![Fig. 3 a Diagram of precipitate with radius rp embedded in a matrix with natural radius of ra, and b diagram of a dislocation bending around a distribution of precipitates, with spacing of L (based on [66])](/figures/fig-3-a-diagram-of-precipitate-with-radius-rp-embedded-in-a-2c7w2mpk.webp)
![Fig. 10 Experimental setup of nonlinear Rayleigh wave measurement with air-coupled transducer detection (reproduced from [41] with permission from Elsevier)](/figures/fig-10-experimental-setup-of-nonlinear-rayleigh-wave-1nmupc4z.webp)
![Fig. 8 Comparison of nonlinear diffraction curve-fit model and linear fit of normalized second harmonic amplitude over propagation distance from Rayleigh wave measurements on thermally aged 2205 duplex stainless steel [43] (reproduced from [40] with permission from Elsevier)](/figures/fig-8-comparison-of-nonlinear-diffraction-curve-fit-model-3swnuvdf.webp)
![Fig. 4 Diagram of rough surface in contact with rigid plane [52]. Adapted with permission from Nazarov and Sutin [51]. Copyright 1997, Acoustical Society of America](/figures/fig-4-diagram-of-rough-surface-in-contact-with-rigid-plane-13cz2f44.webp)
![Fig. 1 Amplitude of first harmonic from Eq. (16) (left) and second harmonic from Eq. (17) (right) Rayleigh wave with propagation distance and sample width, including diffraction and attenuation effects, at y=0 [40]. Results normalized by input first harmonic amplitude (reproduced from [40] with permission from Elsevier)](/figures/fig-1-amplitude-of-first-harmonic-from-eq-16-left-and-second-21a6nrpu.webp)
![Fig. 7 Normalized second harmonic Rayleigh wave amplitude measured over increasing propagation distance, on samples with varying intensities of shot-peening (reproduced from [80] with permission from Elsevier)](/figures/fig-7-normalized-second-harmonic-rayleigh-wave-amplitude-1ks5hvl5.webp)
J Nondestruct Eval (2015) 34:273
DOI 10.1007/s10921-014-0273-5
Review of Second Harmonic Generation Measurement Techniques
for Material State Determination in Metals
K. H. Matlack · J.-Y. Kim · L. J. Jacobs · J. Qu
Received: 11 July 2014 / Accepted: 12 November 2014 / Published online: 25 November 2014
© Springer Science+Business Media New York 2014
Abstract This paper presents a comprehensive review of
the current state of knowledge of second harmonic genera-
tion (SHG) measurements, a subset of nonlinear ultrasonic
nondestructive evaluation techniques. These SHG techniques
exploit the material nonlinearity of metals in order to mea-
sure the acoustic nonlinearity parameter, β. In these measure-
ments, a second harmonic wave is generated from a propagat-
ing monochromatic elastic wave, due to the anharmonicity
of the crystal lattice, as well as the presence of microstruc-
tural features such as dislocations and precipitates. This arti-
cle provides a summary of models that relate the different
microstructural contributions to β, and provides details of the
different SHG measurement and analysis techniques avail-
able, focusing on longitudinal and Rayleigh wave methods.
The main focus of this paper is a critical review of the liter-
ature that utilizes these SHG methods for the nondestructive
evaluation of plasticity, fatigue, thermal aging, creep, and
radiation damage in metals.
