HOP, NI, 2007

The diffraction coefficients for surface-breaking cracks

Larissa FradkinLarissa Fradkin

Waves and Fields Research Group

Faculty of Engineering, Science and Built Environment,

London South Bank University, UK

Funding bodies: IMC, EPSRC,LSBUFunding bodies: IMC, EPSRC,LSBU

HOP, NI, 2007

Professor V.M. Babich Professor V.A. Borovikov Dr V. Kamotski Dr B.A. Samokish

CollaboratorsCollaborators

HOP, NI, 2007

MotivationMotivation

Historical Overview

Statement of the problem

Sommerfeld Integral

Reduction to functional equations

Reduction to a singular integral problem

Numerical schedule

Equivalence of the singular integral problem to the original

Validation of the code

ConclusionsConclusions

Outline of the talk

HOP, NI, 2007

Probe

Motivation

Ultrasonic ray path Surface-breaking crack

Diffracting corner (wedge)

Pulse-echo inspection of a smooth planar defect at the back-wall of the component. When defect is vertical, have the ‘cat-eye’ effect, otherwise corner diffraction can become important.

HOP, NI, 2007

• Sommerfeld: 1896 - diffraction of an electro-magnetic wave by a perfectly conducting semi-infinite screen. Obtained: an exact solution in the form of a Sommerfeld integral which represents the wave field as a superposition of plane waves propagating in complex directions.

• Malyuzhinets: 1955-1958 - diffraction of acoustic plane wave by a wedge with the impedance boundary conditions. Reformulated the boundary conditions in the form of functional equations in F:

F( ) = R() F(- ) , where R = (-sin - a)/(sin -b),with a and b

known constants, and obtained an analytical solution.

Historical overview of wedge diffraction problem

HOP, NI, 2007

• Many worked on the wedge problems throughout the second half of XX century. The problem became a diffractionist's analogue of the famous Fermat's Last Theorem!

• Some relied on potential theory to reduce the problem to integral equations: Gautesen 1985-2002, Fujii 1980 - 1994, Croiselle & Lebeau 1992-2000.

• Budaev, Budaev and Bogy 1985 – 2002 followed the Sommerfeld - Malyuzhinets approach and arrived at another set of singular integral equations. We refine their arguments & develop a new numerical implementation of numerical schedule

Historical overview

HOP, NI, 2007

Equations: Helmholtz eqns for Boundary cdtis: zero-traction on weddge facesRadiation cdtns at infinity & tip conditions of bounded energy

Incident wave: P, S or Rayleigh

Solution exists and is unique (Kamotski and Lebeau 2006)

x

y

r

Statement of the problem

HOP, NI, 2007

In wedges a solution of the Helmholtz equation may often berepresented in the form of the Sommerfeld integral, =

CC(+)eikrcosdcS/cP

or as an asymptotic series in kr (Kondratiev, 1963).If andare known evaluating their Sommerfeld Integrals give us body waves diffracted from wedge tip, multiply reflected, surfaceRayleigh and head waves. If kr large, integrand is HO!

Sommerfeld Integral

2

~C

0

CC

C1

~

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We use two decompositions,

and sing +

and all poles k and k describing the multiplyreflected waves belong to strip

Decompositions of Sommerfeld amplitudes

)();(ResinSko

k

Skg s

where

2

Re

and are regular in this strip

SP

2

2

HOP, NI, 2007

Substituting Sommerfeld Integrals into bdry cdtns & using tip condition we obtain functional eqns

+(g(+) r11() r12() +(g(-)+(+) r21() r22() +(-)

and

+(g(-) r11() r12() +(g(+)+(-) r21() r22() +(+)

and a similar pair for - and -

Function g()=cos-1(-1cos ) transforms P scatter angles into S scatter angles, g( )=2

Reduction to functional equations

[ ]=[ [] ]+c1f1()

[ ]=[ ] [ ]+c1f2()

+

+

...2

2

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The functional equations can be re-arranged to give

+(g(+)++(g(-)+B +(+)++(-) =

R1()+c1()S1()

and

+(g(+)-+(g(-)+ +(+)-+(-) = R2()+c1()tanS2()

and a similar pair for - and -

{

Rearrangement of functional equations

} { }

{ {} }

+

+

+

+

HOP, NI, 2007

If F() is analytic in |Re - /2 |

and F()=O(e-Re p |Im |), Re p > -1, |Im |oo

a Hilbert-type integral transform

has the property

H: F(+)+F(-) -> F(+)-F(-), Re =

A singular integral problem

i

id

FPV

iHF

2/

2/ )(2

sin

)(..

2

1)(

2

2

2

HOP, NI, 2007

Using the Hilbert-type integral transforms the functionaleqns may be transformed into integral problems on a realline,

(H’d + K)y + =q0+

+c1+q1

+

where K is a regular operator,H’ is analytically invertibleIn the space of bounded functions and

The equation is solvable only if (H’d + P1K)y + =P1q0

+

where P1u=

0)qcq)((

dKy

A singular integral problem

0)('

dfH

+1 10

u if

0 if u=q1

0)(

du

+

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Applying (H’) and using a symmetrisation procedure the regularised singular integral equation is y + +L + y + = q +

When they exist, the GE, multiply reflected P and S waves may be found following well defined procedure to give P

k and S

k . Budaev-Bogy numerical schedule involves threemajor steps:

A singular integral problem

-1

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• solve singular integral for y+ and x- equations on line Re =/2, and finding y- and x+ using algebraic equations;• use singular integrals to find amplitudes and in strip

• use functional equations to effect analytical continuation of and to the right and to the left of this strip

Numerical schedule

d

i

fPV

)2

(2

cos

)(..

