1 Introduction

Continuous and reliable monitoring of structures is becoming increasingly important in engineering. The structures and elements to be monitored are as varied and diverse as their areas of application. An important element in many structures is the preloaded bolt [1], whose operational safety is significantly influenced by the level of preload force [2]. For the safety and maintenance of the functionality of structures with high-strength prestressed (HV) bolts, it is, therefore, important to know the current level of prestress in order to be able to intervene in time in the event of critical reduction. The assessment of preload loss can be carried out through various methods, the choice of which depends on specific conditions such as the type of load, stress, and accessibility.

Preload loss can be quantified through various methods. One approach involves indirect assessment via longitudinal strain measurements, using strain gauges positioned around the bolt circumference [3, 4]. An alternative indirect method involves longitudinal strain measurements using an internal strain gauge placed inside the stressed bolt shaft [5, 6]. Ultrasound is another indirect technique for measuring extension [7, 8]. In addition, indirect measurement can be achieved by assessing extension with an external micrometer [9]. For direct measurement, the integration of a load cell [10], load ring, or an “intelligent washer” [11] allows for continuous and accurate quantification of preload.

However, many of the common measurement methods are not suitable for continuous monitoring of bolt preload forces. Therefore, there is a need for a method by which preload forces in bolted joints can be reliably and continuously monitored under real-life conditions. Monitoring using electromechanical impedance (EMI) spectra is growing in importance due to its reliability and cost-effective measurement technology. Investigations [12,13,14,15,16] have demonstrated the feasibility of detecting preload losses through EMI spectra in bolted joints, often using piezoelectric wafer active sensors (PWAS) made of lead zirconate titanate (PZT). The influence of temperature has always been a significant challenge for monitoring methods based on piezoelectric principles. This challenge extends beyond the EMI method to other techniques such as Pitch–Catch [17] or Pulse-Echo [17, 18]. The effect of temperature on EMI spectra has been observed in various applications, including corrosion monitoring [19], monitoring fatigue crack propagation in welded joints [20], detection of clamping force loss in bolted joints of rail supports [21] and monitoring of preload forces in bolts [22, 23]. The authors of [24] used the effect to monitor the soil freeze\(^-\)thaw process. However, most research to date has focused on the statistical analysis of EMI Spectra, with less attention paid to the underlying physical processes of the influencing factors. This publication aims to fill this research gap. In [25, 26] have also investigated the effect of temperature on piezoelectric sensors. In contrast, we explain the effect of temperature on the capacitance and electromechanical impedance of PZT piezoelectric sensors using the phenomenological thermodynamic theory of ferroelectricity developed by Landau, Ginsburg and Devonshire. These findings are then linked to a highly practical application: the detection of preload losses in HV connections (Fig. 1).

The study focuses on an essential compilation of physical phenomena from the fields of electrodynamics, piezo technology and solid-state physics (Sect. 2). The temperature and preload-dependent parameters are analysed and discussed (Sect. 2.2). The information obtained served as a basis to create a temperature and preload-dependent numerical model, which, after experimental validation, was used to explain and illustrate the crucial physical relationships (Sect. 3). The article sequentially examines the effects of temperature and preload force on electromechanical impedance spectra, using theoretical principles supported by experimental findings and numerical simulations. By isolating the effects of temperature and preload force, it is ultimately possible to compensate for temperature in the electromechanical impedance spectra by analytical means.

2 Fundamentals of the measurement method

2.1 Measurement principle

Structures vibrate in a combination of their natural modes, which can provide information about the state of the structure through system-inherent quantities such as natural frequencies, natural mode, modal mass or modal damping. These modal parameters are governed by geometry, support, stiffness and mass occupancy. When a system parameter changes, it is reflected in the modal characteristics and the vibration analysis [27, 28]. To determine the modal characteristics of structures, they are excited and the system response is measured. Excitation can be applied, for example, by means of shock excitation, ambient excitation or by means of hydraulic excitation. The measured system response is often the vibration velocity or vibration acceleration at several positions and in all relevant vibration directions [27]. Appropriate processing algorithms are used to transform the measured signal from the time domain to the frequency domain to determine the underlying natural frequencies. A shift in the natural frequencies indicates a change in the system and allows indirect conclusions to be drawn about damage to the excited structure.

Fig. 1
figure 1

Basic principle of measurement with electromechanical impedance spectra (top left) as well as visualised M16x50 test specimen with attached measurement technology

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Damage detection by means of electromechanical impedance spectra is also based on this principle, where in this case the observed system-inherent quantity is the frequency-dependent mechanical impedance \({\underline{Z}}_\text {m}(\omega )\) of the structure. The mechanical impedance describes the dynamic oscillatory resistance that an elastic structure opposes to the force F acting on it with the frequency \(\omega\) [29]. The method is based on the idea that the mechanical impedance of the structure to be measured will change as a result of damage (e.g., reduction in bolt preload), which allows conclusions to be drawn indirectly about the damage.

