Multiphysics Simulation of Guided Wave Propagation Under Load Condition

8th International Symposium on NDT in Aerospace, November 3-5, 2016

Multiphysics Simulation of Guided Wave Propagation under Load Condition

Lei QIU1,2, Ramanan SRIDARAN VENKAT2, Christian BOLLER2, Shenfang YUAN1

1 Research Center of Structural Health Monitoring and Prognosis, State Key Lab of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics; Nanjing, China E-mail: [email protected], [email protected] 2 Chair of Non-Destructive Testing and Quality Assurance (LZfPQ), Saarland University; Saarbrücken, Germany; [email protected], [email protected]

http://www.ndt.net/?id=20587 Abstract A multiphysics simulation method of Guided Wave (GW) propagation under load condition is proposed. With this method, two key mechanisms of load influence on GW propagation are considered and coupled with each other. The first key mechanism is the acoustoelastic effect which is the main reason of GW phase change. The second key mechanism is the load influence on piezoelectric materials, which results in a change of the GW amplitude. Based on COMSOL multiphysics, a finite element model of GW propagation on an aluminium plate under load condition has been established. The simulation model includes two physical phenomena to be considered represented by simulation modules. The first module is called solid mechanics, which is used to

More info about this article: simulate the acoustoelastic effect being combined with the hyperelastic material property. The second module is called electrostatics, which considers the simulation of the piezoelectric effect for GW excitation and response. To simulate the load influence on piezoelectric materials, a non-linear numerical model of the relationship between load and piezoelectric constant d31 is built. The simulation results under uniaxial load are obtained and they are compared with the data obtained from an experiment of load influence on GW. It shows that the variations of phase and amplitude of GW obtained from the simulation match the experimental results well.

Keywords: Structural health monitoring, time-varying condition, guided wave, multiphysics simulation, acoustoelastic effect

1. Introduction

Real aircraft structures serve under uncertain time-varying conditions such as environmental conditions, load conditions and structural boundary conditions etc. Almost all the damage monitoring features can be directly affected by the time-varying conditions, which leads to low damage monitoring reliability. Among Structural Health Monitoring (SHM) methods, Guided Wave (GW) and piezoelectric sensor based method is a promising one because it is a regional monitoring method and is sensitive to small damage [1]. To deal with the time- varying problem [2], several methods [3-7] such as the environmental compensation method, baseline free method, data normalization method and mixture probability method etc. have been proposed but limitations remain. Thus, for real applications, the problem of reliable damage monitoring under time-varying conditions must be fully studied. Considering that the time-varying conditions are often complicated, and the corresponding experiments are highly costly and time consuming, GW simulation under time- varying conditions is an effective way to study the time-varying problem. Based on the simulation, the GW propagation on complex structure under complicated time-varying conditions can be studied easily and the simulation data can be also used to validate the methods which are aimed to deal with the time-varying problem. Although the simulation of GW propagation has been widely studied [8-11], the GW simulation under time-varying conditions is still rarely reported [12], especially for a simulation method which fully considers the influence of time-varying conditions recorded by piezoelectric sensors adhered on a structure and this under close to real conditions.

Among a lot of time-varying factors, the changing load condition is a main factor, which can introduce large variations to the phase and amplitude of a GW signal recorded. In this paper, an efficient method of multiphysics simulation of GW propagation under load condition is proposed. The two key mechanisms of load influence on GW are the acoustoelatstic effect which is the main reason of GW phase change and the load induced influence on piezoelectric materials which results in a change of the GW amplitude [12]. Thus, the two mechanisms are considered and coupled together in this method. Based on COMSOL multiphysics, a finite element model of GW propagation on an aluminum plate under load condition is established. An experiment of load influence on GW is performed to validate the proposed multiphysics simulation method.

2. The Experiment of Load Influence on Guided Wave

2.1 Experimental Setup

The experimental system is shown in Figure 1. The structure is 2024 aluminium alloy and its dimension is 400mm×200mm×2mm (length×width×thickness). Two piezoelectric sensors (PZT-5A) are placed on the structure which are numbered as PZT 1 and PZT 2. PZT 1 is used to excite GW and PZT 2 is used to be a GW receiver. The distance between PZT 1 and PZT 2 is 200 mm. The structure is fixed on a static tensile machine which is used to apply a uniaxial tension load to the structure. The eleven levels of load (from 0 MPa to 100 MPa with 10 MPa interval) are applied to the structure. For each load level, the GW excitation and response of the two sensors are performed and controlled by a GW based SHM system which is developed by the authors [13]. The excitation signal is a five-cycle sine burst modulated by Hanning window. The central frequency and amplitude of the excitation signal are 200 kHz and ±70V respectively. The sampling rate of GW signal is 10 Msamples/s.

