Acoustoelastic Coefficients in Thick Steel Plates Under Normal and Shear Stresses

Experimental Mechanics DOI 10.1007/s11340-016-0186-6

Acoustoelastic Coefficients in Thick Steel Plates under Normal and Shear Stresses

Z. Abbasi1 & D. Ozevin1

Received: 12 November 2015 /Accepted: 21 June 2016 # Society for Experimental Mechanics 2016

Abstract Ultrasonic monitoring of the integrity of structural increase the cumulative stress in critical bridge components, materials utilizes the acoustoelasticity of materials (stress- which may lead to unexpected failures. The collapse of the I- dependence of ultrasonic wave velocity) obtained from the 35W Mississippi River truss arch bridge in Minnesota is a higher order strain terms in the constitutive equations. This famous case. It was attributed to an increase of the dead load method has been applied to measure residual and applied by 30 % over the original design from concrete deck overlay, stresses in thin metals under principal stresses without shear. plus additional construction-related loads on the day of the Here, acoustoelastic coefficients are determined in thick steel collapse located above the weakest point of the structure, a plates in normal, orthogonal, and angled directions by means pair of gusset plates [1]. As the bridge was a non-redundant, of an array of ultrasonic sensors. A three-dimensional material fracture-critical bridge, the failure of gusset plates caused the model is developed which includes Murnaghan hyper- complete failure of the bridge. Although the I-35W bridge was elasticity and can determine the effects of plate thickness regularly inspected for corrosion and cracks, these were not and excitation frequency on the acoustoelastic coefficients. identified as contributing factors. In order to estimate the over- This model is experimentally validated by tensile loading of load and the remaining strength capacity of any built structural a thick steel plate by measuring the ultrasonic signals in three element, the quantification of the stress state is needed. This directions. Numerical and experimental results agree within study focuses on the acoustoelastic method of non-destructive the measurement uncertainties of each method. The evaluation, which can provide a materials-dependent coeffi- 1.0 MHz ultrasonic frequency has the highest resolution for cient that enables calculation of stress in structures made of measuring normal and shear stresses in structural plates typi- this material. cally used in highway bridges. There are several Nondestructive Evaluation (NDE) methods for measuring stress in structures, including hole- drilling, X-ray diffraction, and magneto-elastic methods. In Keywords Shear stress . Acoustoelasticity . Ultrasonics . concrete structures, the hole-drilling method introduced by Frequency Mathar in 1933 is widely used for residual stresses. This method requires a small hole to be drilled into the material; therefore, it is considered a semi-destructive technique. The resul- Introduction tant deformations are then used to calculate the residual stresses [2]. However, hole-drilling fails to detect high level stress Increase in traffic on bridges as well as upgrades not foreseen gradients or stresses greater than one-third of yield stress. in the original design (which may add to the dead load) can Additionally, the area being measured should be easily acces- sible and the thickness should be approximately 1 to 2 hole diameters [3]. The X-Ray diffraction method is based on the * D. Ozevin [email protected] effect of stress on the atomic spacing structure of the metal specimen. The stress is calculated by measuring the strain for 1 Civil and Materials Engineering, University of Illinois at Chicago, at least two known orientations relative to the materials sur- 842 W Taylor Street ERF 2095, Chicago, IL 60607, USA face and assuming a linear distortion in the crystal lattice [4]. Exp Mech

Fig. 1 Two-dimensional stress plane and three ultrasonic measurement directions indicated with dashed arrows

This method is considered expensive and difficult to apply in method is limited since it can only be applied to ferromagnetic the field and also requires a well-trained operator [5, 6]. materials, and the measurement depth is dependent on the Magneto-elastic methods are based on the stress-magnetic re- permeability of the material. Additionally, the saturation of lationships in ferromagnetic materials. Magnetic Barkhausen the MBN signal limits the stress detection in the material [8]. Noise (MBN) is a magneto-elastic method commonly used in The acoustoelastic effect is a non-destructive method of the NDE assessment of stresses. The stress dependence of determining stresses in structures by means of ultrasonic magnetic properties of ferromagnetic materials such as coer- waves that is defined as the stress-dependence of ultrasonic civity and permeability is the basis for this technique. wave velocity and only fully expressed when the higher order Barkhausen’s demonstration of this technique in 1919 showed strain terms are considered in the constitutive equations of the that magnetizing a ferromagnetic material generates a noise in material’s stress–strain relationship. The classical theory of a neighboring coil. The amplitude of MBN signal is depen- elasticity is based on small deformations (using only second dent on residual and applied stresses. The amplitude value order elasticity constants) and neglects the squares and prod- increases if the elastic tensile stress is in the direction of mag- ucts of the strain terms; therefore, it does not provide the netization and decreases under compressive stress [7]. This necessary means to explain the ultrasonic wave-stress

