PMUTs Arrays for Structural Health Monitoring of Bolted-Joints

Micro-electro-mechanical systems (MEMS) have enabled new techniques for the miniaturization of sensors suitable for Structural Health Monitoring (SHM) applications. In this study, MEMS-based sensors, specifically Piezoelectric Micromachined Ultrasonic Transducers (PMUT), are used to evaluate and monitor the pre-tensioning of a bolted joint structural system. For bolted joints to function properly, it is essential to maintain a suitable level of pre-tensioning. In this work, an array of PMUTs attached to the head and to the end of a bolt, serve as transmitter and receiver, respectively, in a pitch-catch Ultrasonic Testing (UT) scenario. The primary objective is to detect the Change in Time of Flight (CTOF) of the acoustic wave generated by the PMUT array and propagating along the bolt’s axis between a non-loaded bolt and a bolt in service. To model the pre-tensioning of bolted joints and the transmission of the acoustic wave to and from a group of PMUTs through the bolt, a set of numerical models is created. The CTOF is found to be linearly related to the amount of pre-tensioning. The numerical model is validated through comparisons with the results of a preliminary experimental campaign.


Introduction
The primary function of bolts in joints is to generate sufficient clamping force to prevent the separation between joint parts. Appropriate pre-tensioning of a bolt makes it less sensitive to fatigue and prevents it from self-loosening when subjected to extreme vibration, stress, or temperature cycles [1]. Both the initial magnitude and the clamping force's ability to be maintained over time have an effect on the behavior and condition of the bolted joints [2]. As a result, improperly adjusted bolt pre-tensioning can lead to a number of issues in the joints, such as joint separation [1], bolt fatigue failure [3], and nut vibration loosening [4]. Consequently, it is necessary to measure and monitor the pre-tensioning force of the bolts with precision.
Various methods were proposed in the literature to measure the pre-tension of bolts. Strain gauges are a highly accurate method; the use of such technology results in an accuracy of about ± (1-2)% [1,5,6]. They must be installed on a stressed section of the bolt. Therefore, mounting them on an easily accessible end of the bolt is worthless. To take a measurement, there should be easy access to the clamped portion of the bolt shank, which is problematic for many in situ Structural Health Monitoring (SHM) applications. Thus, despite its great level of precision, this approach is best suited for experimental laboratory work. Methods worth mentioning are the Direct Tension Indicators (DTI) [6], the torque control wrench method, the angle control method [7] and Ultrasonic Testing methods (UT) [8]. The DTI are based on measuring the gap between a special washer with bumps on its top surface and the nut. When the gap is smaller than a certain limit, the proper campaign on a standard M12 bolt are presented and compared with the results obtained from the numerical models. The last section is devoted to closing remarks.

