Kolkoori ultasonics(2013)

Ultrasonics xxx (2013) xxx–xxx
Contents lists available at ScienceDirect
Quantitative evaluation of ultrasonic C-scan image in acoustically
homogeneous and layered anisotropic materials using three
dimensional ray tracing method
Sanjeevareddy Kolkoori ⇑, Christian Hoehne, Jens Prager, Michael Rethmeier, Marc Kreutzbruck
Department of Non-Destructive Testing, Acoustical and Electromagnetic Methods Division, Federal Institute for Materials Research and Testing (BAM), Unter den Eichen 87,
D-12205 Berlin, Germany
a r t i c l e i n f o
Article history:
Received 20 March 2013
Received in revised form 9 August 2013
Accepted 10 August 2013
Available online xxxx
Keywords:
Ultrasonic non-destructive evaluation
Ultrasonic C-scan image
Anisotropic austenitic steel
3D ray tracing
Directivity
a b s t r a c t
Quantitative evaluation of ultrasonic C-scan images in homogeneous and layered anisotropic austenitic
materials is of general importance for understanding the influence of anisotropy on wave fields during
ultrasonic non-destructive testing and evaluation of these materials. In this contribution, a three dimen-sional
ray tracing method is presented for evaluating ultrasonic C-scan images quantitatively in general
homogeneous and layered anisotropic austenitic materials. The directivity of the ultrasonic ray source in
general homogeneous columnar grained anisotropic austenitic steel material (including layback orienta-tion)
is obtained in three dimensions based on Lamb’s reciprocity theorem. As a prerequisite for ray trac-ing
model, the problem of ultrasonic ray energy reflection and transmission coefficients at an interface
between (a) isotropic base material and anisotropic austenitic weld material (including layback orienta-tion),
(b) two adjacent anisotropic weld metals and (c) anisotropic weld metal and isotropic base material
is solved in three dimensions. The influence of columnar grain orientation and layback orientation on
ultrasonic C-scan image is quantitatively analyzed in the context of ultrasonic testing of homogeneous
and layered austenitic steel materials. The presented quantitative results provide valuable information
during ultrasonic characterization of homogeneous and layered anisotropic austenitic steel materials.
2013 Elsevier B.V. All rights reserved.
1. Introduction
Austenitic cladded materials, austenitic welds and dissimilar
welds are extensively used in primary circuit pipes and pressure
vessels in nuclear power plants and chemical industries because
of their high fracture toughness and resistance to corrosion. It is
very important to evaluate the structural integrity of these compo-nents.
Ultrasonic non-destructive inspection of austenitic cladded
and welded components is complicated because of anisotropic
columnar grain structure leading to beam splitting and beam
deflection [1–6]. Simulation tools which provide quantitative
information on ultrasonic wave propagation play an important role
in developing advanced and reliable ultrasonic testing techniques
and in optimizing experimental parameters for inspection of aniso-tropic
materials.
The presented research work is motivated by the interest in
quantitative evaluation of an accurate ultrasonic C-scan image in
homogeneous and layered anisotropic austenitic materials which
is of general importance in understanding the wave propagation
behavior and optimization of experimental parameters during
the ultrasonic non-destructive inspection of anisotropic materials.
In this contribution, an attempt has been made to determine the
ultrasonic C-scan image quantitatively in homogeneous and lay-ered
anisotropic austenitic steel materials using 3D ray tracing
method. The influence of 3D columnar grain orientation on ultra-sonic
C-scan image is analyzed in the context of ultrasonic non-destructive
testing and evaluation (NDTE) of layered austenitic
steel materials.
Using surface acoustic wave technique, Curtis and Ibrahim [7]
conducted texture studies in columnar grained austenitic steel
materials and suggested that the austenitic steels exhibit trans-verse
isotropic symmetry. Generally, in an austenitic steel material
three wave modes will exist, one with quasi longitudinal wave
character (qP), one with quasi shear wave character (qSV) and
one pure shear wave (SH). Pure shear horizontal wave (SH) polar-izes
exactly perpendicular to the plane of wave propagation, i.e. in
the plane of isotropy, so that the polarization direction of this
mode is always perpendicular to the wave vector direction.
An ultrasonic C-scan image is defined as an image representing
the ultrasonic amplitude distribution over the sectional area of the
component. Generally, ultrasonic C-scan images are used for the
⇑ Corresponding author. Tel.: +49 8104 4104; fax: +49 8104 4657.
E-mail addresses: sanjeevareddy.kolkoori@bam.de (S. Kolkoori), christian.
hoehne@bam.de (C. Hoehne), jens.prager@bam.de (J. Prager), michael.rethmeier@
bam.de (M. Rethmeier), marc.kreutzbruck@bam.de (M. Kreutzbruck).
0041-624X/$ - see front matter 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.ultras.2013.08.007
Ultrasonics
journal homepage: www.elsevier.com/locate/ultras
Please cite this article in press as: S. Kolkoori et al., Quantitative evaluation of ultrasonic C-scan image in acoustically homogeneous and layered aniso-tropic
materials using three dimensional ray tracing method, Ultrasonics (2013), http://dx.doi.org/10.1016/j.ultras.2013.08.007

