(Received 11 November 2013; revised 3 September 2014; accepted 10 September 2014) Modern rail construction uses continuous-welded rail (CWR). The presence of very few joints leads to an increasing concern due to the large longitudinal loads caused by restrained thermal expansion and contraction, following seasonal temperature variations. The knowledge of the current state of thermal stress in the rail or, equivalently, the rail neutral temperature (corresponding to zero net longitudinal force) is a key need within the railroad transportation community in order to properly schedule slow-order mandates and prevent derailments. This paper presents a nondestructive diagnostic system for measurement of the neutral temperature in CWR based on nonlinear ultrasonic guided waves. The theoretical part of the study involved the development of a constitutive model in order to explain the origin of nonlinear effects arising in complex waveguides under constrained thermal expansion. A numerical framework has been implemented to predict internal resonance conditions of nonlinear waves in complex waveguides. This theoretical/numerical phase has led to the development of an experimental prototype (Rail-NT) that was tested both in the laboratory and in the field. The results of these experimental tests are also summarized. C 2014 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4896463] V PACS number(s): 43.40.Ga, 43.40.At, 43.35.Zc, 43.25.Zx [JAT]

I. INTRODUCTION

Most modern railways use continuous welded rail (CWR), due to the advantages of this design with increasing axle loads and train velocities. However, due to the impossibility to expand or contract lengthwise under thermal excursions, CWR pose severe safety risk from tension stresses in cold weather (potentially leading to broken rail failures) and compression stresses in warm weather (potentially leading to buckling or “sunkink” failure). According to U.S. Federal Railroad Administration (FRA) safety statistics data (Federal-Railroad-Administration, 2011), rail buckling from uncontrolled thermal stresses was the leading cause of train accidents within the track category in recent years. FRA Safety Statistics data between January 1975 and February 2012 highlight the severity of this type of accident. In this period, more than 2000 derailments with associated costs of $300 million were caused by rail buckling. Between January 2010 and December 2012 alone, sunkink caused 126 derailments. Railroads owners and operators manage the thermal stress problem of CWR by installing the rail at a specific level of prestress. This ensures that the rail will stay at relatively safe thermal stress levels throughout the ambient temperature fluctuations. A crucial property of CWR is the so-called rail neutral temperature (TN), defined as the rail temperature at which the thermal longitudinal force (or stress) in the rail is zero.

a)

Author to whom correspondence should be addressed. Electronic mail: [email protected]

J. Acoust. Soc. Am. 136 (5), November 2014

Pages: 2561–2574

The rail neutral temperature is often associated with the rail “laying” or “anchoring” temperature. The relation between current net longitudinal force, P (or net longitudinal stress r) and current rail temperature, T, is given by P ¼ PT þ P0 ¼ EAaDT þ P0 ¼ EAaðT TN Þ þ P0 ;

(1)

where PT is the thermal force at the current rail temperature T, a is the coefficient of thermal expansion of steel, E is the Young’s modulus of steel, A is the rail cross-sectional area, TL is the rail laying temperature, and P0 is the prestress applied when laying the rail at TL. The neutral temperature TN is then taken from Eq. (1) by equating P ¼ 0, i.e., TN ¼

P0 þ TL : EAa

(2)

Unfortunately, the neutral temperature of a rail may change due to numerous mechanisms of regular service (creep, breathing, ballast settlement, etc.), and/or rail maintenance operations (installation, realignment, distressing, broken rail repairs, etc.). Consequently, even for a known rail “laying” or “anchoring” temperature, the neutral temperature for a rail in service is generally unknown. Despite many years of experience with CWR, the in situ measurement of the applied stress (or TN) still represents a long-standing challenge in railroad engineering. While the problem of rail buckling is fairly well understood (Kerr, 1975, 1978; Kish and Clark, 2004; Read and Shust, 2007; Kish, 2011) the issue of the in situ TN measurement, which is an important input to the rail buckling models, is still

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C 2014 Acoustical Society of America V

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unsolved. Several techniques have been examined in the past for this objective, including VERSE (Tunna, 2000), D’Stresen (Read and Shust, 2007), acoustoelasticity (Egle and Bray, 1976), MAPS-SFT (Read, 2010), guided wave phase velocity measurement (Rose et al., 2002; Wilcox et al., 2003; Damljanovic and Weaver, 2004; Chen and Wilcox, 2007; Loveday, 2009; Bartoli et al., 2010), and ultrasonic Rayleigh polarization measurement (Gokhale and Hurlebaus, 2008). However, there is still ample room for improvements vis-a-vis practicality of deployment, noninvasive nature, and accuracy of performance. The measurement of the neutral temperature of rail is still known as the “Holy Grail” of rail maintenance. This work presents the theoretical framework, numerical modeling, and experimental testing for an inspection system (Lanza di Scalea and Nucera, 2013), based on nonlinear ultrasonic guided waves, aimed at nondestructively identifying the neutral temperature of a CWR in service. II. NONLINEAR ULTRASONIC WAVES IN WAVEGUIDES UNDER CONSTRAINED THERMAL EXPANSION

