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Experimental study of thermal degradation in ferritic Cr–Ni alloy steel plates using nonlinear Lamb waves
NDT&E International 44 (2011) 768–774
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Experimental study of thermal degradation in ferritic Cr–Ni alloy steel plates using nonlinear Lamb waves Yanxun Xiang a, Mingxi Deng b, Fu-Zhen Xuan a,n, Chang-Jun Liu a a Key Laboratory of Safety Science of Pressurized System of MOE, School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai 200237, China b Department of Physics, Logistics Engineering University, Chongqing 400016, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 6 September 2010 Received in revised form 27 July 2011 Accepted 5 August 2011 Available online 22 August 2011
The thermal degradation in ferritic Cr–Ni alloy steel plates is measured using the nonlinear effect of Lamb wave propagation. Experiments were carried out to introduce controlled levels of thermal damage to determine the nonlinear response of Lamb waves. A ‘‘mountain-shape’’ change in the normalized acoustic nonlinearity of Lamb wave versus the level of thermal degradation in the specimens has been observed. The variation in the measured acoustic nonlinearity reveals, based on metallographic studies, that the normalized acoustic nonlinearity increases due to the second phase precipitates in the early stage and it decreases as a combined result of dislocation change and microvoid initiation in the material. The results show a potential application of the nonlinear Lamb waves for the quantitative assessment of thermal damage in metallic plates or pipes. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Ultrasonic nonlinearity Lamb wave Thermal degradation Ferritic Cr–Ni steel Dislocations
1. Introduction Heat-resistant steels have been widely applied in high temperature components of petrochemical industry and power plants due to their excellent high-temperature strength and creep resistance. However, when the heat-resistant steels are used for long periods at elevated temperature in practice, they generally experience degradation of thermal damage or creep damage, which would shorten the lifespan of components and would bring about serious disaster due to the unforeseen fracture. Therefore, it is of increasing importance to evaluate such degradation of strength of materials at early stage for structural health monitoring (SHM). In order to conduct a nondestructive evaluation (NDE) on the state of structural material damages, ultrasonic method is often used for its capability of evaluating both the surface and internal damage state of in-service components with the comparatively simple and easy instrumentation. During thermal damage, the formation of various precipitates in heat-resistant steel would cause local strain concentrations and therefore would produce different dislocations concentrated near the precipitates [1]. The volume fraction of the precipitates has a direct influence on the changes of dislocation density and average length. Moreover, the dislocation pileups caused by strain concentration or residual stress would make the precipitates
n
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break off from the matrix and then leads to formation of microvoids. All these microstructure changes during thermal degradation would produce a very clear nonlinear response in material [2,3]. For early detection of damage even before the first crack initiation, there are some ultrasonic techniques used for measuring material degradation, such as the combination of nonlinear elastic wave spectroscopy (NEWS) with time reversal (TR) [4], the measurement of acoustic wave parametric interaction [5] and the higher harmonic generation method [6–11]. In recent years, the higher harmonic generation method has been focused as a potential tool for NDE of the material property degradation [6–11]. This technique relies on measuring the higher order harmonics generated by the intrinsic nonlinearity of materials relating to the anharmonicity and imperfection of atomic lattices. The classical theory of nonlinear wave propagation in elastic solids has been discussed and presented for nearly half a century [12]. Most previous researches have measured the acoustic nonlinearity parameter b using longitudinal waves, which is a function of the second- and third-order elastic constants, and have indicated that b is sensitive to subtle damage in materials and can be correlated with certain microstructural changes during fatigue damage [8,13], thermal damage [3,7], or creep damage [10,11]. Compared with the nonlinear ultrasonic techniques that use bulk acoustic waves, ultrasonic guided waves such as Lamb waves have obvious advantages, such as one-side access of pitch-catch configuration and high efficiency in a long distance propagation, for using as a more suitable candidate to interrogate large plate- and shell-like structures in the field applications.
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Due to the dispersive and multi-mode nature of Lamb wave, recent studies revealed that the generation of cumulative second harmonic of Lamb wave propagation in materials does occur once some conditions are satisfied [14–16]. Deng [14,15] and de Lima and Hamilton [16] described these conditions and experimentally validated the cumulative effect of second harmonics of Lamb wave propagation in an elastic plate [17]. Srivastava et al. [18] performed a theoretical studies on the symmetry characteristics of Rayleigh-Lamb waves in nonlinear isotropic plates, as well as on the higher-order (than double) harmonics of Lamb waves. Bermes et al. [19] and Pruell et al. [20,21] developed a procedure to measure the second harmonic of a Lamb wave by combining a time–frequency process with a hybrid wedge generation and laser interferometric detection system. Besides satisfying two conditions for Lamb wave propagation, i.e., phase velocity matching and nonzero power flux, Jacobs’ group proposed another condition [21,22], namely, the group velocity matching. Most of the past works relating to nonlinear Lamb waves have focused on the plastic deformation [21] and fatigue damage [20,23] in simple metals, and their results also manifested a contradictive tendency (i.e., monotonous increase or decrease) on the variation of the cumulative second harmonic of Lamb wave propagating in the damaged materials. Moreover, there is no detailed explanation on these different experimental results in the aluminum plate materials during the process of fatigue damage. In contrast to the studies of fatigue damage assessments in the simple metals via nonlinear Lamb waves, the current paper deals with the characterization of thermal damage in structure materials (such as HP40Nb material, ASTM Grade, a kind of ferritic Cr–Ni alloy steel) using the nonlinear effect of ultrasonic Lamb waves. Measurements of the second harmonic of Lamb waves have been carried out on the HP40Nb specimens with different thermal damage levels. The influence of microstructure evolution on the nonlinear Lamb wave propagation has also been thoroughly analyzed based on metallographic studies, such as scanning electron microscope (SEM) and transmission electron microscope (TEM). Moreover, an analytical model calculation of precipitate– dislocation interaction has been performed to interpret the correlation between microstructural evolutions and the nonlinear Lamb wave measurements.
2. Determination of nonlinear Lamb waves propagation in an elastic plate For a solid plate, the coordinate origin is located in its central plane, and the oy axis is along the plate thickness and the oz axis is along the direction of wave propagation. Within second-order perturbation, the general solution of the cumulative second harmonic by Lamb wave propagation, generated by a line source parallel to the ox axis, can be expressed as [14,24,25] 8 ðDLÞ 9 q 2
2 sinðDjÞ X
Dj
ðFCÞ UTq
q¼1
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T or L in the subscript means that the corresponding quantity is associated with the transverse or longitudinal wave, respectively. ðFCÞ ðFCÞ ðDLÞ ULq and UTq are proportional to ULq [14,24], x^ is the unit vector along the ox axis, yL or yT is the angle between the oy axis and the KLq or KTq (see Fig. 2 in Ref. [25]), K^ Lq and K^ Tq are the unit vectors of the partial wave vectors KLq and KTq. cL is the longitudinal wave velocity, f is the frequency of primary Lamb wave propagation, k, m, A, B, and C are second- and third-order elastic constants of the solid specimen with the thickness of 2d. c(f,l) is phase velocity of the primary Lamb wave l-mode and c(2f,n) is that of the DFLW n-mode. ðf ,lÞ Zdis is the propagation distance. ULq represents the longitudinal component of the primary Lamb wave of lth mode. From the previous studies [14,15,25], it can be known that if the phase velocity of a DFLW component is exactly or approximately equivalent to that of the primary Lamb wave, the secondharmonic amplitude of Lamb wave propagation increases clearly with propagation distance or within a certain distance. Under the condition of Dj ¼0 or Dj E0, the analytic solution of the cumulative second harmonic of Lamb wave propagation in an elastic plate is expressed in Ref. [14]. As in previous works [8–13], the ratio of A2 =A21 is used to represent the normalized acoustic nonlinearity and is proportional to the acoustic nonlinearity parameter b (e.g., b ¼ 8A2 =ðk2 Zdis A21 Þ for longitudinal waves), where A1 and A2 are the measured amplitudes of the fundamental and the second-harmonic waves, respectively, and k is the wave number. Based on Eqs. (1) and (2), we can formally define the acoustic nonlinearity of Lamb waves as follows
bLamb ¼ FðyT , yL ,KLq ,KTq ,r,E2 ,E3 ÞA2f =ðk2 Zdis A21f Þ
ð3Þ
where A1f and A2f are, respectively, the measured amplitudes of the fundamental and the second-harmonic signals of Lamb wave propagation on the surface of the plate-like specimens, and E2 and E3 are of second- and third-order elastic constants. Note that it is more complicated to give an explicit express of the factor F associated with a specified Lamb wave mode and that the ratio A2f =A21f contains the essential information on the nonlinear wave propagation [19,20], so we select A2f =A21f as the normalized acoustic nonlinearity parameter in the following measurements of nonlinear Lamb waves. It should also be pointed out that since the following measurement of fundamental and second harmonics is not in terms of displacement, the nonlinear response is expressed in terms of A2f =A21f measured in volts. So calibration is necessary for material nonlinearity measurements, which includes confirmation of the linear relationship between the second harmonic and the square of the fundamental frequency response for a fixed distance, and the linear one between the A2f =A21f and the propagation distance [8–11,19–21]. The details of nonlinearity measurement calibration can be seen in Section 4. For observation of an obvious cumulative second-harmonic signal of Lamb wave propagation, a specific Lamb wave pair is selected (e.g., the 7th primary Lamb wave mode and the 15th DFLW mode, see Fig. 1 in Ref. [26]) to satisfy the phase velocity matching between the primary Lamb waves and the DFLWs. For saving the length of paper, the details of dispersion curves can be seen in Ref. [26]. It should be pointed out that the Lamb wave mode pair selected in the present measurements also satisfies the condition of group velocity matching [21,22,27].
