A Review of the Degradation Research on the Single-Lap Bolted Joint

Abstract

Bolted joints, with their advantages of simple structure and convenient disassembly and assembly, are widely used in complex equipment fields such as aerospace systems and weaponry. Subject to complex mechanical loads, the contact surfaces may undergo nonlinear behaviors such as contact–separation and viscous slip, leading to the nonlinear degradation of the connection stiffness, which severely threatens the safety and reliability. These have driven research on bolted joints to span multiple disciplines, from interfacial micro-friction to macro-structural dynamics. Therefore, from the field of micro-friction to macro-dynamics, this review summarizes and analyzes three major degradation models, outlines the experimental development in connected structures, and provides an overview of the numerical analysis methods for degradation simulation. This paper also looks forward to the development directions for future research on the degradation of connected structures.

1. Introduction

For sophisticated equipment, high performance requirements, complex structural composition, and a multitude of components are the norm, with a significant number of mechanical assembly structures present internally. As a quintessential method of mechanical assembly, bolted joints play an essential role in the design and operation of large-scale complex equipment such as machinery and aerospace systems. Bolted joints achieve the fastening of two or more structures with through-holes by the engagement of the external and internal threads, offering advantages such as high load-bearing capacity, simple structure, and ease of assembly and disassembly [1,2].

Gaining a comprehensive understanding of the dynamic characteristics and degradation patterns is essential for enhancing the reliability and safety of equipment under complex conditions, making it a critical task in the dynamic response analysis of equipment. The nonlinear contact of bolted joints leads to the nonlinear degradation of connection stiffness and preload, posing challenges for the dynamic response analysis of equipment. Moreover, due to the difficulty in conducting dynamic experiments on equipment, the analysis primarily relies on numerical simulation methods. However, the multi-scale phenomena triggered by bolted joints can result in an excessively large scale of computation that becomes impractical. At present, bolted joints are often simplified to rigid connections to avoid the nonlinearity and multi-scale effects, but this approach cannot ensure the accuracy of computational results.

Therefore, there is an urgent need for research on the dynamic characteristics and degradation patterns of bolted structures. Establishing dynamic degradation models is necessary to accurately describe the degradation behavior. Additionally, exploring equivalent modeling methods will facilitate the engineering application of theoretical models. This will provide a theoretical and methodological foundation for the dynamic response analysis of equipment.

Research teams represented by Sandia and Los Alamos National Laboratories are vigorously advancing the study of dynamic characteristics and degradation patterns of bolted joints [3]. However, there is still no universally recognized theoretical and methodological system. Consequently, there is a continued need to pursue research on the dynamic characteristics and degradation patterns of bolted joints. Among the challenges are the internal, localized, nonlinear, and multi-scale properties of joint contacts, which are the primary obstacles. These challenges specifically manifest as:

(1) Bolted joints exhibit nonlinear and dynamic characteristics, complicating the mechanical behavior at the interface and posing direct challenges to the study of dynamic models.

Taking the Iwan model as an example, the Iwan model is extensively used for the dynamic modeling of degradation phenomena. Iwan models can be categorized into phenomenological models and constitutive models. Phenomenological models reproduce experimental phenomena, without delving into the underlying functionality. Constitutive models are based on contact friction theory and describe the dynamic characteristics of joints. Based on whether the model’s density function changes dynamically with tangential load, constitutive models can be further categorized into static/dynamic constitutive models.

Challenge 1: The research on static constitutive models based on nonlinear pressure distribution.

In recent years, static constitutive models have demonstrated a stronger ability to explain degradation patterns compared to traditional phenomenological models and have preliminarily established the relationship between macroscopic phenomena of connection degradation and contact friction [4,5,6,7]. However, the coupling of contact nonlinearity and pressure nonlinearity increases the difficulty in solving the density function of static-constitutive models, hindering their further development.

Challenge 2: The research on dynamic constitutive models based on the evolution of contact states.

Within connected structures, there is a coupled relationship among contact, friction, and preload. The establishment of a dynamic model is an important direction for the in-depth development of theoretical model research, according to the contact evolution law and mixed-loading effects [8,9]. Under the excitation of dynamic loads, the nonlinear pressure and contact, pressure dynamics, and the boundary conditions are intercoupled, which poses a challenge to the study of dynamic constitutive models, as shown in Figure 1.

