5. Conclusion

5.1 Contributions

In this work the propagation behaviour of ultrasound in austenitic weld metal has been analyzed by the time-harmonic plane wave approach. Bounded beam and pulse propagation as occurring in ultrasonic testing can be sufficiently dealt with by this approach. More sophisticated approaches principally do not offer any improvements in the results of plane wave modeling except ford diffraction and aperture effects and, therefore, the subject matter of this work has been limited to plane wave propagation in the bulk of the medium and at different types of interfaces.

Inspite of the fact, that the individual columnar grains of the weld metal have cubic symmetry, the austenitic weld metal as a whole exhibits cylinder-symmetrical texture, as substantiated by metallurgical examination, and therefore has been treated as an anisotropic poly-crystalline medium with transverse isotropic symmetry.

5.1.1 Wave modes

Generally three wave types occur in anisotropic materials. In media with transverse isotropic symmetry these are one with predominantly longitudinal character, one with predominantly transverse character, and one pure transverse wave. They are found as solutions of the eigenvalue problem represented by the Christoffel equation for the infinite space yielding direction-dependent phase and group velocities, and direction-dependent polarizations of the three wave modes, all of which have been calculated for ultrasound propagation in three-dimensional space:

• Generally in anisotropic materials the group velocity direction deviates from the direction of the wave vector. The practical consequence of this is beam skewing, so that in ultrasonic testing of anisotropic specimens the transducer has to be offset to effectively intercept the beam. Furthermore, an ultrasonic beam entering the anisotropic medium is spreaded due to the effect of beam skewing. It has been shown, that in the range of incidence angles relevant for ultrasonic weld testing
• the divergence of a quasi longitudinal beam is predominantly reduced,
• on the contrast the divergence of a quasi transverse beam is predominantly increased,
• the divergence of a pure transverse beam is least affected compared to the other wave types.
• The particle displacement polarizations form an orthogonal trihedral. If the columnar grain direction is not contained in the plane of wave propagation, the particle displacement polarizations are neither restricted to the plane of propagation nor perpendicular to it, respectively. The directions of polarizations rather vary in the three dimensional space as a function of the wave vector direction. Therefore the character of the waves which is described by the particle displacement polarizations is changed. In particular the character of the transverse waves, viz. vertically and horizontally polarized in the meridian plane, generally is not maintained.

Following details have been shown:

1. The properties of the T2 wave are as follows:
• The polarization direction of the transverse wave T2 is invariant, i.e. always perpendicular to the columnar grain direction Z.
• It is always perpendicular to the k vector. Therefore T2 has been defined as a pure wave.
• The polarization of the T2 wave varies as a function of the k-vector direction in the plane transverse to it. This means that it is not perpendicular to the plane of sound propagation and therefore T2 is generally not horizontally polarized.
2. The polarization direction of the transverse wave qT1 is not in the plane of sound propagation and therefore qT1 generally is no longer vertically polarized.
3. Increasing tilt of the columnar grain direction Z relative to the plane of wave propagation causes the transverse wave polarizations to change their 'roles':
1. If the grain tilt reaches 90°, the polarization of the pure transverse wave T2 is contained in the plane of wave propagation, therefore being now vertically polarized, though as before perpendicular to k-vector and columnar grain direction Z.
2. If the grain tilt reaches 90°, the polarization of the now pure transverse wave T1 becomes independent of the wave vector direction, and will be perpendicular to the plane of wave propagation, i. e. horizontally polarized.

Since polarization determines mode coupling at interfaces it can be concluded that

• in the case of an interface between two isotropic materials, the horizontally polarized wave (T_H) does not couple with the other two waves, viz. with the vertically polarized shear wave (T_V) and the longitudinal wave (L) and vice versa,
• in the case of an interface between two anisotropic materials, always all three wave modes couple.

5.1.2 Reflection and transmission

Plane wave reflection and transmission between two generally anisotropic materials has been analyzed. The analysis was divided into three sections:

• Reflection and transmission at perfect interfaces, which is a defect free rigid contact interface,
• Reflection and transmission at imperfect interfaces including the thin viscoelastic layer between perspex and anisotropic medium (i. e. the coupling of the ultrasonic transducer to the anisotropic weld and cladding metal),
• Reflection of bounded beams at an interface between fluid and anisotropic solid.

