Flaw Scattering Models

Transcription

1 Flaw Scattering Models

2 Learning Objectives Far-field scattering amplitude Kirchhoff approximation Born approximation Separation of Variables Examples of scattering of simple shapes (spherical pore, flat crack, side-drilled hole)

3 Flaw Scattering Incident plane wave p e i Fluid Model e s y p pressure amplitude flaw x s r s n At many wavelengths away from the flaw the scattered waves are spherical waves p scatt ( y, ω ) = p A( e ; e ) i s exp ( ikr ) r s s scattered pressure A is called the plane wave far-field scattering amplitude

4 Flaw Scattering Incident plane wave p e i Fluid Model e s y p pressure amplitude flaw x s r s n plane wave far field scattering amplitude of the flaw 1 p% A ik p ik ds 4π n S f p = p ( x s,ω ) ( e ; e ) = + ( e n) % exp( x e ) ( x ) i s s s s s p

5 Flaw Scattering 1 p% A ik p ik ds 4π n ( e ; e ) = + ( e n) % exp( x e ) ( x ) i s s s s s S f ( ) p = p x s,ω p To obtain the far field scattering amplitude, we need to know the pressure and velocity on the surface of the flaw due to the incident and scattered waves These quantities can be found by solving a flaw scattering boundary value problem.

6 u scatt ( y ) scattered displacement from a flaw Flaw Scattering ( β P) Elastic Solid ( ) ( β S) A ei ; es A ei ; es, ω = U exp ik r + U exp ik r ( ) p s s s rs rs P-wave S-wave U β ei U displacement amplitude for wave of type β (β = P,S) in problem (1) flaw x s n r s α es Scattered wave of type α (α = P,S)

7 Flaw Scattering ( β α ; ) i s Ae e Vector scattering amplitude for a scattered wave of type α due to a plane wave of unit displacement amplitude and type β ( β ) ( ; β ; = ) A e e f e e P P P P i s s s ( β ) ; β ( ; β ; = ) A e e f f e e S S S S S i s s s ( α ) ( ) αβ ; αβ ; lkpj l p α α l l k α sk p j s s s 4πρc α x S j ( β = P, S) elastic constants C e u% f = f e = n + ik e n u exp ik ds 2 % x e x displacements and displacement gradients ( α = P,S) ( β = P, S)

8 Flaw Scattering 3-D scattering amplitude u scatt ( y ) ( β P) ( ) ( β S) A ei ; es A ei ; es, ω = U exp ik r + U exp ik r ( ) p s s s rs rs The received voltage in some measurement models is related to a specific component of the vector scattering amplitude given by ( ω ) = ( e ; ) ( ; )( ) i es = A ei es d A A β α β α α polarization vector of an incident wave from receiving transducer (to be discussed shortly)

9 Flaw Scattering A(ω) can be obtained from: Analytical/Numerical methods Separation of variables Boundary Elements (BEM) Finite Elements (FEM) Method of Optimal Truncation (MOOT) T-matrix Kirchhoff Approximation Born Approximation or A(ω) can be found experimentally through deconvolution: V R ( ω) = E( ω)a ω ( ) A( ω) = VR E * ( ω) E ( ω) 2 2 ( ω) + n measure model, measure

10 Flaw Scattering Kirchhoff Approximation for a Volumetric Flaw incident wave e i tangent plane P n e r D x s reflected wave x O S lit on the "lit" surface: fields are those from plane waves interacting with a plane surface on the rest of the surface the fields are assumed to be zero

11 Flaw Scattering It has been recently shown that the pulse-echo scattering amplitude response, A(ω), for a stress-free flaw (void or crack) in an elastic solid is identical to the scalar (fluid) scattering amplitude: ik A = A = ikβ ds 2π ( ) ( ) β ω e β ; e β ( e β n) exp( 2 x e β ) ( x ) i i i s i s S β ei lit e β s = e β i

