ELASTIC WAVE DIFFRACTION AT CRACKS IN ANISOTROPIC MATERIALS. G.R. Wickham Department of Mathematics University of Manchester, Manchester, M13 9PL, UK.

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1 ELASTIC WAVE DIFFRACTION AT CRACKS IN ANISOTROPIC MATERIALS PA Lewis Division of Mathematics, Bolton Institute, Deane Road, Bolton, BL3, UK JAG Temple AEA Technology, 552 Harwell, Didcot, Oxon, OXll ORA, UK GR Wickham Department of Mathematics University of Manchester, Manchester, M13 9PL, UK INTRODUCTION Ultrasonic inspection is used to confirm that there are no defects of concern in various regions of a nuclear reactor primary circuit All materials are naturally anisotropic, but if the grains are small relative to the ultrasonic wavelength and are also randomly oriented, then the material will appear as homogeneous and isotropic as in ferritic steel The ultrasonic wavelength is chosen as a compromise between resolution of defect size and acoustic noise from grain boundaries In austenitic steel, the wavelength chosen will typically be smaller than the grain size, at least in one direction The grains are not randomly oriented but exhibit macroscopic patterns which depend on the welding process, and the material is neither homogeneous nor isotropic Elastic waves diffracted from crack-like defects are essential to the accurate sizing of such defects The redistribution of energy into a range of diffracted angles is a purely local phenomenon controlled only by the properties of the material at the crack tip and its shape We need to understand how diffraction is modified in an anisotropic material Austenitic weld material never has a symmetry lower than orthorhombic and has often been treated as transversely isotropic [lj We have calculated diffraction coefficients for a crack at an arbitrary orientation in a material of orthorhombic or higher symmetry, for any mode incident at any angle on the crack edge Our results are also of interest for ultrasonic inspection of fibre reinforced composite materials ANISOTROPIC MEDIA The incident displacement field, Aeikr, must satisfy Navier's equation of Review of Progress in Quantitative Nondestructive Evaluation, Vol 15 Edited by DO Thompson and DE Chimenti, Plenum Ptess, New York,

2 motion where r denotes the point of interest, Ui(r) the component of displacement in the ith direction at this point where the density is p(r) and the elastic constants are Ciilc/(r) The notation ~ denotes ()2ui/8t 2 and Ua,/J denotes 8ua /8x{J and the summation convention applies A time harmonic wave with displacement!! given by (1) Up = Apexpiw(mr - t) (2) where Ap is the component of the polarization, m is the slowness vector proportional to the inverse of the phase velocity, w the frequency of the wave and t the time; satisfies equation 1 provided [2] that the Green-Christoffel equation: is satisfied Here the ni are the components of a unit vector in the direction of m and the eigenvalues Vp are the allowed phase velocities, usually three distinct values The eigenvectors associated with V p, p = 1, 3 and denoted A can be used to determine the group velocities V9 for each mode [3] Vi 9 = Ciilc/AiA/mlc p The group and phase velocity directions are not necessarily parallel, an effect known as beam skewing IT the allowed slownesses are plotted as functions of propagation direction they form three surfaces which can intersect In general the eigenvectors are not purely longitudinal (P) or shear (S) DIFFRACTION The component of the slowness vector along the crack edge governs the spread of diffracted energy and the amplitudes of diffracted rays For a ray incident at angles other than 90 relative to the edge of a crack-like defect, the diffracted rays in an isotropic material form circular cones with axis along the tangent to the crack edge The amplitude of rays varies around the circumference of the cone of diffracted rays for a given angle of -incidence and also varies as the angle of incidence changes In an anisotropic material, the diffracted rays do not generally form circular cones The cross-section of the envelopes of diffracted rays will be closed curves with considerably less symmetry than a circle and may well have the crack edge not at the 'centre' of the curve As the incident ray approaches grazing incidence on the crack edge, the diffracted rays may split up into cones propagating in distinctly different directions In general this is not a typical inspection geometry CONES OF DIFFRACTED RAYS IN ANISOTROPIC MEDIA Plotting the slowness sheets as isometric plots about the crack edge and slicing these at different components of the slowness projection defines contours of the envelopes of diffracted slowness vectors Figure 1 shows this for the quasi-s2 mode in a material representing austenitic steel whose density is p = 79 x 10 3 kgm- 3 and with elastic constants in Voigt notation given by [4]: (3) (4) 250 x 109Nm-2, 112 x 10 9 Nm- 2, 117 x 109Nm-2, 250 x 10 9 Nm- 2, 180 x 10 9 Nm- 2, 92 x 109Nm-2, 250 x 10 9 Nm- 2, 138 x 10 9 Nm- 2, 70 x 109Nm-2, (5) 42