For this purpose, the system is locally excited in the high-frequency kilohertz range by means of PWAS applied to the structure (frequency sweep, frequency-modulated signal). The excitation of the structure is based on the inverse piezoelectric effect [30, 31]. The system response, which depends on the mechanical impedance, is measured in parallel with (the same) piezoelectric transducer using the direct piezoelectric effect. The electromechanical impedance determined from the excitation and response of the structure indirectly reflects the mechanical impedance \({\underline{Z}}_\text {m}(\omega )\) of the structure. The electromechanical impedance or a quantity derived from it (such as the admittance \({\underline{Y}}(\omega )\)) is evaluated. This admittance is then compared with the original admittance \({\underline{Y}}_\text {baseline}(\omega )\) (baseline or “0-measurement”) in the undamaged state:

$$\begin{aligned} {\underline{Y}}\left( \omega \right) =i\omega a_\text {p}\left[ {\varepsilon _{ij}}^T\left( 1-i\delta \right) -\frac{Z_\text {m}(\omega )}{Z_\text {p}\left( \omega \right) +Z_\text {m}(\omega )}{d_{ij}}^2{{{\bar{Y}}}_\text {p}}^E\right] . \end{aligned}$$
(1)

Equation 1 contains the parameters \(a_\text {p}\), \(d_{ij}\), \(\delta\) and \({{{\bar{Y}}}_\text {p}}^E\), which represent the geometric/piezoelectric constant, the dielectric loss factor, and the complex elastic modulus of the PWAS, respectively. Furthermore, \(Z_\text {m}(\omega )\) is the sought-after mechanical impedance of the structure, \(Z_\text {p}(\omega )\) is the mechanical impedance of the PWAS, and \({\varepsilon _{ij}}^T\) is the dielectric coefficient in the polarity direction with the applied electric field.

PWAS are cost-effective transducers and have already been used to monitor various engineering structures [32,33,34,35,36,37,38,39,40,41].

2.2 Physical correlations

The electrotechnical parameters essential for the measuring method result from a network operated with alternating current. The evaluation of these complex electrotechnical parameters is essential for the EMI methodology.

Fig. 2
figure 2

AC network with an ideal condensation

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In particular, the phase shift \(\varphi\) between the voltage U and the electric current I resulting from a capacitance C (see Fig. 1 top left) is of great importance for the measurement of electromechanical impedance spectra, since the PWAS used in the study are very similar to a capacitor due to their design and their materials. The phase shift arising from a capacitance is shown schematically in Fig. 2. Two areas are particularly important for the detection of electromechanical impedances:

  • Electrical impedance of the PWAS

  • Mechanical impedance of the structure \({\underline{Z}}_\text {m}\) (changes directly as a result of damage or loss of preload force)

The mechanical impedance \({\underline{Z}}_\text {m}\) is the ratio between an excitation force \({\underline{F}}\) and the resulting wave with the oscillatory velocity \({\underline{v}}\) [42, 43]:

$$\begin{aligned} {\underline{Z}}_\text {m}=\frac{{\underline{F}}}{{\underline{v}}}. \end{aligned}$$
(2)

Here, \({\underline{v}}\) is the surface vibration velocity at the location and direction where the causing force \({\underline{F}}\) acts. Also applies:

$$\begin{aligned} F=f_0 m, , \end{aligned}$$
(3)

where \(f_0\) is the external excitation and m is the excited mass. The vibration velocity is defined as

$$\begin{aligned} {\underline{v}}\left( \omega _e\right) =\frac{f_0\omega _e}{2\omega _e\left( \frac{\delta }{2m}\right) +i\left( \omega ^2_e-{\omega _0}^2\right) }. \end{aligned}$$
(4)

Equations 3 and  4 can be substituted into Eq. 2. After transformation, this results in:

$$\begin{aligned} {\underline{Z}}_\text {m}\left( \omega _e\right) =\delta +i\frac{\left( \omega ^2_e-{\omega _0}^2\right) }{\omega _e}m. \end{aligned}$$
(5)

According to Eq. 5, the mechanical impedance of the forced oscillation depends on the mass m, the damping \(\delta\) and the natural frequency \(\omega _0\) of the excited system.

However, applying the EMI method, the mechanical impedance is not measured directly, but indirectly through complex electrotechnical characteristics. The interaction between the mechanical and the electrical impedance is the basis for the application of the measurement method and is explained in more detail in Sect. 3.1.

2.2.1 PWAS

The processes in a PWAS can be described by means of an electrical equivalent circuit (Fig. 3). The process mapping as an equivalent circuit is used to identify potential temperature and preload dependent parameters. It also enables the analytical calculation of the complex electrotechnical parameters (e.g., in Fig. 4 the real and imaginary part of the complex admittance \({\underline{Y}}\)). This allows a deeper understanding of the processes. Comparison with experimental results allows the numerical model to be calibrated and results of the analytical calculation to be checked.

The capacitance of the dielectric is symbolised in Fig. 3 by the parallel resistor \(C_0\). The resistors \(R_1\), \(C_1\) and \(L_1\) connected in series represent the ideal ohmic resistance in the electric resonant circuit, the dynamic capacitance and the mechanically vibrating mass of the resonator.

