Experiment and numerical simulation for laser ultrasonic measurement of residual stress

Ultrasonics xxx (2016) xxx–xxx

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Experiment and numerical simulation for laser ultrasonic measurement of residual stress Yu Zhan a, Changsheng Liu b,⇑, Xiangwei Kong c, Zhongya Lin a a

College of Sciences, Northeastern University, Shenyang 110819, China Key Laboratory for Anisotropy and Texture of Materials Ministry of Education, Northeastern University, Shenyang 110819, China c School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China b

a r t i c l e

i n f o

Article history: Received 4 May 2016 Received in revised form 17 August 2016 Accepted 18 August 2016 Available online xxxx Keywords: Laser ultrasonic Surface wave Residual stress Finite element model

a b s t r a c t Laser ultrasonic is a most promising method for non-destructive evaluation of residual stress. The residual stress of thin steel plate is measured by laser ultrasonic technique. The pre-stress loading device is designed which can easily realize the condition of the specimen being laser ultrasonic tested at the same time in the known stress state. By the method of pre-stress loading, the acoustoelastic constants are obtained and the effect of different test directions on the results of surface wave velocity measurement is discussed. On the basis of known acoustoelastic constants, the longitudinal and transverse welding residual stresses are measured by the laser ultrasonic technique. The finite element method is used to simulate the process of surface wave detection of welding residual stress. The pulsed laser is equivalent to the surface load and the relationship between the physical parameters of the laser and the load is established by the correction coefficient. The welding residual stress of the specimen is realized by the ABAQUS function module of predefined field. The results of finite element analysis are in good agreement with the experimental method. The simple and effective numerical and experimental methods for laser ultrasonic measurement of residual stress are demonstrated. Ó 2016 Published by Elsevier B.V.

1. Introduction Residual stress exists in material that is produced by nearly every mechanical, chemical, and thermal process. It is an important performance index of material because residual stress affects the material fatigue life, strength and structure. The conventional methods of measuring residual stress include hole-drilling method, cutting groove method, X-ray method and traditional ultrasonic method, etc. [1–3]. However, mechanical method is complicated, time consuming, and will induce damages to the material. X-ray method has the disadvantage of damaging human health and being sensitive to the direction of detection. Traditional ultrasonic method is not suitable for some circumstances, such as hightemperature, radioactive, and toxic environment. Laser ultrasonic technique has several advantages such as: non-contact, highprecision, non-destructive, and high adaptability [4]. In recent years, laser ultrasonic has been widely used in defect detection [5,6], elastic constant measurement [7–10], residual stress measurement [11–14] and many other applications [15,16].

⇑ Corresponding author. E-mail address: [email protected] (C. Liu).

The research of laser ultrasonic measurement of residual stress has attracted a great deal of attention of researchers. Sanderson [11] used the simplified finite element modeling to determine the capability and sensitivity of the technique for residual stress measurement. A clear correlation between the magnitude of the residual stress and the surface wave behavior was presented. Wang [17] Analyzed laser-generated ultrasonic force source at specimen surface and displayed of bulk wave in transversely isotropic plate with numerical method. Daniel [18] used surface skimming longitudinal wave to detect the residual stress of friction stir welds and the results are in good agreement with the finite element method. The surface residual stress in steel rods with different heat treatments was measured by laser generated ultrasonic surface wave by Duquennoy [19]. Achenbach analyzed the process that ultrasonic wave was excited by using the force dipole model on the surface of a homogeneous and isotropic linear elastic body [20]. The mass spring lattice modeling was used to simulate laser ultrasonic technique for the detection of small surface defects by Sohn and Krishnaswamy [21]. Karabutov [13] designed a special optoacoustic transducer which was used both for the excitation and detecting of the ultrasonic pulses. This technique was used for measurement of welding residual stress and the results are in good agreement with the traditional method. Moreau [14] used

http://dx.doi.org/10.1016/j.ultras.2016.08.013 0041-624X/Ó 2016 Published by Elsevier B.V.