Fig.1. The experiment system of load influence on guided wave.

2.2 Experimental Results

The acquired GW signals at all load levels are de-noised by a method [14] based on complex continuous Shannon wavelet transform first. The de-noised GW signals are displaced in Figure 2 (a). The amplitude and phase variation of S0 mode of the GW signals are enlarged to be a better observation as well. There should be noted that in this paper, only S0 mode is considered. The reason will be explained in section 4. Figure 2 (b) gives the quantitative variation of the amplitude and phase. It can be noted that the phase velocity decreases linearly and the amplitude increases non-linearly accompanying with the increasing of the load. For measuring the change in phase velocity,

2 equation (1) is used [14]. The slope of phase velocity change is -0.576m·s-1·MPa-1. For measuring the change in amplitude, equation (2) is used. The change in amplitude is fitted by a mixed exponential equation shown in equation (3) and the parameters are obtained as a = 0.1328, b = 0.0022, c = -0.1328 and d = -0.0476.

0 6 0MPa Experimental result 10MPa Linear fit 5 20MPa -10 30MPa 4 40MPa 50MPa -20 60MPa 3 70MPa

Relative Relative amplitude 80MPa 2 -30 90MPa 100MPa 1 1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2 2.25 2.3 -40 Time(s) -5 x 10 15 Phase velocity changes(m/s) -50 10

5 -60 0 20 40 60 80 100 0 External load(MPa)

-5 20 Relative Relative amplitude

-10

-15 0 1 2 3 4 5 6 7 8 15 Time(s) -5 x 10

0MPa 0.1 10MPa 10 20MPa 0.05 30MPa 40MPa 50MPa

0 Amplitude changes(%) 60MPa 5 70MPa

Relative Relative amplitude -0.05 80MPa Experimental result 90MPa Exponential fit 100MPa -0.1 0 0 20 40 60 80 100 2.1 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 Time(s) -5 External load(MPa) x 10 (a) GW signals under all load levels (b) Phase and amplitude variation of GW signals

Fig.2. The experimental results of load influence on GW signals.

2 VP Vtp   (1) l

Amplevel  Amp0 Amp 100% (2) Amp0

b load d load Ampfit  a  e  c  e (3)

Where, Vp is the phase velocity of S0 mode and l is the GW propagation distance. Δt is the time shit of constant phase of GW signal. Amplevel is the GW amplitude at the corresponding load level and Amp0 is the amplitude at level 0. load is the load induced stress (Unit: MPa).

3. The Mechanism of Load Influence on Guided Wave

3.1 Acoustoelastic Effect

Acoustoelastic effect refers to the stress-dependence of acoustic bulk wave velocity in solid media [15]. When a structure is stress-free, the longitudinal wave velocity and transverse wave velocity of a non-dispersion elastic wave propagating in a solid structure can be expressed as equation (4) and (5) by using the second order Lame constants and . When the structure is in a stressed condition because of external load, the above two velocities can be expressed as equation (6) and (7) by combining with the third order Murnaghan constants l, m and n. In these two equations, T denotes the external load and K is the bulk modulus.

2 0CL  2  (4)

2 0C T  (5)

T C2 2 2 l 4 m 4 10 0 L       (6) 3K 

Tn Cm2  2 0 L    (7) 34K  It can be known that the two velocities become to be stress-independence because of the external load, and the relationship between the velocity change and the stress is linear. Although GW is a dispersion and multi-mode wave, the abovementioned equation can be also applicable to describe the acoustoelastic effect of GW because the GW is composed by the two wave components of longitudinal wave and transverse wave [16]. Thus, it is clear that if the load influence on GW phase change needs to be simulated, the acoustoelastic effect should be considered combined with the third order elastic constants. Specifically, the third order Murnaghan constants l, m and n should be included in the simulation model.

