Elastic waves guided by a welded joint in
a plate
By Zheng Fan and Mike J. S. Lowe
Department of Mechanical Engineering, Imperial College, London SW7 2AZ, UK
The inspection of large areas of complex structures is a growing interest for indus-
try. An experimental observation on a large welded plate found that the weld can
concentrate and guide the energy of a guided wave traveling along the direction of
the weld. This is attractive for NDE since it offers the potential to quickly inspect
for defects such as cracking or corrosion along long lengths of welds. In this paper,
a two dimensional Semi Analytical Finite Element (SAFE) method is applied to
provide a modal study of the elastic waves which are guided by the welded joint in
a plate. This brings understanding to the compression wave which was previously
observed in the experiment. However, during the study, a shear weld-guided mode,
which is non-leaky and almost non-dispersive has also been discovered. Its char-
acteristics are particularly attractive for NDE, so this is a significant new finding.
The properties for both the compression and the shear mode are discussed and
compared, and the physical reason for the energy trapping phenomena is then ex-
plained. Experiments have been undertaken to validate the existence of the shear
weld-guided mode and the accuracy of the FE model, showing very good results.
Keywords: feature guiding, SAFE, shear mo de
1. INTRODUCTION
Guided waves are interesting for large area inspections since they offer the potential
for rapid screening from a single transducer position. Several successful applications
have been made on one dimensional structures such as pipelines (Alleyne et al. 2001)
and rails (Wilcox et al. 2003; Rose et al. 2002). Research work has also been carried
out to study the possibility of applying the guided wave inspection to two dimen-
sional plate-like structures such as storage tanks, pressure vessels and airframes
(Lowe & Diligent 2002; Lee & Staszewski 2003; Wilcox et al. 2005), although this
has resulted in little commercialization so far. This is mainly because in a two di-
mensional structure waves can propagate in an infinite number of directions from a
single transducer position, and in each direction the energy of the spreading wave
decays with the distance away from the source. Another challenge to inspect real
plate-like structures is that there always exists some features such as welds and ribs,
which may cause extra coherent noise which interferes with the inspection signal.
However, a recent experimental observation (Sargent 2006) on a large welded
plate revealed that there existed ’weld-guided’ modes which can propagate along
the weld and concentrate the energy in and around the weld. Similar trapped modes
have also been discovered by Postnova and Craster (2008) from an analytical calcu-
lation on a welded plate structure based on the long-wave theory. Fig. 1 schemati-
cally shows the propagation of a feature-guided wave on a welded plate from a pulse
Article submitted to Royal Society T
E
X Paper
2 Z. Fan and M. J. S. Lowe
excitation. As can be seen from the figure, due to the geometry change part of the
energy is trapped in and around the weld and propagates along the weld. This is
very attractive for NDE as we know defects frequently occur preferentially in or
near the weld, and the same is true for other geometric features, such as joints and
stiffeners. Therefore, instead of seeing the features as a problem, it may be possible
to exploit them as waveguides to focus the energy of the guided wave, which offers
the potential to quickly inspect for defects such as corrosion along long lengths of
features on plate-like structures.
In order to exploit this feature-guided wave, it is necessary to understand its
nature and propagation characteristics. Juluri et al. (2007) performed a three di-
mensional time step finite element simulation on an idealized welded plate geometry,
and demonstrated the existence of the compression weld-guided mode (similar to
the Lamb S0 mode in the plate), which had been experimentally detected (Sargent
2006). However these simulations are very time consuming, and can only model the
chosen frequency and wave mode, thus they are not sufficient to investigate the
guided wave properties over ranges of parameters and feature geometries. In order
to further understand how the guiding is affected by the geometry and frequency,
it is therefore necessary to perform a modal study of the welded-plate, in order to
fully predict the properties of the waves which are guided by the features. Such a
model may then also create the possibility of finding other feature-guided modes
which could be candidates for inspection but have not yet been discovered.
A Semi Analytical Finite Element (SAFE) method, which uses finite elements
to represent the cross section of the waveguide, plus a harmonic description along
the propagation direction, has become popular in recent years for the modal study
of waveguides with irregular cross section, and has the potential for application to
the case of feature guided waves. The SAFE method was first demonstrated in 1972
(Lagasse 1972), and has been used to calculate the properties of guided waves in
railway lines (Gavri´c 1995; Hayashi et al. 2003), beams (Wilcox et al. 2002) and
stiffened plates (Orrenius & Finnveden 1996). Recently the SAFE method has been
developed by Castaings and Lowe (2008) to study leaky waves which propagate
along an elastic waveguide with arbitrary cross section and radiate into a solid of
infinite extent. They also presented a model of two large plates connected by a
rectangular butt weld, and showed results for the compression weld-guided mode
which had been found experimentally (Sargent 2006) and demonstrated by the
time-step finite element simulation (Juluri et al. 2007).