Nonlinear ultrasonic techniques are highly sensitive to structural damage and load conditions (Dace et al., 1991; Zaitsev et al., 1995; Jhang, 2009), and are used to assess fatigue damage of metals (Yost and Cantrell, 1992; Cantrell and Yost, 2001; Cantrell, 2006) and concrete (Shah and Ribakov, 2009), efficient location of internal cracks and dislocations (Arias and Achenbach, 2004; Bermes et al., 2008; Kuchler et al., 2009; Kim et al., 2010), and contact stresses (Biwa et al., 2006; Nucera and Lanza di Scalea, 2011; Yan et al., 2012). The method examined in this work, relies on nonlinearities arising in waveguides subjected to constrained thermal expansion. Among the possible manifestations of nonlinear behavior, the proposed system exploits the higher-harmonics generation mechanism (de Lima and Hamilton, 2003). However, the resulting higher-harmonic generation is, in this

case, due to potential energy stored in the material due to the prevented thermal expansion, rather than due to finite strains as in classical nonlinear hyperelasticity (e.g., acoustoelasticity). Accordingly, a new model explaining the origin of nonlinear effects in constrained solids subjected to thermal excursions is described in the next section. A. Proposed theoretical model for nonlinearities in waveguides under constrained thermal variations

Nonlinear phenomena arising in wave propagation have been classically treated using acoustoelasticity (Egle and Bray, 1976) or finite amplitude wave theory (de Lima and Hamilton, 2003). According to these frameworks, applied finite strains (or, similarly, finite amplitude waves) constitute a requirement for the occurrence of nonlinearity. However, the generation of nonlinear effects (higher harmonics in particular) in solids subjected to constrained thermal expansion (as it happens in CWR) requires a different theoretical perspective. When the constrained structure experiences temperature changes, it cannot globally deform because of the boundaries, while at the same time, the lattice particles acquire an increased energy of vibration (proportional to temperature) (Tilley, 2004). The starting point is the concept of interatomic potential. Its general form for a pair of lattice particles can be written as (Mie, 1903) m=ðnmÞ " n m # n n q q ; w VMIE ðrÞ ¼ nm m r r (3) where r is the interatomic distance, w is the potential well depth, q is the van der Waals radius, n and m are material coefficient parameters. The minimum of the V(r) curve is the position of equilibrium of the atoms, r0, where the repulsive force is equilibrated by the attractive force and the net force is zero. A specific form of the interatomic potential most often used for

FIG. 1. (Color online) Average bonding distance (ABD) curve for free expansion and residual potential for perfectly constrained thermal expansion.

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its computational efficiency was proposed by Lennard-Jones (Lennard-Jones, 1924c,b,a), where n ¼ 12 and m ¼ 6. Assuming that the solid is free to expand, an increase in temperature produces a stretch of the atoms’ equilibrium distance r0, following the dashed curve in Fig. 1. This curve is the so-called average bonding distance (ABD), or the loci of midpoints between repulsive and attractive branches of the potential. The ABD curve represents the well-known thermal expansion of the material, which directly results from the “anharmonicity” of the lattice. Following conventional linearized thermal expansion theory, the rABD curve can be also simply written as a function of temperature as rABD ðTÞ ¼ r0 ½1 þ aðT T0 Þ;

(4)

where a is the thermal expansion coefficient of the material, and DT ¼ TT0 is the temperature change from the initial interatomic distance r0. For free thermal expansion, the minima of the interatomic potential curve at the various temperatures lie on the rABD curve, and the potential well depth rises to reflect the additional kinetic energy imparted by the temperature increase. In this case, the new positions of the atoms at all temperatures are still at zero net force (strain without stress). If, instead, the solid is prevented from thermally expanding due to external constrains, it is known that it develops thermal stresses. This fact, in turn, implies that the interatomic potential at T does not have a minimum point (zero force), but it assumes a value that corresponds to the V(r) curve for the original T0 temperature, calculated at the current “free expansion” position rABD(T) (see Fig. 1). For such cases of constrained thermal expansion, the general form of the current interatomic potential at T can be found by expanding Eq. (3) in Taylor’s series around the rABD(T) value. The result is @V ½r rABD ðT Þ V ðr; T Þ ¼ V ½rABD ðT Þ þ @r r¼rABD ðT Þ 1 @ 2 V 2 þ ½r rABD ðT Þ 2 @r2 r¼rABD ðT Þ 1 @ 3 V 3 þ ½r rABD ðT Þ þ 6 @r3 r¼rABD ðT Þ ¼ AðT Þ þ BðT Þ ½r rABD ðT Þ 1 2 þ CðT Þ ½r rABD ðT Þ 2 1 3 þ DðT Þ ½r rABD ðT Þ þ ; 6

(5)

where A(T) is an initial energy, B(T) ¼ @V/@r, C(T) ¼ @ 2V/ @r2, and D(T) ¼ @ 3V/@r3, with the derivatives calculated at r ¼ rABD(T). The main difference with classical finitedeformation nonlinear elasticity is that the nonlinearity [arising from the O(r3) term in the potential, or the term D], in this case does not arise from applied finite deformations but rather from the “residual” strain energy stored as internal forces from the prevented thermal expansion. Classical equilibrium considerations lead, after some algebraic manipulations (Nucera and Lanza di Scalea, 2014), to the nonlinear partial differential equation governing the J. Acoust. Soc. Am., Vol. 136, No. 5, November 2014

propagation of longitudinal waves in solids subjected to constrained thermal expansion: 2 @ 2 u1 @u1 @ u1 2 ¼ V1 1 c 1 ; (6) @t2 @x1 @x1 2 where two new definitions for longitudinal bulk wave velocity and nonlinear parameter are introduced: sﬃﬃﬃﬃﬃﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ C2 CðT Þ h1 Longitudinalwavespeed; (7) V1 ¼ ¼ q qS1 c1 ¼

