- Email: [email protected]
Random multi-phase medium model and its application in analysis of ultrasonic propagation characteristics for AlSi-polyester abradable seal coating
NDT&E International 108 (2019) 102173
Contents lists available at ScienceDirect
NDT and E International journal homepage: http://www.elsevier.com/locate/ndteint
Random multi-phase medium model and its application in analysis of ultrasonic propagation characteristics for AlSi-polyester abradable seal coating Lin Li a, Zhang Wei a, Ma Zhiyuan a, *, Mingkai Lei b a b
NDT & E Laboratory, Dalian University of Technology, Dalian, 116024, China Surface Engineering Laboratory, School of Materials Science and Engineering, Dalian University of Technology, Dalian, 116024, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Abradable seal coating Elastic spatial distribution Random multi-phase medium model Velocity Attenuation
Ultrasonic nondestructive characterization of multi-phase heterogeneous coatings is challenging due to the complex elastic spatial distribution. In this paper, a two-dimensional Random Multi-phase Medium Model (RMMM) for heterogeneous coatings containing solid-solid-gas three phases is developed, which is able to simulate the elastic spatial distribution of coatings by taking into account material microstructure. This model is then applied to correlate ultrasonic wave velocity and attenuation coefficient with component contents and microstructural characteristics. Microscopic observations were performed for AlSi-polyester abradable seal coatings to examine the morphology and content of AlSi, polyester, and pore. Further statistical analysis of the autocorrelation lengths, orientation angle, mean value, standard deviation, and roughness factor of the coating density distributions was carried out. These statistical characteristics provided quantitative constraints on the construction of a random medium, which were then taken as the basis to establish the RMMM by virtue of a bidirectional peak-valley searching algorithm. The RMMMs for AlSi-polyester abradable seal coating with a thickness of 1.0 mm, polyester content of 47.6%–44.1%, porosity of 0.3%–4.0% were presented. Based on these RMMMs, the effects of component contents, sizes, and spatial distributions on velocity and attenuation were analyzed using two-dimensional Finite-Difference Time-Domain (2D-FDTD) method. Results indicate that when porosity increases from 0.3% to 4.0%, the longitudinal wave velocity decreases from 2251 m/s to 2150 m/s whose relative variation ratio is about 4.5%, while the attenuation coefficient increases from 4.45 dB/mm to 6.56 dB/mm whose relative variation ratio is up to 47.4%. The simulated results show a good consistency with those of the experiments. The modeling method is expected to be effective for the precise interpretation of ul trasonic propagation in other multi-phase heterogeneous coatings.
1. Introduction Abradable seal coating has been widely used in aircraft engines, which can significantly improve engine efficiency and save fuel by reducing the radial air gap between rotating and stationary parts [1,2]. In order to balance the abradability and the erosion resistance, abrad able seal coatings commonly consist of three components: metal matrix, non-metal lubricant, and a certain amount of pores [3]. Taking AlSi-polyester abradable seal coating as an example, the polyester and pore are randomly distributed in the AlSi matrix. The content of poly ester and pore is up to 50% and the sizes of them range from 10 μm to 150 μm. A desired abradable seal coating is one within which the
non-metal lubricant and pores are dispersed and evenly embedded in the metal matrix. If the non-metal lubricant crowds together, the erosion resistance will be reduced, while the local aggregation of pores will easily raise large voids or even expand to cracks, damaging the integrity of coatings [4–6]. Aiming at improving the comprehensive service per formance of abradable seal coatings by precisely controlling coating microstructures, the quantitative description of microstructures was listed as an essential issue in the framework of a European Contract (FP5 research and development framework plan) [7]. Several works have been conducted to investigate the microstructures of heterogeneous materials by using both destructive and non-destructive testing, high lighting the ultrasonic nondestructive testing [8–13]. These ultrasonic
* Corresponding author. E-mail address: [email protected] (M. Zhiyuan). https://doi.org/10.1016/j.ndteint.2019.102173 Received 6 June 2019; Received in revised form 25 September 2019; Accepted 26 September 2019 Available online 27 September 2019 0963-8695/© 2019 Published by Elsevier Ltd.
L. Li et al.
NDT and E International 108 (2019) 102173
techniques have been proven to be effective in the characterization of porosity in CFRP composite [10,11], the measurement of polycrystalline metal grain size [12], and the prediction of cancellous bone density [13]. Different from the above heterogeneous materials, the abradable seal coating coupled with solid-solid-gas three phases has more com ponents and higher heterogeneity. The non-metal lubricant and pores in abradable seal coatings have the feature of irregular shape, large range of size, and random distribution on the microscopic scale. How to effectively describe the complex elastic spatial distribution, and then establish the quantitative correlation between the component contents, microstructural characteristics and the ultrasonic propagation charac teristics (such as velocity and attenuation) is the premise for ultrasonic nondestructive characterization of abradable seal coatings. With the development of advanced aircraft engine to ultra-high speed, large thrust, and long endurance, more attentions have been paid to research coating macroscopic mechanical properties based on the microstructural characteristics. To investigate the effects of micro structures on the macroscopic mechanical properties, Faraoun et al. [14, 15] used quantitative metallographic method to quantify the micro structure characteristics of the AlSi-hBN and NiCrAl-bentonite abrad able seal coatings. The non-metal lubricant and pores in coatings were simplified to ellipses. The shape factor, size factor, and slope factor were used to quantitatively characterize the morphology, size, and orienta tion of the non-metal lubricant and pores. On the basis of these simpli fied models, the elastic modulus and Poisson’s ratios of coatings were calculated utilizing finite element numerical simulation. The simulation results were in good agreement with the theoretical prediction. Dura mou et al. [16] identified the AlSi matrix, polyester, and pore through digital image processing technology based on the metallographic pho tographs of AlSi-polyester abradable seal coating, and directly endowed them with corresponding material properties. The microstructural finite element models, which can accurately reflect the microscopic morphology of polyester and pore, has been constructed. The relative error is about 13% between the predicted elastic modulus and experi mental values. It is still necessary to have suitable models that allow for a more precise description of microstructural characteristics as ultra sonic waves propagation through the multi-phase heterogeneous mate rials. This description must take into account the influence of component contents, as well as component sizes, irregular morphologies, and spatial distributions if component sizes are comparable to the ultrasonic wavelength [17,18]. Therefore, a precise description of the micro structural characteristics is key to the ultrasonic nondestructive char acterization of abradable seal coatings. Lin et al. [19,20] proposed a
Random Void Method (RVM) to describe the random voids in solid-gas two-phase CFRP composites and thermal barrier coatings based on random medium theory. The RVM has advantages of significant en hancements in the description of void morphology and the quantitative correlation between void content and ultrasonic attenuation coefficient. These features make the random medium theory a good alternative to simulate the complex irregular morphologies and spatial distributions of non-metal lubricant and pores in the three-phase abradable seal coatings. To describe the complex elastic spatial distribution of the AlSipolyester abradable seal coatings containing solid-solid-gas three pha ses, a random multi-phase medium modeling method based on random medium theory and statistical method has been developed. Statistical parameters of autocorrelation function such as autocorrelation lengths, orientation angle, mean value, standard deviation, and roughness factor are calculated utilizing the coating density spatial distribution, and are then taken as constraints for the establishment of RMMMs. Subse quently, the effects of constituent contents, sizes, and distributions on the longitudinal wave velocity and attenuation coefficient are analyzed utilizing 2D-FDTD numerical simulation. The validity of RMMM is verified by ultrasonic experiments. 2. Principles 2.1. Random medium theory The random medium theory is appropriate to describe heterogeneous mediums consisting of large- and small-scale heterogeneity [21]. The large-scale heterogeneity is characterized by mean value, which reflects the average characteristics of the entire medium. The small-scale het erogeneity is characterized by autocorrelation lengths and standard deviation, which are used to describe the random disturbances imposed on the average characteristics. For a two-dimensional random medium, Lam�e parameters, λ(x, z) and μ(x, z), and density ρ(x, z) at a spatial coordinate point (x, z) can be expressed as a superposition of a large-scale heterogeneous medium background and a small-scale random disturbance: 8 < λðx; zÞ ¼ λ0 ðx; zÞ þ σ λ ðx; zÞ μðx; zÞ ¼ μ0 ðx; zÞ þ σμ ðx; zÞ (1) : ρðx; zÞ ¼ ρ0 ðx; zÞ þ σ ρ ðx; zÞ where, λ0(x, z), μ0(x, z), and ρ0(x, z) are Lam� e parameters and density of the background medium, respectively, which can be either constant or slowly varying with spatial coordinate (x, z). σλ(x, z), σμ(x, z) and σρ(x, z) represent small disturbances superimposed on the background. For an isotropic elastic material, the relationship between Lam� e parameters and ultrasonic velocity, υL and υS , can be expressed as λ ¼ ρυ2L 2μ and μ ¼ ρυ2S . For simplicity, one usually assumes that the fluctuations of Lam�e parameters are correlated and that the density depends linearly on them according to Birch’s law [22]. In this paper, the density of the heterogeneous material is discussed as a disturbance parameter in the random medium:
ρðx; zÞ ¼ ρ0 ðx; zÞ þ σρ ðx; zÞ
(2)
where, ρ0(x, z) is the average density in the large-scale, which can be either constant or slowly varying with spatial coordinate (x, z). σρ(x, z) is the density fluctuation on the small-scale. On assumption of secondorder spatial stationary random process, the σρ(x, z) can be decom posed into the following formula:
σ ρ ðx; zÞ ¼ δρ ðx; zÞ⋅f ðx; zÞ
(3)
where δρ(x, z) corresponds to the standard deviation of ρ(x, z), f(x, z) is a two-dimensional random sequence which has a mean value equals of zero and standard deviation of 1. The spatial distribution characteristic
Fig. 1. Autocorrelation lengths and orientation angle of 2-D autocorrela tion function. 2
L. Li et al.
NDT and E International 108 (2019) 102173
Fig. 2. Flow chart of random multi-phase medium modeling.