Please cite this article in press as: Y. Zhan et al., Experiment and numerical simulation for laser ultrasonic measurement of residual stress, Ultrasonics (2016), http://dx.doi.org/10.1016/j.ultras.2016.08.013

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laser ultrasonic measurements of residual stress in a 7075-T651 aluminum sample surface-treated with low plasticity burnishing. Based on the achievement of former researchers, the systematic work of laser ultrasonic measurement of residual stress is demonstrated in this paper. The pre-stress loading device is designed and fixed on the laser ultrasonic experimental platform, which can easily realize the control of tension stress. The acoustoelastic constants of the material are obtained by measuring the velocity of the surface wave in the known condition of stress, which solve the problem that the residual stress can’t be measured accurately due to unknown acoustoelastic constants. In addition, considering the direction of the principal stress is unknown under normal circumstances, we investigate the effect of different test directions on the results of surface wave velocity measurement. On the basis of known acoustoelastic constants, the welding residual stress is measured by the laser ultrasonic technique. The results are consistent with drilling method. A finite element analysis model of laser ultrasonic surface wave detection of welding residual stress is established. In order to avoid the large amount of calculation caused by thermo-mechanical coupling and transient analysis, the pulsed laser is equivalent to the related load of Gaussian profile and the welding residual stress is realized by the ABAQUS function module of predefined field. The results show that the experimental method and finite element model in this paper are effective for laser ultrasonic measurement of residual stress.

stress direction but also on the propagation direction. Considering the research for the welding residual stress of the uniform and isotropic thin plate, the stress in the thickness direction is very small compared to the other two directions, we can approximate r3 ¼ 0 and the problem is simplified into plane stress state. In order to obtain the stresses using Eqs. (1) and (2), the constants A1 and A2 must be determined in advance. These constants can be calculated if the second-order and third-order elastic constants are known. If the elastic constants are not available, the constants A1 and A2 can be determined by the calibration experiment under the condition of uniaxial stress. When the specimen is in the state of uniaxial stress, the Eqs. (1) and (2) can be simplified as:

Dv 1

v1  v0 ¼ ¼ A1 r1 v0 v0 Dv 2 v 2  v 0 ¼ ¼ A2 r1 v0 v0

ð3Þ ð4Þ

Eqs. (3) and (4) show that if the loading stress is known and the surface wave propagation velocity along the stress direction and vertical stress direction is measured respectively, we can get the acoustoelastic constants. It is assumed that the residual stress is 600 MPa. The typical acoustoelastic constant value is about 106 MPa1 order of magnitude. Therefore, the maximum change rate is Dvv01  103 . This requires that the measurement of time must be of high accuracy.

2. Theory 3. Experiment The dependency of the acoustic wave velocity on the value of applied stress can be obtained from the acoustoelastic theory [22]. A Rayleigh wave propagating on the free surface of a homogeneous, elastic and half-space material under uniform static deformation is considered. When the principal directions of strain coincide with the symmetry axes of the material, and when the displacement due to the propagation of the wave is infinitesimal, then the relative variations of the velocity can be expressed in terms of static stresses as [23]:

The measurement sample is a thin 4140 steel plate which is fixed on the pre-stress loading device by bolts. Sample size is 400 mm  15 mm  1:8 mm. The laser ultrasonic system includes three parts: the Nd:YAG laser (for ultrasonic generation), the laser Doppler vibrometer (for ultrasonic detection) and the pre-stress loading device (for acoustoelastic constant measurement). The experimental system is shown in Fig. 1.

Dv 1

3.1. Experimental setup

v1  v0 ¼ ¼ A1 r1 þ A2 r2 v0 v0 Dv 2 v 2  v 0 ¼ ¼ A1 r2 þ A2 r1 v0 v0

ð1Þ ð2Þ

where v 1 ; v 2 and v 0 are the surface wave velocities in stress state and unstressed state, respectively, v 1 is the velocity of the surface wave propagating along the ‘‘1” direction and v 2 is the velocity of the surface wave propagating along the ‘‘2” direction. r1 and r2 are the stress in the ‘‘1” and ‘‘2” direction, respectively. A1 and A2 are the acoustoelastic constants, which not only depend on the

