
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12400-0
10.1016/j.heliyon.2024.e36369
e36369
Research Article
Study on interfacial mechanical properties of bonded pipe joint under dynamic load
Lin Zhenhao
Li Shanqing lishanqing09@163.com
⁎
MOE Key Laboratory of Disaster Forecast and Control in Engineering, School of Mechanics and Construction Engineering, Jinan University, Guangzhou, 510623, China
⁎ Corresponding author. lishanqing09@163.com
15 8 2024
30 8 2024
15 8 2024
10 16 e363699 3 2024
7 8 2024
14 8 2024
© 2024 The Authors. Published by Elsevier Ltd.
2024

https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
This study analyses the interfacial mechanical behaviour of adhesive-bonded pipe joints under dynamic loading conditions. First, a mechanical model of the bonding interface of a pipe joint was established. Next, computational formula for the interfacial slip and shear stress in the pipe joints were obtained using variable separation, eigenfunction expansion, and Laplace transform methodologies. Further, the effects of various parameters, including the loading duration, bond length, adhesive layer thickness, stress rate, adhesive shear modulus, elastic moduli of the main pipe and coupler pipe, and adhesive-layer thickness, on the mechanical response of the pipe joints were assessed. The goal is to study the impact of these parameters on the maximum shear stress, interfacial slip, and normal stress in the main and coupler pipes, their effects on the pipe joint strength can be determined, thereby providing a theoretical basis for the design of practical pipe–joint configurations. The innovation of this study is as follows: it considers high-stress-rate dynamic loads for formulating complete mathematical equations. Further, it employs theoretical methods to calculate the relative slip and shear stress in pipe joints, through which theoretical formulas for engineering applications of adhesive-bonded pipe joints were derived. Additionally, the study analyses the impact of various parameters on the dynamic mechanical behaviour of the adhesive layer at the pipe joint interface, leading to a deeper understanding of the interface mechanical behaviour characteristics and key parameters to be considered while receiving dynamic loads in pipe joints.

Keywords

Bonded joints
Dynamic loads
Pipe joints
Interfacial mechanical properties
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pmc1 Introduction

Bonded pipe joints possess notable hermeticity and corrosion resistance, as well as exhibit uniform stress distribution. In use, there is no need to open holes or welds, thereby avoiding stress concentration. With rapid advancements in polymer chemistry, the widespread application of novel synthetic adhesives has become prevalent, particularly in bonded pipe joints. The consequent enhancement in adhesive properties has resulted in the augmented longevity, heightened strength, improved stability, and cost-effectiveness of bonded pipe joints.

Investigations on bonded joints have a rich historical background. In 2018, Amidi et al. [1] analysed the stress redistribution caused by the viscoelastic behaviour of adhesives. A three-parameter viscoelastic model was introduced to simulate the viscoelastic behaviour of the adhesive layer and derive closed-form solutions for the sectional stress. These solutions were subsequently compared with the finite element analysis results obtained using Abaqus. Starting with the assumption of a linear distribution of the longitudinal normal stress along the thickness, Zhao and Lu [2] proposed a general analytical approach for obtaining two-dimensional elastic stress solutions for single-lap adhesive joints. Comparative assessments were performed against the existing computational formula and finite element analysis results. Wang et al. [3] conducted a study on composite-bonded joints comprising carbon fibre-reinforced polymer (CFRP) and aluminium alloy (AI) plates bonded together. They established a mechanical model of the joint and derived the stress at the bonding interface through theoretical modelling. The effect of the bending moments generated by non-axial tensile loads on the bonded joint was considered, and the results were compared with those from the finite element analysis. Yu et al. [4] experimentally investigated the influence of material parameters and bond thickness on the interfacial mechanical properties and found that the interfacial fracture energy is the most critical parameter. Pan et al. [5,6] described the interfacial behaviour using four parameters and comprehensively studied the effects of different parameters such as fiber-reinforced polymer (FRP) plate stiffness, interfacial fracture energy, and interfacial shear strength on the effective bond length. Biscaia et al. [7,8] numerically simulated a bonded joint using the discrete element method and finite difference method and studied the bond stress and slip of FRP composite bonded joints under various temperatures. They also analysed bonded structures composed of three different substrates—wood, concrete, and steel—with CFRP laminates. Through shear tests on 40 bonded joints, they found that the CFRP–wood combination had the longest effective bond length and exhibited the highest fracture energy; moreover, the local nonlinear bond–slip curve of CFRP-concrete could be approximated as an exponential curve. It can be observed that for bonded joints, the mechanical properties of the bonding interface are not only related to the material parameters of the interface but are also greatly influenced by the material parameters of the adherents. Ma et al. [9] investigated the mechanical behaviour of adhesive joints under dynamic loading. They employed the separation of variables method to derive the interfacial slip and stress distributions of adhesive joints made of the same material. Numerical simulations of asymmetric joints were conducted using Laplace transforms, and the results were compared with the finite element analysis outcomes. Haider et al. [10] conducted research on adhesive joints under varying materials and temperatures, discovering that the addition of cork powder can enhance the strength of the adhesive joints.