12 Flaw Scattering-Spherical Void For the pulse-echo response of a spherical void of radius a a A( ω) A( ei; ei, ω) = exp( ika) exp( ika) 2 sin ( ka) ka.7.6 2a A(ω) a = 1 mm c = 59m/s frequency, MHz

13 Flaw Scattering-Spherical Void Comparison with Kirchhoff solution with exact separation of variables solution for the P-P scattering amplitude of a void in an elastic solid (pulse-echo) leading edge response exact solution (solid line) 1 "creeping" wave 2 A /a Kirchhoff solution (dotted line) frequency, MHz

14 Flaw Scattering-Spherical Void Inverting the scattering amplitude (fluid model) into the time domain a 2a c 2a A() t A( ei; ei, t) = δ t U,; t 2 + c 2a c (, ; t) U t t = t < t < t 1 2 otherwise "lit" surface response A(t) c 4 leading edge response 2a c t

15 Flaw Scattering-Spherical Void Exact solution for a void in an elastic solid (time domain) vs the Kirchhoff approximation solution creeping wave time, μsec

16 Flaw Scattering-Spherical Void Although the Kirchhoff approximation is formally a high frequency approximation ( ka >>1), numerical tests have shown it capable of producing good agreement (<1 db difference) for the peak-to-peak time domain response of the spherical void down to ka =1 provided the bandwidth is sufficient white: differences <1 db gray: >1 db but < 1.5 db black: > 1.5 db ka bandwidth, %

17 Flaw Scattering-Inclusion Leading Edge Response - General Convex Inclusion (scattered mode same as incident mode) e s tangent plane n α θ i e i x s stat α e s d S lit High frequency, stationary phase approximation θi = α RR 1 2 A( ei; es, ω ) = R12 exp ik i s d 2 e e Plane wave reflection coefficient Gaussian curvature of flaw surface at stationary phase point

18 Flaw Scattering-Inclusion Leading Edge Response - General Convex Inclusion (scattered mode same as incident mode) e s tangent plane n α θ i x s stat α e s S lit e i In the time domain d + A () t 1 = ( ) exp( ) 2 A ω i ω t d ω π RR A i s t R12 t i s d c 2 ( e ; e, ) = 1 2 δ ( + e e / )

19 Flaw Scattering-Crack Kirchhoff Approximation Flat Elliptical Cracks e i x 3 e s n e a q r e a 2 1 u x 2 u 2 1 plane normal to e q e q = e i e s e i e s x 1 ( ) A( e i ;e s )= ia 1a 2 e i n e i e s r e J ( 1 k e i e s r ) e r e = a 2 ( 1 e q u ) a 2 ( 2 e q u ) 2 2

20 Flaw Scattering-Crack General pitch-catch crack response-circular crack frequency, MHz The magnitude of the P-wave pulse-echo far-field scattering amplitude versus frequency for a 1 mm radius circular crack in steel with an angle of incidence of from the crack normal.

21 Flaw Scattering-Crack Comparison of the Kirchoff approximation and MOOT for a.381 mm radius circular crack at an incident angle of 45 degrees (pulse-echo).25 inc = 45 KIR MOOT.2 amplitude frequency, MHz

22 When e q is parallel to the crack normal, n : A( e i ;e s )= ik ( e i n)a 1 a 2 2 Special case - pulse echo: A( e i ; e i )= ika 1 a Flaw Scattering-Crack frequency, MHz The magnitude of the P-wave pulse-echo far-field scattering amplitude versus frequency for a 1 mm radius circular crack in steel at normal incidence. e i e i e q n e s n e s

23 Flaw Scattering-Crack Comparison of the Kirchoff approximation and MOOT for a.381 mm radius crack at normal incidence (pulse-echo) KIR MOOT amplitude inc= numerical errors in MOOT frequency, MHz

