applied sciences

Article Investigation of Uncertain Factors on Measuring Residual with Critically Refracted Longitudinal Waves

Shunmin Yang , Mingquan Wang * and Lu Yang

Science and Technology on Electronic Test and Measurement Laboratory, North University of China, Taiyuan 030051, China; [email protected] (S.Y.); [email protected] (L.Y.) * Correspondence: [email protected]; Tel.: +86-139-3454-8995

 Received: 12 December 2018; Accepted: 24 January 2019; Published: 31 January 2019 

Abstract: Critically refracted longitudinal (LCR) waves are commonly used to evaluate the residual stress of a material. The utilization of LCR waves is advantageous in that these waves are not sensitive to the texture of the material. Thus, LCR can be considered as a bulk longitudinal mode and can penetrate into the material well below its surface. However, while measuring the residual stress, the precision of the LCR wave travel- is influenced by several uncertain factors. In order to further improve the accuracy of test results, we developed a measurement approach based on three aspects. First, the distances between the transmitter and the receivers were determined by transducer signal analysis. Second, for the residual stress detection to be consistent, transducers with different presented similar beam angles. Finally, three different frequencies were used to evaluate the residual stress throughout the plate thickness. Based on the results of the above analysis, we used LCR waves to test 304 stainless steel plates. The detection error of residual stress measurement was ca. ±23 MPa. When compared with the X-ray diffraction approach, our method showed similar trends for the same regions of the specimens.

Keywords: critically refracted ; residual stress; uncertain factors; X-ray diffraction

1. Introduction Residual stresses are created during most manufacturing processes involving thermal treatment, , or other operations required to reshape materials or change their properties. In addition, residual stresses may exist in the raw materials before processing or may arise during in-service loading operations [1]. The properties of engineering materials and their structural components (e.g., distortion, fatigue life, corrosion resistance, dimensional stability, and brittle fracture) can be greatly influenced by residual stress [2,3]. The effects of residual stress on these properties increase the repairing/restoring cost. Thus, residual stress analysis is an indispensable design tool when working with structural elements or evaluating their reliability under actual service conditions. Residual stress evaluation methods can be destructive, semidestructive, or nondestructive [4]. The main destructive and semidestructive techniques include contour, sectioning, hole-drilling, ring-core, and deep-hole methods [5–9], among others. However, these methods are complicated, time-consuming, and will cause damage to the material. Typical nondestructive methods include neutron diffraction, X-ray, ultrasound, and magnetic-based techniques [10–12]. Neutron diffraction requires several minutes to over an hour and the measurements are costly. The X-ray method is harmful to human health and sensitive to the direction of detection. The magnetic-based method is only applied to measure residual stresses of ferromagnetic materials. The ultrasonic method is advantageous because it is portable, not expensive, harmless to operators, and provides real-time results. In recent

Appl. Sci. 2019, 9, 485; doi:10.3390/app9030485 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 485 2 of 12

Appl. Sci. 2018, 8, x FOR PEER REVIEW 2 of 12 years, ultrasound has been widely used in defect detection, residual stress measurement, and many other applications. There are numerous types of ultrasonicultrasonic testing methods for determining residual stresses in engineering materials, including ultrasonic longitudinal wave, ultrasonic shear wave (and their combination with the former), ultrasonic surfacesurface wave, ultrasonic guided wave, and ultrasonic critically refractedrefracted longitudinal longitudinal (LCR (L) waveCR) wave methods methods [13–15 ].[13–15]. The latter The method latter has method been demonstrated has been todemonstrated possess the highestto possess sensitivity the highest to strain sensitivity among to the strain ultrasonic among wavethe ultrasonic methods wave [16]. methods [16]. Many researchers havehave workedworked onon detectingdetecting residualresidual stressstress by by L LCRCR waves. Jia et al. [[17]17] analyzed the temperaturetemperature effecteffect andand foundfound thatthat thethe LLCRCR waves propagating in the stress direction is more sensitive than those propagating inin the thickness directiondirection ofof materials.materials. Habibalahi et al. [[18]18] proved that it isis feasiblefeasible toto useuse thethe neuralneural networknetwork toto improveimprove the accuracy and reliability of residual stress measurement by the ultrasonic method. The The results of Javadi et al. [[19]19] showedshowed thatthat longitudinallongitudinal residual stresses stresses were were increased increased by by using using a clamp a clamp during during the welding the welding of stainl of stainlessess steel plates. steel plates. Xu et Xual. [20] et al. proposed [20] proposed the method the method of calibration of calibration of a stress of a coefficient stress coefficient to improve to improve detection detection precision. precision. Zhan Zhanet al. et[21] al. discussed [21] discussed the effect the effect of different of different test test directions directions on on the the result result of of surface surface wave velocity σ measurement. Zhu Zhu et et al. al. [22] [22] stud studiedied the the effects effects of of initial initial stress stress ddσ0 and and microstructuremicrostructure onon thethe stress K t coefficientcoefficient Kand and flight flight time time 0 tin0 in stress-free stress-free conditions. conditions. Liu Liu et et al. al. [ 23[23]] discusseddiscussed thethe optimaloptimal step length and grain size for residual stress measurement by L waves. Ramasamy et al. [24] proved length and grain size for residual stress measurement by LCRCR waves. Ramasamy et al. [24] proved the the effectiveness of L waves by comparing the finite element simulation method with the L effectiveness of LCR CRwaves by comparing the finite element simulation method with the LCRCR experimental method. Wang et al. [25] presented the velocity variations of L waves induced by experimental method. Wang et al. [25] presented the velocity variations of LCRCR waves induced by orthotropic anisotropy andand internalinternal stresses.stresses. Among thethe aboveabove references, references, several several uncertain uncertain factors factors involved involved in measuring in measuring residual residual stress stress with L wave have been analyzed, such as temperature, neural network, clamping effect, stress coefficient, withCR LCR wave have been analyzed, such as temperature, neural network, clamping effect, stress testcoefficient, direction, test microstructure, direction, microstructure, and grain size.and gr Forain the size. purpose For the of purpose further of improving further improving the accuracy the of residual stresses evaluation by L waves, another three factors have been studied in this paper. accuracy of residual stresses evaluationCR by LCR waves, another three factors have been studied in this First,paper. the First, distances the distances between between the transmitter the transmitter and the and receivers the receivers have beenhave discussedbeen discussed from from sound sound field andfield signal and signal processing processing perspectives. perspectives. Second, Second, the the frequency and diameter and diameter of the of transducer the transducer have beenhave analyzed.been analyzed. Finally, Finally, three differentthree different frequencies frequencies of transducers of transducers were used were to used evaluate to evaluate the residual the residual stresses throughoutstresses throughout the 304 stainless the 304 stai steelnless plate steel thickness. plate thickness. 2. Theoretical Background 2. Theoretical Background Stress measurement by ultrasonic methods is based on the linear variation of the velocity of Stress measurement by ultrasonic methods is based on the linear variation of the velocity of an an ultrasonic wave with the stress. This relationship is denoted as an acoustoelastic effect within ultrasonic wave with the stress. This relationship is denoted as an acoustoelastic effect within the the elastic limit, and it provides the flight time variation of the ultrasonic wave with the stress [26]. elastic limit, and it provides the flight time variation of the ultrasonic wave with the stress [26]. As As indicated in Figure1, the L CR method uses a special longitudinal bulk wave, which propagates indicated in Figure 1, the LCR method uses a special longitudinal bulk wave, which propagates parallel to the surface and penetrates below the surface. parallel to the surface and penetrates below the surface.