(2) The multi-scale characteristics make the finite element analysis of bolted joints extremely challenging.

The finite element method is a crucial approach for the dynamic response analysis of equipment. There are primarily two levels of multi-scale challenges. The first is the multi-scale challenge between the contact and non-contact areas within the bolted joints. The second is the multi-scale challenge between the connected structures and the components within the equipment.

Challenge 3: A multi-scale finite element analysis of bolted joints.

The nonlinear dynamic characteristics of bolted joints primarily stem from contact nonlinearity, with the finite element method being one of the main research methodologies [10,11]. There exists a multi-scale feature between the contact area and the connected structure as shown in Figure 2, where the micro-surface profile exhibits distinct morphological characteristics on scales ranging from 10⁻³ to 10⁻⁹ m. To accurately reflect the interfacial contact behavior, extremely small element sizes are required for this region, whereas the scale of the connected structure is typically on the scale of 10⁻² m. The multi-scale characteristics can lead to a sharp increase in the number of computational nodes, causing an expansion of the computational scale to the point where calculations become unfeasible, posing a significant challenge to the finite element analysis of bolted joints.

Challenge 4: A multi-scale finite element analysis of equipment.

In the dynamic response analysis of equipment, there are multiple cross-scale phenomena between components, bolted connection structures, and contact areas, making calculations difficult. These are mainly manifested in the following:

• The characteristic length of components is in the order of 10⁰ to 10¹ m, while the scale of connection structures is typically in the order of 10⁻² m.
• The time for equipment to reach steady-state response under dynamic load is in the order of 10⁰ to 10¹ s, while the time step describing the slip behavior of the joint surface is in the order of 10⁻⁹ s.
• There are a large number of bolted joints in the equipment, leading to widespread and more severe cross-scale phenomena. Therefore, the multiple cross-scale phenomena have made the numerical solution of the dynamic response of equipment challenging.

(3) The intrinsic and localized properties of the contact area significantly amplify the complexity.

Challenge 5: An experimental study on the evolution of contact status and pressure distribution.

The nonlinearity of the contact area is the primary source of the dynamic nonlinear characteristics of the bolted structure. However, all contact areas are located internally within the structure, making the detection of the contact status extremely challenging. In addition, under the action of the bolt preload, the connected structure will warp to a certain extent, leading to local contact characteristics in the contact area. The degree of warping is influenced by factors such as the shape, size, material properties, and preload, further increasing the difficulty of the experiment.

2. Nonlinear Degradation in Bolted Joints

Bolted joints are typically subjected to the combined action of bolt preload and tangential loads, as depicted in Figure 3. There are three contact surfaces: the bolt head-plate surface, the plate-to-plate surface, and the nut–flat surface. When the lower plate is fixed and the upper plate is subjected to the tangential load T, movement occurs, resulting in relative displacement δ between the plates, and tangential stick–slip behavior at the plate-to-plate surface. In addition to the tangential stick–slip behavior, the contact surfaces also exhibit normal contact–separation behavior. These are collectively referred to as contact nonlinear characteristics [12].

To categorize the contact characteristics of stick–slip behavior under various load conditions, Gropper and Hemmye introduced the concepts of microslip (micro-sliding) and macroslip (macro-sliding) [13,14]. Microslip refers to partial sliding that occurs on the contact surface under the action of relatively small loads, while macroslip denotes the overall sliding that takes place on the contact surface under the influence of relatively large loads. For bolted joints, the pressure is typically concentrated near the bolt hole. If the applied tangential load is not substantial enough to cause sliding in the area around the bolt hole, then partial sliding will occur, as shown in Figure 4a. As the tangential load increases, the joint surface may undergo complete sliding.

Brake et al. [15] have divided the degradation process into four stages based on the stick-slip condition: sticking, microslip, macroslip, and pinning, as illustrated in Figure 4b. During the microslip phase, the connected structure experiences significant nonlinear stiffness degradation. In the pinning phase, due to the contact and elastic deformation between the bolt shaft and the bolt hole, the stick–slip behavior is no longer the primary source of the degradation nonlinearity.