Acoustoelastic Effect: Bolt Pre-Tensioning SHM Application
The fastening of a bolt occurs as a result of the tensile stress along the axis of the bolt. The infinitesimal elastic theory fails to express the influence of the state of stress the material is experiencing on the velocity of elastic wave propagation. The acoustoelastic effect explains the dependency of the velocity of an elastic wave in a stressed solid material not only on the second-order elastic constants and density but also on higher-order elastic constants and stress. According to this, the velocity of the wave propagating will change as the stresses changes inside the material. The acoustoelastic effect is also known as the "small on large" theory which considers the propagation of an elastic wave in the material as a small localized state of deformation in addition to the relatively larger deformation caused by the applied stress [36]. Hughes et al. [11,37] were the first to analyze and study the relationship between the stress and the velocity of an elastic wave in an isotropic medium under hydrostatic pressure. Their work was based on the Murnaghan's theory of finite deformations and third-order elastic constants in the energy. For isotropic materials, three additional constants, l, m, and n are required to describe the material in addition to the Lamé constants λ and µ. Takahashi and Motegi's work [13,14], derived an equation of motion for an elastic wave in a finitely deformed state using Lagrangian description and the Murnaghan's finite deformation theory for a unidirectional deformed isotropic solid. Their work shows the need of using the third-order elastic constants in estimating the unknown stresses in structural materials. Considering a longitudinal wave propagating along the same direction as the applied stress, representing a 1D problem, in a uniform isotropic material, a first-order approximation of the velocity relation with the stress can be derived [13]: in which, where ρ is the mass density, λ, and µ are the 2nd-order elastic constants, and l, m, and n, are the 3rd-order elastic constants. For the case of a pre-tensioned bolt, V 0 and V 11 represent the stress-independent or zero-load velocity, and stress-dependent acoustic wave propagation velocity in the solid domain, respectively. T 11 represents the stress along the bolt axis. Keeping track of the behavior of the P-wave, the only source of stresses affecting the wave velocity in the same direction of propagation of the P-wave, i.e., along the bolt axis, is the stress level T 11 . This allows for a reduction of the problem from 3D to 1D, see Figure 1. The relationship presented in Equation (1), which shows the direct and linear relationship between wave velocity and stress, provides a theoretical basis for the axial stress estimation and the assessment of a bolted joint. The most common available technique adopted in ultrasonic testing is based on measuring the TOF of a traveling acoustic ultrasonic wave either in a pitch catch or a pulse-echo scenario. In the pitch-catch scenario, adopted in this work, the transmitting transducer is usually mounted on top of the bolt head, while the receiving transducer is mounted on the far end of the bolt. The TOF for a pre-tensioned bolt differs from the one of the reference, unloaded, bolt for two reasons.

1.
The bolt stretches as it is tightened, thus the path length of the acoustic wave increases.

2.
The reduction in the acoustic wave velocity due to the acoustoelastic effect.
For a bolt loaded in an elastic regime, both changes are linear functions of the level of tension the bolt is experiencing. Hence, the CTOF is also a linear function of the bolt pre-tensioning [1].
The TOF in the pre-tensioned bolt, in a pitch-catch scenario, is defined as follows [38]: where z is the direction along the bolt axis, L is the length of the bolt, ε(z) is the strain along the bolt axis, and V 11 (z) is the stress-dependent acoustic wave velocity given in Equation (1). Based on linear elasticity, the following relation holds: Substituting Equations (1) and (5) into Equation (4), the TOF can be expressed as: By assuming that (ε(z) α 11 ) 1, using the Maclaurin expansion of 1/(1 + ε(z)α 11 ) ≈ 1 − (ε(z)α 11 )) + (ε(z)α 11 ) 2 − (ε(z)α 11 ) 3 , and neglecting the higher order terms, the following expression can be obtained: By neglecting the term ε(z) 2 α 11 and integrating: δ is the elongation of the bolt due to pre-tensioning, while TOF 0 is the time of flight before pre-tensioning. The CTOF can therefore be computed as: For a bolt loaded elastically, and reduced to a 1D problem, the value of δ is defined as: where P is the pre-tensioning force and K is the equivalent stiffness of the pre-tensioned bolt, equal to EA/L in which A is an equivalent cross-sectional area and L is the deformed length. Equations (10) and (11), show that the CTOF is linearly proportional to the bolt pre-tensioning. To develop this relationship, a set of numerical models are needed. The first model simulates the pre-tensioning of a bolted joint under several loading conditions. For different levels of pre-tensioning forces P, the axial stress T 11 (z) along the bolt axis and the elongation δ can be evaluated. A second model is created to simulate a pitch-catch UT scenario using an array of PMUTs for transmission and reception. From this model, the value of the Reference Time of Flight (TOF 0 ) can be evaluated. Finally, a MATLAB code is needed to combine the acoustoelastic effect with the results obtained from the numerical models.