C3 DðT Þ h1 ¼ C ðT Þ C2

Nonlinear parameter:

(8)

As expected, Eq. (7) shows that the wave speed depends on the quadratic term O(r2) of the interatomic potential, C(T). This term reflects the curvature of the interatomic potential, which is changing with interatomic distance. In classical linear Hooke’s theory, this curvature (hence material stiffness) is approximated constant through the deformation range, hence the wave velocity is a constant. By considering the full asymmetric potential as a function of the prevented thermal expansion, the wave velocity is found to change with accumulating thermal stresses. Trends of V1 from Eq. (7) are plotted in Figs. 2(a) and 2(b) assuming a Lennard-Jones interatomic potential (n ¼ 12, n ¼ 6) and sample values of Van Der Waals radius q ¼ 4 angstroms, and potential well depth w ¼ 40 kJ/mol, without loss of generality for the trends. The material density is assumed to be that of steel, q ¼ 7800 kg/m3. Specifically, Fig. 2(a) plots the relative change in wave speed, as a function of interatomic distance rABD. The temperature T0 is the stress-free value ˚ —mini(corresponding to equilibrium distance r0 ¼ 4.489 A mum of the assumed Lennard-Jones potential). The trend in Fig. 2(a) clearly indicates a decrease in wave speed with increasing (prevented) thermal expansion. Therefore a material “softening” effect takes place, consistently with the decrease in curvature of the interatomic potential, C(T), when moving slightly to the right of the equilibrium position r0 (see, for example, Fig. 1). Figure 2(b) plots the same velocity change as a direct function of the temperature change, DT ¼ TT0. For the case considered, for example, the longitudinal wave velocity is expected to decrease by about 1% for a temperature increase of 100 C in the fully constrained solid. Regarding the nonlinear parameter of Eq. (8), it contains the nonlinear portion of the interatomic potential through the cubic term O(r3), D(T). Again, the difference from classical nonlinear wave theory is that the cubic O(r3) energy term arises from the prevented thermal expansion due to the asymmetry of the interatomic potential, rather than from applied finite deformations. The solution of Eq. (6) can be obtained using perturbation analysis (de Lima and Hamilton, 2003), thereby decomposing the displacement field into the linear portion, u1(1), and the nonlinear portion, u1(2), with u1(1) u1(2). The final solution to the nonlinear wave equation can be written in the classical form as u1 ¼ u1 ð1Þ þ u1 ð2Þ ¼ A1 cosðkx1 xtÞ 1 c1 k2 A1 2 x1 sin 2ðkx1 xtÞ; 8

C. Nucera and F. Lanza di Scalea: Ultrasonic measurement of temperature

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FIG. 2. (Color online) Relative change in longitudinal wave velocity as a function of (a) “prevented” thermal expansion of interatomic distance and (b) temperature change from stress-free position. Normalized nonlinear parameter b as a function of (c) “prevented” thermal expansion of interatomic distance and ˚ , w ¼ 40 kJ/mol, a ¼ 11 106/ C, q ¼ 7800 kg/m3). (d) temperature change from stress-free position (Lennard-Jones potential, n ¼ 12, n ¼ 6, q ¼ 4 A

where x1 is the wave propagation distance and k is the wavenumber. It can be seen that the nonlinearity generates a secondharmonic contribution at 2x under a fundamental excitation at x. The magnitude of the second-harmonic is proportional to the nonlinear parameter given in Eq. (8), and also proportional to the wave propagation distance, x1 [the “cumulative” behavior (de Lima and Hamilton, 2003)]. Experimentally, it is common to directly measure the amplitudes of the second-harmonic, A2, and that of the fundamental frequency, A1. Therefore, an “experimental” nonlinear parameter can be defined from the “theoretical” nonlinear parameter in Eq. (8) as b¼

jA2 j 1 ¼ c 1 k 2 x1 8 A21 2

¼

III. NUMERICAL MODELING

2

p f c1 2 x1 Exper: nonlinear parameter; 2 V1

(10)

where f is the excitation wave frequency (fundamental), V1 is the longitudinal bulk wave speed, and x1 is the wave propagation distance. Trends of the nonlinear parameter b from Eq. (10) are plotted in Figs. 2(c) and 2(d) assuming a Lennard-Jones interatomic potential (n ¼ 12, n ¼ 6) and sample values of ˚ , and w ¼ 40 kJ/mol, without loss of generality for the q¼4A nonlinear trends. The b values in this figure have been normalized to all terms independent of temperature, to highlight the effect of the temperature-dependent terms C(T) and D(T). 2564

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Figure 2(c) plots b as a function of interatomic distance rABD. The nonlinear parameter increases monotonically with increasing “prevented” thermal expansion, i.e., increasing thermal stress absorbed by the constrained solid. Figure 2(d) plots the same nonlinear parameter b as a direct function of the temperature change, DT ¼ TT0, from the initial, stressfree temperature T0 (corresponding to equilibrium distance r0). It is clear that the nonlinear parameter monotonically increases with increasing temperature, as the constrained thermal expansion builds nonlinear effects through thermal stresses. The slope of the hb vs Ti curve will, of course, depend on the coefficient of thermal expansion of the material, with larger slopes expected for larger a’s.