½cos yT z þ ð1Þq1 sin yT yð1Þq ½x^ K^ Tq exp½2jKTq r þjDj ð1Þ where Dj is the phase matching degree between the primary Lamb wave and the double frequency Lamb wave (DFLW), that is Dj ¼ 2pfZdis ðcð2f ,nÞ cðf ,lÞ Þ=ðcð2f ,nÞ Ucðf ,lÞ Þ with the unit in rad, and ðDLÞ ¼ ULq
2 ðf ,lÞ 2 ½ULq ½3k þ 4m þ2A þ 6B þ 2C 2pf d 4½k þ 4m=3 cL d
ð2Þ
3. Experimental setup and specimens preparation 3.1. Experimental setup The experimental setup for Lamb wave measurement is illustrated in Fig. 1. The Ritec SNAP system (model RAM-5000) is used to generate tone-burst voltages for excitation of the
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and their microstructural observations were performed using a SEM (secondary electron SEM, Vega& Tescan) and a TEM (H-800 Hitachi). The secondary electron SEM images can provide a clear microstructural morphology of the precipitates in samples and the TEM analysis can provide an observation of the dislocations and a confirmation of the crystal structure of precipitated phases. Also, the major elements and phase transitions of the carbide precipitates in thermal degraded materials were observed by X-ray energy dispersive spectrometry (EDS) and electron diffraction.
4. Experimental results and analytical model calculations 4.1. Nonlinear Lamb wave measurements
Fig. 1. Experimental setup for nonlinear ultrasonic Lamb waves.
transmitting wedge transducer T and to perform the signal receiving from the receiving wedge transducer R. The wedge angle of T and R is adjustable (with organic glass whose longitudinal wave velocity is 2720 m s 1) and the wedges are coupled to the specimen using light lubrication oil. The longitudinal PZT narrow-band transducer T has a central frequency of 2.2 MHz and the receiving PZT broad-band transducer R has a central frequency of 4.9 MHz. The tone-burst duration t is 8 ms. The spatial separation Zdis between the transmitting and receiving wedge transducers is kept the same (Zdis ¼11 cm) during the experimental measurements of second harmonics of Lamb wave propagation in various thermal damaged specimens. Here, the Lamb mode pairs (i.e., the 7th primary Lamb wave mode and the 15th DFLW mode) can be excited and received via wedge transducer T and R with an oblique angle 19.71 and an excitation frequency of about 2.45 MHz (at this certain excitation angle and frequency, it would satisfy cðf ,7Þ ¼ cð2f ,15Þ or Dj ¼0 in Eq. (2)). Due to a narrow-band transducer excitation and an exactly oblique incidence of the longitudinal wave, the transmitting signal propagating in plate specimen contains a nearly pure mode of the 7th primary Lamb wave, and the receiving signals contain both the primary wave and the higher order harmonics, which can be processed by a fast Fourier transform (FFT) to extract the amplitude of the fundamental (A1f) and second harmonic (A2f) signal in the frequency domain.
Before the measurements of the cumulative second-harmonic generation in the HP40Nb plates, it is necessary to calibrate the response function for the fundamental and the second-harmonic waves associated with the experimental system Ritec SNAP and two transducers. The calibration method adopted here is the same as that described by Deng et al. [28]. Note that the second harmonic generated in the organic glass material and the two transducers would be constant in the process of nonlinear Lamb wave measurements in the specimens with different thermal damaged levels. Therefore, after normalization, the contribution of second harmonic from the organic glass material and the two transducers would not change the trend of the nonlinear Lamb wave response to the damaged specimens. To confirm that the second-harmonic signals of Lamb wave being measured are due to the material nonlinearity rather than the spuriousness of the linear signal or the nonlinearity from the experimental instrumentation, the nonlinear signature is measured for an increasing input voltage at a fixed propagation distance and is shown in Fig. 2a with a propagation distance of 11 cm. Fig. 2a clearly shows a linear relationship between the second-harmonic amplitude A2f and the square of the fundamental amplitude (A1f)2 for increasing the amplitude of excitation voltage (f¼2.45 MHz). It is verified
3.2. Specimens preparation The ferritic Cr–Ni alloy steel examined here is HP40Nb material, which was obtained from furnace tubes (f130 11 mm2) of engineering practice in a plant with 130,000 h and 160,000 h under the operating temperature of 900 1C [26]. Artificially early-stage degraded materials were acquired by high-temperature aging tests based on as-cast HP40Nb alloy tempered at 950 1C72 1C in a high temperature furnace for different holding time intervals, which are 600 h, 900 h, and 2,000 h, respectively. The total service life of HP40Nb alloy was estimated to be about 180,000 h by a rupture testing based on a service-exposed sample of 160,000 h at a temperature of 950 1C and a stress of 25 MPa, in which the extrapolation time of the sample is to be about 20,000 h obtained by Larson–Miller parameter method [26]. Therefore, the six plate-like specimens (200 40 5 mm3) with different thermal aging time (i.e., as-cast material, 600 h, 900 h, 2,000 h, 130,000 h, and 160,000 h, respectively) were available for following experiments. For observation of microstructural evolution during thermal aging, another six samples were cut from the corresponding tubes
Fig. 2. (a) Second-harmonic amplitude A2f versus the square of the amplitude of the fundamental wave A21f for increasing input voltage; (b) Curve of A2f =A21f versus propagation distance Zdis.
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that A2f =A21f ¼ constant for a fixed distance, which is in accordance with the Eq. (3). Some parameter settings of the Ritec SNAP system are indicated in Fig. 2a. Then, the amplitudes of the fundamental Lamb wave and the corresponding second harmonic are measured as a function of propagation distance by keeping the transmitting transducer T fixed and moving the receiving transducer R along oz direction, away from the excitation source. Fig. 2b shows the curve of A2f =A21f with respect to propagation distance Zdis. The quasi-linear cumulative growth effect of the second harmonic indicates that the nonlinear signals are generated by the material nonlinearity, and not by instrumentation, transducers, or other nonlinearity. Measurements of nonlinear Lamb wave signals were performed on the surface of the plate-like specimens. Fig. 3 represents frequency contents of the receiving signals where the toneburst duration maintains a width of 8 ms and the separation between the two transducers is kept fixed (Zdis ¼11 cm). The fundamental frequency amplitudes (namely A1f) change slightly with increment of the thermal damage time, except for 160,000 h specimen whose fundamental amplitude decreases about 15% when compared with those of the others. However, the second harmonic signals (namely A2f) change clearly versus the levels of thermal degradation. It is noted that the magnitude of the frequency content between 0.5 MHz and 1.5 MHz is at the order of that of the second harmonics, which may be caused by two reasons. One reason is that, except the frequency of about 2.45 MHz, the other frequency signatures also can be excited by the transmitting transducer due to impure vibration of the encapsulated PZT transducer. The other one is that the low frequency signal may have higher amplitude compared with that of the high frequency one (for instance, 3.5 or 4.0 MHz) during the propagation due to the scattering of the coarse grain size in HP40Nb material. Fig. 4 depicts the normalized acoustic nonlinearity A2f =A21f at a fundamental frequency of 2.45 MHz as a function of thermal degradation time. It can be inferred that the curve of A2f =A21f exhibits three stages, namely, first a significant increase within the early stage of about t¼1,000 h (stage I), then a relatively flat in a long-term service between about t¼1,000 h and t¼105,000 h (stage II), and finally a gradual decrease in the region after t¼105,000 h (stage III). It should be pointed out that the degradation time t¼105,000 h between stages II and III is a reasonable estimation from the experimental results given in
Fig. 3. Fourier spectra (FFT) of the different damaged specimens showing the amplitudes of the fundamental frequency and the second-harmonic frequency (the ordinate is in logarithmic scale).