A laser pulse (pulse width s ¼ 8 ns, wavelength k ¼ 1064 nm, repetition frequency f ¼ 20 Hz, single pulse energy E0 ¼ 100 mJ) with Gaussian profile sent by Nd:YAG laser (Beamtech Dawa100) is focused by a cylindrical lens as a line source with 0.6 mm width to generate the surface wave. The ultrasonic vibration information is received by laser Doppler vibrometer (Sdptop LV-S01, frequency band 5 MHz). The digital oscilloscope (Tektronix Dpo4102, frequency band 1 GHz) collects electrical signal from laser Doppler vibrometer and photoelectric detector. A part of

Fig. 1. Schematic of experimental system: (a) laser ultrasonic system, (b) pre-stress loading device.

Please cite this article in press as: Y. Zhan et al., Experiment and numerical simulation for laser ultrasonic measurement of residual stress, Ultrasonics (2016), http://dx.doi.org/10.1016/j.ultras.2016.08.013

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the laser beam is detected by a photoelectric detector (Thorlabs Det10A/M, wavelength range 200–1100 nm), which achieves perfect synchronization of the oscilloscope with the laser shots. The computer processes and analyzes the data from oscilloscope. In the experiment, two methods are used to avoid the influence of other signals on the surface wave reception. Firstly, the signal receiving position is far away from the excited position. Due to the different propagation velocities of various waves in the specimen, enough distance can make the different waves separated, to better identify the surface wave. In addition, the filtering function of the laser Doppler vibrometer is used, and the high frequency wave is received by the setting of the high frequency. When measuring the acoustoelastic constants, one end of the sample is fixed on the experimental platform (Thorlabs Nexus), the other end is connected with the weight, and the different tension stresses are realized by changing the weight. According to Saint Venant principle, away from the loading area, we can get the tensile stress field with uniform distribution.

Fig. 2. Travel time of surface wave in the unstressed steel plate Dl ¼ 245 mm.

3.2. Experimental results and discussion

A1 ¼ 1:5793  106 MPa1 ; A2 ¼ 0:1809  106 MPa1 . It is con1 For interpretation of color in Fig. 2, the reader is referred to the web version of this article.

experiment linear fitting

200

l, mm

150

100

50

0

0

20

40

60

t, s

80

Fig. 3. Linear fitting curve of experimental results.

1.00000

0.99995

0.99990

v/v0

The whole experiment is divided into three steps to complete. First of all, the ultrasonic surface wave velocity is measured in the unstressed reference sample. Then, measurement is conducted on the sample of uniform tensile stress state. The velocities in stressed sample and the acoustoelastic constants of surface waves are obtained at this stage. The effect of different test directions on the results of surface wave velocity measurements is studied. Finally, the welding residual stress is measured by laser ultrasonic method. In these three cases the velocities are determined by the same scheme. Surface wave velocity is determined by travel distance and time. The distance Dl of between the excitation position and the receiving position is measured with vernier caliper. The travel time Dt is obtained by oscilloscope as shown in Fig. 2. The measurement is performed on the same side of the sample. The two channels of the oscilloscope are used in the experiment. Channel 1 record the vibration signal measured from laser Doppler vibrometer (yellow1 curve). The signal of the surface wave is very obvious, and the characteristic of the surface wave is satisfied. Channel 2 records the time reference signal provided by the photoelectric detector (blue curve). The propagation time t between the excitation and the detection position can be read by the cursor. In the experiment, the measurement errors of the ultrasonic surface wave travel time and travel distance are unavoidable. Therefore, the method of linear fitting experimental data is used to solve the problem. As shown in Fig. 3, the slope of the linear fitting curve is calculated as a1 ¼ 2:887, which is converted to the velocity of v 0 ¼ 2887 m=s, the theoretical solution is v ¼ 2907, the error is 0.68% [24]. Understandably, the difference is caused by different sample, environment temperature and other factors. The different stress fields are generated in the sample by the pre-stress loading device. By changing the weight, the magnitude of the different tensile stress can be obtained. In each uniaxial tensile loading stage, the surface wave propagation velocities parallel to the loading direction and perpendicular to the loading direction are measured. The relationship between the surface wave velocity ratio and stress is shown in Fig. 4. As can be seen from Fig. 4, the velocity ratio and tensile stress showed excellent linear relationship, which is consistent with the acoustoelastic theory. By linear fitting, the slope of the curve is calculated, which is converted to the acoustoelastic constant of