At the interfaces columnar grain orientations in the anisotropic weld metal were chosen as encountered in typical non-destructive testing problems. The computer codes to calculate reflection and transmission energy coefficients are written in FORTRAN 77 with graphics integrated. The types of interfaces considered were:

• Isotropic base metal - anisotropic weld metal, representing the weld fusion face and the cladding interface.

The transparency of the perfect interface for all three wave modes is fairly high and approximately independent of the columnar grain orientation. Mode conversion only marginally exceeds 10%, generally it is much lower. However, transverse waves exhibit mutual mode conversion at the fusion face, which depends on the columnar layback angle and can reach up to 100%. The energy of the transmitted transverse waves depends on the polarization of the incident transverse wave, because only modes with (at least partially) identical polarizations couple to the incident wave.

Conventional ultrasonic probes generate either vertically polarized transverse waves or horizontally polarized transverse waves. These waves incident from the isotropic face are split at the fusion face. In the anisotropic medium only two orthogonally polarized transverse waves, qT1 and T2, exist. In the case that the columnar grains are tilted relative to the plane of incidence, this decreases the energy of transverse waves considerably during examination of austenitic welds and cladded components. Both the transverse waves do not superpose due to different refraction angles.

In the isotropic base material, however, the both transmitted components, TH and TV, are degenerated and therefore form a resulting transverse wave T with polarization according to the intensities of both superposed components.

• Anisotropic base metal - anisotropic weld metal, represent the weld fusion face in welded austenitic cast components and the interface between adjacent columnar grain bundles.

Again for all three wave modes the transparency of the perfect interfaces made up by adjacent different crystallographic orientations is fairly high and is approximately independent of the columnar grain orientation. Mode conversion of the incident energy exceeds 10% only at certain points, generally it is much lower.

However, direction dependent mutual mode conversion of transverse waves can reach 100% as before.

At boundaries between two general transverse isotropic media with different grain orientations, when the propagation plane is not the meridian plane but an arbitrary plane, always both transverse wave modes couple simultaneously - with complementary energy distributions in reflection and transmission -, both with considerable energy, because the particle displacement polarization direction of the incident wave is not restricted by any crystallographic symmetry conditions.

Whereas the grain angle, which denotes the tilt of the columnar grains in the incidence plane, does not influence the transverse wave mode conversion at large scales, it is the layback angle denoting the tilt of the columnar grains out of the incidence plane, which governs the transverse wave mode conversion. In austenitic stainless steel weld testing with layback angles in the welding direction up to 20° and even more between adjacent dendritic bundles, large transverse wave mode conversion rates are to be predicted even in the low angle range, attenuating considerably the transverse wave actually used for testing.

In the special case where the columnar grains are contained in the meridian plane, i. e. no layback of the columnar grains in the incident plane, ( medium 1), mutual mode conversion of the transverse waves in medium 2 is similar to what is observed at the fusion face at wave incidence from the isotropic face.

Also the transparency of solid imperfect interfaces for all wave modes is still fairly high, though the crack area fraction comprises 75% of the interface. However, in contrast to the solid perfect interface a larger portion of the incident energy is reflected, mainly in the range below 45° incident angle . In this region reflection increases with frequency, and may be used to characterize the interface.

Similarly it is observed that a larger portion of the incident wave energy, quasi longitudinal and quasi transverse, is mode converted reaching 15\% at incidence angles larger than 45°, both, in reflection and refraction. However, mode conversion to the pure transverse wave T2 generally does not exceed 1%.

• The transparency of thin viscoelastic layers between isotropic and anisotropic materials, which represents the coupling layer between ultrasonic probe and austenitic material, basically is similar to that obtained during coupling on isotropic material. The columnar grain direction in the weld metal relative to the coupling surface only has little influence on the echo transparency.