3. Contact Models

Contact models can be categorized into static-friction models, dynamic-friction models, and friction-constitutive models. The static- and dynamic-friction models describe the frictional phenomenon as a function of velocity and displacement, respectively. Friction-constitutive models, based on the physical properties of the contact area interface, describe the friction phenomenon in a localized manner. The following sections will introduce these three types of contact models separately.

3.1. Static-Friction Models

Static-friction models are primarily categorized into the Coulomb model, the Coulomb viscous friction model, the static-friction Coulomb viscous friction model, and the Stribeck model.

The Coulomb model is the earliest friction model, where the frictional force is related only to the normal load and is independent of velocity. This model cannot describe the behavior of frictional forces near non-zero velocities. The Coulomb viscous friction model, developed based on the partial fluid viscous characteristics in fluid mechanics, posits a linear relationship between frictional force and relative velocity, which aligns more closely with the experimentally observed variations in friction.

The static-friction Coulomb viscous friction model better reflects the phenomenon observed in experiments, where static friction exceeds dynamic friction, and it can describe the abrupt change in frictional force during the transition from static to sliding at the contact surface. Stribeck noted that when overcoming maximum static friction, the frictional force exhibits a continuous downward trend with only a slight change in relative velocity. Thus, the Stribeck model was proposed. The relationship between frictional force and velocity in the four models is illustrated in Figure 5.

3.2. Dynamic-Friction Models

Dynamic-friction models characterize the frictional force as a function of relative velocity and displacement and are widely applied in the field of control engineering.

(1) Dahl model

Dahl conducted dynamic-friction experiments on servo systems incorporating rolling bearings and proposed a theoretical model that describes the frictional force, as illustrated in Figure 6a. Prior to reaching the maximum static-friction force, there is no relative movement between the two contact surfaces. However, there exist minor displacements between the contact peaks, termed pre-slip [16]. The model is mathematically expressed as:

$${\displaystyle \frac{d{F}_c(x)}{dx}}=\sigma {\left|1-{\displaystyle \frac{F_c}{f_{critical}}}\mathrm{sgn}\dot{x}\right|}^i\mathrm{sgn}\left(1-{\displaystyle \frac{F_c}{f_{critical}}}\mathrm{sgn}\dot{x}\right)$$

(1)

where σ represents the stiffness coefficient, $f_{critical}$ is the critical friction force when the displacement increases monotonically, and x denotes the relative displacement.

(2) Bristle model

The bristle model is a microscopic description of the random motion at the contact points of surfaces in contact. Due to the inherent roughness at the microscale, contact occurs at the asperities of the surfaces. The contact is conceptualized as occurring between two rigid bodies connected by bristles, as depicted in Figure 6b, which represents the interaction between elastic and rigid bristles [17]. When a tangential force is applied, the bristles deflect like springs, thereby generating frictional force. The expression of this model is as follows:

$$F_c={\displaystyle \sum _{i=1}^{n_b}{k}_b(X_{E,i}-X_{R,i})}$$

(2)

where $n_b$ represents the number of bristles, $k_b$ is the stiffness of the elastic bristles, X_E,i denotes the position of the elastic bristles, and X_R,i is the position of the rigid bristles.

(3) LuGre model

Based on the refinement of the Dahl model and the bristle model, the LuGre model can characterize static friction and the Stribeck phenomenon. The irregular shapes of contacting surfaces are highly stochastic at the microscale. Haessig and Friedland proposed the bristle model to represent the random behavior and the reset integral model to describe the collective behavior of bristles [18]. The average deflection of bristles is denoted by z, and the core modeling function is as follows:

$${\displaystyle \frac{dz}{dt}}=v-{\displaystyle \frac{\left|v\right|}{h(v)}}z$$

(3)

where h is a function influenced by material properties, lubrication, and temperature.

(4) Valanis model

The Valanis model [19] portrays the micro- and macro-slippage at the contact interface and its response under dynamic loading, finding extensive application in the field of plasticity [20,21,22]. Gaul and Lenz utilized the Valanis model to ascertain the nonlinear behavior of load transfer in joint structures during both micro- and macro-slippage states, simulating their response to cyclic and transient loading, and successfully replicating these responses experimentally. The model expression is given by Equation (4), where $E_0$, $E_t$, and $\lambda$ are material parameters, and $\mathit{\ell }$ is a dimensionless parameter [23].