Fig. 3
figure 3

Equivalent circuit diagram of a PWAS according to [44]

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From the equivalent circuit in Fig. 3, a series impedance \({{\underline{Z}}}_{\textrm{ser}}\) with components \(R_1\), \(C_1\), and \(L_1\) is obtained as follows:

$$\begin{aligned} {{\underline{Z}}}_{\textrm{ser}}\left( \omega \right) =R_1+i{\omega L}_1+\frac{1}{i\omega C_1} . \end{aligned}$$
(6)

In addition, the parallel impedance \({{\underline{Z}}}_{\textrm{par}}\) can also be determined:

$$\begin{aligned} {{\underline{Z}}}_{\textrm{par}}\left( \omega \right) =\frac{{{\underline{Z}}}_{\textrm{ser}}\frac{1}{i\omega C_0}}{{{\underline{Z}}}_{\textrm{ser}}+\frac{1}{i\omega C_0}}. \end{aligned}$$
(7)

Substituting Eq. 6 into Eq. 7 gives:

$$\begin{aligned} {{\underline{Z}}}_{\textrm{par}}\left( \omega \right) =\frac{\frac{L_1}{C_0}-\frac{1}{\omega ^2C_0C_1}+R\frac{1}{i\omega C_0}}{i\left( \omega L_1-\frac{1}{\omega C_1}-\frac{1}{\omega C_0}\right) +R} . \end{aligned}$$
(8)

The application of the conjugate complex expansion

$$\begin{aligned} \psi =\omega L_1-\frac{1}{\omega C_1}-\frac{1}{\omega C_0}, \end{aligned}$$
(9)

yields the term:

$$\begin{aligned}&{{\underline{Z}}}_{\textrm{par}}\left( \omega \right) \nonumber \\&\quad =\frac{\frac{{RL}_1}{C_0}-R\left( \frac{1}{\omega ^2C_0C_1}+\frac{1}{\omega C_0}\psi \right) +i\left( \frac{1}{\omega ^2C_0C_1}\psi -\frac{L_1}{C_0}\psi -R^2\frac{1}{\omega C_0}\right) }{R^2+\psi ^2} . \end{aligned}$$
(10)

Furthermore, the real and imaginary parts can be written independently:

$$\begin{aligned} \textrm{Re}\left( {{\underline{Z}}}_{\textrm{par}}\right)&=\frac{R\left[ \frac{L_1}{C_0}-\frac{1}{\omega C_0}\left( \frac{1}{\omega C_1}+\psi \right) \right] }{R^2+\psi ^2}, \end{aligned}$$
(11)
$$\begin{aligned} \textrm{Im}\left( {{\underline{Z}}}_{\textrm{par}}\right)&=\frac{\psi \left( \frac{1}{\omega ^2C_0C_1}-\frac{L_1}{C_0}-\frac{R^2}{\omega C_0}\right) }{R^2+\psi ^2}. \end{aligned}$$
(12)

The study demonstrates that Eq. 10 resp. Equations 11 and  12 can accurately reproduce the experimental or numerical impedance spectrum (or in this case, admittance spectrum). Figure 4 shows the experimentally, numerically and analytically determined conductance spectrum \(G\left( \nu \right)\) and susceptance spectrum \(B\left( \nu \right)\) of a PWAS (\(G=\textrm{Re}({\underline{Y}}), B=\textrm{Im}({\underline{Y}}))\). The correlation between the curves from the numerical simulation and the experiments is very good. The analytically determined spectrum differs slightly from the other two spectra. This is partly due to the fact that the analytical solution is based on an idealised system in which, for example, no mechanical damping is implemented. Nevertheless, the analytical solution is very close to the numerical and experimental results, the position of the resonance peak as well as the height of the amplitude agree very well.

Fig. 4
figure 4

Experimental, numerical and analytically determined conductance spectrum \(G\left( \nu \right)\) and susceptance spectrum \(B\left( \nu \right)\) of an excited piezoelement

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Furthermore Eqs. 11 and  12 can be used to determine the position of the resonance \(\nu _{\textrm{ser}}\) of the series resonant circuit when \({{\underline{Z}}}_{\textrm{par}}\rightarrow 0\) and the position of the resonance of the parallel resonant circuit \(\nu _{\textrm{par}}\) can be determined when \({{\underline{Z}}}_{\textrm{par}}\rightarrow \infty\)

$$\begin{aligned} \nu _{\textrm{ser}}&=\frac{1}{2\pi \sqrt{C_1L_1}}, \end{aligned}$$
(13)
$$\begin{aligned} \nu _{\textrm{par}}&=\frac{1}{2\pi \sqrt{C_1L_1}}\sqrt{1+\frac{C_1}{C_0}}, \end{aligned}$$
(14)
$$\begin{aligned}&=\nu _{\textrm{ser}}\sqrt{1+\frac{C_1}{C_0}}. \end{aligned}$$
(15)

The yellow circles in Fig. 4 (lower diagram) indicate the locations of the resonances from the series and parallel resonant circuits.

2.2.2 Bolts

The study found that the EMI spectrum of a PWAS coupled to a structure can be well-mapped using an electrical series resonant circuit. The magnitude of the complex impedance \(|{\underline{Z}}|\) is obtained for a series resonant circuit with:

$$\begin{aligned} |{\underline{Z}}|&=\sqrt{R^2+\left( X_\text {L}-X_\text {C}\right) ^2}, \end{aligned}$$
(16)
$$\begin{aligned} |{\underline{Z}}|&=\sqrt{R^2+\left( \omega L-\frac{1}{\omega C}\right) ^2}. \end{aligned}$$
(17)

The study has shown that in particular the reactance from capacitance \(X_\text {C}\) plays a significant role in the formation of the electromechanical impedance spectra, since the reactance from an inductance \(X_\text {L}\) is significantly lower than the reactance from a capacitance (\(X_\text {L}\ll X_\text {C}\)).