Figure 1. Generation of LCR waves. PMMA: polymethyl methacrylate. Figure 1. Generation of LCR waves. PMMA: polymethyl methacrylate. Reflection or refraction occurs when an ultrasonic wave propagates from one medium to another. For example,Reflection when or refraction a longitudinal occurs when wave an is ultrason transmittedic wave from propagates a polymethyl from one methacrylate medium to (PMMA) another. For example, when a longitudinal wave is transmitted from a polymethyl methacrylate (PMMA) wedge to steel (Figure 1), shear and longitudinal waves are both created in the steel. The propagation angles of the waves follow Snell's law: Appl. Sci. 2019, 9, 485 3 of 12 wedge to steel (Figure1), shear and longitudinal waves are both created in the steel. The propagation angles of the waves follow Snell’s law:

C C C L1 = L2 = S2 (1) sin αCR sin βL sin βS where CL1 is the longitudinal velocity in the PMMA and CL2 and CS2 are the longitudinal and shear velocities in the steel, respectively. As suggested by Snell’s law, when the incident angle αCR increases ◦ ◦ ◦ from 0 to 27.8 , the refraction angle βL reaches 90 (i.e., first critical angle), and the wave is called the LCR wave. The velocity of the LCR waves propagating parallel to the load direction correlates with the strain (σ) according to the equation below:

2 ρ0V11 = λ + 2µ + (2l + λ)θ + (4m + 4λ + 10µ)σ1 (2) where ρ0 represents the initial density and the first and the second subscript of the velocity (V) indicate the propagation and polarization direction of the wave, respectively. V11 is the speed of the LCR wave traveling parallel to the applied stress; λ and µ are both second-order elastic constants (Lame constant); l and m represent the third elastic constants (Murnaghan constant); θ = σ1 + σ2 + σ3, and σ1, σ2, and σ3 are the homogeneous triaxial principal strains. When dealing with uniaxial stress conditions, we know that σ1 = ε, σ2 = σ3 = −υ × ε is the strain along the normal direction (i.e., parallel to the propagation direction), and υ is the Poisson’s ratio. With these values, Equation (2) becomes:

2λ ρ V2 = λ + 2µ + [4(λ + 2µ) + 2(µ + 2m) + νµ(1 + )]·ε (3) 0 11 µ

A list of materials and their elastic constants are given in Table1[27].

Table 1. Lame and Murnaghan constants (in GPa) for some representative materials.

Material λ 1 µ 1 l 2 m 2 n 2 Steel (0.12%) 115 82 −301 ± 37 −666 ± 6.5 −716 ± 4.5 (99%) 61 ± 1 25 −47 ± 25 −342 ± 10 −248 ± 10 Copper (99%) 104 46 −542 ± 30 −372 ± 5 401 ± 5 1 The second-order elastic constants (Lame constants). 2 The third-order elastic constants (Murnaghan constants).

The relative sensitivity measures the degree to which the velocity varies with the strain. This parameter is estimated by Equation (4), where K is a dimensionless constant for LCR waves, which can be evaluated during a uniaxial tensile test or calculated from Equation (4). For anisotropic elastomers, the independent second- and third-order elastic constant are 21 and 56, respectively. For isotropic elastomers, there are two independent second-order elastic constants (Lame constants λ and µ) and three independent third-order elastic constants (Murnaghan constants l, m, and n)[28].