of f(x, z) is subject to the spatial autocorrelation function R(x, z), which describes the correlation degree of physical parameters between two locations. The commonly used intermixed elliptic autocorrelation function is defined as follows: 2 6 Rðx; zÞ ¼ exp6 4
ðcos θ⋅x þ sin θ⋅zÞ2 ðsin θ⋅x þ cos θ⋅zÞ2 þ a2 b2
1 !1þr
2.2. Random multi-phase medium model 2.2.1. Characteristic parameters extraction It can be seen from the theoretical analysis in section 2.1 that the characteristic parameters of autocorrelation function such as mean value, standard deviation, autocorrelation lengths, orientation angle, and roughness factor are important indexes reflecting the elastic spatial distribution of heterogeneous materials. It is the basis of establishing an accurate RMMM to census and extract these characteristic parameters. The mean value ρ0(x, z) and standard deviation δρ(x, z) can be calculated through Eqs. (5) and (6). It is assumed that the density fluctuation of heterogeneous material σρ(x, z) obeys a second-order stationary random process σρ(x, z). The power spectrum density Г (kx, kz) of σρ(x, z) can be obtained according to Eqs. (7)–(9). Where, kx and kz are the spatial frequencies, M and N represent the numbers of sampling points in the x and z directions, respectively. According to Wiener-Khintchine theorem, the spatial autocorrelation function of σρ(x, z) is the inverse Fourier transform of Г(kx, kz). Referencing to Zhang et al. [23], the autocorre lation length is the distance from the maximum value 1 to e 1 for the
3 7 7 5
(4)
where a and b are the autocorrelation length in elliptic major axis and minor axis directions, respectively. r is the roughness factor, describing the microscopic roughness of the boundary [18]. r ¼ 0 corresponds to Gaussian elliptic autocorrelation function and r ¼ 1 corresponds to exponential elliptic autocorrelation function. The larger the roughness value, the rougher the microstructure. For a heterogeneous material, its small-scale heterogeneity usually has a preferred orientation. θ defines the angle between the direction of autocorrelation length a and the x-axis, as shown in Fig. 1. 3
L. Li et al.
NDT and E International 108 (2019) 102173
Fig. 4. Normalized autocorrelation function Rρ(x, z).
2.2.2. Bidirectional peak-valley searching method The random medium model is essentially a continuous medium. However, the abradable seal coatings sprayed by Atmospheric Plasma Spraying (APS) technique are commonly composed of metallic phase, non-metal lubricant, and pores. There is a significant difference in elastic properties and obvious micro-interfaces between constituent phases, which leads to a discrete elastic spatial distribution of hetero geneous materials. The random medium model cannot reflect this key feature. In this work, a new bidirectional peak-valley searching algo rithm is developed to transform the continuous random medium into a discrete RMMM. A RMMM with given polyester content and porosity could be established according to the following principle: all the density values in the random medium were sorted from large to small. Then, a bidirectional searching process was started with peak and valley values, respectively. When the amount percent of the peaks and valleys reached given AlSi content and porosity respectively, the searching process stopped. Thus the threshold ρTH1 and ρTH2 could be determined by reading the density values corresponding to the valley and peak posi tions where the searching program stopped. All areas with ρ(x, z) smaller than ρTH1 are set as pores, the areas with ρ(x, z) greater than ρTH2 are set as AlSi matrix, and the remaining areas are set as polyester. A RMMM with given polyester content and porosity is thus established. Flow chart of random multi-phase medium modeling is shown in Fig. 2.
Fig. 3. SEM micrograph of AlSi-polyester abradable seal coating (a) and its density spatial distribution (b).
normalized autocorrelation function Rρ(x, z). The characteristic pa rameters of autocorrelation function such as autocorrelation lengths (a, b), orientation angle θ, and roughness factor r can be obtained by fitting the experimental autocorrelation function Rρ(x, z) with the theoretical autocorrelation function Re(x, z) utilizing the Least Square Method (LSM). 1 XM 1 XN 1 ρ0 ðx; zÞ ¼ ρ ðx; zÞ z¼0 m M � N x¼0 δ2ρ ðx; zÞ ¼
1 XM 1 XN 1 ðρm ðx; zÞ z¼0 M � N x¼0
Iðkx ; kz Þ ¼
1 2 jIðkx ; kz Þj M�N
Rðx; zÞ ¼ IFFT2ðΓðkx ; kz ÞÞ
3.1. Microscopic observation and characteristic parameter statistics Microscopic observation was carried out through the Scanning Electron Microscope (SEM). Fig. 3(a) shows one typical cross-section SEM micrograph of AlSi-polyester abradable seal coatings. The gray matrix is AlSi, the black particles are polyesters, and the dark black parts are pores. Based on the difference of grayscale level of each component in the SEM micrograph, the threshold segmentation technology was applied to identify and reconstruct the AlSi matrix, polyester, and pores. Then the corresponding densities were endowed. Subsequently, the coating density spatial distribution ρm(x, z) was obtained (Fig. 3(b)). The mean density ρ0(x, z) ¼ 1.941 g/cm3 and standard deviation δρ(x, z) ¼ 0.998 g/cm3 were calculated according to Eqs. (5) and (6). Fig. 4 shows the normalized autocorrelation function Rρ(x, z) of coating den sity spatial distribution. Autocorrelation length a ¼ 13 μm, b ¼ 12 μm, orientation angle θ ¼ 0, and roughness factor r ¼ 0.52 were calculated using the LSM. Table 1 shows the statistical results of characteristic parameters in 5 groups of coating specimens with different levels of
(5)
ρ0 ðx; zÞÞ
� 1 XM 1 XN 1 ρ ðx; zÞ � exp m z¼0 M � N x¼0
Γðkx ; kz Þ ¼
3. Modeling and numerical simulation
2
(6) �
j2π
xkx zkz þ M N
�� (7) (8) (9)
4
L. Li et al.
NDT and E International 108 (2019) 102173
Table 1 Constituent contents and statistical characteristics of coating density spatial distribution. Specimen
1# 2# 3# 4# 5#
Density of coatings (g/cm3)
Parameters of autocorrelation function
Mean value
Standard deviation
Autocorrelation length a (μm)
Autocorrelation length b (μm)
Oriented angle θ (rad)
Roughness factor r
Polyester
Pore
AlSi
1.941 1.931 1.913 1.903 1.886
0.998 1.012 1.022 1.029 1.033
13 14 15 16 19
12 13 14 14 17
0 0 0 0 0
0.52 0.53 0.52 0.49 0.53
47.6 46.7 46.0 45.1 44.1
0.3 1.1 2.1 2.9 4.0
52.1 52.2 51.9 52.0 51.9
Volume fraction (%)
coating specimens as described in section 3.1, each group contains 5 random elastic spatial distributions) have been constructed according to the statistical parameters in Table 1. Fig. 5(a) shows the random me dium with a ¼ 13 μm, b ¼ 12 μm, θ ¼ 0, r ¼ 0.52, and the corresponding RMMM (Fig. 5(b)) with a 47.6% polyester and 0.3% porosity is con structed using the bidirectional peak-valley searching algorithm. Fig. 6 shows the SEM micrographs and corresponding RMMMs with different levels of porosity P. It is evident that RMMM is suitable to describe the random spatial distributions and irregular morphologies of polyester and pores in the AlSi-polyester abradable seal coatings. 3.3. 2D-FDTD numerical simulation Based on the 5 groups of RMMMs constructed in section 3.2, 2DFDTD numerical simulation [24] was used to calculate the wave prop agation in AlSi-polyester abradable seal coatings. The 2D-FDTD model based on RMMM is essentially a mathematical description of the microstructure statistical characteristics and the elastic spatial distri bution of the abradable seal coatings. Different from the previous model which assumed that the scatterers shape is regular, uniform in size and uniformly distributed, the 2D-FDTD model based on RMMM can reflect the irregular morphology of polyester and pores, meanwhile describe the complex spatial distributions of polyester and pores in the three-phase coatings. To obtain accurate and stable calculation results, a small mesh grid size is very important for the numerical simulation. In this paper, the uniform squared grid division in cartesian coordinate system has been used. Typically, the mesh grid size in 2D-FDTD should be less than a fifteenth of the shortest wavelength for numerical convergence [25]. With the consideration of the numerical convergence limit of about 48 μm and the smallest pore limit of about 10 μm, the grid width was set as 1 μm. Fig. 7 shows an example of the numerical simulation model, where the coating thickness is 1.0 mm and the water layer thickness is 3.0 mm. The left and right boundaries were set as symmetry boundaries to avoid converted waves. The bottom boundary was set as a Perfectly Matched Layer (PML). The top boundary loads the excitation source and the receiving probe. Ultrasonic longitudinal wave excitation signal, as shown in Fig. 9(a), perpendicularly propagated into the multi-layer water/coating/alloy substrate structure. AlSi matrix, polyester, and pores were set as isotropic homogeneous materials, whose materials properties listed in Table 2 [16]. 4. Experiments 4.1. Specimen preparation Rectangular GH4169 superalloy specimens, with dimensions 50 mm � 25 mm � 5 mm were used as substrates. Prior to the coating operation, the substrates were grit-blasted. Then the AlSi-polyester abradable seal coatings with thicknesses of 1.5–1.8 mm were sprayed using UNICOAT APS system under optimized spray parameters. The content of AlSi is about 52% and that of polyester is about 48% in the original specimens. In order to obtain coating specimens with different porosity, the original specimens were heat-treated at different times. Polyester was partially ablated above its decomposition temperature,
Fig. 5. Random medium (a) and random multi-phase medium model (b).