Fig. 2 3D view of numerical model indicating loading direction, boundary condition, and ultrasonic excitation in the direction (a) parallel, (b) orthogonal, and (c) angled at 30° to the stress direction Exp Mech

Table 1 The material constants used in the numerical simulations [46]

Property A572 Grade 50 Steel

Density [kg/m3]7850 Young’s modulus [MPa] 200E03 Poisson’s ratio 0.33 Lame constants [MPa] λ ¼ 150E03 μ ¼ 75E03 Murnaghan constants l ¼À300E03 m ¼À620E03 n ¼À720E03 [MPa] relationship. Consequently, the classical theory has to be ex- coefficient depends on wave mode [27], frequency [28, 29], tended to include the finite deformations. Murnaghan [9]de- material texture [30], and temperature [31, 32]. veloped the hyper-elastic materials model that represents non- Other studies focus on the application of acoustoelasticity linear ultrasonic behavior of materials and introduced the third to complex loaded structures. Jassby and Saltoun [33]imple- order elasticity constants (TOE) known as l, m and n for iso- mented Rayleigh waves to measure the biaxial surface stresses tropic materials. Hughes and Kelly [10] used Murnaghan’s in 2024-T352 aluminum plates and proposed the biaxial model and calculated the third order elasticity constants for acoustoelasticity equation. Todaro and Capsimalis [29]stud- different materials and loading conditions. ied the frequency dependence of Rayleigh wave velocity un- Numerous studies have implemented the acoustoelastic der non-uniform stress fields in steel cylinders. Husson et al. method with an emphasis on uniaxial stress measurement. [34] used an electromagnetic acoustic transducer (EMAT) to For instance, acoustoelasticity has been applied to measure measure the acoustoelastic coefficients for stainless steel. stress at prestressed steel strands [11–13], longitudinal stress Vangi [35] used longitudinal waves in pulse-echo technique in rails [14, 15], and uniaxial residual stress in metals [16–18] to calculate the biaxial stress field near notches and fatigue and in welds [19–22]. Wave modes such as longitudinal [23], cracks. Duquennoy [36] determined the biaxial stress field shear [24], and surface waves [25, 26] can be used to measure along the thickness of aluminum alloy sheets by variations stress with ultrasonic wave velocity. The acoustoelasticity of Rayleigh wave velocity using wedge transducers.

Fig. 3 Frequency spectra of 9.53 mm plate using the 0.5 MHz ultrasonic frequency when the ultrasonic excitation is, (a) parallel, (b) orthogonal, and (c) angled (30°) to the stress direction Exp Mech

Fig. 4 Unwrapped phase angle corresponding to the frequency spectra of 9.53 mm plate using the 0.5 MHz ultrasonic frequency when the ultrasonic excitation is, (a) parallel, (b) orthogonal, and (c) angled (30°) to the stress direction

Akhshik et al. [37] used the Rayleigh wave method to measure Hu et al. [38] used the digital correlation method to identify residual stresses of circumferential welds in thin walled pipes. the differences in the time of flight of Rayleigh waves and

Fig. 5 Frequency spectra of 12.7 mm plate using the 0.5 MHz ultrasonic frequency when the ultrasonic excitation is, (a) parallel, (b) orthogonal, and (c) angled (30°) to the stress direction Exp Mech

Fig. 6 Unwrapped phase angle corresponding to the frequency spectra of 12.7 mm plate using the 0.5 MHz ultrasonic frequency when the ultrasonic excitation is, (a) parallel, (b) orthogonal, and (c) angled (30°) to the stress direction calculated the acoustoelastic coefficients for two directions. acoustoelastic Lamb waves in thin aluminum plates under Gandhi [39] developed the theoretical equations of biaxial loading.

Fig. 7 Frequency spectra of 9.53 mm plate using the 1 MHz ultrasonic frequency when the ultrasonic excitation is, (a) parallel, (b) orthogonal, and (c) angled (30°) to the stress direction Exp Mech

Fig. 8 Unwrapped phase angle corresponding to the frequency spectra of 9.53 mm plate using the 1 MHz ultrasonic frequency when the ultrasonic excitation is, (a) parallel, (b) orthogonal, and (c) angled (30°) to the stress direction

In contrast to the above, there are limited studies on the shear stress. Rogerson [40] and Destrade [41, 42]investigated estimation of the acoustoelastic coefficient in the presence of the effect of wave propagation in the direction of non-

Fig. 9 Frequency spectra of 12.7 mm plate using the 1 MHz ultrasonic frequency when the ultrasonic excitation is, (a) parallel, (b) orthogonal, and (c) angled (30°) to the stress direction Exp Mech