2l dV11 µ + 2m + νµ(1 + /λ) /V11 = 2 + = K (4) dε λ + 2µ

As shown in Equation (4), the acoustoelastic constant K is related to the Lame constant and Murnaghan constant. From Equation (4), it can be deduced that the calculation formula of K is

−2V (3λ + 2µ) K = 0 4λ+10µ+4m 2l−3λ−10µ−4m (5) ( µ + λ+2µ )L where V0 is the velocity of the material and L is the travel distance for the ultrasonic wave. For steel, V0 = 5790 m/s and L = 37.6 mm. The value of K can be calculated from the parameters in Table1: K = 11.47. Compared with the result of 9.68 obtained by tensile test later in this paper, there are Appl. Sci. 2019, 9, 485 4 of 12 Appl. Sci. 2018, 8, x FOR PEER REVIEW 4 of 12 someThe differences. values of However, the acoustoelastic for the residual constants stress in detection other directions of in-service can components,be obtained itby is ofa similar certain referencemethod. Stress value can because be estimated the acoustoelastic by the axial constants application cannot of be the obtained stress– bystrain tensile relationship test. to elastic

The values of the acoustoelastic constants in other directions can be obtained by a similar method.dt solids. Equation (4) can be reorganized to describe the variation of stress with time-of-flight ( t0 ), as Stress can be estimated by the axial application of the stress–strain relationship to elastic solids. indicated in the following equation: Equation (4) can be reorganized to describe the variation of stress with time-of-flight ( dt ), as indicated t0 E()dV11 in the following equation: dσ ==V11 E ()dt (6) KKt0 E(dV11/V11) E dσ = = (dt/t0) (6) σ K K where d represents the stress variation, E is the , and t0 is the time whererequireddσ forrepresents the wave the to stress cover variation, a stress Efreeis thepath elastic in the modulus, studied andmaterial.t0 is the For time a given required transducer for the wave to cover a stress free path in the studied material. For a given transducer distance, the L wave distance, the LCR wave travels faster in a compressive stress field as compared to a tensile stress.CR The travelsacoustoelastic faster in constant a compressive ( K ) describes stress field how as the compared velocity toor athe tensile travel stress. time Thevaries acoustoelastic with the stress. constant (K) describes how the velocity or the travel time varies with the stress. 3. Experiment 3. Experiment

3.1. Description of the Samples The specimens studied were 304 stainless steel steel bars, bars, with with dimensions dimensions of of 390 390 mm mm ×× 6060 mm mm × ×6 6mm mm (15.35 (15.35 in in ×× 2.362.36 in in × ×0.240.24 in) in) (length (length ×× widthwidth × ×thickness).thickness). To To facilitate facilitate the the experiments, experiments, the the front surfaces were polished with a milling machine to a roughness ( R ) of 2.83 μm. The calibration sample surfaces were polished with a milling machine to a roughness (Raa) of 2.83 µm. The calibration sample underwent a heat treatment (4 h at 600 degrees Celsius,Celsius, followedfollowed byby slowslow cooling)cooling) forfor stressstress relief.relief.

3.2. Measurement Device The ultrasonic ultrasonic system system (Figure (Figure 2)2) consisted consisted of of se severalveral functional functional units: units: three three transducers transducers (one (onetransmitter transmitter (T) and (T) two and receivers two receivers (R1 and (R1 R2)), and R2)),a pulser/receiver, a pulser/receiver, and a and computer a computer (PC). (PC).The Thepulser/receiver pulser/receiver generates generates high-voltage high-voltage electrical electrical pulses. pulses. Driven Driven by by the the pu pulser,lser, the the transmitting transducer produces high ultrasonic energy with a high frequency. The sound energy is introduced and emitted through thethe 304304 stainlessstainless steelsteel inin thethe formform ofof anan L LCRCR wave.wave. The LLCRCR wavewave is is converted converted by the receivingreceiving transducerstransducers (R1 (R1 and and R2) R2) into into an electricalan electrical signal, signal, which which is shown is shown on the on computer the computer screen. Thescreen. pulse The received pulse received at R2 appears at R2 appears on the digitalon the screendigital later screen than later the than pulse the received pulse received at R1. Double at R1. receivingDouble receiving transducers transducers are advantageous are advantageous in that they in eliminatethat they environment eliminate environment effects (e.g., temperatureeffects (e.g., andtemperature coupling and conditions, coupling among conditions, others). among Time-of-flight others). wasTime-of-flight measured usingwas measured the zero-crossing using the method zero- betweencrossing themethod two receiverbetween echoes the two R1 receiver and R2. echoes This measuring R1 and R2. device This showedmeasuring a time device resolution showed of a 0.5 time ns. resolution of 0.5 ns.

FigureFigure 2. 2.Schematic Schematic of the the experimental experimental critically critically refracted refracted longitudinal longitudinal (LCR)(L waveCR) wavesystem system (T: (T:transmitting transmitting transducer, transducer, R1: R1: the the first first receiving receiving tr transducer,ansducer, R2: R2: the the second second receiving receiving transducer, transducer, and and PC: personal computer). PC: personal computer).

Three groups of transducers with different frequenciesfrequencies were used to detect residual stresses of 10 samples. This means that there there were three grou groups,ps, and each group group consiste consistedd of of three three transducers. transducers. The specimens were numbered from S1 to S10, with S1 being the reference specimen free of stresses determined in advance. Samples S2 through S10 were tested for travel time using the L technique, determined in advance. Samples S2 through S10 were tested for travel time using the LCRCR technique, and each travel time was compared to the travel time of S1. The residual stresses were evaluated using Equation (5). The parameters of the different frequency transducers are shown in Table 2. Appl. Sci. 2019, 9, 485 5 of 12

Appl. Sci. 2018, 8, x FOR PEER REVIEW 5 of 12 and each travel time was compared to the travel time of S1. The residual stresses were evaluated using Equation (5). TheTable parameters 2. Measurement of the different parameters frequency of the different transducers frequency are showntransducers. in Table 2.

