Nonlinear Ultrasonic Guided Wave Method Using Semi-Analytical Finite Element (SAFE) Technique on a Damaged SWO-V Spring Coil

: This work presents a non-destructive method for quantitative essessment of fatigue damage of materials with linear elastic properties using nonlinear ultrasonic techniques. A nonlinear study was conducted on these materials with fatigue and shot peening processing using a nonlinear ultrasonic technique. A numerical method based on the semi-analytical ﬁnite element (SAFE) technique, was used to obtain the phase-matching modes of the specimens. Experiments conﬁrm that the nonlinearity for shot peening and samples with a certain level of fatigue shows a tendency to increase with levels of fatigue.


Introduction
Driving Safety is seriously affected by malfunction or damage of materials with linear elastic properties, such as tension rods, ton-shoulders, and springs used in vehicles. Research on the safety of the structure and research on the optimization and weight reduction of parts are becoming more important. Therefore, research on these materials is of a great concern.
In general, the life evaluation of materials with linear elastic properties is performed through the Nakamura fatigue test, which is one of the destructive evaluation techniques. However, this technique consumes considerable time and cost. In addition to life evaluation, to improve the durability, life, and performance of a spring, a cold working process called shot peening is performed on these materials. Shot peening leads to plastic deformation on the surface and changes the mechanical properties of the material. Surface cracks do not propagate well in materials subjected to compressive stress, so the propagation of micro-cracks on the surface of the material can be prevented through shot peening.
In addition, the plastic deformation caused by shot peening results in residual compressive stress on the surface and can impart resistance to metal fatigue and some forms of stress corrosion. However, if residual stress is generated while impurities are contained within the material, it may break along grain boundaries that are weakened by impurities. Currently, preventing the accidents induced by damaged springs becomes a significant issue, and the soundness of spring coils has a great influence on vehicle performance and reliability. Therefore, it is necessary to strictly apply safety management of a structure before an accident occurs, and to improve the life evaluation by detecting micro-cracks or defects in an early stage by a non-destructive method.
For life evaluation and safety management, a nonlinear guided wave technique can be used. It has been explored for many years as a promising tool that can diagnose microdefects or microscopic structural changes in micro-and nano-units through the selection of various guided wave modes [1,2]. Buck et al. [3] conducted a study on stress and changes in the second harmonic function. Later, Jacobs and colleagues [4] studied the plasticization of materials using nonlinear ultrasonic guided waves [5][6][7][8][9][10][11][12][13]. They investigated the nonlinear shear horizontal wave and the cumulative second harmonic.
Guided waves were applied to efficiently inspect slender samples such as wire rods [14,15]. Gazis performed basic ultrasonic guided wave analysis on cylindrical structures [16]. Other related research was mainly studied by Rose and Cowley [17,18]. Recently, Cho et al. conducted an inspection by using a PVDF sensor to detect localized fatigue damage on an aluminum plate [19]. Guan and Su compared the differences in fatigue crack detection using nonlinear guided ultrasound between plate and pipe structures [20]. Li et al. conducted a nonlinear guided ultrasonic assessment of laminated composites subjected to low-speed impact and improved the signal-to-noise ratio (S/N ratio) by using a phase inversion method to measure the second harmonic [21]. Recently, many researchers have conducted research on the relationship between nonlinearity and linear elastic materials such as pipes using advanced technology [22][23][24].
This study presents an experimental approach to assess the local micro-grain structure change of materials with linear elastic properties based on the nonlinear features corresponding to the damage. In linear elasticity, there is a proportional constant representing the slope of a straight line relating with load and deformation. For example, in the case of a simple unidirectional tensile test, the slope between the axial stress and the corresponding axial strain is expressed as Young's modulus E, and this is one of the linear elastic moduli. It is a well-known fact that when a material is subjected to microdamage, such as grain structure change, the slope tends to show a tiny nonlinearity. The nonlinearity can be linked to the higher harmonic amplitude of ultrasound, providing a specific feature of the corresponding microdamage. NDE was performed for a wire rod specimen subjected to shot peening using the nonlinear guided ultrasonic method in the comparison of the Nakamura fatigue test. In this study, the second harmonic of the longitudinal wave was generated, and the reliability of the experimental result was confrimed by controlling the fatigue variables.