3.2 Load Influence on Piezoelectric Material

Piezoelectric material works based on the piezoelectric effect which is controlled by the piezoelectric constitutive equation. However, according to the studies of Hall [17] and Lynch [18] et al., the linear behaviour of piezoelectric constitutive relationships is generally confined to relatively low levels of applied electric field and stress, under which the dielectric, elastic and piezoelectric constant keep unchanged. When the piezoelectric material is loaded, the external load can change the polarization state of the piezoelectric material so as to change the piezoelectric constitutive relationships. According to the experimental results given by Lynch [18], the change of polarization state happens when the stress of piezoelectric material PZT- 5A is only 5MPa or larger. Besides, the experimental results also show that the dielectric, elastic constant and piezoelectric constant d33 are non-linear stress-dependence. Kang et al. [19] gave that the compression in the longitudinal direction of PZT-5A causes the non-linear decreasing of the piezoelectric constant d31 but does not affect the elastic constant. Thus based on the abovementioned research, it is clear that if the load influence on GW amplitude change needs to be simulated, the load influence on piezoelectric materials should be considered. However, the main obstacle for the simulation is that the numerical models which can be used to describe load influence on different piezoelectric parameters in the simulation model are unknown. Actually speaking, the GW signal waveform get from simulation model should match the real GW signal waveform get from real world. However, for the research on time-varying problem, it is enough if the variation trend of the simulated GW features can match that of the real GW features. Considering this point, some simplications are made as follows. (1) The load influence on the piezoelectric constant d31 is only considered because GW excitation and response by adhered piezoelectric sensors are mainly controlled by this parameter. (2) Considering that the load influence on piezoelectric material is the main factor that changes the GW amplitude, a numerical model is established to be equation (8) based on the amplitude change obtained from the experiment results in section 2.

dloadd00 d aeb load ce d load 31  31    31        (8) Where, a = 0.1328, b = 0.0022, c = -0.1328 and d = -0.0476, and load is the actual load induced stress (Unit: MPa). 4

4. The Multiphysics Simulation Model

4.1 The Simulation Physics

The physics of simulating GW under load condition are simplified shown in Figure 3. An aluminum plate is loaded at one end and is fixed on the other end. Two piezoelectric sensors (PZT 1 and PZT 2) are placed on the plate surface and are coupled with the aluminum plate directly. The adhesive layer is ignored because the load introduce little influence on it [12]. Thus, the acoustoelastic effect, piezoelectric effect and the load influence on piezoelectric sensors should be integrated into one simulation model. In addition, the load may change with time. However, in terms of short duration of GW, the load can be considered as static load. Thus, the process of GW excitation and response needs to be simulated under static load condition in the simulation model. In this paper, the multiphysics simulation is realized by using COMSOL Multiphysics 5.0. Considering the above discussed physics, the physic module called Piezoelectric Devices is adopted. It is a kind of multiphysics model shown in Figure 4 which includes the physic modules of Solid Mechanics and Electrostatics.

Fig.3. The physics diagram of GW propagation under load condition.

Fig.4. The simulation physics in COMSOL of GW propagation under load condition The Solid Mechanics is used to simulate mechanics feature of the aluminum plate and the piezoelectric sensors. The Electrostatics is used to simulate the electric feature of the piezoelectric sensors. The two physics are coupled by the Multiphyiscs-Piezoelectric Effect and they are described as follows, 1) In Solid Mechanics, Fixed Constraint and Boundary Load are used to simulate the fixed end and the external load of the aluminum plate respectively. The external load is applied to the plate from 0 MPa to 100 MPa with 20 MPa interval (Six load levels totally). 2) Hyperelastic material is used to realize the simulation of acoustoelastic effect. In the material model of Hyperelastic material, the Murnaghan model is adopted and the values of the third order Murnaghan constants will be given out in section 4.2. 5

3) Piezoelectric material combinded with Electrostatics is used to realize the simulation of the piezoelectric sensors. The property of Mechanical Damping is adopted, in which the Rayleigh damping is used and the parameters are set to be α = 0 and β = 2.2×10-8. The material property of the piezoelectric sensors will be given out in section 4.2. 4) Low-Reflecting Boundary is used to reduce the GW boundary reflection. The damping type is set to be ‘P and S waves’. 5) In Electrostatics, Ground is the electric ground of the piezoelectric sensors and it is set to their lower surface. 6) Electric Potential 1 and Electric Potential 2 are a zero potential and a voltage waveform of the GW excitation respectively. They are set to the upper surface of PZT 1 but they are mutually exclusive. This point will be explained in section 4.3.