A multitude of studies have been conducted on the interface mechanisms of bonded pipe joints. Zhao et al. [11] conducted a strength analysis of epoxy adhesive-bonded joints in aluminium plate configurations using cohesive elements. This study simulated the failure propagation of individual adhesive elements during the loading of the adhesive layer and compared the results with experimental data to validate the accuracy of their numerical models. This study provides valuable insights for the rational selection of adhesive materials and enables rapid estimation of the load-carrying capacity of adhesive-bonded joints in practical engineering applications. Pugno [12] investigated the torsional behaviour of adhesive-bonded pipe joints and solved the stress distribution of shear stresses at the adhesive interface by employing constitutive, equilibrium, and compatibility equations. Han [13] conducted a comprehensive investigation on the mechanical behaviour of adhesive interfaces in pipe joints. Nonlinear fracture mechanics was employed to derive analytical expressions for the distribution of shear stresses and displacement relationship at the adhesive interface under torsional loading. This study focused on examining the effects of various adhesive lengths on the load–displacement curve and ultimate load. Furthermore, it aimed to elucidate the stress transfer mechanism, interface crack propagation, and ductile behaviour of pipe joints. In a similar vein, Zou et al. [14] delved into the mechanical response of pipe joints under torsional loading. A theoretical analysis of the stress field of the adhesive layer in adhesive-bonded pipe joints was conducted. Specifically, this study evaluates the impact of parameters such as the overlap length and adhesive-layer thickness on the stress concentration at the joint ends and adjacent regions. These findings provide a solid theoretical basis for optimising the design of pipe joints. Basit et al. [15] investigated concrete pipe joints with varying material parameters and found that cement mortar has a higher load-bearing capacity than non-shrinkage grouting while also being more cost-effective. Zhao et al. [16] analysed three types of bonded joint configurations, including pipe joints, and assessed the damage evolution patterns under different loads, evaluating the joints' load-carrying capacity. Yuan et al. [17] derived the failure process of adhesive pipe joints under tensile load and validated it through finite element modeling.

Griffin [18] conducted experimental investigations to examine the mechanical performance of pipe joints. This study analysed the influence of parameters such as the elastic modulus, shear modulus, bond length, and bond thickness on the load-carrying capacity. Additionally, this study assessed the effects of bond length and adhesive-layer thickness on the magnitude of the ultimate load-carrying capacity, providing valuable insights into the behaviour of pipe joints under various loading conditions. Notably, the findings indicated that the shear modulus of the adhesive layer significantly influenced the ultimate load-carrying capacity. Nemes et al. [19] defined distinct failure criteria and derived computational formula. By leveraging the principle of minimum potential energy, a comprehensive study on the shear stress distribution in pipe joints subjected to axial tension was conducted. The computational formula elucidated the stress intensity and stress distribution within the pipe joints, offering valuable theoretical insights into the mechanical behaviour of these joints under axial loading.

