The diffraction coefficients for surface-breaking cracks Larissa Fradkin

1. The diffraction coefficients for surface-breaking cracks Larissa Fradkin Waves and Fields Research Group Faculty of Engineering, Science and Built Environment, London South Bank University, UK Funding bodies: IMC, EPSRC,LSBU

2. Collaborators • Professor V.M. Babich • Professor V.A. Borovikov • Dr V. Kamotski • Dr B.A. Samokish

3. Outline of the talk Motivation 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 Conclusions

4. Motivation Probe Surface-breaking crack Ultrasonic ray path 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.

5. Historical overview of wedge diffraction problem • 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 inF: F(w  a) = R(w) F(-w a) , whereR = (-sin w - a)/(sin w-b),withaandbknown constants, and obtained an analytical solution.

6. Historical overview • Many worked on the wedge problems throughout the second half of XX century. The problem became adiffractionist's analogue of the famous Fermat's Last Theorem! • Some relied on potential theory to reduce the problem to integral equations:Gautesen1985-2002,Fujii1980 - 1994,Croiselle & Lebeau1992-2000. • Budaev, Budaev and Bogy1985 – 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

7. r q Statement of the problem Equations: Helmholtz eqns forf, y Boundary cdtis: zero-traction on weddge faces Radiation cdtns at infinity & tip conditions of bounded energy Incident wave: P, S or Rayleigh Solution exists and is unique (Kamotski and Lebeau 2006) y a a x

8. Sommerfeld Integral In wedges a solution of the Helmholtz equation may often be represented in the form of the Sommerfeld integral, • F= CCF(w +q)eigkrcoswdw, g=cS/cP or as an asymptotic series in kr (Kondratiev, 1963). If F and Y are known evaluating their Sommerfeld Integrals give us body waves diffracted from wedge tip, multiply reflected, surface Rayleigh and head waves. If kr large, integrand is HO! ~ (w)

9. 2a (w) Decompositions of Sommerfeld amplitudes We use two decompositions, Y= Y+ + Y- andY = Ysing + Y and all poleswkandwkdescribing the multiply reflected waves belong to strip where P S andYare regular in this strip

10. ]= ] [ ]+c1f1(w) [ [ [ ]= [ ] [ ]+c1f2(w) (w) Reduction to functional equations Substituting Sommerfeld Integrals into bdry cdtns & using tip condition we obtain functional eqns F+(g(w) +a) r11(w) r12(w) F+(g(w) -a) Y+(w +a) r21(w) r22(w) Y+(w -a) and F+(g(w) -a) r11(w) r12(w) F+(g(w) +a) Y+(w -a) r21(w) r22(w) Y+(w +a) and a similar pair forF-andY- Function g(w)=cos-1(g-1cos w) transforms P scatter angles into S scatter angles, g( )= + +

11. Rearrangement of functional equations The functional equations can be re-arranged to give F+(g(w) +a)+F+(g(w) -a)+BY+(w +a)+Y+(w -a) = R1(w)+c1(w)S1(w) and AF+(g(w) +a)-F+(g(w) -a)+ Y+(w +a)-Y+(w -a) = R2(w)+c1(w)tana S2(w) and a similar pair forF-andY- + + + +

12. 2a (w) A singular integral problem If F(w) is analytic in |Re w - p/2 |a and F(w)=O(e-Re p |Im w |), Re p > -1, |Im w|oo a Hilbert-type integral transform has the property H: F(w +a)+F(w -a) -> F(w +a)-F(w -a), Re w=

13. + 1 0 1 A singular integral problem Using the Hilbert-type integral transforms the functional eqns may be transformed into integral problemson a real line, (H’d + K)y + =q0++c1+q1+ where Kis a regular operator,H’is analytically invertible In the space of bounded functions and The equation is solvable only if • (H’d + P1K)y + =P1q0+ whereP1u= + u if 0 if u=q1

14. A singular integral problem 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 givewPk andwSk . Budaev-Bogy numerical schedule involves three major steps: -1

15. 2a (w) Numerical schedule • solve singular integral for y+andx-equations on line Rew=p/2, and findingy-andx+using algebraic equations; • use singular integrals to find amplitudesFandYin strip • use functional equations to effect analytical continuation ofF andYto the right and to the left of this strip

16. (w) Code testing The computed Sommerfeld amplitudes appear to • exhibit the correct behaviour at infinity (decrease as correct exponents); • be analytic functions satisfying the corresponding functional equations, i.e. are continuous on the boundaries of strip • possess physically meaningful singularities (by constructions) and no physically meaningless singularities (because they possess the correct symmetries).

17. (w) 2. FandYare analytic functions satisfying the corresponding functional equations 3. F-is even andY- ---odd

18. (w) 2. FandYare analytic functions satisfying the corresponding functional equations 3.F+ isevenandY +isodd FandYare analytic functions satisfying the corresponding functional equations

19. Equivalence of the singular integral problem to the original Since the computed Sommerfeld amplitudes • exhibit the correct behaviour at infinity; • are analytic functions satisfying the corresponding functional equations; • possess physically meaningful and no physically meaningless singularities. The corresponding Sommerfeld integralsf(kr, q)andy(kr, q)satisfy • the Helmholtz equations and correct tip condition; • zero stress boundary conditions; • radiation conditions at infinity.

20. Code validation 2DWeD, Budaev and Bogy (1994) computations and Fujii (1994) numerical (solid line) and experimental (dots) Rayleigh reflection and transmission coefficients.Amplitudes.

21. Code validation 2DWeD, Budaev and Bogy (1994) computations and Fujii (1994) numerical (solid line) and experimental (dots) Rayleigh reflection and transmission coefficients.Phases.

22. Back scatter diffraction coefficients If kr large, integrand is HO and can use the steepest descent method where P or S (inc) 1/2 q r=

23. Code validation (P-P amplitudes)

24. Code validation (P-P phases)

25. Code validation (S-S amplitudes)

26. Code validation (S-S phases)

27. Gautesen’s approach • 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

28. Conclusions

29. Conclusions • The code for modelling surface-breaking cracks has been • validated against other codes and experimental data. Limits of • applicability: 400 < 2a < 1780 • The code is now used by British Energy Plc in design of new • inspections of nuclear power plants, e.g. Sizewell,and to provide • evidence of detection capability • The Gautesen code has been extended to simulate250 < 2a <1780 • The Gautesen technique has been applied to evaluating • diffraction coefficients for planar cracks in TI media