2 S. Kolkoori et al. / Ultrasonics xxx (2013) xxx–xxx
non-destructive testing (NDT) of the volumetric defects in engi-neering
materials [8–12]. In case of isotropic materials, the calcu-lation
of an ultrasonic C-scan image is straightforward approach
because the wave phase velocity and group velocity directions
are equal. Whereas in homogeneous and layered anisotropic mate-rials,
due to the directional dependency of ultrasonic wave propa-gation,
it is very important to consider all the anisotropic effects of
the material while calculating an accurate ultrasonic C-scan image.
Schmitz et al. [13] presented the 3D RaySAFT algorithm to calcu-late
the direction of the ultrasound beam and the deformation of
the transmitted ultrasonic field qualitatively in inhomogeneous
weld material and discussed the C-scan representation of ray trac-ing
results in unidirectional weld structure qualitatively. They con-sidered
the 2D representation of the columnar grain orientation
(i.e. grain orientation is confined to the meridian plane) in the
austenitic material. With this particular assumption, the quasi lon-gitudinal
(qP) and quasi shear vertical (qSV) waves couple at inter-faces,
whereas the horizontally polarized pure transverse wave
(SH) decouples.
Generally, columnar grains in the austenitic weld material are
not only tilted in the plane of sound propagation but also perpen-dicular
to it. The columnar grain orientation along the weld run
direction is called as layback orientation. If the columnar grain ori-entation
of the austenitic steel material is oriented in 3D space, the
particle displacement polarizations of incident, reflected and trans-mitted
rays are neither restricted to the plane of wave propagation
nor perpendicular to it. Consequently, coupling between all the
three wave modes (i.e. qP, qSV and SH waves) exist.
The propagation of acoustic waves within an ideal isotropic
multilayered plate structure using a 2D ray technique was pre-sented
by Sadler and Maev [14]. A computationally efficient Gauss-ian
beam superposition approach to calculate the transducer fields
three dimensionally in layered materials, immersed components
and inhomogeneous anisotropic materials have been presented
by Spies [15–18]. The commercially available ultrasonic modeling
and simulation tool CIVA [19–22] is able to calculate the ultrasonic
wave propagation in homogeneous and heterogeneous materials,
where ultrasonic wave propagation is modeled using semi-analyt-ical
approximated methods. A ray theory based homogenization
method for simulating ultrasonic transmitted fields in multilay-ered
composites was presented by Deydier et al. [23]. According
to this homogenization method, the parallel regions are simplified
with one homogeneous medium whereas non-parallel regions are
replaced by progressively rotated homogeneous media. A compar-ison
of simulated 2D ultrasonic wave fields in a homogeneous sin-gle
crystal transversal isotropic austenitic steel material (X5 CrNi
18 10) with experiments were presented by Ernst et al. [24] and
they assumed 2D representation of columnar grain orientation.
Based on the reciprocity theorem, the directivity patterns for nor-mal
and transverse point sources in the unidirectional grain struc-tured
austenitic steel (308) were presented by Spies [25]. Recently,
a 2D ray tracing method for evaluating ultrasonic field profiles in
general inhomogeneous anisotropic austenitic steel materials was
presented in [26]. A three dimensional ultrasonic probe model
based on the Fourier transform technique in a homogeneous, linear
elastic, anisotropic half space was presented by Niklasson [27].
In the presented research work, the ultrasonic ray theory is ex-tended
to the more general case in austenitic materials, where the
columnar grains are not only tilted in the plane of sound propaga-tion
but also along the weld run direction. The ultrasonic C-scan
image in homogenous and layered anisotropic austenitic steel
materials is quantitatively evaluated and its importance to the
practical ultrasonic NDT of anisotropic austenitic steel and clad
materials is discussed. The influence of columnar grain orientation
and layback orientation on an ultrasonic C-scan image is quantita-tively
analyzed in an anisotropic austenitic steel material.
The aim of the present paper is three fold. First, we present the
theoretical description of evaluating ultrasonic C-scan image quan-titatively
in homogeneous and layered anisotropic materials using
a 3D ray tracing model. Ultrasonic ray reflection and transmission
behavior at the interface of two general anisotropic solids is solved
three dimensionally and the influence of layback orientation on en-ergy
reflection and transmission coefficients at an interface be-tween
two columnar grained austenitic steel materials is
presented. The ultrasonic beam directivity for three wave modes
(i.e. qP, qSV and SH waves) in a columnar grained austenitic steel
material is obtained three dimensionally using Lambs reciprocity
theorem. The influence of 3D columnar grain orientation on ultra-sonic
ray source directivity patterns for qP, qSV and SH waves un-der
the excitation of normal and tangential forces on semi-infinite
columnar grained austenitic steel material is investigated. Second,
the effect of columnar grain orientation and layback orientation on
an ultrasonic C-scan image in homogeneous austenitic steel mate-rial
is quantitatively analyzed. Third, the quantitative results for
ultrasonic C-scan image evaluation in homogeneous and multilay-ered
austenitic steel material with different crystallographic orien-tations
are presented.
2. Importance of layback orientation in the modeling of
ultrasonic wave propagation in austenitic welds
The assumption of two dimensional columnar grain orientation
(i.e. no layback orientation) is valid only for ultrasonic testing of lon-gitudinal
defects (i.e. defects oriented parallel to the weld run direc-tion)
in austenitic weld materials. In this ideal case, the ray tracing
method can be restricted into two dimensions (i.e. xz plane) in order
to evaluate the ultrasonic wave propagation in austenitic welds
[26,28–30]. Whereas for the ultrasonic inspection of transverse de-fects
(i.e. defects oriented perpendicular to the weld run direction)
in austenitic welds, it is essential to consider layback orientation
since the ultrasonic ray propagation is no longer in the xz plane
but in 3D space. Furthermore, a spatially separated ultrasonic sen-der
and receiver arrangement in a V-shaped form is generally used
for transverse defects detection [31] and consequently 3D ultra-sonic
ray paths are strongly influenced by the layback orientation
along the weld run direction. Recently, a Synthetic Aperture Focus-ing
Technique (SAFT) was applied for imaging transverse defects in
austenitic welds and concluded that the quality of the reconstructed
image and the determination of defect location and sizing can be
improved by assuming proper layback orientation in the austenitic
weld [32]. Chassignole et al. [33] investigated the crystallographic
texture of the multipass austenitic welds using X-ray diffraction
(XRD) analysis and observed a layback angle of 9 along the plane
parallel to the welding direction. Additionally, the layback angle
in the weld run direction varies with the welding speed.
3. Theoretical procedure: ultrasonic C-scan Image evaluation in
homogeneous and layered anisotropic materials
In this section, the evaluation of three dimensional ultrasonic
ray energy paths in general anisotropic solids, ray transmission
and reflection coefficients at an interface between two columnar
grained austenitic steel materials and ray directivity factor are dis-cussed.
These important features of the ray will play an important
role in quantitative evaluation of ultrasonic C-scan image in general
homogeneous and layered anisotropic austenitic steel material.
3.1. Evaluation of 3D ultrasonic energy ray path
In order to set a desired plane of anisotropic medium as an
incident plane, we need to rotate the coordinate system by