Several studies have investigated the dynamic loads on pipe joints. Kaya et al. [20] employed the finite element analysis method to investigate the impact of various dynamic performance factors on adhesive joints under dynamic loading conditions, thereby obtaining the natural frequencies and mode shapes. Whitcomb et al. [21] employed linear and geometrically nonlinear finite element analyses to investigate the interface mechanics of bonded pipe joints subjected to axial tension and bending moments. Notably, the mechanical behaviour of the interface exhibited more pronounced nonlinear characteristics when the effects of bending moments were considered. Mertiny et al. [22] conducted experiments to compare the mechanical performance of different pipe joints under internal pressure and axial traction and employed finite element simulations to analyse the influence of geometric parameters on the joint performance. Throughout the study, all analyses were conducted within the framework of linear elastic material behaviour, with initial crack initiation occurring at both ends of the bonded interface. Yang et al. [23] established a finite element analysis model based on first-order laminated shell theory to study the distribution of shear stresses in pipe joints subjected to axial tension and torsional moments, exploiting symmetrical considerations. Das and Pradhan [24,25] proposed a finite-element-based simulation technique for analysing adhesive pipe joints in composite structures. They determined the effective overlap length required for an appropriate performance based on the Tsai–Wu failure criterion and conducted a three-dimensional stress analysis as well as an in-depth study on the occurrence and propagation of delamination damage in pipe joints.

The existing research on adhesive joints has predominantly focused on single-lap joints, with relatively few studies dedicated to pipe joints. Most studies on pipe joints have concentrated only on those under static loading conditions, relying primarily on finite element analysis methods and experimental approaches. In contrast to previous studies on pipe joints, the current study targeted pipe joints under dynamic loading conditions and employed theoretical methods for analysis, with finite element analysis methods utilised for validation. The innovation of this study lies in the incorporation of high-stress-rate loading into established static models, thereby enabling a theoretical analysis of the dynamic mechanical characteristics of bonded pipe joints. By employing high-rate dynamic loads, a comprehensive examination of the mechanical behaviour of the joints was conducted, resulting in the formulation of complete mathematical equations. Subsequently, methodologies such as the separation of variables, eigenfunction expansion, and Laplace transforms were employed to derive solutions for the relative slip and shear stresses in the pipe joints. The objective and most significant outcome of this study is the theoretical derivation of formulas for calculating interfacial shear stress and slip in adhesive pipe joints under dynamic loads with high stress rates. Utilizing these formulas, the study has assessed the impact of key parameters such as bond length and adhesive thickness on the mechanical performance of the joints. This research furnishes a theoretical foundation for the practical computation and design of adhesive pipe joints, thereby enhancing the theoretical study of adhesive joint dynamics.

2 Interface mechanical model of pipe joints

2.1 Models and control equations

The pipe-joint model encompasses two main pipes and one coupler pipe, as shown in Fig. 1. The main and coupler pipes are thin-walled circular tubes. The leftmost end of the main pipe was subjected to fixed-end constraints, and the main pipe on the right side was subjected to a time-dependent dynamic tensile force. The interconnection of the two main pipes is achieved using a coupler pipe, and an adhesive layer is employed to establish a bond between the main pipes and the coupler pipe.Fig. 1 Schematic diagram of a pipe joint.

Fig. 1

Before the theoretical derivation, the following assumptions were made in the present study.(1) The primary and coupler pipes exhibited uniformity as linearly elastic materials.

(2) The adhesive layer experienced only shear forces, neglecting the bending effects of the joint [26].

(3) Under the influence of high stress rates, the adhesive layer exhibited a linear elastic behaviour [27].

(4) The interfacial shear stress remains constant throughout the thickness of the adhesive layer [28].

