NDT&E International 72 (2015) 41–49
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Creep degradation characterization of titanium alloy using nonlinear ultrasonic technique Yanxun Xiang a, Wujun Zhu a, Chang-Jun Liu a, Fu-Zhen Xuan a,n, Yi-Ning Wang b, Wen-Chuan Kuang c a Key Laboratory of Pressure Systems and Safety, MOE. School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai 200237, China b Special Equipment Safety Supervision Inspection Institute of Jiangsu Province, Nanjing 211178, China c Shanghai Volkswagen Automotive Corporation Limited, Shanghai 201805, China
art ic l e i nf o
a b s t r a c t
Article history: Received 26 November 2014 Received in revised form 11 February 2015 Accepted 13 February 2015 Available online 23 February 2015
A through transmission nonlinear ultrasonic measurement has been proposed to characterize the creep degradation of titanium alloy that was conducted by creep tests at temperature of 600 1C. The experimental results show a change of “N”-like shape of the acoustic nonlinearity versus the creep loading time, which reveal based on metallographic studies that the variation of acoustic nonlinearity is closely related to the microstructure evolutions. An analytical model calculation has revealed a good agreement with the measured result, which indicates that the precipitate–dislocation interaction is likely the dominant mechanism responsible for the change of acoustic nonlinearity in the crept materials. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Nonlinear ultrasonics Titanium alloys Creep degradation Precipitation Dislocations
1. Introduction Structural metals used in some ﬁelds such as petrochemical industry, fossil power plants and aerospace area are usually subjected to complex service environments, in which fatigue damage, creep damage, or corrosion would occur . Creep is one of the most critical factors for determining the structural integrity of components, which is a potential threat on the safety operation of key equipments. At an elevated temperature, creep deformation is associated with the results of both microstructural changes and strain accumulation, which leads to formation of microvoids, multi-poles, micro-cracks, macro-cracks and gradually to failure. There are usually three stages of creep can be identiﬁed during the creep-induced damage process , which constitutes a primary stage (characterized by a decreased strain rate), secondary stage (or stationary stage with a nearly constant strain rate) and tertiary one (characterized by an acceleration of strain rate). The primary and secondary stages generally account for the longest part of the creep life, especially in the cases of low stress level, which would be more than 0.7 creep life fraction for micro-cracks turning into macro-cracks [3,4]. Therefore, it is of increasing importance to establish an effective non-destructive characterization method of creep degradation at early stage for structural health monitoring (SHM). In
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http://dx.doi.org/10.1016/j.ndteint.2015.02.001 0963-8695/& 2015 Elsevier Ltd. All rights reserved.
recent years, it is noted that nonlinear ultrasonic technique, using the amplitude ratio of primary frequency to higher harmonic frequencies, has been found to be considerably sensitive to microstructure evolution of materials [5–15]. The nonlinear ultrasonic technique relies on measuring the higher order harmonics generated by the intrinsic nonlinearity of materials relating to the anharmonicity and imperfection of atomic lattices. In the ﬁeld of micro-damage evaluation, this technique has experienced a rapid development in recent years. Nagy  applied cyclic bending to generate fatigue damage in the specimens including plastics, metals, composites and adhesives, and reported that nonlinear measurements can be provided as an earlier and more sensitive indicator for fatigue damage characterization in materials. Cantrell  experimentally observed the change in the acoustic nonlinearity parameter resulting from effect of fatigue-induced dislocations, and presented an analytical model that suggested a strong nonlinear interaction of ultrasonic wave with dislocation dipoles during metal fatigue. Walker et al.  used nonlinear Rayleigh waves to characterize the plastic deformation in A36 steel specimens, and the results show an increase in the measured acoustic nonlinearity, which indicates that the acoustic nonlinearity for A36 steel is highly dependent on plastic deformation. Recently, Metya et al.  applied higher order harmonic to study the ageing behavior of C-250 grade maraging steel, whose results showed that the nonlinear parameter β increases with the formation of ﬁner precipitates at the initial stage of ageing and then