Fig. 10 Unwrapped phase angle corresponding to the frequency spectra of 12.7 mm plate using the 1 MHz ultrasonic frequency when the ultrasonic excitation is, (a) parallel, (b) orthogonal, and (c) angled (30°) to the stress direction principal stresses for incompressible materials using the steel truss bridges, where shear stress is significant. The ultra- Mooney-Rivlin material model and showed its complex na- sonic velocity and stress equation is modified in order to add ture compared to the principal stress measurement. Connor the shear effect in addition to normal stresses. The outline of [43, 44] developed the theoretical equations for the influence this paper is as follows. Numerical demonstration to extract of simple shear on the propagation of surface waves in a the acoustoelastic coefficients in a three-dimensional structur- prestressed incompressible material. Shi et al. [45] studied al geometry using Murnaghan’s hyper-elastic materials model the dependence of the acoustoelastic coefficient for lamb is described in section BNumerical Study^. The experimental waves on propagation direction and wave mode and frequen- component is presented in section BExperimental Study^, cy in a biaxially stressed aluminum plate. which consists of loading a thick plate while measuring the The present research is focused on applying the ultrasonic wave in three directions in order to simulate differ- acoustoelastic method for stress analysis to thick steel plates ent stress states (normal as well as shear stress), measure the (9–12 mm) of the type commonly used for gusset plates of acoustoelastic coefficients, and compare the results with the

Fig. 11 Relative time of flight (TOF) change with stress for 9.53 mm plate when the ultrasonic measurement is (a) parallel, (b) orthogonal, and (c) angled (30°) to the stress direction Exp Mech

Fig. 12 Relative time of flight (TOF) change with stress for 12.7 mm plate when the ultrasonic measurement is (a) parallel, (b) orthogonal, and (c) angled (30°) to the stress direction

B ^ v−v 0 0 0 numerical simulations. Section Conclusions includes the o ¼ K σ þ K σ þ K σ ð Þ 1 11 2 22 3 12 2 conclusions of this study. vo

where K3 is defined as the acoustoelastic coefficient due to Numerical Study σ0 shear stress and 12 is the shear stress in the plane of the measurement direction. Figure 1 shows the stresses at a given Acoustoelastic Equations to Measure Stress cross section for three measurement directions at three angles as 0°, 90° and 30° in a given plane. There are three unknowns The acoustoelastic equation for a biaxial stress solution is 0 0 0 σ σ σ presented by Jassby and Saltoun [33]as: ( 11, 22 and 12 ); therefore, three measurements are required to solve three equations simultaneously. v−vo ¼ K1σ11 þ K2σ22 ð1Þ vo Numerical Model where v and vo are the wave velocity in the stressed and stress- free body, K1 and K2 are acoustoelastic coefficients parallel Numerical study includes generating Rayleigh waves in a and orthogonal to the measurement direction, and σ11 and σ22 thick steel plate in order to understand the effects of the are the principal stresses in two directions. This equation can ultrasonic frequency and the plate thickness on the be solved for two material dependent coefficients, K1 and K2, acoustoelastic coefficients. COMSOL Multiphysics soft- using the analytical solution of plates or experimental mea- ware (COMSOL 4.2a) is used for numerical simulation surement. Since there are two unknowns (σ11 and σ22Þ in a in the present study, Fig. 2. The distance between the biaxial loading condition in the principal stress direction, two transmitter (S1) and the receiver (S2)is2.54cm(1in.). separate measurements are required in order to solve the The numerical model includes a three-dimensional steel stresses. As the measurement is taken in the principal stress plate, which is loaded by a two-step process. In the ini- direction, equation (1) does not include the shear stress. tial step, the structure is statically loaded at three levels When the measurement is rotated from the principal stress (0, 50 and 100 MPa) to generate a stressed medium. The direction and taken from an arbitrary direction, shear stress second step consists of applying frequency domain finite exists in addition to normal stresses. The acoustoelastic equa- element analysis (available in COMSOL) within a fre- tion should consider the multiple stress effects in the measure- quency range near perturbation frequency (i.e. central ment as: frequency of the excitation signal) when the structure is