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Fig. 4. Curve of the normalized acoustic nonlinearity A2f =A21f versus the thermal degradation time at the fundamental frequency of 2.45 MHz (the abscissa is in logarithmic scale).
Ref. [1] (reported about 105,000 h). Furthermore, the following microstructure analyses of these specimens would help to understand the approximate time of inflexion and the decrease in the final stage. 4.2. Microstructural evolution For further understanding the change tendency of the normalized acoustic nonlinearity A2f =A21f , the microstructural evolutions of the thermal degraded materials were observed and analyzed by SEM, TEM, EDS, and electron diffraction. Fig. 5 shows the typical SEM micrographs of four different thermal degraded specimens. The microstructure of the as-cast alloy is shown in Fig. 5a, where the grain boundary is known to be of niobium carbides and chromium carbides by EDS analysis of SEM. As for the specimen with the thermal degradation time of 2,000 h, its microstructure can be seen from Fig. 5b, some fine chromium carbides precipitates are observed in the matrix nearby the grain boundary (e.g., in the circled region). Also, some new precipitates occur in the vicinity of niobium carbides structures, which are identified to be equilibrium Z carbide (M6C) [1] by EDS analysis and electron diffraction patterns of TEM (see Fig. 6). Fig. 5b reveals a rapid precipitate and coalescence of carbides in the grain boundary, which may cause a significant increase of A2f =A21f in the stage I (see Fig. 4). It deserved to note that the precipitated phases such as Z carbide or the fine chromium carbides play an important role in the increase of the nonlinearity signals. Firstly, these precipitates would lead to a local micro-strain due to the precipitate–matrix lattice misfit, which would cause the increase of dislocations density and material nonlinearity [3,6,12]. Secondly, the coalescence of precipitated carbides such as increase of grain size would intensify the local stress concentration and therefore promote the rise of material nonlinearity [3,6]. Furthermore, the formation of the Z carbide during high temperature exposure made the microstructure more stable and better mechanical properties than the raw material in a long service life [1], which accounts for the relatively flat section of A2f =A21f in the stage II (see Fig. 4). Fig. 5c shows the SEM image of material subjected to a long-term service of 130,000 h, where some carbide precipitates drastically diminished or vanished. After more than 105,000 h high temperature exposure, the Crdepletion in the matrix of HP40Nb steel would cause a gradual decrease of matrix strength and the generation of micro-voids. It can be seen from the micrograph of 160,000 h thermal damaged specimen in Fig. 5d, both the size of the micro-voids and volume fraction increased. In order to identify the structures of precipitate phase and their morphologies, Fig. 6 shows the TEM morphologies of several
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Fig. 5. Secondary electron micrographs of different thermal damage specimens; (a) as-cast material specimen, (b) with 2,000 h thermal aging, (c) with 130,000 h thermal aging, and (d) with 160,000 h thermal aging.
Fig. 6. TEM morphologies of several types of carbides and their SADPs (in top right corner); (a) a MC precipitate (niobium carbides) in an as-cast specimen and its SADP with beam direction [011]MC and [035]matrix, (b) a M23C6 precipitate (chromium carbides) in an as-cast specimen and its SADP with beam direction [013]M23C6 and [013]matrix, and (c) a M6C precipitate (Z carbide) in a 2,000 h thermal aging specimen and its SADP with beam direction [111]M6C and [111]matrix.
types of carbides and their diffraction patterns. Fig. 6a and b show the morphologies of niobium carbides and chromium carbides of an as-cast HP40Nb alloy and select area diffraction patterns (SADP) of the precipitate region. Diffraction analysis reveals that the niobium carbides have an ordinary face-center-cubic (fcc) ˚ This result structure and their lattice constant is about 4.52 A. indicates that the niobium carbides should be the structure of MC phase (i.e. NbC). Moreover, SADP in Fig. 6b reveals that the chromium carbides are known to be the fcc M23C6 carbide phase ˚ With increasing high with a lattice constant of about 10.7 A. temperature exposure, the carbides in HP40Nb alloy change dramatically. In a 2,000 h specimen, some fine chromium carbides precipitating in the matrix nearby the grain boundary (e.g. the circled region of Fig. 5b) are consistent with orthorhombic M7C3, which are unstable carbide phase and would transform to relatively stable phase M23C6 carbides during long-term high temperature loading [1]. Fig. 6c shows the morphology of the new precipitates occurring in the vicinity of NbC of a 2,000 h specimen and a SADP of the precipitate region, which indicates that the new phase should be equilibrium M6C (i.e. Z carbide)
˚ It should be pointed out with a lattice constant of about 10.84 A. that a misfit parameter d can be used to describe the mismatch degree between the precipitated phase and the matrix, where d ¼2(ap am)/(ap þam), ap and am are the lattice parameters of the precipitate and the matrix, respectively. The precipitated phase gives rise to a lattice strain due to the mismatch and can influence the distortion of ultrasonic waves. So, if the volume fraction of the equilibrium M6C (i.e. Z carbide) keeps rising and that of the NbC phase decreases, the local micro-strain would be intensified between the matrix and the precipitated phases since the misfit parameter d of M6C is greater than that of NbC phase (can see in Table 1). Also, the local micro-strain and dislocations changes would have a great influence on the nonlinear effect in damaged material. 4.3. Analytical model calculation and discussions In the present study, it can be seen that the dominant microstructural changes are the precipitate carbides generation and coalescence, and the micro-voids initiation and increase, which play
Y. Xiang et al. / NDT&E International 44 (2011) 768–774
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Table 1 Microstructure parameters of the HP40Nb specimens with different thermal degradation time. HP40Nb specimens with different thermal degradation time
Dislocation density L ( 1013 m 2) Dislocation length L (mm) ˚ Lattice parameter of the precipitate ap (A)
Misfit parameter d (the lattice parameter ˚ of the matrix am is 3.6 A)
Volume fraction of the precipitates fp (%)
0h
600 h
2,000 h
130,000 h
5.0 0.40 NbC: 4.52 M23C6: 10.7
5.5 0.42 M7C3: 8.71 M6C: 10.84 NbC: 4.52 M23C6: 10.7
8.0 0.38 M7C3: 8.71 M6C: 10.84 NbC: 4.52 M23C6: 10.7
4.0 0.46 M6C: 10.84 M23C6: 10.7
dNbC: 0.23 dM23C6: 0.99
dM7C3: 0.83 dM6C: 1.0 dNbC: 0.23 dM23C6: 0.99
dM7C3: 0.83 dM6C: 1.0 dNbC: 0.23 dM23C6: 0.99
dM6C: 1.0 dM23C6: 0.99 (Voids dV: 0.05)
fp(NbC): 1.7 fp(M23C6): 2.8
fp(M7C3): 0.2 fp(M6C): 0.45 fp(NbC): 1.4 fp(M23C6): 2.6
fp(M7C3): 0.9 fp(M6C):1.2 fp(NbC): 0.95 fp(M23C6): 2.5
fp(M6C):1.3 fp(M23C6): 2.7 (Voids fp : 5)
an important role in the distortion of ultrasonic wave propagation during thermal degradation of HP40Nb steels. According to the studies on the acoustic harmonic generation resulting from dislocation under a micro-strain due to the precipitate–matrix lattice misfit [6,12], the expression of the change in the acoustic nonlinearity parameter (in the form of longitudinal wave or longitudinal stress– strain relation) resulting from the application of a stress s on a pinned dislocation can be given by
Db 24 OLL4 R3 E22 ¼ 9s9 5 b0 m3 b2 b0
ð4Þ
where b0 ¼14.1 is the initial value of nonlinearity parameter [25], Db ¼(b b0) is the change of nonlinearity parameter caused by microstructure evolution in material, L is the dislocation density, L is the dislocation length, E2 is the second-order elastic constant, m is the shear modulus of the matrix, b is the Burgers vector, R is the resolving shear factor, and O is the conversion factor from shear strain to longitudinal strain. The strain field resulting from the precipitate–matrix lattice misfit may increase the stress, which can be given by s E2mefp [3], where fp is the volume fraction of the dispersed precipitate phase and e is the coherency strain. For assuming the precipitates to be spherical and elastically isotropic, the coherency strain resulting from a spherical precipitate embedded in a finite matrix is presented as follows [3]
e¼
3K d 3K þ 2Eð1 þ nÞ
ð5Þ
where K is the bulk modulus of the precipitate, and E and n are the Young’s modulus and Poisson’s ratio in the matrix. From Eqs. (4) and (5), it can be seen that Db ¼ CðO,R, m,b, K,E, nÞE22 LL4 dfp is proportional to the product of E22 , L, L4, d, and fp if the coefficient of C(O,R,m,b,K,E,n) kept unchanged during thermal degradation in HP40Nb specimens. Table 1 gives the values of microstructure parameters of the HP40Nb specimens with different thermal degradation time. The density of dislocations was measured by the method proposed by Keh and Weissmann [29] and the average dislocation length was obtained by analyzing the TEM micrograph in computer. Fig. 7 shows the normalized Db (with respect to b0) as a function of the thermal degradation time, which can be obtained by substituting the data of Table 1 into Eqs. (4) and (5). It is clear that the change tendency of the stages I and II of the normalized nonlinear parameter of Lamb wave propagation in Fig. 4 is in good accordance with that predicted by theoretical model. However, there is a little difference between the stage III in Fig. 4 and the
Fig. 7. Graph of normalized Db as a function of thermal degradation time of HP40Nb materials (the abscissa is in logarithmic scale).