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decreases at longer ageing time due to the coarsening of precipitates. Baby et al.  conducted experimental measurements of the higher order harmonics in titanium alloy in order to evaluate their creep damages. A 200% change in β is observed as a function of creep fraction life, which clearly indicates that β is sensitive to damage accumulation during creep deformation. More recently, a low-amplitude nonlinear ultrasonic technique was employed by Balasubramaniam et al.  for characterizing creep damage in copper. Their results show that the nonlinear responses are greater as creep damage accumulates and, among these nonlinear responses, the third harmonic is more sensitive to creepgenerated dislocations than either the second harmonic or static component nonlinear parameter. During creep loading, the formation of various precipitates in heat-resistant alloy would cause local strain concentrations and therefore would produce different dislocations concentrated near the precipitates . The volume fraction of the precipitates has a direct inﬂuence on the changes of dislocation density and average length. Moreover, when the precipitates grow to a critical size, then further growth of the precipitates would lead to the nucleation of misﬁt dislocations and possibly structural phase transformations of the precipitates with an associated change in the misﬁt parameter . These events will cause losses of coherency and material strength, and then give rise to a formation of micro-voids. All these microstructure evolution during creep loading process would produce a very clear nonlinear response in materials [4,8–10]. Though recently some investigations have been made to study thermal ageing behavior in various heat-resistant alloys using nonlinear ultrasonic, there is a lack of information with respect to systematic work and analysis on precipitation kinetics study of creep damage in complex titanium alloy. The present study deals with the characterization of creep damage in heat-resistant titanium alloy Ti60 using the nonlinear effect of ultrasonic longitudinal wave. Measurements of the nonlinear parameter have been carried out on the Ti60 specimens with different creep loading levels. The inﬂuence of microstructure evolution on the ultrasonic nonlinearity has also been thoroughly analyzed based on metallographic studies, such as scanning electron microscope (SEM) and transmission electron microscope (TEM). Moreover, an existing analytical model calculation of precipitate–dislocation interaction  has been performed to interpret the correlation between microstructural evolutions and the measured ultrasonic nonlinearity.
2. Theoretical consideration In order to explain the generation of higher order harmonic waves in solid material, we can introduce the nonlinear stress– strain relationship for describing the material nonlinearity, which can be described by [19,20] 1 σ ij ¼ Aijkl ukl þ Aijklmn ukl umn þ ⋯; 2
represented by (neglecting the attenuation) ∂2 u ∂σ ij ρ 2i ¼ ; ∂xj ∂t
where t is time, ρ is the density of material. Assuming the ultrasonic longitudinal wave propagating along the [1 0 0] direction of the crystal and substituting Eq. (1) into Eq. (2), it can get ∂2 u ∂ 2 ui ∂u ∂2 ui ρ 2i ¼ C 11 þ ð3C 11 þ C 111 Þ i ; 2 2 ∂x ∂x ∂t 1 ∂x1 1
where we considered Eq. (1) up to the second order term. A secondorder perturbation solution of Eq. (3) is u ¼ u0 þu1 , where u0 is a solution with β ¼ 0 and u1 is the ﬁrst order perturbation solution. Then, the solution of Eq. (3) can be obtained by an iterative process 2
uðx; t Þ ¼ A1 sin ðkx ωt Þ
A21 k βx cos 2ðkx ωt Þ; 8
where A1 is the amplitude of the fundamental frequency component, k is the wave number, ω is the angular frequency, β is the acoustic nonlinear parameter which is related to the third-order elastic constants of the material. Due to the material nonlinearity, it can be seen from Eq. (4) that the initial single frequency sinusoidal wave generates higher harmonics components in nonlinear medium. For the second harmonic wave, the amplitude A2 is 2 A21 k βx=8. However, the issue with using Eq. (4) to measure β is that it is based on a plane wave assumption, and ignores the effect of diffraction of the acoustic ﬁeld . Considering a typical experimental measurement with circular transducer coupled to the surface of specimen with a relatively thin thickness, the acoustic ﬁeld is likely to exhibit diffraction and near ﬁeld effect, which would make the nonlinear generation occurred in the specimen to be signiﬁcantly more complex. The consideration of diffraction and near ﬁeld correction of the nonlinear acoustic ﬁeld is explained and detailed by Best et al. . Here, for simplicity of experimental measurement, the amplitude ratio of the measured fundamental and second-harmonic signals, i.e., A2 =A21 is used to describe the change of the nonlinear behavior of a creep degraded material in the following analyses. In general, there have been identiﬁed two primary sources contributed to the nonlinearity of the stress–strain relation. One source is the anharmonicity of the lattice and has been veriﬁed extensively in experiments on single crystals. Another source is the contribution of the nonlinearity in the dislocation movement such as a single dislocation between pinning points under an applied stress, which was predicted by Suzuki et al.  by including nonlinear terms into the dislocation string model. Therefore, the damages accumulated during the high-temperature loading process could be relatively characterized by measuring A2 =A21 .