porosity. In each specimen, the parameter statistical process was carried out on 20 SEM micrographs to ensure the reliability of data. 3.2. Random multi-phase medium modeling In this section, 5 groups of RMMMs (corresponding to the 5 groups of 5
L. Li et al.
NDT and E International 108 (2019) 102173
Fig. 6. Comparison of SEM micrographs (a, c, e, g) and RMMMs (b, d, f, h): (a, b) P ¼ 1.1%; (c, d) P ¼ 2.1%; (e, f) P ¼ 2.9%; (g, h) P ¼ 4.0%.
Fig. 7. 2D-FDTD numerical simulation model.
Fig. 9. Excitation source time-domain waveform (a) and its amplitude spec trum (b).
and thus form pores. In this way, 5 groups of coating specimens with different contents of polyester and pore were obtained under different temperatures. The prepared AlSi-polyester abradable seal coating specimens are shown in Fig. 8, the left is the coating sample, and the
Fig. 8. Photo of the coating sample and the SEM observation sample.
6
L. Li et al.
NDT and E International 108 (2019) 102173
Table 2 Material properties used in the simulation. Materials Water (20 C) AlSi Polyester Air(20 � C) GH4169 �
Elastic modulus (GPa)
Poisson’s ratio
Density (g/cm3)
— 71.0 3.5 1.0 � 10 199.9
0.50 0.32 0.38 — 0.30
1.00 2.41 1.44 1.20 � 10 8.24
4
3
Fig. 10. Time-domain waveforms of experiments and numerical simulations.
right is the SEM microscopic observation sample. Fig. 11. Comparison of experiments and numerical simulations: (a) longitu dinal wave velocity; (b) attenuation coefficient.
4.2. Ultrasonic experiments The longitudinal wave velocity and attenuation coefficient have been measured using water immersion pulse-echo method. Ultrasonic experiments were carried out on the AlSi-polyester specimens using a short-duration pulse generated by GE USIP40 ultrasonic flaw detector and broadband, water immersion, flat transducer with a nominal fre quency of 5.0 MHz. The effective broadband range ( 6 dB) of the transducers was 2.95–8.35 MHz, as shown in Fig. 9. The received signals were recorded and stored by DPO4032 digital oscilloscope with a 200 MHz sampling frequency. Each measurement was averaged 64 times for improving the signal-to-noise ratio. Due to the large roughness of the coating surface, 1000# sandpaper was used to properly polish the coating before ultrasonic experiments. Each specimen has a measured coating thickness of 1.0 mm. The ultrasonic testing system was fully automated and controlled by a computer. The accuracy of the control system was 0.01 mm, which was enough to ensure the displacement reproducibility. By scanning each specimen along the direction parallel to the coating cross-section with a resolution of 1.0 mm, 25 waveforms were collected. Then the longitudinal wave velocity υL and attenuation coefficient αL in different coating specimens were calculated by Eqs. (10) and (11) as given by:
υL ¼
2d Δt
" ! 10 A1 lg þ lg 1 αL ¼ d A2
þ ρw*υLw) and R23¼(ρs*υLs – ρc*υLc)/(ρs*υLs þ ρc*υLc) are the acoustic reflection coefficients at the water/coating and coating/substrate in terfaces, respectively. Where ρw and ρs are the densities of water and substrate. υLw and υLs are the longitudinal wave velocities of water and substrate. ρc and υLc are the mean density and the mean longitudinal wave velocity of AlSi-polyester abradable seal coating specimens, respectively. 5. Results and discussion 5.1. Comparison of waveforms The simulated and experimental time-domain waveforms were shown in Fig. 10 (coating thickness of 1.0 mm, polyester content of 46.7%, porosity of 1.1%). It is found that the simulated waveforms show a good consistency with those from experiments. The FDTD numerical simulation based on RMMM reproduces the scattering echoes appeared in experiments, and the fluctuations of interface echoes are also observed under the same polyester content and porosity. Due to the limitation of manufacturing and microscopic observation technology, it is difficult to prepare a series of coating specimens with specified component contents. The RMMM method can flexibly simulate the elastic spatial distribution that conforms to the specified statistical characteristics. Simulated time-domain signals from the RMMM could carry a lot of information about the component contents and micro structural characteristics. So a further study of ultrasonic propagation characteristics in multi-phase heterogeneous materials can be realized utilizing the RMMM.
(10) !# 1
R12 � R212 R23
(11)
where d is the thickness of the coating. A1 and A2 are the amplitude of coating surface and interface echoes respectively. Δt is the time of flight between the surface and interface echoes. R12¼(ρc*υLc – ρw*υLw)/(ρc*υLc 7
L. Li et al.
NDT and E International 108 (2019) 102173
Fig. 12. RMMMs of AlSi-polyester abradable seal coating (polyester content 44.1%, porosity 4.0%).
5.2. Analysis of ultrasonic propagation characteristics
numerical simulation. The dispersion of attenuation coefficients in creases with the increase of porosity. When porosity increases up to 4.0%, the fluctuation between experimental attenuation coefficients reaches 1.85 dB/mm. Fig. 12 shows the RMMMs with same polyester content of 44.1% and porosity of 4.0%. Fig. 12(a)–(d) corresponds to the different attenuation coefficients of 5.8 dB/mm, 6.8 dB/mm, 6.9 dB/ mm, and 7.3 dB/mm, respectively, and the maximum relative change rate is up to 25.9%. The corresponding longitudinal wave velocities are 2145 m/s, 2153 m/s, 2150 m/s, and 2137 m/s, respectively, and the maximum relative change rate is less than 1%. Compared with the longitudinal wave velocity, the attenuation coefficient is more sensitive to the microscopic morphology and spatial distribution of pores. It is found that local aggregation of pores appeared in Fig. 12(b)–(d), and some small pores in the coating interconnected to form large voids. According to elastic wave scattering theory [26], wave scattering can be divided into Rayleigh scattering (kD≪1), random scattering (0.1 < kD < 10), and diffuse scattering (kD≫1) depending on the normalized wave number kD (D is the mean diameter of the scatters, k is the wave number, k ¼ 2π/λ). The diameter of pores in AlSi-polyester coating ranges from 10 μm to 50 μm. At the nominal frequency of 5.0 MHz, the average longitudinal velocity in the coating is about 2000 m/s–2300 m/s, so kD is between 0.1 and 0.8. When the ultrasonic wave interacts with the pores in AlSi-polyester coatings, intense random scattering will occur, and the scattered energy is in direct proportion to the size of pores [11,18,19]. The pores aggregation state and the pres ence of large voids will further intensify the ultrasonic attenuation. It should be pointed out that the types, sizes, orientations, shapes,
5.2.1. Effect of component contents on ultrasonic characteristics The longitudinal wave velocity and attenuation coefficient as a function of porosity and polyester content obtained from numerical calculations and experiments are shown in Fig. 11. It is found that, with the increase of porosity, both the experimental and simulated longitu dinal wave velocities decrease slowly, while the attenuation coefficients tend to increase gradually. When the porosity increases from 0.3% to 4.0%, the average longitudinal wave velocity decreases from 2251 m/s to 2150 m/s whose relative variation ratio is about 4.5%. Meanwhile, the average attenuation coefficient increases rapidly from 4.45 dB/mm to 6.56 dB/mm whose relative variation ratio is up to 47.4%. Since the velocity of air is much less than that of polyester, the equivalent velocity of the coating gradually decreases with the increase of porosity. The pores in the coating are mainly formed by high-temperature ablation of polyester, and the total content of polyester and pores in the coating remains unchanged. However, since the acoustic impedance difference between pore and AlSi matrix is much greater than that between poly ester and AlSi matrix, the appearance of pores intensifies the multiple scattering of scatterers, resulting in a gradual increase of the attenuation coefficient with the increase of porosity. 5.2.2. Effect of microstructure on ultrasonic characteristics It can also be seen from Fig. 11 that there is an obvious multi-value correspondence relationship between the velocity or attenuation coef ficient and porosity both in the experimental measurement and RMMM 8
L. Li et al.
NDT and E International 108 (2019) 102173
spatial distribution of scatterers, and other microscopic structural characteristics will all have impacts on the ultrasonic propagation characteristics. Furthermore, the coupling of various factors leads to the ultrasonic propagation more complicated. In the following works, we will use RMMM to quantitatively analyze the influence of the micro structures of abradable seal coatings on ultrasonic propagation characteristics.