24 Inverting these results into the time domain e i n () ( ) aac ( e n) 1 2 i t t < e i 2 i; s, π ei es re i s e / ( ) At Ae e t = e e r c t otherwise flash points Flaw Scattering-Crack e s re c Example: pulse-echo t

25 Flaw Scattering-Crack At normal incidence, for pulse-echo A e ; e, t = ( ) i i aa 2c 1 2 dδ dt t ( ) A(e i ; e s ) t

26 Flaw Scattering-Crack A planar crack is a very specular reflector θ β ei angle, degrees

27 Flaw Scattering-Crack Although the Kirchhoff approximation is formally a high frequency approximation ( ka >>1), numerical tests have shown it capable of producing good agreement (<1 db difference) for the peak-to-peak time domain response of a circular crack at normal incidence down to ka =1.5 provided the bandwidth is sufficient white: differences <1 db gray: >1 db but < 1.5 db black: > 1.5 db ka bandwidth, %

28 Flaw Scattering-Crack Comparison of synthesized waveforms scattered from a.381 mm radius crack using the Kirchhoff approximation and MOOT using frequencies from -25 MHz approximately (pulse echo) amplitude frequency, MHz

29 Flaw Scattering-Crack -15 degrees from crack normal 2 5 KIR MOOT

30 Flaw Scattering-Crack 2-35 degrees from crack normal KIR MOOT

31 Flaw Scattering-Crack 4-55 degrees from crack normal KIR MOOT

32 Flaw Scattering-Crack 6-7 degrees from crack normal KIR MOOT

33 Flaw Scattering-Crack degrees from crack normal KIR MOOT

34 Flaw Scattering-Crack Comparison of peak-to-peak values of MOOT and Kirchhoff versus angle of incidence for the.381mm radius crack (pulse-echo) The arrow shows where agreement is within 1 db 25 MOOT KIR

35 Now consider the scattering amplitude with narrow band Gaussian window The central frequency is 1 MHz, and the bandwidth is 1 MHz Radius of the crack a =.381mm Flaw Scattering-Crack amplitude frequency, MHz

36 Flaw Scattering-Crack the range of angles where the agreement is good is significantly reduced 3.5 KIR MOOT

37 Now consider the scattering amplitude with wider band Gaussian window The central frequency is 1 MHz, and the bandwidth is 6 MHz Radius of the crack a =.381mm Flaw Scattering-Crack amplitude frequency, MHz

38 Flaw Scattering-Crack The range of excellent agreement now is back to angles as great as 45 degrees or more KIR MOOT Amplitude Angle, degree

39 Flaw Scattering-Crack Maximum incident angle where the peak-to-peak time domain agreement between the Kirchhoff approximation and the exact solution for a circular crack is less than 1 db for ka =5. (other ka values shown same trend). max. incident angle, degrees bandwidth, %

40 Flaw Scattering-SDH Kirchhoff approximation for pulse-echo scattering of a side-drilled hole (incident direction in plane perpendicular to the hole axis) β ei L 2b ( β ) kbl β β A( ei ; ei ) = J1( 2kβb) is1( 2kβb) + 2 ( β ) ikbl π Bessel function Struve function

41 Flaw Scattering-SDH Comparison with exact 2-D separation of variables solution for P-waves (pulse-echo) The three-dimensional normalized pulse-echo P-wave scattering amplitude versus normalized wave number for a side drilled hole in the Kirchhoff approximation (solid line) and from the exact two-dimensional separation of variables solution (dashed line).

42 Flaw Scattering-SDH Comparison with exact 2-D separation of variables solution for S-waves (pulse-echo_) The three-dimensional normalized pulse-echo SV-wave scattering amplitude versus normalized wave number for a side drilled hole in the Kirchhoff approximation (solid line) and from the exact two-dimensional separation of variables solution (dashed line).