In this paper, we apply the SAFE modeling on a geometry of a real welded plate
and provide further investigation on the properties of feature guided waves and
the physical reason for the energy trapping effect. Significantly, during the modal
study, another interesting shear mode, which is similar to the SH0 mode in the
plate, has been discovered. The particle displacement of this mode is perpendicular
to plane of propagation and therefore it is expected to be more sensitive than the
compression mode to the fatigue cracks that are typically aligned along the weld in
the heat affected zone. In addition this shear mode has no leakage to the side plates
and is almost non-dispersive, thus it is very interesting as a candidate mode for
industrial inspection. Experiments have been set up to demonstrate the existence
of this shear weld guided mode and the accuracy of the SAFE model, showing very
good agreement.
This paper starts with a brief review of the SAFE theory in Sec.2 and follows
Article submitted to Royal Society
Feature guided waves in a welded plate 3
with the modal study of the actual welded-plate in Sec.3. In this section, both com-
pression and shear weld-guided waves are investigated and compared at a single
frequency. Then the dispersion curves (frequency-velocity relationships), the possi-
ble attenuation by leakage of energy into the adjacent plates, and the mode shapes
(distributions of stress and displacement across the section of the waveguide) are
calculated and discussed for both modes. In Sec.4, the guiding phenomena are ex-
plained by comparing the mode shapes and the phase velocity of the weld cap and
the adjacent plate. Finally, a validation experiment is introduced, and results of
measurements are shown in Sec.5.
2. THEORY
Since the derivation of the SAFE theory in solids has been detailed in previous
research work (Predoi et al. 2007), only the equations used in this paper will be
briefly explained. The mathematical model of the SAFE method in solids is based
on the three dimensional elasticity approach. Consequently, the displacement vector
in the waveguide can be written:
u
i
(x
1
, x
2
, x
3
, t) = U
i
(x
1
, x
2
)e
I(kx
3
−ω t)
, I =
√
−1, (2.1)
in which k is the wavenumber, ω = 2πf is the angular frequency, f being the
frequency, t is the time variable and the subscript i = 1, 2, 3. The harmonic guided
waves are assumed to propagate along the x
3
direction. For general anisotropic
material, the equation of dynamic equilibrium can be written in the following form
of an eigenvalue problem:
C
ikjl
∂
2
U
j
∂x
k
∂x
l
+ I(C
i3jk
+ C
ikj3
)
∂(kU
j
)
∂x
k
− kC
i3j3
(kU
j
) + ρω
2
δ
ij
U
j
= 0, (2.2)
with summation over the indices j = 1, 2, 3 and k, l = 1, 2. The coefficients C
ijkl
are the stiffness moduli and δ
ij
is the Kronecker symbol. In the commercial FEM
code (COMSOL 2008) used in this study, the formalism for eigenvalue problems
has the general expression:
∇ · (c∇U + αU − γ) − β∇U − aU + λd
a
U − λ
2
e
a
U = 0, (2.3)
in which all matrix coefficients are given by Predoi et al (2007).
The nature of the solution is thus to find eigenvalues of wavenumber k for chosen
values of angular frequency ω. Each solution at a chosen frequency will reveal the
wavenumbers of all of the possible modes at that frequency; then the full dispersion
curve spectrum can be found by repeating the eigenvalue solutions over the desired
range of frequencies.
3. MODAL STUDY OF THE WELDED PLATE
Compared to the conventional three dimensional time step finite element simula-
tion, the two dimensional SAFE model has many advantages. With this method,
only the cross section which is normal to the direction of the wave propagation has
to be meshed by finite elements. A typical calculation (calculation of all the prop-
agation wave numbers at one frequency) in our model presented here only takes
Article submitted to Royal Society
4 Z. Fan and M. J. S. Lowe
approximately one minute on a Pentium 4 PC with 2Gbyte memory, while it takes
several hours to calculate one specific mode propagation at one frequency in the
3D time step FE model on the same computer. Therefore the SAFE model is much
more convenient to obtain the dispersion curve of the weld guided mode, and is
more flexible to study different geometries and parameters.
(a) Model description
The schematic of the model is shown in Fig. 2, in which the profile has been
measured from an actual welded 6-mm-thick steel plate. Stress free conditions are
imposed at the outer limit of the system. The material properties are given in
Table 1, and the weld is assumed to have the same properties as the adjacent
plate. In order to model the wave propagation along the weld and leaking into
the side plates, an absorbing region has been attached at each side of the plate
to avoid reflections from the edges (Castaings & Lowe 2008). This region, shown
in Fig. 2, has the same mass density and elastic properties as the side plate, but
its viscoelasticity gradually increases with the distance away from the central axis
of the system. To achieve this, the imaginary parts of its elastic moduli gradually
increase according to the following law:
C
ija
= C
0
ij
[1 + Iα
1
(
|r − r
a
|
L
a
)
3
], I =
√
−1, (3.1)
where C
0
ij
represents the stiffness of the side plate, r
a
is the distance between the
inner border of the absorbing region and the central axis, L
a
is the length of the
absorbing region, r is the position in the absorbing region with respect to the central
axis, and C
ija
are the resulting viscoelastic moduli of the absorbing region. α
1
is
a coefficient that defines the proportion of the viscoelasticity at the outer limit of
the absorbing region.