Pre-Tensioned Bolt Modeling
An M12 steel bolt is chosen and the FE code COMSOL Multiphysics V5.5 (COMSOL, Inc. Stockholm, Sweden) is used to create and analyze the bolted joint system. The FE model consists of the following elements: the bolt, made of a bolt-head and a bolt-shank, a nut, and a clamped element represented by the plates; see Figure 2. For the sake of simplicity, the threaded part of the bolt shank extending along the bolt length is not depicted. Additionally, the hexagonal shape of the bolt head is replaced by a cylinder with a diameter equal to the standard nominal diameter of an M12 bolt. The cylinder representing the bolt head has a height of 7.5 mm and a radius of 10 mm while the bolt shaft has a height of 100 mm and a radius of 6.85 mm. The length of the clamped region, represented by the height of the two plates, is equal to 40mm. The numerical modeling of a bolted-joint under pre-tensioning is carried out under the following hypotheses: material and geometrical nonlinearities, implementation of hyperelastic Murnaghan's constitutive model, contact conditions between surfaces of the clamped elements with the upper surface of the nut and the lower surfaces of the head of the bolt, and contact conditions between the bolt-shank and the hole in the clamped elements; see Figure 3. All structural elements are modeled using structural steel having Young's modulus and Poisson's ratio of 200 GPa and 0.3, respectively, and with values of the 2nd and 3rd elastic constants equal to (λ = 1.15 × 10 11 , µ = 7.69 × 10 10 , l = −3.0 × 10 11 , m = −6.2 × 10 11 , n = −7.2 × 10 11 ) Pa [14,39]. The system of bolted joint, shown in Figure 2, is entirely meshed using quadratic wedge elements. The mesh size is selected with reference to the dimensions of the discretized bolt portion; for example, the bolt head mesh is generated by meshing to the top surface of the bolt head with an element size equal to the radius of the bolt head divided 8. The surface mesh is then swept along the height of the bolt head creating wedge elements. A similar meshing process is adopted for the bolt shank, the nut and the plates.
To apply the prescribed pre-tensioning, a surface boundary A P , see Figure 3, is created in the bolt-shank perpendicular to the bolt axis at the level the two clamped plates are meeting. A pre-tensioning load is applied on A P in the form of pre-tensioning stress σ P . This method ensures that the prescribed pre-tensioning is attained inside the bolt. The exterior boundary of the plates is modeled as fixed boundaries. This model involves a high number of contact surfaces, such as the contact problem at the bolt head/plate, nut/plate, and the plate and nut hole/thread interaction surfaces. A fine-scale computation of the bolted joint assemblies, in which the geometry of the threaded regions is modeled, is too computationally expensive. Hence, more simplified computational methods for accurately solving contact problems may be necessary to handle such computationally demanding problems. A non-penetration condition is enforced at the contacting boundaries. Both contributions of the normal and friction tractions are modeled at the contacting surface. A friction coefficient of 0.1 is used. Additional treatment is devoted to the contacting surfaces in the threaded regions of the plates and nut threads and bolt threads. For the threaded region, the actual geometry of the threads is handled by a mathematical formulation of the contact condition. Hence, some additional geometrical design parameters can be numerically implemented at the mating threads. A 30 • half-angle of a single thread is used with a lead length of 1.25 mm. Adopting this simplified modeling technique of the pre-tensioned bolt is computationally very efficient and provides accurate results [40]. A spring foundation BC is enforced on the bottom end of the shank, primarily to reduce instability, which is a common concern in numerical modeling of contact problems. To solve the contact problem at the contacting surfaces, the Augmented Lagrangian technique is adopted. The technique is implemented through the algorithmic Uzawa's Method, which provides an iterative update of the contact tractions at the contacting surface [41]. The non-linear contact problem of the pre-tensioning of bolted joints is numerically solved by adopting the double dogleg method, which is highly suitable for non-linearly constrained contact problems. The method is based on a combination of the Steepest descent and Newton-Raphson methods, which guarantee a good convergence of the solution.   Table 1 is obtained by integrating the strains along the bolt axis. Figure 4 shows the T 11 stress distribution for two of the loading cases simulated, namely, the case of a pretension level of 200 and 400 MPa. However, only the variations of T 11 along the bolt axis are of interest for the particular application presented in this work, since the problem is reduced to 1D; see Figure 1. The stresses along the bolt axis T 11 (z) for different pre-tensioning forces are shown in Figure 5. The bolt shank stress is observed to differ from zero by negligible amounts. These forces result from the spring foundation BC assigned to the end surface of the shank. It is possible to adjust the spring foundation's stiffness to minimize the effect on the bolt shank while preserving the solution's stability. From the stress T 11 (z), the stress-dependent acoustic wave velocity V 11 (z) is computed along the length of the bolt using Equation (1). The acoustoelastic, or small-on-large, theory is based on the assumption that the time-dependent deformation due to the wave propagation is much smaller than the static state of deformation due to the applied stress, which indeed holds for this work [36]. This makes the mapping of the velocity on the deformed length, as presented in Figure 6, of the bolt, admissible.