Nonlinear Wave Theory
When the higher-order elastic constants of solid materials increase, the relationship of stress and strain is nonlinear rather than linear. With linear ultrasonic theory, it is difficult to describe the behavior of a material that does not follow the plastic region or elastic motion of the stress-strain curve, but applying nonlinear supersonic theory allows for more accurate behavior [25]. In the nonlinearity measurement method, ultrasonic waves of a single frequency are incident on the sample and propagate for a certain distance. The signal is distorted due to the third-order modulus of elasticity of the material, which is considered to be the second harmonic. The second harmonic occurs when there is a change in micro-structure [26], such as plastic deformation and corrosion, and it is relatively small compared to the first harmonic.
The nonlinear factor indicates the relationship between the microstructural changes of these materials and the denominator and is defined by Equation (1): where C 111 is the cubic elastic modulus, c l is the velocity of the longitudinal wave, ρ is the density of the material, and A 1 and A 2 are the amplitude of the primary and second harmonic, respectively. k is the wave number, and x is distance of the propagation. Under certain conditions, the parameter β is measured for a fixed wavenumber and propagation distance:

General Solution of Nonlinear Wave Equation in a Solid Cylindrical Rod
To obtain a general solution for a nonlinear wave equation in a solid cylindrical rod, elastic waveguides with an arbitrary cross-section area are theoretically studied. The harmonic generation of ultrasonic guided wave propagation in cylindrical rods was reported by de Lima et al. [12]. Practically, in tube-like structures, the guided wave propagation provides a more efficient approach to detect material nonlinearity due to its lower energy diffusion [27].
In Lagrangian coordinates, the nonlinear wave equation is expressed as in Equation (3): where u is the particle displacement, λ and µ are the Lamé constants, ρ 0 is the initial density of the material, and f (u) is the nonlinear term. The perturbation method is used to solve the nonlinear wave equation, and the solution is taken from Equation (4): where u 1 is the primary wave, u 2 is the second harmonic (double frequency component), and u 2 is assumed to be much smaller than u 1 . The primary wave with frequency ω and the wavenumber k is written as in Equation (5): Constructing the second-order solution using model expansion [21], the total secondharmonic field can be written as Equation (6): where u(2,m) is the displacement field function. v = du dt and v(2,m) are the particle velocity, and s(2,m) is the stress tensor for the mth double frequency components. r is the thickness of the waveguide. A m (z) is the modal amplitude: A m (z) is the solution of the following ordinary differential Equation (8) with the condition u 2 = 0 at z = 0: where: In Equation (9), Ω is the waveguide cross-sectional area and Γ is the curve enclosing Ω. The terms f surf n and f vol n are respectively defined as the complex external power due to the surface traction s 2ω and the volume force f 2ω . k n is the wavenumber of the mode identified by index n, which is not orthogonal (p mn = 0) to the mode with wave number k m . This analysis is applied to a cross-sectional area of any shape. For the cylindrical rod shown in Figure 1, the expressions of p mn , f surf n , and f vol n are [12]: The assumed primary wavefield of particle displacement components is [26]: where the circumferential order is n = 0, 1, 2,···. For axis-symmetric modes, n = 0 and n ≥ 1 for the flexural modes. Generally, the terms 1 r u , θ 1 u , and 1 z u depend on the type of waveguide and family of modes. In practice, second harmonic generation with an accumulative effect is of great interest. The reason is that the cumulative effect of the second harmonic amplitude has a great advantage for detection in experimental work to measure the nonlinear effect with a sufficient signal-to-noise ratio. From the analysis, specific conditions must be satisfied for second harmonic generation with accumulative effect [3,14,28].   The assumed primary wavefield of particle displacement components is [26]:

Phase Matching Mode
where the circumferential order is n = 0, 1, 2, · · · . For axis-symmetric modes, n = 0 and n ≥ 1 for the flexural modes. Generally, the terms u r 1 , u θ 1 , and u z 1 depend on the type of waveguide and family of modes.
In practice, second harmonic generation with an accumulative effect is of great interest. The reason is that the cumulative effect of the second harmonic amplitude has a great advantage for detection in experimental work to measure the nonlinear effect with a sufficient signal-to-noise ratio. From the analysis, specific conditions must be satisfied for second harmonic generation with accumulative effect [3,14,28].

1.
Nonzero power flux from the primary to the secondary wave: Matching phase and group velocities require equal phase and group velocities of the primary and the secondary waves when k * n = 2k (synchronism).

Phase Matching Mode
Due to the dispersion and multi-mode characteristics of guided ultrasound, it is difficult to efficiently generate the second harmonic. Therefore, one of the numerical analysis methods, the SAFE technique, was used to induce the phase-matching mode of the longitudinal wave mode of the specimen. Figure 2 shows a SAFE model of a rod.

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Due to the dispersion and multi-mode characteristics of guided ultrasound, it is difficult to efficiently generate the second harmonic. Therefore, one of the numerical analysis methods, the SAFE technique, was used to induce the phase-matching mode of the longitudinal wave mode of the specimen. Figure 2 shows a SAFE model of a rod.
In Equation (12), T is the transpose of the matrix, and the differential operator L in the cylindrical coordinate system is: The relationship between strain and displacement is expressed as Equation (14): where b1, b2, b3, and b4 are:  Figure 3 shows a method of performing discretization along the radial direction using a one-dimensional three-node element. The shape function is an interpolation function defined as Equation (16): In Equation (12), T is the transpose of the matrix, and the differential operator L in the cylindrical coordinate system is: The relationship between strain and displacement is expressed as Equation (14): where b 1 , b 2 , b 3 , and b 4 are: Figure 3 shows a method of performing discretization along the radial direction using a one-dimensional three-node element. The shape function is an interpolation function defined as Equation (16): The displacement field of each discrete element is described as a shape function matrix and a node displacement vector after finite element discretization.
In Equation (17), m is the circumferential order. The shape function is:  The displacement field of each discrete element is described as a shape function matrix and a node displacement vector after finite element discretization.
In Equation (17), m is the circumferential order. The shape function is: Since each node has 3 degrees of freedom (DOFs) in different directions, the vector q (e) is denoted as Equation (19): Therefore, the strain can be expressed as Equations (20) and (21): Here, the geometric matrix B1 is: N is the derivative term of the shape function matrix for the e-direction in the local coordinate system. By applying the traction-free boundary condition for the SAFE model, the guided ultrasound equation for the wave motion of each element can be obtained based on Hamilton's principle [29,30]: where: Figure 3. Discretization of a pipe with 1D three-node elements.
Since each node has 3 degrees of freedom (DOFs) in different directions, the vector q (e) is denoted as Equation (19): Therefore, the strain can be expressed as Equations (20) and (21): Here, the geometric matrix B 1 is: N is the derivative term of the shape function matrix for the e-direction in the local coordinate system. By applying the traction-free boundary condition for the SAFE model, the guided ultrasound equation for the wave motion of each element can be obtained based on Hamilton's principle [29,30]: where: k A standard finite element assembly for all elements can be simplified to a common eigenvalue problem in the global coordinate system as in Equation (25): It can also be marked as a first-order native system as in Equation (26): where: Given the frequency values of the wave model, a series of wavenumbers can be calculated by solving this typical eigenvalue problem. The dispersion relationship can be derived later.

Material Properties
The material properties of the specimen are shown in Tables 1 and 2 [31]. The phase and group velocity dispersion curves of the longitudinal mode are shown in Figure 4. This information can be obtained directly by replacing this information with the SAFE code. Single-frequency and double-frequency dispersion diagrams are indicated by solid and dotted lines, respectively.

Experimental Design
In Figure 4, it can be seen that the same values of the phase and group velocity appeared in the red region. Therefore, 1.4 MHz in mode L (0,2) and 2.8 MHz in L (0,3) were selected. Figure 5 shows a diagram of the experimental setup. The tone burst ultrasonic device was a RITEC PRP-4000, which was used to generate and receive ultrasonic waves. The transducer and the acrylic wedge generated longitudinally induced ultrasonic waves, and the angle of incidence was derived by Snell's law [32]: In this equation, c acrylic = 2670 m/s is the propagation velocity of ultrasonic waves in acrylic, c p = 6500 m/s is the phase velocity in L (0,2) mode at 1.4 MHz, and theta is the incident angle. The diameter of the specimen is 3.1 mm, and it was cut to a length of 946 mm. Shot peening was performed for 40 min with a 0.6 mm diameter shot ball made of SWRH82A and then shot peening was performed again with a 0.3 mm shot ball.