250

0.99985

v1 (loading) direction v2 (normal to loading) direction

0.99980

0.99975

linear fitting curve

0

30

60

90

120

150

, MPa Fig. 4. The relationship between velocity ratio and tensile stress.

sistent with the result in the literature [24], which shows that the experimental scheme is reasonable and accurate. When the acoustoelastic constants are measured by the experiment, the welding residual stress can be evaluated by Eqs. (1) and (2). When the residual stress is measured by laser ultrasonic, under normal circumstances, the magnitude and direction of principal stress is uncertain. However, we are much more in the hope of

Please cite this article in press as: Y. Zhan et al., Experiment and numerical simulation for laser ultrasonic measurement of residual stress, Ultrasonics (2016), http://dx.doi.org/10.1016/j.ultras.2016.08.013

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1.00000

=72.6MPa =145.2MPa

0.99996

vR/v 0

0.99992 0.99988 0.99984 0.99980 0.99976

0

30

60

90

120

150

180

, degree Fig. 5. The relationship between velocity ratio and test directions.

700

laser ultrasonic hole-drilling

600

, MPa

500

x

400

y

300

welding seam

200 100 0 -16

-12

-8

-4

0

4

8

12

16

distance from welding center, mm Fig. 6. The longitudinal residual stress distribution.

250

laser ultrasonic hole-drilling

200

x

, MPa

150

y

100

50

4. Numerical simulation

welding seam

0

-50

-16

-12

-8

-4

0

4

surface wave velocity measurements is discussed in Fig. 5. The angle a is the included angle of between the propagation direction of surface wave and the principal stress direction. Fig. 5 shows that there is a very obvious nonlinear relationship between the surface wave velocity ratio and test directions. When a ¼ 0 is expressed, the propagation direction of the wave is consistent with the principal stress direction, and it is the most reasonable to evaluate the principal stress by using the measured surface wave velocity. With the increase of the angle, the error caused by the measurement will become larger. When a ¼ 90, v R ¼ v 0 , which means that there is no stress in the propagation direction of surface wave, and the measurement error is the largest. By comparing the two curves, it can be seen that the larger the stress, the greater the relative error caused by the angle. Therefore, the method proposed in this paper can be used to determine the optimal detection direction in the case that the residual stress is larger and the principal direction is unknown. The 4140 steel specimens are butt welding and the welding residual stress is measured by laser ultrasonic method. The welding is completed by CO2 laser (maximum power 3 kW, focal depth 3.2 mm, welding speed 1.0 m=min). In order to verify the reliability of the results, hole-drilling method is applied for the same sample after laser ultrasonic measurement. The acoustoelastic constants of the 4140 steel and the surface wave velocity in unstressed state have been obtained by previous experiments. The width of the laser focal line is 0.6 mm which remains constant during the welding stress measurement. The problem is simplified to the plane stress state and the velocity v x and v y are measured separately. Solving Eqs. (1) and (2), longitudinal stress rx and transverse stress ry are obtained. The longitudinal propagation distance is Dlx ¼ 160 mm, the transverse propagation distance is Dly ¼ 2 mm, and we do so because we are more concerned about the overall situation of the longitudinal residual stress. The distribution of the longitudinal and transverse residual stresses as shown in Figs. 6 and 7. It can be seen from the figs, the large tensile stress area is in the welding seam and around the welding seam there is a small compressive stress area. This is a typical distribution of welding residual stress, which is similar with the results in Ref. [13]. The longitudinal residual stress is much larger than the transverse residual stress, so we should mainly check the longitudinal residual stress for laser welding structure simple checking. The results of the two methods are consistent which indicating the residual stress of laser ultrasonic measurement is feasible and accurate. In the large stress area, the result of hole-drilling method is bigger than laser ultrasonic method because of the stress concentration effect of small hole is more obvious.