S. Kolkoori et al. / Ultrasonics xxx (2013) xxx–xxx 3
transforming the elastic stiffness matrix from crystallographic
coordinate system to the calculated coordinate system. The bond
matrix multiplication method [34] is used to obtain the stiffness
matrix in the calculated coordinate system and is represented as:
CN ¼ M C MT ; ð1Þ
where C and CN are the matrices of stiffness constants in the old and
new coordinate systems, respectively. M stands for the bond trans-formation
matrix and MT represents its transposed pair. Fig. 1
shows the columnar grain axis z which is arbitrarily rotated around
y-axis by columnar grain angle h and x-axis by layback angle w.
The Christoffel equation [34] of ultrasonic plane wave propaga-tion
in an anisotropic solid is given as:
k2Cij qx2dij
h i

mj
¼ 0; ð2Þ
where C is the Christoffel tensor, q the density of the material, x
the angular frequency, m the particle displacement, k the wave num-ber
and i, j take the values 1, 2, 3.
dij is the Kronecker symbol with the property:
dij ¼ dji ¼ 0 ði–jÞ; dij ¼ 1 ði ¼ jÞ:
Eq. (2) has non-trivial solutions for the particle displacement m,
only if the determinant equals zero. The eigenvalues of the deter-minant
are processed yielding three phase velocity magnitudes
which correspond to the quasi longitudinal (qP), quasi shear verti-cal
(qSV) and quasi shear horizontal (qSH) waves. Slowness vector
(S) is defined as the reciprocal of the phase velocity vector. Particle
polarization components for the three wave modes can be ob-tained
by substituting the phase velocity magnitudes in Eq. (2)
and solving the eigenvector problem.
The analytical expressions for the Poynting vector along x, y and
z directions for the three wave modes present in general aniso-tropic
solids are given by [26]:
Px ¼
1
2
A2x
Vi
!
#
Ay
Ax
ða1Þþ
Az
Ax
ða2Þþ

Az
Ax

Ay
Ax
ða3Þþ
2
Ay
Ax
ða4Þþ
2
Az
Ax
ða5Þþa6
;
ð3Þ
Py ¼
1
2
A2x
Vi
!
#
Ay
Ax
ðb1Þþ
Az
Ax
ðb2Þþ

Az
Ax

Ay
Ax
ðb3Þþ
2
Ay
Ax
ðb4Þþ
2
Az
Ax
ðb5Þþb6
;
ð4Þ
Pz ¼
1
2
A2x
Vi
!
#
Ay
Ax
ðc1Þþ
Az
Ax
ðc2Þþ

Az
Ax

Ay
Ax
ðc3Þþ
2
Ay
Ax
ðc4Þþ
2
Az
Ax
ðc5Þþc6
;
ð5Þ
where the coefficients am, bm and cm with m = 1, 2, 3, 4, 5, 6 are ex-pressed
in terms of elastic stiffness components of the material and
directional cosines of the propagating wave. The explicit expres-sions
for the am, bm, cm, Ay
Ax
and Az
Ax
were found in [26].
Averaged stored energy density in a general anisotropic med-ium
is expressed as:
Uav ¼
q
2
A2x
1 þ
2
Ay
Ax
þ
2
Az
Ax
!
: ð6Þ
The energy velocity in a general homogenous anisotropic med-ium
is defined as the ratio of the averaged complex pointing vector
and the averaged stored energy density. The energy velocity in a
lossless general anisotropic medium is represented as:
Vg ¼
Pav
Uav
: ð7Þ
Introducing Eqs. (3)–(6) into the Eq. (7) yields the energy veloc-ity
components along x, y and z directions respectively.
The magnitude of the energy velocity for an incident ultrasonic
wave mode in a general anisotropic medium is expressed as:
q
Vg ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
V2
gx þ V2
gy þ V2
gz
: ð8Þ
The above explicit analytical expressions for energy velocity
components are used to evaluate the 3D ray energy direction and
velocity magnitudes in the ray tracing calculation. The ray trajec-tory
in a general inhomogeneous anisotropic medium is related
to the energy velocity. In the present article, ray tracing calcula-tions
are performed in three dimensions by taking into account
3D columnar grain orientation (including layback orientation) of
the anisotropic austenitic steel material. The analytical expressions
presented in this section play an important role in evaluating 3D
ray reflection and transmission coefficients, 3D ray directivity fac-tor
which will be discussed in the Sections 3.2 and 3.3.
3.2. 3D Ultrasonic ray reflection and transmission coefficients at an
interface between two differently oriented austenitic steel materials
The behavior of the reflected and transmitted rays at an inter-face
between two general anisotropic solids is obtained by solving
the problem of reflection and transmission analytically based on
the approach presented by Rokhlin et al. [35,36]. A review of the
approach was presented in [26].
The resulting six degree polynomial equations in the modified
Rz
Rz
Rz
Christoffel equation for evaluating unknown vertical slowness
components for the reflected and transmitted waves in medium
1 and medium 2 are expressed as follows:
ARðS6
5
Rz
4
3
2
Rz
Þ
þBRðSÞ
þCRðSÞ
þDRðSÞ
þERðSÞ
þFRSþGR ¼ 0; ð9Þ
Rz
AT ðSTz
Þ
6
þBT ðSTz
5
þCT ðSTz
Þ
4
þDT ðSTz
Þ
3
þET ðSTz
Þ
Þ
2
þFTSTz
þGT ¼ 0:
ð10Þ
The coefficients in the Eqs. (9) and (10) are expressed in terms
of horizontal slowness components (SX, SY) and material elastic
constants of the medium 1 and medium 2. The coefficients in the
Eqs. (9) and (10) are presented in [26]. Six solutions are found in
Fig. 1. Coordinate system used to represent the three dimensional crystal orien-tation
of the transversal isotropic austenitic steel material. h represents the
columnar grain orientation and w represents the lay back orientation.