Experimental Design
In Figure 4, it can be seen that the same values of the phase and group velocity appeared in the red region. Therefore, 1.4 MHz in mode L (0,2) and 2.8 MHz in L (0,3) were selected. Figure 5 shows a diagram of the experimental setup. The tone burst ultrasonic device was a RITEC PRP-4000, which was used to generate and receive ultrasonic waves. The transducer and the acrylic wedge generated longitudinally induced ultrasonic waves, and the angle of incidence was derived by Snell's law [32]:

Experimental Set up
As seen in Figure 5, the signal was generated at a distance of 460 mm from the end of the sample and received at intervals of 20 mm from a distance of 60 mm. A total of 7 sections were inspected, including the center of the specimen. Each section was measured 5 times. The total length of the flaw was 180 mm, and the area where the maximum fatigue stress occurred was inspected. The signal was measured to investigate the correlation of the nonlinearity according to the degree of fatigue. A signal in longitudinal mode was generated for the specimen using the RITEC RPR-4000 tone burst ultrasonic device.
As shown in Figure 6, a motor was installed at the left end of the rod, and a fatigue test was conducted by supporting the other end of the rod [33]. While the test was being carried out, the largest load was applied to the center of the specimen. A separate recording device was connected to the device, and the number of rotations until the rod was

Experimental Set Up
As seen in Figure 5, the signal was generated at a distance of 460 mm from the end of the sample and received at intervals of 20 mm from a distance of 60 mm. A total of 7 sections were inspected, including the center of the specimen. Each section was measured 5 times. The total length of the flaw was 180 mm, and the area where the maximum fatigue stress occurred was inspected. The signal was measured to investigate the correlation of the nonlinearity according to the degree of fatigue. A signal in longitudinal mode was generated for the specimen using the RITEC RPR-4000 tone burst ultrasonic device.
As shown in Figure 6, a motor was installed at the left end of the rod, and a fatigue test was conducted by supporting the other end of the rod [33]. While the test was being carried out, the largest load was applied to the center of the specimen. A separate recording device was connected to the device, and the number of rotations until the rod was damaged during the experiment was recorded. In this study, to manufacture damaged specimens due to fatigue load, fatigue tests were conducted by dividing the degree of fatigue into 4 types. The number of revolutions per minute was 5000 rpm, and based on 100% being 10 million cycles, 0% (0 cycle), 30% (3 million cycles), 60% (6 million cycles), 100% (10 million cycles), 130% (13 million cycles), and 160% (16 million cycles) were applied. Michihiko (1992) also showed the result of the test with the number of cycles up to 10 7 for SWO-V during the Nakamura fatigue test. Therefore, in this paper, 10 7 cycles were the 100% standard [34].
of the sample and received at intervals of 20 mm from a distance of 60 mm. A total of 7 sections were inspected, including the center of the specimen. Each section was measured 5 times. The total length of the flaw was 180 mm, and the area where the maximum fatigue stress occurred was inspected. The signal was measured to investigate the correlation of the nonlinearity according to the degree of fatigue. A signal in longitudinal mode was generated for the specimen using the RITEC RPR-4000 tone burst ultrasonic device.
As shown in Figure 6, a motor was installed at the left end of the rod, and a fatigue test was conducted by supporting the other end of the rod [33]. While the test was being carried out, the largest load was applied to the center of the specimen. A separate recording device was connected to the device, and the number of rotations until the rod was damaged during the experiment was recorded. In this study, to manufacture damaged specimens due to fatigue load, fatigue tests were conducted by dividing the degree of fatigue into 4 types. The number of revolutions per minute was 5000 rpm, and based on 100% being 10 million cycles, 0% (0 cycle), 30% (3 million cycles), 60% (6 million cycles), 100% (10 million cycles), 130% (13 million cycles), and 160% (16 million cycles) were applied. Michihiko (1992) also showed the result of the test with the number of cycles up to 10 7 for SWO-V during the Nakamura fatigue test. Therefore, in this paper, 10 7 cycles were the 100% standard [34].        Figure 9 shows the nonlinear result of shot peening. For the two specimens, it was confirmed that the nonlinearity increased with the distance, and the nonlinearity was dominantly influenced by the overlapping effect. However, the nonlinearity of the shot peening specimen marked with a red dotted line was higher by about 0.5 to 1 than that of the general specimen marked with a solid black line. Although there is a slight difference in nonlinearity, it could be seen that in shot peening and general specimens, the nonlinearity tended to increase as the measurement distance increased.  Figure 9 shows the nonlinear result of shot peening. For the two specimens, it was confirmed that the nonlinearity increased with the distance, and the nonlinearity was dominantly influenced by the overlapping effect. However, the nonlinearity of the shot peening specimen marked with a red dotted line was higher by about 0.5 to 1 than that of the general specimen marked with a solid black line. Although there is a slight difference in nonlinearity, it could be seen that in shot peening and general specimens, the nonlinearity tended to increase as the measurement distance increased.  The results of the nonlinear experiment according to the degree of fatigue are shown in Figure 10. For the specimen (0%) not subjected to the fatigue test, it was confirmed that the nonlinearity increased as the measurement distance increased, and the overlapping effect was dominantly affected. For specimens with a fatigue degree of 30% to 100%, the nonlinear tendency was dominated by the superposition effect. The degree of fatigue was a nonlinear result for 130% and 160% specimens. With a specimen with a degree of fatigue exceeding 10 million times, and it can be seen that the nonlinear tendency gradually decreased from around 120 mm, the point where the fatigue load was the largest. The results of the nonlinear experiment according to the degree of fatigue are shown in Figure 10. For the specimen (0%) not subjected to the fatigue test, it was confirmed that the nonlinearity increased as the measurement distance increased, and the overlapping effect was dominantly affected. For specimens with a fatigue degree of 30% to 100%, the nonlinear tendency was dominated by the superposition effect. The degree of fatigue was a nonlinear result for 130% and 160% specimens. With a specimen with a degree of fatigue exceeding 10 million times, and it can be seen that the nonlinear tendency gradually decreased from around 120 mm, the point where the fatigue load was the largest.

