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Theoretical and experimental investigation on interference fit in electromagnetic riveting
International Journal of Mechanical Sciences 156 (2019) 261–271
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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Theoretical and experimental investigation on interference fit in electromagnetic riveting Xu Zhang a, Hao Jiang b, Tong Luo b, Lin Hu a, Guangyao Li b, Junjia Cui b,∗ a b
College of Automotive and Mechanical Engineering, Changsha University of Science and Technology, Changsha 410114, China State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, China
a r t i c l e
i n f o
Keywords: Electromagnetic riveting Interference-fit model Residual stress Stress wave
a b s t r a c t As an advanced interference-fit joining technology, electromagnetic riveting (EMR) has wide engineering application prospects in manufacturing and assembly fields. In this paper, a theoretical model on interference fit for EMR process is derived based on the stress wave theory and thick wall cylinder theory. EMR experiments are conducted to verify the analytical model. The compared results show that the analytical solutions agree well with experimental values. The interference-fit model synthetically considers many process parameters factors, not only the material properties and size of rivet, sheets and punch, but also the riveting force. The residual stress distribution, which is an important factor on fatigue life, can be predicted by this model. In addition, this can provide scientific guidance for the process parameter design and riveting setup optimization in engineering application.
1. Introduction In the transportation industry, the safety of aircraft and automobile is significantly important. After long-time service, fatigue cracks are easily produced in aircraft and car structures, especially at the joints. According to the previous reports [1], more than 60% of failures occur at the joints. To solve this issue, interference-fit joining technology was proposed in assembly of aircraft and automobile parts, and it had been increasingly applied in automobile and aerospace manufacturing industry [2,3]. Based on the advantage of improving the fatigue life of interferencefit joints, many studies have been conducted on interference-fit joining techniques [4]. The effect of interference-fit values on the fatigue properties of bolted joints in composite and hybrid composite/metal structures were respectively investigated by Wei et al. [5] and Li et al. [6]. It was found that the interference-fit could remarkedly improve the fatigue life of joints compared with non-interference joints. Specifically, for composite structures, the joints with 1.8% interference-fit values under higher cyclic stress and 3% interference-fit values under lower cyclic stress had better fatigue properties. For hybrid composite/metal structures, the joints with 2.1% interference-fit values under higher cyclic stress and 1.2% interference-fit values under lower cyclic stress had better fatigue properties. In addition, Skorupa et al. [4] comprehensively researched the effect of interference-fit values on the fatigue behavior of riveted joints in metal structures. The results showed that riveted
∗
joints with higher squeeze riveting forces had better fatigue properties. Higher squeeze riveting forces improved the interference-fit values. Consequently, the higher interference-fit values could not only prevent rivet from inclining under fatigue loading, but also cause a pretension stress around the hole wall. Abovementioned studies proved that the interference-fit joining techniques could effectively enhance joining strength. The riveting and bolting techniques were the most widely used in the interference-fit assembling processes. However, the interferencefit values were formed during traditional riveting techniques with hydraulic or direct hammers [7–9]. It was not only easy to cause damage on the hole wall, but also cause the uneven interference-fit values. Besides, traditional interference-fit bolting joints were relatively easy to loosen and required designated bolts, which would increase the cost. Therefore, more advanced joining techniques have been urgently needed to realize uniform interference-fit quality and lower costs. Electromagnetic riveting (EMR), also known as stress-wave riveting, is developed by electromagnetic forming technology [10] and traditional riveting process. As a relatively new joining technique, it has great technical advantages (high efficiency, fast speed and large loading forces) to ensure the uniformity of interference-fit values, which can significantly improve the fatigue life of riveted joints compared with traditional riveting process [11]. Moreover, it does not require designated rivets. EMR technique has been applied in aircraft manufacturing of Boeing and Airbus aeries products [12,13], and has wide application prospects in other manufacturing industries. Based on the even interference-fit advantage of EMR joints, many researchers had
Corresponding author. E-mail address: [email protected] (J. Cui).
https://doi.org/10.1016/j.ijmecsci.2019.04.002 Received 31 December 2018; Received in revised form 1 March 2019; Accepted 1 April 2019 Available online 4 April 2019 0020-7403/© 2019 Elsevier Ltd. All rights reserved.
X. Zhang, H. Jiang and T. Luo et al.
International Journal of Mechanical Sciences 156 (2019) 261–271
Fig. 1. Schematic diagram of the EMR process.
also studied the effect of interference-fit quality on EMR joints [14– 16]). Cao et al. [14] compared the interference-fit qualities between EMR and traditional riveting in composite joints by experiment investigations. The results showed that the interference-fit distribution of EMR joints was not only uniform, but also had higher average values. Zhang et al. [15] investigated riveted structures with multi-layer sheets through numerical simulations and experiments. It was found that EMR technique can also ensure relatively higher interference-fit values. The relative interference-fit values of EMR joints for three, four, five and seven-layer aluminum sheets could reach to 3.4%, 3.3%, 3.1% and 2.5%, respectively. In addition, Jiang et al. [16] experimentally studied the relationship between the interference- fit values and discharge energies. The results showed that the interference-fit values of EMR joints could be well controlled by discharge energies, and the interference-fit size increased with the increasing of discharge energy. During the electromagnetic riveting process, the radial expansion occurred when the rivet was axially compressed. Plastic deformations of rivet completed under the high-speed impact, and were similar to the cylinder upsetting process with drop hammer [17]. Wang et al. [18] the stress wave significantly influenced the plastic flow and deformed geometry during the dynamic upsetting process of the cylinder. Many studies showed that the impact compressive issues of cylinders could be solved using the elastic-plastic stress wave theory [19,20]. The hole wall of riveted sheets was squeezed and expanded by the rivet cylindrical surface. The plastic deformations of riveted sheets could be considered as thick-walled cylinder under internal pressure [21,22]. Interference-fit during electromagnetic riveting was formed by the mutual squeezing effect between hole wall and rivet. At present, studies on interference-fit quality of EMR joints were conducted by experimental measurements. And interference-fit values were measured under destroying the riveted samples. Nondestructive testing technique was difficult to use because interference-fit parts located inside samples. In addition, interference-fit quality was affected by many process parameters (such as loading velocity, materials properties and rivet size). Multiple measurements would waste materials and human resources. The purpose of this paper was to investigate the interference-fit model of EMR process through mathematical physics methods. The electromagnetic riveting process was completed within some milliseconds, causing that the riveting process belonged to dynamic impact loading. In this work, stress wave and thick-wall cylinder theories were comprehensively used to establish the interference-fit model of EMR joints. Some important process parameters affecting interference-fit quality were considered in this model. The boundary conditions required for solving model could be experimentally measured, and the analytical
model was finally verified by experiments. This model can be extended to other interference-fit processes formed by the dynamic impact. 2. An analytical model of interference fit 2.1. The principle of electromagnetic riveting process The schematic diagram of the EMR process is shown in Fig. 1. The EMR equipment can be divided into two parts: the setup (namely electromagnetic pulse generator) and riveting mold. The capacitor bank imbedded in the setup stores electric energies by a charging process. When the capacitors are charged to the preset energy value, closing the discharge switch makes these energies unleash to the flat spiral coil (placed in the riveting mold). A RLC oscillation circuit is constituted among resistance (R) of whole system, inductance (L) of coil and capacitance (C) of setup. Consequently, a strong pulse current flows into the coil and then a strong electromagnetic field is generated around it. Meanwhile, a reversed eddy current is induced in the driver plate (copper plate with high electric conductivity) closed the coil. The other electromagnetic field is also produced by the eddy current. The repulsion forces from the two reversed electromagnetic fields will drive the punch impact the rivet. The rivet shaft outside the hole is deformed into a driven head, locking the two sheets together. In addition, the rivet shaft inside the hole is expanded to form reference fit with hole walls. The existence of residual stress field in riveted structures can effectively improve the fatigue life and tensile strength of structures. The riveted joints are usually used to join structures which bear strong shear loads. Fig. 2 shows stress distributions of riveted sheets under the shear loading. If there is no residual stress (namely clearance fit), stress concentration occurs inside the riveted sheets under the action of external loads. And the maximum stress located on the edge of the holes, leading to significant decrease in fatigue life of riveted structures. When interference fit is formed between the rivet and hole wall, there is residual stress inside the riveted sheets (especially around the hole wall). The residual compressive stress can relieve tensile stress caused by the external loads and reduce the stress concentration. Thus, the fatigue life of riveted structures can be remarkably improved. Consequently, interference-fit values will directly affect distributions of residual stress, and then determine the fatigue life of riveted structures. According to the principle of EMR, it is a high-speed joining technology. The whole riveting process is employed under the impact loads and completed within some milliseconds. The impact loading causes that the effect of stress wave on interference fit is not neglected during the EMR process. 262
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International Journal of Mechanical Sciences 156 (2019) 261–271
Fig. 2. Stress distributions of riveted structures under the shear loading.
Fig. 3. The stress state in the deformation process of riveted sheets.
2.2. Relative interference-fit model
the physical equation under plane strain, it can be known that:
2.2.1. Stress state of riveted sheets The hole walls of riveted sheets are subjected to radial pressures of rivets during the riveting process. Consequently, the deformation of the riveted sheets can be simplified to deformation issue of thick cylinders under the inner pressures (𝜎). The simplified model of deformed sheets and stress state are shown in the Fig. 3. It is assumed that the inner and outer radius of the cylinder are Ri and Ro , respectively. The micro-element (ABNM) on the random radius (r) position is selected to stress analysis. The thickness of sheets is relatively smaller than outer radius of them, causing that the deformation of sheets can be regarded as plane stress. According to the theory of thick cylinders [23,24], the radial stress (𝜎 r ) and circumferential stress (𝜎 𝜃 ) on the random positions can be expressed as Eqs. (1) and (2) in the cylindrical polar coordinate system. ) ( 𝑅𝑖 2 𝑅𝑜 2 1 1 𝜎𝑟 = 𝜎 − (1) 𝑅𝑜 2 − 𝑅𝑖 2 𝑅𝑜 2 𝑟2
𝜀𝑟 =
𝜕 𝑢𝑟 𝜕𝑟
(3)
𝜀𝑟 =
( ) 1 − 𝜈2 𝜈 𝜎 𝜎𝑟 − 𝐸 1−𝜈 𝜃
(4)
𝜎𝜃 =
𝑅𝑖 2 𝑅𝑜 2 𝑅𝑜 2 − 𝑅𝑖 2
𝑢𝑟 =
1 1 + 𝑅𝑜 2 𝑟2
∫
) ( 𝑅𝑖 2 𝑅𝑜 2 1 + 𝜈 1 − 2𝜈 1 𝜎 − 𝑑𝑟 ∫ 𝑟2 𝑅𝑜 2 − 𝑅𝑖 2 𝐸 𝑅𝑜 2 ] [ 𝑅𝑖 2 𝑅𝑜 2 1 + 𝜈 (1 − 2𝜈) 1 = 𝜎 𝑟+ 𝑟 𝑅𝑜 2 − 𝑅𝑖 2 𝐸 𝑅𝑜 2
𝜀𝑟 𝑑𝑟 =
(5)
Substituting the interference-fit value (Δ = r − Ri = ur ) into the Eq. (5), the inner pressures (𝜎) on the hole walls can be calculated as: ( ) Δ Δ + 𝑅𝑖 𝑅𝑜 2 − 𝑅𝑖 2 𝐸 𝑟 𝐸 = [ ] [ ] ( ) 𝑅𝑖 2 𝑅𝑜 2 1 + 𝜈 (1−22𝜈) 𝑟2 + 1 𝑅𝑖 2 𝑅𝑜 2 1 + 𝜈 (1−22𝜈) Δ + 𝑅 2 + 1 𝑖 𝑅𝑜 𝑅𝑜 ( ) Δ 1 + 𝑅Δ 𝑅𝑜 2 − 𝑅𝑖 2 𝐸 𝑅𝑖 𝑖 (6) = [ ] ( )2 𝑅𝑖 2 𝑅𝑜 2 1 + 𝜈 (1−2𝜈) Δ 1 1 + + 2 2 𝑅 𝑅 𝑅
𝜎 = 𝑢𝑟
)
( 𝜎
Substituting the Eqs. (1) and (2), The radial displacement (ur ) can be obtained by solving the Eqs. (3) and (4).