PMUT Arrays for SHM of Bolted Joints
Smart designing and miniaturizing of sensors have led to the realization of transducers that have low power consumption with high accuracy and efficiency, which made them suitable to be used in smart, wireless, and highly reliable real-time monitoring and communication applications. Various SHM applications rely on transducers that utilize bulky PZT materials to emit and detect acoustic waves. Replacing traditional portable transducers with smart and miniaturized transducers for SHM, such as PMUT arrays, will provide many advantages such as improving the accuracy and sensitivity in addition to the possibility of smart integration with on-chip embedded systems and IoT frameworks. A single PMUT, which are diaphragm-like shaped multi-layered systems, has an overall thickness of 7.445 µm. The active PZT layer has a thickness of 2 µm and a radius of 69 µm. The structural layer made of silicon has a thickness of 4 µm with a radius of 94 µm. The multi-layered system of each PMUT is modeled using shell elements, except for the active PZT layer, see Figure 7, thus leading to a great reduction of the number of Degrees of Freedom (DOF) and of the computational burden [22]. The PMUT array adopted in this work consists of 228 individual PMUTs, in which every 19 PMUTs is electronically connected in parallel, creating a cluster. A total of 12 electrically independent clusters arranged uniquely generate an array, see Figure 8. The PMUTs cavities are considered to be fully released with a total thickness of 400 µm. This eliminates any possibility of squeeze-film damping from the backside of the PMUTs. Hence, the array of PMUTs that is modeled and simulated for this application is a reduced-order model that considers only the vibrating plates. Details about the modeling, simulation and sensitivity analysis of this array have been discussed in our previous work [22].  The wave generated from the transmitting PMUT will propagate along the bolt until it reaches the end of the bolt where the PMUT receiver is placed, see Figure 9. The receiver will read a voltage generated as a consequence of the mechanical vibrations in the PZT active layers in the PMUTs produced by the acoustic wave. The difference between the time of transmission and reception defines the TOF. A very high computational burden is faced when simulating this model. Starting from modeling two large arrays of PMUTs attached to the huge solid domain of the bolt, having the same elastic material properties adopted in the pre-tensioned bolt model, the number of DOF solved for it will be very high. Hence, the modeling and simulation of this problem are divided into three different phases, governed by three models, as described here below; see Figure 10. MHz. This frequency corresponds to the fundamental frequency of the PMUT in contact with the silicone layer. The acoustic pressure at the bolt head/silicone layer interface A1, see Figure 11, is computed at every shared mesh node of the interface between the two domains, and stored. • In model (b), the acoustic pressure history computed and stored from the model (a), is assigned as a boundary load at a surface area A2, which is equivalent to A1, at the bolt head. This model simulates the acoustic wave propagation into the solid bolt domain only. Figure 12 shows the acoustic pressure history that represents the output of the model (a) and the input for the model (b). The acoustic pressure history arriving at the bolt end is then computed (and stored) at the surface A3 at the bolt end, which corresponds to the bolt end/silicone layer interface. • Similarly, in model (c), the acoustic pressure computed at surface A3 is assigned as a boundary load at the top surface of the silicone layer A4. Figure 13 shows the acoustic wave velocity history that represents the output of the model (b) and the input for the model (c). This model simulates the reception phase of the acoustic wave arriving at the PMUT array in receiving mode. In Figure 10, the exterior surface of the hemisphere representing the truncated domain of the bolt head is assigned with an Absorbing Boundary Condition (ABC). This BC is added to simulate an infinite domain, with respect to the minimum wavelength (λ m ) of the acoustic wave in steel, by allowing an almost negligible reflection of an incident wave. At the bolt head/silicone layer interface, an Acoustic-Structure Interaction (ASI) condition is enforced. In the silicone layer (acoustic domain), the pressure equation is solved, while in the steel domain (solid domain), the equation of motion is solved. The equations governing this study are discussed thoroughly in a previous work [22].     The reference value of TOF 0 needed to compute the CTOF can be computed from Figures 12 and 13, considering the difference between the time at which the first evident peak of the acoustic pressure shows, indicated by the red line shown in Figure 12, and the time at which the first evident peak of the structural velocity shows in Figure 13. The obtained value is equal to 1.8357 × 10 −5 s. The length of the bolt is equal to 107.5 mm, while the elastic wave propagation velocity in the steel bolt is 5856.4 m/s. Hence, the theoretical value of TOF 0 is 1.83559 × 10 −5 s. Comparing the computed and the theoretical value of TOF 0 , the numerical model simulates accurately the elastic wave propagation problem. Additionally, Figure 14, shows that the electric potential amplitude gained at the receiving PMUT array is enough to sense the TOF. The slight time shift of the red line shown in Figure 14, from the one in Figure 13, is equal to 3.3752 × 10 −7 s, which is equivalent to the time needed for an acoustic pressure wave to travel through the thickness of the silicone layer. Hence, the models show high accuracy in simulating the emission, propagation, and reception phases of the acoustic wave.