The previous section discussed how the (at least) cubic dependence on strain of the residual strain energy that is stored in a constrained waveguide under thermal variations gives raise to second-harmonic generation of the propagating elastic waves. In order to maximize this nonlinear response and, consequently, gain sufficient sensitivity to thermal stress measurement, it is crucial to identify certain mode-frequency combinations that indeed support cumulative higher-harmonic generation in the waveguide. Such mode identification process was achieved by a computational algorithm (CO.NO.SAFE) that is discussed in the present section. The CO.NO.SAFE algorithm, originally introduced by Nucera and Lanza di Scalea (2012) allows to perform such C. Nucera and F. Lanza di Scalea: Ultrasonic measurement of temperature

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nonlinear mode identification in waveguides with complex cross-sectional geometries (such as a rail), where analytical guided wave propagation solutions are either nonexistent or very difficult to obtain (at high frequencies). CO.NO.SAFE is based on the implementation of a nonlinear semianalytical finite element formulation into a commercial finite element package (COMSOLV). The numerical framework was applied to a 136 RE rail, with the objective of identifying suitable combinations of primary and second-harmonic wave modes. A further constraint was that these modes should be primarily propagating within the rail web, with little leakage into the rail head or the rail foot. The reason for this constraint lies in the necessity for the waves to minimize influences of residual stresses and wear (more pronounced in the rail head) and influences of the rail supports (through the rail foot). Both of these influences, in fact, have been main limitations of other techniques explored in the past for in situ measurement of the neutral temperature in CWR (Read and Shust, 2007).

resulting twin-parameter EVP admits nontrivial eigensolutions by solving for k and x. The present partial differential equations (PDE) system has been solved using the so-called COMSOL coefficient form PDE Interface. The COMSOL input formalism to model the most general PDE problem reads (MATLAB, 2012)

R

A. CO.NO.SAFE analysis—resonant mode identification in the rail

The field equations are first characterized enforcing the SAFE approximation, that consists in assuming the displacement field harmonic along the wave propagation direction, z and represented by finite element spatial shape functions in the cross section of the waveguide (Bartoli et al., 2006) (Fig. 3). Making use of the SAFE assumption for the displacement field, and applying intermediate algebraic manipulations, the guided wave propagation problem can be formulated as a twin-parameter eigenvalue problem (EVP) using the following system of partial differential equations with Neumann boundary conditions (waveguides are treated as domains with stress-free boundary conditions): Cijkl Nj;kl þ iðCi3jk þ Cikj3 ÞðkNj Þ;k kCi3j3 ðkNj Þ þ qx2 dij Nj ¼ 0 in X; Cikjl Nj;l nk þ iCikj3 ðkNj Þnk ¼ ^t i on Cr ;

(11)

ea

@2u @u þ r ðcru au þ cÞ þ da @t2 @t þ b ru þ au ¼ f in X;

(13)

n ðcru þ au cÞ þ qu g ¼ hT l on Cr ;

(14)

hu ¼ r on Cu ;

(15)

where X is the computational domain (union of all subdomains) corresponding to the meshed 2D cross-section of the waveguide, Cr is the portion of the domain boundary C where surface tractions are prescribed, Cu is the remaining part of the domain boundary where displacement are prescribed, n is the outward unit normal vector on C, ea is the mass coefficient, da is the damping/mass coefficient, c is the diffusion coefficient, a is the conservative flux convection coefficient, b is the convection coefficient, a is the absorption coefficient, c is the conservative flux source term, f is the source term and u represent the set of variables to be determined. The above coefficients must be established via an identification procedure. Their identification depends on the physical problem under investigation. Equation (13) is the PDE, which must be satisfied in X. Equations (14) and (15) represent the natural (generalized Neumann BC) and essential (Dirichlet BC) boundary conditions, respectively, which must hold in C. Considering Eqs. (13) and (14), the original PDE problem can be reformulated as a scalar eigenvalue problem via the correspondence @/@t $ t, linking the time derivative to the eigenvalue k. The result of this manipulation, dismissing unnecessary forcing terms, reads –k2 ea þ kda u þ r ðcru þ au cÞ bru au ¼ 0 in X;

(16)

(12) n ðcru þ au cÞ þ qu ¼ 0 on Cr :

where Cijkl is the fourth-order elasticity tensor, X is the volume of the waveguide, Cr is the portion of the exterior surface C, where surface tractions are prescribed, i ¼ 1,2,3, and summation is implied over the indices j, k, and l. The

(17)

If c ¼ ea ¼ 0, Eqs. (16) and (17) can be rewritten as Cijkl uj;kl þ ðaijk bijk Þuj;k aj uj þ kdij uj ¼ 0 in X; (18) Cikjl uj;l nk þ aijk uj nk þ qij uj ¼ 0 on C:

FIG. 3. (Color online) Generic eth finite element on the waveguide crosssection for the SAFE modeling of ultrasonic guided waves. J. Acoust. Soc. Am., Vol. 136, No. 5, November 2014

(19)

It is evident how the EVP formulated by Eqs. (18) and (19) effectively represents the original EVP in Eqs. (11) and (12) once all the coefficients have been correctly defined. Nontrivial solutions can be found by solving this twinparameter generalized EVP in k and x. The frequency x is a real positive quantity. The wavenumber k can be real, complex, or imaginary and can have both positive and negative signs, associated with so-called right-propagating and leftpropagating waveguide modes, respectively.