Keywords Second harmonic generation · Microstructural
evolution in metal · Nonlinear ultrasonic measurements
K. H. Matlack (
B
)
Department of Mechanical and Process Engineering, Swiss Federal
Institute of Technology (ETH Zurich), Zurich, Switzerland
e-mail: matlackk@ethz.ch
J.-Y. Kim · L. J. Jacobs
School of Civil and Environmental Engineering,
Georgia Institute of Technology, Atlanta, GA, USA
J. Qu
Department of Civil and Environmental Engineering,
Northwestern University, Evanston, IL, USA
1 Introduction
Nonlinear ultrasonic methods have the powerful ability to
characterize microstructural features in materials. Compared
to more conventional linear ultrasonic methods that can
detect cracks or features on the order of the wavelength
of the ultrasonic wave, nonlinear methods are sensitive to
microstructural features that are orders of magnitude smaller
than the wavelength. Second harmonic generation (SHG) is a
type of nonlinear ultrasonic method that has been shown to be
capable of detecting and monitoring microstructural changes
in metals. The physical mechanism of this is as follows: as
a sinusoidal ultrasonic wave propagates through a mater-
ial, the interaction of this wave with microstructural features
generates a second harmonic wave. This effect is quantified
with the measured acoustic nonlinearity parameter, β.SHG
measurement methods have received significant focus and
attention in the literature in recent decades, as the reliability
and integrity of structural components becomes increasingly
important to ensure safe operation of critical structures in,
for example, the energy, transportation, and aviation indus-
try. This paper presents a review of SHG measurements and
their applications, to provide a comprehensive summary of
these techniques, to thoroughly explain these measurements
to new comers in this field of research, and to show where
advances in this field of research are needed.
SHG methods were first reported on back in the 1960s,
with a series of papers by Breazeale et al. [1,2], work by
Gedroitz and Krasilnikov [3], and another series by Hikata
et al. [4,5]. Some of the first reported SHG measurements of
anharmonicity were conducted by both Gedroitz and Krasil-
nikov [3], as well as Breazeale and Thompson [1]. In the
latter, the second harmonic wave was measured in polycrys-
talline aluminum over increasing source voltage of the fun-
damental wave with quartz crystal transducers, showing the
123
273 Page 2 of 23 J Nondestruct Eval (2015) 34 :273
linear dependence of the generated second harmonic wave on
the squared amplitude of the fundamental wave. Other ini-
tial experimental studies primarily focused on single crystals
(e.g. copper [2,6–10], germanium [11], aluminum [5,12]),
and on fused silica [12–15].
There are a multitude of other nonlinear ultrasonic
NDE techniques. Acousto-elasticity measurements are based
on the phenomenon that changes in the stress state of
a nonlinear medium cause a change in the wave veloc-
ity [16], and measurements exploiting the acousto-elastic
effect have shown the sensitivity of material nonlinearity
to fatigue microcracks [17]. Nonlinear elastic wave spec-
troscopy (NEWS) methods—such as nonlinear resonant
ultrasound spectroscopy (NRUS) and nonlinear wave modu-
lation spectroscopy (NWMS) techniques—have been exten-
sively used to characterize a variety of materials such as geo-
materials, rock, and concrete [18]. These techniques exploit
the nonlinear hysteretic nature of these materials. NRUS
techniques look at a resonance frequency shift with increas-
ing amplitude excitation [18–20]. NWMS techniques typi-
cally look at modulation frequencies as a result of mixing
a very low frequency (sometimes induced by a shaker or
mechanical tapping) and a high frequency ultrasonic wave
[21,22]. Nonlinear time reversal methods have been used
in geophysics applications and detecting surface defects
[23,24]. Nonlinear mixing techniques [25,26] utilize two
input waves at different ultrasonic frequencies, and have
recently gained attention in the literature. Material nonlinear-
ity causes the generation of sum or difference frequencies,
depending on input wave polarities, where the two waves
interact. Nonlinear mixing techniques have the potential to
be isolated from equipment and system nonlinearities inher-
ent in harmonic generation experiments.
Second harmonic generation measurements have shown
to be very applicable in detecting microstructural changes in
metals prior to macroscopic damage and/or microcracking.
Some experimental techniques have the unique advantage
of simplicity in utilizing commercial transducers and stan-
dard ultrasonic testing equipment. Much of the current work
on SHG techniques is focused on how these measurements
can be applied to interrogate real materials under realistic
loading conditions. A variety of different measurement tech-
niques have been developed to interrogate different geome-
tries and damage types. This review is intended to extend pre-
vious reviews [27,28], and will specifically focus on second
harmonic generation experiments and applications explored
particularly within the recent years. This review begins with
a theoretical overview of SHG of longitudinal and Rayleigh
waves, followed by a review of different microstructural con-
tributions to β. Then, the different SHG measurement tech-
niques reported throughout the literature are reviewed, fol-
lowed by a discussion on how these measurements have been
applied to monitor microstructural evolution during material
damage. Finally, an outlook on potential future directions of
SHG measurement research is given.
