where
P is the longitudinal force in the rail,
E is the modulus,
A is the cross-sectional area,
α is the coefficient of thermal expansion,
TR is the rail temperature, and
TN is the rail neutral temperature.
However, inferring the longitudinal stress directly using
Equation 1 is generally not practical since the RNT may change
over time and is affected by the track constraints and boundary
conditions that are difficult to characterize.
The overall goal of the research reported herein is to develop
a technology for in situ in-motion measurement of longitudinal
stresses in rails. The research objectives reported in this paper
are to establish the stress-birefringence relationship in common
rail materials and assess the potential variation of fundamental
acoustoelastic properties between different weight rail sections.
This paper is organized to present readers with a brief background
on the RNT measurement conducted by other researchers, the
underlying theory for the ultrasonic stress measurement (USM)
approach, and results from a series of laboratory tests. Conclusions
and future outlook are discussed at the end of this paper.
Background
Several studies have addressed the issue of assessing the RNT
and controlling axial stresses in railroad rails. These studies have
investigated many destructive techniques, semi-destructive tech-
niques, and nondestructive evaluation (NDE) methods. Before
the progression of the semi-destructive and nondestructive
stress evaluation techniques, early tests relied on the destruc-
tive method of rail cutting for measuring the axial forces. These
tests included cutting the rail in two places and measuring the
cut length change to assess the axial forces and RNT (Johnson
2004). A semi-destructive approach for measuring the thermal
stresses is the rail uplift method, which is the basis of contem-
porary technologies such as the VERSE® technology by Vortok
International and AEA technology in the UK (Shrubsall and
Webber 2001). This method is considered semi-destructive
because it involves unclipping part of the rail to perform the
measurement. The method is based on the beam-column
deflection theory by applying a vertical load to the rail and mea-
suring the resulting deflection to calculate the axial force (Kish
et al. 1993 Kish and Samavedam 1987). The rail-cutting method
and the semi-destructive procedure of rail uplift are consid-
ered the most reliable methods to monitor rail buckling force
with sufficient accuracy and reliability (Kish et al. 2013). Strain
gages have also been implemented to evaluate longitudinal
rail stresses in rails but the installation of many strain gages is
required if a long rail path is to be monitored (Kish et al. 1982
Liu et al. 2018 Weaver 2006). For tracks in service, strain gage
usage requires either the knowledge of the longitudinal forces in
the rail by other means before installing the strain gages, or the
zero-force condition has to be established by cutting the rails
before installing the strain gages (Enshaeian and Rizzo 2021).
Several NDE methods have been investigated to estimate
longitudinal rail stresses and RNT. Some of these include
magnetic-based methods that rely on the ferromagnetic prop-
erties of rail steel, such as the magnetic Barkhausen noise
(MBN) effect (Posgay and Molnár 2011 Wang et al. 2013),
acousto-magnetic interactions (Burkhardt and Kwun 1988
Kwun et al. 1990), and variations in magnetic properties of the
steel (Utrata et al. 1995). However, the outcome of these efforts
reported that this technique is confined to assessing surface
stresses and is also affected by localized variation in metal-
lurgy. X-ray, or the Bragg diffraction technique, has similar
limitations of assessing only a small volume of the rail on a
clean surface, and it is usually performed with bulky measur-
ing equipment that makes it inconvenient for field application
(Kelleher et al. 2003 Ruud 1979 Turan et al. 2019).
Vibrational methods have also been examined where a
small part of the rail is excited laterally by a shaker source and
the generated wave is recorded and analyzed. The frequency
and modal parameter changes are related to the axial forces in
the rails. Different forms of this technique have been explored,
including the use of mechanical shakers like a hammer (Béliveau
1997 Zhang et al. 2021) or the use of constant-frequency or
variable-frequency piezoelectric excitation (Kjell and Johnson
2009 Read and Shust 2007). Those studies have reported that the
major challenge with this technique is that the lateral stiffness
from railroad crossties significantly impacts the wave properties
and the resonant frequency more than the axial forces in the rail.
Another study extensively investigated the use of non-
linear coefficients related to the nonlinear guided waves
(Phillips et al. 2014). The method showed promising results,
but field applications were challenging, and the approach
requires sophisticated, nonlinear semi-analytical finite element
modeling for the rail being tested to yield results.
Acoustoelasticity using different ultrasonic wave modes,
including longitudinal and reflected waves (Szelaz˙ek ˛ 1992),
Rayleigh waves (Gokhale and Hurlebaus 2008 Hurlebaus 2011),
transverse shear waves (Szelaz˙ek ˛ 1998), and surface-skimming
shear horizontal waves (Alers and Manzanares 1990) have also
been explored by various researchers. In some experiments, good
linear agreement has been reported for the relation between the
rail stress and the wave speed change in the laboratory. However,
the challenges included acoustic coupling variations and surface
effects when piezoelectric sensors were used, effects of surface
conditions and surface stresses, and plastic deformation in the
head of the rails affecting wave properties. Additionally, varia-
tions in texture, metallurgy, and composition in the materials
being tested affected the acoustoelastic properties of the material
and reduced the accuracy of results. Temperature effects that
change the velocity of the ultrasonic waves also present a chal-
lenge for the practical implementation of the technologies.
Recent approaches that are still under development and labo-
ratory investigation include the use of nonlinear solitary waves
(Nasrollahi and Rizzo 2019), the photoluminescence piezo-
spectroscopy method (Yun et al. 2019), and the deformation-
based method using StereoDIC technology (Knopf et al. 2021).