(2)
𝑅𝑜 2 − 𝑅𝑖 2
𝑜
2.2.2. Interference-fit model under the elastic limit of riveted sheets Absolute interference-fit value (Δ) can be defined as Δ = r − Ri , in which the r − Ri is just the radial displacement of hole walls. Consequently, the calculation of interference-fit value can be changed into solution issue of this displacement. According to strain definition and
𝑖
𝑖
Consequently, taking the inner pressures (𝜎) into the Eqs. (1) and (2), the radial stress (𝜎 r ) and circumferential stress (𝜎 𝜃 ) can be further solved. The outer edge of sheets can be considered as infinity relatively to the diameter of holes. When the outer radius is infinity (namely 263
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International Journal of Mechanical Sciences 156 (2019) 261–271
Ro → ∞), the inner pressure, radial stress and circumferential stress can be further simplified as: ) ( Δ𝐸 Δ 𝜎= 1+ (7) 1+𝜈 𝑅𝑖
The other case: on the elastic deformation zone (r > rp ). The elastic radial stress and circumferential stress in this zone can be obtained by
( )( )2 𝑅𝑖 𝐸 Δ Δ 𝜎𝑟 = − 1+ 1 + 𝜈 𝑅𝑖 𝑅𝑖 𝑟
𝜎𝑒𝑟 = −
( )( )2 𝑅𝑖 𝐸 Δ Δ 𝜎𝜃 = 1+ 1 + 𝜈 𝑅𝑖 𝑅𝑖 𝑟
replacing Ri and 𝜎 with rp and −𝜎𝑝𝑟 (𝑟𝑝 ) =
(8)
(9)
𝜎𝑠 1 2 𝑟 2 𝑅𝑖 2
(10)
Δ𝑒𝑙 (1 + 𝜈)𝜎𝑠 = 𝑅𝑖 2𝐸 − (1 + 𝜈)𝜎𝑠
(11)
𝑢𝑟 =
𝑑𝑟
+
𝜎𝑝𝑟 − 𝜎𝑝𝜃 𝑟
=0
The radial stress (𝜎 pr ) and circumferential stress (𝜎 p𝜃 ) in the plastic deformation zone can be obtained by integrating the equilibrium differential equation and substituting boundary conditions (the radial stress 𝜎 pr = −𝜎 when the r = Δ + Ri ).
) (15)
𝑅𝑖 + Δ
𝜎𝑠 𝑅 𝑜 − 𝑅 𝑖 2 𝑅𝑜 2 2
)
(22)
𝑅0 →∞
𝑑 𝑢𝑝𝑟
(23)
Δ 𝑅𝑖
+1 𝑟𝑝 𝑅𝑖
⎞ 1 − ⎟+ 2⎟ ⎠
( 𝑟 )2 𝑝
𝑅𝑖 Δ 𝑅𝑖
⎫ (1 − 𝜈) ⎪ ⎬ +1 ⎪ ⎭𝑅𝑜 →∞
𝑖
if the radius (rp ) of plastic deformation zone is known. According to the Eq. (17), the radius (rp ) can be expressed as: ) ( 𝜎 1 𝑟𝑝 = 𝑅𝑖 + Δ exp( − )||𝑅𝑜 → ∞ 𝜎𝑠 2
(26)
Substituting the Eq. (26) into the Eq. (25), the relative interferencefit value can be described as: ( ) { } Δ Δ (1 + 𝜈) 2𝜎 𝜎 = 1+ 𝜎𝑠 (1 − 𝜈) exp( − 1) − (1 − 2𝜈) (27) 𝑅𝑖 𝑅𝑖 𝐸 𝜎𝑠 𝜎𝑠 Making 𝜉 =
2
(1+𝜈) 𝜎𝑠 { ( 1 − 𝐸
𝜈) exp( 2𝜎𝜎 − 1) − (1 − 2𝜈) 𝜎𝜎 }, the relative 𝑠
interference-fit value can be simplified as:
(17)
𝜉 Δ = 𝑅𝑖 1−𝜉
The plastic radial stress and circumferential stress on the boundary position between the plastic and elastic deformation zone (r=rp ) can be obtained: ( ) 𝑟 1 𝜎𝑝𝑟 = 𝜎𝑠 ln − (18) 𝑟𝑝 2 𝑅 →∞
𝑠
(28)
2.3. The plastic deformation and propagation of stress wave in the rivet 2.3.1. The propagation of stress wave in the rivet The impact process can be regarded as a collision of two bars (the punch and rivet) at the initial riveting moment. Fig. 4 shows the colliding schematic diagram of two bars on a common axis. The bar A with
0
( ) 𝑟 1 𝜎𝑝𝜃 = 𝜎𝑠 ln + 𝑟𝑝 2 𝑅
1+𝜈 𝐸𝑟
The yield stress (𝜎 s ) and Poisson’s ratio (v) of the riveted are usually given. Therefore, the relative interference-fit value ( 𝑅Δ ) can be obtained
Besides, the inner pressure can be obtained according to previous analysis. +
𝑟𝑝 2
(21) 𝑅0 →∞
(25)
The yield stress is far less than modulus (E) of elasticity. Consequently, it is known that: ( )2 2𝐸 ≈1 2𝐸 − (1 + 𝜈)𝜎𝑠
𝑟𝑝
𝑟
)2
⎧ ( )⎛ Δ (1 + 𝜈) ⎪ Δ ⎜ = 𝜎𝑠 ⎨(1 − 2𝜈) 1 + ln 𝑅𝑖 𝐸 𝑅𝑖 ⎜ ⎪ ⎝ ⎩
According to the continuity condition, the elastic deformations also occur outsider the plastic deformation zone of riveted sheets. So, there are the main cases to discuss. One case: on the boundary position between the plastic and elastic deformation zone (r = rp ). The plastic radial stress is equal to the inner pressures (𝜎 el ). ( )2 𝜎 𝑅 2 − 𝑅 𝑖 2 2 𝜎𝑠 𝑅 𝑜 2 − 𝑅 𝑖 2 2𝐸 𝜎𝑝𝑟 = −𝜎𝑒𝑙 = 𝑠 𝑜 𝑟 = (16) 2 2 2 2 𝑅𝑖 𝑅𝑜 2 2𝐸 − (1 + 𝜈)𝜎𝑠 𝑅𝑜
𝜎 = 𝜎𝑠 ln
𝑟𝑝
Substituting the conditions (r = Δ + Ri and upr = Δ) mentioned above, the relationship between absolute interference-fit value (Δ) and the radius of plastic deformation zone.
(14)
𝑟 𝑅𝑖 + Δ
(20) 𝑅0 →∞
Integrating this equation and substituting boundary conditions (the plastic radial displacement upr = ur when the r = rp ). It is obtained as: the relationship between radial displacement of hole walls and the radius of plastic deformation zone. { ( ) 𝑅 2 − 𝑅𝑖 2 (1 + 𝜈) 𝑟 𝑢𝑝𝑟 = 𝜎𝑠 (1 − 2𝜈)𝑟 ln − 𝑜 𝐸 𝑟𝑝 2𝑅𝑜 2 [ ( ) ]} 𝑟𝑝 2 𝑟𝑝 2 − 𝑅𝑖 2 + +1 +1 (24) (1 − 2𝜈) 2𝑟 𝑅𝑜 2
(12)
𝑟 𝑅𝑖 + Δ
)2
𝑢𝑝𝑟 + 𝑑𝑟 𝑟 ) (1 − 2𝜈)(1 + 𝜈) ( ) 1 𝑑 ( 𝑟𝑢𝑝𝑟 = 𝜎𝑝𝑟 + 𝜎𝑝𝜃 = 𝑟 𝑑𝑟 𝐸 ) (1 − 2𝜈)(1 + 𝜈) ( = 2𝜎𝑝𝑟 + 𝜎𝑠 𝐸 ( ) 𝑅𝑜 2 − 𝑅𝑖 2 (1 − 2𝜈)(1 + 𝜈) 𝑟 𝜎𝑠 2 ln − +1 = 𝐸 𝑟𝑝 𝑅𝑜 2
(13)
( 𝜎𝑝𝜃 = −𝜎 + 𝜎𝑠 1 + ln
2
(
𝑟𝑝 𝑟
𝜀𝑝𝑟 + 𝜀𝑝𝜃 =
𝜎𝑝𝜃 − 𝜎𝑝𝑟 = 𝜎𝑠
𝜎𝑝𝑟 = −𝜎 + 𝜎𝑠 ln
𝑠
(
respectively.
According to strain definition and the physical equation under plane strain, the following Eq. (23) can be obtained by combined he plastic radial stress and circumferential stress.
2.2.3. Interference-fit model the radius (rp ) of plastic deformation zone of riveted sheets The riveted sheets will have plastic deformations when inner pressure exceeds the elastic limit (𝜎 el ). The plastic deformation zone is approximately in the state of plane strain (𝜏 𝜃r = 0). Thus, the equilibrium differential equation and yield criterion are as follows: 𝑑 𝜎𝑝𝑟
𝜎𝑠 2 (𝜎
𝜎𝑒𝑟 =
According to Tresca yield criterion (𝜎 𝜃 − 𝜎 r = 𝜎 s ) [25], the inner pressures (𝜎 el ) and interference-fit value (Δel ) under elastic limit can be calculated by substituting Eqs. (8) and (9). 𝜎𝑒𝑙 =
𝜎𝑠 2
𝜎𝑠 𝑅 𝑜 2 − 𝑅 𝑖 2 , 2 𝑅𝑜 2
(19) 0 →∞
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International Journal of Mechanical Sciences 156 (2019) 261–271
Fig. 5 shows the simplified model of the EMR process. The riveting model is regarded as the process which a bar (punch) with the velocity of v1 and length of l1 impacts a fixed bar (rivet) with length of l2 . l1 and l2 represent the length of punch and rivet shaft, respectively. Firstly, according to the principle of momentum conservation, the velocity of the punch can be calculated by Eq. (33). 𝑣1 =
∫ 𝐹 𝑑𝑡 𝑀
(33)
where M and F represent the mass of the punch and the riveting force, respectively. Subsequently, the punch can be regarded as a rigid rod during the EMR process. And the punch impacts the rivet at the velocity of v1 . So its wave impedance (𝜌C)1 is infinite. The rivet is stationary, causing that its velocity is 0 m/s. Substituting the above values into Eq. (32), the stress and velocity of the impact surface at the impact moment are expressed as following: } 𝜎= −(𝜌𝐶 )2 𝑣1 (34) 𝑣 = 𝑣1 where (𝜌C)2 is the wave impedance of the rivet. In the subsequent rivet deformation process, the velocity of the impact surface is constantly changing under the action of stress wave. Fig. 6 presents the schematic diagram of the stress wave propagation in the rivet and punch. Stress wave mainly includes the elastic and plastic waves. Elastic wave has faster propagation speed in the rivet, so it first reaches the fix end and then reflects back. According to the reflection law [20] of stress wave, the particle velocity becomes zero and the stress doubles when elastic wave reaches the fixed end. As for the punch (rigid materials), it has no deformations during the riveting process. Therefore, it is assumed that only elastic wave exists in the punch. The elastic wave bounces off the free end of the punch and its speed will be doubled. The speed changing value and time for elastic wave in a round trip in the punch can be calculated by:
Fig. 4. The collision schematic of two bars.
the speed of VA strikes the bar B with speed of VB . At the moment of striking, two stresses are generated at the impact surface of bar A and B, respectively. The stresses in the form of waves with the speeds of CA and CB propagate to the other end. According to the continuity requirement, the stress 𝜎 and velocity v of bar A and B at the impact surface are identical. Based on the stress wave theory reported by Wang [26], the basic characteristic line differential equation and its compatibility relation can be expressed as: 𝑑𝑥 = 𝐶𝑑𝑡
(29)
𝑑 𝜎 = ±𝜌𝐶𝑑 𝑣
(30)
2(𝜌𝐶 )2 𝑣1 ⎫ Δ𝑣 = ( ) 𝜌𝐶𝑒 1 ⎪ ⎬ 2𝑙1 ⎪ Δ𝑡 = ⎭ 𝐶𝑒
So according to the above characteristic line compatibility conditions, the relationship between stress 𝜎 and velocity v is depicted as follows: ( ) ( ) 𝜎 = (𝜌𝐶 )𝐴 𝑣 − 𝑣𝐴 = (𝜌𝐶 )𝐵 𝑣𝐵 − 𝑣 (31)
where Ce is the spread speed of the elastic wave in the punch, and l1 is the length of the punch. Then the elastic wave spreads back and forth n times in the punch until the punch speed drops to zero. Thus, n is expressed as:
where (𝜌C)A and (𝜌C)B are the wave impedance of bar A and B, respectively. 𝜌 and C respectively represent the materials density and stress wave velocity. Further, the stress 𝜎 and velocity v could be obtained by Eq. (32): 𝑣𝐵 − 𝑣𝐴 ⎫ 1∕(𝜌𝐶 )𝐴 + 1∕(𝜌𝐶 )𝐵 ⎪ (𝜌𝐶 )𝐴 𝑣𝐴 + (𝜌𝐶 )𝐵 𝑣𝐵 ⎬ 𝑣= ⎪ (𝜌𝐶 )𝐴 + (𝜌𝐶 )𝐵 ⎭
(35)
𝑛=
𝑣1 Δ𝑣
(36)
Therefore, the collision time between the punch and rivet is calculated by:
𝜎=
(32) 𝑡 = 𝑛Δ𝑡 =
𝑙 1 𝜌1 (𝜌𝐶 )2
(37) Fig. 5. The simplified model of the EMR process.