$${\dot{F}}_c={\displaystyle \frac{E_0\dot{\delta }\left[1+\frac{\lambda }{E_0}\mathrm{sgn}(\dot{\delta })(E_t\delta -F_c)\right]}{1+\mathit{\ell }\frac{\lambda }{E_0}\mathrm{sgn}(\dot{\delta })(E_t\delta -F_c)}}={\dot{F}}_c(\delta ,\dot{\delta },F_c)$$

(4)

(5) Iwan model

In 1930, Timoshenko introduced a hysteresis system composed of elastoplastic elements with varying yield levels. However, at that time, it was considered that this system would complicate the force–displacement relationship and did not receive sufficient attention. It was not until 1966 that Iwan conducted a theoretical analysis of this system, deriving a relatively simple hysteresis expression for the Iwan model. This demonstrated that the model could be applied to study the dynamic response of hysteresis systems [24]. The Iwan model is divided into parallel–series and series–series configurations, as shown in Figure 7.

Argatov, Butcher, and Wentzel have conducted research indicating that the parallel–series model has better applicability [25,26]. Segalman et al. [27,28] pointed out that any generalized constitutive model satisfying the Masing criterion can be represented as a parallel–series Iwan model, demonstrating the superiority of the parallel–series Iwan model. The traditional parallel–series Iwan model is based on two core theories: the Jenkins element, composed of a linear spring and a slider in series, describes the viscous, sliding, and mutual transformation phenomena of the connection interface; the yield force of the Jenkins element follows a uniform distribution. The tangential force function expression of the Iwan model is as follows:

$$T={\displaystyle {\int }_0^{k_t\delta }{f}^{*}\phi (f^{*})d{f}^{*}}+k_t\delta {\displaystyle {\int }_{k_t\delta }^{\infty }\phi (f^{*})d{f}^{*}}$$

(5)

where $k_t$ represents the total stiffness of the Jenkins element springs, $f^{*}$ is the yield force of the Jenkins element, $\delta$ denotes the tangential relative displacement, and $\phi (f^{*})$ signifies the yield force density function.

Gaul designed an experimental setup featuring a longitudinal resonant cavity chamber and an isolated lap joint structure. By applying tangential harmonic loads of varying amplitudes, the force–displacement response of the jointed structure and the residual stiffness in the macroslip is obtained [23]. Building on this, Song developed a new model based on the parallel–series Iwan model that incorporates linear springs to characterize residual stiffness [29]. Zhang et al. derived a normalized periodic energy dissipation calculation formula for the parallel–series Iwan model using a uniform density function. The variation in the normalized periodic energy dissipation value with the displacement amplitude is consistent with the experimental results in the literature [30], but not with those in the literature [31,32,33]. Under the dynamic load, the bolt preload also varies. Han and Gilmore, through the theoretical analysis and finite element methods, have conducted research and proposed different dynamic models [34,35,36,37]. Rajaei considered the effect of normal load on tangential stiffness and proposed a dynamic model capable of replicating asymmetric hysteretic behavior. A dynamic relationship between normal pressure and reference pressure was established, and an expression for the force–displacement relationship under variable normal load was derived and experimentally verified [38]. Ouyang et al. [39,40] used finite element and experimental methods to study the connection structure and replicated the experimental results through the Iwan model. Segalman proposed a reduced-order model on the basis of the Song–Iwan model. According to the power–law relationship between energy dissipation and load amplitude, a truncated power–law density function was proposed [30,41,42]. This function, composed of Dirac delta function and Heaviside function, can describe the abrupt change in the density function [43]. The four-parameter model proposed by Segalman cannot represent residual stiffness and assumes that there is always slip at the interface, without constraining the density function near the origin. The Sandia Laboratory points out that when the tangential load is small, the restoring force is linearly related to displacement. Li et al. [44] proposed a six-parameter model with a truncated power–law distribution and dual pulses based on the Segalman–Iwan model. Li et al. [6] proposed a new method for solving the Iwan model density function through frictional contact. Zhao et al. [7], based on Li’s method, extended the Hertz pressure distribution to a linear distribution, proposed a new Iwan model, and carried out experimental verification.