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In operational terms, for each frequency x, the algorithm produces wavenumbers and modeshapes of propagative (real wavenumber) and nonpropagative (complex wavenumber) guided modes. However, for each frequency x, a relatively complex second-order polynomial eigenvalue problem needs to be solved. This scenario is computationally optimized using a classic technique consisting in recasting the original EVP to a first-order eigensystem by introducing a new vector variable v defined as M v ¼ kM u;

(20)

where M is an arbitrary diagonal matrix. In order to correctly formulate the original problem, the following set of variables can be introduced: ¼ ½ u1 u

u2

u3

v1

v2

v3 T :

(21)

With this new set of variables, the coefficients appearing in the SAFE formalism discussed above must be

0 iA a¼ ; M 0 0 0 C 0 M 0 c¼ ; a¼ : 0 0 0 M

da ¼

0

D

;

b¼

0

iB

0

0

; (22)

All the coefficient matrices are detailed in Nucera and Lanza di Scalea (2012). The discussed manipulation doubles the algebraic size of the original eigensystem. This size depends on the finite element mesh used to discretize the cross section of the waveguide and, consequently, on the number of degrees of freedom of the finite element model. As a result, being 2M the size of the linearized eigensystem, at each frequency x, 2M eigenvalues km and 2M associated eigenvectors are obtained. The eigenvectors are the M-forward and the corresponding M-backward waveguide modes. Further details of the proposed algorithms can be found in Nucera (2012).

TABLE I. Linear and nonlinear material properties assumed for the railroad track analysis. q (kg/m3)

k (GPa)

l (GPa)

A (GPa)

B (GPa)

C (GPa)

7932

116.25

82.754

340

646.667

16.667

In terms of flowchart, the CO.NO.SAFE code first computes the guided wave propagation properties in the linear regime (dispersion curves and waveguide mode shapes). In a second step, the nonlinear part of the algorithm uses the above eigensolutions for the mode expansion to obtain the nonlinear solution with a perturbative approach and identifies the potential combinations of resonant modes. As mentioned before, the 136 RE rail geometry was studied by this approach. This rail size is often used in the U.S. and is also used in the large-scale experimental testing that will be discussed later. In order to correctly explore the displacement field and all the derived quantities (essential for the calculation of all the terms during the nonlinear post-processing), 351 cubic Lagrangian 10-node triangular isoparametric elements (Onate, 2009) were employed to construct the numerical model. Material properties used in the analysis are given in Table I. The Landau-Lifshitz third-order elastic constants are given by Sekoyan and Eremeev (1966). Wavenumber and phase velocity dispersion curves for the rail model calculated with the SAFE method are shown in Fig. 4 in the DC-600 kHz frequency range. Each of these curves is a particular vibrating mode of guided waves propagating along the rail. Also, these curves include all of the possible modes of vibration of the rail in this frequency range. The inset in Fig. 4(a) shows the rail geometry and FE mesh (average quality index equal to 0.9319). The complexity of the guided wave propagation for this particular waveguide is apparent, considering the abundance of propagative modes present and their dispersion characteristics (especially at higher frequencies) (see Fig. 5).

FIG. 4. (Color online) (a) Wavenumber spectrum and (b) phase velocity dispersion curves calculated with the SAFE-COMSOL algorithm for a 136 RE rail. These curves represent all of the possible vibrating modes of the rail in the DC-600 kHz frequency range. The inset in (a) shows the finite element mesh adopted for the nonlinear analysis. 2566

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FIG. 5. (Color online) Zoomed views on rail phase velocity dispersion curve around 80 and 330 kHz, respectively.

Figure 6 shows some propagative modes found inside the range (80–160) kHz. It can be noted how differently each mode concentrates the energy within the waveguide and how significant is the difference between these modes in terms of in-plane (cross section) and out-of-plane (along the wave propagation direction) displacement distributions. In view of a wayside installation for the proposed system (without interfering with trains transit), waveguide modes with energy propagation mainly confined in the rail web are of particular interest. These specific propagative web-modes were found at relatively high frequency (roughly above 150–200 kHz), and the numerical analysis was therefore performed for combinations of primary and secondary frequencies in this range.