3 04 05 > ' '-- ---' X 05 Figure 1: Projection of the quasi-s2 slowness surface of austenitic steel along the tangent to the crack edge when this is the z-axis Contours obtained from slicing this at 01, 02 and 03 sm- 1 provide an indication of the changing shape of the envelopes of diffracted slowness vectors as the incident ray approaches grazing incidence The crack edge is along the z-axis and we have cut the surface at slownesses of 01, 02 and 03 x 1O-3m-ls to produce the envelopes of diffracted slowness vectors For the quasi-s2 mode at higher incident slowness projections, 03 X 1O- 3 m- 1 s, the diffracted slowness vectors are confined to two separate cones, neither of which is circular Of course these slices show only the phase velocity directions and the energy flows along the group velocity direction which is the normal to the slowness surface From figure 1 we see that the group velocity will point along the crack edge at three values of slowness, leading to complicated wave decay properties: in directions corresponding to cuspidal edges of the group velocity surface as O(r- S / 6 ) and as O(r- 1 / 2 ) along directions corresponding to conical points [5J This behaviour has recently been demonstrated in finite element numerical models of anisotropic media [6J THE MODEL The Geometrical Theory of Diffraction (GTD) is the most useful approach for calculating diffraction coefficients [7J The theory of wave propagation in anisotropic materials is well known [8,9,lOJ GTD can be useful only if the diffraction coefficients are known for arbitrary angles of incidence on defects in arbitrary orientations located in materials with cubic or lower symmetries Norris and Achenbach [11,12J calculated diffraction coefficients for a crack located in the plane of isotropy in a transversely isotropic material One of us previously undertook purely numerical calculations in an attempt to determine diffraction coefficients for more general anisotropic material, using a 3D time dependent finite difference simulation (13], but these were unsuccessful The geometry suggests a Wiener-Hopf problem which is, in general, a 3x3 43

4 Our backscatter u 1 com onent inc-sv NA backscatter u1 com onent inc-sv NA backscatter u 1 com onent inc-sv Our bockscattered uj com---'lonent inc-sv 04 Q2 Ii il - 1 -~ 00 -ISO ISO Our bockscottered u1 com onent inc-p Q : i: NA backscottered u3 co~nent inc-sv 04 : : : i 1 02 : :r 00 j: :\ - -ISO ISO NA bockscottered u1 com onent inc-p D" NA backscottered u3 co~nent inc-sv NA backscattered u1 com onent inc-p 04 Our bockscottered u3 com onent inc-p NA bockscottered u3 com onent inc-p \ NA bockscottered u3 com anent inc-p Figure 2: Comparison of our new results with those of Norris and Achenbach [11,12] for a tranvsersley isotropic material with the crack in the plane of isotropy Our results are in the left hand column with all three diffracted modes shown, but with the quasi-sh mode identically zero The Norris and Achenbach results are shown as separate modes in columns two and three 44

5 matrix problem In elasticity, mode conversion makes these matrix problems difficult to solve We introduce the slowness tensor whose cofactors and determinant are respectively, Bkm( a, w) and S( a, w) We take the X 2 crack axis to be tangent to the crack with the Xl axis in the plane of the crack and normal to the crack edge and X3 normal to the crack plane and normal to the crack edge The condition of vanishing stress on the upper surface of the crack leads to the Wiener-Hopf integro-differential equation with U the fundamental Green's function: (6) For an incident plane wave of slowness vector (kl' k2' k3 ), the solution is translationally invariant parallel to the crack edge Fourier transformation shows that the boundary condition is equivalent to the Wiener-Hopf functional equation where T+ is the transformation of the unknown surface traction on the complementary half-plane to the crack, U- is the transform of ~ the crack opening displacement, A is the amplitude of the incident wave and K is a tensor kernel function of al with k2 appearing as a parameter As usual, the solution depends on the determination of matrix factors K+, K- such that [K+tl K = K- The mathematical argument proceeds by taking out the principal contributions which occur on the main diagonal (9) with Kj = O( Val) as lall ~ 00 Performing a sum and product split on H gives with H = 0(1) as lall ~ 00 An intricate argument is used to show that H- satisfies the non-singular second kind integral equation: (8) (10) in which and H-(a) = 1- G-(a) H-(() K,((, a)d( 2?rz -00 K,((,a) = (H-I(() - I) T((,a) T(Ca) = G-((~ = ~-(a) (11) (12) (13) The integral equation acts as a continuation formula for H-, once its values are known on the real line they are known everywhere The equations are solved by using an efficient sum splitter based on a conformal mapping in the strip of analyticity of K, followed by Gaussian quadrature along a finite contour Scalar product splits are divided by a gauge function with known factors to remove bad behaviour at infinity The integral equation is solved for the unknown matrix factor using the Nystrom method with Gaussian block elimination Kelvin's method of stationary phase is used 45