Fig. 1 illustrates a schematic of the stress model for the pipe joint, and Fig. 2 depicts a cross-sectional view of the pipe joint. Fig. 3 presents the stress microelement, and Fig. 4 displays a high-strain-rate triangular pulse load, where the symbol σ˙ represents the expression for the strain rate. Considering the symmetrical nature of the pipe joint, the analysis focused on the right side of the joint for convenience in the derivation. In this study, the bonding length L denotes the distance from the leftmost end of the inner pipe on the right side to the leftmost end of the coupler pipe. The bonding length of the entire pipe joint was defined as 2L. The dynamic loading applied in this study encompassed a representative triangular pulse that is commonly employed in laboratory experiments. Table 1 presents a comprehensive description of the parameter symbols.Fig. 2 Cross-sectional diagram of a pipe joint.

Fig. 2

Fig. 3 Force element diagram.

Fig. 3

Fig. 4 Loading path.

Fig. 4

Table 1 Symbols used in the derivation and their brief explanation.

Table 1Variable	Brief description	Variable	Brief description	
Fi	Tensile force	τ	Interfacial shear stress	
h	Thickness of the adhesive layer	Ai	Cross-sectional area of the pipe i	
G	Shear modulus of adherend	δ	Relative slip	
E	Young's modulus of the main pipe and coupler	ρ	Density of the main pipe and coupler	
ui	Displacement of pipe i	σi	Normal stress of pipe i	
ti	Thickness of pipe i	L	Bonded length	
Note: i = 1,2; 1 represents the main pipe, and 2 represents the coupler.

For the force analysis of a loaded microelement, the inertial force terms are introduced by considering the dynamic load applied to the adhesive component and expressed as:(1) σ1(x+Δx,t)−σ1(x,t)+τ⋅2πR⋅Δx=m1∂2u1∂t2,

(2) σ2(x+Δx,t)−σ2(x,t)−τ⋅2πR⋅Δx=m2∂2u2∂t2.

Dividing both sides of the equation by AiΔx,we obtain equilibrium Equations (1), (2), where σ represents the normal stress generated in the pipe under tensile force, and τ denotes the shear stress in the adhesive layer. According to D'Alembert's principle, in dynamic problems, the inertial force terms represented by the acceleration also appear and are expressed as on the right-hand side of the equation. The following equilibrium equations are thus derived.(3) ∂σ1∂x−τ2πRA1=ρ∂2u1∂t2.

(4) ∂σ2∂x+τ2πRA2=ρ∂2u2∂t2.

The main pipe and coupler exhibited relative displacement, which was facilitated by the transfer of shear forces through the bonding layer.(5) τ=G(u1−u2)h.

Stress can be expressed in terms of displacement as(6) σ1=E∂u1∂x,

(7) σ2=E∂u2∂x.

By combining Equations (2), (3), (4), (5), (6), (7), the governing equation can be expressed as follows:(8) E∂2δ∂x2−2πRA2(1A1+1A2)δ=ρ∂2δ∂t2,

where R is the average radius of the pipe joint and determines using Equation (9).(9) R=12[(R1+t12)+(R2−t22)].

The relative displacement of the adhesive layer is given by(10) δ=u1−u2.

Due to the equilibrium conditions, we analyse only the right half of the structure. The main pipe on the right side is subjected to a force of magnitude F(t), and the sleeve pipe on the left side will also be subjected to a force of the same magnitude F(t). Thus, the boundary conditions are(11) ∂δ(0,t)∂x=−F(t)EA2,∂δ(L,t)∂x=F(t)EA1.

As shown in Fig. 4, under the high stress rate load at t = 0, F(0)=0,Thus, initial conditions are(12) δ(x,0)=0,∂δ(x,0)∂x=0.

2.2 Derivation of the analytical solution

Let's assume that δ(x,t)=V(x,t)+W(x,t), where W(x,t) satisfies the boundary conditions:(13) ∂W(0,t)∂x=−FEA2,

(14) ∂W(L,t)∂x=FEA1.