3. Specimens preparation and experimental setup 3.1. Creep damaged specimens preparation
where σ ij is the stress tensor, uij ¼ ∂ui =∂xj is the strain tensor, ui and xj are the components of the displacement and the position vector. Aijkl and Aijklmn are the second- and third-order Huang coefﬁcients, respectively, which can be expressed in the form of higher order elastic constants as Aijkl ¼ T jl δik þC ijkl and Aijklmn ¼ C jlmn δik þ C ijnl δkm þ C jnkl δim þ C ijklmn , where T jl are the initial stresses in the material at uij ¼ 0 and δik is the Kronecker delta, C ijkl and C ijklmn are the second- and third-order Brugger elastic constants, respectively. Considering the case of a single frequency ultrasonic longitudinal wave propagating in a degraded material, the equation of motion for the longitudinal wave in the material can be
A new near-α high-temperature titanium alloy Ti60 having the composition of Ti–5.8Al–4.0Sn–3.5Zr–0.4Mo–0.4Nb–1.0Ta–0.4Si–0.06C was used in this study. The β transformation temperature of Ti60 was measured to be 1049 1C. The as-received Ti60 alloy was ﬁrst solution treated at 1020 1C for 2 h followed by air cooling and then heated up to 700 1C for 2 h followed by air cooling. For fabricating creep specimens with different loading time fraction, dog-boneshaped Ti60 specimens prepared from the heat-treated material by wire-cut electrodischarge machining (EDM), just as shown in Fig. 1. Accelerated creep tests of constant load were carried out in air at a temperature of 600 1C and a stress of 240 MPa. The time to rupture tr of the Ti60 specimen was determined to be about 210 h
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Fig. 1. Dog-bone-shaped specimen used for creep degradation studies. The dimensions are in mm.
Fig. 3. Experimental setup for nonlinear ultrasonic measurements.
Fig. 2. The creep tension curve of Ti60 specimen.
by a complete creep test until fracture, in which the creep curve exhibits a clear two stages as shown in Fig. 2. It should be pointed out that the primary stage is difﬁcult to be strictly determined in Fig. 2, which however could be a reasonable estimation of about 20 h by referring the proportions of the primary stage reported in literatures [2,4,9].The fracture points are usually located between point 1 and point 2 as indicated in Fig. 1. Two different creep tests, called interrupted tests and continuous tests, were conducted. In the interrupted tests, only one Ti60 sample was used for creep loading. When the loading time reaches a target value of 30 h, we interrupted creep loading and furnace cooled the sample. Then, nonlinear ultrasonic measurements have been carried out on this sample. After measuring the ultrasonic signals, we restarted the creep test and repeated this procedure for every 30 h until the rupture. In the continuous tests, 8 specimens with different creep holding time intervals were prepared for testing at 240 MPa, which were 0 h, 40 h, 80 h, 100 h, 120 h, 140 h, 160 h, 180 h and 200 h. For microstructural characterization, another ﬁve samples with different creep life fractions including 0, 0.3, 0.57, 0.75 and 0.95 were prepared at the same loading conditions and were cut for examination of creep damage. A SEM with type of ZEISS EVO MA15 was used for micrograph observations. A TEM with type of H-800 Hitachi operating at 200 kV was used to observe the morphology of the precipitated phases and the dislocation variations within thin ﬁlm specimens that were prepared by twin jet electronpolishing in an electrolyte containing 10% perchloric acid and 90% ethanol (by volume) at 10 oC and 35 V. In addition, the diffraction patterns of the precipitates also have been performed to identify the lattice parameters and microstructures. 3.2. Experimental setup The experimental setup for nonlinear ultrasonic measurement is illustrated in Fig. 3, which is primarily composed of a high