Generally, as a matter of fact, transverse waves do not pass through a fluid coupling layer due to its vanishing viscosity. However, the viscosity of glycerine and even fresh water, though very small, is not zero. Therefore transverse waves are transmitted at a corresponding low level into the weld metal. High viscosity couplants provide transparencies up to 50% for transverse waves, whereas solid coupling between perspex and weld metal only yields slightly increased transparency.

Splitting of the slowness surface domain of permissible wave vector angles of the quasi transverse wave: Due to the concave parts of the slowness surface of the quasi transverse wave in anisotropic austenite for certain incidence planes the slowness surface splits into disjoint sectors of "permissible" wave vectors.

This is because the existence criterion for reflected and transmitted ultrasonic waves is, that their energy flow direction (group velocity) vectors should be real and point away from the boundary, thus defining the sectors of the slowness surface containing the permissible wave vectors. The remaining sectors of the slowness surface contain "forbidden" wave vectors, because reflected and transmitted sound rays would be directed towards the interface: such rays do not exist.

This splitting phenomenon occurs both in transmission and reflection, when upon incidence of the quasi transverse wave qT1 and also of the pure transverse wave T2 the wave vector of the mode converted quasi-longitudinal wave qL reaches its critical angle. The incident energy then is redistributed and a second qT1(2) wave is transmitted instead of the qL wave, the phase velocity having continuously decreased as a function of the incidence angle from the value of the qL wave to the value of the quasi-transverse qT1(2) wave. Both qT1 waves have almost the same phase and group velocities with slightly different

polarization directions. This poses special problems in testing of materials with transverse waves, making them less suitable for inspection.

The Schoch-effect, viz. the lateral displacement and splitting of an reflected ultrasound beam upon incidence of a beam onto a liquid-solid interface at an angle around the Rayleigh angle has been calculated for a beam with Gaussian profile for the liquid anisotropic solid. Apart from the effect that the Rayleigh angle and the phase velocity of Rayleigh waves vary with the columnar grain orientation, lateral displacement of the reflected beam relative to its position predicted by geometrical acoustics and splitting of the reflected beam are observed qualitatively similar to the isotropic case.

5.1.3 Ray tracing

Determining the columnar grain distribution by an empirical function, interfaces could be defined between neighbouring grain boundaries. A numerical procedure has been developed describing the transmission of ultrasound as it propagates through numerous grain boundaries and the energy flow direction associated with the wave of interest. Since the energy flow direction (direction of group velocity) is skewed with respect to the wave vector direction, ray paths generally are three dimensional in nature. The procedure developed allows to trace the most probable paths of ultrasound in anisotropic weld metal three dimensionally. The ray tracing code is written in FORTRAN 77 with graphics integrated.

Results for quasi longitudinal, quasi transverse and pure transverse waves are presented for different incidence angles, transducer positions, and microstructures (textures). To simulate a beam, seven rays are assumed to be generated at the probe index point. The divergence of the rays increases (decreases) by one degree steps with respect to the central ray direction.

Beam paths of quasi longitudinal and pure transverse waves are generally more straightforward than the paths of quasi transverse waves. However, it can be observed that beam paths are generally highly sensitive to the weld texture. This is due to the form of the slowness surfaces. The slowness surface of the quasi transverse wave exhibits concave and convex areas with cusps which results in largely varying group velocity (energy flow) directions and ray splitting.

When the quasi transverse wave is incident obliquely at the parent-weld metal interface, for a particular angle of incidence, the transmitted quasi longitudinal wave may not be propagating (evanescent). Then, due to the energy balance criterion the incident energy would be redistributed to the other propagating waves. Under these conditions there are two quasi transverse waves and one pure transverse wave propagating. The two quasi transverse waves, however, have different phase velocities, polarization directions and energy contents.

At every iterative step, there could be two quasi transverse waves branches, one with higher energy than the other. In this work both the possible rays have been traced. If the ray paths of both the quasi transverse waves are not significantly apart, then at the receiving end of the transducer, both the waves could {\em interfere} making it experimentally difficult to identify them.