There are a number of factors that influence the shape of electromechanical impedance spectra. The influence of the preload force forms the basis of the measurement concept: the difference in shape of the spectra between individual measurements should indicate whether or not there is a loss of preload force in a bolt. Other influences are undesirable and must be taken into account in the evaluation. This is particularly the case for the influence of temperature on the spectra. When interpreting the spectra, it is therefore important to know what influence the ambient temperature and the temperature of the specimen have on the development of the spectra.

2.3 Influence of temperature

The study found that the effect of temperature on the (temperature dependent) parameters of the PWAS has the greatest influence on the generation of the electromechanical impedance spectra:

$$\begin{aligned} T\downarrow \ \Rightarrow {\textbf {?}} \Rightarrow |{\underline{Z}}|\uparrow . \end{aligned}$$
(18)

The key parameter is the capacitance. The detection of the quantitative influence of the temperature on the capacitance of the PWAS is extremely complex due to the tetragonal formation of the unit cell, as it is a non-linear dielectric in this case (Fig. 5). Above the Curie temperature \(T_c\) (paraelectric, linear dielectric), the electrical susceptibility \(\chi _e\) (and hence the capacitance) can be described using the Curie–Weiss law. This case is usually not relevant in practical applications.

Fig. 5
figure 5

Schematic crystal structure of lead zirconate titanate (PZT), energy diagram (left), face-centred cubic system for \(T>T_c\) (paraelectric, middle), tetragonal system for \(T<T_c\) (ferroelectric, right) [45]

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Below the Curie temperature, the electrical susceptibility depends on many factors. The temperature dependent behaviour of dielectric media has been discussed in many publications. For example, [46,47,48] show that the capacitance or permittivity increases with increasing temperature, although the reasons for this are often not mentioned. The temperature dependent behaviour of dielectric media below \(T_c\) can be explained, for example, by the Landau-Ginsburg-Devonshire theory. This phenomenological thermodynamic theory of ferroelectricity, developed by Devonshire [49,50,51], describes the transition from the disordered paraelectric phase to the ordered ferroelectric phase. Considering an isothermal-isochoric process \(\left( \text {d}T=\text {d}V=0\right)\), the spontaneous polarisation \(P_s\) is given by

$$\begin{aligned} P_{s,i}=-\frac{1}{V}\left( \frac{\partial F}{\partial E_i}\right) _{V,T}. \end{aligned}$$
(19)

This results in the electrical susceptibility:

$$\begin{aligned} \chi _{ij}&=\frac{1}{\varepsilon _0}\left( \frac{\partial P_{s,i}}{\partial E_i}\right) _{V,T}, \end{aligned}$$
(20)
$$\begin{aligned}&=-\frac{1}{{V\varepsilon }_0}\left( \frac{\partial ^2F}{\partial E_i\partial E_j}\right) _{V,T}. \end{aligned}$$
(21)
Fig. 6
figure 6

Schematic representation of the influence of temperature on the polarisation potential \(\psi _\text {pol}\) (left) and the corresponding electric field \(E_z\) parallel to the polarisation (right)

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The partial derivative of the free energy F as a function of the electric field \(E_{ij}\) gives the course of polarisation \(P_{s,i}\) for electric field strengths. The relative permittivity \(\varepsilon _r\) or electric susceptibility \(\chi _{ij}\) can be understood as the slope of polarisation hysteresis at electric field \(E=0\) (Fig. 6, right diagram, red circle) [52]. The relationship between the capacitance of an inhomogeneous field, the permittivity and the electric susceptibility can be interpreted from Fig. 6, respectively. Equations 22 and  23

$$\begin{aligned} C&=\varepsilon _0\varepsilon _r\frac{\oint _{A}{\vec {E}\cdot \text {d}\vec {A}}}{\oint _{s}{\vec {E}\cdot \text {d}\vec {s}}}, \end{aligned}$$
(22)
$$\begin{aligned}&=\varepsilon _0\left( 1+\chi \right) \frac{\oint _{A}{\vec {E}\cdot \text {d}\vec {A}}}{\oint _{s}{\vec {E}\cdot \text {d}\vec {s}}}. \end{aligned}$$
(23)

In Fig. 6 (right diagram, red circle), it can be seen that the capacitance (and the slope) of the dielectric decreases with decreasing temperature. As a consequence, the impedance increases (Fig. 7 (Case 2a)). For the present case, the following relationship could be found:

$$\begin{aligned} T\downarrow \ \Rightarrow C\downarrow \ \Rightarrow \left| X_\text {C}\right| \uparrow \ \Rightarrow \left| X\right| \uparrow \ \Rightarrow |{\underline{Z}}|\uparrow . \end{aligned}$$
(24)

Since the temperature dependence of the EMI spectra can mainly be attributed to the temperature dependence of the capacitance of the dielectric, according to Eq. 24 and Fig. 7, the characteristic increase of \(|{\underline{Z}}|\) with decreasing temperature T can be explained.