[4] Tang JJ, Liu K, Yang QZ, et al. The influence of size and distribution of graphite on the friction and wear behavior of Ni-graphite coatings. Surf Coat Technol 2014; 252:48–55. https://doi.org/10.1016/j.surfcoat.2014.04.063. [5] Johnston RE. Mechanical characterisation of AlSi-hBN, NiCrAl-Bentonite, and NiCrAl-Bentonite-hBN freestanding abradable coatings. Surf Coat Technol 2011; 205:3268–73. https://doi.org/10.1016/j.surfcoat.2010.11.044. [6] Lotfi B, Shipway PH, McCartney DG, et al. Abrasive wear behaviour of Ni(Cr)TiB2coatings deposited by HVOF spraying of SHS-derived cermet powders. Wear 2003;254(3–4):340–9. https://doi.org/10.1016/S0043-1648(03)00014-0. [7] European Commission FP5 Growth Program. Abradable seal coatings and claddings for compressor applications. Seal Coat Proj No GRD1-2001-40124. [8] Ma ZY, Zhang W, Du PC, et al. Nondestructive measurement of elastic modulus for thermally sprayed WC-Ni coatings based on acoustic wave mode conversion by small angle incidence. NDT E Int 2018;94:38–46. https://doi.org/10.1016/j. ndteint.2017.11.003. [9] Lin L, Cao HQ, Luo ZB. Total focusing method imaging of multidirectional CFRP laminate with model-based time delay correction. NDT E Int 2018;97:51–8. https://doi.org/10.1016/j.ndteint.2018.03.011. [10] Ding SS, Jin SJ, Luo ZB, et al. Researches on the ultrasonic scattering attenuation of carbon fibre reinforced plastics with 2D real morphology void model. Acoust Phys 2017;63(4):490–5. https://doi.org/10.1134/s1063771017040029. [11] Kim KB, Hsu DK, Barnard DJ. Estimation of porosity content of composite materials by applying discrete wavelet transform to ultrasonic backscattered signal. NDT E Int 2013;56:10–6. https://doi.org/10.1016/j.ndteint.2013.01.014. [12] Zhang C, Li XB, Song YF, et al. Evaluating the grain size in curved components using the ultrasonic attenuation method with diffraction correction. NDT E Int 2016;84:20–6. https://doi.org/10.1016/j.ndteint.2016.07.004. [13] Wear KA, Nagaraja S, Dreher ML, et al. Relationships of quantitative ultrasound parameters with cancellous bone microstructure in human calcaneus in vitro. J Acoust Soc Am 2012;131(2):1605–12. https://doi.org/10.1121/1.3672701. [14] Faraoun HI, Grosdidier T, Seichepine JL, et al. Improvement of thermally sprayed abradable coating by microstructure control. Surf Coat Technol 2006;201: 2303–12. https://doi.org/10.1016/j.surfcoat.2006.03.047. [15] Faraoun HI, Seichepine JL, Coddet C, et al. Modelling route for abradable coatings. Surf Coat Technol 2006;200:6578–82. https://doi.org/10.1016/j. surfcoat.2005.11.105. [16] Duramou Y, Bolot R, Seichepine JL, et al. Relationships between microstructural and mechanical properties of plasma sprayed AlSi-Polyester composite coatings: application to abradable materials. Key Eng Mater 2014;606:155–8. https://doi. org/10.4028/www.scientific.net/kem.606.155. [17] Barguet L, Pezerat C, Bentahar M, et al. Ultrasonic evaluation of the morphological characteristics of metallic powders in the context of mechanical alloying. Ultrasonics 2015;60:11–8. https://doi.org/10.1016/j.ultras.2015.02.012. [18] Lin L, Chen J, Zhang X, et al. A novel 2-D random void model and its application in ultrasonically determined void content for composite materials. NDT E Int 2011;44 (3):254–60. https://doi.org/10.1016/j.ndteint.2010.12.003. [19] Lin L, Zhang X, Chen J, et al. A novel random void model and its application in predicting void content of composites based on ultrasonic attenuation coefficient. Appl Phys Mater Sci Process 2011;103(4):1153–7. https://doi.org/10.1007/ s00339-010-6061-x. [20] Zhao Y, Li XM, Ma ZY, et al. Correlating ultrasonic velocity and porosity using FDTD method based on random pores model. Mater Sci Forum 2011;675–677: 1221–4. https://doi.org/10.4028/www.scientific.net/msf.675-677.1221. [21] Ikelle LT, Yung SK, Daube F. 2-D random media with ellipsoidal autocorrelation functions. Geophysics 2002;58(9):1359–72. https://doi.org/10.1190/1.1443518. [22] Birch F. The velocity of compressional waves in rocks to 10 kilobars: 2. J Geophys Res 1961;66(7):2199–224. https://doi.org/10.1029/jz066i007p02199. [23] Zhang Y, Sundararajan S. Generating random surfaces with desired autocorrelation length. Appl Phys Lett 2006;88(14):16–9. https://doi.org/10.1063/1.2191882. [24] Schechter RS, Chaskelis HH, Mignogna RB, et al. Real-time parallel computation and visualization of ultrasonic pulses in solids[J]. Science 1994;265(5176): 1188–92. https://doi.org/10.1126/science.265.5176.1188. [25] Chen Y, Luo ZB, Zhou Q, et al. Modeling of ultrasonic propagation in heavy-walled centrifugally cast austenitic Stainless steel based on EBSD analysis. Ultrasonics 2015;59:31–9. https://doi.org/10.1016/j.ultras.2015.01.009. [26] Reynolds WN, Smith RL. Ultrasonic wave attenuation spectra in steels. J Phys D Appl Phys 2000;17(1):109–16. https://doi.org/10.1088/0022-3727/17/1/015.
6. Conclusions In this paper, we developed a two-dimensional random multi-phase medium model, which was used to simulate the complex elastic spatial distribution of the AlSi-polyester abradable seal coatings containing solid-solid-gas three phases. Based on the statistical characteristics of the coating density spatial distribution, efficient RMMMs for the AlSipolyester abradable seal coating with a thickness of 1.0 mm, polyester content of 47.6%–44.1%, porosity of 0.3%–4.0% were constructed. Then the RMMMs were applied to correlate velocity and attenuation coefficient with component contents and microstructural characteristics through 2D-FDTD numerical simulation. It is found that the RMMMs present good similarities with the SEM micrographs. The maximum relative errors between the fitting curve of velocity and attenuation coefficient predicted by the model and the mean value measured by the experiments are 1.01% and 3.97%, respectively. The 2D-FDTD simu lated velocity and attenuation coefficient based on RMMMs agree well with those of experiments, which indicates that this method is a good candidate for accurately analyzing ultrasonic propagation characteris tics in other multi-phase heterogeneous coatings. Declaration of competing interest We declare that we do not have any commercial or associative in terest that represents a conflict of interest in connection with the work submitted. Acknowledgments This work is supported by National Natural Science Foundation of China (NSFC) under Grant No. 51675083, No. 51805072 and National Basic Research Program of China under Grant No. 2015CB057306. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.ndteint.2019.102173. References [1] Bill RC, Ludwig LP. Wear of seal materials used in aircraft propulsion systems. Wear 1980;59(1):165–89. https://doi.org/10.1016/0043-1648(80)90277-X. [2] DeMasi-Marcin JT, Gupta DK. Protective coatings in the gas turbine engine. Surf Coat Technol 1994;68–69:1–9. https://doi.org/10.1016/0257-8972(94)90129-5. [3] Chupp RE, Ghasripoor F, Turnquist NA, et al. Advanced seals for industrial turbine applications: dynamic seal development. Int J Turbomach Propuls Power 2002;18 (6):1260–6. https://doi.org/10.2514/2.6061.