43 Flaw Scattering Kirchhoff approximation - Summary For volumetric flaws -the Kirchhoff approximation properly models the leading edge signal as long as ka >1 approximately but does not model other waves (creeping waves, etc.) For cracks the Kirchhoff approximation models the flash point signals properly in pulse-echo as long as ka > 1 approximately and the incident angle is less than about 5 degrees for wide band responses. For narrow band responses this angle is considerably reduced to as little as 15-2 degrees.

44 Flaw Scattering -Inclusion The Born Approximation The Born approximation assumes that the material of an inclusion differs little from the host material so that to first order the incident wave passes through the inclusion unchanged (weak scattering, low frequency approximation) e in c incident wave

45 Flaw Scattering -Inclusion The Born approximation generally is developed from a volume integral expression for the scattering amplitude density difference difference in elastic constants α d u% A u ik e C ik dv 4πρcα x V f j β α q ( ) 2 α m α ei ; es = exp 2 Δ ρω % q + α skδ kqm j ( αx es ) ( x) fields in the flaw replaced by incident fields u% = u% q incident q u% m u% = x x j incident m j

46 Flaw Scattering -Inclusion For a spherical inclusion in pulse-echo A ( β β e ) i ei ; = 4k b F ( k b) j β β 2kb β spherical Bessel function where F 1 Δρ = + 2 ρ Δc c β relative density difference relative wave speed difference

47 Flaw Scattering -Inclusion Corresponding time domain impulse response for a spherical inclusion (pulse-echo) front surface response back surface response 2b c α 2b c α t

48 Comparison with "exact" separation of variables solution for the pulse-echo response of a spherical inclusion by synthesizing a time-domain response from frequencies ranging from -2 MHz approximately for a 1 mm radius inclusion (1 % differences in properties) 1.5 Flaw Scattering -Inclusion 1 amplitude time, μsec The time domain pulse-echo P-wave response of a 1 mm radius spherical inclusion in steel where the density and compressional wave speed are both ten percent higher than the host steel. Solid line: Born approximation, dashed line: separation of variables solution.

49 Flaw Scattering -Inclusion Comparison with "exact" separation of variables solution for the pulse-echo response of a spherical inclusion by synthesizing a time-domain response from frequencies ranging from -2 MHz approximately for a 1 mm radius inclusion (5 % differences in properties) 8 amplitude errors amplitude time of arrival errors time, μsec The time domain pulse-echo P-wave response of a 1 mm radius spherical inclusion in steel where the density and compressional wave speed are both fifty percent higher than the host steel.solid line: Born approximation, dashed line: separation of variables solution.

50 A Doubly Distorted Born Approximation β β ( ) replace wave speeds of host material by that of the flaw in the Born approximation for the pulse-echo response of a spherical inclusion β β ( ) ( kb β ) j β 2 β e ; e = 4k b F 1 Δρ kb F = + 2 ρ Born i i Flaw Scattering -Inclusion ( f β ) j k b i ; i = 4 DDBA fβ 2kf β A e e k b F% b Δc c β where F% 1 Δρ = + ρ Δc 2 f c f β

51 Front surface amplitude response is improved, and relative time of arrival of back surface response is correct, but there is an error in absolute time of arrivals (5% differences in properties) 8 6 Flaw Scattering -Inclusion amplitude time, μsec The time domain pulse-echo P-wave response of a 1 mm radius spherical inclusion in steel where the density and compressional wave speed are both fifty percent higher than the host steel. Solid line: Doubly Distorted Born approximation, dashed line: separation of variables solution.