2 Review of SHG Theory
2.1 Longitudinal Waves
Consider longitudinal wave propagation through an isotropic
medium with a quadratic nonlinearity, which results from a
non-quadratic interatomic potential in crystalline materials.
The equation of motion, simplified to one-dimension is:
ρ
∂
2
u
∂t
2
=
∂σ
xx
∂x
(1)
where ρ is the material density, u is the particle displacement,
σ
xx
is the normal stress in the x-direction, x is the material
coordinate, and t is time. The constitutive equation for a
quadratic nonlinearity is given as:
σ
xx
= σ
0
+ E
1
∂u
∂x
+
1
2
E
2
∂u
∂x
2
+ ... (2)
where E
1
and E
2
are the appropriate second- and third-order
elastic constants. The nonlinear wave equation can thus be
derived as
∂
2
u
∂t
2
= c
2
1 − β
∂u
∂x
∂
2
u
∂x
2
(3)
where β is the nonlinearity parameter, and c is the longitudi-
nal wave velocity in the material. For a material in its virgin
state, β is equivalent to the lattice anharmonicity component,
β
0
, and is a function of second- and third-order elastic con-
stants of the material:
β
0
=−
3C
11
+ C
111
σ
0
+ C
11
(4)
where C
11
and C
111
are the second- and third-order Brugger
elastic constants, respectively, written in Voigt notation, and
σ
0
is the initial stress in the material. Equation (4) assumes
wave propagation in the (100) direction, and is also an exact
solution for isotropic materials. Note that β depends on the
crystalline structure and symmetry of the material, which
was shown in [29] through calculations of β for pure mode
propagation for various single-crystals.
The time harmonic solution to Eq. (3), assuming plane
wave propagation, has the form:
u = A
1
sin(κ x − ωt) +
β A
2
1
xκ
2
8
cos(2κ x − 2ωt) + ...
(5)
123
J Nondestruct Eval (2015) 34 :273 Page 3 of 23 273
where ω(= 2π f )is the radial frequency of the wave at fre-
quency f , κ(= ω/c) is the wavenumber of the propagating
wave, A
1
is the amplitude of the first harmonic wave. The
coefficient in front of the second term is A
2
, the amplitude
of the second harmonic wave, which assumes the absence of
attenuation, diffraction, scattering, and assumes plane wave
propagation. By simply rearranging this amplitude term, the
nonlinearity parameter can be expressed in terms of acoustic
quantities, i.e.:
β =
8A
2
A
2
1
xκ
2
(6)
When written in this form, β is generally referred to as
the acoustic nonlinearity parameter. Thus by measuring the
second harmonic wave amplitude, along with the first har-
monic amplitude, wavenumber, and propagation distance,
one can determine the acoustic nonlinearity parameter, β.
This derivation can be expanded to three dimensions [30],
and has been derived for Rayleigh waves [31] and explored
for Lamb waves [32]. The same general form of A
2
has
been shown for Rayleigh waves, in terms of dependence on
propagation distance, wavenumber, and first harmonic wave.
Experimental results have shown the same relationship of
A
2
proportional to A
2
1
holds true for Lamb waves as well
[82–85].
Note that the energy transfer from the first to second har-
monic wave in SHG is very small compared to the energy of
the propagating first harmonic wave, such that the decrease in
A
1
due to the energy transfer is insignificant for small propa-
gation distances. Further, in most experiments, the amplitude
A
2
is orders of magnitude smaller than A
1
. In the present
considerations, we assume the wave propagation distance is
small enough such that the energy loss of A
1
is negligible
compared to the total energy of the propagating first har-
monic wave.