Fig. 7
figure 7

Case discrimination for the influence of temperature on electrical impedances (upper row), qualitative evolution of the electrical impedance for increasing (2a) and decreasing (2b) temperatures for the case \(\chi _\text {L}<\chi _\text {C}\)

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2.4 Influence of preload

High-strength bolts according to DIN EN 14399 [53] with a yield strength of \(f_\text {y}=900\ \mathrm {N/mm}^2\) were used in the study. According to DIN EN 1090-2 [54], the high yield strength allows a maximum preload force of \(F_\text {p,C}=110\ \textrm{kN}\) for the M16 test series and \(F_\text {p,C}=172\ \textrm{kN}\) for the M20 test series.

Figure 8 shows the FE plot of the strain transverse to the longitudinal axis of the bolt (\(\varepsilon _{22}\)). The cross-section is split in two, with the right half showing the compression strain and the left half showing the tension strain in the HV set at maximum preload \(F_\text {p,C}\). It can be seen that the PWAS1 is compressed at the bolt head when a preload is present in the HV set. The high mechanical stress also affects the spacing of the atoms. Figure 9 shows a two-part cross section in which the von Mises equivalent stress for the range \(\sigma _v > 3.0\ \mathrm {kN/cm}^2\) at full prestress is shown in colour on the left-hand side; the right-hand side shows the unstressed, loose HV set. In the right part of Fig. 9, the schematic evolution of the atomic distance r for the case \(F=F_\text {p,C}\) as well as \(F=0\) is shown within the circle highlighted in grey. It can be seen that the atomic distance increases in the longitudinal direction of the bolt at full preload force compared to a loose HV set (no preload). The preload force also causes a decrease in the atomic distance in the radial direction of the PWAS, which is indirectly reflected in the EMI spectra.

Fig. 8
figure 8

Schematic representation of the change in stiffness C resulting from the atomic bonding potential U and the strain distribution in the radial direction of the PWAS at full preload force \(F_\text {p,C}\) for an M16x50 HV set according to DIN EN 14399

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Fig. 9
figure 9

Schematic representation of the force acting on the crystal lattice with no preload and with full preload using an M16x50 HV set according to DIN EN 14399 as an example

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According to Fig. 8, a reduction of the atomic distance tends to increase the stiffness in this direction. The interaction between the preload force and the quantities derivable from the atomic bond U(r) is shown schematically in Fig. 8. In the top graph the internal interaction force \(F_i\) is plotted on the ordinate and the atomic distance r is plotted on the abscissa. Using a spring model for illustration, the “springs” between the atoms are compressed when an external compressive force is applied and the distance between the atoms decreases. In the diagram, this situation is represented by the shift of the yellow dot towards the green dot. The yellow dot is intended to symbolise the actual rest position \(r_0\) of the atoms without the influence of temperature and preload force. If, on the other hand, a tensile force is applied, the matter behaves approximately the other way around. In the present case, therefore, the bolt is stretched, which can easily be demonstrated experimentally. The increase in the magnitude of the internal interaction force leads to a change in the stiffness C (note: not capacity here). The lower diagram in Fig. 8 shows that the stiffness decreases when a tensile force is applied, whereas the system becomes stiffer when a compressive force is applied. Consequently, it is expected that the application of a preload force will also result in a shift of the resonances. It is believed that due to the high sensitivity of the piezoelectric parameters to deformation, the influence of the change in (local) stiffness due to the applied preload force is reflected in the EMI spectra, even though the stresses are within the linear elastic range. Therefore, the stiffness has an effect on the EMI spectra as the radial displacement of the PWAS is increasingly restrained at higher preload force. The results from Sect. 3 show that the radial displacement of the PWAS has a significant influence on the formation of the EMI spectra.

In addition, applying and increasing the preload force results in a greater surface pressure between the individual HV components. This also changes the vibration behaviour of the structure as the increased excitation of the structure results in a larger excitable mass. There is also a change in the vibration due to the change in stiffness of the structure and the change in bearing conditions, which makes the quantification of the continuum vibrations quite complex. However, it can be seen from [55] that a constant tensile force leads to an increase in the natural frequencies of the structure due to the influence of strain. However, a constant tensile force is assumed here, which is not the case for a prestressed HV set. Nevertheless, it has been shown that a prestress shifts the resonances to higher frequency ranges. It has also been shown that a preload not only shifts the resonances, but also causes a significant rotation of the entire EMI spectrum. As will be shown later, this behaviour results from the different stress or strain ratios caused by different preload forces.

3 Numerical simulation

A geometrically nonlinear analysis (GNA) was performed using the program Comsol Multiphysics. The simulation included all components of the HV set and the PWAS, which were mapped, as shown in Fig. 10. The coupling (adhesive layer) between the PWAS and the structure was described as a thin elastic layer with a spring constant per unit area. To account for the damping of the excited structure, Rayleigh damping was used. As contact condition between the individual HV components (bolt-washer, washer-plate, plate-washer, washer-nut and nut-bolt), a fixed continuity condition proved to be sufficiently accurate in the present case, so more detailed modelling of the contact conditions was not performed. The investigations were carried out based on PWAS from PI Ceramic made of Pb[Zr,Ti]O\(_3\)-lead zirconate titanate (PZT) (Fig. 10, top right).

Fig. 10
figure 10

Transfer of an M16x50 HV set from experiments into a numerical model [45]

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The simulation of the HV set was performed in two steps: First, a step bolt preload was carried out to apply the preload force, taking into account geometric nonlinearities. Based on this, a frequency domain preload analysis was performed in the next step, capturing both the mechanical and electrical responses of the system over a defined frequency range. The frequency step size of 0.05 kHz proved to be sufficiently accurate to reproduce the amplitudes in the resonant regions in accordance with the experimental results.