2

2

HOP, NI, 2007

The computed Sommerfeld amplitudes appear to

1. exhibit the correct behaviour at infinity (decrease as correct exponents);

2. be analytic functions satisfying the corresponding functional equations, i.e. are continuous on the boundaries of strip

3. possess physically meaningful singularities (by constructions) and no physically meaningless singularities (because they possess the correct symmetries).

Code testing

2

Re

2

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2. and are analytic functions satisfying the corresponding functional equations

3. is even and --- odd

-4

-3

-2

-1

0

1

2

3

4

-4 -2 0 2 4

Im PhiIm PsiRe PhiRe Psi

The computed Sommerfeld amplitudes. Wedge angle - 70o, magic

strip [0.96, 2.18], Poisson's ratio =0.25, incident wave - S and inc=0 o.

2

HOP, NI, 2007

and are analytic functions satisfying the corresponding functional equations

2. and are analytic functions satisfying the corresponding functional equations

3.is even and is odd

-4

-3

-2

-1

0

1

2

3

4

-4 -2 0 2 4

Im PhiIm PsiRe PhiRe Psi

The computed Sommerfeld amplitudes. Wedge angle - 70o, magic strip [0.96, 2.18], Poisson's ratio =0.25,

incident wave - P and inc=0o.

2

HOP, NI, 2007

Equivalence of the singular integral problem to the original

Since the computed Sommerfeld amplitudes1. exhibit the correct behaviour at infinity; 2. are analytic functions satisfying the corresponding functional

equations; 3. possess physically meaningful and no physically meaningless

singularities.

The corresponding Sommerfeld integrals (kr, ) and (kr, ) satisfy 1. the Helmholtz equations and correct tip condition; 2. zero stress boundary conditions; 3. radiation conditions at infinity.

HOP, NI, 2007

Code validation

Re fle c tio n&Tra nsm issio nC o e ffic ie nts(a m p litud e s)

w e d g e a n g le

2DWeD, Budaev and Bogy (1994) computations and Fujii (1994) numerical (solid line) and experimental (dots) Rayleigh

reflection and transmission coefficients. Amplitudes.

HOP, NI, 2007

2DWeD, Budaev and Bogy (1994) computations and Fujii (1994) numerical (solid line) and experimental (dots) Rayleigh

reflection and transmission coefficients. Phases.

Code validation

Re fle c tio n&Tra nsm issio n C o e ffic ie nts(p ha se s)

we d g e a ng le

HOP, NI, 2007

If kr large, integrand is HO and can use the steepest descent method

where P or S

Back scatter diffraction coefficients

~),( kru )( D)(kr 1/2

ikreu

(inc)

kr

diff)(

r=

HOP, NI, 2007

Code validation (P-P amplitudes)

P-wave diffraction coefficients - 100 degree wedge

0.01

0.1

1

10

0 5 10 15 20 25 30 35 40 45 50

Angle to wedge bisector (deg)

Diff

ract

ion

coef

ficie

ntSBU Theory

Wedge 2 MHz (+ve angles)

Wedge 2 MHz (-ve angles)

Wedge 5 MHz (+ve angles)

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P-wave phases of diffraction coefficients - 100 degree wedge

-180

-120

-60

0

60

120

180

0 5 10 15 20 25 30 35 40 45 50

Angle to wedge bisector (deg)

Ph

ase

(deg

)

SBU Theory

Wedge 2MHz (+ve angles)

Wedge 2 MHz (-ve angles)

Wedge 5 MHz (+ve angles)

Code validation (P-P phases)

HOP, NI, 2007

Angle

LSBU

Theory

Gau

Theory

Birch

Exper

150 0.10 0.11 0.08

200 0.22 0.23 0.14

250 0.51 0.51 0.47

Code validation (S-S amplitudes)

HOP, NI, 2007

Code validation (S-S phases)

Angle

LSBU

Theory

Gau

Theory

Birch

Exper150 340 340 130

200 -430 -430 -600

250 -340 -340 -450

HOP, NI, 2007

• Start with the Green’s formula in the form of Extinction Theorem (eqtn and bdry cdtns)

• Use the Fourier Transform, radiation cdtns and tip cdtns to obtain functional eqns for the Wiener-Hopf type unknowns

• Represent solution as a sum of geometrical contributions, Rayleigh waves and an analytical unknown

• Use the Cauchy integrals to reduce the functional equations for the analytical parts to regular integral equations

Gautesen’s approach

HOP, NI, 2007

Conclusions

HOP, NI, 2007

• The code for modelling surface-breaking cracks has been validated against other codes and experimental data. Limits ofapplicability: 400 < 2 < 1780

• The code is now used by British Energy Plc in design of new inspections of nuclear power plants, e.g. Sizewell, and to provideevidence of detection capability

• The Gautesen code has been extended to simulate 250 < 2a <1780

• The Gautesen technique has been applied to evaluatingdiffraction coefficients for planar cracks in TI media

Conclusions