Implementation of the Acoustoelastic Effect
The acoustoelastic effect is implemented in a MATLAB code following the hyperelastic Murnaghan's constitutive model. For each pre-tensioning scenario, the CTOF is obtained following the steps below.

1.
The deformed length L f is divided into (N) segments of equal length denoted by ∆x.

3.
The average velocity in each segment is obtained as an average velocity of the two nodal velocity values of the segment. For a generic (i-th) segment: with i having values between 1 and N + 1.

4.
The time needed for the acoustic wave with an average velocity V i,avg to travel through an element of the length ∆x is given as:

5.
The TOF is simply the sum of the N values of t i : 6. Finally, the CTOF is given by: Since this numerical model is implemented on the deformed length L f , the obtained CTOF contains the contributions to the change in the time of flight due to both the physical elongation and the acoustic wave velocity reduction, i.e., the acoustoelastic effect. Table 2 contains, for each pre-tensioning scenario, the value of the TOF computed using the above procedure. A linear relationship between the CTOF and the corresponding level of the bolt pre-tensioning is obtained.

Experimental Validation
This section presents the results of a preliminary experimental campaign on the use of UT for an M12 bolted joint. The objective is to measure the CTOF of an ultrasonic acoustic wave propagating inside a clamped bolt. Figure 15 demonstrates the experimental setup used for ultrasonic testing of an M12 bolted joint. Both the head and the end surfaces of the bolt are reworked using an Electrical Discharge Machining (EDM) process to decrease the surface roughness and enhance the adhesion of the transducers to the bolt. The clamped length of the threaded region of the bolt shaft is 47.5 mm. The transducers used for transmission and reception are the UTensor II from MicroControl, which are 3 mm in length × 3 mm in width × 0.1 mm in height piezoelectric ceramics. They are adhered to the bolt surface using Loctite 680. The substitution of the Silicone intermediate material with Loctite 680 will have two levels of impact. TOF is expected to change because Loctite has a different characteristic value of wave velocity propagation than Silicone. However, this will not affect the primary sought-after results of CTOF. The second consideration is the impact of the mismatch in acoustic impedance between Loctite 680 and the steel solid domain. Depending on the acoustic impedance mismatch, this could result in a higher or lower acoustic pressure being transmitted into the bolt. The transmitting transducer is excited with an impulsive input at a central frequency of 7.5 MHz with a 2 V peak-to-peak amplitude from an Arbitrary Waveform Generator (AWG). A magnetic pickup unit is mounted on top of the transducer with a cable connecting it to the ultrasonic measurement unit. The experiment is done in a pitch-catch UT scenario with the transmitter located at the bolt head and the receiver at the bolt end. The setup allows for real-time monitoring of the CTOF of the ultrasonic wave propagating in the bolted joint during the tightening process. The experimental campaign is performed at this stage solely to measure the CTOF with respect to the variation of the axial stress of the bolt. The results of the experiments are not dependent on the use of piezoelectric ceramics or the rest of the equipment. They only provide the experimental data with which the numerical results are compared and validated on the basis of the acoustoelastic effect in steel bolts. Figure 16 illustrates the outcomes of the experimental campaign performed on three different M12 bolts: the TOF is plotted versus the load cell signals, the