FIG. 6. (Color online) Propagative modes in the (80–160) kHz frequency range. (a) Flexural vertical mode (energy mainly concentrated in the rail’s head). (b) Flexural horizontal mode (energy exclusively confined in the rail’s web). (c) Axial mode. (d) Complex mode involving a mixture of axial, torsional, and flexural displacements. J. Acoust. Soc. Am., Vol. 136, No. 5, November 2014

As evident from Fig. 4(b), at high frequencies the phase-velocity dispersion curves become extremely complicated and a large number of propagative modes coexists. Consequently, the complexity and the numerical size of the governing eigensystem are dramatically increased (thousands of eigenvalues need to be calculated at each frequency). In order to streamline the computational demand of this process without compromising the precision of the results, a slightly coarser mesh than the one used for the computation of the dispersion curves was created via the COMSOL advancing front algorithm and adopted for the nonlinear part of the analysis. The internal resonance calculations have been developed between 200 kHz (primary mode) and 400 kHz (double harmonic). Several complex propagative modes were discovered in this frequency range. A particular propagating mode at 200 kHz showing a strong energy concentration in the rail web was selected as input for the nonlinear internal resonance analysis [Figs. 7(a) and 7(b)]. The analysis revealed the presence of a resonant secondary mode propagating at 400 kHz [Fig. 7(c) and7(d)]. It is characterized by a similar displacement field and it exhibits strong energy concentration in the web area like the fundamental mode at 200 kHz. The secondary modal amplitude plot (Nucera and Lanza di Scalea, 2012) shown in Fig. 8 emphasizes the clear predominance of a single secondary mode at 400 kHz (the resonant one) when compared to all the other modes propagating at the same frequency. The inset shows small modal amplitude values associated with some synchronous propagating modes. They match (numerically) the phase velocity of the primary mode but their particular power transfer characteristics do not generate internal resonance. The identification of this particular combination of resonant modes propagating at relatively high frequency and, most importantly, concentrating the energy in the rail web

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constitutes an important result in the development of the neutral temperature detection system. IV. Rail-NT PROTOTYPE DEVELOPMENT A. Large-scale validation tests at University of California San Diego

FIG. 7. (Color online) Selected combination of waveguide modes propagating in the AREMA 136 RE rail web. (a) Contour plot of out-of-plane displacement field for the primary mode at 200 kHz. (b) Vector plot of in-plane displacement field for the primary mode at 200 kHz. (c) Contour plot of outof-plane displacement field for the secondary mode at 400 kHz. (d) Vector plot of in-plane displacement field for the secondary mode at 400 kHz.

Hand-in-hand with the modeling study, a comprehensive program of experimental testing has been conducted at UCSD’s large-scale Rail NT test-bed (Fig. 9). This facility is a unique 70-ft long CWR track, that allows to impose thermal loads in a highly controlled laboratory environment, and yet in a quite realistic manner. The track can be pre-stressed at varying rail installation stresses to achieve different values of neutral temperature. The existing track was installed with 90 F as the TN value. The heating is performed by a customized rail switch heater wire that is able to heat up the rail to temperatures in excess of 150 F in about 1 h. The test-bed is heavily instrumented in order to explore efficiently and exhaustively the full static and dynamic response of the track. The instrumentation includes eight, full-bridge temperature compensated strain gauge locations (four on each rail) and eight temperature sensing locations (four on each rail). The temperature compensated strain gauges provide the measurement of the true thermal stress in the rail (Ajovalasit, 2008). In conjunction with the temperature

FIG. 8. (Color online) Resonant combination of modes propagating at relatively high frequencies in the AREMA 136 RE rail web. (a) Selected primary mode at 200 kHz. (b) Resonant secondary mode at 400 kHz. (c) Modal amplitude plot for secondary propagative modes. 2568

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FIG. 9. (Color online) The large-scale Rail NT measurement test-bed at UCSD’s Powell Structural Laboratories constructed with the technical assistance of FRA and BNSF.

readings, the stress measurements allow to independently obtain the true neutral temperature of the rail according to Eq. (1). An infrared camera was also used to map graphically the temperature distribution during the thermal cycles. The large-scale testbed was designed and constructed with the financial support of the FRA and in-kind support of the Burlington Northern Santa Fe (BNSF) railway. Instrumentation Services of Pueblo, CO provided the strain gauge and temperature instrumentation and measurements according to railroad standards. The experimental prototype for the measurement of the rail TN consists of an ultrasonic transmitter and an ultrasonic receiver that are mounted on a case that is magnetically attached to the rail web for a wayside installation. The nonlinear parameter (quantifying higher-harmonic generation phenomena) of the selected ultrasonic guided modes is measured as a function of rail temperature. This parameter is calculated according to Eq. (9) by measuring the ratio

FIG. 10. (Color online) Thermal test protocol. Ultrasonic nonlinear features recorded at each measurement point during the heating cycle. J. Acoust. Soc. Am., Vol. 136, No. 5, November 2014

between the amplitude of the second harmonic and the square of the amplitude of the primary excitation. These amplitudes were determined from the fast-Fourier transform amplitude spectrum of the pitchcatch ultrasonic signal. The ultrasonic excitation was a 10-cycle toneburst. Since the interest of these measurements was tracking trends of the nonlinear parameter as a function of thermal stress in the rail, no effort was made to eliminate instrumentation nonlinearities (including electronic and transducer-to-structure coupling), which were not expected to change during the test. A minimum of the ultrasonic nonlinear parameter indicates the rail TN (zero stress temperature). Starting from the initial pretensioned state (at rail laying temperature), several heating tests were performed increasing progressively the rail temperature in successive steps via the heating system. In this way, repeated ultrasonic nonlinear measurements were systematically recorded at each temperature step, passing through the neutral temperature state in both heating phase and subsequent cooling phase (Fig. 10). A typical result from the Rail-NT prototype is shown in Fig. 11 for a 2-day test where the TN was crossed eight times (four heating phases and four cooling phases). In this figure the measured nonlinear parameter of the selected guided modes is plotted as a function of time. The vertical black lines correspond to the TN points, as identified by the strain gauges. As expected, it can be seen that the nonlinear parameter has a minimum value corresponding to the zero stress state of the rail. The independent determination of the rail TN, at the location where Rail-NT was installed, was provided by a zero reading of the thermal stress (or force) from the strain gauge instrumentation at that location. The accuracy of the result is very satisfactory (to within 62 F) considering a thermal expansion coefficient for steel of 6.7 microstrain/ F. Figure 11 also shows that the nonlinear measurement indeed follows the evolution of the thermal stresses in the rail, with the slopes of the measurement