In recent years, numerous scholars have conducted in-depth research on the Iwan model. Wang et al. [45] utilized the parallel–series Iwan model along with numerical integration methods to investigate the impact of excitation with varying amplitudes on the nonlinear dynamic behavior. Liu et al. [46] proposed a parameterized finite element modeling approach for the Iwan model using the COMBIN40, COMBIN14, and MASS21 elements. Gong et al. [47] introduced an Iwan model suitable for local slippage on threaded surfaces.

In addition to the dynamic models mentioned above, there are also reset integral friction models and Leuven models, among others [48,49]. Static-friction models, while relatively simple in form, can only describe the friction force as a function of velocity. Dynamic models, on the other hand, address the shortcomings of static-friction models but also present challenges such as higher model complexity and greater difficulty in parameter identification.

3.3. Friction-Constitutive Model

Many scholars have been committed to establishing a connection between the dynamic characteristics of bolted joint structures and their microscopic mechanisms. This part of the research can be divided into two categories: statistical summation models and contact fractal models.

(1) Statistical summation model

Real surfaces exhibit inherent roughness, marked by unpredictable bumps and recesses, with interactions primarily occurring at the micro-asperity level. The statistical summation model posits that surfaces comprise countless spherical elements, sharing an equivalent radius of curvature R. Utilizing Hertzian contact theory, the interplay between the normal load, the extent of penetration, and the contact area for an individual micro-asperity is formulated. In 1949, Mindlin introduced a solution for the contact issue involving elastic spheres under a mixed-loading condition. This model delineates the contact zone between two spheres as encompassing a central viscous region encircled by an annular region prone to sliding, which contracts progressively with the application of tangential loads and vanishes once the load reaches a certain threshold [41,42]. Based on the Mindlin model, the tangential load during sliding friction can be determined using Coulomb friction law. To address the issue of unbounded shear stress at the contact edge when the entire contact area is in a viscous state, Mindlin proposed a method to limit the shear stress according to Coulomb’s friction law [50].

The statistical summation model, grounded in Mindlin’s elastic shear theory and the constant shear strength at the contact edge, establishes a tangential load–displacement relationship [51]. Drawing on the statistical model of rough surfaces to obtain the distribution of asperity heights, it characterizes the normal and tangential contact forces through statistical summation, as depicted in Figure 8 [52]. The core of this model lies in the density function of the heights of the rough surface asperities, typically assumed to follow a Gaussian distribution. However, it can also be empirically derived from specific rough surfaces. Subsequently, the forces and deformations of all asperities are statistically summed to yield the frictional force.

The statistical summation model, originally introduced by Greenwood and Williamson, is known as the GW model [51]. Chang et al. advanced this model to account for plastic deformation, resulting in the CEB model [53]. Kougut, Etsion, and Brizmer refined the deformation of micro-asperities, removing the constraint of volume conservation during their deformation, and introduced the KE and BKE models [54,55]. Building on contact mechanics theory and the elastic deformation of micro-asperities, Zhao et al. proposed a novel micro-contact model for rough surfaces that exhibit elastoplastic behavior [56].

The physical characteristics of real surfaces are highly complex, and the modeling process inevitably introduces numerous simplifications. Since the true contact area is much smaller than the nominal contact area, it leads to high normal pressures locally, which may cause plastic deformation of the micro-asperities. The model incorporates a plasticity index to simplify this aspect. Repeated contacts reduce the elastic load that micro-asperities can withstand in both normal and tangential directions; hence, the model assumes the contact surface is initial contact [57]. In addition, the statistical summation model is also based on five fundamental assumptions: it assumes ideal elastic contact for metal surfaces; the surface is isotropic; the micro-asperities are spherical; it disregards the coupling effect of micro-asperities; surface parameters remain constant; and there is no lubrication, with the surface exhibiting dry friction behavior.

(2) Contact fractal model

Mandelbrot introduced the concept of fractal geometry in 1967, noting that when the measurement unit is reduced, the length of a coastline does not converge but instead increases monotonically [58]. A typical case of fractal geometry is shown in Figure 9.

Majumdar and Bhushan applied the fractal models to describe the contact mechanism, employing the Weierstrass–Mandelbrot function to characterize the statistical self-affine fractal nature of surface profiles. Based on the assumption of elastic contact, the relationship between the fractal dimension and characteristic length scale parameter with surface pressure is established. The surface pressure at contact points is calculated by utilizing the empirical scaling laws, Hertz theory, and curvature [59]. The fractal model modeling method is depicted in Figure 10.