5.1.4 Scattering of ultrasound

By assuming the weld metal to be mono-crystalline with transverse isotropic symmetry, the attenuation which is inherent in such materials can not be accounted for. The weld metal, therefore, has been assumed to be an anisotropic, polycrystalline material with cylinder-symmetric texture (transverse isotropy). Such material exhibits grain scattering depending on elastic anisotropy and geometric features of the grains and on the grain boundaries. To determine attenuation for an arbitrarily oriented columnar grain texture three-dimensionally the unified theory on elastic wave propagation in polycrystalline materials as proposed by Stanke and Kino in the Keller's approximation for equiaxed grains has been extended to austenitic weld metal. No restrictive assumptions are made with respect to the polarization direction of waves, viz. polarization deviation is taken into account stringently.

Attenuation coefficients in an austenitic CrNi 18 12 stainless steel have been calculated as a function of the wave vector to Z-direction of the crystallographic system X,Y,Z, and as a function of frequency. The computer code for evaluation of reflection and transmission energy coefficients is written in FORTRAN 77 which uses mathematical routines from the commercially available "International Mathematical Society Library" (IMSL).

Whereas for quasi-longitudinal and pure transverse waves attenuation theoretically vanishes when propagating in the columnar grain direction and reaches a maximum when propagating perpendicular to the columnar grain direction, for the quasi transverse wave maximum attenuation occurs at about 45° and, both, at normal incidence and at 90°, no attenuation is predicted.

Generally, attenuation by scattering reaches a high level with the grain size to wavelength ratio increasing and becomes independent on this ratio in the stochastic region though on a high level.

Attenuation of the quasi transverse wave is lesser than for quasi longitudinal and pure transverse waves related to the same wavelength. With increasing frequency the pure transverse wave reaches a level of attenuation, which is three times higher than those of the both other waves. Therefore it is not attenuation that renders the quasi transverse wave inappropriate for austenitic weld testing. It is rather because of the beam splitting this wave type undergoes in austenitic weld metal.

In the presence of ultrasound scattering the phase velocity varies by a maximum of 7% at 2 MHz and 100 µm grain size. Also dispersion of phase velocity occurs, which reaches 6% in the range up to 5 MHz at 100 µm grain size. Furthermore, in the presence of ultrasound scattering the polarization deviation is changed. Whereas the polarization of the pure transverse (T2) wave remains unaffected by scattering being always perpendicular to the wave vector, the other waves (qL and qT1) exhibit an extra deviation up to 2°.

5.2 Areas for continued research

5.2.1 Modeling

The theory of scattering in spherical grains has been extended to ellipsoidal grains using the correlation function suggested by Ahmed. The necessary mathematical programming has been done and is in final stages of implementation.

5.2.2 Software

For the present work ultrasound propagation softwares have been developed which use a macroscopic material model based on the result from averaging the microscopic anisotropy of the single grains. However, since grain growth simulation software is available, which uses the welding input parameters and weld pool data, as well as the data of 'Orientation Imaging Microscopy', it would be worthwhile to integrate this simulation software with the software dealing with ultrasound propagation in such simulated structures in order to validate the predictions of ray tracing.

5.2.3 Experimental validation

Measurements of the attenuation of the three wave modes in real materials comprising the full scale of industrially relevant stainless steels (austenitic stainless CrNi-steels, fully austenitic stainless steels with increased Ni-content, Nickel based-alloys, and Duplex steels (Ferritic-austenitic steels)) can determine how generally useful the plane wave ansatz and the present material model is. The theoretically predicted attenuation of quasi longitudinal waves could be already verified. The experience gained hereby showed that any procedure that does not fully account for the anisotropic nature of the material is bound to yield disputable results: if energy losses due to mode conversion, due to beam skewing and beam spreading, which are characteristic for the anisotropic nature of the media considered in this work, are not taken into account, apparent attenuation due to these effects adds to the scattering-induced attenuation.It is this apparent attenuation that renders measurement and evaluation of scattering-induced attenuation of shear waves in austenitic steel samples more difficult, because it may reduce the amplitude of the reflected signal to the noise level.Experimental ray tracing also can determine how useful the plane wave ansatz and the present material model is. This needs a series of weld metal specimens, which allow to measure contour maps of the sound field distribution at increasing sound path lengths. Specimens of an austenitic stainless CrNi-weld are already available.Both these experimental research works are currently going on.