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International Journal of Mechanical Sciences 156 (2019) 261–271
Fig. 6. Schematic diagram of the stress wave propagation in the rivet and punch.
as: 𝜕𝑢 ⎧ ⎪𝜀𝑟 = 𝜕𝑟 ⎨ 𝑢 ⎪ 𝜀𝜃 = ⎩ 𝑟
According to the Hooke’s law, the relationship between the stress and strain is described as: ( ) ⎧ 𝜀𝑟 = 1 𝜎𝑟 − 𝜇𝜎𝜃 𝐸 ⎪ ) ⎪ 1( (41) 𝜎 − 𝜇𝜎𝑟 ⎨𝜀 𝜃 = 𝐸 𝜃 ⎪ 1 ⎪ 𝛾𝑟𝜃 = 𝜏𝑟𝜃 ⎩ 𝐺 where 𝜇 is the Poisson’s ratio. The radial deformation of rivet is caused by the compression of rivet material. As the deformation behavior of rivet is relatively complex, the rule of radial displacement is first considered. Assuming that the cylinder is subjected to a uniform internal pressure p. The inner diameter is a0 and the outer diameter is b0 . The outer surface of the cylinder is free surface, and the pressure is zero. Moreover, rivet deformation follows the principle of volume invariance:
Fig. 7. Cross section of the rivet and the stress state of its element.
Based on the previous study [27], it could be seen that velocity value of the punch almost changes linearly. Consequently, the displacement and average stress at the impact surface are respectively expressed as: 𝑣1 𝑡 𝑙 𝜌 𝑣 ⎫ =− 1 1 1⎪ 2 2(𝜌𝐶 )2 (𝜌𝐶 )2 𝑣1 ⎬ ⎪ 𝜎𝑎𝑣𝑔 = − ⎭ 2
𝑈0 =
(38)
𝜕 𝑢𝑟 𝑢𝑟 𝜕 𝑢𝑧 + + =0 (42) 𝜕𝑟 𝑟 𝜕𝑧 The coordination equation is obtained by combining geometric function. 𝑑 𝜀𝜃 𝜀 − 𝜀𝑟 + 𝜃 =0 (43) 𝑑𝑟 𝑟 As seen from the above equation, there are two unknowns. During the process of rivet deformation, the stress and strain of the inner particle of the cylinder are affected by external forces. Consequently, the boundary conditions are introduced to solve the problem. The boundary conditions of the inner and outer diameter of cylinder are expressed as:
2.3.2. Plastic deformation of rivet The deformation of the rivet is caused by the axial impact riveting force. During riveting process, a corresponding positive stress is generated in the radial direction due to the Poisson ratio effect. Moreover, the upsetting of the rivet is directly related to radial stress. As the radical displacement of the rivet is caused by Poisson’s ratio effect, it can also be regarded as the deformation of the cylinder caused by the inner pressure. Based on the thick cylinder theory, it is assumed that the rivet is a thick-walled cylinder with an infinitely small inner diameter. Fig. 7 shows the schematic of cross-section of the rivet and the stress state of its element. The section is symmetric and a sector element ABCD is selected for analysis. It is supposed that the axial thickness of the rivet is 1, then the equilibrium differential equation of the element is expressed as: 𝜎 − 𝜎𝜃 𝜕 𝜎𝑟 1 𝜕 𝜏𝑟𝜃 + 𝑟 + + 𝑓𝑟 = 0 𝜕𝑟 𝑟 𝜕𝜃 𝑟
(40)
{ 𝜎𝑟 =
( ) −𝑝 𝑟 = 𝑎0 ( ) 0 𝑟 = 𝑏0
(44)
The axial end face of cylinder is under the pressure of T, so the boundary condition of end face of cylinder body is expressed as: 𝑏0
𝑇 = 2𝜋 ∫ 𝜎𝑧 𝑟𝑑𝑟 𝑎0
(39)
(45)
Combining the equilibrium equation Eq. (39) and stress-strain relationship Eq. (41) of elastic deformation, the stress coordination equation is obtained as: ( ) 𝑑 𝜎𝑟 + 𝜎𝜃 𝑑𝜎 −𝜇 𝑟 =0 (46) 𝑑𝑟 𝑑𝑟
where 𝜎 r and 𝜎 𝜃 represent positive stress along radial and tangential directions, respectively. 𝜏 r𝜃 and fr represent the tangential stress and body force, respectively. The radial deformation of the rivet is symmetric, so the annular displacement is zero. The geometric equation is expressed 266
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So the radical displacement for elastic deformation is obtained by the geometric function Eq. (40) as: ] ( ) [ 𝑏 2 1+𝜇 𝑢𝑟 = 𝑝̄ (1 − 2𝜇)𝑟 + 0 − 𝜇𝜀0 𝑟 (47) 𝐸 𝑟
It can be found that the relationship between the A1 and C1 is obtained. Subsequently, the axial and radial displacements are substituted into the basic geometric equation as follows: 𝜕 𝑢𝑧 ⎧ 2 ⎪𝜀𝑧 = 𝜕𝑧 = −2𝐴1 𝑧 − 2𝐶1 ⎪ 𝜕 𝑢𝑟 ⎪ 2 ⎪𝜀𝑟 = 𝜕𝑟 = 𝐴1 𝑧 + 𝐶1 ⎪ 𝑢 ⎨𝜀 𝜃 = 𝑟 = 𝐴 1 𝑧 2 + 𝐶 1 𝑟 ⎪ ⎪ 𝜕𝑢 𝜕𝑢 ⎪𝛾𝑧𝑟 = 𝑟 + 𝑧 = 2𝐴1 𝑧𝑟 𝜕𝑧 𝜕𝑟 ⎪ ⎪𝛾 = 𝛾 = 0 𝑟𝜃 𝜃𝑧 ⎩
During the process of rivet deformation, the area in inner diameter first enters the plastic stage. As for plastic region, the stress should satisfy both the yield condition and the equilibrium equation. When r = a0 , radial stress is maximized. So the ultimate elastic pressure is calculated as: ( ) 𝜎 𝑎 2 𝑝𝑒 = 𝑠 1 − 0 (48) 2 𝑏0 2 If the internal stress is less than the ultimate elastic pressure, there is only elastic deformation inside the cylinder. While the internal stress is greater than the ultimate pressure, plastic deformation in the cylinder occurs. There has plastic zone and elastic zone interface in the cylinder (the plastic zone is inside). Assuming that the diameter is c and stress is continuous at the interface of plastic zone and elastic zone, the plastic radial stress at the particle point of r = c can be taken as the internal pressure of the elastic zone. So the stress in the plastic region can be calculated as: ( ) ⎧ 𝜎𝑠 𝑐 2 𝑏0 2 ⎪ 𝜎𝑟 = 2 𝑏 0 2 1 − 𝑟 2 ( ) (49) ⎨ 𝜎 𝑐2 𝑏 2 ⎪𝜎𝜃 = 2𝑏𝑠 2 1 + 𝑟02 0 ⎩
The simplified rivet constitutive equation is introduced into the above geometric equation: ( ) ( ) 𝜎𝑧 𝑧 = 𝑙2 = 𝐸1 −2𝐴1 𝑧2 − 2𝐶1
( ) 𝜎𝑧 𝑧 = 𝑙2 = 𝜎𝑎𝑣𝑔
⎧ 3𝜎𝑎𝑣𝑔 3𝑈0 ⎪𝐴 = − ⎪ 1 4𝐸 𝑙 2 4𝑙23 1 2 ⎨ 𝜎𝑎𝑣𝑔 3 𝑈 0 ⎪𝐶 = − ⎪ 1 4𝐸1 4𝑙2 ⎩
Substituting the above results into the Eqs. (43) and (45), the axial and radial displacements on the outside surface of the rivet are obtained as follows: (( ) ( ) ) 𝑙1 𝜌1 3𝑙1 𝜌1 (𝜌𝐶 )2 ⎧𝑢 = 3 + (𝜌𝐶 )2 − − 𝑧 𝑧 ∫ 𝐹 𝑑𝑡 2 4𝐸1 𝑀 4𝑀 𝑙2 (𝜌𝐶 )2 ⎪ 𝑧 4𝑀𝑙23 (𝜌𝐶 )2 (( 4𝐸1 𝑀𝑙2 ) (62) ( )) ⎨ 3𝑙1 𝜌1 3(𝜌𝐶 )2 3𝑙1 𝜌1 (𝜌𝐶 )2 2+ ⎪ 𝑢𝑟 = − 𝑧 − 𝑑𝑡 𝑟𝐹 ∫ 8𝑀 𝑙2 (𝜌𝐶 )2 8𝐸1 𝑀 8𝐸1 𝑀𝑙22 8𝑀𝑙23 (𝜌𝐶 )2 ⎩
1 + 𝑁 (𝑧)𝑟 (51) 𝑟 where z and r are the axial and radial displacement, respectively. M(z) and N(z) are the expressions associated with z. When the rivet is deformed, the particles at the center axis position do not shift in the radical direction. When r = 0, ur = 0 and M(z) = 0, the actual deformation results of rivet deformation are approximately parabola at the free boundary [28]. Based on the upper bound methods [29–31], it is setting the expression N(z) as: 𝑢𝑟 (𝑧, 𝑟) = 𝑀 (𝑧)
The corresponding axial stress 𝜎 z and radial stress 𝜎 r on the outside surface of the rivet are expressed as follows: ( ) ( ) ⎧𝜎 = 3(𝜌𝐶 )2 − 3𝐸1 𝑙1 𝜌1 𝑧2 + (𝜌𝐶 )2 − 3𝐸1 𝑙1 𝜌1 ∫ 𝐹 𝑑𝑡 𝑧 2 3 4𝑀 4𝑀 𝑙2 (𝜌𝐶 )2 ⎪ 4𝑀𝑙2 4𝑀𝑙2 (𝜌𝐶 )2 (( ) (63) ( )) ⎨ 3𝐸1 𝑙1 𝜌1 3(𝜌𝐶 )2 3𝐸 𝑙 𝜌 (𝜌𝐶 ) ⎪𝜎 𝑟 = − 𝑧2 + 8𝑀 𝑙1 (1𝜌𝐶1) − 8𝑀2 ∫ 𝐹 𝑑𝑡 3 2 8𝑀𝑙2 8𝑀𝑙2 (𝜌𝐶 )2 ⎩ 2 2
(52)
It is known that deformations of riveted sheets occur under the radial compressions of the rivet. Consequently, the radial stress (𝜎 r ) of the rivet is just inner pressure (𝜎) during the deformation process of hole walls. Consequently, the equation 𝜉 in the relative interference-fit model Eq. (28) can be changed into:
where A1 , B1 and C1 are the constant. It is assumed that the radial displacement near the driven head is 0 mm (z = 0 mm), as shown in Fig. 3. Moreover, due to the symmetry of the rivet, B1 = 0. The radical displacement function can be simplified as: ( ) 𝑢𝑟 (𝑧, 𝑟) = 𝐴1 𝑧2 + 𝐶1 𝑟 (53)
{ } ⎧𝜉 = (1+𝜈) 𝜎 (1 − 𝜈) exp( 2𝜎 − 1) − (1 − 2𝜈) 𝜎 𝑠 𝐸 𝜎 𝜎 ⎪ 𝑠) 𝑠 (( ( )) ⎨ 3𝐸1 𝑙1 𝜌1 3(𝜌𝐶 )2 3𝐸 𝑙 𝜌 (𝜌𝐶 ) − 𝑧2 + 8𝑀 𝑙1 (1𝜌𝐶1) − 8𝑀2 ∫ 𝐹 𝑑𝑡 ⎪𝜎 = 3 2 8𝑀𝑙2 8𝑀𝑙2 (𝜌𝐶 )2 2 2 ⎩
According to the rivet volume invariant principle: 𝜕 𝑢𝑟 𝑢𝑟 𝜕 𝑢𝑧 + + =0 𝜕𝑟 𝑟 𝜕𝑧 The axial displacement function can be obtained as:
(54)
Substituting to the Eq. (55), it can be calculated as: 3𝑈0 + 6𝐶1 𝑙2 2𝑙23
(64)
where v and 𝜎 s is the Poisson’s ratio and yield strength of the riveted sheets, respectively. The E is the elasticity modulus of the riveted sheets. The M is the weight of the punch. The 𝜌1 and l1 is the density and length of the punch, respectively. The E1 and (𝜌C)2 is the elasticity modulus and wave impedance of the rivet, respectively. The l2 is the density and length of the rivet shaft. The F means the impact forces and z represents the coordinate change (0 ∼ l2 ) along the rivet shaft. It can be found that the relative interference-fit model of EMR joints is related to many factors, not only the material properties and size of rivet, sheets and punch, but also the riveting force. These parameters can be obtained by actual measurements.