power tone-burst generator, i.e. the Ritec SNAP system with model of RAM-5000, a high power attenuator, a pre-ampliﬁer, low and high pass ﬁlters, transducers, oscilloscope and computer. Amplitudes of the fundamental and second harmonic wave were measured by through transmission method in the present measurements (7.5 mm thickness). A narrow-band longitudinal piezoelectric transducer with a central frequency of 5.0 MHz was used to generate the fundamental wave, labeled by T. A broad-band 10.0 MHz transducer was used to pick up the signals, labeled by R. Fig. 4a–c give the frequency–amplitude response curves of the high-pass ﬁlter with a cutoff frequency of 9.4 MHz, the transmitter transducer T with a 6 dB bandwidth of 2.42 MHz and the receiver transducer R with a 6 dB bandwidth of 8.96 MHz, respectively. Here, the transmitter and receiver transducers both have the same diameters of 6 mm. A special ﬁxture was designed to maintain alignment parallelism between the transmitting and receiving transducers during the whole measurement, where commercial coupling gel (glycerin) was used as a couplant. A Hanning-windowed 10-cycles tone burst signals with a frequency of 5 MHz were used to excite the transmitter. The received signal was ﬁltered and captured by a 300 MHz oscilloscope and then was processed using an amplitude spectrum Fast Fourier Transformation (FFT) to extract the amplitudes of the fundamental (A1) and second harmonic (A2) signals in the frequency domain.
4. Results and discussions 4.1. Ultrasonic nonlinearity Using the above experimental setup of nonlinear ultrasonic measurement, a typical time-domain signal can be picked up as shown in Fig. 5, in which the fundamental and harmonic waves can be easily observed. Then, 3450 points were extracted from the signal as shown between the two dash lines in Fig. 5 and then was zero-padded into 4096 points for FFT processing. Also, Hanningwindow was applied in FFT processing. Fig. 6 represents frequency
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Fig. 4. (a) The frequency–amplitude response curve of high-pass ﬁlter with a cutoff frequency of 9.4 MHz; (b) the frequency–amplitude response curve of the transmitter transducer T; (c) the frequency–amplitude response curve of the receiver transducer R.
Fig. 5. A typical receiving time-domain signal. The data between the two dash lines was extracted for FFT processing.
contents of the receiving signal via FFT, in which the amplitude of the second-harmonics is near a half of that of the fundamentals due to the inﬂuences of ﬁlters and transducers used in the experiments. There are some points, we assume, should be made to interpret the results of Figs. 5 and 6. First, in order to improve signal-to-noise ratio, the low-pass ﬁlter with cutoff frequency of 6.3 MHz and high-pass ﬁlter with cutoff frequency of 9.4 MHz (see Fig. 4a) were used in the present ultrasonic measurement. When the signals of the second-harmonics and fundamentals pass through the high-pass ﬁlter, the A1 would be suppressed with about 37 dB
Fig. 6. The amplitudes of the fundamental (A1) and second-harmonic (A2) in the frequency domain of the receiving signals for an original specimen.