stage ‘‘C–D’’ in Fig. 7, that is the measured normalized acoustic nonlinearity A2f =A21f of Lamb wave monotonically decreased in the final stage, whereas the normalized Db of theoretical calculation displayed a hook-shape change in the same stage (see Fig. 7). From Table 1 and Fig. 7, it can be known that the decrease of Db in the stage ‘‘C’’ may be caused by the variances of second phase precipitates, dislocation density and length since there are no other factors causing a reduction of nonlinearity (the factor of change of longitudinal wave velocity has been already taken into account in Fig. 7). However, in the stage ‘‘D’’ of Fig. 7 the microvoids initiation and coalescence make a contribution to the rise in the Db of theoretical calculation from 130,000 h to 160,000 h in the damaged time. As for the experimental measurements of Lamb wave, it can be indicated from the analytical model and Fig. 5c that the decrease of the second phase precipitates and the microstructural changes in materials would play an important role in the drop of the A2f =A21f of Lamb wave in the final stage. Moreover, another important reason affecting the monotonic reduction of nonlinear Lamb waves may be the increasing mismatch of the phase velocity between the primary Lamb wave and the DFLW after long-term high temperature exposure [15,25,26]. In Fig. 8, we present the evolution of longitudinal wave velocity versus the thermal degradation time. It clearly indicates that the material properties relating to ultrasonic velocity such as elastic modulus and Poisson’s ratio definitely changed in the process of thermal damage, which would worsen the matching degree of phase
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Acknowledgements This work is supported by the National Natural Science Foundation of China under grant numbers (50835003, 10972078, 10974256, and 11004056), Natural Science Foundation of Shanghai (09JC1404400), and the National High Technology Research and Development Program (‘‘863’’Program) of China under grant number 2009AA04Z421. References
Fig. 8. Variation in the ultrasonic longitudinal velocity with the thermal degradation of HP40Nb specimens (the abscissa is in logarithmic scale).
velocity [25] and cause the decline of the value of A2f =A21f for Lamb wave propagation in the final stage. According to these analyses, it is a reasonable deduction that a further increase in the scales of the creep micro-voids would give rise to a gradual decrease in the response of A2f =A21f in the stage III in Fig. 4 and finally induce a fracture. It is also worth noting that the obvious rising of the normalized acoustic nonlinearity A2f =A21f of Lamb wave propagation in the early and middle stages is in good agreement with the results of increasing acoustic nonlinearity of Lamb wave with increasing fatigue damage or plasticity deformation in aluminum alloys [20,21]. However, as the thermal exposure progresses, the generation and coalescence of creep micro-voids would have a clear influence on the change of the material properties, which would break the original phase velocity matching condition and thus lower the generation of the second harmonic by Lamb wave propagation [25]. Note that Deng and Pei [23] recently performed nonlinear Lamb wave measurements for fatigue loading in pure aluminum specimens and gave the result of monotonic decrease in the acoustic nonlinearity of Lamb waves as loading cycle progresses, which may also be caused by the continuing worsening of the phase velocity matching condition.
5. Conclusions Thermal degradation of ferritic Cr–Ni alloy plates has been characterized using the nonlinear effect of Lamb waves for the purpose of SHM and life prediction application. The experimental result shows a clear relationship between the normalized acoustic nonlinearity A2f =A21f of Lamb wave propagation and the thermal degradation time during the total damage life in HP40Nb specimens. The variation in the measured acoustic nonlinearity is in good agreement with the analytical model calculation of precipitate–dislocation interaction based on metallographic studies. These results reveal that the acoustic nonlinearity of Lamb wave increases due to the second phase precipitates in the early stage, and it decreases as a combined result of dislocation change and micro-void initiation in the material that cause the phase velocity mismatching between the primary Lamb wave and the double frequency one after long-term high temperature exposure. Consequently, the results from this study show that ultrasonic nonlinearity of Lamb wave has been found to be sensitive to the microstructure evolution in materials during the thermal degradation process, and it could be a potential application for the quantitative assessment of thermal damage in metallic plates or pipes.
[1] Kenik EA, Maziasz PJ, Swindeman RW, Cervenka J, May D. Structure and phase stability in a cast modified-HP austenite after long-term ageing. Scr Mater 2003;49:117–22. [2] Dobmann G, Kroening M, Theiner W, Willens H, Fiedler U. Nondestructive characterization of materials (ultrasonic and micromagnetic techniques) for strength and toughness prediction and the detection of early creep damage. NDT and E Int 1997;30:326. [3] Kim CS, Park IK, Jhang KY. Nonlinear ultrasonic characterization of thermal degradation in ferritic 2.25Cr–1Mo steel. NDT and E Int 2009;42:204–9. [4] Goursolle T, Dos Santos S, Bou Matar O, Calle S. Non-linear based time reversal acoustic applied to crack detection: simulations and experiments. Int J Non Linear Mech 2008;43:170–7. [5] Stratoudaki T, Ellwood R, Sharples S, Clark M. Measurement of material nonlinearity using surface acoustic wave parametric interaction and laser ultrasonics. J Acoust Soc Am 2011;129:1721–8. [6] Cantrell JH, Yost WT. Effect of precipitate coherency strains on acoustic harmonic generation. J Appl Phys 1997;81:2957–62. [7] Jhang KY. Applications of nonlinear ultrasonics to the NDE of material degradation. IEEE Trans Ultrason Ferroelectr Freq Control 2000;47(3):540–8. [8] Kim JY, Jacobs LJ, Qu J, Littles JW. Experimental characterization of fatigue damage in a nickel-base superalloy using nonlinear ultrasonic waves. J Appl Phys 2006;99:124913. [9] Metya A, Ghosh M, Parida N, Palit Sagar S. Higher harmonic analysis of ultrasonic signal for ageing behavior study of C-250 grade maraging steel. NDT and E Int 2008;41:484–9. [10] Baby S, Kowmudi BN, Omprakash CM, Satyanarayana DVV, Balasubramaniam K, Kumar V. Creep damage assessment in titanium alloy using a nonlinear ultrasonic technique. Scr Mater 2008;59:818–21. [11] Kim CS, Lissenden CJ. Precipitate contribution to the acoustic nonlinearity in nickel-based superalloy. Chin Phys Lett 2009;26:086107. [12] Hikata A, Chick BB, Elbaum C. Dislocation contribution to the second harmonic generation of ultrasonic waves. J Appl Phys 1965;36:229–36. [13] Buck O, Morris WL, Richardson JM. Acoustic harmonic generation at unbonded interfaces and fatigue cracks. Appl Phys Lett 1978;33:371–3. [14] Deng MX. Cumulative second-harmonic generation of Lamb-mode propagation in a solid plate. J Appl Phys 1999;85:3051–8. [15] Deng MX. Analysis of second-harmonic generation of Lamb modes using a modal analysis approach. J Appl Phys 2003;94:4152–9. [16] de Lima WJN, Hamilton MF. Finite-amplitude waves in isotropic elastic plates. J Sound Vib 2003;265:819–39. [17] Deng M, Wang P, Lv X. Experimental verification of cumulative growth effect of second harmonics of Lamb wave propagation in an elastic plate. Appl Phys Lett 2005;86:124104. [18] Srivastava A, di Scalea FL. On the existence of antisymmetric or symmetric Lamb waves at nonlinear higher harmonics. J Sound Vib 2009;323:932–43. [19] Bermes C, Kim JY, Qu J, Jacobs LJ. Experimental characterization of material nonlinearity using Lamb waves. Appl Phys Lett 2007;90:021901. [20] Pruell C, Kim JY, Qu J, Jacobs LJ. Evaluation of plasticity driven material damage using Lamb waves. Appl Phys Lett 2007;91:231911. [21] Pruell C, Kim JY, Jacobs LJ, Qu J. Evaluation of fatigue damage using nonlinear guided waves. Smart Mater Struct 2009;18:035003. ¨ [22] Muller MF, Kim JY, Qu J, Jacobs LJ. Characteristics of second harmonic generation of Lamb waves in nonlinear elastic plates. J Acoust Soc Am 2010;127:2141–52. [23] Deng MX, Pei JF. Assessment of accumulated fatigue damage in solid plates using nonlinear Lamb wave approach. Appl Phys Lett 2007;90:121902. [24] Xiang YX, Deng MX. Cumulative second-harmonic generation of Lamb waves propagating in a two-layered solid plate. Chin Phys B 2008;17:4232–41. [25] Xiang Y, Deng M, Xuan FZ. Analysis of second-harmonic generation of Lamb waves using a combined method in a two-layered solid waveguide. J Appl Phys 2009;106:024902. [26] Xiang YX, Xuan FZ, Deng MX. Evaluation of thermal degradation induced material damage using nonlinear Lamb waves. Chin Phys Lett 2010;27:016202. [27] Lee TH, Choi IH, Jhang KY. The nonlinearity of guided wave in an elastic plate. Mod Phys Lett B 2008;22(11):1135–40. [28] Deng M, Wang P, Lv X. Experimental observation of cumulative secondharmonic generation of Lamb-wave propagation in an elastic plate. J Phys D: appl Phys 2005;38:344–53. [29] Keh AS, Weissmann S. Electron Microscopy and Strength of Crystal. New York: interscience; vol. 963, p. 231.