Fig. 2. Graphical representation of evaluating particle displacement directivity of the (a) quasi longitudinal (qP) wave, (b) quasi shear vertical wave (qSV) and (c) pure shear
horizontal wave (SH) using Lamb’s reciprocity theorem.
each medium from which only three correspond to physically real
solutions. Energy flow directions for the reflected and transmitted
waves are the criterion for the selection of valid roots [37]. The
roots of six degree polynomial are generally complex. Purely real
roots correspond to propagating waves, purely imaginary roots
correspond to evanescent waves whose amplitude decay in the
direction perpendicular to the energy propagation and complex
roots represent inhomogeneous waves.
Energy reflection and transmission coefficients are obtained by
solving the boundary conditions for traction force components and
particle velocity components at the interface. The same procedure
with appropriate changes is used for the case of wave incidence
from an isotropic medium to an anisotropic medium and from an
anisotropic medium to an isotropic medium. In case of wave inci-dence
on a free surface boundary of anisotropic medium, the
refraction coefficients will be equated to zero. In case of wave inci-dence
from fluid medium into anisotropic solid, there are no re-flected
shear waves at the smooth planar interface.
3.3. 3D Ultrasonic ray directivity evaluation in columnar grained
austenitic steel material
In this section, the ray directivity in a general anisotropic med-ium
is obtained three dimensionally based on Lamb’s reciprocity
theorem [25,26,38,39]. The theory is applied for evaluating the
ray directivity in general anisotropic austenitic steel material
exhibiting three dimensional columnar grain orientation. Consider
a radial force F, which is applied at a point remote from the origin
(see Fig. 2a). This force is directed parallel to the polarization vec-tor
of the quasi longitudinal wave (qP) and it is associated with a
wave vector kqP and phase angle hqP. The qP wave propagates in
the direction of the radius vector pointing towards the origin (i.e.
free surface boundary). When a qP wave is incident onto the free
surface boundary of a semi infinite anisotropic medium, it converts
into three reflected waves (i.e. RqP, RqSV and RSH). The tangential
(x, y) or normal (z) displacements at the origin are expressed as
follows:
ua;IðhÞ ¼ Aa;I ua;IðhÞ þ Ab1;I ua;b1 þ Ab2;I ua;b2 þ Ab3;I ua;b3 ð11Þ
where a represents the normal (z) or tangential components (x, y),
I = qP, qSV, SH representing the type of incident wave, Ab1 ; Ab2 ; Ab3
represent the reflected wave amplitudes from the stress free surface
boundary of an anisotropic medium and ua;I ; ua;b1 ; ua;b2 ; ua;b3 are the
particle polarization components for incident and reflected waves,
respectively.
In Fig. 2a–c, the red circle1 at the origin represents the displace-ment
or force acting perpendicular to the xz-plane. Lamb’s reciproc-ity
theorem states that if a normal force Fn is applied at the origin,
then the same displacements (given by Eq. (11)) are generated along
the radial direction at a point R(x, y, z) in the semi infinite anisotropic
medium. Similarly, the theorem can be applied for x-direction tan-gential
force (see Fig. 2b) and y-direction tangential force excitation
(see Fig. 2c) on a free surface boundary of a semi infinite anisotropic
medium. A generalized form to represent the displacement directiv-ity
factor Da,I for the quasi longitudinal, quasi shear vertical and
shear horizontal waves under the excitation of normal (z) or tangen-tial
(x, y) forces is given as:
Da;IðhÞ ¼ ua;IðhÞ þ Rb1;I ua;b1 þ Rb2;I ua;b2 þ Rb3;I ua;b3 ð12Þ
where Rb1 ; Rb2 ; Rb3 represent the reflection coefficients at a free sur-face
boundary of an anisotropic medium. The reflected waves parti-cle
polarization components, amplitudes and energy coefficients at
a free surface boundary of a general anisotropic medium are ob-tained
based on the elastic plane wave theory [34], as described
in Section 3.2. In case of anisotropic medium, the wave vector direc-tion
does not coincide with the energy direction. While evaluating
ultrasonic wave propagation in general anisotropic medium, it is
very important to consider the energy angle instead of wave vector
angle [25,40]. The procedure for evaluating directivity pattern
based on Lamb’s reciprocity theorem is employed for the general
austenitic materials exhibiting both columnar grain angle and lay-back
orientation.
1 (For interpretation of the references to colour in this figure legend, the reader is
referred to the web version of this article.)