For real materials and finite propagation distances,
attenuation (dissipation, scattering, diffraction) will further
decrease the amplitudes of the first and second harmonic
waves with increasing propagation distance. The effect on the
generated second harmonic wave when attenuation effects
are non-negligible is derived elsewhere for longitudinal
waves [5,16,33]. Attenuation effects are non-negligible at
larger propagation distances, i.e. when x(α
2
− 2α
1
)<<1
does not hold, and the acoustic nonlinearity parameter for
this case is given as [5,16]:
β
atten
= β
x(α
2
− 2α
1
)
{
1 − exp
[
−(α
2
− 2α
1
)x
]
}
(7)
where α
1
is the attenuation coefficient at the first harmonic
frequency, and α
2
is the attenuation coefficient at the second
harmonic frequency. Note that in Eq. (7)itisassumedthat
SHG and attenuation effects occur independently, and in the
limit α
2
→ 2α
1
, the measured β equals the actual β.
A more accurate expression for the acoustic nonlinearity
parameter can be found by accounting for the on-axis dif-
fraction effects of both the first and second harmonic wave.
A diffraction correction
D
β
to β was introduced by Hurley
et al. [14]as:
D
β
=
|
D(ω)
|
2
|
D(2ω)
|
(8)
where
|
D(ω)
|
and
|
D(2ω)
|
are the diffraction corrections
to the first and second harmonic waves, respectively. The
acoustic nonlinearity parameter scaled by this diffraction cor-
rection is thus β
D
= β D
β
.
The linear diffraction correction, i.e. the diffraction cor-
rection for the propagating first harmonic wave, has been
derived in full previously [34] for a piston source such that
the amplitude is constant across the source. This diffraction
correction is given by:
D(ω, x, a) =1−exp(−iκa
2
/x)
J
0
(κa
2
/x)+iJ
1
(κa
2
/x)
(9)
where a is the transducer radius, and J
0
and J
1
are Bessel
functions of the first kind. In actuality, transducers are not a
perfect piston source and there is some spatial distribution
of amplitude over the surface of the transducer face, which
could potentially approximate a Gaussian distribution [35].
The diffraction of the second harmonic wave is spatially
different than that of the first harmonic. The wave is gener-
ated not by the transducer (in a perfect system at least), but
by the propagating first harmonic wave, which is diffracting
over propagation distance. This nonlinear diffraction can be
physically interpreted as follows: at each instance that a por-
tion of the second harmonic wave is generated, that portion
will then diffract linearly over the remainder of the propa-
gation distance to the receiving transducer. This nonlinear
diffraction effect has been derived as [14,36]:
D(2ω, x, a) =
x
0
[
D(ω, x − σ/2, a)
]
2
dσ
x
(10)
where D(ω, x, a) is the linear diffraction correction given in
Eq. (9).
2.2 Rayleigh Surface Waves
Consider a Rayleigh surface wave propagating in the positive
x direction in an isotropic infinite half-space, with direction
z pointing into the half-space, with a weak quadratic nonlin-
earity. The Rayleigh wave motion along a stress-free surface,
123
273 Page 4 of 23 J Nondestruct Eval (2015) 34 :273
assuming plane wave propagation, can be decomposed in
terms of its shear and longitudinal wave components, which
for a propagating sinusoidal wave is given as:
u
x
= A
1
e
−pz
−
2 ps
κ
2
R
+ s
2
e
−sz
e
i(κ
R
x−ωt)
(11)
u
z
= iA
1
p
κ
R
e
−pz
−
2κ
2
R
κ
2
R
+ s
2
e
−sz
e
i(κ
R
x−ωt)
(12)
where p
2
= κ
2
R
− κ
2
p
, s
2
= κ
2
R
− κ
2
s
, and κ
R
, κ
P
, and κ
S
are
the wavenumbers for the Rayleigh, longitudinal, and shear
waves, respectively. The second harmonic wave displace-
ment components can be approximated at a sufficiently far
distance as [37,38]:
u
x
(2ω) = A
2
e
−2 pz
−
2 ps
κ
2
R
+ s
2
e
−2sz
e
i2(κ
R
x−ωt)
(13)
For isotropic materials, the acoustic nonlinearity due to shear
waves vanishes due to symmetry conditions, so the gener-
ated second harmonic wave is purely due to the longitudi-
nal wave component of the Rayleigh wave. Herrmann et al.
