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HOP, NI, 2007 The diffraction coefficients for surface-breaking cracks Larissa Fradkin Waves and Fields Research Group Faculty of Engineering, Science.

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Presentation on theme: "HOP, NI, 2007 The diffraction coefficients for surface-breaking cracks Larissa Fradkin Waves and Fields Research Group Faculty of Engineering, Science."— Presentation transcript:

1 HOP, NI, 2007 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 HOP, NI, 2007 Professor V.M. Babich Professor V.A. Borovikov Dr V. Kamotski Dr B.A. Samokish Collaborators

3 HOP, NI, 2007 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 codeConclusions Outline of the talk

4 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.

5 HOP, NI, 2007 Sommerfeld: 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: 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

6 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 , Fujii , Croiselle & Lebeau 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

7 HOP, NI, 2007 Equations: Helmholtz eqns for  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)   x y  r Statement of the problem

8 HOP, NI, 2007 In wedges a solution of the Helmholtz equation may often be represented in the form of the Sommerfeld integral,  =  C  C  (  +  )e  i  krcos  d  c S /c P or as an asymptotic series in kr (Kondratiev, 1963). If  and  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! Sommerfeld Integral ~ 

9 HOP, NI, 2007 We use two decompositions,     and  sing +  and all poles  k and  k describing the multiply reflected waves belong to strip Decompositions of Sommerfeld amplitudes where and  are regular in this strip S P  22

10 HOP, NI, 2007 Substituting Sommerfeld Integrals into bdry cdtns & using tip condition we obtain functional eqns   + (g(  +  ) r 11 (  ) r 12 (  )  + (g(  -  )  + (  +  ) r 21 (  ) r 22 (  )  + (  -  ) and  + (g(  -  ) r 11 (  ) r 12 (  )  + (g(  +  )  + (  -  ) r 21 (  ) r 22 (  )  + (  +  ) and a similar pair for  - and  - Function g(  )=cos -1(  -1 cos  ) transforms P scatter angles into S scatter angles, g( )= Reduction to functional equations [ ]=]= [ []]+c1f1()]+c1f1() []=]= [][]+c1f2()]+c1f2() + + 

11 HOP, NI, 2007 The functional equations can be re-arranged to give  + (g(  +  )+  + (g(  -  )+B  + (  +  )+  + (  -  ) = R 1 (  )+c 1 (  )S 1 (  ) and  + (g(  +  )-  + (g(  -  )+  + (  +  )-  + (  -  ) = R 2 (  )+c 1 (  )tan  S 2 (  ) and a similar pair for  - and  - Rearrangement of functional equations

12 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  22

13 HOP, NI, 2007 Using the Hilbert-type integral transforms the functional eqns may be transformed into integral problems on a real line, (H’d + K)y + =q 0 + +c 1 + q 1 + where K is a regular operator,H’ is analytically invertible In the space of bounded functions and The equation is solvable only if  (H’d + P 1 K)y + =P 1 q 0 + where P 1 u= A singular integral problem u if 0 if u=q 1 +

14 HOP, NI, 2007 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 three major steps : A singular integral problem

15 HOP, NI, 2007 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  22

16 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 

17 HOP, NI,  and  are analytic functions satisfying the corresponding functional equations 3.   is even and   --- odd 

18 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 

19 HOP, NI, 2007 Equivalence of the singular integral problem to the original Since the computed Sommerfeld amplitudes 1. 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.

20 HOP, NI, 2007 Code validation 2DWeD, Budaev and Bogy (1994) computations and Fujii (1994) numerical (solid line) and experimental (dots) Rayleigh reflection and transmission coefficients. Amplitudes.

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

22 HOP, NI, 2007 If kr large, integrand is HO and can use the steepest descent method where P or S Back scatter diffraction coefficients 1/2 (inc)  r=

23 HOP, NI, 2007 Code validation (P-P amplitudes)

24 HOP, NI, 2007 Code validation (P-P phases)

25 HOP, NI, 2007 AngleLSBU Theory Gau Theory Birch Exper Code validation (S-S amplitudes)

26 HOP, NI, 2007 Code validation (S-S phases) AngleLSBU Theory Gau Theory Birch Exper

27 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

28 HOP, NI, 2007 Conclusions

29 HOP, NI, 2007 The code for modelling surface-breaking cracks has been validated against other codes and experimental data. Limits of applicability: 40 0 < 2  < 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 simulate 25 0 < 2a <178 0 The Gautesen technique has been applied to evaluating diffraction coefficients for planar cracks in TI media Conclusions


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