3.4. Quantitative evaluation of ultrasonic C-scan image in a layered
anisotropic material
The developed 3D ray tracing approach for homogeneous and
layered anisotropic material is schematically illustrated in Fig. 3.
For simplicity, a 2D slice of 3D problem is shown in Fig. 3. The fol-lowing
primary steps have to be taken to calculate the ultrasonic C-scan
image in a general homogeneous and layered anisotropic
material:
Step 1: A diverged ray bundle is considered at the transducer
emitting point. Each ray in the diverged ray bundle is associated
by a polar angle a and an azimuthal angle d. Stepping forward
along the ray in the direction of Poynting vector (i.e. group velocity
direction), the ray’s new position is calculated. The procedure for
evaluating 3D energy ray path in an anisotropic medium is pre-sented
in Section 3.1. The determined group velocity direction is
characterized by both polar and azimuthal angle which are used
for calculating new position of the ray. The ray directivity factor
is evaluated based on the Lamb’s reciprocity theorem, as described
in Section 3.3.
Step 2: Increase the step along z-direction and determine the
new x, y positions and check the new ray position material
properties.
Step 3: If the ray reaches the layer boundary, the transmitted ray
energy direction and coefficients are evaluated based on the plane
wave theory as stated in Section 3.2. Previous ray amplitudes are
multiplied by the present ray transmission coefficients and direc-tivity
factors. It is important to notice that in the presented model
the coupling between all the three wave modes (i.e. qP, qSV and SH
waves) are considered while the anisotropy of the austenitic steel
material is defined in 3D space.
Step 4: Return to Step 2 and the iteration is continued till the ray
reaches the back wall of the layered austenitic steel material. The
final ray amplitude is obtained by introducing phase factor and
an inverse distance factor which accounts the drop of displacement
amplitudes due to the ultrasonic beam spread in the layered aniso-tropic
material. The final ray amplitude and corresponding x and y
positions are stored in an array.
Step 5: The above steps are repeated for all azimuthal angles i.e.
range between 180 and 180 in steps of e.g. 0.5. The transducer
incident angle is varied between 1 and 45 in steps of e.g. 1 and
for all azimuthal angles, the final ray amplitude and corresponding
x and y position data are stored in an array.
Step 6: An ultrasonic C-scan image in homogeneous and layered
anisotropic material is quantitatively visualized by plotting final
ray amplitudes over the XY-plane. The stored group velocity direc-tions
are plotted to visualize the 3D energy ray paths.
In case of finite dimension transducer ultrasonic fields, the sur-face
of a transducer is discretized into several point sources. From
each point source in the finite aperture transducer, a diverged
ultrasonic ray bundle with a single fixed frequency is considered
and allowed to propagate into the spatially varying layered austen-itic
steel material. The energy loss due to the ray transmission as
well as mode conversion at each interface is calculated. The total
ultrasonic field due to the finite dimension transducer is obtained
by the cumulative effect of displacements produced by the each
element source. With this implementation, the constructive and
destructive interferences are achieved. A more detailed quantita-tive
analysis on influence of columnar grain orientation and lay-back
orientation on an ultrasonic C-scan image in a
homogeneous and layered anisotropic austenitic material will be
presented in the next section.
4. Numerical results and discussion
For the presented quantitative results, an inhomogeneous re-gion
of the austenitic weld material is discretized into several
homogeneous layers and each layer exhibits 3-D columnar grain
orientation and domain of the austenitic weld material is consid-ered
for the calculations. The elastic constants Cab (GPa) and den-sity
q (kg/m3) of the austenitic weld material (X6 Cr Ni 18 11)
and isotropic steel material used for the ray tracing calculations
[41] are given in Table 1.
4.1. Influence of the layback orientation on the ultrasonic ray energy
reflection and transmission coefficients at an interface between two
columnar grained austenitic steel materials
The selected configuration is typically encountered in the ultra-sonic
investigation of austenitic clad components and dissimilar
welds with buffering where an ultrasonic wave propagates be-tween
two adjacent columnar grained austenitic materials. The
elastic anisotropy of the two adjacent columnar grained austenite
regions plays the important role because the density of both the
media is equal. As an illustration, Fig. 4 shows the influence of lay-back
orientation on energy reflection and transmission coefficients
when a quasi longitudinal (qP) wave is incident at an interface be-tween
arbitrarily oriented austenitic steel materials. The selected
columnar grain orientation (h1) in the medium 1 is 75 and layback
orientation (w1) is 20. The selected columnar grain orientation (h2)
in the medium 2 is 50 and layback orientation (w2) of the medium
2 is varied in between 0 and 90 with a step size of 25. For the
selected configuration, the polarizations of all the six waves (i.e.
three reflected and three transmitted waves) couple together for
all layback orientations.
Fig. 3. Schematic of the 3D ray tracing method for ultrasonic field evaluation in a
general layered anisotropic material.
Table 1
Material properties for the isotropic steel and austenitic steel (X6 Cr Ni 18 11)
material. q (kg/m3), Cij (GPa).
Material parameter Isotropic steel Austenitic steel (X6 Cr Ni 18 11)
q 7820 7820
C11 272.21 241.1
C12 112.06 96.91
C13 112.06 138.03
C33 272.21 240.12
C44 80.07 112.29
C66 80.07 72.092

Fig. 4. Energy reflection and transmission coefficients for the reflected and transmitted waves when a quasi longitudinal (qP) wave is incident at an interface between two
anisotropic columnar grained austenitic steel materials. The layback angle of the austenitic steel material in medium 2 is varied between 0 and 90 with a step size of 25. (a)
Reflected quasi longitudinal wave (RqP), (b) reflected quasi shear vertical wave (RqSV), (c) reflected pure shear horizontal wave (RSH), (d) transmitted quasi longitudinal wave
(TqP), (e) transmitted quasi shear vertical wave (TqSV) and (f) transmitted pure shear horizontal wave (TSH).
The transmission coefficients for the transmitted quasi longitu-dinal
waves (TqP) are influenced by the anisotropic properties of
medium 1 and medium 2. The mode conversion of incident qP
wave energy into reflected quasi shear vertical wave (RqSV)
reaches 10% and that of transmitted quasi shear vertical wave
(TqSV) reaches 20% and even less for layback orientations other
than 0. The energy coefficients for the reflected pure shear hori-zontal
wave (RSH) stay below the 2.5% and that of refracted shear
horizontal waves (TSH) reach up to 12% of the incident energy. For
the wide range of incidence angles, the energy coefficients for the
reflected quasi longitudinal wave (RqP) are negligible. It can be
seen from Fig. 4, thus the energy coefficients for the reflected
and transmitted shear horizontal waves decreases with increasing
layback orientation. Beyond the critical angle for the transmitted
quasi longitudinal wave, the incident energy is redistributed
among the other propagating reflected and transmitted waves.
The quantitative results for ray reflection and transmission
coefficients will be employed in the 3D ray tracing method in order
to calculate the accurate ultrasonic C-scan image in homogeneous
and layered anisotropic austenitic steel materials.
4.2. Point source directivity pattern
In this section, numerical results are presented for analytically
evaluated three dimensional directivity patterns for the three wave
modes under the excitation of normal (z) and tangential (x, y)
forces on a free surface boundary of a columnar grained austenitic
steel material (X6 Cr Ni 18 11). For the presentation of the quanti-tative
results, the selected columnar grain orientation in the
austenitic steel material is 45 and the lay back orientation is
15. In the presented case, all three wave modes namely quasi lon-gitudinal
(qP), quasi shear vertical (qSV) and pure shear horizontal
(SH) waves couple together.
Fig. 5. (a) Amplitude coefficients and (b) energy coefficients for the three reflected waves when a quasi longitudinal (qP) wave is incident at a free surface boundary of a
columnar grained austenitic steel material exhibiting 45 columnar grain orientation and 15 layback orientation.