compared with the A2 due to the frequency–amplitude response of high-pass ﬁlter. Second, both of the second-harmonics and fundamentals can be picked by the broad-band transducer R, but with a different frequency response of the transducer, by which the A1 would again be suppressed with about 8 dB compared with the A2 , as shown in Fig. 4c. Therefore, the total difference of signal suppression caused by receiver transducer and high-pass ﬁlter is about 45 dB, which means that the amplitude of the receiving fundamental signal should be suppressed about 178 times than that of the receiving second-harmonics. Now, in our ultrasonic measurements of Figs. 5 and 6, the fundamental A1 should multiply 178 compared with
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the second-harmonic A2, by which the real amplitude ratio of the fundamental A1 to the second-harmonic A2 would be about 300. The result is in line with those reported in Refs. [6,23], that is the amplitude of second-harmonics is usually 102–103 times less than that of the fundamentals. Additionally, although the thickness of the Ti60 specimen is only 7.5 mm due to the limitation of creep tests, the sound ﬁeld still satisﬁes the far ﬁeld condition because the frequency and element diameter of the transmitting transducer are 5.0 MHz and 6.0 mm, respectively. According to the critical distance N ¼ D2 =4λ, where D is the excitation element diameter and λ is longitudinal wave length in Ti60, and the longitudinal velocity of Ti60 is 6272 m/s, it can be calculated that N is 7.17 mm, which is less than the thickness of the specimen. Note that only the ﬁrst 10 cycles longitudinal tone burst was received and processed, the receiving time-domain signals of Fig. 5 could be a relatively pure longitudinal wave propagating in the Ti60 material. In order to conﬁrm that the second harmonic of the receiving signal is due to the material nonlinearity rather than the spuriousness of the linear signal or the nonlinearity from the experimental instrumentations, the fundamental and second-harmonic signals as a function of propagation distance increment from 20 mm to 50 mm were measured, whose result can be seen in Fig. 7 and shows a linear increase of the A2 =A21 with respect to the distance. Note that there have inevitable second-harmonic signals generated from coupling layer and transducer themselves under the condition of contact excitation and receiving, which can be indicated from Fig. 7 with a nonzero “intercept” at zero distance. However, we kept the same measurement system for the whole ultrasonic wave measurements, and only changed the Ti60 specimens. So, it can be concluded that the measured A2 =A21 as a normalized value could be used to relatively characterize the material nonlinearity, which would not be inﬂuenced by the initial extra nonlinearity (a constant value). Also, it should be pointed out that Fig. 7 was measured by a slice of the intact Ti60 material with a initial distance of 50 mm and then shortening the material at a 10 mm interval after each measurement. As for the creep specimens of continuous test, Fig. 8a shows the variation of A2 =A21 with respect to the creep life fraction, which is carried out at point 3 and point 4 as indicated in Fig. 1 because the upper and lower surfaces of the specimens in these two points can keep more parallel than those in the point 1 and 2 (near the vicinity of necking and fracture in specimen) and are more suitable for measuring the acoustic nonlinearity during the creep loading process. For ease of observation, a B-spline curve was plotted based on the average values of points 3 and 4, as shown in blue line in Fig. 8a. The error bars of the standard deviation are determined by repeating the measurements three times on the specimens. It can be seen that the curve of A2 =A21 exhibits a gradual increase with increasing creep loading time. After showing a peak at a creep life fraction of about 0.6, it quickly decreases to a creep
Fig. 8. (a) The variation of the measured A2 =A21 with respect to the creep life fraction in a continuous test; (b) the variation of the measured A2 =A21 with respect to the creep life fraction in an interrupted test. Here, the blue line is a B-spline ﬁtting curve plotted based on the average values of points 3 and 4. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)
life fraction of 0.8. Afterward, it rises sharply to the rupture. Note that the change of A2 =A21 in Fig. 8a is consistent with the earlier observations reported in Baby et al.  for Ti alloys and Valluri et al.  for pure copper. By the same way, it can be seen from Fig. 8b that the variation of A2 =A21 of the interrupted test specimens is similar to those of the continuous tests. Note that there is a little difference between the curves of Fig. 8a and b, which may be attributed to the different creep loading modes and the slight variations between the initial Ti60 specimens. 4.2. Microstructural evolution analysis
Fig. 7. Curve of A2 =A21 versus the propagation distance.