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Experimental study of thermal degradation in ferritic Cr–Ni alloy steel plates using nonlinear Lamb waves Yanxun Xiang a, Mingxi Deng b, Fu-Zhen Xuan a,n, Chang-Jun Liu a a Key Laboratory of Safety Science of Pressurized System of MOE, School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai 200237, China b Department of Physics, Logistics Engineering University, Chongqing 400016, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 6 September 2010 Received in revised form 27 July 2011 Accepted 5 August 2011 Available online 22 August 2011
The thermal degradation in ferritic Cr–Ni alloy steel plates is measured using the nonlinear effect of Lamb wave propagation. Experiments were carried out to introduce controlled levels of thermal damage to determine the nonlinear response of Lamb waves. A ‘‘mountain-shape’’ change in the normalized acoustic nonlinearity of Lamb wave versus the level of thermal degradation in the specimens has been observed. The variation in the measured acoustic nonlinearity reveals, based on metallographic studies, that the normalized acoustic nonlinearity increases due to the second phase precipitates in the early stage and it decreases as a combined result of dislocation change and microvoid initiation in the material. The results show a potential application of the nonlinear Lamb waves for the quantitative assessment of thermal damage in metallic plates or pipes. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Ultrasonic nonlinearity Lamb wave Thermal degradation Ferritic Cr–Ni steel Dislocations
1. Introduction Heat-resistant steels have been widely applied in high temperature components of petrochemical industry and power plants due to their excellent high-temperature strength and creep resistance. However, when the heat-resistant steels are used for long periods at elevated temperature in practice, they generally experience degradation of thermal damage or creep damage, which would shorten the lifespan of components and would bring about serious disaster due to the unforeseen fracture. Therefore, it is of increasing importance to evaluate such degradation of strength of materials at early stage for structural health monitoring (SHM). In order to conduct a nondestructive evaluation (NDE) on the state of structural material damages, ultrasonic method is often used for its capability of evaluating both the surface and internal damage state of in-service components with the comparatively simple and easy instrumentation. During thermal damage, the formation of various precipitates in heat-resistant steel would cause local strain concentrations and therefore would produce different dislocations concentrated near the precipitates [1]. The volume fraction of the precipitates has a direct influence on the changes of dislocation density and average length. Moreover, the dislocation pileups caused by strain concentration or residual stress would make the precipitates
n
Corresponding author. Tel.: þ86 21 6425 2819; fax: þ 86 21 6425 3135. E-mail address: [email protected] (F.-Z. Xuan).
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break off from the matrix and then leads to formation of microvoids. All these microstructure changes during thermal degradation would produce a very clear nonlinear response in material [2,3]. For early detection of damage even before the first crack initiation, there are some ultrasonic techniques used for measuring material degradation, such as the combination of nonlinear elastic wave spectroscopy (NEWS) with time reversal (TR) [4], the measurement of acoustic wave parametric interaction [5] and the higher harmonic generation method [6–11]. In recent years, the higher harmonic generation method has been focused as a potential tool for NDE of the material property degradation [6–11]. This technique relies on measuring the higher order harmonics generated by the intrinsic nonlinearity of materials relating to the anharmonicity and imperfection of atomic lattices. The classical theory of nonlinear wave propagation in elastic solids has been discussed and presented for nearly half a century [12]. Most previous researches have measured the acoustic nonlinearity parameter b using longitudinal waves, which is a function of the second- and third-order elastic constants, and have indicated that b is sensitive to subtle damage in materials and can be correlated with certain microstructural changes during fatigue damage [8,13], thermal damage [3,7], or creep damage [10,11]. Compared with the nonlinear ultrasonic techniques that use bulk acoustic waves, ultrasonic guided waves such as Lamb waves have obvious advantages, such as one-side access of pitch-catch configuration and high efficiency in a long distance propagation, for using as a more suitable candidate to interrogate large plate- and shell-like structures in the field applications.
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Due to the dispersive and multi-mode nature of Lamb wave, recent studies revealed that the generation of cumulative second harmonic of Lamb wave propagation in materials does occur once some conditions are satisfied [14–16]. Deng [14,15] and de Lima and Hamilton [16] described these conditions and experimentally validated the cumulative effect of second harmonics of Lamb wave propagation in an elastic plate [17]. Srivastava et al. [18] performed a theoretical studies on the symmetry characteristics of Rayleigh-Lamb waves in nonlinear isotropic plates, as well as on the higher-order (than double) harmonics of Lamb waves. Bermes et al. [19] and Pruell et al. [20,21] developed a procedure to measure the second harmonic of a Lamb wave by combining a time–frequency process with a hybrid wedge generation and laser interferometric detection system. Besides satisfying two conditions for Lamb wave propagation, i.e., phase velocity matching and nonzero power flux, Jacobs’ group proposed another condition [21,22], namely, the group velocity matching. Most of the past works relating to nonlinear Lamb waves have focused on the plastic deformation [21] and fatigue damage [20,23] in simple metals, and their results also manifested a contradictive tendency (i.e., monotonous increase or decrease) on the variation of the cumulative second harmonic of Lamb wave propagating in the damaged materials. Moreover, there is no detailed explanation on these different experimental results in the aluminum plate materials during the process of fatigue damage. In contrast to the studies of fatigue damage assessments in the simple metals via nonlinear Lamb waves, the current paper deals with the characterization of thermal damage in structure materials (such as HP40Nb material, ASTM Grade, a kind of ferritic Cr–Ni alloy steel) using the nonlinear effect of ultrasonic Lamb waves. Measurements of the second harmonic of Lamb waves have been carried out on the HP40Nb specimens with different thermal damage levels. The influence of microstructure evolution on the nonlinear Lamb wave propagation has also been thoroughly analyzed based on metallographic studies, such as scanning electron microscope (SEM) and transmission electron microscope (TEM). Moreover, an analytical model calculation of precipitate– dislocation interaction has been performed to interpret the correlation between microstructural evolutions and the nonlinear Lamb wave measurements.
