattered field on a measurement grid around obstacles like a hole with and without a defect was measured, gaming an understanding of

the geometry of the scattered

by

a

waves

FFT gives the local values of

The evaluation of the measured time

amplitude

series

and allows the direct

com¬ phase like multicomplicated signals, appropriate digital signal processing

parison to the theoretical calculations Even

mode

and

more

signals could be evaluated using the study in this thesis was confined to the first antisymmetric Lamb wave mode Good experimental experience and know-how for this mode exists The mea¬ A0 surement of changes in the amplitude and phase of the scattered field overcomes the problems associated with the dispersive propagation characteristics The A0mode below the cutoff frequencies of the higher wave modes is easily excited using piezoelectric transducers, as the mam displacement is out-of-plane The energy transferred to the symmetric mode and shear mode is negligible The selective excitation of the first symmetric mode S0 or one of the higher modes would be more difficult and require the use of specialized excitation transducers, e g wedge transducers, selecting the angle of incidence according to Snell's law It would be interesting to apply the measurement with a laser interferometer to other wave types, like Rayleigh waves, possibly improving the sensitivity to The

,

small

defects,

characteristics

as

very local variations of the

can

be observed

wave

propagation and scattering

Conclusions and Outlook

6.2

101

Theoretical calculations

In the context of Lamb

theories for the

waves in

homogeneous, isotropic plates,

of flexural

the approximate

reviewed The scattering at

a description calculated and the different was plate analytically approaches using classical plate theory, Mmdlm's theory, and an asymptotic expansion were compared to experimental results Excellent agreement between the measure¬ ments and the analytical calculations was obtained Care has to be taken concern¬ ing the validity of the different approximations, as not only the ratio from wavelength to plate thickness, but also the ratio of hole radius to plate thickness define the validity of the approximations Several models for the complicated problem of the scattering at a hole with a crack at its boundary were examined Different analytical approaches were tried, that proved to be rather complex in their application and no generally valid ana¬ lytical solution could be found The superposition of two separate problems for a

circular hole

in a

circular hole and theoretical

waves were

a

crack shows the most promise and would be

an

interesting

study further However, the scattering of a flexural wave incident on a crack at an arbitrary angle would have to be solved beforehand Therefore, the combined scattered field was calculated numerically, using finite difference methods to discretize Mmdlm's equations on a staggered Cartesian grid Through explicit time integration a fast and stable algorithm was achieved The crack was modelled as a through notch, without considering the sharp edge and effects like crack closure For the model system of a through notch at a hole in a large plate, good agreement with experiments was achieved for the whole range of parameter variations The effect of a fatigue grown crack on the scat¬ tered field around a hole in a tensile specimen was also well predicted The com¬ plicated geometry of tensile specimens with multiple scattering at the hole and specimen boundaries poses no problem Accurate predictions on the detectabihty of a defect were made Conducting a numerical study for a variation of all geo¬ metrical parameters, it was found that the mam influence on the detectabihty is the ratio from wavelength to defect size The excitation frequency should be selected high enough, so that the wavelength is not more than eight times larger than the defect length to allow a reliable detection The finite difference modeling might be improved, using a radial grid in the vicinity of the hole and incorporating the sharp edge of the crack Alternatively, a finite element modeling might be investigated For further studies, it might be advantageous to investigate the applicability of other wave modes, especially at higher frequencies, corresponding to shorter wavelengths and possibly improvproblem

to

102

Conclusions and Outlook

mg the

sensitivity of the

initial stage of the

measurement method For the

cyclic

tensile

loading,

the mode

part-through

conversion

cracks

in

the

to other modes

should be studied

6.3

Application

to NDT

Building on the measurement of the model geometries and the description of the results by theoretical calculation, realistic cases were studied experimentally, involving more complicated geometries and fatigue grown cracks Multiple scat¬ tering at a line of holes, simulating a line of rivets in an aircraft fuselage was measured Good description of the combined scattered field by a superposition of the scattered fields at the single holes, taking the complete complex magnitude into account, was obtained Fast measurements at only a line or a few points of the structure

were

made using broadband excitation, minimizing the time for

defect detection In

cooperation with the fatigue engineering

the

applicability

aluminum

center of RUAG

Aerospace,

of the measurement method for the detection of

specimens

was

investigated

The

substantial

fatigue

geometric

Emmen

cracks

in

relation

between hole radius and specimen thickness

was selected as in a fighter plane generated by cyclic tensile loading in a servo-hydraulic material testing machine Experimental know-how on the line excitation in the strip-like specimen, the measurement using different measure¬ ment heads, and the implementation and carrying out of measurements in the harsher environment of an aircraft hangar was gained Two measurement series at RUAG Aerospace and intermediate measurements at the laboratory were per¬ formed, increasing the efficiency of the setup and reducing external effects on the

Fatigue

cracks at circular holes

were

measurements

The scattered field around the described

by

damaged

holes

was

measured and found to be well

the numerical model using finite difference methods

monitoring of the crack length during the cyclic tensile loading

An on-line

implemented experimentally Good correlation between measured and calculated change in signal and the optically measured crack length was found, allowing a sizing of the defect However, the variation in the measured signal due to noise and exter¬ nal influences was still rather high, making a reliable detection of cracks at an early stage of the damaging process impossible Higher excitation frequencies and the use of other wave types, e g Rayleigh waves, would allow the detection of smaller defects Additional measurement series should be performed to sys,

was

Conclusions and Outlook

103

eliminate errors, optimize the

tematically teners

or

rivets

the holes and the

in

aircraft parts should be

6.4

excitation and propagation, and Furthermore, the influence of fas¬

wave

test out the boundaries of the detection limit

applicability

to the

complicated geometry

of

investigated

Outlook

Insight gained

the mechanics of the scattering of flexural

on

From the exact measurement of the

at defects

waves

complex magnitude

was

of the scattered

field, the influence of the defect could be accurately described and modelled the¬ Further fundamental research would be necessary for

oretically

model, allowing lem, the

i e

use

mental

,

a

faster calculation and

possibly

a

solution of the

the localization and sizing of the crack from

of

higher

excitation

frequencies

a

an

analytical prob¬

inverse

By resulting experi¬ Alternatively, differ¬

remote measurement

and the solution of the

obstacles, smaller cracks could be reliably detected

modes might be employed, applying the exact measurement of amplitude and phase variations for an improved resolution of small defects Building on the knowledge gained from the fundamental research, the nonde¬ ent

wave

structive

testing

approached

of

real

aircraft

or

other

technical

systems

can

now

be

104

Conclusions and Outlook

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Curriculum vitae

Paul Fromme Born 28 November Citizen of

1977

1981

1990

-

-

-

1971, Schwemfurt, Germany

Germany

1981

Public

1990

Rhon-Gymnasium, Bad Neustadt/ Saale, Germany Graduation ('Abitur')

1996

Studies at the Mechanical

school, Bischofsheim, Germany

University 1993

-

1994

University

of

Waterloo, Canada

thesis at

University

of

Waterloo, Canada

Graduation

1996

University 1996

-

2001

PhD

('Diplom')

with honors

of Karlsruhe

in

Mechanical

Engineering,

(TH), Germany

student, reasearch and teaching assistant,

Institute of Mechanical

2000

Engineering Department,

(TH), Germany

at

Exchange year Diploma

1996

of Karlsruhe

Lecturer for first year

Department,

ETH

Systems,

ETH

Zurich, Switzerland

mechanics, Mechanical Engineering Zurich, Switzerland