3.1 Relationship between electrodynamic and mechanical parameters

In the previous sections, it has already been shown by means of figures and equations that the electrical impedance depends on various parameters such as capacitance or polarisation (e.g., Eqs. 20, 21 and Fig. 6). Therefore, the relationship between the electrodynamic parameters essential to the measurement method has been studied within the framework of the numerical simulation. To assign certain behavioural patterns of the electromechanical impedance spectra to the individual components, non-coupled PWAS were investigated first.

Fig. 11
figure 11

Notation and labelling of the evaluated parameters using the example of the M16x50 bolt with milled bolt shank [45]

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3.1.1 Non-coupled PWAS

First, the relationship between the polarity \(P_z\) (in z direction = thickness direction, polarisation direction) and the electric charge Q was investigated. In the upper diagram of Fig. 12 (frequency domain) it can be seen that the course of the polarity (yellow dashed line) qualitatively coincides with the course of the electric charge (red line). It can be said that a large polarisation causes a substantial electric charge. However, it should be noted that the polarisation in the dielectric is usually not homogeneously distributed. An inhomogeneous distribution results, for example, from zones of compression and tension (bending) within the dielectric. In this simple case, however, it can be assumed that the polarisation is approximately homogeneous in the z-direction. Furthermore, in the frequency domain, it can be seen that the trajectories of \(P_z(\nu )\) and \(Q(\nu )\) correspond qualitatively both to the trajectory of the susceptance \(B(\nu )\) and to the inverse trajectory of the radial displacement \(x(\nu )\) of the PWAS, thus illustrating an essential relationship between mechanical and electrical parameters. Consequently, the radial displacement of the PWAS has a significant influence on the polarisation, electric charge and impedance spectra.

Fig. 12
figure 12

Correlation of selected electrodynamic parameters in the resonance range of the PWAS

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It can also be seen from the top diagram in Fig. 12 that the electric charge and polarity to the left of the resonant frequency of the series resonant circuit \(\left( \nu \lesssim 200\ \textrm{kHz}\right)\) is positive. In contrast, the radial displacement of the PWAS is negative. This can be explained by looking at it in the time domain. In the lower diagram of Fig. 12 (time domain) it can be seen that the amplitudes of the terminal voltage \({\hat{U}}\) and the electric charge \({\hat{Q}}\) are approximately in phase at excitation frequencies below the resonant frequency of the series resonant circuit (\(\nu \lesssim 200\ \textrm{kHz}\), here 190 kHz, purple lines), while the amplitude of the radial dilatation \({\hat{x}}\) is phase-shifted by about \(\pm 90^\circ\) with respect to the voltage amplitude. Between the resonant frequency of the series and parallel resonant circuit (\(200\ \textrm{kHz}\lesssim \nu \lesssim 245\ \textrm{kHz}\), here 210 kHz, red lines) the relationship between the amplitudes under consideration changes, which is noticeable among other things by the fact that at positive voltage amplitudes the amplitude of the radial displacement of the PWAS corresponding to the same time is now positive and that of the electric charge is negative. Above the resonance frequency of the parallel resonant circuit, however, this behaviour changes again. The process can also be mapped using the electrical parameter of phase shift \(\varphi\). In the range \(\nu \lesssim 200\ \textrm{kHz}\) the capacitive resistances in the electric resonant circuit predominate, which is why the phase shift \(\varphi\) here is approximately \(-90^\circ\). At excitation frequencies between the resonant frequencies of the series and parallel resonant circuits \(\left( 200\ \textrm{kHz}\lesssim \nu \lesssim 245\ \textrm{kHz}\right)\), a visible change in the phase shift \(\varphi\) occurs (Fig. 12, upper diagram, purple dashed line). At these frequencies, the inductive part of the electric resonant circuit dominates and a phase shift of \(\varphi \approx +90^\circ\) occurs. Above the resonant frequency of the parallel resonant circuit, the capacitive part of the resonant circuit becomes dominant again and the phase shift \(\varphi\) returns approximately to the original state \((\varphi \approx -90^\circ )\). The middle diagram of Fig. 12 (time domain) also shows that the amplitude of the radial displacement is largest at 200 kHz, which is also consistent with the information obtained from the upper diagram in Fig. 12 (frequency domain). An important result of the study is the relationship between the electrical and mechanical parameters of the measurement method. For example, a large radial displacement is associated with a large amount of electric charge. Consequently, the radial displacement \(\left| \delta _x\right|\) has a decisive influence on the formation of the electromechanical spectra \(|{\underline{Z}}|\):

$$\begin{aligned} \left| \delta _x\right| \uparrow \Rightarrow \left| Q\right| \uparrow \Rightarrow C\uparrow \Rightarrow |{\underline{Z}}|\downarrow . \end{aligned}$$
(25)

Note: For the numerical simulation in the small-signal model region, the actual remanent polarisation \(P_R\) of the ferroelectric was not taken into account, since only the relative change is important here. It should also be noted that this evolution was observed for a freely oscillating PWAS with the parameters given in Fig. 10. However, it can be assumed that the result is valid for all radial oscillators.