Fig. 6. (a) Amplitude coefficients and (b) energy coefficients for the three reflected waves when a quasi shear vertical (qSV) wave is incident at a free surface boundary of a
Fig. 7. (a) Amplitude coefficients and (b) energy coefficients for the three reflected waves when a pure shear horizontal (SH) wave is incident at a free surface boundary of a
4.2.1. Amplitude and energy reflection coefficients for the reflected
waves at a free surface boundary of an austenitic steel material
Fig. 5 shows the amplitude and energy reflection coefficients for
the three wave modes when a quasi longitudinal wave (qP) is inci-dent
at a free surface boundary of a columnar grained austenitic
material. It can be seen from Fig. 5 that the amplitude and energy
coefficients for the three reflected waves are influenced by the
anisotropic properties of the austenitic material. No critical angles
for reflected and transmitted waves are observed and all the angles
are real.
The numerical results for the dependence of amplitude and en-ergy
reflection coefficients on the incident quasi shear vertical
(qSV) wave angle for three different reflected ultrasonic waves
are shown in Fig. 6. Complicated critical angle phenomena for
the reflected waves are observed. From Fig. 6, the mode converted
reflected quasi longitudinal waves (RqP) are capable of propagating
for the incident angles from 36 to 32.5. The critical angles for
the reflected quasi longitudinal wave (RqP) are 36.5 and 33.
For the wide range of incident angles, the reflected shear horizontal
(RSH) waves are permeable. The critical angles for the RSH wave
are 68 and 67.5. Depending on the magnitudes of the reflection
coefficients for the reflected waves, maxima and minima in the
directivity patterns occur. This will be explained in the next
section.
Fig. 7 shows the angular dependency of amplitude and energy
coefficients when a pure shear horizontal (SH) wave is incident
at a free surface boundary of an austenitic steel material. The re-flected
quasi longitudinal (RqP) wave can propagate between the
incident angles of 29.7 and 37.1. Beyond these angles, the RqP
wave becomes evanescent. It can be seen from Fig. 7 that the
amplitude coefficient for the reflected quasi longitudinal wave
rises sharply at the critical angles. Generally, the evanescent waves
do not carry any energy but its amplitudes decay exponentially
away from the boundary. From Fig. 7, it can be observed that for
a wide range of incident angles, the reflected quasi shear vertical
(RqSV) and shear horizontal (RSH) waves are permeable. The crit-ical
angle for the incident shear horizontal wave occurs at an inci-dent
angle of 77.9.
4.2.2. Influence of 3D columnar grain orientation on ultrasonic ray
directivity patterns in an anisotropic austenitic steel material
Fig. 8 shows the point source directivity patterns for the quasi
longitudinal (qP), quasi shear vertical (qSV) and shear horizontal
(SH) waves for the normal (z) and tangential (x, y) force excitation
on a columnar grained austenitic steel material. The considered
columnar grain orientation and layback orientation in the aniso-tropic
austenitic steel material are 45 and 15 respectively. As ex-pected,
the directivity patterns for the three waves are
nonsymmetrical (see Fig. 8).
In case of normal force excitation (see Fig. 8a), the directivity
pattern for the quasi longitudinal wave contains one principal lobe
with maximum amplitude close to the normal direction (i.e. near

Fig. 8. Directivity patterns of the (a) quasi longitudinal wave (qP), (b) quasi shear vertical wave (qSV) and (c) shear horizontal wave (SH) for the normal (z) and tangential (x,
y) point source excitation on a columnar grained austenitic steel material exhibiting 45 columnar grain orientation and 15 lay back orientation.
0 angle) and zero directivity along the tangential direction. For the
tangential force excitation in x-direction, the qP wave directivity
pattern contains two principal lobes and equals zero in direction
either parallel or perpendicular to the free surface. The directivity
pattern of the quasi longitudinal wave under tangential force exci-tation
in y-direction shows one principal lobe in the positive angu-lar
region and side lobes with less displacements in the negative
angular region. Interesting is that the displacements for the qP
waves produced by the y-direction tangential force are much less
than as compared to x-direction tangential force. As expected,
the quasi shear vertical wave directivity patterns are strongly
influenced by the anisotropy of the columnar grained austenitic
steel material.
In case of x-direction tangential force excitation, the directivity
pattern of qSV wave exhibits two maxima and the reason for these
maxima can be explained from the reflection coefficients when a
qSV wave is incident at a free surface boundary of an austenitic
steel material (see Fig. 6). The two maxima in qSV wave pattern oc-cur
at regions of critical angles for the reflected quasi longitudinal
waves. The qSV wave directivity pattern under the excitation of
normal (z) force is highly deviated from the isotropic case. Interest-ing
is that the focusing effects are observed for the qSV wave direc-tivity
patterns when a y-direction tangential force excited on a free
surface of an austenitic steel material. At incident angles close to
the 45, a pronounced maximum is observed (see Fig. 8b). Depend-ing
on the layback angle of the austenitic steel material, the focus-sing
effects in the directivity pattern vary.
Fig. 8c shows directivity pattern for the pure shear horizontal
waves (SH) under the excitation of normal force (z) on a free sur-face
of an austenitic steel material. As can be seen from Fig. 8c, that
the SH wave directivity pattern contains one major lobe and a side
lobe in the positive incident angular region and a side lobe forma-tion
in the negative incident angular region. These behavior of
showing predominant amplitudes in the positive angular region
can be explained based on the energy reflection coefficients for
the reflected waves when a SH wave is incident at a free surface
boundary of a columnar grained austenitic material (see Fig. 7).
From Fig. 7, the energy coefficients for the reflected shear horizon-tal
(RSH) wave carries most of the incident SH wave energy for the
incident angles between 75 and 35 and consequently shear
horizontal wave amplitudes produced by the normal (z) and x-direction
tangential forces decreases. While on the other hand,
for positive incident angles between 35 and 60, the RSH wave
carries minimum energy. Consequently, predominant SH wave dis-placements
in positive angles of incidence are resulted for the y-direction
tangential force (see Fig. 8c). In case of SH wave directiv-ity
pattern for the tangential force in y-direction shows diverging
behavior for a wide range of incident angles. In case of x-direction
tangential force excitation, the SH wave directivity pattern con-tains
two principal maxima. Interesting is that non zero directivity
of the SH waves occurs in the tangential direction to the free sur-face.
As expected, for a point source, the directivity patterns are
independent of frequency. Furthermore, the quasi shear vertical
(qSV) wave directivity pattern for the x-direction tangential force
excitation shows a sharp maximum in the negative angles of inci-dence
(see Fig. 8b). These features lead to qSV waves generally not
considered for the ultrasonic investigation of austenitic weld mate-rials.
As expected, the directivity patterns show non-symmetrical
behavior about the direction of the excitation force. The results
of this section will be employed in Section 4.3 in order to evaluate
the accurate ultrasonic C-scan image in homogeneous and layered
anisotropic austenitic steel materials.