[38] derived the acoustic nonlinearity parameter in terms of
out-of-plane components of the first and second harmonic
amplitude as:
β =
¯u
z
(2ω)
¯u
2
z
(ω)x
i8 p
κ
2
P
κ
R
1 −
2κ
2
R
κ
2
R
+ s
2
(14)
where the overbar indicates the displacement is evaluated at
z = 0.
As can be seen in Eq. (14), the dependence of β on first
and second harmonic amplitude, as well as on propagation
distance is the same for nonlinear Rayleigh waves as it is for
nonlinear longitudinal waves, at least under the plane wave
assumption. The dependence on propagation distance is gen-
erally exploited in nonlinear Rayleigh wave measurements—
the wave propagation distance is varied over multiple mea-
surements of first and second harmonic wave amplitude, and
a relative measure of β can thus be made.
Since nonlinear Rayleigh waves are typically used for
longer propagation distances, accounting for attenuation [39]
and diffraction effects [35,39,40] are crucial for accurate
measurements of β. Shull et al. [35] explored diffraction
effects of nonlinear Rayleigh waves both theoretically and
numerically for a general forcing function, a uniform line
source, and a Gaussian source. For example, a line source
with a Gaussian distribution along the y-axis is defined by:
w(y) = v
0
e
−(y/a)
(15)
where v
0
is the source amplitude and a is the half-length of
the line source. The first harmonic wave velocity component
propagation distance [m]
Sample/source width [m]
−0.05 0 0.05
0
0.1
0.2
0.3
0.4
0.5
0
0.2
0.4
0.6
0.8
1
propagation distance [m]
Sample/source width [m]
−0.05 0 0.05
0
0.1
0.2
0.3
0.4
0.5
0
1
2
3
4
5
x 10
−3
Fig. 1 Amplitude of first harmonic from Eq. (16)(left) and second har-
monic from Eq. (17)(right) Rayleigh wave with propagation distance
and sample width, including diffraction and attenuation effects, at y=0
[40]. Results normalized by input first harmonic amplitude (reproduced
from [40] with permission from Elsevier)
for this forcing function has been derived as [35]:
v
1
(x, y) =
v
0
e
−(α
1
x)
√
1 + ix/x
0
exp(−
(y/a)
2
1 + ix/x
0
) (16)
where x
0
= κ
R
a
2
/2 is the Rayleigh distance that marks the
transition from the near field to the far field diffraction region
of the source, where κ
R
is the Rayleigh wavenumber. The
second harmonic wave velocity component can be written
as:
v
2
(x, y) = β
11
v
2
0
D
R
(2ω) (17)
where β
11
is a nonlinearity parameter defined elsewhere
[35,37], and D
R
(2ω) is the complex diffraction coefficient
for the propagating and generated second harmonic Rayleigh
wave, which is given in full in [35]. The first and second har-
monic Rayleigh wave amplitude with propagation distance
including diffraction and attenuation effects, i.e. Eqs. (16)
and (17), are plotted in Fig. 1 [40]. These plots are normal-
ized to the input first harmonic amplitude at x=0. In these
plots, it is assumed that α
n
= n
4
α
1
, and note that the prop-
agation distance, x, is plotted on the vertical axis on these
plots, and the sample/source width, y, is plotted on the hori-
zontal axis.
The proportionality of the generated second harmonic
wave remains the square of the fundamental at a given propa-
gation distance x, i.e. v
2
(y) ∝ v
2
1
(y), though the dependence
of v
2
(y)on propagation distance is obscured. While the linear
dependence of v
2
(y) on propagation distance has been shown
to be a good approximation for short propagation distances
in many metals [38,41–43], the full solution incorporating
attenuation, diffraction, and source effects will result in a
more accurate determination of β.
123
J Nondestruct Eval (2015) 34 :273 Page 5 of 23 273
2.3 Microstructural Contributions to β
The parameter β depends on the crystalline structure of the
material, and also on localized strain present in the mate-
rial. This strain arises from microstructural features such
as dislocations and precipitates, and dislocation contribu-
tions to β can greatly exceed that of the lattice anhar-
monicity. This section provides a comprehensive review
of different microstructural contributions to the magnitude
of the acoustic nonlinearity parameter. Theoretical deriva-
tions of effects of dislocation pinning [4,16,44], dislocation
dipoles [45,46], precipitate-pinned dislocations [16,47–49],
and cracks [50–52] are reviewed.