2 𝑢𝑧 (𝑧, 𝑟) = − 𝐴1 𝑧3 − 2𝐶1 𝑧 (55) 3 At the impact surface z = l2 (l2 represents the length of screw rod as shown in Fig. 5), the displacement of the punch is U0 . So the boundary condition is: ( ) 𝑢𝑧 𝑧 = 𝑙2 = 𝑈0 (56)
𝐴1 = −
(61)
(50)
where c1 is the constant. Based on the above analysis, no matter according to the radial displacement expression in the plastic zone or elastic zone, the radical displacement function can be expressed as:
Δ = 𝜀𝑟 + 𝜀𝜃 + 𝜀𝑧 =
(60)
The A1 and C1 can be calculated by the following:
Further, the radical displacement in the plastic region is expressed
𝑁 (𝑧) = 𝐴1 𝑧2 + 𝐵1 𝑧 + 𝐶1
(59)
where E1 is the elasticity modulus of rivet. According to the boundary condition:
as:
[ ] 𝑐 (1 − 2𝜇)(1 + 𝜇) 𝑟 𝑐2 𝑟 𝑟 𝑢𝑟 = 𝜎𝑠 + 𝑟 ln − − 𝜇𝜀𝑧 𝑟 + 1 𝐸 2 𝑏0 2 𝑐 2 𝑟
(58)
(57)
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Fig. 8. The measured equipment of riveting forces: (a) whole composition, (b) local piezoelectric sensor. Fig. 9. The measured and analyzed results of the riveting forces: (a) the measured results under the varying discharge energies, (b) the integral of riveting force under the discharge energy of 4.7 kJ.
3. Results and discussion
In addition, the main material parameters are shown in Table 1. The compared EMR experiments were conducted with 6082-T6 aluminum alloy sheets and 2A10 aluminum alloy rivets. The thickness of the sheets is 4 mm. The diameter and length of the rivets are 5 mm and 14 mm, respectively. According to the universal riveting standard [28], the diameter of riveted hole is set to 5.1 mm. The punch in the electromagnetic equipment was manufactured by 45# steel. Its length and mass are 150 mm and 7 kg, respectively. The propagation velocity of stress wave in aluminum alloy (Al) is 5300 m/s. To obtain the qualified riveted joints, the discharge energy of 4.7 kJ was used to conduct the EMR experiments. Fig. 10 shows the riveted specimen. Subsequently, seven points were uniformly selected on the rivet shaft to measure the diameter after the riveting. The end surface of manufactured head is regarded as the axial distance z = 0 mm. The rivet diameter was measured by the Vernier caliper with accuracy of 0.01 mm. In order to avoid the accidental error, five-time measurements for every position are averaged to use the validation analysis. It can be observed from Fig. 9(b) that the integral of riveting force is 15.17 under the discharge energy of 4.7 kJ. The above results are substituted into Eq. (64) and the theoretical results of interference-fit model are obtained. The results of actual measurements are shown in Table 2. The comparison results between experimental and theoretical values are presented in Fig. 11. During the high-speed EMR process, the axial compression stress wave of rivets propagates from the loading
3.1. Model solution and validation The riveting force should be firstly measured before calculating the interference-fit model. Fig. 8 shows the measured equipment. It mainly consists of piezoelectric sensor, charge amplifier and oscilloscope. When the punch impacts the piezoelectric sensor, the sensor is pressurized to produce a charge signal. The charge signal is amplified and changed into a voltage signal by the charge amplifier. Finally, the measured results are transmitted and displayed on the oscilloscope. Because there is a strong electromagnetic field around the coil (riveting equipment) during the EMR process, which would produce electromagnetic waves. The charge amplifier and the oscilloscope should keep enough distance with riveting equipment so as to affect the operation of charge amplifier and oscilloscope. Fig. 9 shows the measured results of the riveting forces using the varying discharge energies. It can be seen that the peak force increases with the increasing of discharge energies. The integral of riveting force can be described by the Eq. (65), and the area under the curves can be obtained by the drawing software. ∫ 𝐹 𝑑𝑡 = Area
(65)
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Table 1 Material parameters of critical components. Components Punch Sheet Rivet
Density (kg/m3 )
Materials
3
45# steel 6082 T6 Al 2A10 Al
7.5 × 10 2.7 × 103 2.7 × 103
Elasticity modulus (Pa) 11
2.0 × 10 7.2 × 1010 7.2 × 1010
Poisson’s ratio
Yield strength (Pa)
0.30 0.35 0.35
3.55 × 108 2.6 × 108 2.5 × 108
Table 2 Experiment results of interference-fit values and rivet diameter after riveting. Axial distance z (mm)
Measured diameters of rivets (mm)
Average interference-fit values (%)
1 2 3 4 5 6 7 8
(5.17 + 5.17 + 5.18 + 5.18 + 5.18) / 5 = 5.176 (5.17 + 5.17 + 5.18 + 5.18 + 5.19) / 5 = 5.178 (5.17 + 5.17 + 5.18 + 5.19 + 5.19) / 5 = 5.180 (5.17 + 5.19 + 5.19 + 5.19 + 5.19) / 5 = 5.185 (5.19 + 5.19 + 5.20 + 5.21 + 5.21) / 5 = 5.200 (5.20 + 5.22 + 5.22 + 5.22 + 5.24) / 5 = 5.220 (5.25 + 5.25 + 5.25 + 5.25 + 5.25) / 5 = 5.250 (5.26 + 5.27 + 5.27 + 5.27 + 5.28) / 5 = 5.270
1.49 1.53 1.57 1.69 1.96 2.35 2.94 3.33
ably predict the interference-fit value of EMR joints. This interference-fit model can also guide the engineering design of EMR joints. 3.2. The residual stress distributions on the riveted sheeted According to the Eq. (11), the relative interference-fit value under the limit elasticity was 0.24% by substituting the property parameters of riveted sheets. When the interference-fit values are above 0.24%, the hole wall of the riveted sheets will have plastic deformations. It can be seen in Fig. 11 that interference-fit values of all measured positions are over 0.24%. Consequently, plastic deformations of hole walls occur along the whole thickness direction of riveted sheets. Because the outer boundary of sheets is far bigger than the radius of prefabricated holes, the size (rP ) of plastic deformation areas can be obtained by Eq. (25). The calculated results of plastic areas are shown in Fig. 12. It can be seen that the deformation degree of hole walls is closely related to interferencefit values. The size of plastic deformation areas is the larger under the bigger interference-fit value. In addition, the changing law of the size along the thickness of the riveted sheets is the same as the interferencefit distribution. Assuming that the outer boundary of riveted sheets is infinity, the Eqs. (18) and (19) can be changed into the following Eqs. (66) and (67). The residual stress distributions (including radial stress 𝜎 pr and circumferential stress 𝜎 p𝜃 ) can be calculated by substituting the radius (rp ) of plastic deformation areas, as shown in Fig. 13. The radius (Ri ) of holes is 5.1 mm before the riveting in this paper. For the thickness direction of riveted sheets, both radial stress and circumferential stress have the same changing trend. These stresses decrease from the driven head to the manufactured head. For the radial direction of riveted sheets, the radial compression stress (𝜎 pr ) decrease with the increase of radial positions (r). However, the circumferential tensile stress (𝜎 p𝜃 ) increase with the increase of radial positions (r). The riveted sheets is assumed as a plane strain state during the derivation process of interference-fit model, leading to a contrary tendency between radial stress and circumferential stress. ( ) 𝜎𝑝𝑟 𝑟 1 = ln − (66) 𝜎𝑠 𝑟𝑝 2 𝑅 →∞
Fig. 10. The EMR riveted specimen.
Fig. 11. The comparison between experimental and theoretical results of interference-fit values.
end to the other end. The upsetting deformation preferentially occurs in the position close the loading, and the position preferentially contact with the hole wall. Consequently, the interference fit is also gradually formed from the loading end to the other end. The partial energies in the preferential positions are transferred to the hole wall, leading to gradual energy loss from the loading end to the other end. It can be seen that the interference-fit values of the riveted joints gradually increase with the axial distance (from manufactured head to driven head). The variation trend of interference-fit values is consistent with the previous studies [14,15]. In general, the theoretical values agree well with experimental values. Therefore, the analytical model is accurate and can commend-
0
𝜎𝑝𝜃 𝜎𝑠
( ) 𝑟 1 = ln + 𝑟𝑝 2 𝑅
(67) 0 →∞
3.3. The effect of process parameters on interference-fit values The interference-fit model (Eq. (64)) shows that the process parameters affecting this model include riveting forces, material properties and structure dimensions of rivets, riveted sheets and punch. The material 269
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Fig. 12. The size of plastic deformation areas for riveted sheets: (a) the schematic diagram of deformation areas, (b) the radius values of plastic area.
Fig. 13. The residual stress distribution around hole walls of riveted sheets: (a) radial stress, (b) circumferential stress. Fig. 14. The relationship between parameters and interference-fit values: (a) the rivet shaft effect, (b) discharge energy effect.
properties and structure dimensions of the punch have no choice if the existing EMR equipment is used. In addition, the material properties of rivets and riveted sheets are also given before the riveting. Consequently, the length of rivet shaft and riveting forces can usually be adjusted according to the joined structures. Fig. 14 shows the relationship between the two parameters and relative interference-fit values. In this paper, the rivets with the diamater of 5 mm are used as an example. In order to insure the dismension of driven head, the rivet shaft must ex-
ceed 1.2 × 5 mm than the total thickness of all riveted sheets before the riveting pocess. Consequently, the length of rivet shaft increases as total thickness of all riveted sheets increases. It can be seen in Fig. 14(a) that the whole relative interference-fit values decrease with the increase of length of rivet shaft. The rivet shaft with a larger length-diameter ratio can be regarded as a slender rod. For the slender rod, stress wave effect is very signifanctly and can be considered as one-dimension stress. Due to transverse dispersion of stress wave, plastic deformations maily
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International Journal of Mechanical Sciences 156 (2019) 261–271
concentrate in the part closing the loading end of rivet. Therefore, The rivet shafts with the larger length have the smaller relative interferencefit values. Only riveting forces can be adjusted to achieve the thicksheet structure with enough interference-fit values. The EMR process with higer discharge energy has larger riveting forces. The precise control of discharge energy is one advantage of EMR. Fig. 14(b) demonstrates increasing the discharge energies can remarkably improve the interference-fit values. Moveover, EMR riveted sheets prepared under high discharge energy have relatively poorer uniform interference-fit. However, the increase of rivet shaft can improve the uniform, which can just compensate for this shortage.
[5] Wei JC, Jiao GQ, Jia PR, Huang T. The effect of interference fit size on the fatigue life of bolted joints in composite laminates. Compos Part B 2013;53:62–8. [6] Li J, Zhang KF, Li Y, Liu P, Xia JJ. Influence of interference-fit size on bearing fatigue response of single-lap carbon fiber reinforced polymer/Ti alloy bolted joints. Tribol Int 2016;93:151–62. [7] Manes A, Giglio M, Vigano F. Effect of riveting process parameters on the local stress field of a T-joint. Int J Mech Sci 2011;53:1039–49. [8] Ofsthun M. When fatigue quality enhancers do not enhance fatigue quality. Int J Fatigue 2003;25:1223–8. [9] Abe Y, Mori K, Kato T. Joining of high strength steel and aluminum alloy sheets by mechanical clinching with dies for control of metal flow. J Mater Process Technol 2012;212:884–9. [10] Psyk V, Risch D, Kinsey BL, Tekkaya AE, Kleiner M. Electromagnetic forming-a review. J Mater Process Technol 2011;211:787–829. [11] Li GY, Jiang H, Zhang X, Cui JJ. Mechanical properties and fatigue behavior of electromagnetic riveted lap joints influenced by shear loading. J Manuf Process 2017;26:226–39. [12] Huffer BW. HH54 rugged and reliable handheld EMR. Aerospace technology conference and exposition; 2009. [13] Hartmann J, Brown T. Integration and qualification of the HH500 hand operated electromagnetic riveting system on the 747 Section 11. Wichita: SAE aerofast conference; 1993. [14] Cao ZQ, Cardew-Hall M. Interference-fit riveting technique in fiber composite laminates. Aerosp Sci Technol 2006;10:327–30. [15] Zhang X, Zhang MY, Sun LQ, Li CF. Numerical simulation and experimental investigations on TA1 titanium alloy rivet in electromagnetic riveting. Arch Civ Mech Eng 2018;18:887–901. [16] Jiang H, Li GY, Zhang X, Cui JJ. Fatigue and failure mechanism in carbon fiber reinforced plastics/aluminum alloy single lap joint produced by electromagnetic riveting technique. Compos Sci Technol 2017;152:1–10. [17] Yang HW, Guan WB, Lu GY. Experimental and numerical investigations into collapse behavior of hemispherical shells under drop hammer impact. Thin Wall Struct 2018;124:48–57. [18] Wang ZJ, Cheng LD. Experimental research and numerical simulation of the dynamic cylinder upsetting. Mater Sci Eng A 2009;499(1-2):138–41. [19] Wang H, Ying XC, Deng QM, Yu B, Hao QM, Dong XY. Experimental and theoretical analyses of elastic-plastic repeated impacts by considering wave effects. Eur J Mech A-Solid 2017;65:212–22. [20] Kocakaplan S, Tassoulas JL. Wave propagation in initially-stressed elastic rods. J Sound Vib 2019;443:293–309. [21] Dávalos-Orozco LA. Sideband thermocapillary instability of a thin film flowing down the outside of a thick walled cylinder with finite thermal conductivity. Int J Nonlinear Mech 2019;109:15–23. [22] Zare HR, Darijani H. Strengthening and design of the linear hardening thick-walled cylinders using the new method of rotational autofrettage. Int J Mech Sci 2017;124–125:1–8. [23] Collina F, Caillerie D, Chambon R. Analytical solutions for the thick-walled cylinder problem modeled with an isotropic elastic second gradient constitutive equation. Int J Solids Struct 2009;46:3927–37. [24] Eriksson TSE, Watton PN, Luo XY, Ventikos Y. Modelling volumetric growth in a thick walled fiber reinforced artery. J Mech Phys Solids 2014;73:134–50. [25] Nelli A. Over-critical load analysis for residual stresses and displacement around fastener-holes based on the decohesive failure mechanism and non-linear Mises yield criterion. Eur J Mech A/Solids 2019;73:373–80. [26] Wang LL. Foundations of stress waves. Amsterdam: Elsevier BV; 2007. [27] Cui JJ, Qi L, Jiang H, Li GY, Zhang X. Numerical and experimental investigations in electromagnetic riveting with different rivet dies. Int J Mater Form 2018;11(6):839–53. [28] Zhang X, Yu HP, Su H, Li CF. Experimental evaluation on mechanical properties of a riveted structure with electromagnetic riveting. Int J Adv Manuf Technol 2016;83:2071–82. [29] Avitzur B, Sauerwine F. Limit analysis of hollow disk forging-part 1: upper bound. J Eng Indus 1978;100:340. [30] Darani HR, Ketabch M. Simulation of ‘‘L’’ section extrusion using upper bound method. Mater Des 2004;25:535–40. [31] Yeh WC, Wu MC. A variational upper-bound method for analysis of upset forging of rings. J Mater Process Technol 2005;170:392–402.
4. Conclusion This paper investigated the interference fit in electromagnetic riveted joint using the theoretical methods. The analytical model of interference fit is derived based on the stress wave and thick wall cylinder theory. The accuracy of the model is well verified by EMR experiments. The residual stress distribution law of riveted sheets is obtained by this model. And the relationship between relatve interference-fit values and two critical parameters (riveting forces and rivet shaft length) is also estibulished. This can guide the interference-fit design of EMR joining structures according to the existing equipments and materials. In additon, this model include many factors (not only the material properties and size of rivet, sheets and punch, but also the riveting force). Consequently, the residual stress distribution law and interference-fit values can be predicted to reduce the waste during repeated tests. Interferencefit value is most impantant factor affacting riveted structures. Taking the interference-fit value as the ultimate optimization objective, this model can provide scientific basis for improving material properties and size of rivet and sheet, and designing riveting setup. Conflict of interest None. Acknowledgements This project is supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (51621004); the Key Research and Development Program of Hunan Province (2017GK2090). References [1] Skorupa A, Skorupa M. Riveted lap joints in aircraft fuselage: design, analysis and properties. Dordrecht, Heidelberg, New York, London: Springer; 2012. [2] Hu JS, Zhang KF, Yang QD, Cheng H, Liu P, Yang Y. An experimental study on mechanical response of single-lap bolted CFRP composite interference-fit joints. Compos Struct 2018;196:76–88. [3] Binnur GK. Effect of the clearance and interference-fit on failure of the pin-loaded composites. Mater Des 2010;31:85–93. [4] Skorupa M, Skorupa A, Machniewicz T, Korbel A. Effect of production variables on the fatigue behaviour of riveted lap joints. Inter J Fatigue 2010;32:996–1003.
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Contents lists available at ScienceDirect
International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Theoretical and experimental investigation on interference fit in electromagnetic riveting Xu Zhang a, Hao Jiang b, Tong Luo b, Lin Hu a, Guangyao Li b, Junjia Cui b,∗ a b
College of Automotive and Mechanical Engineering, Changsha University of Science and Technology, Changsha 410114, China State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, China
a r t i c l e
i n f o
Keywords: Electromagnetic riveting Interference-fit model Residual stress Stress wave
a b s t r a c t As an advanced interference-fit joining technology, electromagnetic riveting (EMR) has wide engineering application prospects in manufacturing and assembly fields. In this paper, a theoretical model on interference fit for EMR process is derived based on the stress wave theory and thick wall cylinder theory. EMR experiments are conducted to verify the analytical model. The compared results show that the analytical solutions agree well with experimental values. The interference-fit model synthetically considers many process parameters factors, not only the material properties and size of rivet, sheets and punch, but also the riveting force. The residual stress distribution, which is an important factor on fatigue life, can be predicted by this model. In addition, this can provide scientific guidance for the process parameter design and riveting setup optimization in engineering application.
1. Introduction In the transportation industry, the safety of aircraft and automobile is significantly important. After long-time service, fatigue cracks are easily produced in aircraft and car structures, especially at the joints. According to the previous reports [1], more than 60% of failures occur at the joints. To solve this issue, interference-fit joining technology was proposed in assembly of aircraft and automobile parts, and it had been increasingly applied in automobile and aerospace manufacturing industry [2,3]. Based on the advantage of improving the fatigue life of interferencefit joints, many studies have been conducted on interference-fit joining techniques [4]. The effect of interference-fit values on the fatigue properties of bolted joints in composite and hybrid composite/metal structures were respectively investigated by Wei et al. [5] and Li et al. [6]. It was found that the interference-fit could remarkedly improve the fatigue life of joints compared with non-interference joints. Specifically, for composite structures, the joints with 1.8% interference-fit values under higher cyclic stress and 3% interference-fit values under lower cyclic stress had better fatigue properties. For hybrid composite/metal structures, the joints with 2.1% interference-fit values under higher cyclic stress and 1.2% interference-fit values under lower cyclic stress had better fatigue properties. In addition, Skorupa et al. [4] comprehensively researched the effect of interference-fit values on the fatigue behavior of riveted joints in metal structures. The results showed that riveted
∗
joints with higher squeeze riveting forces had better fatigue properties. Higher squeeze riveting forces improved the interference-fit values. Consequently, the higher interference-fit values could not only prevent rivet from inclining under fatigue loading, but also cause a pretension stress around the hole wall. Abovementioned studies proved that the interference-fit joining techniques could effectively enhance joining strength. The riveting and bolting techniques were the most widely used in the interference-fit assembling processes. However, the interferencefit values were formed during traditional riveting techniques with hydraulic or direct hammers [7–9]. It was not only easy to cause damage on the hole wall, but also cause the uneven interference-fit values. Besides, traditional interference-fit bolting joints were relatively easy to loosen and required designated bolts, which would increase the cost. Therefore, more advanced joining techniques have been urgently needed to realize uniform interference-fit quality and lower costs. Electromagnetic riveting (EMR), also known as stress-wave riveting, is developed by electromagnetic forming technology [10] and traditional riveting process. As a relatively new joining technique, it has great technical advantages (high efficiency, fast speed and large loading forces) to ensure the uniformity of interference-fit values, which can significantly improve the fatigue life of riveted joints compared with traditional riveting process [11]. Moreover, it does not require designated rivets. EMR technique has been applied in aircraft manufacturing of Boeing and Airbus aeries products [12,13], and has wide application prospects in other manufacturing industries. Based on the even interference-fit advantage of EMR joints, many researchers had
Corresponding author. E-mail address: [email protected] (J. Cui).
https://doi.org/10.1016/j.ijmecsci.2019.04.002 Received 31 December 2018; Received in revised form 1 March 2019; Accepted 1 April 2019 Available online 4 April 2019 0020-7403/© 2019 Elsevier Ltd. All rights reserved.
X. Zhang, H. Jiang and T. Luo et al.
International Journal of Mechanical Sciences 156 (2019) 261–271
Fig. 1. Schematic diagram of the EMR process.
also studied the effect of interference-fit quality on EMR joints [14– 16]). Cao et al. [14] compared the interference-fit qualities between EMR and traditional riveting in composite joints by experiment investigations. The results showed that the interference-fit distribution of EMR joints was not only uniform, but also had higher average values. Zhang et al. [15] investigated riveted structures with multi-layer sheets through numerical simulations and experiments. It was found that EMR technique can also ensure relatively higher interference-fit values. The relative interference-fit values of EMR joints for three, four, five and seven-layer aluminum sheets could reach to 3.4%, 3.3%, 3.1% and 2.5%, respectively. In addition, Jiang et al. [16] experimentally studied the relationship between the interference- fit values and discharge energies. The results showed that the interference-fit values of EMR joints could be well controlled by discharge energies, and the interference-fit size increased with the increasing of discharge energy. During the electromagnetic riveting process, the radial expansion occurred when the rivet was axially compressed. Plastic deformations of rivet completed under the high-speed impact, and were similar to the cylinder upsetting process with drop hammer [17]. Wang et al. [18] the stress wave significantly influenced the plastic flow and deformed geometry during the dynamic upsetting process of the cylinder. Many studies showed that the impact compressive issues of cylinders could be solved using the elastic-plastic stress wave theory [19,20]. The hole wall of riveted sheets was squeezed and expanded by the rivet cylindrical surface. The plastic deformations of riveted sheets could be considered as thick-walled cylinder under internal pressure [21,22]. Interference-fit during electromagnetic riveting was formed by the mutual squeezing effect between hole wall and rivet. At present, studies on interference-fit quality of EMR joints were conducted by experimental measurements. And interference-fit values were measured under destroying the riveted samples. Nondestructive testing technique was difficult to use because interference-fit parts located inside samples. In addition, interference-fit quality was affected by many process parameters (such as loading velocity, materials properties and rivet size). Multiple measurements would waste materials and human resources. The purpose of this paper was to investigate the interference-fit model of EMR process through mathematical physics methods. The electromagnetic riveting process was completed within some milliseconds, causing that the riveting process belonged to dynamic impact loading. In this work, stress wave and thick-wall cylinder theories were comprehensively used to establish the interference-fit model of EMR joints. Some important process parameters affecting interference-fit quality were considered in this model. The boundary conditions required for solving model could be experimentally measured, and the analytical
model was finally verified by experiments. This model can be extended to other interference-fit processes formed by the dynamic impact. 2. An analytical model of interference fit 2.1. The principle of electromagnetic riveting process The schematic diagram of the EMR process is shown in Fig. 1. The EMR equipment can be divided into two parts: the setup (namely electromagnetic pulse generator) and riveting mold. The capacitor bank imbedded in the setup stores electric energies by a charging process. When the capacitors are charged to the preset energy value, closing the discharge switch makes these energies unleash to the flat spiral coil (placed in the riveting mold). A RLC oscillation circuit is constituted among resistance (R) of whole system, inductance (L) of coil and capacitance (C) of setup. Consequently, a strong pulse current flows into the coil and then a strong electromagnetic field is generated around it. Meanwhile, a reversed eddy current is induced in the driver plate (copper plate with high electric conductivity) closed the coil. The other electromagnetic field is also produced by the eddy current. The repulsion forces from the two reversed electromagnetic fields will drive the punch impact the rivet. The rivet shaft outside the hole is deformed into a driven head, locking the two sheets together. In addition, the rivet shaft inside the hole is expanded to form reference fit with hole walls. The existence of residual stress field in riveted structures can effectively improve the fatigue life and tensile strength of structures. The riveted joints are usually used to join structures which bear strong shear loads. Fig. 2 shows stress distributions of riveted sheets under the shear loading. If there is no residual stress (namely clearance fit), stress concentration occurs inside the riveted sheets under the action of external loads. And the maximum stress located on the edge of the holes, leading to significant decrease in fatigue life of riveted structures. When interference fit is formed between the rivet and hole wall, there is residual stress inside the riveted sheets (especially around the hole wall). The residual compressive stress can relieve tensile stress caused by the external loads and reduce the stress concentration. Thus, the fatigue life of riveted structures can be remarkably improved. Consequently, interference-fit values will directly affect distributions of residual stress, and then determine the fatigue life of riveted structures. According to the principle of EMR, it is a high-speed joining technology. The whole riveting process is employed under the impact loads and completed within some milliseconds. The impact loading causes that the effect of stress wave on interference fit is not neglected during the EMR process. 262
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International Journal of Mechanical Sciences 156 (2019) 261–271
Fig. 2. Stress distributions of riveted structures under the shear loading.