3.1.2 PWAS coupled with bolt

Figure 13 shows in the upper diagram (time domain) the time history of the radial displacement x(t) (green line) and the electric charge Q(t) (red line) of the point Pkt1b as well as the excitation voltage U(t) (purple line) using the example of a M16x50 bolt (for notation see Fig. 11) and an excitation frequency of 85 kHz. The lower diagram in Fig. 13 (frequency domain) shows the evolution of the radial displacement \(x(\nu )\) (green line) as well as the evolution of the electric charge \(Q(\nu )\) (red line) in an excitation frequency range from 50 to 200 kHz. Even when the PWAS is coupled to a structure, the electric charge is directly related to the radial displacement of the PWAS, which is consistent with the finding from Fig. 12. Furthermore, it can be seen in the upper diagram that at an excitation frequency of 85 kHz, the amplitude of the electric charge and the excitation voltage are approximately in phase, while the radial displacement is out of phase by approximately \(\pm 90^\circ\). This can also be seen in the lower diagram (frequency domain). Here, at a frequency of 85 kHz, the radial displacement is negative by about – 35 nm and the electric charge is positive by about 160 nC.

Fig. 13
figure 13

Development of the radial displacement \(x(\nu )\) at the point Pkt1b using the example of an M16x50 bolt

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From both diagrams it can be seen again that a large radial displacement of the PWAS is accompanied by a large electric charge. Due to the proportionality between electric charge and capacitance \((Q=CU)\) and the coupling between capacitance and impedance (Eq. 17), the radial displacement has a direct (and decisive) influence on the EMI spectra.

3.2 Influence of the preload force

Figure 14 shows the numerically and experimentally determined impedance spectrum \(|{\underline{Z}}|\) at 20\(^{\circ }\)C under various preload levels using an M16x80 HV set as an example.

Fig. 14
figure 14

Comparison of the experimentally and numerically determined impedance spectrum \(\left| {\underline{Z}}\right|\) at different preload levels for an M16x80 HV set [45]

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Again, the resonances shift, but this time not as a result of different temperature levels, but as a result of stress or stiffness ratios. A key feature of impedance is the phenomenological rotation caused by a preload, which can also be seen in Fig. 14. A decrease in preload is visible in the impedance spectrum, for example by a counterclockwise rotation \((\text {point of rotation }|{\underline{Z}}|:\nu \rightarrow \infty )\). This behaviour can be very well explained by the findings in Sect. 2.4, as the PWAS at the bolt head is increasingly compressed as the preload force is increased. As a result, the radial displacement is increasingly reduced due to the increasing local stiffness perpendicular to the longitudinal axis of the bolt (Fig. 16). This in turn leads to a decrease in electric charge and capacitance (Fig. 15), which increases the impedance \(\left| {\underline{Z}}\right|\) (Fig. 14).

Fig. 15
figure 15

Normalised radial deformation at point Pkt1b x(F), normalised electrical charge Q(F) and normalised polarisation \(P_\text {z}(F)\) of the lead zirconate titanate piezoelectric sensor

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Fig. 16
figure 16

Schematic representation of the relative polarisation within PWAS1 at different levels of preload

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Figure 15 shows the relationship between electrical charge Q(F), polarisation \(P_z(F)\) and radial deformation x(F) of the piezoelectric sensor at point Pkt1b. Since the qualitative evolution of the parameters is of primary importance and all parameters are strongly dependent on the applied electrical voltage, the values have been normalised for better clarity. The radial deformation has been normalised to the range from 1 to 2, the electrical charge to the range from 2 to 3, and the polarisation to values between 3 and 4. It can clearly be seen that the values decrease linearly with increasing preload force. An increased preload force results in a reduced radial deformation of the sensor, a reduced polarisation and a reduced electrical charge.

3.3 Influence of temperature

In Fig. 17 it can be seen that the qualitative progression of the spectra at different temperatures is approximately the same as for the case of preload loss, with a slight rotation of the spectra at different temperatures. It can also be seen that the resonances shift towards lower frequency regions with increasing temperature, which is consistent with the physical principles discussed in Sect. 2.3 and confirmed by the work of [20, 56, 57].

Fig. 17
figure 17

Numerically determined impedance spectrum \(\left| {\underline{Z}}\right| (\nu ,T)\) at 100 % \(F_\text{p,C}\) and varying sample temperature using the example of an M16x80 HV set

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It can also be observed that the impedance spectrum at – 20 \(^{\circ }\)C has a higher level than the spectrum at 20\(^{\circ }\)C in the frequency range considered. This is in line with the finding in Sect. 2.3 resp. Figure 7. It is also clear from the temperature plots that the distance between the impedance spectra decreases with increasing frequency. This behaviour can be explained, for example, by Eq. 17. The capacitive resistance \(X_\text {C}\) is the inverse of the product of frequency and capacitance, where the influence of the capacitance on the capacitive resistance decreases with increasing frequency because the capacitance does not increase to the same extent at higher frequencies. In the limiting case, the function approaches zero, regardless of the value of C. In the present case, this mathematically highly simplified relationship serves to illustrate that the capacitance decreases its contribution to the formation of the capacitive resistance as the frequency increases. This explains the slight rotation of the spectra in Figs. 14,  17 and  18. In the course of \(|{\underline{Z}}|\left( \nu ,T\right)\), a hyperbolic drop in impedance with increasing frequency can be seen. This can be explained by the following proportionality:

$$\begin{aligned} |{\underline{Z}}|\backsim \dfrac{1}{\omega }, \end{aligned}$$
(26)

(the contributions of \(\omega L\) and R in Eq. 17 are negligible). This gives the impedance spectrum the characteristic 1/x shape, which is not directly apparent due to the limited frequency range shown here \((50 \text { kHz} \leqslant \nu \leqslant 200 \text { kHz})\).