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The method proposed in this study utilizing acoustic bire-
fringence continues the effort to find an effective nondestruc-
tive technique that can overcome the challenges encountered
with the ultrasonic techniques investigated earlier. The acoustic
birefringence approach has several advantages over other
USM approaches. For example, using ultrasonic bulk shear
waves propagating through the rail web provides an averaged
through-thickness measurement that represents the stress
state in the entire thickness of the rail rather than stresses in
the surface only. Because the acoustic birefringence method
compares the wave velocities of orthogonally polarized shear
waves propagating through the same volume of material, the
texture effects are minimized compared to other acoustoelastic
approaches. The temperature effect on wave velocity is natu-
rally eliminated because the birefringence uses a ratio between
the time of flight (ToF) of orthogonally polarized shear waves,
each equally affected by temperature changes. Variations in
the thickness of the material under test do not affect the bire-
fringence measurement for the same reason. EMATs provide
contactless measurement and eliminate errors due to coupling
variations problematic with piezoelectric transducers. In
addition, the noncontact nature of the EMAT technology may
be suitable for an in-motion stress measurement because the
sensor can be positioned adjacent to the rail without contact,
launching and receiving ultrasonic waves through electromag-
netic interactions in the rail material itself.
Acoustoelastic Theory
Evaluating the longitudinal thermal stresses in railroad rails
based on the acoustoelastic effect relates the change in the
ultrasonic wave velocity with the change of applied stress or
strain. In linear elasticity theory, the stress-strain relationship
is assumed to be linear and consists of first- and second-order
Lamé coefficients (λ, µ), which usually appear in engineering
applications as elastic and shear moduli. Under this isotro-
pic and homogeneous solid media theory, wave propagation
velocity is constant and is not a stress function. When con-
sidering the nonlinear effect of the constitutive relationship,
a third-order elastic constant appears in the relationship that
defines the change in wave velocity due to the applied stress/
strain field, commonly known as the acoustoelastic effect.
Acoustic anisotropy resulting from stresses in the rail
material shows variations in the shear wave velocities. The
change in shear wave velocity depends on the orientation of
the shear wave polarization direction to the direction of the
applied loads/stresses and results in a fast shear wave velocity
in one polarization direction and a slow wave velocity in the
other direction (i.e., birefringence). The acoustic birefrin-
gence used here can be defined as the ratio of the velocity
differences between the fast and slow directions to the
average shear wave velocity. When the thickness of the rail
section is unknown, or for the sake of direct application of
the concept, the acoustic birefringence can also be expressed
in terms of the fast and slow time of travels for the waves as
presented in Equation 2: ​ ​ ​
(2)​ B =​ ​ V​fast​​​ − ​ V​slow​​ _
Vavg​​ ​ ​
=​ ​ t​slow​​ − ​ t​fast​​ _​
t​avg​​ ​ ​ ​
where
B is the acoustic birefringence,
Vfast and Vslow are the shear wave velocities polarized in the
fast and slow directions, respectively,
tfast and tslow are the ToFs for the shear wave polarized in the
fast and slow directions, respectively, and
Vavg and tavg are the average shear wave velocity and the
average ToFs from both directions.
Two sources of material anisotropy manifest as birefrin-
gence. The first source is the residual stresses and texture
from the manufacturing process including the shaping and
straightening of rails. The rolling and straightening process of
rails causes crystallographic textures that result in shear waves
having fast travel velocity when polarized along the rolling
directions and slow velocity when polarized in the direction
perpendicular to the rolling direction. This birefringence is
referred to as the “in situ birefringence” or “unstressed birefrin-
gence” and can be quantified using Equation 2 by measuring
the shear wave velocities or the ToFs polarized in the rolling as
well as in the direction perpendicular to the rolling direction
when the rails are in a stress-free condition.
The second source of anisotropy is the stresses induced
on the rails when the rail is in service. The general equation
for acoustic birefringence accounts for stresses in the
section under the biaxial loading condition, as presented in
Equation 3. The general birefringence equation includes the in
situ birefringence due to residual stresses, the stresses applied,
the stress-acoustic constants, and the orientations of the
stresses with respect to the rolling direction (Okada 1980). ​ ​
(3)​ B =​ ​ {​​[​B​0​​ +​ m​1​​​(​σ​1​​ +​ σ​2​​)​ +​ m​2​​​(​σ​1​​ − ​ σ​2​​)​cos2θ]​​​2​
+​ ​ [​m​3​​​(​σ​1​​ − ​ σ​2​​)​sin2θ]​​​2}​​​2​​​1​​
where
B0 is the in situ birefringence,
σ1 and σ2 are the biaxial in-plane stresses in the rolling direc-
tion and the direction perpendicular to rolling, respectively,
m1, m2, and m3 are the stress-acoustic constants, and
θ is the orientation of rolling direction with respect to the
in-plane stress (σ1).
The rail in service is under primarily uniaxial loading con-
ditions from the thermal stresses and, as such, σ2 in Equation 3
is assumed to be zero. The axial stresses resulting from thermal
load coincide with the rail’s rolling direction, resulting in θ =0.
To simplify the calculations and since m2 is very small, it can
also be assumed to be zero, which puts the general equation in
the following form: ​ ​
(4)​ B =​ B​0​​ +​ m​1​​ ​ σ​1​​​
Equation 4 can be rearranged and written in terms of the
longitudinal thermal stresses (σ) as shown in Equation 5:
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