For further understanding the variation of the A2 =A21 with respect to the creep life fraction, the microstructural evolutions of the creep damaged materials were observed and analyzed by SEM, TEM and electron diffraction. The specimens with different creep life fractions including 0, 0.3, 0.57, 0.75 and 0.95 have been carried out to correlate changes in microstructural parameters with the measured nonlinear longitudinal signals. Fig. 9 represents the typical secondary electron SEM images of the specimens corresponding to an original state and a creep fraction life of 0.57. In the original
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Fig. 9. The typical secondary electron SEM micrographs of Ti60 specimens with different creep damage levels, (a) original state, and (b) creep life fraction of 0.57.
specimen of Fig. 9a, the basic structure consists of not only equiaxed primary α phase and lamellar α phase (transformed β phase), but also some ﬁne white color precipitates congregating in the grain boundaries of the lamellar α phase and in the inner grain of the primary α phase. The selected area electron diffraction (SAED) from the sample reveals that the ﬁne precipitates are identiﬁed to be the α2 ordered phases with chemical constituent of Ti3Al and silicides with chemical constituent of Ti3Si. As for the specimen with creep fraction life of 0.57, its microstructure can be seen from Fig. 9b which reveals a rapid homogenous precipitate and coalescence of the α2 ordered phase in the grain boundaries of the lamellar α phase and in the inner grain of the primary α phase. The further growth and coalescence of the α2 precipitation would cause a signiﬁcant increase of A2 =A21 during the creep damage stage up to about 0.6 life fraction as seen in Fig. 8a because of the fact that, ﬁrst 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 [18,24], second the coalescence of precipitated phases would intensify the coherency stress concentration and therefore promote the increment of material nonlinearity. Actually, for a given creep loading temperature of 600 1C, the coarsening rate of the α2 phase would reach to a maximum one. Therefore, the α2 phase would play a dominant role in the variation of the precipitated phases in Ti60 during the creep loading process. As creep loading proceeds, the precipitation of α2 phase would reach to a saturated state, in which the distributions and sizes of the α2 phase would keep a dynamic balance and stable. Then, further growth of the precipitates would lead to losses of coherency and
Fig. 10. (a) A TEM image of dislocation pile-up in the 0.95 creep life fraction specimen, and (b) a SEM image of micro-void in the 0.95 creep life fraction specimen.
material strength. Subsequent coarsening of the precipitates would result in a clear decrease of the precipitated phases and a release of local strain in material, which is due to the break-off of the precipitations from the matrix under the back stress of the dislocations pile-up and then the generation of micro-voids, just as shown in a typical TEM image of Fig. 10a in the 0.95 creep life fraction specimen. Fig. 10b shows a SEM image with a micro-void in the grain boundary of the lamellar α phase and the primary α phase in the 0.95 creep life fraction specimen. Therefore, the reduction of the α2 phase and silicide and the generation of micro-void seem to exert an inﬂuence on the generation of acoustic nonlinearity. For observing the variation of dislocations in specimens, Fig. 11 shows the bright ﬁeld TEM images of specimens corresponding to an original state and a creep fraction life of 0.57, in which the morphologies of the silicides and α2 phases and the evolutions of the dislocations can be observed. It can be seen from Fig. 11 that the dislocation density of the specimen at 0.57 t=t r is much denser than that of the original specimen, which is caused by the loading stress and the precipitates coalescence during the creep damage process. The purpose of using large scale in Fig. 11b is to make a full view of the dislocation evolution, in which it seems that the inner of lamellar α phase exhibits a sparse dislocation structure with most dislocations concentrated near the coarse silicide precipitates. The variation of the dislocations would also play a signiﬁcant effect on the material nonlinearity, just like the precipitated secondary phases. It deserved
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Fig. 11. The typical TEM micrographs of Ti60 specimens with different creep damage levels, (a) original state, and (b) creep life fraction of 0.57.