Results
in Figure 10. For the specimen (0%) not subjected to the fatigue test, it was confirmed that the nonlinearity increased as the measurement distance increased, and the overlapping effect was dominantly affected. For specimens with a fatigue degree of 30% to 100%, the nonlinear tendency was dominated by the superposition effect. The degree of fatigue was a nonlinear result for 130% and 160% specimens. With a specimen with a degree of fatigue exceeding 10 million times, and it can be seen that the nonlinear tendency gradually decreased from around 120 mm, the point where the fatigue load was the largest.

Conclusions
In this study, a SAFEM was used to calculate the dispersion curve of a guided ultrasonic wave. One of the NDT methods, the nonlinear guided ultrasonic flaw detection method, was used to investigate the nonlinear tendency of a shot peening specimen according to the surface change, and it was confirmed that the nonlinear tendency increase when the sub surface grain structure changed due to the shot peening. For the specimens subjected to the Nakamura fatigue test, in the case of 0% to 100%, the damping effect was

Conclusions
In this study, a SAFEM was used to calculate the dispersion curve of a guided ultrasonic wave. One of the NDT methods, the nonlinear guided ultrasonic flaw detection method, was used to investigate the nonlinear tendency of a shot peening specimen according to the surface change, and it was confirmed that the nonlinear tendency increase when the sub surface grain structure changed due to the shot peening. For the specimens subjected to the Nakamura fatigue test, in the case of 0% to 100%, the damping effect was not large, so nonlinearity was dominant; however, when the fatigue cycle exceed the fatigue limit, plastic deformation occurres relatively greater attenuation. In Figure 10, the 120 mm point was the point where the fatigue load was the largest of the length of the specimen, and since the plastic deformation occurred the most, it was estimated that the nonlinearity decreased in the experimental results of 130% and 160%. This was assumed to be the result of plastic deformation in the microstructure due to fatigue.