Fig. 3. The stress state in the deformation process of riveted sheets.
2.2. Relative interference-fit model
the physical equation under plane strain, it can be known that:
2.2.1. Stress state of riveted sheets The hole walls of riveted sheets are subjected to radial pressures of rivets during the riveting process. Consequently, the deformation of the riveted sheets can be simplified to deformation issue of thick cylinders under the inner pressures (𝜎). The simplified model of deformed sheets and stress state are shown in the Fig. 3. It is assumed that the inner and outer radius of the cylinder are Ri and Ro , respectively. The micro-element (ABNM) on the random radius (r) position is selected to stress analysis. The thickness of sheets is relatively smaller than outer radius of them, causing that the deformation of sheets can be regarded as plane stress. According to the theory of thick cylinders [23,24], the radial stress (𝜎 r ) and circumferential stress (𝜎 𝜃 ) on the random positions can be expressed as Eqs. (1) and (2) in the cylindrical polar coordinate system. ) ( 𝑅𝑖 2 𝑅𝑜 2 1 1 𝜎𝑟 = 𝜎 − (1) 𝑅𝑜 2 − 𝑅𝑖 2 𝑅𝑜 2 𝑟2
𝜀𝑟 =
𝜕 𝑢𝑟 𝜕𝑟
(3)
𝜀𝑟 =
( ) 1 − 𝜈2 𝜈 𝜎 𝜎𝑟 − 𝐸 1−𝜈 𝜃
(4)
𝜎𝜃 =
𝑅𝑖 2 𝑅𝑜 2 𝑅𝑜 2 − 𝑅𝑖 2
𝑢𝑟 =
1 1 + 𝑅𝑜 2 𝑟2
∫
) ( 𝑅𝑖 2 𝑅𝑜 2 1 + 𝜈 1 − 2𝜈 1 𝜎 − 𝑑𝑟 ∫ 𝑟2 𝑅𝑜 2 − 𝑅𝑖 2 𝐸 𝑅𝑜 2 ] [ 𝑅𝑖 2 𝑅𝑜 2 1 + 𝜈 (1 − 2𝜈) 1 = 𝜎 𝑟+ 𝑟 𝑅𝑜 2 − 𝑅𝑖 2 𝐸 𝑅𝑜 2
𝜀𝑟 𝑑𝑟 =
(5)
Substituting the interference-fit value (Δ = r − Ri = ur ) into the Eq. (5), the inner pressures (𝜎) on the hole walls can be calculated as: ( ) Δ Δ + 𝑅𝑖 𝑅𝑜 2 − 𝑅𝑖 2 𝐸 𝑟 𝐸 = [ ] [ ] ( ) 𝑅𝑖 2 𝑅𝑜 2 1 + 𝜈 (1−22𝜈) 𝑟2 + 1 𝑅𝑖 2 𝑅𝑜 2 1 + 𝜈 (1−22𝜈) Δ + 𝑅 2 + 1 𝑖 𝑅𝑜 𝑅𝑜 ( ) Δ 1 + 𝑅Δ 𝑅𝑜 2 − 𝑅𝑖 2 𝐸 𝑅𝑖 𝑖 (6) = [ ] ( )2 𝑅𝑖 2 𝑅𝑜 2 1 + 𝜈 (1−2𝜈) Δ 1 1 + + 2 2 𝑅 𝑅 𝑅
𝜎 = 𝑢𝑟
)
( 𝜎
Substituting the Eqs. (1) and (2), The radial displacement (ur ) can be obtained by solving the Eqs. (3) and (4).
(2)
𝑅𝑜 2 − 𝑅𝑖 2
𝑜
2.2.2. Interference-fit model under the elastic limit of riveted sheets Absolute interference-fit value (Δ) can be defined as Δ = r − Ri , in which the r − Ri is just the radial displacement of hole walls. Consequently, the calculation of interference-fit value can be changed into solution issue of this displacement. According to strain definition and
𝑖
𝑖
Consequently, taking the inner pressures (𝜎) into the Eqs. (1) and (2), the radial stress (𝜎 r ) and circumferential stress (𝜎 𝜃 ) can be further solved. The outer edge of sheets can be considered as infinity relatively to the diameter of holes. When the outer radius is infinity (namely 263
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International Journal of Mechanical Sciences 156 (2019) 261–271
Ro → ∞), the inner pressure, radial stress and circumferential stress can be further simplified as: ) ( Δ𝐸 Δ 𝜎= 1+ (7) 1+𝜈 𝑅𝑖
The other case: on the elastic deformation zone (r > rp ). The elastic radial stress and circumferential stress in this zone can be obtained by
( )( )2 𝑅𝑖 𝐸 Δ Δ 𝜎𝑟 = − 1+ 1 + 𝜈 𝑅𝑖 𝑅𝑖 𝑟
𝜎𝑒𝑟 = −
( )( )2 𝑅𝑖 𝐸 Δ Δ 𝜎𝜃 = 1+ 1 + 𝜈 𝑅𝑖 𝑅𝑖 𝑟
replacing Ri and 𝜎 with rp and −𝜎𝑝𝑟 (𝑟𝑝 ) =
(8)
(9)
𝜎𝑠 1 2 𝑟 2 𝑅𝑖 2
(10)
Δ𝑒𝑙 (1 + 𝜈)𝜎𝑠 = 𝑅𝑖 2𝐸 − (1 + 𝜈)𝜎𝑠
(11)
𝑢𝑟 =
𝑑𝑟
+
𝜎𝑝𝑟 − 𝜎𝑝𝜃 𝑟
=0
The radial stress (𝜎 pr ) and circumferential stress (𝜎 p𝜃 ) in the plastic deformation zone can be obtained by integrating the equilibrium differential equation and substituting boundary conditions (the radial stress 𝜎 pr = −𝜎 when the r = Δ + Ri ).
) (15)
𝑅𝑖 + Δ
𝜎𝑠 𝑅 𝑜 − 𝑅 𝑖 2 𝑅𝑜 2 2
)
(22)
𝑅0 →∞
𝑑 𝑢𝑝𝑟
(23)
Δ 𝑅𝑖
+1 𝑟𝑝 𝑅𝑖
⎞ 1 − ⎟+ 2⎟ ⎠
( 𝑟 )2 𝑝
𝑅𝑖 Δ 𝑅𝑖
⎫ (1 − 𝜈) ⎪ ⎬ +1 ⎪ ⎭𝑅𝑜 →∞
𝑖
if the radius (rp ) of plastic deformation zone is known. According to the Eq. (17), the radius (rp ) can be expressed as: ) ( 𝜎 1 𝑟𝑝 = 𝑅𝑖 + Δ exp( − )||𝑅𝑜 → ∞ 𝜎𝑠 2
(26)
Substituting the Eq. (26) into the Eq. (25), the relative interferencefit value can be described as: ( ) { } Δ Δ (1 + 𝜈) 2𝜎 𝜎 = 1+ 𝜎𝑠 (1 − 𝜈) exp( − 1) − (1 − 2𝜈) (27) 𝑅𝑖 𝑅𝑖 𝐸 𝜎𝑠 𝜎𝑠 Making 𝜉 =
2
(1+𝜈) 𝜎𝑠 { ( 1 − 𝐸
𝜈) exp( 2𝜎𝜎 − 1) − (1 − 2𝜈) 𝜎𝜎 }, the relative 𝑠
interference-fit value can be simplified as:
(17)
𝜉 Δ = 𝑅𝑖 1−𝜉
The plastic radial stress and circumferential stress on the boundary position between the plastic and elastic deformation zone (r=rp ) can be obtained: ( ) 𝑟 1 𝜎𝑝𝑟 = 𝜎𝑠 ln − (18) 𝑟𝑝 2 𝑅 →∞
𝑠
(28)
2.3. The plastic deformation and propagation of stress wave in the rivet 2.3.1. The propagation of stress wave in the rivet The impact process can be regarded as a collision of two bars (the punch and rivet) at the initial riveting moment. Fig. 4 shows the colliding schematic diagram of two bars on a common axis. The bar A with
0
( ) 𝑟 1 𝜎𝑝𝜃 = 𝜎𝑠 ln + 𝑟𝑝 2 𝑅
1+𝜈 𝐸𝑟
The yield stress (𝜎 s ) and Poisson’s ratio (v) of the riveted are usually given. Therefore, the relative interference-fit value ( 𝑅Δ ) can be obtained
Besides, the inner pressure can be obtained according to previous analysis. +
𝑟𝑝 2
(21) 𝑅0 →∞
(25)
The yield stress is far less than modulus (E) of elasticity. Consequently, it is known that: ( )2 2𝐸 ≈1 2𝐸 − (1 + 𝜈)𝜎𝑠
𝑟𝑝
𝑟
)2
⎧ ( )⎛ Δ (1 + 𝜈) ⎪ Δ ⎜ = 𝜎𝑠 ⎨(1 − 2𝜈) 1 + ln 𝑅𝑖 𝐸 𝑅𝑖 ⎜ ⎪ ⎝ ⎩
According to the continuity condition, the elastic deformations also occur outsider the plastic deformation zone of riveted sheets. So, there are the main cases to discuss. One case: on the boundary position between the plastic and elastic deformation zone (r = rp ). The plastic radial stress is equal to the inner pressures (𝜎 el ). ( )2 𝜎 𝑅 2 − 𝑅 𝑖 2 2 𝜎𝑠 𝑅 𝑜 2 − 𝑅 𝑖 2 2𝐸 𝜎𝑝𝑟 = −𝜎𝑒𝑙 = 𝑠 𝑜 𝑟 = (16) 2 2 2 2 𝑅𝑖 𝑅𝑜 2 2𝐸 − (1 + 𝜈)𝜎𝑠 𝑅𝑜
𝜎 = 𝜎𝑠 ln
𝑟𝑝
Substituting the conditions (r = Δ + Ri and upr = Δ) mentioned above, the relationship between absolute interference-fit value (Δ) and the radius of plastic deformation zone.
(14)
𝑟 𝑅𝑖 + Δ
(20) 𝑅0 →∞
Integrating this equation and substituting boundary conditions (the plastic radial displacement upr = ur when the r = rp ). It is obtained as: the relationship between radial displacement of hole walls and the radius of plastic deformation zone. { ( ) 𝑅 2 − 𝑅𝑖 2 (1 + 𝜈) 𝑟 𝑢𝑝𝑟 = 𝜎𝑠 (1 − 2𝜈)𝑟 ln − 𝑜 𝐸 𝑟𝑝 2𝑅𝑜 2 [ ( ) ]} 𝑟𝑝 2 𝑟𝑝 2 − 𝑅𝑖 2 + +1 +1 (24) (1 − 2𝜈) 2𝑟 𝑅𝑜 2
(12)
𝑟 𝑅𝑖 + Δ
)2
𝑢𝑝𝑟 + 𝑑𝑟 𝑟 ) (1 − 2𝜈)(1 + 𝜈) ( ) 1 𝑑 ( 𝑟𝑢𝑝𝑟 = 𝜎𝑝𝑟 + 𝜎𝑝𝜃 = 𝑟 𝑑𝑟 𝐸 ) (1 − 2𝜈)(1 + 𝜈) ( = 2𝜎𝑝𝑟 + 𝜎𝑠 𝐸 ( ) 𝑅𝑜 2 − 𝑅𝑖 2 (1 − 2𝜈)(1 + 𝜈) 𝑟 𝜎𝑠 2 ln − +1 = 𝐸 𝑟𝑝 𝑅𝑜 2
(13)
( 𝜎𝑝𝜃 = −𝜎 + 𝜎𝑠 1 + ln
2
(
𝑟𝑝 𝑟
𝜀𝑝𝑟 + 𝜀𝑝𝜃 =
𝜎𝑝𝜃 − 𝜎𝑝𝑟 = 𝜎𝑠
𝜎𝑝𝑟 = −𝜎 + 𝜎𝑠 ln
𝑠
(
respectively.