4.3. Quantitative determination of ultrasonic C-scan image in
homogeneous anisotropic austenitic steel material
4.3.1. Effect of columnar grain orientation on ultrasonic C-scan image
Fig. 9 shows the ultrasonic transmitter–receiver set-up used for
calculating ultrasonic C-scan image for a normal beam contact
transducer (2.25 MHz frequency, 0.1 mm length) in columnar
grained anisotropic austenitic steel material using 3D ray tracing
method. The considered thickness of the austenitic steel material
is 32 mm. According to the 3D ray tracing method, a diverged ray
bundle is considered at the transducer excitation point and it is al-lowed
to propagate along its energy direction into the anisotropic
material. The 3D ray directivity in an isotropic material (or) aniso-tropic
material is calculated using Lamb’s reciprocity theorem as
explained in Section 3.3. The ultrasonic ray amplitudes along the
back wall of the anisotropic austenitic material are calculated by
incorporating inverse distance and phase factors as described in
Section 3.4. The ultrasonic C-scan image is obtained by plotting
the amplitudes over the calculated XY-plane.
Fig. 10 shows the simulated quasi longitudinal (qP) normal
beam C-scan images in the xy-plane for selected columnar grain
orientations of the austenitic material which are 0, 15, 45, 75
and 90. The definitions of columnar grain orientation and lay back
orientation are illustrated in Fig. 1. For the numerical calculations,
a cone of ray bundles with polar angular range from 0 to 50 with
0.5 step size and azimuthal angular range between 0 and 360
with 2 step size is considered. The presented ray amplitudes in
the Fig. 10 are normalized to their respective maxima. It can be
seen from Fig. 10 that the ultrasonic C-scan images are strongly
influenced by the columnar grain orientation of the austenitic
material. The reasons for deformation of the circular shape of an
ultrasonic C-scan image can be explained due to the effect of
anisotropy on the ray bundle propagation. From Fig. 10, it is ob-served
that the pattern of the ultrasonic C-scan image for the 0
columnar grain orientation of the austenitic steel material is simi-lar
to that generally found in isotropic steel material. Whereas for
the other columnar grain orientations, the ultrasonic C-scan
images are deformed from the isotropic case and some unusual
beam focusing and beam spreading phenomena are observed.
The ultrasonic C-scan image for the 45 columnar grain shows
strong focusing effects of the ultrasonic beam in the negative half
of x-positions and highly divergent in the positive half of x-posi-tions
(see Fig. 10). The ultrasonic C-scan images for the 15 and
75 columnar grain orientations are largely deviated from the iso-tropic
behavior because for these columnar grain orientations large
beam skewing angles are observed.
4.3.2. Effect of layback orientation on ultrasonic C-scan image
Fig. 11 shows the calculated quasi longitudinal (qP) ultrasonic
C-scan images in the xy-plane for different layback orientations
(i.e. grain tilt along the xy-plane) of the austenitic steel material
Fig. 9. Geometry used for evaluating the ultrasonic C-scan image in 32 mm thick
anisotropic columnar grained austenitic steel material using 3D ray tracing method.
Fig. 10. Ultrasonic quasi longitudinal (qP) wave C-scan images along the back wall surface of the 32 mm thick anisotropic columnar grained austenitic steel material using
normal beam longitudinal contact transducer (2.25 MHz centre frequency, 0.1 mm width). ‘h’ represents the columnar grain orientation of the austenitic steel material. The
layback orientation is assumed as 0.