Table 2 The Table 3 The numerically obtained Thickness 9.53 mm 12.7 mm numerically obtained Thickness 9.53 mm 12.7 mm acoustoelasticity acoustoelasticity −1 − − −1 − − coefficients (MPa )for K1 4.92E-06 9.42E-06 coefficients (MPa )for K1 3.04E-06 1.83E-06 different thicknesses K2 −3.32E-06 3.53E-06 different thicknesses K2 −3.01E-06 7.20E-06 using the 0.5 MHz K 1.44E-05 9.36E-06 using the 1 MHz K 6.76E-06 −6.95E-5 ultrasonic frequency 3 ultrasonic frequency 3 Exp Mech

used here. Piersol [47] demonstrated a method to calculate the time delay based on the phase change in the frequency spectrum. The shift in the phase angle in the frequency domain can be correlated to the time delay in the time domain data as Fig. 13 UT device as signal generator and oscilloscope as receiver ΔϕðÞ¼f 2πfΔt ð3Þ under each stress level. The direction of excitation signal is varied to obtain parallel, orthogonal, and angled mea- In order to calculated the time delay, the following equation surements with respect to uniaxial stress direction as can be used. shown in Fig. 2. The linear and nonlinear elastic proper- Δt ΔϕðÞf ties assigned to the model are summarized in Table 1. ¼ ð4Þ to 2πfd=v0 The purpose of the numerical simulation is to understand the effects of plate thickness and ultrasonic frequen- Δt where to is the relative time delay due to the presence of cy on the acoustoelastic coefficients. Two different plate stress, d is the distance between the transmitting and receiving thicknesses of 9.53 and 12.7 mm (the most common points, f is the peak frequency, and ΔϕðÞf is the shift in phase thicknesses used in steel truss highway bridges) are con- in the frequency spectrum of the signal (corresponding to the sidered. To understand the acoustoelastic coefficient and peak frequency). ultrasonic frequency relationship, two different excitation The acoustoelastic equation is divided into three separate frequencies of 0.5 and 1 MHz are considered. Higher equations to calculate three acoustoelastic coefficients, K1, K2 excitation frequency would be influenced more by the and K3: surface texture; lower excitation frequency would produce plate waves that result in thickness-dependent Δt1 ¼ K1σ11 þ K2σ22 ð5Þ acoustoelastic coefficients. The process consists of three to measurement directions with a direct surface load pertur- Δt2 ¼ K2σ11 þ K1σ22 ð6Þ bation in the direction of wave propagation. The receiving to points located at three angles of 0, 30 and 90° resemble Δt the rosette geometry. The mesh size is gradually reduced 3 ¼ K σ0 þ K σ0 þ K σ0 ð Þ t 1 11 2 22 3 12 7 near the wave propagation region until no significant o change in the result is observed. σ0 σ0 σ0 where 11, 22 and 12 are the rotated stresses according to the In order to calculate the change in wave velocity due to measurement direction and can be calculated using stress stress as shown in equation (2), time delay has to be transformation at θ, the selected angle direction, as shown in measured. There are various methods of time delay esti- Fig. 1: mation in the literature, such as direct arrival time estima- 0 σ þ σ σ −σ tion, cross correlation, and measuring phase shift in the σ ¼ 11 22 þ 11 22 cos2θ ð8Þ frequency domain. The selection of arrival time to indi- 11 2 2 rectly calculate wave velocity introduces higher error in 0 σ11 þ σ22 σ11−σ22 σ ¼ − cos2θ ð9Þ measurement due to imprecision in determining the actual 22 2 2 wave arrival time. However, the selection of phase shift 0 σ11−σ22 σ ¼ − sin2θ ð10Þ introduces less error in measurement and is accordingly 12 2

Numerical Results

The variations in the phase shift corresponding to peak frequency with different levels of stress for parallel, orthogonal, and angled directions with respect to the uniaxial stress direction are obtained for different plate thicknesses as well as different perturbation frequencies. Figure 3 and 4 show the results of frequency domain analyses for 9.53 mm thick plate and 0.5 MHz excitation Fig. 14 Test sample and dimensions frequency. Shifts in the peak frequency as well as phase Exp Mech

Fig. 15 The ultrasonic test set-up (a) sensors oriented in orthogonal to stress direction, (b) details of measurement system when sensors oriented parallel to stress direction

angle with the presence of stress are observed. Figures 5 Similar behaviors of phase angle shift with stress as the and 6 show the similar results for 12.7 mm thick plate 0.5 MHz excitation frequency are again observed. and support the same conclusion. The simulations are As Fig. 2 indicates, the plate is uniaxially stressed by repeated for the 1 MHz excitation frequency for two restraining one side of the plate and loading the other side with plate thicknesses, and presented in Figs. 7, 8, 9 and 10. a uniform stress in the x direction. When the ultrasonic