Number CenterTable 2.FrequencyMeasurement f/MHz parameters Detect ofed the Depth different D/mm frequency Crystal transducers. Dimension /mm 1 5 1.28 10 Number Center Frequency f/MHz Detected Depth D/mm Crystal Dimension/mm 2 2.5 2.48 10 3 11 5 5.98 1.28 10 10 2 2.5 2.48 10 3 1 5.98 10 The results obtained with ultrasonic and X-ray diffraction methods were compared in order to validate our measurements. After the measurement of residual stress using the X-ray diffraction method,The the results sample obtained was electrochemically with ultrasonic andpolished X-ray (b diffractionecause the methodsX-ray beam were penetrates compared to ina depth order of to onlyvalidate a few our microns), measurements. and the Aftermeasurement the measurement was repeated. of residual The thickness stress using of each the etched X-ray diffractionlayer was approximatelymethod, the sample 20 μm. was The electrochemically surface was etched polished 10 , (because which enabled the X-ray us beam to reach penetrates a depth toof a200 depth μm belowof only the a fewmachined microns), surface. and the measurement was repeated. The thickness of each etched layer was approximately 20 µm. The surface was etched 10 times, which enabled us to reach a depth of 200 µm below the machined surface.Table 3. Main parameters of the X-ray stress analyzer. An X-ray stress analyzer from Proto Corporation was used herein, and the main parameters of Name Parameter the device are shown in Table3. Before the experiment, the X-ray stress analyzer was used to detect the residual stress in the referenceTube voltage bar (S1). A value of 30− kV0.78 (fixed)± 3.45 MPa was obtained, indicating that the analyzer was sufficientlyTube accurate current for our25 mA purposes. (continuously adjustable) X-ray type Ka TableTarget 3. Main parameters of the X-rayMn stress analyzer. Focusing area 4 mm × 4 mm Name Parameter An X-ray stress analyzerTube from voltage Proto Corporation was 30 kV used (fixed) herein, and the main parameters of the device are shown in TableTube 3. currentBefore the experime 25 mAnt, (continuously the X-ray stress adjustable) analyzer was used to detect X-ray type Ka the residual stress in the reference bar (S1). A value of −0.78 ± 3.45 MPa was obtained, indicating that Target Mn the analyzer was sufficientlyFocusing accurate area for our purposes. 4 mm × 4 mm

3.3. Distance between the Transmitter and the Receiver 3.3. Distance between the Transmitter and the Receiver A wave propagates from the transmitter to the base surface (the 1st leg). The wave is A wave propagates from the transmitter to the base surface (the 1st leg). The wave is subsequently subsequently reflected back, continuing its path until reaching the surface wall (the 2nd leg), as reflected back, continuing its path until reaching the surface wall (the 2nd leg), as shown in Figure3. shown in Figure 3. The 1st and 2nd legs generate a sound path ( SP ). The skip distance ( SKD ) is a The 1st and 2nd legs generate a sound path (SP). The skip distance (SKD) is a term that measures the term that measures the distance between the point of excitation (beginning of SP ) and the end of the distance between the point of excitation (beginning of SP) and the end of the second leg (end of SP). second leg (end of SP ).

Figure 3. Distance between the transmitter and the receiver (β : the refraction angle, l: the distance Figure 3. Distance between the transmitter and the receiver S( β : the refraction angle, l : the between transmitting transducer and the first receiving transducer,S and D: the plate thickness). distance between transmitting transducer and the first receiving transducer, and D : the plate According to Snell’s law, the refraction anglethickness).βS is determined by the following equation:

According to Snell's law, the refractionCS2 angle3100 β is determined by ◦the following equation: sin βS = = = 0.5354S ⇒ βS = 32.4 (7) CL2 5790 C  sinββ==S 2 3100 = 0.5354 = 32.4 (7) SSCL 2 5790

The depth of the 304 stainless steel specimens ( D ) is 6 mm. The following equation is adapted: Appl. Sci. 2019, 9, 485 6 of 12

Appl.The Sci. depth 2018, of8, xthe FOR304 PEER stainless REVIEW steel specimens (D) is 6 mm. The following equation is adapted:6 of 12

◦ = × × = × × ( ) = SKD 2=×D ×tan ββS =××2 6 tan 32.4 = 7.62mm (8) SKD2 D tanS 2 6 tan(32.4 ) 7.62 mm (8) The length of the surface distance (SD) is 3.81 mm, which is half that of SKD. The segments L , The length of the surface distance ( SD ) is 3.81 mm, which is half that of SKD . The segments L1 1 L2, and, LL, 3andof theL SPof thein FigureSP in3 Figure correspond 3 correspond to the to 1st, the 2nd, 1st, 2nd, and 3rdand legs,3rd legs, respectively. respectively.SKD SKD, SD, SD, L1, L2, 2 3 ◦ and L3 can be found using trigonometric functions by setting βS as equalβ to 32.4 . The 1st leg can be , L1 , L2 , and L3 can be found using trigonometric functions by setting S as equal to 32.4 . The calculated as follows: 1st leg can be calculated as follows:SD 3.81 3.81 L1 = = = = 7.12mm (9) sin==β SD sin(3.8132.4◦ =) 3.810.5354 = L S β  7.12mm 1 sinS sin(32.4 ) 0.5354 (9) The distance between the transmitter and receiver (l) depends on the first leg (L1) and SKD. The The distance between the transmitter and receiver ( l ) depends on the first leg ( L ) and SKD . received waves for different transmitter–receiver distances are displayed in Figure4.1 The number of The received waves for different transmitter–receiver distances are displayed in Figure 4. The number received waves is proportional to the transmitter–receiver distance. This can be explained by Snell’s of received waves is proportional to the transmitter–receiver distance. This can be explained by law; thus, when the refracted shear waves reflect from the back wall to the front wall, numerous Snell's law; thus, when the refracted shear waves reflect from the back wall to the front wall, reflected shear waves turn into longitudinal waves. However, the amplitude of the L wave is numerous reflected shear waves turn into longitudinal waves. However, the amplitude of CRthe LCR inverselywave relatedis inversely to therelated transmitter–receiver to the transmitter–receiver distance, distance, since asince larger a larger distance distance attenuates attenuates the the sonic energy.sonic Therefore, energy. Therefore, the distance the distance between between the transmitting the transmitting probe probe and and the the receiving receiving probe should should be as largebe as as large possible as possible to avoid to avoid integer integer multiples multiples of SKD of SKD, so, as so to as achieve to achieve accurate accurate identification identification ofof LCR waves.LCR In waves. thispaper, In this 37.6paper, mm 37.6 was mm selected was selected as the as distancethe distance from from the the transmitter transmitter to to the the first first receiverreceiver and 7.6 mmand was7.6 mm selected was selected as the distanceas the distance between between two receivers.two receivers.