Fig. 11. Ultrasonic quasi longitudinal (qP) wave C-scan images along the back wall surface of the 32 mm thick anisotropic columnar grained austenitic steel material using
normal beam longitudinal contact transducer (2.25 MHz centre frequency, 0.1 mm width). ‘w’ represents the layback orientation of the austenitic steel material. The
columnar grain orientation is assumed as 45 which is kept constant.
using 3D ray tracing method. The columnar grain is oriented in 3D
space of the laboratory coordinate system of the anisotropic
austenitic steel material (see Fig. 1). The assumed columnar grain
orientation of the austenitic steel material is 45 and it is kept con-stant.
The selected layback orientations for the present investiga-tion
are 0, 15, 30, 45, 75 and 90. The beam deflection and
beam distortion phenomena in the ultrasonic C-scan images can
be observed in Fig. 11. From the quantitative analysis on ultrasonic
fields, it has been observed that the presence of layback orientation
reduces the ultrasonic beam coverage and sound field intensity in
an austenitic steel material. The presented ray tracing method is
also capable of calculating ultrasonic C-scan image for the other
two wave modes namely quasi shear vertical (qSV) and pure shear
horizontal (SH) waves in an anisotropic austenitic steel material.
4.3.3. Quantitative determination of ultrasonic C-scan image in
layered anisotropic austenitic steel material
A 3D ray tracing method is developed for evaluating ultra-sonic
C-scan image in a layered anisotropic austenitic steel
material. In the presented investigation, ultrasonic C-scan image
is quantitatively determined in 32 mm thick layered austenitic
clad material where 16 mm as isotropic steel material, 8 mm
as austenitic steel material with 20 columnar grain orientation
and 8 mm as austenitic steel material with 45 columnar grain
orientation.
A normal beam ultrasonic transducer (2.25 MHz frequency,
0.1 mm width) is excited on the surface of the isotropic steel mate-rial
and the ultrasonic field distribution is evaluated over the XY
plane at a depth of 32 mm from the top surface. Fig. 12 shows
the geometry used for calculating ultrasonic C-scan image in lay-ered
austenitic clad material. Fig. 13 shows 3D ultrasonic ray prop-agation
in an anisotropic layered austenitic steel material for the
20 longitudinal wave incidence. The 3D ray tracing model is capa-ble
to simulate the ray paths in both side view (C-scan) and top
view (B-scan) representation (see Fig. 13a–c). As expected from
the Fig. 13, the ultrasonic ray paths are bended due to the ray
skewing at anisotropic layer boundaries of the austenitic clad
material. The calculated ultrasonic C-scan image for the normal
beam contact transducer (with centre frequency 2.25 MHz) in the
anisotropic layered austenitic clad material is shown in Fig. 14.
For the numerical calculations, a cone of ray bundles with polar
angular range from 0 to 55 with 0.5 step size and azimuthal
angular range between 0 and 360 with 2 step size is considered.
The presented ray amplitudes in the Fig. 14 are normalized to their
respective maxima. The ray energy transmission coefficients at the
interface between isotropic and austenitic materials as well as the
interface between two austenitic steel materials are taken into
Fig. 12. Geometry used for evaluating ultrasonic C-scan image in 32 mm thick
layered anisotropic austenitic steel material using 3D ray tracing method.

Fig. 13. 3D ultrasonic ray propagation in 32 mm thick multi layered austenitic steel material. Calculated ray pattern along the (a) XY plane, (b) XZ plane and (c) YZ plane. The
polar angle for the ray tracing calculation is 20 and the azimuthal angular range is between 0 and 360 with a step size of 5.
account in the ray tracing model while calculating the ultrasonic C-scan
image. It is observed that the shape of the ultrasonic C-scan
image is deformed from circular shape (i.e. isotropic behavior) into
an elliptical shape (see Fig. 14).
Generally, a ray tracing method is based on the high frequency
and far field approximation. Hence, the modeling of ultrasonic
wave propagation in the near-field region is limited. The novelty
of the presented quantitative ray tracing approach in this paper
is that the ray amplitudes and phase information are incorporated
and employed the superposition phenomenon which takes into ac-count
the interference effects. Consequently, the ray tracing model
simulates the near-field effects too. By taking into account the
interference effects in the near-field region, the ultrasonic ray
source can be placed close to the first interface. The presented
ray tracing model is valid for both near field as well as far-field re-gion
of the source/receiver.
5. Concluding remarks
In this paper, a 3D ray tracing method is presented for evaluating
ultrasonic C-scan image quantitatively in homogeneous and lay-ered
anisotropic austenitic steel materials. The ultrasonic ray theory
is extended to the more general case of anisotropic austenitic steel
materials where the columnar grain orientations are not only tilted
in the plane of sound propagation but also perpendicular to it (i.e.
layback orientation). The ultrasonic ray reflection and transmission
coefficients at an interface between two arbitrarily oriented colum-nar
grained austenitic steel weld metal areas are solved three
dimensionally. The influence of layback orientation on ray energy
reflection and transmission coefficients at an interface between
two columnar grained austenitic steel materials is discussed.
The directivity patterns for three wave modes (i.e. qP, qSV and
SH waves) under the excitation of normal (z) and tangential (x, y)
forces on a semi-infinite columnar grained austenitic steel material
are obtained three dimensionally using Lamb’s reciprocity theo-rem.
The influence of 3D columnar grain orientation (including
lay-back orientation) on point source directivity patterns for three
wave modes is quantitatively analyzed. The quantitative results on
influence of columnar grain orientation and layback orientation in
homogeneous and layered anisotropic austenitic steel materials re-veals
that the ultrasonic C-scan image exhibits non-symmetrical
wave field distribution, strong beam focusing and beam divergence
phenomena. The presented a priori quantitative information on
ultrasonic C-scan images in homogeneous and layered anisotropic
materials helps in deep understanding of wave filed distribution,
optimization of experimental parameters and reliable interpreta-tion
of the experimental results during ultrasonic non-destructive
inspection of columnar grained austenitic steel materials. The pre-sented
3D ray tracing model has the potential to evaluate the ultra-sonic
C-scan image quantitatively for the quasi shear vertical and
Fig. 14. The ultrasonic quasi longitudinal (qP) wave C-scan image calculated for a
normal beam contact transducer (2.25 MHz centre frequency and 0.1 mm width)
along the back wall surface of the multi layered anisotropic austenitic steel material
using 3D ray tracing method.

shear horizontal waves with no additional difficulties. Experimen-tal
comparison of 3D ray based ultrasonic C-scan images in inho-mogeneous
austenitic weld materials with spatially varying
columnar grain orientation are planned for the future work.
Acknowledgements
This work is financially supported by BMWi (Bundesministeri-um
fur Wirtschaft und Technologie) under the Grant 1501365
which is gratefully acknowledged. The authors would like to thank
R. Boehm and M.U. Rahman from Federal Institute for Materials Re-search
and Testing, Berlin for many helpful discussions.
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