4.2 The Simulation model

The simulation model is decribed as follows based on the modeling process of COMSOL. 1) Geometry: 3D model is used in this paper including the aluminum plate and the pieozelectric sensors. The dimension of the aluminum plate is 400mm×200mm×2mm (length×width×thickness). For the piezoelectric sensor, the diameter is 8mm and the thickness is 0.48mm. These diemnsions are the same with those of experiment. 2) Definitions: The definitions include two parts. The first part is the GW excitation waveform which is expressed as equation (9). The corresponding parameters are set to be A = 35 (Unit: V), f = 200 (Unit: kHz) and N = 5. The second part is the GW observation probe. It is set to the upper surface of PZT 2. The type of the probe is voltage and the average voltage of the whole surface will be output. EA1 cos 2 ftN sin 2 fttNf 1          (9) 3) Material: The material of the aluminum plate is shown in table 1 which contains the third order Murnaghan constants for the simualtion of acoustoelastic effect. The material of the piezoelectric sensors is shown in table 2. It is PZT-5A and the d31 parameter is set to be eqaution (9) for the simulation of load influence on the pieozelectric sensors. Table 1. Material property of the aluminum plate (2024 aluminum alloy). Parameter Value Density (kg/m3) ρ 2700 26 Lame constant (GPa) 51 l -250 Murnaghan constant (GPa) m -330 n -350 Table 2. Material property of piezoelectric sensor (PZT-5A).

Parameter Value ε 1730 Relative permittivity 11 ε33 1700 d -1.71-1.71×(a·eb·load + c·ed·load ) Piezoelectric constant 31 d 3.74 (×10-10 C/N) 33 d15 5.84 sE11 16.4 sE12 -5.74 Compliance coefficient sE13 -7.22 -12 2 (×10 m /N) sE33 18.8 sE55 4.75 sE66 4.43

4) Physics: The physics have been given out in section 4.1. 5) Mesh: The mesh type is Free Tetrahedral. For the plate, the mesh size depends on GW wavelength. To the GW of 200 kHz on the plate of 2mm, the wavelength of S0 mode is nearly 27mm. According to some research on GW simulation [9-11], the largest mesh length is recommended to be smaller than 1/6 of the wavelength. Thus, the largest mesh and smallest mesh size are set to be 3mm and 2mm respectively. The wavelength of A0 mode is only 8.6mm and the mesh size should be less than 1.4mm. It will lead to a huge amount of computation and a normal computer with 16 GB RAM cannot support such computation. That is reason why this paper only considers S0 mode. For the piezoelectric sensors, the largest and smallest mesh size are set to be 2 mm. The complete mesh consists of 83373 domain elements, 55246 boundary elements and 996 edge elements. There are 501876 degrees of freedom.

4.3 The solver of the Simulation Model

For solving the multiphysics simulation model given above, two study steps are adopted to construct the Study of the simulation model, as shown in Figure 5. The first step is Stationary which is used to perform the static load. After that, Time Dependent is performed to simulate the process of GW excitation-propagation-response under the static load condition. The results of Step 1 are used to be the initial values of Step 2.

Fig.5. The solver settings of the Multiphysics simulation model For each step, the physics settings are also shown in Figure 5. In Step 1, the Electric Potential 1 of zero potential is enabled to disable the PZT 1 but the Electric Potential 2 of GW excitation is disabled. This is because of two reasons. First, if the PZT 1 is not disabled in Step 1, the static load would introduce stress to PZT 1 and makes it generate a DC bias voltage. In Step 2, the DC bias voltage would act as a step excitation to introduce wideband GW propgating on the aluminium plate so as to lead a false GW response signal output from PZT 2. The second reason is that Step 1 is stationary study, there is no need to excite GW. In Step 2, Electric Potential 1 is disabled and Electric Potential 2 is enabled for GW excitation. The solver run one time for each load level mentioned in section 4.1. Thus, the GW signals under all six load levels can be obtained. The time step is set to be 1×10-7s in Step 2.

5. Simulation Results of Guided Wave Propagation under Load Condition

In the simulation model, there is no charge amplifier but the experimental system has one. So under this situation, the simulation signal under load 0 MPa and the experimental signal under 0 MPa are compared with each other as shown in Figure 6 (a). For a better comparsion, the two signals, which are normalized based on the amplitude of S0 mode, are given in Figure 6 (b). As it can be seen that the S0 mode of the two signals are matched well but there is a large error in A0 mode. This is because of the large mesh size. The simulation GW signals under all load conditions are displayed in Figure 7 (a). The amplitude and phase variation of S0 mode of the GW signals are enlarged to a better observation as well. Figure 7 (b) gives the quantitative variation of the amplitude and phase.