CTOF and the wave propagation velocity are plotted against the pre-tensioning axial stress. The difference in TOF measurements between bolts is due to the varying thickness of the intermediate layer utilized in each measurement. This, however, has no impact on the CTOF measurement. For comparison with the numerical results, the mean value curve of the CTOF vs. bolt pre-tensioning derived from the three bolt measurements is chosen, see Figure 17. Both experimental and numerical results predicted a level of CTOF ranging between 100-450 µs for the investigated pre-tensioning loading range. The differences between the numerical and experimental results are due to the uncertainty in the characteristic values of the 3rd-order elastic constants (l, m, n) of the bolt used in the experimental campaign.

Closing Remarks
Bolted joints are critical structural elements that support the functionality of many structural systems, such as large tanks, containers, bridges, and complex mechanical systems, such as wind turbines and moving vehicles. Monitoring and assessment of the bolt pre-tensioning are crucial for the integrity of the bolted joints.
In this work, a MEMS-based UT technique was investigated for its suitability of the real-time monitoring of bolt pre-tensioning. The work aimed at investigating the possibility of the replacement of bulk-PZT-based transducers with PMUT arrays.
A pitch-catch UT technique was used to measure the bolt pre-tensioning. The TOF which the ultrasonic wave needs to propagate along the pre-tensioned bolt axis is linearly proportional to the level of pre-tensioning. The TOF increases with the pre-tensioning due to the physical elongation of the bolt and to the acoustoelastic effect. Both reasons are linearly proportional to the pre-tensioning force.
A set of numerical models was created to reproduce this linear relationship. A model was created to study the behavior of an M12 bolted joint system under different loading conditions. Two arrays, each consisting of 228 PMUTs, were modeled to transmit and receive acoustic waves in a pitch-catch UT scenario, with the transmitting array located at the bolt head and the receiving array at the bolt end.
The transmitting array was found to be able to emit a pressure level of 30 kPa into the bolt, using a single cycle sinusoidal input signal with an amplitude of 5 V, that reaches the far end with enough energy that can generate an electric potential gain, at the receiving transducer, of an amplitude of more than 2 mV Peak-to-Peak.
The values of the acoustic pressure transmitted and received and the electric potential gained by the PMUT array indicate the suitability of using PMUT arrays for SHM applications of bolted joints. The applicability is not restricted to M12 bolts only. Higher density arrays are anticipated to generate greater pressure intensities that propagate over greater distances before losing energy and becoming difficult to detect. Consequently, inspection of bigger bolt sizes may be possible.
The numerical simulations of the transmission and repetition phases of the acoustic wave allowed for an accurate estimation of the reference TOF 0 . The hyperelastic constitutive model proposed by Murnaghan was implemented in a MATLAB code and a linear relationship was obtained between the CTOF of the ultrasonic wave and the level of pre-tensioning in the bolt.
A preliminary experimental UT was conducted on a clamped M12 Bolt. The numerical and experimental results were found to be in good agreement.
Future work will involve experimental tests to precisely measure the characteristic values of the elastic constants of the steel material and an extensive campaign with the application of PMUT arrays.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: