Journal of Mechanical Science and Technology 27 (11) (2013) 3207~3214 www.springerlink.com/content/1738-494x
DOI 10.1007/s12206-013-0843-y
Numerical simulation of ultrasonic minimum reflection for residual stress evaluation in 2D case† Maodan Yuan1, To Kang1,2, Jianhai Zhang1, Sung-Jin Song1,* and Hak-Joon Kim1 1
School of Mechanical Engineering, Sungkyunkwan University, 300 Chunchun-dong, Jangan-gu, Suwon, 440-746, Korea Nuclear Convergence Technology Development Division, Korea Atomic Energy Research Institute (KAERI), 150 Duckjin Dong, Yusung Gu, Daejeon, Korea
2
(Manuscript Received August 7, 2012; Revised January 9, 2013; Accepted April 20, 2013) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Abstract A number of interesting phenomena at fluid-solid interfaces can be observed when the incident angle approaches the Rayleigh angle, including Schoch displacement and leaky Rayleigh waves. Besides the experimental and theoretical research on these problems, numerical tools have been more and more widely used for these complex problems. Based on previous experimental and numerical researches, a 2D finite element model has been built to reproduce the Schoch effects. With the same model, the minimum reflection profile is investigated for the feasibility of material characterization, especially for residual stress evaluation. Residual stress is one of the important properties for structures, and its measurement is a popular research topic in nondestructive evaluation. However, it is not possible to put the residual stress into the numerical model directly. According to the relation of residual stress with mechanical properties, the material damping and wave speed have been alternatively adopted in this work. The influence of minimum reflection profile by residual stress has been shown by the change of wave speed and damping factor. Simulation results show that the minimum reflection profile is a potential method for residual stress evaluation. Keywords: Rayleigh angle; Schoch effect; Minimum reflection profile; Material damping; Residual stress ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1. Introduction When a wave is incident into a solid surface at the Rayleigh angle, the Rayleigh surface wave induced by the surface particle perturbation will propagate along the surface of the solid body. Similarly, at the liquid-solid interface, because of the presence of the liquid, particle disturbance not only in the solid but also in the liquid will carry some energy. In this case, a Rayleigh-like wave, also known as leaky Rayleigh wave, will be generated. The interaction of the reflected wave and leaky Rayleigh wave will redistribute the wave field in the fluid, as shown in Fig. 1. A variety of studies have focused on several interesting phenomena, including a separation of reflected beams, a null field near original positions, and a backward beam and a trailing leaky wave due to the energy dissipating into the fluid medium. In the Fig. 1, the incident angle θi is equal to the reflected angle θr, and Δs is the Schoch displacement, which was first discovered by Schoch using Schlieren photograph technology [1]. Much work has been done since then and some ideas have been proposed to explain this phenomenon. Bertoni and Tamir have built unified theory *
Corresponding author. Tel.: +82 31 290 7451, Fax.: +82 31 290 5276 E-mail address:
[email protected] † Recommended by Editor Yeon June Kang © KSME & Springer 2013
for the liquid and solid interface and explain the Schoch effect very well theoretically [2]. Based on that, Breazeale et al. take the ultrasonic wave propagation distance into consideration to include the diffraction effect [3]. Ngoc and Mayer have developed a numerical integration method to calculate the intensity profile of the reflected beam, which can be applicable at other angles of incidence as well as Rayleigh angle [4]. Zeroug and Felsen have removed the limitations for special incident beam and interface condition in previous researches [5]. With the sharp increase of computation power, numerical methods, especially finite element method, are playing more and more important roles in complicated problems. Recently, Declercq and Lamkanfi have done some new researches on the liquid-solid interface experimentally and theoretically [68]. Through the analysis of transmission of leaky Rayleigh waves at the extremity of a plate, they have confirmed that Rayleigh waves are primarily stimulated by the edge of an incident bounded beam rather than the middle [6]. With the analysis the Rayleigh wave transmission an acoustic barrier experimentally and numerically, they have verified that the Schoch effect results from the combination effect of the reflected wave field and another wave field caused by leaky Rayleigh waves [7, 8]. These researches have well explained the physical meaning of the Schoch effect at liquid-solid inter-
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Table 1. Chemical composition (wt%) of the common IN718 [10]. Ni
Fe
Cr
Mo
Nb
Ti
Al
C
Mn
Si
52.5
18.5
19
3.0
5.1
0.9
0.5
0.04
0.2
0.2
Table 2. The general mechanical properties of IN718 [10]. 8190 kg/m3
Density Young’s modulus
200 GPa
Poisson ratio
0.294 (room temperature)
Longitudinal wave velocity
5685.6 m/s
Shear wave velocity
3071.2 m/s
Fig. 1. The schematic of the reflection around critical Rayleigh angle. With attenuation Without attenuation 1.00
Reflection coefficient
0.98 0.96 0.94 0.92 0.90 0.88 0
10
20
30
40
Incident angle/Degree
Fig. 3. The reflection profile of water-In718 interface.
Fig. 2. The different displacement potentials at the interface.
face. Based on that, some features can be adopted as tools for material characterization in nondestructive evaluation. If a plane wave propagates into an interface of two solid media, generally, both reflected and refracted longitudinal and shear waves are produced according to Snell’s law. The displacement potentials have been expressed in Fig. 2. Based on the boundary conditions containing the continuity of both stresses and displacement, when the solid is assumed to be an elastic isotropic material, the reflection coefficients R for the liquid-solid interface have been shown [9]: 2
æc ö cos 2 2g 22 + ç 22 ÷ sin 2g 12 sin 2g 22 è c21 ø R P ; P (w ) = 2 æc ö cos 2 2g 22 + ç 22 ÷ sin 2g 12 sin 2g 22 + è c21 ø
r1c11 sin g 12 r 2c12 sin g 11 r1c11 sin g 12 r 2c12 sin g 11
(1) cij =
Vij 1 - ia ijVij / w
(i, j=1,2)
(2)
where I = 1 and 2 represent longitudinal and shear wave; j = 1, 2 represent liquid and solid media, respectively. V is the wave speed and γ is the angle; α is the attenuation coefficient; P in RP;P means the longitudinal wave.
According to Eqs. (1) and (2), the attenuation coefficient α has a significant influence on the reflection coefficient profile. To observe the difference due to α, some assumptions should be made for the simplification in the case of water-In718 interface. First, the In718 is assumed as an elastic isotropic material; the chemical composition and mechanical properties are shown as Tables 1 and 2. As the ratio of the longitudinal-to-shear wave speed ratio is about 2, the shear wave is four-to-ten times more attenuated than longitudinal wave since the attenuation coefficient is mainly dependent on the grain size-to-wavelength ratio. Here we consider the ratio of attenuation ratio of shear-tolongitudinal wave is 10, and compare the minimum reflection profiles (shown in Fig. 3) at f = 5 MHz. Since the early 1950s, interesting phenomena at the liquidsolid interface have been reported by a number of theoretical and experimental studies [1-8]. These various features, reflecting the combination effect of both the liquid and the specimen, have provided greater insight into the ultrasonic research field in characterizing the engineering materials. Experiments by Kim et al. [11] show that backscattered Rayleigh surface waves could be a potential tool for nondestructive evaluation of corrosion degradation of aged material. Leaky Rayleigh wave was verified effectively for the evaluation of imperfect bonding quality in a CVD diamond coating layer [12, 13]. The minimum reflection profile was applied to evaluate the subsurface material properties and bonding quality, since it is less
M. Yuan et al. / Journal of Mechanical Science and Technology 27 (11) (2013) 3207~3214
sensitive to the surface roughness of specimens than the backward radiation profile [14]. These engineering researches remind us that it is feasible to develop new NDE methods for material characterization for new application such as residual stress evaluation. Residual stress is the remaining stress in work pieces after external loads are removed. Residual stress measurement is still one of the most difficult and challengeable topics in nondestructive evaluation field [15]. Ultrasonic method for residual stress evaluation, based on acoustoelasticity, is one of the popular approaches [16]. As an alternative method, SAW velocity dispersion is always applied for surface residual stress evaluation [17]. However, the conventional methods have some limitations, such as the remarkable surface roughness effect and stress profiling. As mentioned by Kim et al. [14], minimum reflection profile is less sensitive to surface roughness and the feature of focusing at specimen surface; it could be a potential method for residual stress evaluation. Therefore, it will be meaningful for us to discuss the feasibility of residual stress evaluation using this method at first. According to the research of acoustoelasticity, residual stress can change the velocity of the materials or the elastic constants, inducing some anisotropy according to the birefringence phenomena [18]. Also, based on Granato-Lucke dislocation damping theory combined with transmission electron microscopy observations, conclusions have been made that mobile dislocations in tangled networks near interfaces contribute to energy dissipation at vibration strain levels [19]. The research result infers that the residual stress could have an influence on the damping response. Some studies have been done by Singh that show that the damping factor has a linear relation with the compressive residual stress field [20]. For this reason, the influence from the residual stress could be reflected in the damping factor. Based on the above conclusions, although it is not convenient to add the residual stress directly into finite element models, it is still possible to relate the residual stress with some other parameters, such as damping factors and wave speed. It is the first time to discuss the feasibility of residual stress evaluation using minimum reflection profile. In this work, the minimum reflection profile variation due to the change of these parameters resulting from residual stress will be predicted numerically and could be helpful for further experimental verification. The specimen investigated here is In718, which is considered as an isotropic elastic material. Numerical simulation for the interface phenomena and discussion about the Schoch displacement will be given in part 2. The minimum reflection profile can be obtained with this model, and parametric studies for the damping and wave speed will be shown in part 3.
2. Numerical simulation for the water-IN718 interface phenomena To study the phenomena at liquid-solid interfaces, a 2D
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Fig. 4. The FEM model for water-In718 interface.
model is built, as shown in Fig. 4, and frequency domain analysis is applied. This numerical simulation is implemented in the commercial finite element package Comsol multiphysics [21]. Considering that the wave propagation problem using finite element method requires very fine mesh, a miniature model is built and composed of a water layer and 30 × 5 mm2 In718 layer, wrapped by an absorbed layer to reduce the reflection effect as much as possible. T is the transducer at an incident angle θ with a pressure p0 = 1 Pa. R is the receiver transducer. Boundary 1 is the acoustic-structure boundary. For the fluid and solid interfaces, the coupling equation [21] is: æ 1 ö - n × ç - ( Ñpt - q ) ÷ = - n × utt . r è ø s × n = pt n
(3)
ρ refers to the density; n is the outward-pointing unit normal vector seen from inside the solid domain; utt the second derivatives of the structural displacements u with respect to time; q is the dipole source (here is zero); pt is the pressure incident at the boundary; σ is the stress acting on the solid material. From these, we can obtain the approximate Rayleigh velocity using the following empirical equation: VR ;
0.862 + 1.14n VS . 1 +n
(4)
ν is Poisson's ratio and from the Snell's law, the Rayleigh angle can be calculated as æ VWater ö ÷ = 31.5° . è VR ø
q R = sin -1 ç
(5)
The mesh density N in the finite element model plays a critical role in the accuracy of the wave propagation results. The mesh density N is defined as:
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same direction. Away from Rayleigh angle, these phenomena can hardly be observed, which is coincident with the theoretical analysis [2]. Bouzidi and Schmitt discussed the relation between the width of the transducer and the wavelength [23]. The width of the incident beam should be larger than the wavelength. The transducer aperture with the wavelength is defined as a parameter:
(a) θ = 27°
a=
l w
.
(7)
When the ratio is large, due to the dispersive effect in wave number domain, the energy will spread randomly, which makes separation of the reflected beam not apparent. However, when this parameter is smaller than a certain value, the Schoch displacement is well exhibited. This phenomenon could be simulated by changing the frequency and the transducer width. The results of several different cases are shown in Fig. 6. From the above results, it is concluded that in order to obtain well-behaved Schoch displacement, the ratio a should be chosen as small as possible. Considering the mesh size problem in wave propagation using finite element method, a = 0.0075 is a sufficient choice. To reduce the computation time, a relatively small model is chosen as in Fig. 4, with w = 4 mm as the width of the transducer, thus setting the excitation frequency to f = 5 MHz.
(b) θ = 31.5°
3. Minimum reflection profile due to residual stress 3.1 Numerical simulation for minimum reflection profile (c) θ = 37° Fig. 5. The simulation results for different incident angles at waterIN718 interface: (a) θ = 27°; (b) θ = 31.5°; (c) θ = 37°.
N=
l Dx
=
V . f Dx
(6)
λ is the wavelength, V is the wave speed, f is the frequency and Δx is the element size. For a given mesh density, quadratic quadrilateral elements have been proved to strongly reduce the wave velocity error to less than 0.5% for a node density higher than 8 [22]. The excitation is 5 MHz, so the mesh size of the model is Δx = 0.03 mm (select N = 10). The absolute pressures of water at different angles are shown in Fig. 5. From the simulation results, we can see that when the incident angle is near the Rayleigh angle, the reflected beam is separated into two parts, and a null (or minima) field appears. Also, the main beam is shifted along the interface, which is called Schoch displacement. Furthermore, to the right of the incident point, some trailing fields that come from the leaky energy of the Rayleigh wave into water are shown along the
To obtain the minimum reflection profile from the numerical simulation, the incident angle should be changed around the Rayleigh angle. In our simulation, an angle range from 27 to 37 degrees is analyzed. The receiving position, shown in Fig. 4, is set as a perfect mirror to calculate the reflection coefficient. The reflection amplitude is the average value of the receiver. The simulation result is shown in the following: Compared with the analytical result Fig. 3, the main difference is due to the analytical model referring to a plane which is just an ideal case. The angle corresponding to minimum reflection is about 32º, which is near the Rayleigh angle 31.5º but there is a small difference. To some degree, the minimum reflection profile is a comprehensive effect of the material properties, ranging from the attenuation (equivalent to damping in mechanics) to elastic constants. Therefore, it is necessary for us to do some parametric studies to investigate the minimum reflection phenomena. Several features are necessary to describe a Gaussian profile. The width of the beam, the minimum value, and its position are the most basic characteristics. To explain this profile, we will focus on these parameters. As mentioned in the first part, damping and wave speed could change with the existence of
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Table 3. The relation among different damping parameters [19]. Parameters
Symbol
Unit
Loss factor
η
1
Relationship with η η
Quality factor
Q
1
1/η
Damping ratio
ζ
1
η/2
Imaginary of E
Ei
Pa
Er*η
Frequency band
Δf
Hz
f*η
0.9
Minimum reflecion profile
Reflected absolute pressure
(a) f = 2 MHz, a = 0.1875 0.8
0.7
0.6
0.5
26
28
30
32
34
36
38
Angle/degree
Fig. 7. The numerical reflection profile for water-In718 interface. (b) f = 5 MHz, a = 0.0750
3.2 Minimum reflection profile variation with damping factor
(c) f = 8 MHz, a = 0.0469
(d) f = 10 MHz, a = 0.0375 Fig. 6. The simulation results for different excitation frequencies at Rayleigh angle: (a) f = 2 MHz; (b) f = 5 MHz; (c) f = 8 MHz; (d) f = 10 MHz.
residual stress. Damping will also influence the wave energy and the amplitude of the beam. Due to the relation between the Rayleigh speed and the shear wave, the variation of wave speed would mainly change the position of the profile.
From Eq. (1), we can see that the attenuation is one of the main factors which can contribute considerably to the profile. Attenuation is a complex parameter still studied by researchers. In summary, the factors which induce energy loss of the ultrasonic wave traveling in the object can be concluded as geometrical and intrinsic effects [24]. In finite element simulation, the geometrical effect can be taken into consideration automatically, providing an effective approach for the complex geometry problems. The intrinsic attenuation is introduced by damping. Alternatively, there are several different parameters corresponding to different application fields to represent the same physical meaning, and they can replace each other. We can choose the proper parameters for our calculation as is convenient. The relation is shown in Table 3. The isotropic loss factor ηs is adopted in Comsol, and several values have been investigated. The results are shown in Fig. 8. From the result, we can see when the damping changes, the minimum reflection profile would change the amplitude and the width of the profile rather than position. Moreover, when more different damping factors are investigated, it is interesting to find that the minimum amplitude does not change monotonically. This minimum value will have a critical point at a certain damping factor. This point means that the suppression of the geometrically reflected beam reaches the highest degree. This phenomenon has been verified in an experiment by Bertoni and Tamir [2]. In their research, this corresponding
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h =0 h =0.1 h =0.01
0.9
1000MPa -1000MPa Stress free
0.58 0.56
Reflected Amplitude
Reflected amplitude
0.8
0.7
0.6
0.5
0.54 0.52 0.50 0.48 0.46 31.0
0.4 26
28
30
32
34
36
38
31.5
Fig. 8. The reflection profile for different damping factors.