52 Flaw Scattering -Inclusion Why is the front surface response amplitude improved by the Doubly Distorted Born Approximation? F R 12 F % Δρ Δc = ρ c ; Answer: because F % R ββ (plane wave reflection coefficient) 12

53 Flaw Scattering -Inclusion This suggests we apply a phase correction to the doubly distorted Born approximation and replace F % ; by R β β 12 resulting in the modified Born approximation (MBA) for the pulse-echo response of a spherical inclusion β β 2 3 β; β ( ei ; ei ) = 4 2 f 12 exp β 2 f β ( 1 fβ / β ) A k b R ik b c c MD 1 ( 2 f β ) j k b 2k f β b phase correction

54 Flaw Scattering -Inclusion The MBA (5 % differences in properties) amplitude time, μsec The time domain pulse-echo P-wave response of a 1 mm radius spherical inclusion in steel where the density and compressional wave speed are both fifty percent higher than the host steel.solid line: MBA approximation, dashed line: separation of variables solution.

55 Flaw Scattering -Inclusion The MBA (1 % differences in properties) 2 15 amplitude time, μsec The time domain pulse-echo P-wave response of a 1 mm radius spherical inclusion in steel where the density and compressional wave speed are both one hundred percent higher than the host steel. Solid line: MBA approximation, dashed line: separation of variables solution.

56 Flaw Scattering The Method of Separation of Variables The sphere and the cylinder are the only two geometries where we can obtain exact separation of variables solutions for elastic wave scattering problems. These are commonly used as "exact" solutions to test more approximate theories and numerical methods.

57 Flaw Scattering-Void Example 1: pulse-echo P-wave scattering of a spherical void A 1 ; 1 n p p ( e e ) = ( ) A i i n ik p n= A n = E E E E E E E E { } ( 2 1 ) ( /2) s n( pb) 2 p n+ ( p ) ( 2 1) {( 1) n( pb) p n+ ( p )} () 1 () 1 ( /2) s n ( p ) 2 p n+ ( p ) ( ) () 1 () 1 1 n ( p ) p n+ ( p ) E = n+ n n k b j k + k bj kb 3 1 E = n+ n j k k bj k b 4 1 E = n n k b h k b + k bh k b 31 1 E = n h k b k bh k b 41 1 ( 1) ( 1) () ( ) () ( ) ( ) () ( ) 1 1 E32 = n n+ n hn ksb ksbhn+ 1 ksb () 1 1 s /2 n s s n+ ( kb s ) E = n k b h k b k bh 42 1

58 Flaw Scattering-Void The normalized P-wave pulse-echo scattering amplitude 2A/b for a spherical void of radius b A / b frequency, MHz

59 Flaw Scattering-Void Using this separation of variables solutions at many frequencies to synthesize a P-wave impulse time domain solution (pulseecho) 6 amplitude time, μsec The time-domain pulse-echo P-wave response of a.5 mm radius spherical void in steel ( c p = 59 m/s, c s = 32 m/sec) obtained by applying a low-pass cosine-squared windowing filter between 1 and 2 MHz to the separation of variables solution and then inverting the result into the time domain with the inverse Fourier transform.

60 Flaw Scattering-Void Example 2: pulse-echo SV-wave scattering of a spherical void A 1 = ik s s ( ei; ei ) B n s n= 1 n ( 1) ( 2n+ 1) 2 H J H J J = H H H H H ( 1) ( 1) ( ) ( ) J12 = n n+ n jn ksb ksb jn+ 1 ksb ( 1) ( 1) () ( ) B n () ( ) 1 1 H12 = n n+ n hn ksb ksbhn+ 1 ksb () 1 () 1 ( /2) s n ( p ) 2 p n+ ( p ) H13 = n n kb h kb + kbh 1 kb J = n j k b k b j k b ( 1) n( s ) s n+ ( s ) ( ) () 1 () 1 1 ( ) ( ) 41 1 H = n h k b k bh k b 41 n s s n+ 1 s ( ) s n( s ) s n+ ( s ) J42 = n 1 k b /2 j k b + k b j 1 k b H 42 n 1 ks b /2 hn ksb + ksbhn+ 1 ksb () = ( ) ( ) ( ) () 1 () 1 1 n ( p ) p n+ ( p ) H = n h k b k bh k b 43 1 () ( )