that the α2 phase has a cubic structure with lattice constants of l1 ¼5.98 Å and l2 ¼4.88 Å and that the silicide has a complex cubic structure with lattice constants of l1 ¼9.82 Å and l2 ¼5.67 Å, where l1 means the side length of the square bottom surface and l2 means the edge length of the cuboid . Note that a misﬁt parameter δ can be used to describe the mismatch degree between the precipitated phase and the matrix, where δ ¼ 2 ap am = ap þ am , ap and am are the lattice parameters of the precipitate and the matrix [25–26], respectively. The precipitated phase gives rise to a lattice strain due to the mismatch and has an inﬂuence on the distortion of ultrasonic waves. Therefore, the microstructure evolution of creep-damaged Ti60 alloys indicates that the acoustic nonlinearity increases may be due to the rise of the α2 precipitation volume fraction and the dislocation density in the early stage (i.e., before of about 0.6 creep life fraction), and it decreases between the 0.6 and 0.8 creep life fraction because of the reductions of the precipitates volume fraction and the dislocation density. The ﬁnal quick increment of acoustic nonlinearity after 0.8 creep life fraction is due to the increase and coalescence of creep-voids in volume fraction after a further creep loading. In Fig. 12, we present the evolution of the normalized attenuation coefﬁcient of longitudinal wave at a frequency of 5 MHz versus the creep life fraction. It clearly indicates that the acoustic attenuation coefﬁcient sharply rises after about 0.8 creep life fraction, which means the creep microvoid playing a dominant role in the increasing attenuation coefﬁcient and making a major contribution to the increase of the acoustic nonlinearity in Fig. 8a and b. The result of the increase of acoustic attenuation as the creep progresses after 0.8 creep life fraction is consistent with the model calculation reported by Caleap et al., , where the longitudinal and transverse attenuations increase clearly with increasing cavity volume fraction. For being convenient to the analytical model calculation, Table 1 gives the values of microstructure parameters of the Ti60 specimens with different creep life fractions. The density of dislocations was measured by the method of Ref.  and the pinning dislocation length was obtained by analyzing the TEM micrographs, and the volume fraction of the precipitates was obtained by analyzing the SEM micrographs in computer. Note that each mean value in Table 1 related to dislocations or precipitates fraction is determined by an average of ﬁve micrographs from different locations of the specimens. 4.3. Comparison of theoretical calculations to experiments In order to understand the inﬂuence of the microstructure evolution of the creep damaged specimens on the nonlinear ultrasonic response, an analytical model calculation of precipitate–dislocation interaction  could be performed to describe the correlation between them. It can be seen from Figs. 9 and 10 that the dominant microstructure evolutions are the generation and coalescence of the precipitated phase, and the initiation and increase of the micro-voids, which play an important role in the distortion of ultrasonic wave propagation during creep degradation of Ti60 alloys. The expression of the change of the acoustic nonlinearity parameter in the form of longitudinal stress–strain relation resulting from the application of a stress σ on a pinned dislocation can be given by [18,24]
Fig. 12. The evolution of the normalized attenuation coefﬁcient of longitudinal wave versus the creep life fraction.
to point out that the precipitates of α2 phase are coherent with the αTi matrix, which can be identiﬁed by the superlattice spots of the selected area diffraction patterns (SADP). Diffraction analyses reveal
Δβ 24 ΩΛL4 R3 E22 jσ j ¼ β0 5 β0 μ3 b2
nonlinearity parameter of where β0 ¼ 4:67 is the initial acoustic titanium alloy , Δβ ¼ β β0 is the change of nonlinearity parameter caused by microstructure evolution in material, Λ is the dislocation density, L is the dislocation length, E2 is the second-order elastic constant, μ is the shear modulus of the matrix, b is the
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Table 1 Microstructure parameters of the Ti60 specimens with different creep life fractions. Ti60 specimens with different creep life fraction 0
Dislocation density Λ ( 1013 m 2) Pinning dislocation segment L (μm)
Lattice parameter ap (Å) Misﬁt parameter δ (am is a¼ 2.935 Å; c ¼ 4.851 Å)
α2 phase: l1 ¼5.981; l2 ¼4.880; Silicide: l1 ¼ 9.824; l2 ¼ 5.670 δα2: 0.347; δSilicide: 0.507; δVoids:0.15 (only for 0.95 creep life fraction) 
Volume fraction of the precipitates fp (%)
fp(α2): 2.06 fp(Silicide): 7.58
fp(α2): 5.55 fp(Silicide): 8.05
fp(α2): 9.85 fp(Silicide): 8.35
fp(α2): 8.64 fp(Silicide): 8.59
fp(α2): 2.85 fp(Silicide): 6.8 fp(Voids): 12.0
dislocation segment length and the volume of the creep-voids may cause the clear increase of the acoustic nonlinearity after 0.8 creep life fraction until rupture. The results indicate that the effect of the precipitate–dislocation interactions on the nonlinear ultrasonic is likely the dominant mechanism responsible for the change of acoustic nonlinearity. However, note that the predicted results by Eq. (5) are a little larger than those experimental measurements, which we assume may be attributed to the underestimation of the contribution of voids to the generation of acoustic nonlinearity by the analytical model. This underestimation is also reported in Ref.  and means that a more accurate modeling of the voids contribution to the nonlinearity should be considered in the further work.