8

12

16

distance from welding center, mm Fig. 7. The transverse residual stress distribution.

getting the principal stress, which requires us to measure the surface wave velocity along the principal direction. The reason for the concern of the principal stress is that the strength theory of judging whether the material is invalid is based on the principal stress. Therefore, the effect of different test directions on the results of

The finite element analysis model of laser ultrasonic detection of residual stress is established. There are two key problems in the process of establishing model. The first one is the low computational efficiency for the thermo-mechanical coupling and transient analysis. The pulsed laser is equivalent to the related mechanic load of Gaussian profile, then the computational efficiency is greatly improved and the accuracy is satisfactory. The other is how to apply the welding residual stress, which is realized by cutting the sample into some uniform width perpendicular strips with different stress. 4.1. Model establishment and parameter setting A homogeneous and isotropic thin plate model is established. The pulsed laser irradiates equably on the surface of the specimen along the width direction. For facilitating the finding of main peaks

Please cite this article in press as: Y. Zhan et al., Experiment and numerical simulation for laser ultrasonic measurement of residual stress, Ultrasonics (2016), http://dx.doi.org/10.1016/j.ultras.2016.08.013

Y. Zhan et al. / Ultrasonics xxx (2016) xxx–xxx

where r0 is the Gaussian beam radius, g ¼ 15:003 is the correction factor,which can be determined by the extremum of the function Pðr; tÞ. The function Pðr; tÞ constructed by the above method has two characteristics, one is the same distribution characteristics in time and space as the pulse laser energy, the other is the maximum value of function Pðr; tÞ is Pmax . According to the finite element analysis experience of laser ultrasonic, the total time of the pulse load

2 Node 1 Node 2 Node 3 Node 4 Node 5 Node 6

Vertical displacement, m

1

5

excitation is t ¼ 3s ¼ 2:4  108 s.

0

4.3. Results and discussion Firstly, the propagation of surface wave is simulated in the stress free specimen and the wave velocity is obtained. Six reception points are set up on the surface of the model away from the excitation position. By observing the displacement history over time in Fig. 8, the moments when each node reaches its maximum displacement and the minimum displacement are found. The values can then be used to calculate the velocity of the surface wave.

-1

-2 2.0

2.5

3.0

t, s

3.5

4.0

4.5

Fig. 8. The displacement-time curves of nodes.

of wave propagation, the infinite element is used to eliminate the reflections caused by boundaries. The finite element model has been established and it agrees with the experiment conditions in the case of ignoring minor factors. The parameters are as follows: elastic modulus E ¼ 199:5 GPa; Poisson’s ratio l ¼ 0:29, density

q ¼ 7:85  10 kg=m , yield stress rs ¼ 930 MPa. 3

3

Generally speaking, the division of mesh and the choice of time step size have a great effect on the accuracy and stability of laser ultrasonic finite element analysis results. First, the maximum element size must be set as less than a quarter of the elastic wave length to ensure the accuracy. In addition, factors such as the size of the model, the surface wave velocity, and computational efficiency should also be considered. The type of the finite element is CPE4 and the size of the mesh is 0:045 mm  0:013 mm. The selected step is ABAQUS/Explicit, the total time for the analysis step is 5  105 s, the time of the pulse load excitation is 2:4  108 s, and the fixed incremental step is 1  109 s. 4.2. Application of equivalent load The pulse load of the Gauss distribution in both time and space domain is applied to generate ultrasonic surface wave. Moreover, the quantitative relationship between the physical parameters of the laser and load size is established by introducing the correction coefficient. The equivalent surface load corresponding to the pulse laser is the following:

Pðr; tÞ ¼ gP max f ðrÞgðtÞ

ð5Þ

where Pðr; tÞ is a surface load normal to the surface of elastomer, g is the correction coefficient, P max is the peak load and it can be calculated by the following [25,26]: 3