= =+1 (a) lL01 (b) lL112 SKD

=+ =+3 (c) lLSKD21 (d) lL312 SKD

=+ =+5 (e) lL412 SKD (f) lL512 SKD

=+ =+7 (g) lL613 SKD (h) lL712 SKD

=+ =+9 (i) lL814 SKD (j) lL912 SKD

Figure 4. Received waves for different distances between the transmitter and the receiver (test conditions: frequency of transducers: 5 MHz, diameter of transducers: 6 mm. l: the distance between

transmitting transducer and the first receiving transducer, L1 the first leg, SKD: the skip distance). Appl. Sci. 2019, 9, 485 7 of 12

3.4. Frequency and Diameter of the Transducer Appl. Sci.The 2018 beam, 8, x angleFOR PEER plays REVIEW an important role in the selection of transducers for several reasons.7 First, of 12 because of the low concentration of the sound field, the propagation of the beam reduces the amplitude Figure 4. Received waves for different distances between the transmitter and the receiver (test of the reflection. Second, the light beam’s spreading may hinder the interpretation of the signal due to conditions: frequency of transducers: 5 MHz, diameter of transducers: 6 mm. l : the distance between other features outside the reflective or examination region fromL the side ofSKD the test subject. Therefore, transmitting transducer and the first receiving transducer, 1 : the first leg, : the skip distance). representing the sound field generated by the transducer is a prerequisite for understanding the observed signal. 3.4. Frequency and Diameter of the Transducer The beam spread depends to a large extent on the frequency and diameter of the transducer. The largerThe beam the angle beam plays angle, an the important poorer the role directivity in the selection and the of transducers more dispersive for several the acoustic reasons. energy. First, becauseThe directivity of the oflow the concentration sound field can of be the improved sound fi byeld, decreasing the propagation frequency of and the increasingbeam reduces diameter. the amplitudeThe specific of relationship the reflection. is as Second, follows: the light beam’s spreading may hinder the interpretation of the signal due to other features outside the reflective or examination region from the side of the test V subject. Therefore, representing the soundsin fieldθ = 1.2generated× by the transducer is a prerequisite (10)for understanding the observed signal. D f whereTheθ is beam the angle spread between depends the to tangent a large ofextent the main on th lobee frequency beam and and the diameter adjacent of secondary the transducer. lobe beam The largerand the the axis beam of the angle, main the beam, poorerV is the the directivity sound velocity and the in themore material dispersive (inch/sec the acoustic or m/sec), energy.D is The the directivitydiameter of of the the transducer sound field (inch can orbe mm),improved and fbyis decreasing the frequency frequency of the transducer and increasing (Hz). diameter. The specificFigure relationship5a–c shows is as the follows: received waves for a frequency of 2.5 MHz and transducer diameters of 6 mm, 8 mm, and 10 mm, respectively. Theθ beam=× divergenceV angle of each sample was calculated sin 1.2 Df ◦ ◦ (10)◦ according to Equation (9), and the respective results were θa = 27.6 , θb = 20.3 , and θc = 16.1 . The waveformwhere θ is shown the angle in Figure between5c was the clearlytangent observed of the main and lobe possessed beam and a clear the outline.adjacent Therefore,secondary lobefor the beam 2.5 and MHz the transducer, axis of the themain optimum beam, V diameter is the sound was 10 velocity mm. This in the means material that (inch/sec the suitable or m/sec), beam ◦ divergenceD is the diameter angle of of the the 2.5 transducer MHz transducer (inch or was mm), ca. and 16 . f is the frequency of the transducer (Hz).

(a) The diameter of transducer is 6 mm (b) The diameter of transducer is 8 mm

(c) The diameter of transducer is 10 mm (d) The frequency of transducer is 1 MHz

(e) The frequency of transducer is 2.5 MHz (f) The frequency of transducer is 5 MHz

FigureFigure 5. 5. ReceivedReceived waves waves for for various various diameters diameters and frequencies of the transducer.