According to strain definition and the physical equation under plane strain, the following Eq. (23) can be obtained by combined he plastic radial stress and circumferential stress.
2.2.3. Interference-fit model the radius (rp ) of plastic deformation zone of riveted sheets The riveted sheets will have plastic deformations when inner pressure exceeds the elastic limit (𝜎 el ). The plastic deformation zone is approximately in the state of plane strain (𝜏 𝜃r = 0). Thus, the equilibrium differential equation and yield criterion are as follows: 𝑑 𝜎𝑝𝑟
𝜎𝑠 2 (𝜎
𝜎𝑒𝑟 =
According to Tresca yield criterion (𝜎 𝜃 − 𝜎 r = 𝜎 s ) [25], the inner pressures (𝜎 el ) and interference-fit value (Δel ) under elastic limit can be calculated by substituting Eqs. (8) and (9). 𝜎𝑒𝑙 =
𝜎𝑠 2
𝜎𝑠 𝑅 𝑜 2 − 𝑅 𝑖 2 , 2 𝑅𝑜 2
(19) 0 →∞
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Fig. 5 shows the simplified model of the EMR process. The riveting model is regarded as the process which a bar (punch) with the velocity of v1 and length of l1 impacts a fixed bar (rivet) with length of l2 . l1 and l2 represent the length of punch and rivet shaft, respectively. Firstly, according to the principle of momentum conservation, the velocity of the punch can be calculated by Eq. (33). 𝑣1 =
∫ 𝐹 𝑑𝑡 𝑀
(33)
where M and F represent the mass of the punch and the riveting force, respectively. Subsequently, the punch can be regarded as a rigid rod during the EMR process. And the punch impacts the rivet at the velocity of v1 . So its wave impedance (𝜌C)1 is infinite. The rivet is stationary, causing that its velocity is 0 m/s. Substituting the above values into Eq. (32), the stress and velocity of the impact surface at the impact moment are expressed as following: } 𝜎= −(𝜌𝐶 )2 𝑣1 (34) 𝑣 = 𝑣1 where (𝜌C)2 is the wave impedance of the rivet. In the subsequent rivet deformation process, the velocity of the impact surface is constantly changing under the action of stress wave. Fig. 6 presents the schematic diagram of the stress wave propagation in the rivet and punch. Stress wave mainly includes the elastic and plastic waves. Elastic wave has faster propagation speed in the rivet, so it first reaches the fix end and then reflects back. According to the reflection law [20] of stress wave, the particle velocity becomes zero and the stress doubles when elastic wave reaches the fixed end. As for the punch (rigid materials), it has no deformations during the riveting process. Therefore, it is assumed that only elastic wave exists in the punch. The elastic wave bounces off the free end of the punch and its speed will be doubled. The speed changing value and time for elastic wave in a round trip in the punch can be calculated by:
Fig. 4. The collision schematic of two bars.
the speed of VA strikes the bar B with speed of VB . At the moment of striking, two stresses are generated at the impact surface of bar A and B, respectively. The stresses in the form of waves with the speeds of CA and CB propagate to the other end. According to the continuity requirement, the stress 𝜎 and velocity v of bar A and B at the impact surface are identical. Based on the stress wave theory reported by Wang [26], the basic characteristic line differential equation and its compatibility relation can be expressed as: 𝑑𝑥 = 𝐶𝑑𝑡
(29)
𝑑 𝜎 = ±𝜌𝐶𝑑 𝑣
(30)
2(𝜌𝐶 )2 𝑣1 ⎫ Δ𝑣 = ( ) 𝜌𝐶𝑒 1 ⎪ ⎬ 2𝑙1 ⎪ Δ𝑡 = ⎭ 𝐶𝑒
So according to the above characteristic line compatibility conditions, the relationship between stress 𝜎 and velocity v is depicted as follows: ( ) ( ) 𝜎 = (𝜌𝐶 )𝐴 𝑣 − 𝑣𝐴 = (𝜌𝐶 )𝐵 𝑣𝐵 − 𝑣 (31)
where Ce is the spread speed of the elastic wave in the punch, and l1 is the length of the punch. Then the elastic wave spreads back and forth n times in the punch until the punch speed drops to zero. Thus, n is expressed as:
where (𝜌C)A and (𝜌C)B are the wave impedance of bar A and B, respectively. 𝜌 and C respectively represent the materials density and stress wave velocity. Further, the stress 𝜎 and velocity v could be obtained by Eq. (32): 𝑣𝐵 − 𝑣𝐴 ⎫ 1∕(𝜌𝐶 )𝐴 + 1∕(𝜌𝐶 )𝐵 ⎪ (𝜌𝐶 )𝐴 𝑣𝐴 + (𝜌𝐶 )𝐵 𝑣𝐵 ⎬ 𝑣= ⎪ (𝜌𝐶 )𝐴 + (𝜌𝐶 )𝐵 ⎭
(35)
𝑛=
𝑣1 Δ𝑣
(36)
Therefore, the collision time between the punch and rivet is calculated by:
𝜎=
(32) 𝑡 = 𝑛Δ𝑡 =
𝑙 1 𝜌1 (𝜌𝐶 )2
(37) Fig. 5. The simplified model of the EMR process.
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Fig. 6. Schematic diagram of the stress wave propagation in the rivet and punch.
as: 𝜕𝑢 ⎧ ⎪𝜀𝑟 = 𝜕𝑟 ⎨ 𝑢 ⎪ 𝜀𝜃 = ⎩ 𝑟
According to the Hooke’s law, the relationship between the stress and strain is described as: ( ) ⎧ 𝜀𝑟 = 1 𝜎𝑟 − 𝜇𝜎𝜃 𝐸 ⎪ ) ⎪ 1( (41) 𝜎 − 𝜇𝜎𝑟 ⎨𝜀 𝜃 = 𝐸 𝜃 ⎪ 1 ⎪ 𝛾𝑟𝜃 = 𝜏𝑟𝜃 ⎩ 𝐺 where 𝜇 is the Poisson’s ratio. The radial deformation of rivet is caused by the compression of rivet material. As the deformation behavior of rivet is relatively complex, the rule of radial displacement is first considered. Assuming that the cylinder is subjected to a uniform internal pressure p. The inner diameter is a0 and the outer diameter is b0 . The outer surface of the cylinder is free surface, and the pressure is zero. Moreover, rivet deformation follows the principle of volume invariance:
Fig. 7. Cross section of the rivet and the stress state of its element.
Based on the previous study [27], it could be seen that velocity value of the punch almost changes linearly. Consequently, the displacement and average stress at the impact surface are respectively expressed as: 𝑣1 𝑡 𝑙 𝜌 𝑣 ⎫ =− 1 1 1⎪ 2 2(𝜌𝐶 )2 (𝜌𝐶 )2 𝑣1 ⎬ ⎪ 𝜎𝑎𝑣𝑔 = − ⎭ 2
𝑈0 =
(38)
𝜕 𝑢𝑟 𝑢𝑟 𝜕 𝑢𝑧 + + =0 (42) 𝜕𝑟 𝑟 𝜕𝑧 The coordination equation is obtained by combining geometric function. 𝑑 𝜀𝜃 𝜀 − 𝜀𝑟 + 𝜃 =0 (43) 𝑑𝑟 𝑟 As seen from the above equation, there are two unknowns. During the process of rivet deformation, the stress and strain of the inner particle of the cylinder are affected by external forces. Consequently, the boundary conditions are introduced to solve the problem. The boundary conditions of the inner and outer diameter of cylinder are expressed as:
2.3.2. Plastic deformation of rivet The deformation of the rivet is caused by the axial impact riveting force. During riveting process, a corresponding positive stress is generated in the radial direction due to the Poisson ratio effect. Moreover, the upsetting of the rivet is directly related to radial stress. As the radical displacement of the rivet is caused by Poisson’s ratio effect, it can also be regarded as the deformation of the cylinder caused by the inner pressure. Based on the thick cylinder theory, it is assumed that the rivet is a thick-walled cylinder with an infinitely small inner diameter. Fig. 7 shows the schematic of cross-section of the rivet and the stress state of its element. The section is symmetric and a sector element ABCD is selected for analysis. It is supposed that the axial thickness of the rivet is 1, then the equilibrium differential equation of the element is expressed as: 𝜎 − 𝜎𝜃 𝜕 𝜎𝑟 1 𝜕 𝜏𝑟𝜃 + 𝑟 + + 𝑓𝑟 = 0 𝜕𝑟 𝑟 𝜕𝜃 𝑟
(40)
{ 𝜎𝑟 =
( ) −𝑝 𝑟 = 𝑎0 ( ) 0 𝑟 = 𝑏0
(44)
The axial end face of cylinder is under the pressure of T, so the boundary condition of end face of cylinder body is expressed as: 𝑏0
𝑇 = 2𝜋 ∫ 𝜎𝑧 𝑟𝑑𝑟 𝑎0
(39)
(45)
Combining the equilibrium equation Eq. (39) and stress-strain relationship Eq. (41) of elastic deformation, the stress coordination equation is obtained as: ( ) 𝑑 𝜎𝑟 + 𝜎𝜃 𝑑𝜎 −𝜇 𝑟 =0 (46) 𝑑𝑟 𝑑𝑟
where 𝜎 r and 𝜎 𝜃 represent positive stress along radial and tangential directions, respectively. 𝜏 r𝜃 and fr represent the tangential stress and body force, respectively. The radial deformation of the rivet is symmetric, so the annular displacement is zero. The geometric equation is expressed 266
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So the radical displacement for elastic deformation is obtained by the geometric function Eq. (40) as: ] ( ) [ 𝑏 2 1+𝜇 𝑢𝑟 = 𝑝̄ (1 − 2𝜇)𝑟 + 0 − 𝜇𝜀0 𝑟 (47) 𝐸 𝑟
It can be found that the relationship between the A1 and C1 is obtained. Subsequently, the axial and radial displacements are substituted into the basic geometric equation as follows: 𝜕 𝑢𝑧 ⎧ 2 ⎪𝜀𝑧 = 𝜕𝑧 = −2𝐴1 𝑧 − 2𝐶1 ⎪ 𝜕 𝑢𝑟 ⎪ 2 ⎪𝜀𝑟 = 𝜕𝑟 = 𝐴1 𝑧 + 𝐶1 ⎪ 𝑢 ⎨𝜀 𝜃 = 𝑟 = 𝐴 1 𝑧 2 + 𝐶 1 𝑟 ⎪ ⎪ 𝜕𝑢 𝜕𝑢 ⎪𝛾𝑧𝑟 = 𝑟 + 𝑧 = 2𝐴1 𝑧𝑟 𝜕𝑧 𝜕𝑟 ⎪ ⎪𝛾 = 𝛾 = 0 𝑟𝜃 𝜃𝑧 ⎩
During the process of rivet deformation, the area in inner diameter first enters the plastic stage. As for plastic region, the stress should satisfy both the yield condition and the equilibrium equation. When r = a0 , radial stress is maximized. So the ultimate elastic pressure is calculated as: ( ) 𝜎 𝑎 2 𝑝𝑒 = 𝑠 1 − 0 (48) 2 𝑏0 2 If the internal stress is less than the ultimate elastic pressure, there is only elastic deformation inside the cylinder. While the internal stress is greater than the ultimate pressure, plastic deformation in the cylinder occurs. There has plastic zone and elastic zone interface in the cylinder (the plastic zone is inside). Assuming that the diameter is c and stress is continuous at the interface of plastic zone and elastic zone, the plastic radial stress at the particle point of r = c can be taken as the internal pressure of the elastic zone. So the stress in the plastic region can be calculated as: ( ) ⎧ 𝜎𝑠 𝑐 2 𝑏0 2 ⎪ 𝜎𝑟 = 2 𝑏 0 2 1 − 𝑟 2 ( ) (49) ⎨ 𝜎 𝑐2 𝑏 2 ⎪𝜎𝜃 = 2𝑏𝑠 2 1 + 𝑟02 0 ⎩
The simplified rivet constitutive equation is introduced into the above geometric equation: ( ) ( ) 𝜎𝑧 𝑧 = 𝑙2 = 𝐸1 −2𝐴1 𝑧2 − 2𝐶1
( ) 𝜎𝑧 𝑧 = 𝑙2 = 𝜎𝑎𝑣𝑔
⎧ 3𝜎𝑎𝑣𝑔 3𝑈0 ⎪𝐴 = − ⎪ 1 4𝐸 𝑙 2 4𝑙23 1 2 ⎨ 𝜎𝑎𝑣𝑔 3 𝑈 0 ⎪𝐶 = − ⎪ 1 4𝐸1 4𝑙2 ⎩
Substituting the above results into the Eqs. (43) and (45), the axial and radial displacements on the outside surface of the rivet are obtained as follows: (( ) ( ) ) 𝑙1 𝜌1 3𝑙1 𝜌1 (𝜌𝐶 )2 ⎧𝑢 = 3 + (𝜌𝐶 )2 − − 𝑧 𝑧 ∫ 𝐹 𝑑𝑡 2 4𝐸1 𝑀 4𝑀 𝑙2 (𝜌𝐶 )2 ⎪ 𝑧 4𝑀𝑙23 (𝜌𝐶 )2 (( 4𝐸1 𝑀𝑙2 ) (62) ( )) ⎨ 3𝑙1 𝜌1 3(𝜌𝐶 )2 3𝑙1 𝜌1 (𝜌𝐶 )2 2+ ⎪ 𝑢𝑟 = − 𝑧 − 𝑑𝑡 𝑟𝐹 ∫ 8𝑀 𝑙2 (𝜌𝐶 )2 8𝐸1 𝑀 8𝐸1 𝑀𝑙22 8𝑀𝑙23 (𝜌𝐶 )2 ⎩