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FIG. 11. (Color online) Laboratory result showing the nonlinear parameter of the ultrasonic guided wave identifying the rail NT with high accuracy during several heating and cooling cycles of the rail.

following the expected slopes of the thermal cycles (faster heating phase from the active heating vs slower cooling phase from ambient convection into air). Motivated by the promising laboratory results, the prototype was tested in the field as discussed in the next section.

B. Field test at the Transportation Technology Center, Pueblo, CO

The first field tests of the Rail-NT prototype were conducted at the Transportation Technology Center (TTC) in

FIG. 12. (Color online) The test site at TTC with instrumentation.

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Pueblo, CO, in June 2012. The tests were aimed at assessing the performance of the proposed system on a real railroad track and the influence of rail size and support type. Personnel from BNSF, ISIRail Instrumentation, and the Volpe National Transportation Systems Center participated in the tests. The track section tested was a 141 RE rail, with a transition between concrete and wood ties (Fig. 12). A total of two Rail-NT prototypes were installed on the same rail, on either side of the transition, the first one in the concrete tie section and the second one in the wood tie section. Both locations were also independently instrumented by conventional temperature-compensated strain gauges and thermocouples, installed and monitored by ISIRail Instrumentation. The rail was cut and then welded at a known temperature to obtain a known TN value by appropriately zeroing the strain gauge readings under free rail conditions. The results of 2 days of monitoring following the rail cut and weld are shown in Figs. 13 and 14 for the concrete tie location and the wood tie location, respectively. The top trace in these figures illustrates the thermal force vs time independently measured by the strain gauges. The bottom trace in the figures is the output of the Rail-NT prototypes. It can be seen that the Rail-NT output follows closely the evolution of the thermal force in both concrete ties and wood ties. The four NT points (zero force) are properly identified as four Minima in the Rail-NT output with very good accuracy. Specifically, the Rail-NT accuracy was within 2 F for the concrete tie location and within 5 F for the wood tie location. The accuracy values shown in Figs. 13 and 14 were

computed by comparing the TN readings from the strain gauges to the rail temperatures corresponding to the minima of the Rail-NT output. These results successfully confirm the repeatability and accuracy that was observed in the laboratory tests conducted at UCSD’s large-scale test-bed. The temporary change in force observed around 3:00 a.m. of day 3 in both Figs. 13 and 14 is due to the passage of a train. Correspondingly, an isolated change is also seen in the nonlinear parameter curves. A train pass brings a transient increase in rail temperature (from the wheelrail interaction), as well as a similarly transient change in mechanical force applied to the rail. The change in mechanical force on the rail can last for several minutes due to the temporary “unsettling” of the rail from the constraints induced by the large train loads. The natural temperature fluctuations, following the passage of a train, eventually resettles the rail at its “baseline” force level, as seen in the force-temperature plots. The nonlinear parameter is unable to distinguish the load applied by a passing train from the thermal load in the rail, and therefore tracks the total load in the rail. It should be also noted that the absolute values of the nonlinear parameter shown in the field tests results of Figs. 13 and 14 are different from those of the laboratory results of Fig. 11. This difference can be due to different excitation amplitudes used in the two tests (which will alter the nonlinear parameter in the presence of damping), and different rail materials and age (with different interatomic potential curve affecting beta according to Sec. II). However, these

FIG. 13. (Color online) Field test results (concrete tie location). The four minima correspond to actual rail NT of 90.2 F, 95.4 F, 91.5 F, and 92.6 F, respectively.

J. Acoust. Soc. Am., Vol. 136, No. 5, November 2014

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FIG. 14. (Color online) Field test results (wood tie location). The four minima correspond to actual rail NT of 94.2 F, 97.1 F, 93.8 F, and 95.9 F, respectively.

differences do not alter the trend of the curve and, specifically, the identification of the minimum point that is of interest to this study. V. CONCLUSIONS

The problem of the nondestructive measurement of the neutral temperature of rail is a long standing challenge in railroad maintenance engineering. The problem arises from the thermal stresses that develop in a constrained rail due to temperature excursions. In this paper, an approach is presented based on ultrasonic guided waves utilized in their nonlinear regime. A theoretical model based on interatomic potential is first developed to justify why a solid that is constrained and subjected to a thermal load is hyperelastic. Consequently, in the presence of thermal stresses, waves propagating in the solid exhibit nonlinear behavior (specifically double harmonic generation). The issue of identifying suitable guided wave modes to exploit the nonlinear double-harmonic generation in the rail waveguide is examined by a numerical framework that combines semi-analytical finite element formulation with COMSOL finite element formalism. The framework allows us to identify combinations of primary excitation and double harmonic response that offer the desired “resonant” behavior for practical measurements. Based on the theoretical and the numerical work, an experimental prototype for the measurement of the rail neutral temperature (Rail-NT system) has been designed, developed and tested both in the laboratory and in the field. Results are shown in this paper from testing in the large-scale testbed of 2572