Fig. 16 Stress (y component) distribution of the experimental sample (a) parallel, (b) orthogonal, and (c) angled placement of the ultrasonic transducers with respect to stress direction Exp Mech

shear stresses exist. Once K1 and K2 are obtained and K3 is extracted using equation (7). The vertical axis of angled direc- Δtmeasured ′ ′ tion is calculated by −K1σ −K2σ such that the t0 11 22 slope of the data results in the K3 coefficient. Figures 11 and 12 show the relative TOF changes with stress for two excitation frequencies and 9.53 mm and 12.7 mm thick steel plates, respectively. Tables 2 and 3 summarize the acoustoelastic coefficients obtained using the slopes of curves shown in Figs. 11 and 12. Fig. 17 Stress (y component) change along the measurement lines for (a) parallel, (b) orthogonal, and (c) angled placement of the ultrasonic The acoustoelastic coefficients of normal stress components transducers with respect to stress direction for the same plate thickness have similar values for two frequencies. However, the acoustoelastic coefficients show significant change with the plate thickness. The plate with excitation is in the x direction (parallel to stress), equation (5) Δt 12.7 mm thickness has acoustoelastic coefficients in agree- is simplified as 1 ¼ K σ . When the ultrasonic excitation is to 1 11 ment [48] with the theory for surface waves where K1 coeffi- in y direction (orthogonal to stress), equation (6)issimplified cient has negative sign, and K2 coefficient has positive sign. Δt 2 ¼ K σ as to 2 11. For the angled direction, both normal and When the plate thickness decreases, different wave modes

Fig. 18 Waveforms recorded using four different stress states and the 0.5 MHz ultrasonic transducer parallel to the stress direction, (a) time domain histories, (b) amplitude and phase of frequency spectra Exp Mech

Fig. 19 Waveforms recorded using four different stress states and the 1 MHz ultrasonic transducer parallel to the stress direction, (a) time domain histories, (b) amplitude and phase of frequency spectra may affect the measurement and the acoustoelastic coefficient. more repeatable output signals than the hand-held device, it Therefore, it is concluded that the 1 MHz excitation frequency is utilized to record ultrasonic waves in this study. Figure 13 is more suitable to measure stresses in thick plates in order to displays the layout of the measurement technique, which con- reduce the effect of plate thickness in the acoustoelastic sists of two types of data acquisition systems, a hand-held coefficients. ultrasonic device manufactured by Mistras Group working as signal transmitter and a MSO2014 oscilloscope manufactured by Tektronix as signal receiver, which has Experimental Study 100 MHz frequency resolution. The excitation signal is a spike signal. The piezoelectric Measurement Equipment ultrasonic transducers with resonant frequencies are utilized in through-transmission mode, and the experiments are repeat- To validate the numerical results a laboratory scale test is ed for two different frequency transducers at 0.5 MHz and required. Error in the measurement chain is investigated. 1 MHz. The sensors are placed on wedges and angled at the The study shows major differences between the frequency second critical angle in order to generate Rayleigh surface spectra of a hand-held ultrasonic device and oscilloscope mea- waves. The wedges are bridged with a Plexiglas plate in order surement. Significant differences are observed when the hand- to keep the distance between transducers fixed (at 6.5 cm held ultrasonic device is utilized to record the signal [49]. On apart) while the steel plate is elongated under loading. the other hand, an oscilloscope has more consistent results in Figure 14 shows the specimen used, with the dimensions as time and frequency domains. As the oscilloscope provides 18 × 12 × 0.953 cm and the material as A572 grade 50 steel Exp Mech

Fig. 20 Waveforms recorded using four different stress states and the 1 MHz ultrasonic transducer orthogonal to the stress direction, (a) time domain histories, (b) amplitude and phase of frequency spectra plate with two 2.4 × 4 cm prongs in the middle of the width of (7) and the experimental results, the acoustoelasticity coeffi- the plate for connecting to the loading machine. The transduc- cients are obtained. ers are coupled to the steel plate using oil, and held in place using a magnetic holder. Experimental Results

In order to extract the accurate stress values at the ultrasonic Experimental Procedure measurement location a numerical model is built using COMSOL Multiphysics software (COMSOL 4.2a). The exact The testing procedure consists of an incremental increase of dimensions of the plate (including the prongs) are modeled. A tensile vertical loading from 1 to 50 kN. Ultrasonic data is simple static loading condition similar to the experiment is recorded at eleven evenly distributed load steps. Figure 15 carried out to get the stress values at the measurement loca- shows the test set-up. The ultrasonic transducers with wedges tions. Figure 16 shows the stress distribution and the locations and Plexiglas bridge as shown in Fig. 13 are oriented at three of the measurement lines, and Fig. 17 shows the stress varia- different directions (parallel, orthogonal, and angled to the tion along the measurement lines for different placements of stress direction) similar to the numerical models. The resulting the ultrasonic transducers. The average stress between the time domain waveform is converted into the frequency do- measurement points is used in order to compare the ultrasonic main using a Matlab script with a windowed FFT over the data with the stress level. Rayleigh wave arrival window to calculate the phase shift The time domain waveform is transformed into the fre- for different directions on the plate. Using equations (5)to quency domain. Figure 18 shows the time domain waveforms Exp Mech