The minimalistic function that fulfils Equations (13), (14) can be expressed as(15) W(x,t)=FA1+FA22EA1A2Lx2−FEA2x.

By substituting the aforementioned equations into Equations (6), (7), (8), (9), (10), we derive the following expressions:(16) {∂2V∂t2=Eρ∂2V∂x2−2πRGρh(1A1+1A2)V+H∂V(0,t)∂x=∂V(L,t)∂x=0V(x,0)=−F(0)(A1+A2)2EA1A2Lx2+F(0)EA2x,∂V(x,0)∂t=−F′(0)(A1+A2)2EA1A2Lx2+F′(0)EA2x

Here,(17) H=(A1+A2)F(t)ρA1A2L+πRG(A1+A2)2F(t)ρhEA12A22Lx2+2πRG(A1+A2)F(t)ρhEA1A22Lx−(A1+A2)F″(t)2EA1A2Lx2+F′(t)EA2x

By setting V=V1+V2, Equation (16) is split into a homogeneous partial differential equation with nonhomogeneous initial conditions (Equation(18)) and a non-homogeneous partial differential equation with homogeneous initial conditions (Equation (19)).(18) {∂2V1∂t2=Eρ∂2V1∂x2−2πRGρh(1A1+1A2)V1∂V1(0,t)∂x=∂V1(L,t)∂x=0V1(x,0)=−F(0)(A1+A2)2EA1A2Lx2+F(0)EA2x,∂V1(x,0)∂t=−F′(0)(A1+A2)2EA1A2Lx2+F′(0)EA2x

(19) {∂2V2∂t2=Eρ∂2V2∂x2−2πRGρh(1A1+1A2)V2+H∂V2(0,t)∂x=∂V2(L,t)∂x=0V2(x,0)=0,∂V2(x,0)∂t=0

Equation (18) can be solved using the separation of variables. The resulting solution is as follows:(20) V1(x,t)=∑n=0∞(Ancosαt+Bnsinαt)cosnπLx,

whereAn=2L∫0L[−F(0)(A1+A2)2EA1A2Lx2+F(0)EA2x]cosnπxLdxBn=2αL∫0L[−F′(0)(A1+A2)2EA1A2Lx2+F′(0)EA2x]cosnπxLdxα2=2πRG(A1+A2)ρhA1A2+Eρn2π2L2

Equation (17) can be solved using the eigenfunction expansion method, considering that the solution is characterised by the following expression:(21) V2=∑n=0∞vn(t)cosnπLx.

We can also expand H(x,t) as follows.(22) H(x)=∑n=0∞hn(t)cosnπLx,

where(23) hn(t)=2L∫0LH(x,t)cosnπLxdx.

Substituting Equation (21) into the initial condition of Equation (15), we obtain(24) vn(0)=0,vn′(0)=0.

Hence, to ascertain the function of vn, it is necessary to solve the following problem:(25) {vn″(t)+α2vn(t)−hn(t)=0vn(0)=0,vn′(0)=0.

By employing the Laplace transform technique in Equation (26) and performing a Laplace transform with respect to the variable t at both ends, we derived the subsequent expression.(26) p2Vn(p)+α2Vn(p)=Hn(p).

Here, Hn(p) represents the Laplace transform of hn(t), and V(p) represents the Laplace transform of vn(t) and can be expressed as:(27) Vn(p)=Hn(p)p2+α2.

Further, taking the Laplace transform of Equation (26), we can obtain(28) vn(t)=1α∫0thn(τ)sin[α(t−τ)dτ]cosnπLx.

Thus, the derived expression for V2 is as follows(29) V2(x,t)=∑n=0∞{1α∫0thn(τ)sin[α(t−τ)]dτ}cosnπLx.

Based on the aforementioned analysis, the derivation of the expression for interface slippage revealed a tripartite composition.(30) δ(x,t)=W(x,t)+V1(x,t)+V2(x,t).