FigureThe signals 5a–c shownshows inthe Figure received5d–f waves were obtainedfor a frequency for a diameter of 2.5 MHz of 6 mmand andtransducer varying diameters frequencies of 6(1, mm, 2.5, 8 and mm, 5 MHz,and 10 respectively). mm, respectively. Similarly, The beam since sindivergenceθd > 1, θangled does of not each exist, sample in other was words,calculated the θ =  θ =  θ = ◦ accordingenergy of to this Equation signal scattered,(9), and the and respective the signals results cannot were be a interpreted.27.6 , b θ20.3e and, andθ f were c 16.1 27.6 . andThe ◦ 13.4waveform, respectively. shown in For Figure a given 5c was transducer clearly observed diameter ofand 6 mm,possessed only thea clear waves outline. received Therefore, by the 5for MHz the 2.5transducer MHz transducer, were available. the optimum Thus, for diameter the 6 mm was transducer, 10 mm. theThis optimum means that frequency the suitable was 5 beam MHz. ◦ divergenceTherefore, the angle proper of the beam 2.5 MHz divergence transducer angle was of the ca. 616 mm . transducer was approximately 13 . The signals shown in Figure 5d–f were obtained for a diameter of 6 mm and varying frequencies θ > θ (1, 2.5, and 5 MHz, respectively). Similarly, since sind 1, d does not exist, in other words, the θ θ  energy of this signal scattered, and the signals cannot be interpreted. e and f were 27.6 and 13.4 , respectively. For a given transducer diameter of 6 mm, only the waves received by the 5 MHz Appl. Sci. 2018, 8, x FOR PEER REVIEW 8 of 12

Figure 5a–c shows the received waves for a frequency of 2.5 MHz and transducer diameters of 6 mm, 8 mm, and 10 mm, respectively. The beam divergence angle of each sample was calculated θ =  θ =  θ =  according to Equation (9), and the respective results were a 27.6 , b 20.3 , and c 16.1 . The waveform shown in Figure 5c was clearly observed and possessed a clear outline. Therefore, for the 2.5 MHz transducer, the optimum diameter was 10 mm. This means that the suitable beam divergence angle of the 2.5 MHz transducer was ca. 16 . The signals shown in Figure 5d–f were obtained for a diameter of 6 mm and varying frequencies Appl. Sci. 2019, 9, 485 θ > θ 8 of 12 (1, 2.5, and 5 MHz, respectively). Similarly, since sind 1, d does not exist, in other words, the θ θ  energy of this signal scattered, and the signals cannot be interpreted. e and f were 27.6 and 13.4According, respectively. to the For above a given analysis, transducer in order diameter to ensure of 6 themm, consistency only the waves of residual received stress by the detection, 5 MHz transducerstransducer withwere differentavailable. frequencies Thus, for shouldthe 6 mm have transducer, similar beam the divergence optimum angles.frequency In this was paper, 5 MHz. we selectedTherefore, 16 ◦theas proper the beam beam divergence divergence angle. angle This of meansthe 6 mm that transducer the diameter was of approximately the 1 MHz transducer 13°. is 25 mm,According while the to diameters the above of analysis, 2.5 and 5in MHz order transducers to ensure the can consistency be 10 and 5 of mm, residual respectively. stress detection, transducers with different frequencies should have similar beam divergence angles. In this paper, we 3.5. Evaluation of the Calibration Constants selected 16 as the beam divergence angle. This means that the diameter of the 1 MHz transducer is 25 mm,The while calibration the diameters constants of must2.5 and be 5 evaluated MHz transducers before measuring can be 10 and the residual5 mm, respectively. stress. Calibration can be performed in a lab by using a tension–compression testing machine providing standard stress values.3.5. Evaluation All calibration of the Calibration samples have Constants been stress-relieved (the oven is heated at 600 degrees Celsius for 4 h, thenThe cooledcalibration naturally constants in the must furnace). be evaluated before measuring the residual stress. Calibration can beThe performed value of t 0inwas a lab determined by using a directly tension–compression from the stress-free testing samples, machine while providingK was experimentallystandard stress derivedvalues. fromAll calibration a uniaxial samples tensile test have involving been stress-relieved ultrasonic measurements. (the oven is heated As described at 600 bydegrees Equation Celsius (6), Kforis 4 the h, slopethen cooled of the relativenaturally variation in the furnace). curve of the time of flight:

The value of t0 was determined directly from the stress-free samples, while K was 1 t − t0 experimentally derived from a uniaxial tensileK = − test (involving) ultrasonic measurements. As described(11) dσ t0 by Equation (6), K is the slope of the relative variation curve of the time of flight: where t and t0 are the time of flight estimated between− the two receivers for stressed and unstressed =− 1 tt0 K σ () (11) samples, respectively, and dσ represents the applieddt stress.0 The relationship between the tensile and the transit time is demonstrated in Figure6. t t Axialwhere loadings and of0 upare tothe 40% time of of the flight elastic estimated limit were between applied the using two areceivers tensile testingfor stressed machine. and Theunstressed tensile testingsamples, machine respectively, load was and controlled dσ represents in steps the of applied 20 MPa stress. at a rate The of relationship 2 MPa/s. The between time delaythe tensile (10 min) and the between transit two time steps is demonstrated was required in Figure for installing 6. the transducer and recording the ultrasonic waveforms.

FigureFigure 6.6. TensionTension versusversus transittransit time.time. (Test(Test conditions: conditions: frequencyfrequencyof oftransducers: transducers: 55 MHz,MHz, thethe distancedistance betweenbetween transmittingtransmitting transducertransducer andand thethe firstfirstreveiving reveiving transducer: transducer:37.6 37.6mm, mm,specimen: specimen: 304304 stainlessstainless K steel,steel, environmentenvironment temperature:26 temperature:26 degrees degrees Celsius, Celsius, :K the: the acoustoelastic acoustoelastic constant.) constant.)