2. Determination of nonlinear Lamb waves propagation in an elastic plate For a solid plate, the coordinate origin is located in its central plane, and the oy axis is along the plate thickness and the oz axis is along the direction of wave propagation. Within second-order perturbation, the general solution of the cumulative second harmonic by Lamb wave propagation, generated by a line source parallel to the ox axis, can be expressed as [14,24,25] 8 ðDLÞ 9 q 2
2 sinðDjÞ X
Dj
ðFCÞ UTq
q¼1
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T or L in the subscript means that the corresponding quantity is associated with the transverse or longitudinal wave, respectively. ðFCÞ ðFCÞ ðDLÞ ULq and UTq are proportional to ULq [14,24], x^ is the unit vector along the ox axis, yL or yT is the angle between the oy axis and the KLq or KTq (see Fig. 2 in Ref. [25]), K^ Lq and K^ Tq are the unit vectors of the partial wave vectors KLq and KTq. cL is the longitudinal wave velocity, f is the frequency of primary Lamb wave propagation, k, m, A, B, and C are second- and third-order elastic constants of the solid specimen with the thickness of 2d. c(f,l) is phase velocity of the primary Lamb wave l-mode and c(2f,n) is that of the DFLW n-mode. ðf ,lÞ Zdis is the propagation distance. ULq represents the longitudinal component of the primary Lamb wave of lth mode. From the previous studies [14,15,25], it can be known that if the phase velocity of a DFLW component is exactly or approximately equivalent to that of the primary Lamb wave, the secondharmonic amplitude of Lamb wave propagation increases clearly with propagation distance or within a certain distance. Under the condition of Dj ¼0 or Dj E0, the analytic solution of the cumulative second harmonic of Lamb wave propagation in an elastic plate is expressed in Ref. [14]. As in previous works [8–13], the ratio of A2 =A21 is used to represent the normalized acoustic nonlinearity and is proportional to the acoustic nonlinearity parameter b (e.g., b ¼ 8A2 =ðk2 Zdis A21 Þ for longitudinal waves), where A1 and A2 are the measured amplitudes of the fundamental and the second-harmonic waves, respectively, and k is the wave number. Based on Eqs. (1) and (2), we can formally define the acoustic nonlinearity of Lamb waves as follows
bLamb ¼ FðyT , yL ,KLq ,KTq ,r,E2 ,E3 ÞA2f =ðk2 Zdis A21f Þ
ð3Þ
where A1f and A2f are, respectively, the measured amplitudes of the fundamental and the second-harmonic signals of Lamb wave propagation on the surface of the plate-like specimens, and E2 and E3 are of second- and third-order elastic constants. Note that it is more complicated to give an explicit express of the factor F associated with a specified Lamb wave mode and that the ratio A2f =A21f contains the essential information on the nonlinear wave propagation [19,20], so we select A2f =A21f as the normalized acoustic nonlinearity parameter in the following measurements of nonlinear Lamb waves. It should also be pointed out that since the following measurement of fundamental and second harmonics is not in terms of displacement, the nonlinear response is expressed in terms of A2f =A21f measured in volts. So calibration is necessary for material nonlinearity measurements, which includes confirmation of the linear relationship between the second harmonic and the square of the fundamental frequency response for a fixed distance, and the linear one between the A2f =A21f and the propagation distance [8–11,19–21]. The details of nonlinearity measurement calibration can be seen in Section 4. For observation of an obvious cumulative second-harmonic signal of Lamb wave propagation, a specific Lamb wave pair is selected (e.g., the 7th primary Lamb wave mode and the 15th DFLW mode, see Fig. 1 in Ref. [26]) to satisfy the phase velocity matching between the primary Lamb waves and the DFLWs. For saving the length of paper, the details of dispersion curves can be seen in Ref. [26]. It should be pointed out that the Lamb wave mode pair selected in the present measurements also satisfies the condition of group velocity matching [21,22,27].
½cos yT z þ ð1Þq1 sin yT yð1Þq ½x^ K^ Tq exp½2jKTq r þjDj ð1Þ where Dj is the phase matching degree between the primary Lamb wave and the double frequency Lamb wave (DFLW), that is Dj ¼ 2pfZdis ðcð2f ,nÞ cðf ,lÞ Þ=ðcð2f ,nÞ Ucðf ,lÞ Þ with the unit in rad, and ðDLÞ ¼ ULq
2 ðf ,lÞ 2 ½ULq ½3k þ 4m þ2A þ 6B þ 2C 2pf d 4½k þ 4m=3 cL d
ð2Þ
3. Experimental setup and specimens preparation 3.1. Experimental setup The experimental setup for Lamb wave measurement is illustrated in Fig. 1. The Ritec SNAP system (model RAM-5000) is used to generate tone-burst voltages for excitation of the
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and their microstructural observations were performed using a SEM (secondary electron SEM, Vega& Tescan) and a TEM (H-800 Hitachi). The secondary electron SEM images can provide a clear microstructural morphology of the precipitates in samples and the TEM analysis can provide an observation of the dislocations and a confirmation of the crystal structure of precipitated phases. Also, the major elements and phase transitions of the carbide precipitates in thermal degraded materials were observed by X-ray energy dispersive spectrometry (EDS) and electron diffraction.
4. Experimental results and analytical model calculations 4.1. Nonlinear Lamb wave measurements
Fig. 1. Experimental setup for nonlinear ultrasonic Lamb waves.
transmitting wedge transducer T and to perform the signal receiving from the receiving wedge transducer R. The wedge angle of T and R is adjustable (with organic glass whose longitudinal wave velocity is 2720 m s 1) and the wedges are coupled to the specimen using light lubrication oil. The longitudinal PZT narrow-band transducer T has a central frequency of 2.2 MHz and the receiving PZT broad-band transducer R has a central frequency of 4.9 MHz. The tone-burst duration t is 8 ms. The spatial separation Zdis between the transmitting and receiving wedge transducers is kept the same (Zdis ¼11 cm) during the experimental measurements of second harmonics of Lamb wave propagation in various thermal damaged specimens. Here, the Lamb mode pairs (i.e., the 7th primary Lamb wave mode and the 15th DFLW mode) can be excited and received via wedge transducer T and R with an oblique angle 19.71 and an excitation frequency of about 2.45 MHz (at this certain excitation angle and frequency, it would satisfy cðf ,7Þ ¼ cð2f ,15Þ or Dj ¼0 in Eq. (2)). Due to a narrow-band transducer excitation and an exactly oblique incidence of the longitudinal wave, the transmitting signal propagating in plate specimen contains a nearly pure mode of the 7th primary Lamb wave, and the receiving signals contain both the primary wave and the higher order harmonics, which can be processed by a fast Fourier transform (FFT) to extract the amplitude of the fundamental (A1f) and second harmonic (A2f) signal in the frequency domain.
Before the measurements of the cumulative second-harmonic generation in the HP40Nb plates, it is necessary to calibrate the response function for the fundamental and the second-harmonic waves associated with the experimental system Ritec SNAP and two transducers. The calibration method adopted here is the same as that described by Deng et al. [28]. Note that the second harmonic generated in the organic glass material and the two transducers would be constant in the process of nonlinear Lamb wave measurements in the specimens with different thermal damaged levels. Therefore, after normalization, the contribution of second harmonic from the organic glass material and the two transducers would not change the trend of the nonlinear Lamb wave response to the damaged specimens. To confirm that the second-harmonic signals of Lamb wave being measured are due to the material nonlinearity rather than the spuriousness of the linear signal or the nonlinearity from the experimental instrumentation, the nonlinear signature is measured for an increasing input voltage at a fixed propagation distance and is shown in Fig. 2a with a propagation distance of 11 cm. Fig. 2a clearly shows a linear relationship between the second-harmonic amplitude A2f and the square of the fundamental amplitude (A1f)2 for increasing the amplitude of excitation voltage (f¼2.45 MHz). It is verified
3.2. Specimens preparation The ferritic Cr–Ni alloy steel examined here is HP40Nb material, which was obtained from furnace tubes (f130 11 mm2) of engineering practice in a plant with 130,000 h and 160,000 h under the operating temperature of 900 1C [26]. Artificially early-stage degraded materials were acquired by high-temperature aging tests based on as-cast HP40Nb alloy tempered at 950 1C72 1C in a high temperature furnace for different holding time intervals, which are 600 h, 900 h, and 2,000 h, respectively. The total service life of HP40Nb alloy was estimated to be about 180,000 h by a rupture testing based on a service-exposed sample of 160,000 h at a temperature of 950 1C and a stress of 25 MPa, in which the extrapolation time of the sample is to be about 20,000 h obtained by Larson–Miller parameter method [26]. Therefore, the six plate-like specimens (200 40 5 mm3) with different thermal aging time (i.e., as-cast material, 600 h, 900 h, 2,000 h, 130,000 h, and 160,000 h, respectively) were available for following experiments. For observation of microstructural evolution during thermal aging, another six samples were cut from the corresponding tubes
Fig. 2. (a) Second-harmonic amplitude A2f versus the square of the amplitude of the fundamental wave A21f for increasing input voltage; (b) Curve of A2f =A21f versus propagation distance Zdis.