By introducing the imaginary part of the stiffness moduli, the propagation wave
numbers, which are eigen solutions of the system, become complex (k = k
0
+ Ik
00
).
The imaginary parts (k
00
) represent the attenuation due to the leakage from the weld
to side plates. The length of the absorbing regions has been chosen to be 800 mm,
which is twice the biggest wavelength of any radiated wave in the whole frequency
range (Castaings & Lowe 2008), and was proven to be efficient by a convergence
check: when the length was increased the same solutions for the propagating modes
in the waveguide was still obtained. The total width of the cross section is 2 m,
including the absorbing region. The whole geometry is meshed by 1072 triangular
elements of the first order, with sidelengths comprised between 1 and 6 mm. These
elements are automatically generated by the software used, and are finer in the
welded zone than in the adjacent plates. The number of degrees of freedom is
15882.
The system is solved using the SAFE method to find values of the wave number
k at different frequencies. For each frequency, several solutions are obtained. For
each solution, the axial component of the energy flow is calculated at each nodal
position of the mesh, and the quantity is expressed by the following formula (Auld
1990):
P
x
3
= −Re[(
Iω
2
)(u
∗
1
σ
31
+ u
∗
2
σ
32
+ u
∗
3
σ
33
)], (3.2)
Article submitted to Royal Society
Feature guided waves in a welded plate 5
Where σ
31
, σ
32
and σ
31
are the axial stress components; u
∗
1
, u
∗
2
and u
∗
3
are the
complex conjugate of the vertical, horizonal and axial displacements respectively.
Solutions with higher axial component of the energy-flow in the weld cap than
in the side plates generally represent modes guided along the weld and possibly
radiating in the plates, while other solutions represent resonances of the whole
system and are unwanted. From our calculation, several weld guided modes have
been found in the low frequency range. In this paper, we will only compare and
discuss two fundamental modes, one is the compression mode, which has previously
been discovered experimentally (Sargent 2006) and the other a new interesting mode
which has a similar mode shape as the SH0 mode in the plate.
(b) Numerical results at single frequency
The compression weld guided mode at 100 kHz is shown in Fig. 3 with the
eigenvalue k = 115.486 − 3.034 × 10
−2
i /m, from which the corresponding phase
velocity is: C
ph
= 5440.6 m/s and the attenuation is: α = 0.263 dB/m. A snapshot
of the axial component of energy flow is shown in Fig. 3(a), which indicates that
the energy is concentrated in and close to the weld. The mode shape of this mode
in the center of the weld along x
2
is shown in Fig. 3(b). From the figure it can
be seen that the mode guided along the weld is dominated by axial displacement
u
3
with respect to the vertical displacement u
2
and horizontal displacement u
1
,
which is similar to a S0 Lamb wave in a plate. According to the Snell-Descartes’
law (Auld 1990), only modes of the lateral plates having smaller phase velocities
than that of the compression weld-guided mode could be radiated in the side plates.
Thus the S0 mode, with its phase velocity of 5441 m/s (Pavlakovic et al. 1997) in
a 6-mm-thick plate, can not radiate, while in principle the other two fundamental
modes, A0 and SH0 could. However, since this compression weld-guided mode is
symmetric with respect to the mid-plane of the plates and weld, the A0 mode,
which is anti-symmetric, cannot be launched. Therefore, the SH0 mode is the only
mode that can be leaked into the plates at 100 kHz, which would be radiated at an
angle equal to θ
leak
= sin
−1
(3260/5440.6) ≈ 36.8
◦
, with respect to the direction
normal to the plates-weld interface. The axial displacement u
3
which dominates
this compression weld guided mode in the center of the plate along x
1
is shown in
Fig. 3(c). From the figure it can be seen that the axial displacement quickly decays
with distance away from the center, which indicates the energy is concentrated in
and around the weld. The oscillation of u
3
represents the leakage of the SH0 wave
in the plate. It can be seen that the separation distance of the oscillation peaks
agrees with the projection of the wavelength of the leaky SH0 wave along the x
1
direction, using λ
proj
= λ
SH0
/cosθ
leak
= 40.7 mm.
The shear mode at 100 kHz is shown in Fig. 4 with the eigenvalue k = 194.86 −
1.3 × 10
−7
i /m. The corresponding phase velocity is: C
ph
= 3224.5 m/s, and the
attenuation is zero which means there is no mode leaking to the lateral plates.
To our knowledge, this non-leaky mode has not been observed before, and it is
very interesting for long distance inspection. A snapshot of the axial component of
energy flow is shown in Fig. 4(a), which indicates that the energy is concentrated
in the weld. The mode shape of this shear mode in the center of the weld along x
2
is shown in Fig. 4(b). From the figure it can be seen that the mode guided along
the weld is dominated by horizontal displacement u
1
with respect to the vertical
Article submitted to Royal Society