9
Contents lists available at ScienceDirect
NDT and E International journal homepage: http://www.elsevier.com/locate/ndteint
Random multi-phase medium model and its application in analysis of ultrasonic propagation characteristics for AlSi-polyester abradable seal coating Lin Li a, Zhang Wei a, Ma Zhiyuan a, *, Mingkai Lei b a b
NDT & E Laboratory, Dalian University of Technology, Dalian, 116024, China Surface Engineering Laboratory, School of Materials Science and Engineering, Dalian University of Technology, Dalian, 116024, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Abradable seal coating Elastic spatial distribution Random multi-phase medium model Velocity Attenuation
Ultrasonic nondestructive characterization of multi-phase heterogeneous coatings is challenging due to the complex elastic spatial distribution. In this paper, a two-dimensional Random Multi-phase Medium Model (RMMM) for heterogeneous coatings containing solid-solid-gas three phases is developed, which is able to simulate the elastic spatial distribution of coatings by taking into account material microstructure. This model is then applied to correlate ultrasonic wave velocity and attenuation coefficient with component contents and microstructural characteristics. Microscopic observations were performed for AlSi-polyester abradable seal coatings to examine the morphology and content of AlSi, polyester, and pore. Further statistical analysis of the autocorrelation lengths, orientation angle, mean value, standard deviation, and roughness factor of the coating density distributions was carried out. These statistical characteristics provided quantitative constraints on the construction of a random medium, which were then taken as the basis to establish the RMMM by virtue of a bidirectional peak-valley searching algorithm. The RMMMs for AlSi-polyester abradable seal coating with a thickness of 1.0 mm, polyester content of 47.6%–44.1%, porosity of 0.3%–4.0% were presented. Based on these RMMMs, the effects of component contents, sizes, and spatial distributions on velocity and attenuation were analyzed using two-dimensional Finite-Difference Time-Domain (2D-FDTD) method. Results indicate that when porosity increases from 0.3% to 4.0%, the longitudinal wave velocity decreases from 2251 m/s to 2150 m/s whose relative variation ratio is about 4.5%, while the attenuation coefficient increases from 4.45 dB/mm to 6.56 dB/mm whose relative variation ratio is up to 47.4%. The simulated results show a good consistency with those of the experiments. The modeling method is expected to be effective for the precise interpretation of ul trasonic propagation in other multi-phase heterogeneous coatings.
1. Introduction Abradable seal coating has been widely used in aircraft engines, which can significantly improve engine efficiency and save fuel by reducing the radial air gap between rotating and stationary parts [1,2]. In order to balance the abradability and the erosion resistance, abrad able seal coatings commonly consist of three components: metal matrix, non-metal lubricant, and a certain amount of pores [3]. Taking AlSi-polyester abradable seal coating as an example, the polyester and pore are randomly distributed in the AlSi matrix. The content of poly ester and pore is up to 50% and the sizes of them range from 10 μm to 150 μm. A desired abradable seal coating is one within which the
non-metal lubricant and pores are dispersed and evenly embedded in the metal matrix. If the non-metal lubricant crowds together, the erosion resistance will be reduced, while the local aggregation of pores will easily raise large voids or even expand to cracks, damaging the integrity of coatings [4–6]. Aiming at improving the comprehensive service per formance of abradable seal coatings by precisely controlling coating microstructures, the quantitative description of microstructures was listed as an essential issue in the framework of a European Contract (FP5 research and development framework plan) [7]. Several works have been conducted to investigate the microstructures of heterogeneous materials by using both destructive and non-destructive testing, high lighting the ultrasonic nondestructive testing [8–13]. These ultrasonic
* Corresponding author. E-mail address: [email protected] (M. Zhiyuan). https://doi.org/10.1016/j.ndteint.2019.102173 Received 6 June 2019; Received in revised form 25 September 2019; Accepted 26 September 2019 Available online 27 September 2019 0963-8695/© 2019 Published by Elsevier Ltd.
L. Li et al.
NDT and E International 108 (2019) 102173
techniques have been proven to be effective in the characterization of porosity in CFRP composite [10,11], the measurement of polycrystalline metal grain size [12], and the prediction of cancellous bone density [13]. Different from the above heterogeneous materials, the abradable seal coating coupled with solid-solid-gas three phases has more com ponents and higher heterogeneity. The non-metal lubricant and pores in abradable seal coatings have the feature of irregular shape, large range of size, and random distribution on the microscopic scale. How to effectively describe the complex elastic spatial distribution, and then establish the quantitative correlation between the component contents, microstructural characteristics and the ultrasonic propagation charac teristics (such as velocity and attenuation) is the premise for ultrasonic nondestructive characterization of abradable seal coatings. With the development of advanced aircraft engine to ultra-high speed, large thrust, and long endurance, more attentions have been paid to research coating macroscopic mechanical properties based on the microstructural characteristics. To investigate the effects of micro structures on the macroscopic mechanical properties, Faraoun et al. [14, 15] used quantitative metallographic method to quantify the micro structure characteristics of the AlSi-hBN and NiCrAl-bentonite abrad able seal coatings. The non-metal lubricant and pores in coatings were simplified to ellipses. The shape factor, size factor, and slope factor were used to quantitatively characterize the morphology, size, and orienta tion of the non-metal lubricant and pores. On the basis of these simpli fied models, the elastic modulus and Poisson’s ratios of coatings were calculated utilizing finite element numerical simulation. The simulation results were in good agreement with the theoretical prediction. Dura mou et al. [16] identified the AlSi matrix, polyester, and pore through digital image processing technology based on the metallographic pho tographs of AlSi-polyester abradable seal coating, and directly endowed them with corresponding material properties. The microstructural finite element models, which can accurately reflect the microscopic morphology of polyester and pore, has been constructed. The relative error is about 13% between the predicted elastic modulus and experi mental values. It is still necessary to have suitable models that allow for a more precise description of microstructural characteristics as ultra sonic waves propagation through the multi-phase heterogeneous mate rials. This description must take into account the influence of component contents, as well as component sizes, irregular morphologies, and spatial distributions if component sizes are comparable to the ultrasonic wavelength [17,18]. Therefore, a precise description of the micro structural characteristics is key to the ultrasonic nondestructive char acterization of abradable seal coatings. Lin et al. [19,20] proposed a
Random Void Method (RVM) to describe the random voids in solid-gas two-phase CFRP composites and thermal barrier coatings based on random medium theory. The RVM has advantages of significant en hancements in the description of void morphology and the quantitative correlation between void content and ultrasonic attenuation coefficient. These features make the random medium theory a good alternative to simulate the complex irregular morphologies and spatial distributions of non-metal lubricant and pores in the three-phase abradable seal coatings. To describe the complex elastic spatial distribution of the AlSipolyester abradable seal coatings containing solid-solid-gas three pha ses, a random multi-phase medium modeling method based on random medium theory and statistical method has been developed. Statistical parameters of autocorrelation function such as autocorrelation lengths, orientation angle, mean value, standard deviation, and roughness factor are calculated utilizing the coating density spatial distribution, and are then taken as constraints for the establishment of RMMMs. Subse quently, the effects of constituent contents, sizes, and distributions on the longitudinal wave velocity and attenuation coefficient are analyzed utilizing 2D-FDTD numerical simulation. The validity of RMMM is verified by ultrasonic experiments. 2. Principles 2.1. Random medium theory The random medium theory is appropriate to describe heterogeneous mediums consisting of large- and small-scale heterogeneity [21]. The large-scale heterogeneity is characterized by mean value, which reflects the average characteristics of the entire medium. The small-scale het erogeneity is characterized by autocorrelation lengths and standard deviation, which are used to describe the random disturbances imposed on the average characteristics. For a two-dimensional random medium, Lam�e parameters, λ(x, z) and μ(x, z), and density ρ(x, z) at a spatial coordinate point (x, z) can be expressed as a superposition of a large-scale heterogeneous medium background and a small-scale random disturbance: 8 < λðx; zÞ ¼ λ0 ðx; zÞ þ σ λ ðx; zÞ μðx; zÞ ¼ μ0 ðx; zÞ þ σμ ðx; zÞ (1) : ρðx; zÞ ¼ ρ0 ðx; zÞ þ σ ρ ðx; zÞ where, λ0(x, z), μ0(x, z), and ρ0(x, z) are Lam� e parameters and density of the background medium, respectively, which can be either constant or slowly varying with spatial coordinate (x, z). σλ(x, z), σμ(x, z) and σρ(x, z) represent small disturbances superimposed on the background. For an isotropic elastic material, the relationship between Lam� e parameters and ultrasonic velocity, υL and υS , can be expressed as λ ¼ ρυ2L 2μ and μ ¼ ρυ2S . For simplicity, one usually assumes that the fluctuations of Lam�e parameters are correlated and that the density depends linearly on them according to Birch’s law [22]. In this paper, the density of the heterogeneous material is discussed as a disturbance parameter in the random medium:
ρðx; zÞ ¼ ρ0 ðx; zÞ þ σρ ðx; zÞ
(2)
where, ρ0(x, z) is the average density in the large-scale, which can be either constant or slowly varying with spatial coordinate (x, z). σρ(x, z) is the density fluctuation on the small-scale. On assumption of secondorder spatial stationary random process, the σρ(x, z) can be decom posed into the following formula:
σ ρ ðx; zÞ ¼ δρ ðx; zÞ⋅f ðx; zÞ
(3)
where δρ(x, z) corresponds to the standard deviation of ρ(x, z), f(x, z) is a two-dimensional random sequence which has a mean value equals of zero and standard deviation of 1. The spatial distribution characteristic
Fig. 1. Autocorrelation lengths and orientation angle of 2-D autocorrela tion function. 2
L. Li et al.
NDT and E International 108 (2019) 102173
Fig. 2. Flow chart of random multi-phase medium modeling.