Zhang et al. established fractal models for the normal contact stiffness, damping, and tangential contact stiffness, and damping of joint surfaces [60,61]. The core of these models lies in the density function of the contact area of micro-asperities, integrating it to find the true contact area, and then calculating the total elastic potential energy during the contact phase and the loss during the plastic deformation phase [52].

The contact fractal model and the statistical summation model each have their own strengths and weaknesses. The contact fractal model has the advantage of being unaffected by sampling length and resolution. However, its weaknesses are that fractal characteristics may not be applicable to all contact surfaces, and it requires the measurement of the fractal dimension. On the other hand, the statistical summation model boasts stronger universality and the ability to explain physical mechanisms. Its disadvantage lies in the fact that the physical parameters of the surface are influenced by sampling length and resolution, and the accuracy of the model is constrained by observation.

4. Experiment of Bolted Joints

Experiment techniques can be broadly classified into indirect and direct methods, based on their ability to directly plot hysteresis loops. The indirect method is employed in scenarios where the direct measurement of the mechanical physical quantities at the connection interface is unfeasible. This approach necessitates subsequent data processing to extract the hysteresis loops that characterize the interface’s mechanical behavior. The direct method has emerged as a result of evolving measurement technologies, enabling the immediate acquisition of hysteresis loop parameters. This method circumvents the need for any post-processing steps, thereby providing a more efficient and accurate means of capturing the hysteretic mechanical properties of the interface.

4.1. Indirect Methods

The Air Force Flight Dynamics Laboratory initiated studies on the slip response and damping of mechanical joint structures as early as 1964, conducting experiments on riveted, bolted, and welded plates. The decay rate of the bolted joint response is used to determine the connection damping. The Sandia Laboratory designed the Big Mass Device (BMD), an experimental setup comprising an initial mass block, load sensors, adjustable nuts, upper and lower rollers, and a vibration table. This device can simulate the bolt pretightening force and minimizing the impact of additional contact surfaces. For the four types of experimental specimens, BMD experiments were conducted, yielding dynamic data such as hysteresis loops. These experiments established a power–law relationship between energy dissipation and the amplitude of the applied load. Gaul designed an experimental setup with longitudinal resonant chambers and isolated lap joints, where the resonator parts were suspended by flexible nylon cords, excited by a vibrator, with the input force of the vibrator measured by a piezoelectric sensor, and the displacement and friction force obtained through the measurement of acceleration data. By applying cyclic loads of different amplitudes, force–displacement experimental data were obtained, revealing the residual stiffness present after macroslip. Rogers and Boothroyd experimentally studied the hysteresis loops, examining the effects of the material, surface finish, and contact area on energy dissipation [62]. Padmanabhan and Ren et al. [63,64] carried out a series of experiments, measuring the energy loss when preloaded planes were subjected to cyclic tangential forces, and studied the impact of related variables on the effectiveness of contact damping. Resor et al. [31,65] designed a quasi-static monotonic tension test for bolted joints, pointing out that the displacement magnitude of macroslip is on the order of microns, and the effective coefficient of friction for the contact surface of the bolted joint is 0.63.

4.2. Direct Methods

In recent years, with the continuous advancement of measurement technology, scholars have utilized direct methods for experimental research on bolted joint structures. Zhao et al. [7] conducted tensile experiments on bolted joints using a universal testing machine, from which they derived the degradation curves of stiffness. The experimental setup comprised a testing machine frame, fixed and mobile ends, laser displacement sensors, and an elevating frame, with the test specimens bolted together without any additional fixtures. This apparatus is user-friendly and capable of simulating the slip and stiffness degradation behavior of contact surfaces under the influence of actual bolt connections and pretightening forces. Eriten [66] designed a lap joint microslip experiment, comparing and analyzing the microslip experiment and device, which exhibited minimal noise, misalignment, stiffness, and damping, enabling the accurate measurement of the microslip behavior without post-processing. Subsequently, Eriten et al. [67,68] used this device to study the hysteresis loops of aluminum and steel connection pieces under cyclic loading, revealing that aluminum connection pieces dissipate more energy than their steel counterparts. Ovcharenko and Varenberg et al. [69,70] conducted experiments with direct measurements of force and displacement, employing a piezoelectric actuator to apply controlled displacement loads, with a proximity sensor for displacement measurement and a load sensor for friction force measurement. Abad et al. [71] performed a quasi-static test on bolted joints using a universal testing machine, applying a periodic triangular signal at 0.6 Hz to the free end, subjecting the connected structure to tangential loading at a constant velocity and minimizing the impact of acceleration. Kartal et al. [72] investigated the slip wear of titanium and nickel alloys, applying normal pretightening loads via a hydraulic actuator, measuring tangential forces with a pressure transducer, and using digital imaging techniques to measure tangential relative displacements, with a linear variable differential transformer (LVDT) measuring displacements between samples and cast-iron blocks.