For the series form of Equation (30), numerical computation can ascertain that the calculated results converge to the same value when n≥10.

3 Numerical computation

3.1 Parameter selection

Considering σ˙=k×103Mpa/ms, where k is the amplification factor, and σ˙ denotes the strain rate. The specific values for the remaining parameters are provided in Table 2.Table 2 Dimensions and material properties for calculation.

Table 2Parameter	Parameter values	Parameter	Parameter values	
G	1.2 GPa	E	120 GPa	
t1	13.5 mm	t2	12.5 mm	
R1	150 mm	R2	163 mm	
ρ	2.81 g/cm3	h	0.5 mm	

3.2 Effect of different parameters on the interface slip and shear stress distribution curves

To gain a comprehensive understanding of the mechanical behaviour of bonded joints under dynamic loads, this study aims to analyse the impact of various parameters on the mechanical responses. These analyses provide valuable insights into the mechanical behaviour of bonded joints with different material compositions and structural configurations, serving as a fundamental theoretical framework for the future design and optimisation of such joints. The findings establish a solid theoretical basis for guiding the selection and optimisation of bonded joint designs in practical applications.

The first aspect of the analysis focused on examining the influence of different external dynamic loads on adhesive-bonded pipe joints, with particular emphasis on the effects of the stress rate and loading duration on the behaviour of the joints. These effects are discussed here in detail.(1) Fig. 5 depicts the relative displacement profiles and shear stress distributions of the bonded pipe joint under different loading durations, namely 0 μs,10 μs,20 μs,30 μs and 40 μs. Fig. 6 illustrates the variation in normal stresses experienced by the main pipe and coupler throughout the loading process, with solid lines representing the main pipe and dashed lines representing the coupler pipe. It is evident that a prolonged loading duration leads to an increase in the magnitude of both the displacement and shear stress. Moreover, within the adhesive region, it can be observed that the stress concentration is prominent near the edge of the coupler, signifying its vulnerability to potential failure initiation. Hence, meticulous attention should be devoted to this specific location during the joint design and operational phases.Fig. 5 Interfacial slip and shear stress of different loading durations.

Fig. 5

Fig. 6 Normal stress of the main pipe (solid line) and coupler pipe (dashed line) of different loading durations.

Fig. 6

(2) Fig. 7 illustrates the relative displacement profiles and shear stress distributions of the interface for various magnitudes of stress rates multiplied by the amplification factor (k). Fig. 8 shows the normal stresses experienced by the main and coupler pipes. It is evident that the magnitude of the stress rate significantly influences the distribution of both the displacement and shear stress at the interface. Therefore, the specific application and operating conditions of the pipe joint should be carefully considered in the design process. Moreover, the expected loads, loading durations, and intensity of potential loadings should be considered to ensure an appropriate joint design.Fig. 7 Interfacial slip and shear stress for different values of amplification factor (k).

Fig. 7

Fig. 8 Normal stress of the main pipe (solid line) and coupler pipe (dashed line) with different amplification factor (k) values.

Fig. 8

The bond length is a highly significant parameter in the design of pipe joints. In practical applications where the pipe dimensions have already been determined, the bond length determines the overall dimensions of the joint and exerts a profound influence on the entire pipe joint system. Therefore, it is one of the most critical parameters in the design process.(3) Fig. 9 shows the simulated curves of the relative interface displacement and shear stress obtained for the bond lengths of 10, 15, 20, and 25 mm. Fig. 10 shows the normal stress distributions experienced by the main and coupler pipes. As depicted in the figures, while maintaining the remaining parameters constant, it becomes apparent that an elongated bond length leads to diminished interface displacement and shear stress. The bond length, which is a crucial parameter, exerts a profound influence on the multitude of mechanical behaviours exhibited by the interface. Therefore, it is of paramount importance to meticulously contemplate and select the appropriate bond length during the design process.Fig. 9 Interfacial slip and shear stress for different bond lengths.