of f(x, z) is subject to the spatial autocorrelation function R(x, z), which describes the correlation degree of physical parameters between two locations. The commonly used intermixed elliptic autocorrelation function is defined as follows: 2 6 Rðx; zÞ ¼ exp6 4
ðcos θ⋅x þ sin θ⋅zÞ2 ðsin θ⋅x þ cos θ⋅zÞ2 þ a2 b2
1 !1þr
2.2. Random multi-phase medium model 2.2.1. Characteristic parameters extraction It can be seen from the theoretical analysis in section 2.1 that the characteristic parameters of autocorrelation function such as mean value, standard deviation, autocorrelation lengths, orientation angle, and roughness factor are important indexes reflecting the elastic spatial distribution of heterogeneous materials. It is the basis of establishing an accurate RMMM to census and extract these characteristic parameters. The mean value ρ0(x, z) and standard deviation δρ(x, z) can be calculated through Eqs. (5) and (6). It is assumed that the density fluctuation of heterogeneous material σρ(x, z) obeys a second-order stationary random process σρ(x, z). The power spectrum density Г (kx, kz) of σρ(x, z) can be obtained according to Eqs. (7)–(9). Where, kx and kz are the spatial frequencies, M and N represent the numbers of sampling points in the x and z directions, respectively. According to Wiener-Khintchine theorem, the spatial autocorrelation function of σρ(x, z) is the inverse Fourier transform of Г(kx, kz). Referencing to Zhang et al. [23], the autocorre lation length is the distance from the maximum value 1 to e 1 for the
3 7 7 5
(4)
where a and b are the autocorrelation length in elliptic major axis and minor axis directions, respectively. r is the roughness factor, describing the microscopic roughness of the boundary [18]. r ¼ 0 corresponds to Gaussian elliptic autocorrelation function and r ¼ 1 corresponds to exponential elliptic autocorrelation function. The larger the roughness value, the rougher the microstructure. For a heterogeneous material, its small-scale heterogeneity usually has a preferred orientation. θ defines the angle between the direction of autocorrelation length a and the x-axis, as shown in Fig. 1. 3
L. Li et al.
NDT and E International 108 (2019) 102173
Fig. 4. Normalized autocorrelation function Rρ(x, z).
2.2.2. Bidirectional peak-valley searching method The random medium model is essentially a continuous medium. However, the abradable seal coatings sprayed by Atmospheric Plasma Spraying (APS) technique are commonly composed of metallic phase, non-metal lubricant, and pores. There is a significant difference in elastic properties and obvious micro-interfaces between constituent phases, which leads to a discrete elastic spatial distribution of hetero geneous materials. The random medium model cannot reflect this key feature. In this work, a new bidirectional peak-valley searching algo rithm is developed to transform the continuous random medium into a discrete RMMM. A RMMM with given polyester content and porosity could be established according to the following principle: all the density values in the random medium were sorted from large to small. Then, a bidirectional searching process was started with peak and valley values, respectively. When the amount percent of the peaks and valleys reached given AlSi content and porosity respectively, the searching process stopped. Thus the threshold ρTH1 and ρTH2 could be determined by reading the density values corresponding to the valley and peak posi tions where the searching program stopped. All areas with ρ(x, z) smaller than ρTH1 are set as pores, the areas with ρ(x, z) greater than ρTH2 are set as AlSi matrix, and the remaining areas are set as polyester. A RMMM with given polyester content and porosity is thus established. Flow chart of random multi-phase medium modeling is shown in Fig. 2.
Fig. 3. SEM micrograph of AlSi-polyester abradable seal coating (a) and its density spatial distribution (b).
normalized autocorrelation function Rρ(x, z). The characteristic pa rameters of autocorrelation function such as autocorrelation lengths (a, b), orientation angle θ, and roughness factor r can be obtained by fitting the experimental autocorrelation function Rρ(x, z) with the theoretical autocorrelation function Re(x, z) utilizing the Least Square Method (LSM). 1 XM 1 XN 1 ρ0 ðx; zÞ ¼ ρ ðx; zÞ z¼0 m M � N x¼0 δ2ρ ðx; zÞ ¼
1 XM 1 XN 1 ðρm ðx; zÞ z¼0 M � N x¼0
Iðkx ; kz Þ ¼
1 2 jIðkx ; kz Þj M�N
Rðx; zÞ ¼ IFFT2ðΓðkx ; kz ÞÞ
3.1. Microscopic observation and characteristic parameter statistics Microscopic observation was carried out through the Scanning Electron Microscope (SEM). Fig. 3(a) shows one typical cross-section SEM micrograph of AlSi-polyester abradable seal coatings. The gray matrix is AlSi, the black particles are polyesters, and the dark black parts are pores. Based on the difference of grayscale level of each component in the SEM micrograph, the threshold segmentation technology was applied to identify and reconstruct the AlSi matrix, polyester, and pores. Then the corresponding densities were endowed. Subsequently, the coating density spatial distribution ρm(x, z) was obtained (Fig. 3(b)). The mean density ρ0(x, z) ¼ 1.941 g/cm3 and standard deviation δρ(x, z) ¼ 0.998 g/cm3 were calculated according to Eqs. (5) and (6). Fig. 4 shows the normalized autocorrelation function Rρ(x, z) of coating den sity spatial distribution. Autocorrelation length a ¼ 13 μm, b ¼ 12 μm, orientation angle θ ¼ 0, and roughness factor r ¼ 0.52 were calculated using the LSM. Table 1 shows the statistical results of characteristic parameters in 5 groups of coating specimens with different levels of
(5)
ρ0 ðx; zÞÞ
� 1 XM 1 XN 1 ρ ðx; zÞ � exp m z¼0 M � N x¼0
Γðkx ; kz Þ ¼
3. Modeling and numerical simulation
2
(6) �
j2π
xkx zkz þ M N
�� (7) (8) (9)
4
L. Li et al.
NDT and E International 108 (2019) 102173
Table 1 Constituent contents and statistical characteristics of coating density spatial distribution. Specimen
1# 2# 3# 4# 5#
Density of coatings (g/cm3)
Parameters of autocorrelation function
Mean value
Standard deviation
Autocorrelation length a (μm)
Autocorrelation length b (μm)
Oriented angle θ (rad)
Roughness factor r
Polyester
Pore
AlSi
1.941 1.931 1.913 1.903 1.886
0.998 1.012 1.022 1.029 1.033
13 14 15 16 19
12 13 14 14 17
0 0 0 0 0
0.52 0.53 0.52 0.49 0.53
47.6 46.7 46.0 45.1 44.1
0.3 1.1 2.1 2.9 4.0
52.1 52.2 51.9 52.0 51.9
Volume fraction (%)
coating specimens as described in section 3.1, each group contains 5 random elastic spatial distributions) have been constructed according to the statistical parameters in Table 1. Fig. 5(a) shows the random me dium with a ¼ 13 μm, b ¼ 12 μm, θ ¼ 0, r ¼ 0.52, and the corresponding RMMM (Fig. 5(b)) with a 47.6% polyester and 0.3% porosity is con structed using the bidirectional peak-valley searching algorithm. Fig. 6 shows the SEM micrographs and corresponding RMMMs with different levels of porosity P. It is evident that RMMM is suitable to describe the random spatial distributions and irregular morphologies of polyester and pores in the AlSi-polyester abradable seal coatings. 3.3. 2D-FDTD numerical simulation Based on the 5 groups of RMMMs constructed in section 3.2, 2DFDTD numerical simulation [24] was used to calculate the wave prop agation in AlSi-polyester abradable seal coatings. The 2D-FDTD model based on RMMM is essentially a mathematical description of the microstructure statistical characteristics and the elastic spatial distri bution of the abradable seal coatings. Different from the previous model which assumed that the scatterers shape is regular, uniform in size and uniformly distributed, the 2D-FDTD model based on RMMM can reflect the irregular morphology of polyester and pores, meanwhile describe the complex spatial distributions of polyester and pores in the three-phase coatings. To obtain accurate and stable calculation results, a small mesh grid size is very important for the numerical simulation. In this paper, the uniform squared grid division in cartesian coordinate system has been used. Typically, the mesh grid size in 2D-FDTD should be less than a fifteenth of the shortest wavelength for numerical convergence [25]. With the consideration of the numerical convergence limit of about 48 μm and the smallest pore limit of about 10 μm, the grid width was set as 1 μm. Fig. 7 shows an example of the numerical simulation model, where the coating thickness is 1.0 mm and the water layer thickness is 3.0 mm. The left and right boundaries were set as symmetry boundaries to avoid converted waves. The bottom boundary was set as a Perfectly Matched Layer (PML). The top boundary loads the excitation source and the receiving probe. Ultrasonic longitudinal wave excitation signal, as shown in Fig. 9(a), perpendicularly propagated into the multi-layer water/coating/alloy substrate structure. AlSi matrix, polyester, and pores were set as isotropic homogeneous materials, whose materials properties listed in Table 2 [16]. 4. Experiments 4.1. Specimen preparation Rectangular GH4169 superalloy specimens, with dimensions 50 mm � 25 mm � 5 mm were used as substrates. Prior to the coating operation, the substrates were grit-blasted. Then the AlSi-polyester abradable seal coatings with thicknesses of 1.5–1.8 mm were sprayed using UNICOAT APS system under optimized spray parameters. The content of AlSi is about 52% and that of polyester is about 48% in the original specimens. In order to obtain coating specimens with different porosity, the original specimens were heat-treated at different times. Polyester was partially ablated above its decomposition temperature,
Fig. 5. Random medium (a) and random multi-phase medium model (b).