Fig. 21 Waveforms recorded using four different stress states and the 1 MHz ultrasonic transducer angled (30°) to the stress direction, (a) time domain histories, (b) amplitude and phase of frequency spectra and their frequency spectra (amplitude and phase) at four dif- Figs. 19, 20 and 21 for three directions. It is important to note ferent load levels when the 0.5 MHz ultrasonic transducer is that the ultrasonic waves penetrate through the thickness of used parallel to the stress direction. The arrival time decreases plate for 0.5 MHz. Therefore, the wave reflections from the when the stress increases; the peak frequency increases when bottom surface interact slightly with surface waves. the stress increases as expected. Similar waveforms are ob- Using equations (5)to(7) and the experimental data, the tained for the 1 MHz excitation frequency, and shown in acoustoelasticity coefficients are calculated. Figures 22, 23

Fig. 22 Relative time of flight (TOF) change with stress using the 1 MHz Fig. 23 Relative time of flight (TOF) change with stress using the 1 MHz ultrasonic transducer parallel to stress direction ultrasonic transducer orthogonal, to stress direction Exp Mech

Fig. 24 Relative time of flight (TOF) change with stress using the 1 MHz Fig. 26 Relative time of flight (TOF) change with stress using the ultrasonic transducer angled (30°) to the stress direction 0.5 MHz ultrasonic transducer orthogonal to stress direction and 24 show the relative time of flight change with stress for frequency in the numerical study are calculated as −4.92 × the parallel, orthogonal, and angled measurements using the 10−6, −3.32 × 10−6 and 1.44 × 10−5 MPa−1. Similar to the 1 MHz sensor. Two methods are used in order to calculate the 1 MHz excitation frequency, the numerical and experimental arrival time variation with stress. In the first method (direct results are in good agreement. method) the second zero crossing after the peak amplitude in time domain waveform is used as seen in Fig. 19(a) [45]. The second method (indirect method) includes the arrival time extracted from the phase data in the frequency domain using Conclusions equation (4). As shown in the figures, both methods agree with each other. The indirect method introduces slightly better In this paper, the influences of plate thickness and excitation correlation coefficient to calculate the acoustoelastic coeffi- frequency on acoustoelastic coefficients of steel material are cients. The calculated acoustoelastic coefficients K1, K2 and numerically and experimentally investigated. The K3 for the 1 MHz excitation frequency (using the indirect acoustoelastic equation for principal stresses is modified to − − − method) are −1.14 × 10 6, −3.97 × 10 6 and 2.29 × 10 5 add the effect of shear stress into the measurement taken in −1 MPa , respectively. The acoustoelastic coefficients K1, K2 an arbitrary angle. Numerical results are obtained in the fre- and K3 for the 9.53 mm thick plate and 1 MHz perturbation quency domain, and the change in wave velocity with stress is − frequency in the numerical study are −3.04 × 10 6, −3.01 × calculated indirectly by measuring the change in phase angle. − − − 10 6 and 6.76 × 10 6 MPa 1. The numerical and experimental The numerical results show the influence of plate thickness methods are in good agreement. and excitation frequency on three acoustoelastic coefficients. The same procedure is repeated for the data obtained using When the plate thickness and the excitation frequency in- the 0.5 MHz ultrasonic transducer. Figures 25, 26 and 27 crease, the acoustoelastic coefficients are similar to the theo- show the relative TOF changes of the parallel, orthogonal, retical values of the Rayleigh wave solution. For the deeper and angled directions with respect to the stress direction. penetration depth of the ultrasonic signal, the acoustoelastic The acoustoelastic coefficients K1, K2 and K3 (using the indi- coefficients become thickness and excitation frequency de- rect method) for the 0.5 MHz excitation frequency are mea- pendent. The experimental data include 9.53 mm plate thick- − − − − sured as −6.85 × 10 6, −5.68 × 10 6 and 1.66 × 10 5 MPa 1 ness and two excitation frequencies, 0.5 MHz and 1 MHz. The respectively. The acoustoelastic coefficients K1, K2 and K3 experimentally obtained acoustoelastic coefficients are in for the 9.53 mm thick plate and 0.5 MHz perturbation good agreement with the numerical results. It is concluded