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that A2f =A21f ¼ constant for a fixed distance, which is in accordance with the Eq. (3). Some parameter settings of the Ritec SNAP system are indicated in Fig. 2a. Then, the amplitudes of the fundamental Lamb wave and the corresponding second harmonic are measured as a function of propagation distance by keeping the transmitting transducer T fixed and moving the receiving transducer R along oz direction, away from the excitation source. Fig. 2b shows the curve of A2f =A21f with respect to propagation distance Zdis. The quasi-linear cumulative growth effect of the second harmonic indicates that the nonlinear signals are generated by the material nonlinearity, and not by instrumentation, transducers, or other nonlinearity. Measurements of nonlinear Lamb wave signals were performed on the surface of the plate-like specimens. Fig. 3 represents frequency contents of the receiving signals where the toneburst duration maintains a width of 8 ms and the separation between the two transducers is kept fixed (Zdis ¼11 cm). The fundamental frequency amplitudes (namely A1f) change slightly with increment of the thermal damage time, except for 160,000 h specimen whose fundamental amplitude decreases about 15% when compared with those of the others. However, the second harmonic signals (namely A2f) change clearly versus the levels of thermal degradation. It is noted that the magnitude of the frequency content between 0.5 MHz and 1.5 MHz is at the order of that of the second harmonics, which may be caused by two reasons. One reason is that, except the frequency of about 2.45 MHz, the other frequency signatures also can be excited by the transmitting transducer due to impure vibration of the encapsulated PZT transducer. The other one is that the low frequency signal may have higher amplitude compared with that of the high frequency one (for instance, 3.5 or 4.0 MHz) during the propagation due to the scattering of the coarse grain size in HP40Nb material. Fig. 4 depicts the normalized acoustic nonlinearity A2f =A21f at a fundamental frequency of 2.45 MHz as a function of thermal degradation time. It can be inferred that the curve of A2f =A21f exhibits three stages, namely, first a significant increase within the early stage of about t¼1,000 h (stage I), then a relatively flat in a long-term service between about t¼1,000 h and t¼105,000 h (stage II), and finally a gradual decrease in the region after t¼105,000 h (stage III). It should be pointed out that the degradation time t¼105,000 h between stages II and III is a reasonable estimation from the experimental results given in
Fig. 3. Fourier spectra (FFT) of the different damaged specimens showing the amplitudes of the fundamental frequency and the second-harmonic frequency (the ordinate is in logarithmic scale).
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Fig. 4. Curve of the normalized acoustic nonlinearity A2f =A21f versus the thermal degradation time at the fundamental frequency of 2.45 MHz (the abscissa is in logarithmic scale).
Ref. [1] (reported about 105,000 h). Furthermore, the following microstructure analyses of these specimens would help to understand the approximate time of inflexion and the decrease in the final stage. 4.2. Microstructural evolution For further understanding the change tendency of the normalized acoustic nonlinearity A2f =A21f , the microstructural evolutions of the thermal degraded materials were observed and analyzed by SEM, TEM, EDS, and electron diffraction. Fig. 5 shows the typical SEM micrographs of four different thermal degraded specimens. The microstructure of the as-cast alloy is shown in Fig. 5a, where the grain boundary is known to be of niobium carbides and chromium carbides by EDS analysis of SEM. As for the specimen with the thermal degradation time of 2,000 h, its microstructure can be seen from Fig. 5b, some fine chromium carbides precipitates are observed in the matrix nearby the grain boundary (e.g., in the circled region). Also, some new precipitates occur in the vicinity of niobium carbides structures, which are identified to be equilibrium Z carbide (M6C) [1] by EDS analysis and electron diffraction patterns of TEM (see Fig. 6). Fig. 5b reveals a rapid precipitate and coalescence of carbides in the grain boundary, which may cause a significant increase of A2f =A21f in the stage I (see Fig. 4). It deserved to note that the precipitated phases such as Z carbide or the fine chromium carbides play an important role in the increase of the nonlinearity signals. Firstly, these precipitates would lead to a local micro-strain due to the precipitate–matrix lattice misfit, which would cause the increase of dislocations density and material nonlinearity [3,6,12]. Secondly, the coalescence of precipitated carbides such as increase of grain size would intensify the local stress concentration and therefore promote the rise of material nonlinearity [3,6]. Furthermore, the formation of the Z carbide during high temperature exposure made the microstructure more stable and better mechanical properties than the raw material in a long service life [1], which accounts for the relatively flat section of A2f =A21f in the stage II (see Fig. 4). Fig. 5c shows the SEM image of material subjected to a long-term service of 130,000 h, where some carbide precipitates drastically diminished or vanished. After more than 105,000 h high temperature exposure, the Crdepletion in the matrix of HP40Nb steel would cause a gradual decrease of matrix strength and the generation of micro-voids. It can be seen from the micrograph of 160,000 h thermal damaged specimen in Fig. 5d, both the size of the micro-voids and volume fraction increased. In order to identify the structures of precipitate phase and their morphologies, Fig. 6 shows the TEM morphologies of several
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Fig. 5. Secondary electron micrographs of different thermal damage specimens; (a) as-cast material specimen, (b) with 2,000 h thermal aging, (c) with 130,000 h thermal aging, and (d) with 160,000 h thermal aging.
Fig. 6. TEM morphologies of several types of carbides and their SADPs (in top right corner); (a) a MC precipitate (niobium carbides) in an as-cast specimen and its SADP with beam direction [011]MC and [035]matrix, (b) a M23C6 precipitate (chromium carbides) in an as-cast specimen and its SADP with beam direction [013]M23C6 and [013]matrix, and (c) a M6C precipitate (Z carbide) in a 2,000 h thermal aging specimen and its SADP with beam direction [111]M6C and [111]matrix.
types of carbides and their diffraction patterns. Fig. 6a and b show the morphologies of niobium carbides and chromium carbides of an as-cast HP40Nb alloy and select area diffraction patterns (SADP) of the precipitate region. Diffraction analysis reveals that the niobium carbides have an ordinary face-center-cubic (fcc) ˚ This result structure and their lattice constant is about 4.52 A. indicates that the niobium carbides should be the structure of MC phase (i.e. NbC). Moreover, SADP in Fig. 6b reveals that the chromium carbides are known to be the fcc M23C6 carbide phase ˚ With increasing high with a lattice constant of about 10.7 A. temperature exposure, the carbides in HP40Nb alloy change dramatically. In a 2,000 h specimen, some fine chromium carbides precipitating in the matrix nearby the grain boundary (e.g. the circled region of Fig. 5b) are consistent with orthorhombic M7C3, which are unstable carbide phase and would transform to relatively stable phase M23C6 carbides during long-term high temperature loading [1]. Fig. 6c shows the morphology of the new precipitates occurring in the vicinity of NbC of a 2,000 h specimen and a SADP of the precipitate region, which indicates that the new phase should be equilibrium M6C (i.e. Z carbide)
˚ It should be pointed out with a lattice constant of about 10.84 A. that a misfit parameter d can be used to describe the mismatch degree between the precipitated phase and the matrix, where d ¼2(ap am)/(ap þam), ap and am are the lattice parameters of the precipitate and the matrix, respectively. The precipitated phase gives rise to a lattice strain due to the mismatch and can influence the distortion of ultrasonic waves. So, if the volume fraction of the equilibrium M6C (i.e. Z carbide) keeps rising and that of the NbC phase decreases, the local micro-strain would be intensified between the matrix and the precipitated phases since the misfit parameter d of M6C is greater than that of NbC phase (can see in Table 1). Also, the local micro-strain and dislocations changes would have a great influence on the nonlinear effect in damaged material. 4.3. Analytical model calculation and discussions In the present study, it can be seen that the dominant microstructural changes are the precipitate carbides generation and coalescence, and the micro-voids initiation and increase, which play
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Table 1 Microstructure parameters of the HP40Nb specimens with different thermal degradation time. HP40Nb specimens with different thermal degradation time
Dislocation density L ( 1013 m 2) Dislocation length L (mm) ˚ Lattice parameter of the precipitate ap (A)
Misfit parameter d (the lattice parameter ˚ of the matrix am is 3.6 A)
Volume fraction of the precipitates fp (%)
0h
600 h
2,000 h
130,000 h
5.0 0.40 NbC: 4.52 M23C6: 10.7
5.5 0.42 M7C3: 8.71 M6C: 10.84 NbC: 4.52 M23C6: 10.7
8.0 0.38 M7C3: 8.71 M6C: 10.84 NbC: 4.52 M23C6: 10.7
4.0 0.46 M6C: 10.84 M23C6: 10.7
dNbC: 0.23 dM23C6: 0.99
dM7C3: 0.83 dM6C: 1.0 dNbC: 0.23 dM23C6: 0.99
dM7C3: 0.83 dM6C: 1.0 dNbC: 0.23 dM23C6: 0.99
dM6C: 1.0 dM23C6: 0.99 (Voids dV: 0.05)
fp(NbC): 1.7 fp(M23C6): 2.8
fp(M7C3): 0.2 fp(M6C): 0.45 fp(NbC): 1.4 fp(M23C6): 2.6
fp(M7C3): 0.9 fp(M6C):1.2 fp(NbC): 0.95 fp(M23C6): 2.5
fp(M6C):1.3 fp(M23C6): 2.7 (Voids fp : 5)
an important role in the distortion of ultrasonic wave propagation during thermal degradation of HP40Nb steels. According to the studies on the acoustic harmonic generation resulting from dislocation under a micro-strain due to the precipitate–matrix lattice misfit [6,12], the expression of the change in the acoustic nonlinearity parameter (in the form of longitudinal wave or longitudinal stress– strain relation) resulting from the application of a stress s on a pinned dislocation can be given by
Db 24 OLL4 R3 E22 ¼ 9s9 5 b0 m3 b2 b0
ð4Þ
where b0 ¼14.1 is the initial value of nonlinearity parameter [25], Db ¼(b b0) is the change of nonlinearity parameter caused by microstructure evolution in material, L is the dislocation density, L is the dislocation length, E2 is the second-order elastic constant, m is the shear modulus of the matrix, b is the Burgers vector, R is the resolving shear factor, and O is the conversion factor from shear strain to longitudinal strain. The strain field resulting from the precipitate–matrix lattice misfit may increase the stress, which can be given by s E2mefp [3], where fp is the volume fraction of the dispersed precipitate phase and e is the coherency strain. For assuming the precipitates to be spherical and elastically isotropic, the coherency strain resulting from a spherical precipitate embedded in a finite matrix is presented as follows [3]
e¼
3K d 3K þ 2Eð1 þ nÞ
ð5Þ
where K is the bulk modulus of the precipitate, and E and n are the Young’s modulus and Poisson’s ratio in the matrix. From Eqs. (4) and (5), it can be seen that Db ¼ CðO,R, m,b, K,E, nÞE22 LL4 dfp is proportional to the product of E22 , L, L4, d, and fp if the coefficient of C(O,R,m,b,K,E,n) kept unchanged during thermal degradation in HP40Nb specimens. Table 1 gives the values of microstructure parameters of the HP40Nb specimens with different thermal degradation time. The density of dislocations was measured by the method proposed by Keh and Weissmann [29] and the average dislocation length was obtained by analyzing the TEM micrograph in computer. Fig. 7 shows the normalized Db (with respect to b0) as a function of the thermal degradation time, which can be obtained by substituting the data of Table 1 into Eqs. (4) and (5). It is clear that the change tendency of the stages I and II of the normalized nonlinear parameter of Lamb wave propagation in Fig. 4 is in good accordance with that predicted by theoretical model. However, there is a little difference between the stage III in Fig. 4 and the
Fig. 7. Graph of normalized Db as a function of thermal degradation time of HP40Nb materials (the abscissa is in logarithmic scale).
stage ‘‘C–D’’ in Fig. 7, that is the measured normalized acoustic nonlinearity A2f =A21f of Lamb wave monotonically decreased in the final stage, whereas the normalized Db of theoretical calculation displayed a hook-shape change in the same stage (see Fig. 7). From Table 1 and Fig. 7, it can be known that the decrease of Db in the stage ‘‘C’’ may be caused by the variances of second phase precipitates, dislocation density and length since there are no other factors causing a reduction of nonlinearity (the factor of change of longitudinal wave velocity has been already taken into account in Fig. 7). However, in the stage ‘‘D’’ of Fig. 7 the microvoids initiation and coalescence make a contribution to the rise in the Db of theoretical calculation from 130,000 h to 160,000 h in the damaged time. As for the experimental measurements of Lamb wave, it can be indicated from the analytical model and Fig. 5c that the decrease of the second phase precipitates and the microstructural changes in materials would play an important role in the drop of the A2f =A21f of Lamb wave in the final stage. Moreover, another important reason affecting the monotonic reduction of nonlinear Lamb waves may be the increasing mismatch of the phase velocity between the primary Lamb wave and the DFLW after long-term high temperature exposure [15,25,26]. In Fig. 8, we present the evolution of longitudinal wave velocity versus the thermal degradation time. It clearly indicates that the material properties relating to ultrasonic velocity such as elastic modulus and Poisson’s ratio definitely changed in the process of thermal damage, which would worsen the matching degree of phase
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Acknowledgements This work is supported by the National Natural Science Foundation of China under grant numbers (50835003, 10972078, 10974256, and 11004056), Natural Science Foundation of Shanghai (09JC1404400), and the National High Technology Research and Development Program (‘‘863’’Program) of China under grant number 2009AA04Z421. References
Fig. 8. Variation in the ultrasonic longitudinal velocity with the thermal degradation of HP40Nb specimens (the abscissa is in logarithmic scale).
velocity [25] and cause the decline of the value of A2f =A21f for Lamb wave propagation in the final stage. According to these analyses, it is a reasonable deduction that a further increase in the scales of the creep micro-voids would give rise to a gradual decrease in the response of A2f =A21f in the stage III in Fig. 4 and finally induce a fracture. It is also worth noting that the obvious rising of the normalized acoustic nonlinearity A2f =A21f of Lamb wave propagation in the early and middle stages is in good agreement with the results of increasing acoustic nonlinearity of Lamb wave with increasing fatigue damage or plasticity deformation in aluminum alloys [20,21]. However, as the thermal exposure progresses, the generation and coalescence of creep micro-voids would have a clear influence on the change of the material properties, which would break the original phase velocity matching condition and thus lower the generation of the second harmonic by Lamb wave propagation [25]. Note that Deng and Pei [23] recently performed nonlinear Lamb wave measurements for fatigue loading in pure aluminum specimens and gave the result of monotonic decrease in the acoustic nonlinearity of Lamb waves as loading cycle progresses, which may also be caused by the continuing worsening of the phase velocity matching condition.
5. Conclusions Thermal degradation of ferritic Cr–Ni alloy plates has been characterized using the nonlinear effect of Lamb waves for the purpose of SHM and life prediction application. The experimental result shows a clear relationship between the normalized acoustic nonlinearity A2f =A21f of Lamb wave propagation and the thermal degradation time during the total damage life in HP40Nb specimens. The variation in the measured acoustic nonlinearity is in good agreement with the analytical model calculation of precipitate–dislocation interaction based on metallographic studies. These results reveal that the acoustic nonlinearity of Lamb wave increases due to the second phase precipitates in the early stage, and it decreases as a combined result of dislocation change and micro-void initiation in the material that cause the phase velocity mismatching between the primary Lamb wave and the double frequency one after long-term high temperature exposure. Consequently, the results from this study show that ultrasonic nonlinearity of Lamb wave has been found to be sensitive to the microstructure evolution in materials during the thermal degradation process, and it could be a potential application for the quantitative assessment of thermal damage in metallic plates or pipes.
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