6 <--_~_~ ~_ o <--_~_~ ~_--' o a d= 10 case L-_~_~_~_----' o L-_~_~_~_--' -180 ~90 o Figure 3: Backscattered diffraction coefficients as a function of increasing anisotropy The crack is in the plane of isotropy and the incident wave is in the plane perpendicular to the plane of the crack (two dimensional problem) to derive the final diffraction coefficient [14] where pj are the solutions of S(a3) = 0 lying in the upper half plane or on the real axis, and 5~ = -al, a2 and pj( -al, a2) for 1 = 1,2,3 respectively: (14) EXAMPLES Norris and Achenbach [11,12] presented results for diffraction from a crack in a transversely isotropic material described by elastic constants which allowed the ratio of C ll to C 33 to vary Their result contains sums of terms in this ratio However some factors were omitted from some terms affecting results for non-normal incidence on the crack Correcting these factors and plotting their results for backscatter in a material with C ll = 5 X C33 and results using our new factorisation yields the comparisons shown in figure 2 where the diffracted Ul and U3 components for incident P and SV modes are shown Our results are in the left hand column and contain all three diffracted modes (the quasi-sh mode is identically zero in this geometry) and the Norris and Achenbach results are in the next two columns, separated by mode The agreement is excellent To illustrate how diffraction coefficients evolve with increasing anisotropy we show in figure 3 the backscatter in the transversely isotropic material with the ratio of C ll/c33 increasing from 40 (isotropy) to 100 (very anisotropic) To show how the general diffraction coefficients behave in an anisotropic 46

7 l~,r- U~I~in~fM~ri~li~c~'~I~~~i~nc~-~p~O~I~S~O~ ~ '----"T:~== -T"'-"''''-'-'!!ipt----, oa ~ U 1 in austenite inc-p at 60 oa ', ' """--: ' F- T'U3"-'in"--'a'-"u"-sl"'e"'n"'il;;e"'in"'c'--!P--"o"-l -'S~O~ ~ " I' 00 r- ~_:u"-l'-i'in,-"a,,,1 "'ho=-"'u"ro"'n"iu"-m!'-'in"'c"-!p--"oe,i-'s"'o'- -, U3 in 01 ho-uranium inc-p 01 SO ', " oa Ii :~ ' c: \ : ' ' -\ ' Figure 4: General diffraction coefficients for a crack in the Xl - X2-plane of isotropic ferritic steel, anisotropic austenitic steel and alpha-uranium with the incident P-wave in the plane perpendicular to the plane of the crack and incidence at 60 47

8 material we consider three materials: isotropic ferritic steel, austenitic steel and alpha-uranium, which is truly orthotropic For a crack in the ZI - z2-plane of the crystal coordinates, with a quasi-p wave incident at 60 0 in the plane perpendicular to the crack plane, the diffraction coefficients are shown in figure 4 The increasing complexity with increasing anisotropy is evident At some angles the coefficients are multi-valued and the diffracted waves will be sensitive to direction It would be wise not to base inspections on elastic waves travelling near these directions, that is at ±45 to the crystal axes CONCLUSIONS There are clearly differences in elastic wave diffraction in anisotropic materials compared with isotropic materials The cones of rays are no longer circular and may even split into separate cones There are some directions, corresponding to cusps and conical points of the velocity surface, which should best be avoided in designing an inspection technique because, at best, these directions are likely to provide very variable responses Apart from these directions, the diffraction coefficients behave smoothly and are similar to those in isotropic materials but with different amplitudes Whether the numerical differences in these amplitudes are important will depend on whether the particular combination of crack orientation relative to the crystal axes and the ultrasonic angle of incidence which produce different results can occur in real cases of practical importance ACKNOWLEDGEMENTS This work was originally funded by the UK Health and Safety Executive as part of a programme of Nuclear Safety Research and has been continued with funding from AEA Technology and the University of Manchester REFERENCES 1 DS Kuperman and KJ Reiman, Ultrasonics, 16(1), 21-27, (1978) 2 JL Synge, JMath Phys 35, , (1956) 3 F1 Fedorov, Theory of elastic waves in crystals, (Plenum, New York, 1968) 4 F Champigny, B Nouilhas, Private communication 5 VT Buchwald, Proc Roy Soc A253, , (1959) 6, Z You, M Lusk, R Ludwig and W Lord, IEEE Trans Ultrasonics Ferroelectrics and Frequency Control, 38(5), , (1991) 7 JD Achenbach, AK Gautesen and H McMaken, Ray Methods for Waves in Elastic Solids, (Pitman, London, 1982) 8 MJP Musgrave, Proc Roy Soc A226, , (1954) 9 MJP Musgrave, Proc Roy Soc A226, , (1954) 10 MJP Musgrave, Reports on Progr in Physics 22, 75-79, (1959) 11 AN Norris, and JD Achenbach, in Review of Progress in Quantitative Non Destructive Evaluation, Edited by DO Thompson and DE Chimenti, Vol 3A, (Plenum, New York, 1984), pp AN Norris, and JD Achenbach, QJ Mech Appl Math 37, , (1984) 13 JAG Temple and L White, in Review of Progress in Quantitative Non Destructive Evaluation, Edited by DO Thompson and DE Chimenti, (Plenum, New York Vol 12, 1993), pp GR Wickham, PA Lewis and E Walker, Proc Roy Soc to be published (1995) 48