Schwingshackl et al. [73] developed the first generation of a friction device, which assumed that the actual loading displacement was equivalent to the relative displacement. A single-laser Doppler velocimeter (LDV) was utilized to measure the relative displacement of the contact surface. The second generation of friction experimental devices, utilizing two LDVs for the accurate measurement of relative displacement, has seen widespread application [74,75,76]. Li et al. [5] proposed a microslip experimental setup for measuring the friction hysteresis characteristics of bolted joints, applying vibratory loads with a piezoelectric actuator, and measuring friction forces and bolted load with force sensors and load pads, establishing a numerical model to extract tangential contact stiffness from measured data. The development of experimental technology cannot be separated from the development of sensor integration technology, and relevant studies [77] have been summarized. The main differences between the direct and indirect method are shown in

Table 1. Differences between direct and indirect method.

Method	Advantages	Disadvantages
Direct method	Real and reliable data	High cost, long experiment period, complex loading equipment and measuring instruments
Indirect method	Low cost, short experiment period	Complex data-parsing and signal-processing techniques, limited information

.

5. Numerical Analysis Methods

Numerical analysis methods are an essential technical approach for integrating theoretical models with engineering applications, and they are primarily divided into time-domain analysis methods and frequency-domain analysis methods [78,79,80,81,82,83,84,85,86,87,88,89,90,91]. Time-domain analysis methods directly solve the system’s finite element equations, including the central difference method, Runge–Kutta method, Newmark method, Wilson-θ method, and Houbolt et al.’s method.

Li et al. [92], based on the time-domain analysis method for multi-degree-of-freedom systems, studied the stiffness characteristics and equivalent dynamic models of bolted joints under impact load, deriving the failure assessment criteria for complex assemblies under impact load. Subbaraj et al. [93] studied the application of the Newmark, Wilson-θ, and Houbolt implicit time-domain solution methods in linear and nonlinear structural dynamics and derived the implicit formula for the Trujilo-modified Newmark-beta method through the weighted residual method. Wang et al. [45] used numerical integration methods to obtain the transmission characteristics of the nonlinear system’s absolute acceleration response, transforming it into the phase-frequency and amplitude-frequency characteristics of relative acceleration, and identified the stiffness and damping characteristics. Xie et al. [94] studied the commonly used time-domain integration methods and Runge–Kutta methods in transient finite element analysis, pointing out that time step can affect the accuracy, and ranked the time-domain analysis methods from most to least efficient as the central difference method, Runge–Kutta method, Newmark method, Houbolt method, and Wilson-θ method. Hagedorn et al. [95] decomposed large systems into linear subsystems and multiple nonlinear subsystems, described the dynamic characteristics of nonlinear systems through time-domain integration, and provided numerical methods for solving integral equations. Oldfield et al. [40] used the finite element method to study the friction contact of bolted joints under harmonic load, applied the parallel–series Iwan model and the Bouc–Wen model to the study of the hysteretic behavior, and solved the control equations of the two models using the fourth-order Runge–Kutta method, with the calculation results being consistent with the theoretical solutions of the models. Gaul et al. [23] used the time-domain integration method to solve the transient dynamic response of bolted joints. Since the time-domain analysis method directly solves the system’s finite element equations, it results in relatively low efficiency and is more suitable for transient responses. To improve the numerical calculation efficiency of large assemblies, order reduction is required before analysis [96,97]. Wang et al. [98] proposed a dynamic order reduction method based on nonlinear transformation, which effectively reduced the computational consumption in the iteration process, increasing the computational efficiency by about 30%. Miller et al. [99] proposed a degradation model for bolted joints and the corresponding reduced-order model, which can accurately obtain energy dissipation results consistent with experiments.