5. Conclusions Fig. 13. The comparison of the model prediction of the acoustic nonlinearity and the measured A2 =A21 as a function of the creep life fraction.
Burgers vector, R is the resolving shear factor, and Ω 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 σ 2μεf p , where fp is the volume fraction of the dispersed precipitate phase and ε 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 ε ¼ 3Kδ=3K þ 2Eð1 þ νÞ, where K is the bulk modulus of the precipitate, and E and ν are the Young’s modulus and Poisson’s ratio in the matrix. It should be pointed out that the contribution to the nonlinearity parameter from voids or vacancies may be estimated from Eq. (5) with substituting the physical parameters of the precipitated phases by those of the voids . From Eq. (5), it can be seen that Δβ ¼ F ðΩ; R; μ; b; K; E; νÞE22 ΛL4 δf p is proportional to the product of E22 , Λ, L4, δ and fp if the coefﬁcient of F ðΩ; R; μ; b; K; E; νÞ keeps unchanged during the creep degradation in Ti60 specimens. Fig. 13 shows the model prediction of the acoustic nonlinearity parameter as a function of the creep life fraction (the red point and its ﬁtting line), which can be obtained by substituting the data of Table 1 into Eq. (5). For ease of comparison, Fig. 13 also gives the measured A2 =A21 of the continuous tests (the black square and its ﬁtting line), whose change tendency to a certain extent is in good accordance with that predicted by theoretical precipitate–dislocation interaction model. It can be inferred, from Table 1 and Figs. 9–11, that the increase of the volume fraction of the precipitates such as α2 phase and the dislocation density in the Ti60 alloys play a dominant effect on the rise of the nonlinear response at the early stage of before 0.6 creep life fraction. The decrease of the nonlinear response between the 0.6 and 0.8 creep life fraction is due to the reductions of the precipitates volume fraction (α2 phase) and the dislocation density. Finally, the increase of the
The creep damage of titanium alloy Ti60 has been characterized by the nonlinear ultrasonic technique for the purpose of creep life prediction. The experimental results of nonlinear measurements on the creep damaged specimens show a clear relationship between the relative acoustic nonlinearity A2 =A21 and the creep loading time during the total damage life, which displays a gradual increase of A2 =A21 in the early stage of about 0.6 creep life fraction due to the growth of α2 phase and the rise of dislocation density, thereafter a decrease between the 0.6 and 0.8 creep life fraction because of the reductions of the precipitates volume fraction and the dislocation density, and then a ﬁnal quick increment after 0.8 creep life fraction for the increase of the dislocation segment length and the volume of creep-voids. The change of A2 =A21 has been veriﬁed by the microstructural evolution analysis. The variation in the measured A2 =A21 is in good agreement with the analytical model calculation of precipitate–dislocation interaction based on metallographic studies, which reveals that the precipitate–dislocation interaction is likely the dominant mechanism responsible for the change of acoustic nonlinearity in the crept materials.
Acknowledgements This work is supported by National Natural Science Foundations of China (grant nos. 51325504 and 11474093), Shanghai Rising-Star Program (grant no. 14QA1401200) and the Fundamental Research Funds for the Central Universities. References  Raj B, Moorthy V, Jayakumar T, Rao KBS. Assessment of microstructures and mechanical behaviour of metallic materials through non-destructive characterization. Int Mater Rev 2003;48(5):273–325.  Sposito G, Ward C, Cawley P, Nagy PB, Scruby C. A review of non-destructive techniques for the detection of creep damage in power plant steels. NDT E Int 2010;43:555–67.
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