Fig. 25 Relative time of flight (TOF) change with stress using the Fig. 27 Relative time of flight (TOF) change with stress using the 0.5 MHz ultrasonic transducer parallel to stress direction 0.5 MHz ultrasonic transducer angled (30°) to the stress direction Exp Mech that the 1 MHz ultrasonic frequency is the most suitable fre- 17. Kino GS, Hunter JB, Johnson GC, Selfridge AR, Barnett DM, quency to reduce the effect of plate thickness on the Hermann G, Steele CR (1978) Acoustoelastic imaging of stress fields. J Appl Phys 50(4):2607–2613 acoustoelastic coefficients. When the ultrasonic data is mea- 18. Allen DR, Sayers CM (1984) The measurement of residual stresses sured from three directions in the kind of steel typically used in textured steel using an ultrasonic velocity combinations tech- in gusset plates in highway bridges, normal and shear stresses nique. J Ultrasonics 22:179–188 can be extracted using the acoustoelastic coefficients obtained 19. Kobayashi H, Arai Y, Ohsawa H, Nakamura H, Todoroki A (1992) Nondestructive measurement of welding residual stresses by in this study. acoustoelastic technique and prediction of fatigue crack growth. ASME J Press Vessel Technol 114:417–421 Acknowledgments This investigation was supported by National 20. Leon-Salamanca T, Bray DE (1996) Residual stress measurement Science Foundation Award No. 133552. The support from the sponsor in steel plates and welds using critically refracted longitudinal is gratefully acknowledged. We acknowledge Dr. Daniel P. Bailey for (LCR) waves. Res Nondestruct Eval 7:169184 detailed copy editing and discussion of the manuscript. Any opinions, 21. Lu H, Liu XS, Yang JG, Zhang SP, Fang HY (2008) Ultrasonic findings, and conclusions or recommendations expressed in this paper stress evaluation on welded plates with Lcr wave. Sci Technol Weld are those of the authors and do not necessarily reflect the views of the Join 13:7075 organization acknowledged above. 22. Walaszek H, Hoblos J, Bourse G, Robin C (2002) Effect of micro- structure on ultrasonic measurements of residual stresses in welded joints. In: Dias AM, Pina J, Batista AC, Diogo E (eds) ECRS 6 : References proceedings of the 6th European Conference on Residual Stresses, Coimbra, Portugal, 10-12 July, 2002. Mater Sci Forum, vol 404. Trans Tech Publications, Switzerland, pp 875–880 1. Liao M, Okazaki T, Ballarini R, Schultz A, Galambos T (2011) 23. Uzun F, Bilge AN (2011) Immersion ultrasonic technique for in- Nonlinear finite-element analysis of critical gusset plates in the I- vestigation of total welding residual stress. Procedia Eng 10: 35W bridge in Minnesota. J Struct Eng 137(1):59–68 30983103 2. Steinzig M, Ponslet E (2003) Residual stress measurement using 24. Fukuhara M, Kuwano Y, Saito K, Hirasawa T, Komura I (1998) the Hole-Drilling method and laser speckle interferometry: PART 1. Performance of nondestructive evaluation by diffracted SH ultra- Exp Tech 27(3):43–46 sonic waves in predicting degree of fatigue in cyclic bending of 3. Withers PJ, Bhadeshia HKDH (2001) Residual stress, part 1 – mea- ferritic steel. J Nondestruct Eval Int 31:211–216 surement techniques. J Mater Sci Technol 17:355–365 25. Berruti T, Gola MM (1996) Acoustoelastic determination of stress- 4. Prevey PS (1996) Current applications of X-ray diffraction residual es in steel using Rayleigh ultrasonic waves. In: Batros AL, Green stress measurement. In: Vander Voort G, Friel J (eds) Developments RE, Ruud CO (eds) Nondestructive characterization of materials in materials characterization technologies. ASM International, VII. Transtec Pub. Ltd, Lebanon, pp 171–178, Part 1 Materials Park, OH, pp 103–110 26. Sanderson RM, Shen YC (2010) Measurement of residual stress 5. Ruud CO (1982) A review of selected non-destructive methods for using laser-generated ultrasound. J Press Vessels Pip 87:762–765 residual stress measurement. NDT Int 15:15–23 27. Kim N, Hong M (2009) Measurement of axial stress using mode- 6. Fitzpatrick ME, Fry AT, Holdway P, Kandil FA, Shakleton J, converted ultrasound. J NDT & E Int 42:164–169 Souminen L (2005) Determination of residual stresses by x-ray 28. Javadi Y, Akhlaghi M, Ahmadi Najafabadi M (2003) Using finite diffraction, measurement good practice guide No 52. National element and ultrasonic method to evaluate welding longitudinal Physics Laboratory, UK residual stress through the thickness in austenitic stainless steel 7. Wang Z, Gu Y, Wang Y (2012) A review of three magnetic NDT plates. J Mater Des 45:628–642 technologies. Magn Magn Mater 324:382–388 29. Todaro ME, Capsimalis GP (1986) Acoustoelastic effect for 8. Rossini N, Dassisti M, Benyounis K, Olabi A (2012) Methods of Rayleigh surface waves in the presence of a non-uniform stress measuring residual stresses in components. Mater Des 35:572–588 field. Proc IEEE Ultrason Symp 1:229232 9. Murnaghan FD (1937) Finite deformations of an elastic solid. Am J 30. Deputat J (1990) Application of acoustoelastic effect in measure- Math 59(2):235–260 ments of residual stresses. Arch Acoust 15(1-2):69–92 10. Hughes DS, Kelly JL (1953) Second order elastic deformation of 31. Sgalla M, Vangi D (2003) A device for measuring the velocity of solids. Phys Rev 92(5):1145–1149 ultrasonic waves: an application to stress analysis. J Exp Mech 11. Lanza di Scalea F, Rizzo P, Seible F (2003) Stress measurement and 44(1):85–90 defect detection in steel strands by guided stress waves. J Mater Civ 32. Buenos AA, Pereira P Jr, Mei PR, dos Santos AA (2014) Influence Eng 15(3):219–227 of grain size on the propagation of LCR waves in low carbon steel. J 12. Chaki S, Bourse G (2009) Stress level measurement in prestressed Nondestruct Eval 33:562–570 steel strands using acoustoelastic effect. Exp Mech 49:673–681 33. Jassby K, Saltoun D (1982) Use of ultrasonic Rayleigh waves for 13. Salamone S, Bartoli I, Srivastava A, Philips R, Nucera C, Scalea the measurement of applied biaxial surface stresses in aluminum FLD (2009) Nonlinear ultrasonic guided waves for stress monitor- 2024-T351 alloy. J Mater Eval 40(2):198–205 ing in Prestressing tendons for post-tensioned concrete structures. 34. Husson D, Bennett SD, Kino GS (1982)Measurement of surface ASNT NDE/NDT for Highways and Bridges, New York stresses using Rayleigh waves. IEEE, San Diego, pp 889–892. 14. Gokhale S (2007) Determination of applied stresses in rails using doi:10.1109/ULTSYM.1982.197962 the acoustoelastic effect of ultrasonic waves. M.SC Thesis. Texas A 35. Vangi D (2001) Stress evaluation by pulse-echo ultrasonic longitu- &MUniversity dinal wave. J Exp Mech 41:277–281 15. Szelazek J (2013) Sets of piezoelectric probeheads for stress eval- 36. Duquennoy M, Ouaftouh M, Ourak M (1999) Ultrasonic evaluation uation with subsurface waves. J Nondestruct Eval 32:188–199 of stresses in orthotropic materials using Rayleigh waves. J NDT & 16. Kato M, Sato T, Kameyama K, Ninoyu H (1995) Estimation of the E Int 32:189–199 stress distribution in metals using nonlinear acoustoelasticity. J 37. Akhshik S, Moharrami R (2009) Improvement in accuracy of the Acoust Soc Am 98:1496–1504 measurement of residual stress Due to circumferential welds in thin- Exp Mech