Fig. 18
figure 18

Analytical temperature compensation using the impedance \(\left| {\underline{Z}}\right| (\nu ,T)\) and susceptance spectrum \(B(\nu ,T)\) of an M16x80 set as an example

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Figure 17 shows that the position of the resonances decreases with increasing temperature. This is due to the fact that as the mean kinetic energy increases, the internal interaction force and the stiffness decrease as shown in Fig. 19. Due to the direct dependence between the stiffness and the modulus of elasticity, there is a resonance shift to lower frequency ranges when the sample temperature increases and thus the modulus of elasticity decreases.

Fig. 19
figure 19

Change in the interaction between two atoms due to thermal energy in the solid state

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Using the knowledge gained from this study, it was possible to perform an analytical temperature compensation by simplification. The essential point is that a PWAS has mainly capacitive resistors and the change in capacitance with temperature is approximately linear in the temperature range considered (– 20 \(^{\circ }\)C \(\le T \le\) 20 \(^{\circ }\)C). As shown in Sect. 2.2.2, the impedance spectrum can be well represented by Eq. 17. The capacitance of the PWAS coupled to the structure can be determined for a reference temperature either experimentally or iteratively based on an EMI measurement. For EMI spectra generated at other temperatures, the capacitance can now be factorised into Eq. 17. This is possible because the temperature mainly affects the capacitance/permittivity of the PWAS. Figure 18 shows the application of analytical temperature compensation to the impedance spectrum as well as to the susceptance spectrum of an M16x80 bolt. The 20 \(^{\circ }\)C spectrum (Ref-T) was selected as the reference (baseline; target temperature). The course of the 20 \(^{\circ }\)C spectrum is shown in Fig. 18 as a black dotted line. The curve shown as a red line represents the uncompensated – 20 \(^{\circ }\)C spectrum. After applying analytical temperature compensation, the – 20 \(^{\circ }\)C spectrum lies very precisely on the desired 20 \(^{\circ }\)C reference temperature spectrum. The possibility of analytical temperature compensation shows that the influences from preload and temperature have been well discussed. However, in the context of structural health monitoring, a statistical evaluation according to [58] is recommended.

4 Summary and conclusions

The objective of the study was to investigate the effects of temperature and preload force on the capacitance and electromechanical impedance spectra of lead zirconate titanate piezoelectric wafer active sensors for structural health monitoring of bolts. To achieve this, a numerical model depending on temperature and preload force was developed and validated against experimental data. The application of this model showed a linear correlation between the radial deflection of the sensor deformation and the preload force within the considered range. Additionally, a proportional relationship was observed between the electrical parameters of polarisation \(P_\text{z}\), electric charge, capacitance, and the mechanical properties of the radial deflection of the PWAS (Fig. 15):

$$\begin{aligned} F\uparrow \ \Rightarrow \left| \delta _x\right| \downarrow \Rightarrow Q\downarrow \Rightarrow C\downarrow \ \Rightarrow \left| X_\text {C}\right| \uparrow \ \Rightarrow \left| X\right| \uparrow \ \Rightarrow \left| {\underline{Z}}\right| \uparrow . \end{aligned}$$
(27)

The equation for calculating the impedance of a series resonant circuit explains the change in slope of the EMI spectra as a function of the preload force. A clear correlation has been established between the mechanical strain in the sensor area (here due to a preload force), the capacitance and the EMI spectra. Lead zirconate titanate piezoelectric sensors are therefore excellent for use in areas of variable mechanical strain. Increasing the preload force restricts the bolt’s freedom of movement locally, resulting in increasingly attenuated amplitudes in the resonance range.

We have also explained the relationship between the temperature and the capacitance of piezoelectric sensors made of lead zirconate titanate using the Landau-Ginsburg-Devonshire theory. It was shown that the temperature has an influence on the polarisation potential \(\psi _\text {pol}\) and the corresponding electric field \(E_\text {z}\) parallel to the direction of polarisation \(P_\text {z}\). At higher temperatures, the same electric field strength results in a greater polarisation of the PZT. Consequently, the electrical susceptibility or permittivity and the capacitance of the PZT increase:

$$\begin{aligned} T\downarrow \ \Rightarrow P_\text {z}\downarrow \ \Rightarrow C\downarrow \ \Rightarrow \left| X_\text {C}\right| \uparrow \ \Rightarrow \left| X\right| \uparrow \ \Rightarrow \left| {\underline{Z}}\right| \uparrow . \end{aligned}$$
(28)

Similar to preload, the change in capacitance results in a change in the slope of the EMI spectrum. Therefore, the effect of temperature and preload force on the EMI spectra is analogous.

In the current study, we have investigated the relationship between crucial factors in monitoring the structural health, including temperature and mechanical strain, and their influence on the electromechanical impedance spectra. The article improves the understanding of the measurement method and facilitates the transfer of the measurement method to other problems in structural health monitoring. In addition, the knowledge gained provides a strong basis for plausibility checking of data sets containing electromechanical impedance spectra.