2.3.1 Dislocation Pinning: Hikata et al. Model
The dislocation motion contribution to acoustic nonlinearity
was first developed by Suzuki et al. [53] and expanded on by
Hikata et al. [4], and this model has been used to interpret a
multitude of experimental results of second harmonic gener-
ation. The model is based on the dislocation string vibration
model of Granato and Lücke [54], and considers dislocation
bowing as a line segment pinned between two points, a dis-
tance 2L apart. These pinning points can be grain boundaries,
other dislocations, or point defects in the material. Assume a
small but non-zero longitudinal stress, σ, with shear compo-
nent τ such that τ = Rσ where R is the resolving shear factor,
is then applied to this dislocation segment such that it bows
out between the two pinning points. An approximation of this
geometry is depicted in Fig. 2, where the radius of curvature,
r, of the bowed segment and angle θ are annotated. Note
that this stress can be thought of as either an internal resid-
ual stress or externally applied stress, but it is small enough
such that the dislocation segment does not break away from
the pinning points. An ultrasonic stress wave sets the pinned
dislocation segment into a vibrational motion. However, the
dynamics of the dislocation motion is neglected by assum-
ing that the mass density of the dislocation line is negligi-
ble. Further, the dislocation dynamics that may occur during
Fig. 2 a Diagram showing geometry of bowed dislocation segment of
length 2L between two pinning points and under an applied shear stress
τ
i
, in terms of radius of curvature r and angle θ. b Diagram showing
movement of dislocation segment with superimposed ultrasonic stress
on top of initial stress τ
transient plastic deformation is not considered since this is
not of the source of ultrasonic nonlinearity considered here.
Assuming the dislocation density is small enough such
that bowed dislocations act independently of each other, the
line tension, T, of this dislocation segment due to the applied
stress was approximated as constant and independent of ori-
entation as T = μb
2
/2, where b is the Burgers vector and μ is
the shear modulus. In the Hikata et al. model, the area swept
out by the dislocation is approximated as a circular arc with
a constant radius.
The total strain in the material can be written as a summa-
tion of the lattice strain plus the strain due to the dislocation
motion, i.e. ε = ε
1
+ γ , where is the conversion factor
between shear and longitudinal strain and the shear strain γ
due to a distribution of bowed dislocations. Assuming the
same form of the stress–strain relation as in Eq. (2)forthe
internal stress, the resulting stress–strain relationship due to
the total strain in the material is:
ε =
1
E
1
+
2L
2
R
3μ
σ +
E
2
E
3
1
σ
2
+
4L
4
R
3
5μ
3
b
2
σ
3
(18)
with bowed dislocations density , dislocation loop length
L. Assuming a small ultrasonic stress σ is superimposed on
the internal stress σ
1
, and assuming the lattice contribution
of the nonlinearity parameter is β
0
=−E
2
/E
1
, the change
in nonlinearity parameter due to dislocation pinning can be
written as:
β
pd
=
24
5
L
4
R
3
C
2
11
μ
3
b
2
σ
1
(19)
Further details of this model can be found elsewhere [4,16].
Note that E
1
is defined as σ
0
+ C
11
in Eq. (4), but here
it is assumed that the initial stress σ
0
is zero. It should be
specifically noted that the internal stress σ
1
in this analysis,
as well as the superimposed ultrasonic stress, is assumed to
be much smaller than the yield stress of the material, such
that dislocation displacement is small.
2.3.2 Dislocation Pinning: Extensions of the Hikata et al.
Model
Later work by Cash and Cai [55] extended the Hikata et al.
model to account for orientation-dependent line energy in
the analytical model and verified with dislocation dynamics
simulations. Results showed that the Hikata et al. model can
accurately predict β only for small values of Poisson’s ratio
for both screw and edge dislocations. However, at values
of Poisson’s ratio greater than about 0.2 (i.e. most metals),
the dislocation dynamics simulations and developed analyt-
ical model show that relationship between β and the applied
stress is not in fact linear, and simulations show that β can
123