3.6. Determination of the LCR Penetration Depth Axial loadings of up to 40% of the elastic limit were applied using a tensile testing machine. The tensileIn testing our experiments, machine load the was time controlled of flight wasin steps measured of 20 MPa for at 10 a samples.rate of 2 MPa/s. The coupling The time fluid delay was (10 reloaded after each measure to recreate the same test conditions. When the LCR technique is used over a limited wall thickness, the LCR wave penetration depth varies as a function of the frequency. Since no relation exists between the LCR depth and the frequency, an experimental measurement should be made. The residual stress of the whole plate thickness is evaluated with three different frequencies. Therefore, the depth should be measured accurately for all frequencies. Three transducers with the same frequency, each acting as a transmitter or receiver, were employed to produce the LCR wave. A slot is dug between the two probes with a milling tools to cut off the LCR wave propagation in the Appl. Sci. 2018, 8, x FOR PEER REVIEW 9 of 12 min) between two steps was required for installing the transducer and recording the ultrasonic waveforms.

Appl. Sci. 2019, 9, 485 9 of 12 3.6. Determination of the LCR Penetration Depth In our experiments, the time of flight was measured for 10 samples. The coupling fluid was detected object. The slot depth was increased in steps and the LCR wave amplitude was measured for reloaded after each measure to recreate the same test conditions. When the LCR technique is used over each step. When the amplitude of the LCR wave was equal to the noise, the milling procedure was a limited wall thickness, the LCR wave penetration depth varies as a function of the frequency. Since stopped. The slot depths correspond to the penetration depths of the LCR wave for the optimum test no relation exists between the LCR depth and the frequency, an experimental measurement should be frequency. The slot depths were 5.98, 2.48, and 1.28 mm for transducers with nominal frequencies of 1, made. 2.5, and 5 MHz, respectively. The residual stress of the whole plate thickness is evaluated with three different frequencies. 4.Therefore, Results andthe depth Discussion should be measured accurately for all frequencies. Three transducers with the same frequency, each acting as a transmitter or receiver, were employed to produce the LCR wave. A slot isResidual dug between stress wasthe two measured probes for wi eachth a specimenmilling tools using to threecut off groups the LCR of transducerswave propagation and an in X-ray the stressdetected analyzer, object. The with slot the depth detection was areaincreased being in divided steps and into the 10 L equalCR wave blocks. amplitude The meanwas measured value of thefor residualeach step. stress When was the obtained amplitude from of 10 the different LCR wave blocks was for equal each to sample. the noise, Figure the7 millinga shows procedure the L CR wave was resultsstopped. for The the slot 1, 2.5, depths and 5 correspond MHz transducers, to the penetration respectively, depths and a comparisonof the LCR wave with for the the X-ray optimum diffraction test datafrequency. for different The slot specimens depths were is also 5.98, given. 2.48, As and shown 1.28 inmm Figure for transducers7b, the difference with nominal between frequencies the X-ray and of ultrasonic1, 2.5, and methods5 MHz, respectively. at different frequencies was more evident when normalized values were used to display residual stress measurements. 4. ResultsThe largest and Discussion deviation of the ultrasonic and X-ray results for the 1 MHz frequency was 123.24 MPa (sample S2). This large deviation resulted from the different thickness of the layers for the ultrasonic Residual stress was measured for each specimen using three groups of transducers and an X-ray and X-ray diffraction methods (ca. 5.98 mm (0.23 in) and 200 µm (0.0078 in)). The deviation of the stress analyzer, with the detection area being divided into 10 equal blocks. The mean value of the ultrasonic and X-ray results for the 2.5 and 5 MHz frequencies were 90.50 and 54.50 MPa, respectively. residual stress was obtained from 10 different blocks for each sample. Figure 7a shows the LCR wave This suggests that the X-ray diffraction is a surface method, which penetrates no more than 1 mm. results for the 1, 2.5, and 5 MHz transducers, respectively, and a comparison with the X-ray The average deviations of all the specimens were 39.90, 56.08, and 79.01 MPa for the tests involving 5, diffraction data for different specimens is also given. As shown in Figure 7b, the difference between 2.5, and 1 MHz transducers, respectively. These results suggested that the L wave provided the mean the X-ray and ultrasonic methods at different frequencies was more evidentCR when normalized values residual stresses within a certain transducer penetration depth. Therefore, where the residual stress is were used to display residual stress measurements. to be measured at an exact distance from the surface, the ultrasonic method is not the ideal option. /MPa residual stress d stress residual normlized value(residual stress)

(a) (b)

FigureFigure 7. 7.Comparison Comparison betweenbetween X-rayX-ray andand LLCRCR waves for for different different specimens. specimens. (a) (a) The The residual residual stress stress for differentfor different specimens; specimens; (b) The (b) normalizedThe normalized value value of residual of residual stress stress for different for different specimens. specimens.