61 Flaw Scattering-Void The SV-wave pulse-echo scattering amplitude amplitude frequency, MHz The magnitude of the pulse-echo SV-wave response,, versus frequency for a.5 mm radius spherical void in steel ( c p = 59 m/s, c s = 32 m/sec) as calculated by the method of separation of variables

62 Flaw Scattering-Void Using this separation of variables solutions at many frequencies to synthesize an SV-wave impulse time domain solution (pulseecho) 6 amplitude time, μsec The time domain pulse-echo SV-wave response for the same void considered in the P-wave case by applying a low-pass cosinesquared windowing filter between 1 and 2 MHz to the separation of variables solution and then inverting the result into the time domain with the inverse Fourier transform.

63 Flaw Scattering- SDH Example 3: pulse-echo P-wave scattering of a cylindrical void A p p ( e ; e ) 3D i i L i = n 2π n= ( 2 δ )( 1) n F n A 2D ( ω) 2iπ = kα 2 1/2 A 3D ( ω) L δ n 1 n = = otherwise F n = 1+ ( ) ( ) ( ) ( ) () ( ) () ( ) ( ) ( ) ( ) ( ) () ( ) () ( ) C k b C k b D k b D k b n p n s n p n s C kb C kb D kb D kb n p n s n p n s 2 () ( ) ( ) () 2 ( ) ( ) () ( ) ( ) () ( ) () () ( ( ) + 1 ( )) i i i i C x = n + n k b 2 H x 2nH x xh x n s n n n () () ( ( ) + 1 ( )) i i i i D x = n n+ 1 H x n 2nH x xh x n n n n

64 Flaw Scattering-SDH Recall, the scattering amplitude for the pulse-echo P-wave case was: A3 D / L kb p

65 Flaw Scattering- SDH Using this separation of variables solutions at many frequencies to synthesize a P-wave impulse time domain solution (pulseecho) 3 amplitude creeping wave time, μsec The time-domain pulse-echo P-wave response of a.5 mm radius cylindrical void in steel (c p = 59 m/s, c s = 32 m/sec) obtained by applying a low-pass cosine-squared windowing filter between 1 and 2 MHz to the separation of variables solution and then inverting the result into the time domain with the inverse Fourier transform.

66 Flaw Scattering - SDH Example 4: pulse-echo SV-wave scattering of a cylindrical void A sv sv ( e ; e ) 3D i i L δ n i = n 2π n= 1 n = = otherwise ( 2 δ )( 1) n G n G n = 1+ ( ) ( ) 2 ( 1 ) ( ) ( 2 ) ( ) ( 1 ) ( ) () 1 ( ) () 1 ( ) () 1 ( ) () 1 ( ) C k b C k b D k b D k b n s n p n s n p C kb C kb D kb D kb n p n s n p n s

67 Flaw Scattering -SDH Again, the scattering amplitude for the pulse-echo SVwave case was: A3 D / L kb s

68 Flaw Scattering - SDH Using this separation of variables solutions at many frequencies to synthesize an SV-wave impulse time domain solution (pulse-echo) 3 2 creeping wave amplitude time, μsec The time-domain pulse-echo SV-wave response of a.5 mm radius cylindrical void in steel (c p =59 m/s, c s = 32 m/sec) obtained by applying a low-pass cosine-squared windowing filter between 1 and 2 MHz to the separation of variables solution and then inverting the result into the time domain with the inverse Fourier transform.

69 Flaw Scattering - SDH Experimentally determined scattering amplitude by deconvolution (side-drilled hole) V R ( ) G( ) ( ω) A ω = ω L G ω = s ω E ω ( ) ( ) ( ) () ( ) ˆ 1 ( ( ) ˆ 2 ) 4πρ2c α 2 E ω = V z, ω V ( z, ω) dz Ta ; ik L α 2Zr

70 Flaw Scattering -SDH A/L frequency, MHz