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UCSD, as well as in the field at the Transportation Technology Center in Pueblo, CO. These tests indicate that the system is able to reliably identify the rail neutral temperature (independently measured by temperature-compensated strain gauges) with a high degree of accuracy. Since the prototype exploits wave modes that propagate predominantly in the rail web, its performance is minimally affected by residual stresses and wear in the rail head, or by changing support conditions in the rail foot (the so-called “tie-to-tie variation” problem of other technologies). One important limitation of the system is the necessity for the rail to cross the neutral temperature, so that a minimum of the nonlinear feature can be identified. This means that the duration of an actual test is dependent on environmental conditions. The nonlinear measurement is currently not able to determine a current, absolute level of thermal stress from one measurement because such determination is dependent on the knowledge of the interatomic potential of the material [at least with a cubic accuracy of O(r3)] which is practically difficult to obtain. In addition, as in any nonlinear ultrasonic test, the nonlinearity of the instrumentation and that of the transducer/structure coupling will affect the absolute measurements. However, the identification of the minimum of the nonlinear curve is not affected by these parameters, and it is therefore possible to identify the point of zero stress of the rail (i.e., the neutral temperature point). Additional field tests are being conducted to further validate the performance of the system vis-a-vis accuracy of neutral temperature identification and practicality of deployment in the C. Nucera and F. Lanza di Scalea: Ultrasonic measurement of temperature

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railroad environment. Additional efforts are being devoted to exploring the option of artificially heating the rail in the field to eliminate the dependence on environmental conditions and accelerate a test. Finally, the effects of passing trains, with the consequent dynamic loads generated on the rail and how these loads may complicate the neutral temperature identification, are still to be understood. ACKNOWLEDGMENTS

This work was supported by the U.S. Federal Railroad Administration under University Grant No. FR-RRD-000910-01-03. Mahmood Fateh (Program Manager), Leith AlNazer and Gary Carr of the FRA provided important technical support and advice throughout the project. Former Ph.D. students Ivan Bartoli (now at Drexel University) and Robert Phillips were instrumental in the first phases of the project, and particularly in the design and construction of the large-scale test-bed at UCSD. BNSF and ISIRail Instrumentation are acknowledged for providing in-kind support for the test-bed design, construction and monitoring

@ 2 VMIE ¼¼ C ðT Þ ¼ @r 2 r¼rðT Þ and @ 3 VMIE D ðT Þ ¼ ¼¼ @r3 r¼rðT Þ

The second-order and third-order coefficients of the interatomic potential for constrained thermal expansion, for the general case of the Mie potential, are given by the following expressions. These expressions are simply obtained by differentiating the potential VMIE in Eq. (3) and calculating the derivatives at the temperature-dependent interatomic position r*, where r* ¼ rABD(T) [Eq. (4)]:

ðn m Þ½ r ðT Þ

(A1)

2

n=ðnmÞ " m n # n q q ð Þ ð Þ ð Þ ð Þ n m 1þm 2þm n 1þn 2þn w m r ðT Þ r ðT Þ ðn m Þ½ r ðT Þ

6

24wq6 ½26q6 7½r ðT Þ

(A3)

½r ðT Þ14

and 6

DLennardJones ðT Þ ¼

APPENDIX

n=ðnmÞ " n m # n q q n n ð1 þ n Þ m ð1 þ m Þ w m r ðT Þ r ðT Þ

For the specific case of the Lennard-Jones potential (n ¼ 12, m ¼ 6), these expressions simplify to CLennardJones ðT Þ ¼

instrumentation. Ph.D. students Stefano Mariani and Peter Zhu, in addition to the authors of the paper, participated to the field tests at TTC and assisted greatly with the elaboration of the field test results. Special thanks are also extended to John Stanford, Thomas Brueske, Nick Dryer, Henry Lees, and Scott Staples of BNSF, Don Rhodes of ISIRail Instrumentation, David Read of TTCI, John Choros of the Volpe National Transportation Systems Center (VNTSC), and Luis Maal of FRA for their participation in the planning and conduction of the field test at TTC.

672wq6 ½13q6 þ 2½r ðT Þ ½r ðT Þ15

:

(A4)

Ajovalasit, A. (2008). Analisi Sperimentale delle Tensioni con gli Estensimetri Elettrici a Resistenza, 2nd ed. (Experimental Analysis of Tensions with the Electrical Resistance Strain Gauges) (Aracne, Italy), 240 pp. Arias, I., and Achenbach, J. D. (2004). “A model for the ultrasonic detection of surface-breaking cracks by the scanning laser source technique,” Wave Motion 39, 61–75. Bartoli, I., Marzani, A., Lanza di Scalea, F., and Viola, E. (2006). “Modeling wave propagation in damped waveguides of arbitrary crosssection,” J. Sound Vib. 295, 685–707. J. Acoust. Soc. Am., Vol. 136, No. 5, November 2014

3

:

(A2)

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