The time-domain method requires the integration of the system over several cycles until the transient response disappears when solving the steady-state response of lightly damped structures, which consumes a significant number of computational resources [100,101]. Therefore, many scholars prefer to use the frequency-domain method to solve nonlinear dynamic equations [102]. Researchers have proposed a series of frequency-domain analysis methods, such as the perturbation method, averaging method, asymptotic method, multi-scale method, harmonic balance method, incremental harmonic balance method, and describing function method, etc. [78,79,80,81,82,83,84,85,86,87,88,89,90,91,103]. The harmonic balance method represents the periodic response of a structure containing nonlinear components in the form of a Fourier series, thereby transforming the nonlinear structural dynamic differential equation into a set of algebraic equations. Ahmadian and Jalali [104] used the incremental harmonic balance method to study the dynamic response of bolted connections, pointing out that the first-order harmonic solution can accurately describe the response of the connected structure. Zhang et al. [105] used the first-order harmonic balance method and the fourth-order Runge–Kutta method to study the free and forced vibrations of the friction dry oscillator model, finding that the model’s equivalent viscous damping has a nonlinear relationship with amplitude. Kim et al. [106] used the multi-harmonic balance method to calculate the nonlinear response, which has the ability of adaptive arc length continuation and stability calculation, improving the convergence of solving periodic solutions.

6. Discussion

(1) Theoretical research on the nonlinear dynamic characteristics of bolted joints

Scholars are dedicated to advancing the theoretical research on the nonlinear dynamic characteristics of bolted joints. The goal is to establish dynamic models that can characterize and predict the degrading properties and patterns of the connections. Among these, the Iwan model, featured for its simplicity and applicability in dynamic friction models, has garnered widespread attention. It can effectively describe the nonlinear dynamic characteristics of connected structures. However, the existing Iwan model has the following shortcomings:

The static phenomenological model, based on the description of physical phenomena, with model parameters lacking physical significance. Moreover, its description of the degradation is still insufficient, failing to reflect the continuity of connection stiffness degradation at the macroslip point. The static–constitutive model, based on certain simplified assumptions, solves the friction shear stress distribution function of the contact surface, maps the density function of the Iwan model, and describes the nonlinear degradation behavior. The existing static constitutive model has preliminarily established the connection between the degradation and the contact mechanism, but some simplified assumptions and pressure distribution functions are not reasonable.

Currently, as the Iwan model faces limitations, the most promising avenues for advancement include not only further refining the model’s intrinsic capabilities but also encompassing precise multi-scale topographical mapping of joint surfaces, ascertaining contact pressures through finite element analysis, and constructing predictive models grounded in statistical aggregation methodologies. It is crucial that these approaches demand sophisticated measurement techniques for joint surface topography and involve a complex modeling process. How to build a set of simple and efficient model is the sustainable development direction of the future.

(2) Experimental study on the degradation pattern of bolted joints

Due to the complexity of the degradation behavior of bolted connections, there may be other nonlinear behaviors under different working conditions and connection forms, and further research is needed on the influencing factors and patterns of connection degradation. In terms of contact experiments, current studies mainly use pressure-sensitive films or ultrasonic methods to measure the pressure distribution under initial preload conditions. However, the degradation process is usually affected by tangential load, and some scholars have studied the contact problems of the contact surface under mixed loads (normal and tangential loads) [6,7,107,108,109]. But due to the limitations of measurement technology, there are still relatively few contact analyses. In addition, the impact of factors such as thread structure, bolt strength, and connection forms on the dynamic characteristics still needs further research.

(3) Equivalent model of the contact surface of bolted joints

Due to the difficulty of conducting overall dynamic experiments on equipment, most studies use numerical analysis methods, with finite element analysis being one of the main methods [110]. The cross-scale is a key difficulty that restricts the accuracy of dynamic response calculations. Scholars are committed to reducing the order of the dynamic model of the bolted joints, using a small number of elements to realize the application of the theoretical model in the finite element, and solving the cross-scale problem in the calculation of equipment dynamic response. However, the existing method of establishing an equivalent model based on dynamic curves has significant limitations. For different load conditions, it is necessary to repeatedly identify model parameters, which greatly affects the applicability and computational efficiency of the equivalent model.