Fig. 9

Fig. 10 Normal stress of the main pipe (solid line) and coupler pipe (dashed line) for different bond lengths.

Fig. 10

The design of pipe joints, including their dimensions and bond length, is not solely determined by the applied load conditions but is also influenced by various contextual factors. The selection of adhesive materials and determination of bond thickness, on the contrary, offer more flexibility than other parameters, thereby providing significant convenience in the design and application of pipe joints. As the adhesive layer predominantly experiences shear stress, the distinct characteristics of different adhesive materials are manifested through variations in the shear modulus and bond thickness. Therefore, it is imperative to comprehensively understand the influence of these two parameters on the mechanical behaviour of the interface.(4) Fig. 11 shows the relative displacement and shear stress curves of the bonded interface in the pipe joint for various adhesive-layer thicknesses. It is evident that the relative displacement of the interface exhibits a positive correlation with increasing thickness of the adhesive layer, whereas the shear stress exhibits a decreasing trend across the entire bonded region. Consequently, when the internal shear stress within the pipe joint exceeds acceptable limits, this issue can be mitigated by increasing the adhesive-layer thickness. Conversely, if the relative displacement of the interface exceeds an acceptable level, it can be alleviated by reducing the adhesive-layer thickness. Fig. 12 shows the distribution of normal stress on the main and coupler pipes for different adhesive-layer thicknesses.Fig. 11 Interfacial slip and shear stress for different bond thicknesses.

Fig. 11

Fig. 12 Normal stress of the main pipe (solid line) and coupler pipe (dashed line) for different bond thicknesses.

Fig. 12

Fig. 13 depicts the relative interfacial displacement and shear stress for bond lengths (L) of 15 and 20 mm, accompanied by corresponding adhesive-layer thicknesses (h) of 0.5 and 0.7, respectively. Notably, as both the bonding length and adhesive-layer thickness increased by approximately one-third, the interfacial displacement experienced a 7 % increase, whereas the shear stress underwent a significant reduction of 23 %. This underscores the importance of considering the interdependence between the bond length and adhesive-layer thickness in practical design. If the objective is to diminish the joint shear stress by augmenting the adhesive-layer thickness, a simultaneous increase in the bond length may mitigate the risk of substantial interfacial slippage.(5) Fig. 14 illustrates the effect of varying the shear modulus of the adhesive on the relative displacement of the bonded interfaces. It can be observed that when the shear modulus of the adhesive increases, the relative displacement of the interface decreases. However, an excessively high shear modulus can result in increased shear stress. However, decreasing the shear modulus led to an increased relative displacement and a more uniform distribution of shear stress.

(6) The selection of materials for main and coupler pipes is a crucial consideration for pipe joints. In other words, the material parameters of the pipes can significantly influence the mechanical behaviour of the interface. Fig. 15 depicts the interface displacement, shear stress, and normal stress curves for different elastic moduli (E, 2E, 3E, and 4E) selected for the pipes. A lower elastic modulus can result in excessive interface displacement and shear stress, which is unfavourable for load transfer between pipes. Fig. 16 shows the stress curves of the main and coupler pipes.

Fig. 13 Interfacial slip and shear.

Fig. 13

Fig. 14 Interfacial slip and shear stress for different shear moduli of the adhesive layer.

Fig. 14

Fig. 15 Interfacial slip and shear stress for different elastic moduli of the main pipe and coupler pipe.

Fig. 15

Fig. 16 Normal stress in the main pipe and coupler pipe for different elastic moduli.

Fig. 16

Fig. 17 illustrates that the shear stress of adhesive layer computed using the approach outlined in this study closely approximates the finite element analysis results, thereby validating the efficacy of the theoretical model.Fig. 17 Shear stress of adhesive layer.

Fig. 17

4 Conclusion

In this study, a theoretical model is introduced for adhesive pipe joints, whereby the dynamic mechanical behaviour is analysed through the application of variational methods and Laplace transforms. Computational formula for the displacement and shear stress were derived, along with a comprehensive examination of the normal stress distribution on the main and coupler pipes.