walled pipe using Rayleigh wave method. J Nucl Eng Des 239(10): 43. Connor P, Ogden RW (1995) The effect of shear on the propagation 2201–2208 of elastic surface waves. J Eng Sci 33(7):973–982 38. Hu E, He Y, Chen Y (2009) Experimental study on the surface 44. Connor P, Ogden RW (1996) The influence of shear strain and stress measurement with Rayleigh wave detection technique. J hydrostatic stress on stability and elastic waves in a layer. J Eng Appl Acoust 70:356–360 Sci 34(4):375–397 39. Gandhi N (2010) Determination of dispersion curves for 45. Shi F, Michaels JE, Lee SJ (2012) In situ estimation of applied acoustoelastic lamb wave propagation, M.SC. Thesis. Georgia biaxial loads with lamb waves. J Acoust Soc Am 133(2):677–687 Institute of Technology University 46. Egle DM, Bray DE (1976) Measurement of acoustoelastic and 40. Rogerson GA, Sandiford KJ (1999) Harmonic wave propagation third-order elastic constants for rail steel. J Acoust Soc Am 60(3): – along a non-principal direction in a pre-stressed elastic plate. J Eng 741 744 Sci 37:1663–1691 47. Piersol AG (1981) Time delay estimation using phase data. J IEEE – 41. Destrade M, Ogden RW (2005) Surface waves in a stretched and Trans Acoust 29(3):471 477 sheared incompressible elastic material. J Non-Linear Mech 40: 48. Makhort FG, Gushcha OI, Chernoochenko AA (1990) Theory of 241–253 acoustoelasticity of Rayleigh surface waves. J Int Appl Mech 26(4): 346–350 42. Destrade M, Otténio M, Pichugin AV, Rogerson GA (2005) Non- 49. Abbasi Z, Cunningham M, Ozevin O (2015) The quantification of principal surface waves in deformed incompressible materials. J errors in the measurement of nonlinear ultrasonics. SAE Eng Sci 43:1092–1106 Conference, Michigan