porosity. In each specimen, the parameter statistical process was carried out on 20 SEM micrographs to ensure the reliability of data. 3.2. Random multi-phase medium modeling In this section, 5 groups of RMMMs (corresponding to the 5 groups of 5
L. Li et al.
NDT and E International 108 (2019) 102173
Fig. 6. Comparison of SEM micrographs (a, c, e, g) and RMMMs (b, d, f, h): (a, b) P ¼ 1.1%; (c, d) P ¼ 2.1%; (e, f) P ¼ 2.9%; (g, h) P ¼ 4.0%.
Fig. 7. 2D-FDTD numerical simulation model.
Fig. 9. Excitation source time-domain waveform (a) and its amplitude spec trum (b).
and thus form pores. In this way, 5 groups of coating specimens with different contents of polyester and pore were obtained under different temperatures. The prepared AlSi-polyester abradable seal coating specimens are shown in Fig. 8, the left is the coating sample, and the
Fig. 8. Photo of the coating sample and the SEM observation sample.
6
L. Li et al.
NDT and E International 108 (2019) 102173
Table 2 Material properties used in the simulation. Materials Water (20 C) AlSi Polyester Air(20 � C) GH4169 �
Elastic modulus (GPa)
Poisson’s ratio
Density (g/cm3)
— 71.0 3.5 1.0 � 10 199.9
0.50 0.32 0.38 — 0.30
1.00 2.41 1.44 1.20 � 10 8.24
4
3
Fig. 10. Time-domain waveforms of experiments and numerical simulations.
right is the SEM microscopic observation sample. Fig. 11. Comparison of experiments and numerical simulations: (a) longitu dinal wave velocity; (b) attenuation coefficient.
4.2. Ultrasonic experiments The longitudinal wave velocity and attenuation coefficient have been measured using water immersion pulse-echo method. Ultrasonic experiments were carried out on the AlSi-polyester specimens using a short-duration pulse generated by GE USIP40 ultrasonic flaw detector and broadband, water immersion, flat transducer with a nominal fre quency of 5.0 MHz. The effective broadband range ( 6 dB) of the transducers was 2.95–8.35 MHz, as shown in Fig. 9. The received signals were recorded and stored by DPO4032 digital oscilloscope with a 200 MHz sampling frequency. Each measurement was averaged 64 times for improving the signal-to-noise ratio. Due to the large roughness of the coating surface, 1000# sandpaper was used to properly polish the coating before ultrasonic experiments. Each specimen has a measured coating thickness of 1.0 mm. The ultrasonic testing system was fully automated and controlled by a computer. The accuracy of the control system was 0.01 mm, which was enough to ensure the displacement reproducibility. By scanning each specimen along the direction parallel to the coating cross-section with a resolution of 1.0 mm, 25 waveforms were collected. Then the longitudinal wave velocity υL and attenuation coefficient αL in different coating specimens were calculated by Eqs. (10) and (11) as given by:
υL ¼
2d Δt
" ! 10 A1 lg þ lg 1 αL ¼ d A2
þ ρw*υLw) and R23¼(ρs*υLs – ρc*υLc)/(ρs*υLs þ ρc*υLc) are the acoustic reflection coefficients at the water/coating and coating/substrate in terfaces, respectively. Where ρw and ρs are the densities of water and substrate. υLw and υLs are the longitudinal wave velocities of water and substrate. ρc and υLc are the mean density and the mean longitudinal wave velocity of AlSi-polyester abradable seal coating specimens, respectively. 5. Results and discussion 5.1. Comparison of waveforms The simulated and experimental time-domain waveforms were shown in Fig. 10 (coating thickness of 1.0 mm, polyester content of 46.7%, porosity of 1.1%). It is found that the simulated waveforms show a good consistency with those from experiments. The FDTD numerical simulation based on RMMM reproduces the scattering echoes appeared in experiments, and the fluctuations of interface echoes are also observed under the same polyester content and porosity. Due to the limitation of manufacturing and microscopic observation technology, it is difficult to prepare a series of coating specimens with specified component contents. The RMMM method can flexibly simulate the elastic spatial distribution that conforms to the specified statistical characteristics. Simulated time-domain signals from the RMMM could carry a lot of information about the component contents and micro structural characteristics. So a further study of ultrasonic propagation characteristics in multi-phase heterogeneous materials can be realized utilizing the RMMM.
(10) !# 1
R12 � R212 R23
(11)
where d is the thickness of the coating. A1 and A2 are the amplitude of coating surface and interface echoes respectively. Δt is the time of flight between the surface and interface echoes. R12¼(ρc*υLc – ρw*υLw)/(ρc*υLc 7
L. Li et al.
NDT and E International 108 (2019) 102173
Fig. 12. RMMMs of AlSi-polyester abradable seal coating (polyester content 44.1%, porosity 4.0%).
5.2. Analysis of ultrasonic propagation characteristics
numerical simulation. The dispersion of attenuation coefficients in creases with the increase of porosity. When porosity increases up to 4.0%, the fluctuation between experimental attenuation coefficients reaches 1.85 dB/mm. Fig. 12 shows the RMMMs with same polyester content of 44.1% and porosity of 4.0%. Fig. 12(a)–(d) corresponds to the different attenuation coefficients of 5.8 dB/mm, 6.8 dB/mm, 6.9 dB/ mm, and 7.3 dB/mm, respectively, and the maximum relative change rate is up to 25.9%. The corresponding longitudinal wave velocities are 2145 m/s, 2153 m/s, 2150 m/s, and 2137 m/s, respectively, and the maximum relative change rate is less than 1%. Compared with the longitudinal wave velocity, the attenuation coefficient is more sensitive to the microscopic morphology and spatial distribution of pores. It is found that local aggregation of pores appeared in Fig. 12(b)–(d), and some small pores in the coating interconnected to form large voids. According to elastic wave scattering theory [26], wave scattering can be divided into Rayleigh scattering (kD≪1), random scattering (0.1 < kD < 10), and diffuse scattering (kD≫1) depending on the normalized wave number kD (D is the mean diameter of the scatters, k is the wave number, k ¼ 2π/λ). The diameter of pores in AlSi-polyester coating ranges from 10 μm to 50 μm. At the nominal frequency of 5.0 MHz, the average longitudinal velocity in the coating is about 2000 m/s–2300 m/s, so kD is between 0.1 and 0.8. When the ultrasonic wave interacts with the pores in AlSi-polyester coatings, intense random scattering will occur, and the scattered energy is in direct proportion to the size of pores [11,18,19]. The pores aggregation state and the pres ence of large voids will further intensify the ultrasonic attenuation. It should be pointed out that the types, sizes, orientations, shapes,
5.2.1. Effect of component contents on ultrasonic characteristics The longitudinal wave velocity and attenuation coefficient as a function of porosity and polyester content obtained from numerical calculations and experiments are shown in Fig. 11. It is found that, with the increase of porosity, both the experimental and simulated longitu dinal wave velocities decrease slowly, while the attenuation coefficients tend to increase gradually. When the porosity increases from 0.3% to 4.0%, the average longitudinal wave velocity decreases from 2251 m/s to 2150 m/s whose relative variation ratio is about 4.5%. Meanwhile, the average attenuation coefficient increases rapidly from 4.45 dB/mm to 6.56 dB/mm whose relative variation ratio is up to 47.4%. Since the velocity of air is much less than that of polyester, the equivalent velocity of the coating gradually decreases with the increase of porosity. The pores in the coating are mainly formed by high-temperature ablation of polyester, and the total content of polyester and pores in the coating remains unchanged. However, since the acoustic impedance difference between pore and AlSi matrix is much greater than that between poly ester and AlSi matrix, the appearance of pores intensifies the multiple scattering of scatterers, resulting in a gradual increase of the attenuation coefficient with the increase of porosity. 5.2.2. Effect of microstructure on ultrasonic characteristics It can also be seen from Fig. 11 that there is an obvious multi-value correspondence relationship between the velocity or attenuation coef ficient and porosity both in the experimental measurement and RMMM 8
L. Li et al.
NDT and E International 108 (2019) 102173
spatial distribution of scatterers, and other microscopic structural characteristics will all have impacts on the ultrasonic propagation characteristics. Furthermore, the coupling of various factors leads to the ultrasonic propagation more complicated. In the following works, we will use RMMM to quantitatively analyze the influence of the micro structures of abradable seal coatings on ultrasonic propagation characteristics.