ForThe thelargest S2 specimen, deviation theof the residual ultrasonic stresses and measured X-ray results by the for 5, the 2.5, 1 andMHz 1 frequency MHz transducers was 123.24 are shownMPa (sample in Figure S2).8. TheThis deviations large deviation of the 10resulted different from specimens the different were ±thickness23, ±19, andof the± 13layers MPa for for the testsultrasonic involving and X-ray 5, 2.5, diffraction and 1 MHz methods transducers, (ca. 5.98 respectively. mm (0.23 The in) and samples 200 µm with (0.0078 different in)). test The frequencies deviation wereof the compared ultrasonic to and show X-ray that results the LCR forwave the 2.5 method and 5 is MHz more frequencies accurate at lowwere frequencies. 90.50 and 54.50 Therefore, MPa, therespectively. deviation This of the suggests 5 MHz that LCR thewave X-ray is higherdiffractio thann is the a surface deviation method, of the which 2.5 MHz penetrates LCR wave, no more and thethan minimum 1 mm. The deviation average deviations happens for of all the the 1 MHz specimens LCR wave. were 39.90, The higher 56.08, and errors 79.01 obtained MPa for for the higher tests frequenciesinvolving 5, can2.5, be and explained 1 MHz bytransducers, the low frequency respectively. transducers, These results which suggested create echoes that inthe the LCR receiver wave thatprovided are sharper the mean and residual more powerful stresses within than the a cert highain frequency transducer transducers. penetration Therefore,depth. Therefore, it is desirable where to increase the “gain” by using higher frequency transducers to avoid low resolutions and higher time-of-flight errors. Appl. Sci. 2018, 8, x FOR PEER REVIEW 10 of 12 the residual stress is to be measured at an exact distance from the surface, the ultrasonic method is Appl. Sci. 2019, 9, 485 10 of 12 not the ideal option. /MPa residual stress d stress residual

Figure 8. Residual stress measured at 5 MHz, 2.5 MHz, and 1 MHz for the S2 specimen (the detection Figure 8. Residual stress measured at 5 MHz, 2.5 MHz, and 1 MHz for the S2 specimen (the area of samples had been divided into 10 equal blocks, and the mean value of the residual stress was detection area of samples had been divided into 10 equal blocks, and the mean value of the residual obtained from 10 different blocks for each sample). stress was obtained from 10 different blocks for each sample). 5. Conclusions For the S2 specimen, the residual stresses measured by the 5, 2.5, and 1 MHz transducers are The uncertain factors involved in measuring residual stress with L waves, such as the distance shown in Figure 8. The deviations of the 10 different specimens wereCR ±23, ±19, and ±13 MPa for the between the transmitter and receiver and the diameter and frequency of the transducer, have been tests involving 5, 2.5, and 1 MHz transducers, respectively. The samples with different test investigated. The distance between the transmitting probe and the receiving probe should be as large frequencies were compared to show that the LCR wave method is more accurate at low frequencies. as possible to avoid integer multiples of SKD, so as to achieve accurate identification of LCR waves. Therefore, the deviation of the 5 MHz LCR wave is higher than the deviation of the 2.5 MHz LCR wave, In order to ensure the consistency of the results of residual stress detection, probes with different and the minimum deviation happens for the 1 MHz LCR wave. The higher errors obtained for higher frequencies and diameters should have the same beam divergence angle. On the basis of obtaining the frequencies can be explained by the low frequency transducers, which create echoes in the receiver acoustoelastic constants with a tensile testing machine and the stress-free sample by annealing process, that are sharper and more powerful than the high frequency transducers. Therefore, it is desirable to the residual stresses of 10 different steel bars have been measured with L waves. The results were increase the "gain" by using higher frequency transducers to avoid low resolutionsCR and higher time- consistent with X-ray diffraction, and the detection error was ca. ±23 MPa. It is an important reference of-flight errors. for further research on the residual stress measurement with LCR waves. 5. Conclusions Author Contributions: Conceptualization, S.Y. and M.W.; Methodology, S.Y.; Software, S.Y. and L.Y.; Validation, L.Y., and S.Y.; Formal Analysis, M.W.; Investigation, L.Y.; Resources, M.W.; Data Curation, L.Y.; Writing—Original The uncertain factors involved in measuring residual stress with LCR waves, such as the distance Draft Preparation, S.Y.; Writing—Review & Editing, S.Y. and M.W.; Visualization, L.Y.; Supervision, M.W.; Project Administration,between the transmitter S.Y. and M.W.; and Funding receiver Acquisition, and the diamet M.W. er and frequency of the transducer, have been investigated. The distance between the transmitting probe and the receiving probe should be as large Funding: This work was supported by the National Key Scientific Instrument and Equipment Development Projectsas possible of China to avoid (Grant integer No. 2013YQ240803), multiples of and SKD the, International so as to achieve Science accurate & Technology identification Cooperation of L ProgramCR waves. of Shanxi,In order China to ensure (Grant No.the 201803D421032).consistency of the results of residual stress detection, probes with different Conflictsfrequencies of Interest: and diametersThe authors should declare have no conflictthe same of interest. beam divergence angle. On the basis of obtaining the acoustoelastic constants with a tensile testing machine and the stress-free sample by annealing Referencesprocess, the residual stresses of 10 different steel bars have been measured with LCR waves. The results were consistent with X-ray diffraction, and the detection error was ca. ±23 MPa. It is an 1. Schajer, G.S. Practical Residual Stress Measurement Methods; John Wiley & Sons Ltd: Chichester, West Sussex, important reference for further research on the residual stress measurement with LCR waves. UK, 2013. 2.AuthorTotten, Contributions: G.E.; Howes, Conceptualization, M.; Inoue, T. Handbook S.Y. ofand Residual M.W.; Stress Method and Deformationology, S.Y.; of Software, Steel; ASM S.Y. International: and L.Y.; Validation,Geauga L.Y., County, and OH,S.Y.; USA, Formal 2002. Analysis, M.W.; Investigation, L.Y.; Resources, M.W.; Data Curation, L.Y.; 3.Writing—OriginalHuang, X.; Sun, Draft J.; Li,Preparation, J. Effect of S.Y.; initial Writing—Review residual stress and& Editing, machining-induced S.Y. and M.W.; residual Visualization, stress on L.Y.; the Supervision,deformation M.W.; of Project aluminium Administration, alloy plate. StrojniškiS.Y. and M.W.; Vestnik-J. Funding Mech. Acquisition, Eng. 2015, 61 M.W., 131–137. 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