Fig. 9. The reflected amplitude with different damping factors at the Rayleigh angle.
point is explained to be the frequency of minimum reflection, which is strongly related to the material properties. If we fix the incident angle at the Rayleigh angle and keep changing the damping factor, the reflection coefficient will change with damping factor. The behavior of variation with damping factor is represented in Fig. 9. 3.3 Minimum reflection profile variation with residual stresses According to the acoustoelasticity, the existence of residual stress will make the elastic tensor of the material different from the natural state. This difference will influence the wave propagation inside the body. In case of the pre-stressed body, the wave equation will change into [18]
(
)
32.5
33.0
Fig. 10. The minimum reflection profile for different residual stress.
i AIJKL = d IK t JL + CIJKL .
2 ¶ é ¶u K ù i i ¶ uI . ê d IK t JL + CIJKL ú=r ¶X J ë ¶X L û ¶t 2
32.0
Incident angle/Degree
Incident angle/Degree
(8)
In this equation, ti is the Cauchy stress upon the body; C is the elastic tensor; u is the particle displacement; X is the configuration coordinate; ρi is the density referring to the prestressed system; and δ is the Kronecker delta. The main difference is the change of the equivalent elastic tensor. For the homogeneous case, the equivalent elastic tensor is
(9)
Due to such a change of the elastic tensor, the wave speed of the solid would suffer a linear relationship with the residual stress for the homogeneous isotropic material [18]. Both the resulting longitudinal and shear wave velocities corresponding to t11 = -1000 MPa, 0 and 1000 MPa are applied for the Comsol frequency domain analysis. The acoustoelastic coefficient is chosen as 1.9 × 10-6/MPa, which is the same level as in the experimental and numerical results [18, 25]. The resulting profiles are shown in Fig. 10. We can see that the variation due the wave speed shows as a shift of the profile, which is coincident with expectations. The wave speed variation resulting from residual stress mainly changes the profile position, which corresponds to the Rayleigh angle.
4. Conclusions To investigate the complicated phenomena at the fluid-solid interface, a numerical simulation model has been built for a water-In718 interface. When the ratio of the wavelength and the transducer width is a = 0.075, Schoch displacement and leaky Rayleigh wave are observed when the incident angle is close to the Rayleigh angle. Furthermore, the ultrasonic minimum reflection profile has been obtained numerically. This profile contains the material information of both water and the specimen, thus providing a possible way for the material characterization. Residual stress measurement is still an important issue, and this ultrasonic method provides a possible approach from methodology. Although residual stress cannot be put into the numerical model, according to Granato-Lucke dislocation damping theory, it is possible to see the influence of the residual stress through damping factor [19]. Moreover, the acoustoelastic effect reveals the variation of elastic tensor and wave speed resulting from the residual stress. For this reason, these two parameters have also been adopted to investigate the residual stress influence on the minimum reflection profile. From the results, the damping factor mainly influences the
M. Yuan et al. / Journal of Mechanical Science and Technology 27 (11) (2013) 3207~3214
Fig. 11. The schematic diagram for variations due to residual stresses.
profile amplitude and width, while the wave speed changes the position of the minimum reflection profile. This relation can be expressed by the following schematic diagram, shown as Fig. 11. These results show that the minimum reflection profile could be a potential method for the residual stress evaluation.
Acknowledgment This work was supported by the Power Generation & Electricity Delivery Program of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Knowledge Economy (No. 20111020400020), in part by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2011-0020024), and in part by the National Research Foundation of Korea (NRF) grant funded by the Ministry of Science, ICT & Furniture Planning (No. 2012M2A8A4013245).
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Maodan Yuan is a Ph.D. student in Graduate School of Mechanical Engineering from Sungkyunkwan University, Suwon, Korea. His current research interests include nondestructive evaluation for material characterization, numerical simulation for wave propagation, surface wave propagation and nonlinear
[25] R. M. Sanderson and Y. C. Shen, Measurement of residual stress using laser-generated ultrasound, International journal of pressure vessels and piping, 87 (12) (2010) 762-765.
Sung-Jin Song received his B.S. in Mechanical Engineering from Seoul National University, Seoul, Korea in 1981, his M.S. in Mechanical Engineering from Korea Advanced Institute of Science and Technology in 1983, and his Ph.D. in Engineering Mechanics from Iowa State University, Ames, Iowa, USA in 1991. He worked at Daewoo Heavy Industries, Ltd., Incheon, Korea for 5 years from 1983, where he was certified as ASNT Level III in RT, UT, MT and PT. He worked at Chosun University, Gwangju, Korea as Assistant Professor from 1993. Since 1998 he has been at Sungkyunkwan University, Suwon, Korea and is currently Professor of Mechanical Engineering.
ultrasonic testing.