[4] Tang JJ, Liu K, Yang QZ, et al. The influence of size and distribution of graphite on the friction and wear behavior of Ni-graphite coatings. Surf Coat Technol 2014; 252:48–55. https://doi.org/10.1016/j.surfcoat.2014.04.063. [5] Johnston RE. Mechanical characterisation of AlSi-hBN, NiCrAl-Bentonite, and NiCrAl-Bentonite-hBN freestanding abradable coatings. Surf Coat Technol 2011; 205:3268–73. https://doi.org/10.1016/j.surfcoat.2010.11.044. [6] Lotfi B, Shipway PH, McCartney DG, et al. Abrasive wear behaviour of Ni(Cr)TiB2coatings deposited by HVOF spraying of SHS-derived cermet powders. Wear 2003;254(3–4):340–9. https://doi.org/10.1016/S0043-1648(03)00014-0. [7] European Commission FP5 Growth Program. Abradable seal coatings and claddings for compressor applications. Seal Coat Proj No GRD1-2001-40124. [8] Ma ZY, Zhang W, Du PC, et al. Nondestructive measurement of elastic modulus for thermally sprayed WC-Ni coatings based on acoustic wave mode conversion by small angle incidence. NDT E Int 2018;94:38–46. https://doi.org/10.1016/j. ndteint.2017.11.003. [9] Lin L, Cao HQ, Luo ZB. Total focusing method imaging of multidirectional CFRP laminate with model-based time delay correction. NDT E Int 2018;97:51–8. https://doi.org/10.1016/j.ndteint.2018.03.011. [10] Ding SS, Jin SJ, Luo ZB, et al. Researches on the ultrasonic scattering attenuation of carbon fibre reinforced plastics with 2D real morphology void model. Acoust Phys 2017;63(4):490–5. https://doi.org/10.1134/s1063771017040029. [11] Kim KB, Hsu DK, Barnard DJ. Estimation of porosity content of composite materials by applying discrete wavelet transform to ultrasonic backscattered signal. NDT E Int 2013;56:10–6. https://doi.org/10.1016/j.ndteint.2013.01.014. [12] Zhang C, Li XB, Song YF, et al. Evaluating the grain size in curved components using the ultrasonic attenuation method with diffraction correction. NDT E Int 2016;84:20–6. https://doi.org/10.1016/j.ndteint.2016.07.004. [13] Wear KA, Nagaraja S, Dreher ML, et al. Relationships of quantitative ultrasound parameters with cancellous bone microstructure in human calcaneus in vitro. J Acoust Soc Am 2012;131(2):1605–12. https://doi.org/10.1121/1.3672701. [14] Faraoun HI, Grosdidier T, Seichepine JL, et al. Improvement of thermally sprayed abradable coating by microstructure control. Surf Coat Technol 2006;201: 2303–12. https://doi.org/10.1016/j.surfcoat.2006.03.047. [15] Faraoun HI, Seichepine JL, Coddet C, et al. Modelling route for abradable coatings. Surf Coat Technol 2006;200:6578–82. https://doi.org/10.1016/j. surfcoat.2005.11.105. [16] Duramou Y, Bolot R, Seichepine JL, et al. Relationships between microstructural and mechanical properties of plasma sprayed AlSi-Polyester composite coatings: application to abradable materials. Key Eng Mater 2014;606:155–8. https://doi. org/10.4028/www.scientific.net/kem.606.155. [17] Barguet L, Pezerat C, Bentahar M, et al. Ultrasonic evaluation of the morphological characteristics of metallic powders in the context of mechanical alloying. Ultrasonics 2015;60:11–8. https://doi.org/10.1016/j.ultras.2015.02.012. [18] Lin L, Chen J, Zhang X, et al. A novel 2-D random void model and its application in ultrasonically determined void content for composite materials. NDT E Int 2011;44 (3):254–60. https://doi.org/10.1016/j.ndteint.2010.12.003. [19] Lin L, Zhang X, Chen J, et al. A novel random void model and its application in predicting void content of composites based on ultrasonic attenuation coefficient. Appl Phys Mater Sci Process 2011;103(4):1153–7. https://doi.org/10.1007/ s00339-010-6061-x. [20] Zhao Y, Li XM, Ma ZY, et al. Correlating ultrasonic velocity and porosity using FDTD method based on random pores model. Mater Sci Forum 2011;675–677: 1221–4. https://doi.org/10.4028/www.scientific.net/msf.675-677.1221. [21] Ikelle LT, Yung SK, Daube F. 2-D random media with ellipsoidal autocorrelation functions. Geophysics 2002;58(9):1359–72. https://doi.org/10.1190/1.1443518. [22] Birch F. The velocity of compressional waves in rocks to 10 kilobars: 2. J Geophys Res 1961;66(7):2199–224. https://doi.org/10.1029/jz066i007p02199. [23] Zhang Y, Sundararajan S. Generating random surfaces with desired autocorrelation length. Appl Phys Lett 2006;88(14):16–9. https://doi.org/10.1063/1.2191882. [24] Schechter RS, Chaskelis HH, Mignogna RB, et al. Real-time parallel computation and visualization of ultrasonic pulses in solids[J]. Science 1994;265(5176): 1188–92. https://doi.org/10.1126/science.265.5176.1188. [25] Chen Y, Luo ZB, Zhou Q, et al. Modeling of ultrasonic propagation in heavy-walled centrifugally cast austenitic Stainless steel based on EBSD analysis. Ultrasonics 2015;59:31–9. https://doi.org/10.1016/j.ultras.2015.01.009. [26] Reynolds WN, Smith RL. Ultrasonic wave attenuation spectra in steels. J Phys D Appl Phys 2000;17(1):109–16. https://doi.org/10.1088/0022-3727/17/1/015.
6. Conclusions In this paper, we developed a two-dimensional random multi-phase medium model, which was used to simulate the complex elastic spatial distribution of the AlSi-polyester abradable seal coatings containing solid-solid-gas three phases. Based on the statistical characteristics of the coating density spatial distribution, efficient RMMMs for the AlSipolyester abradable seal coating with a thickness of 1.0 mm, polyester content of 47.6%–44.1%, porosity of 0.3%–4.0% were constructed. Then the RMMMs were applied to correlate velocity and attenuation coefficient with component contents and microstructural characteristics through 2D-FDTD numerical simulation. It is found that the RMMMs present good similarities with the SEM micrographs. The maximum relative errors between the fitting curve of velocity and attenuation coefficient predicted by the model and the mean value measured by the experiments are 1.01% and 3.97%, respectively. The 2D-FDTD simu lated velocity and attenuation coefficient based on RMMMs agree well with those of experiments, which indicates that this method is a good candidate for accurately analyzing ultrasonic propagation characteris tics in other multi-phase heterogeneous coatings. Declaration of competing interest We declare that we do not have any commercial or associative in terest that represents a conflict of interest in connection with the work submitted. Acknowledgments This work is supported by National Natural Science Foundation of China (NSFC) under Grant No. 51675083, No. 51805072 and National Basic Research Program of China under Grant No. 2015CB057306. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.ndteint.2019.102173. References [1] Bill RC, Ludwig LP. Wear of seal materials used in aircraft propulsion systems. Wear 1980;59(1):165–89. https://doi.org/10.1016/0043-1648(80)90277-X. [2] DeMasi-Marcin JT, Gupta DK. Protective coatings in the gas turbine engine. Surf Coat Technol 1994;68–69:1–9. https://doi.org/10.1016/0257-8972(94)90129-5. [3] Chupp RE, Ghasripoor F, Turnquist NA, et al. Advanced seals for industrial turbine applications: dynamic seal development. Int J Turbomach Propuls Power 2002;18 (6):1260–6. https://doi.org/10.2514/2.6061.
9