Simulation GW response signal 0.4 2 0.2 1.5 0

Voltage(V) -0.2 1

-0.4 0.5 0 100 200 300 400 500 600 700 800 900 1000 Signal dots 0 Experiment GW response signal 2 -0.5 1

Normalized amplitude Normalized -1 0 -1.5 Experiment GW response signal Voltage(V) -1 Simulation GW response signal

-2 -2 0 100 200 300 400 500 600 700 800 900 1000 300 400 500 600 700 800 900 1000 Signal dots Signal dots

(a) Original simulation and experimental signals (b) Normalized comparison (S0 mode) of the two signals Fig.6. The GW signals comparsion between simulation and experiment.

0 0.28 0MPa Simulation result Curve fit 20MPa -10 0.26 40MPa 60MPa 0.24 -20 80MPa 100MPa 0.22 -30 Normalized amplitude Normalized 0.2

5.05 5.1 5.15 5.2 5.25 -40 Time(s) -5 x 10 0.4 Phase velocity changes(m/s) -50 0MPa 0.2 20MPa 40MPa -60 0 20 40 60 80 100 60MPa 0 External load(MPa) 80MPa 20 100MPa -0.2 Normalized amplitude Normalized

-0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 15 -3 Time(s) -4 x 10 x 10

0MPa 5 20MPa 10 40MPa 60MPa 0

80MPa Amplitude changes(%) 5 100MPa Simulation result Normalized amplitude Normalized -5 Exponential fit

5.25 5.26 5.27 5.28 5.29 5.3 0 0 20 40 60 80 100 Time(s) -5 x 10 External load(MPa) (a) Simulation GW signals under all load levels (b) Phase and amplitude variation of signal GW signals

Fig.7. The simulation results of load influence on GW signals. It can be noted from Figure 7 that the phase velocity decreases linearly and the amplitude increases non-linearly accompanying with the increasing of the load. The slope of phase velocity change is -0.467m·s-1·MPa-1. It is a little lower than the experimental result of

-0.576m·s-1·MPa-1. This error may be due to the material difference of the aluminium plate between experiment and simulation. Based on the numerical model shown in equation (9), the simulation of load influence on piezoelectric sensor is realized and the simulation results match the experimental results well.

6. Conclusion

This paper proposes a simple but efficient method for multiphysics simulation of GW propagation under load condition based on COMSOL Multiphysics. The acoustoelastic effect and the load influence on piezoelectric sensor are integrated into one multiphysics simulation model. The simuation of acoustoelastic effect is realized by using Hyperelastic material model which contains the third order Murnaghan constants. The simuation of load influence on piezoelectric sensor is realized by building a d31 numerical model of stress-dependence based on the experimental data. The model solver is constructed by combing the stationary analysis and time-dependent analysis together. The whole GW propagation simulation under load condition is fullfiled in one software platform and in one model solver. The simulation results under uniaxial load from 0 MPa to 100 MPa are obtained and they are compared with experimental data from the two aspects of amplitude change and phase change. It shows that the results obtained from the simulation match the experimental results well which indicates the correctness of the proposed method. According to the simulation results, a preliminary conclusion can be made that a numerical model of load influence on piezoelectric material, which is constructed by a calibration experiment on simple structure and load condtion, can be applied to more complex structure and load condition because the load influence on piezoelectric sensor is structural independent and is only stress-dependence. In the near future, the multiphysics simulation of GW propagation under changing temperature condition will be studied. It will be combined with the method proposed by this paper to achieve a comprehensive multiphysics simulation of GW propagation under time- varying conditions.

7. Acknowledgements

Lei QIU would like to acknowledge the Alexander von Humboldt Research Foundation for its support to undertake scientific collaborations in Germany via a Humboldt Research Award. He also expresses his gratitude to the Chair of Non-Destructive Testing and Quality Assurance of Saarland University for its hospitality in hosting him. This work is supported by National Science Fund for Distinguished Young Scholars of China (Grant No.51225502), Key Program of Natural Science Foundation of China (Grant No. 51635008), National Natural Science Foundation of China (Grant No. 51575263), Qing Lan Project and Young Elite Scientist Sponsorship Program by CAST of China.

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