3

Pmax ¼ 230A16 Z 16 s8 k4  I40 =ð1012 Þ4 7

9

1

1

ð6Þ

where A is the atomic number, Z is the ionization number, s is the laser pulse width, k is the laser wavelength, and I0 is the peak power density. The time and space distribution function of the pulse load is as follows [20]: 2

1 2 2r2 f ðrÞ ¼ pffiffiffiffiffiffiffi e r0 ; 2p r 0

gðtÞ ¼

8t 3

s

4

2t2

e s2

Dl14 ðt 4max  t 1max þ t 4min  t 1min Þ=2 Dl25 V2 ¼ ðt 5max  t 2max þ t 5min  t 2min Þ=2 Dl36 V3 ¼ ðt 6max  t 3max þ t 6min  t 3min Þ=2

V1 ¼

ð7Þ

ð8Þ ð9Þ ð10Þ

The average velocity of the surface wave is obtained.

V0 ¼

V1 þ V2 þ V3 3

ð11Þ

As shown in Fig. 8, the displacement of the nodes is consistent and in accordance with the surface wave characteristics: as they are far from the excitation area, after the complete surface wave is produced, the amplitude of the neighboring nodes will be nearly invariant in the course of wave propagation. According to the results of finite element analysis, the surface wave velocity is obtained. Comparison of V 0 ¼ 2895 and experimental result v 0 ¼ 2887, the error is 0.28%, indicating that the finite element model is reliable. The same method can be used to obtain the surface wave V when different uniaxial tensile stress is applied in the model. The stress is defined according to the load in the experiment of acoustoelastic constant measurement. In the finite element analysis, the stress-strain relationship is defined in accordance with Hooke’s law and the geometric large deformation switch is closed. The relationship between stress and surface wave velocity is shown in Fig. 9. Fig. 9 shows that the simulation results and experimental results are in good agreement. The error is mainly due to the materials being defined as ideal material in numerical simulation. However, there are inevitable defects in the experimental specimen, such as micro crack and residual stress, which will affect the surface wave velocities. The welding residual stress can be achieved through the predefined field function module of ABAQUS. In the test area, the model is cut into seventeen perpendicular strips of uniform width. For every strip, the magnitude of stress is defined corresponding to the experimental test results. Taking the eighth strip as an example, the experimental measured transverse and longitudinal welding stresses are r8y ¼ 138:8 MPa and r8x ¼ 601:4 MPa, respectively. So we can define the stress field s11 ¼ 138:8 MPa, s22 ¼ 601:4 MPa in the eighth strip region. In far away from the weld area, we assume that the stress is zero. Then, the propagation of surface wave in the predefined stress field is simulated, and the results of finite element analysis are shown in Fig. 10. As shown in Fig. 10, the numerical results agree well with experimental results, which show that the finite element model

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1.00000

experiment results FEM results

V/V0

0.99995

References

0.99990

0.99985

0.99980

0.99975

0

20

40

60

80

100

120

140

, MPa Fig. 9. The results of finite element analysis on the effect of stress on wave velocity.

700

FEM results experimental results

600

x

500

, MPa

experimental and numerical methods established in this paper are very simple and effective, and it is an important reference for further research on the welding stress of laser ultrasonic testing.

400

y

300 200

the eighth strip

100 0 -16

-12

-8

-4

0

4

8

12

16

distance from welding center, mm Fig. 10. Numerical results of longitudinal welding residual stress by laser ultrasonic testing.

we established is reasonable and it has the potential to further study the residual stress of laser ultrasonic testing. 5. Conclusions The laser ultrasonic technique for measuring residual stress of 4140 steel is investigated both experimentally and numerically. For experimental measurement, the pre-stress loading method is applied to get the acoustoelastic constant and investigate the effect of different test directions on the results of surface wave velocity measurement. On the basis of known acoustoelastic constant, the longitudinal and transverse welding residual stresses are measured with the laser ultrasonic technique. The results are consistent with hole-drilling method. For numerical simulation, equivalent load model is employed and surface wave is generated by the load of the Gaussian profile in both time and space domain. The welding residual stress of the specimen is defined by cutting into seventeen perpendicular strips of uniform width. The results show that the

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