An investigation was conducted using computational formula to examine the impact of different loading durations, loading rates, bond lengths, and adhesive-layer thicknesses on the interface displacement curve, shear stress curve, and normal stress curve of pipe joints. The findings of this comprehensive analysis provide a solid theoretical foundation for the design of pipe joints in practical scenarios.

In practical design considerations, careful attention must be paid to the operational context in which pipe joints are intended for use. This is because specific loading conditions exert a notable influence on the mechanical behaviour of the joints. Prioritising an accurate understanding of the usage scenarios and anticipated load profiles is imperative for guiding the subsequent joint-design process. By aligning the design parameters with the expected loads, the joints can be engineered to achieve optimised performance and ensure robustness in the intended application.

The loading duration and loading rate significantly impacted the slip and stress on the pipe joints. However, in practical operational domains, as the environmental conditions around pipe joints are often beyond control, the loading time and velocity of force application cannot be predetermined. Therefore, in the design of pipe joints, paramount importance is attributed to the controllable parameters—bond length and adhesive-layer thickness—as these factors have a substantial influence on the joint performance. Careful consideration should be given when selecting bond lengths to ensure an optimal balance. A prolonged bonding length demonstrates a mitigating effect on both the relative slippage and shear stress. Nevertheless, as the length increases beyond a certain threshold, the diminution becomes increasingly marginal. Concurrently, this elongation leads to a concomitant escalation in cost. To strike a balance between performance and cost-effectiveness, the bond length should be determined based on the specific application requirements.

When there is excessive shear stress within the adhesive layer, the internal shear stress can be mitigated by adjusting the adhesive layer thickness. However, increasing the thickness also leads to an increase in the relative slip. Thus, a comprehensive assessment must be conducted to select an appropriate thickness that accounts for both shear stress reduction and a controlled relative slip. In addition, based on theoretical model computations, concurrently augmenting the appropriate bonding length and adhesive-layer thickness yielded a discernibly significant reduction in the shear stress at the bonding interface. However, it increased the relative slippage only marginally. Therefore, in a practical design, the interplay between the bonding length and adhesive-layer thickness can be strategically harnessed.

Furthermore, in practical applications, the selection of suitable pipe materials should consider the overall usage scenario and loading conditions. The choice of adhesive used for bonding also plays a crucial role because different parameters have varying effects on the behaviour of the pipe joint. From the curves, it can be observed that during loading, the slip and stress peaks typically occurred at the ends of the sleeve. Therefore, special attention should be paid to the stress and slip at these positions during the design process.

Pipe joints inevitably encounter dynamic loads with high stress rates in actual working conditions, such as fluid impact within pipelines or the sudden opening and closing of mechanical devices like valves. These situations generate dynamic loads with high stress rates. Currently, there is a lack of theoretical understanding of pipe joints under dynamic loads. Therefore, this study has chosen pipe joints under these load conditions as the subject of research. The derived computational formulas and analysis of various parameters can be applied to the practical design optimisation and calculation of pipe joints. This paper presents a preliminary analysis of the mechanical performance of a fully symmetric pipe joint model under dynamic loading, proposing computational formulas and assessing the impact of various parameters on the joint. Future research could extend to the study of asymmetric pipe joints with varying thicknesses and materials of the main pipes, further investigating the mechanical behavior of adhesive pipe joints under dynamic loads.

Data availability statement

The data used to support the findings of this study are available from the corresponding author upon request. Data included in article/supp. material/referenced in article.

Funding

The authors gratefully acknowledge the financial support provided by the Science and Technology Scheme of Guangzhou City (no.201904010141 ).

CRediT authorship contribution statement

Zhenhao Lin: Writing – original draft, Software, Data curation, Conceptualization. Shanqing Li: Methodology, Investigation, Formal analysis.

Declaration of Competing interest

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
==== Refs
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