1 + 𝑁 (𝑧)𝑟 (51) 𝑟 where z and r are the axial and radial displacement, respectively. M(z) and N(z) are the expressions associated with z. When the rivet is deformed, the particles at the center axis position do not shift in the radical direction. When r = 0, ur = 0 and M(z) = 0, the actual deformation results of rivet deformation are approximately parabola at the free boundary [28]. Based on the upper bound methods [29–31], it is setting the expression N(z) as: 𝑢𝑟 (𝑧, 𝑟) = 𝑀 (𝑧)
The corresponding axial stress 𝜎 z and radial stress 𝜎 r on the outside surface of the rivet are expressed as follows: ( ) ( ) ⎧𝜎 = 3(𝜌𝐶 )2 − 3𝐸1 𝑙1 𝜌1 𝑧2 + (𝜌𝐶 )2 − 3𝐸1 𝑙1 𝜌1 ∫ 𝐹 𝑑𝑡 𝑧 2 3 4𝑀 4𝑀 𝑙2 (𝜌𝐶 )2 ⎪ 4𝑀𝑙2 4𝑀𝑙2 (𝜌𝐶 )2 (( ) (63) ( )) ⎨ 3𝐸1 𝑙1 𝜌1 3(𝜌𝐶 )2 3𝐸 𝑙 𝜌 (𝜌𝐶 ) ⎪𝜎 𝑟 = − 𝑧2 + 8𝑀 𝑙1 (1𝜌𝐶1) − 8𝑀2 ∫ 𝐹 𝑑𝑡 3 2 8𝑀𝑙2 8𝑀𝑙2 (𝜌𝐶 )2 ⎩ 2 2
(52)
It is known that deformations of riveted sheets occur under the radial compressions of the rivet. Consequently, the radial stress (𝜎 r ) of the rivet is just inner pressure (𝜎) during the deformation process of hole walls. Consequently, the equation 𝜉 in the relative interference-fit model Eq. (28) can be changed into:
where A1 , B1 and C1 are the constant. It is assumed that the radial displacement near the driven head is 0 mm (z = 0 mm), as shown in Fig. 3. Moreover, due to the symmetry of the rivet, B1 = 0. The radical displacement function can be simplified as: ( ) 𝑢𝑟 (𝑧, 𝑟) = 𝐴1 𝑧2 + 𝐶1 𝑟 (53)
{ } ⎧𝜉 = (1+𝜈) 𝜎 (1 − 𝜈) exp( 2𝜎 − 1) − (1 − 2𝜈) 𝜎 𝑠 𝐸 𝜎 𝜎 ⎪ 𝑠) 𝑠 (( ( )) ⎨ 3𝐸1 𝑙1 𝜌1 3(𝜌𝐶 )2 3𝐸 𝑙 𝜌 (𝜌𝐶 ) − 𝑧2 + 8𝑀 𝑙1 (1𝜌𝐶1) − 8𝑀2 ∫ 𝐹 𝑑𝑡 ⎪𝜎 = 3 2 8𝑀𝑙2 8𝑀𝑙2 (𝜌𝐶 )2 2 2 ⎩
According to the rivet volume invariant principle: 𝜕 𝑢𝑟 𝑢𝑟 𝜕 𝑢𝑧 + + =0 𝜕𝑟 𝑟 𝜕𝑧 The axial displacement function can be obtained as:
(54)
Substituting to the Eq. (55), it can be calculated as: 3𝑈0 + 6𝐶1 𝑙2 2𝑙23
(64)
where v and 𝜎 s is the Poisson’s ratio and yield strength of the riveted sheets, respectively. The E is the elasticity modulus of the riveted sheets. The M is the weight of the punch. The 𝜌1 and l1 is the density and length of the punch, respectively. The E1 and (𝜌C)2 is the elasticity modulus and wave impedance of the rivet, respectively. The l2 is the density and length of the rivet shaft. The F means the impact forces and z represents the coordinate change (0 ∼ l2 ) along the rivet shaft. It can be found that the relative interference-fit model of EMR joints is related to many factors, not only the material properties and size of rivet, sheets and punch, but also the riveting force. These parameters can be obtained by actual measurements.
2 𝑢𝑧 (𝑧, 𝑟) = − 𝐴1 𝑧3 − 2𝐶1 𝑧 (55) 3 At the impact surface z = l2 (l2 represents the length of screw rod as shown in Fig. 5), the displacement of the punch is U0 . So the boundary condition is: ( ) 𝑢𝑧 𝑧 = 𝑙2 = 𝑈0 (56)
𝐴1 = −
(61)
(50)
where c1 is the constant. Based on the above analysis, no matter according to the radial displacement expression in the plastic zone or elastic zone, the radical displacement function can be expressed as:
Δ = 𝜀𝑟 + 𝜀𝜃 + 𝜀𝑧 =
(60)
The A1 and C1 can be calculated by the following:
Further, the radical displacement in the plastic region is expressed
𝑁 (𝑧) = 𝐴1 𝑧2 + 𝐵1 𝑧 + 𝐶1
(59)
where E1 is the elasticity modulus of rivet. According to the boundary condition:
as:
[ ] 𝑐 (1 − 2𝜇)(1 + 𝜇) 𝑟 𝑐2 𝑟 𝑟 𝑢𝑟 = 𝜎𝑠 + 𝑟 ln − − 𝜇𝜀𝑧 𝑟 + 1 𝐸 2 𝑏0 2 𝑐 2 𝑟
(58)
(57)
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Fig. 8. The measured equipment of riveting forces: (a) whole composition, (b) local piezoelectric sensor. Fig. 9. The measured and analyzed results of the riveting forces: (a) the measured results under the varying discharge energies, (b) the integral of riveting force under the discharge energy of 4.7 kJ.
3. Results and discussion
In addition, the main material parameters are shown in Table 1. The compared EMR experiments were conducted with 6082-T6 aluminum alloy sheets and 2A10 aluminum alloy rivets. The thickness of the sheets is 4 mm. The diameter and length of the rivets are 5 mm and 14 mm, respectively. According to the universal riveting standard [28], the diameter of riveted hole is set to 5.1 mm. The punch in the electromagnetic equipment was manufactured by 45# steel. Its length and mass are 150 mm and 7 kg, respectively. The propagation velocity of stress wave in aluminum alloy (Al) is 5300 m/s. To obtain the qualified riveted joints, the discharge energy of 4.7 kJ was used to conduct the EMR experiments. Fig. 10 shows the riveted specimen. Subsequently, seven points were uniformly selected on the rivet shaft to measure the diameter after the riveting. The end surface of manufactured head is regarded as the axial distance z = 0 mm. The rivet diameter was measured by the Vernier caliper with accuracy of 0.01 mm. In order to avoid the accidental error, five-time measurements for every position are averaged to use the validation analysis. It can be observed from Fig. 9(b) that the integral of riveting force is 15.17 under the discharge energy of 4.7 kJ. The above results are substituted into Eq. (64) and the theoretical results of interference-fit model are obtained. The results of actual measurements are shown in Table 2. The comparison results between experimental and theoretical values are presented in Fig. 11. During the high-speed EMR process, the axial compression stress wave of rivets propagates from the loading
3.1. Model solution and validation The riveting force should be firstly measured before calculating the interference-fit model. Fig. 8 shows the measured equipment. It mainly consists of piezoelectric sensor, charge amplifier and oscilloscope. When the punch impacts the piezoelectric sensor, the sensor is pressurized to produce a charge signal. The charge signal is amplified and changed into a voltage signal by the charge amplifier. Finally, the measured results are transmitted and displayed on the oscilloscope. Because there is a strong electromagnetic field around the coil (riveting equipment) during the EMR process, which would produce electromagnetic waves. The charge amplifier and the oscilloscope should keep enough distance with riveting equipment so as to affect the operation of charge amplifier and oscilloscope. Fig. 9 shows the measured results of the riveting forces using the varying discharge energies. It can be seen that the peak force increases with the increasing of discharge energies. The integral of riveting force can be described by the Eq. (65), and the area under the curves can be obtained by the drawing software. ∫ 𝐹 𝑑𝑡 = Area
(65)
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Table 1 Material parameters of critical components. Components Punch Sheet Rivet
Density (kg/m3 )
Materials
3
45# steel 6082 T6 Al 2A10 Al
7.5 × 10 2.7 × 103 2.7 × 103
Elasticity modulus (Pa) 11
2.0 × 10 7.2 × 1010 7.2 × 1010
Poisson’s ratio
Yield strength (Pa)
0.30 0.35 0.35
3.55 × 108 2.6 × 108 2.5 × 108
Table 2 Experiment results of interference-fit values and rivet diameter after riveting. Axial distance z (mm)
Measured diameters of rivets (mm)
Average interference-fit values (%)
1 2 3 4 5 6 7 8
(5.17 + 5.17 + 5.18 + 5.18 + 5.18) / 5 = 5.176 (5.17 + 5.17 + 5.18 + 5.18 + 5.19) / 5 = 5.178 (5.17 + 5.17 + 5.18 + 5.19 + 5.19) / 5 = 5.180 (5.17 + 5.19 + 5.19 + 5.19 + 5.19) / 5 = 5.185 (5.19 + 5.19 + 5.20 + 5.21 + 5.21) / 5 = 5.200 (5.20 + 5.22 + 5.22 + 5.22 + 5.24) / 5 = 5.220 (5.25 + 5.25 + 5.25 + 5.25 + 5.25) / 5 = 5.250 (5.26 + 5.27 + 5.27 + 5.27 + 5.28) / 5 = 5.270
1.49 1.53 1.57 1.69 1.96 2.35 2.94 3.33
ably predict the interference-fit value of EMR joints. This interference-fit model can also guide the engineering design of EMR joints. 3.2. The residual stress distributions on the riveted sheeted According to the Eq. (11), the relative interference-fit value under the limit elasticity was 0.24% by substituting the property parameters of riveted sheets. When the interference-fit values are above 0.24%, the hole wall of the riveted sheets will have plastic deformations. It can be seen in Fig. 11 that interference-fit values of all measured positions are over 0.24%. Consequently, plastic deformations of hole walls occur along the whole thickness direction of riveted sheets. Because the outer boundary of sheets is far bigger than the radius of prefabricated holes, the size (rP ) of plastic deformation areas can be obtained by Eq. (25). The calculated results of plastic areas are shown in Fig. 12. It can be seen that the deformation degree of hole walls is closely related to interferencefit values. The size of plastic deformation areas is the larger under the bigger interference-fit value. In addition, the changing law of the size along the thickness of the riveted sheets is the same as the interferencefit distribution. Assuming that the outer boundary of riveted sheets is infinity, the Eqs. (18) and (19) can be changed into the following Eqs. (66) and (67). The residual stress distributions (including radial stress 𝜎 pr and circumferential stress 𝜎 p𝜃 ) can be calculated by substituting the radius (rp ) of plastic deformation areas, as shown in Fig. 13. The radius (Ri ) of holes is 5.1 mm before the riveting in this paper. For the thickness direction of riveted sheets, both radial stress and circumferential stress have the same changing trend. These stresses decrease from the driven head to the manufactured head. For the radial direction of riveted sheets, the radial compression stress (𝜎 pr ) decrease with the increase of radial positions (r). However, the circumferential tensile stress (𝜎 p𝜃 ) increase with the increase of radial positions (r). The riveted sheets is assumed as a plane strain state during the derivation process of interference-fit model, leading to a contrary tendency between radial stress and circumferential stress. ( ) 𝜎𝑝𝑟 𝑟 1 = ln − (66) 𝜎𝑠 𝑟𝑝 2 𝑅 →∞
Fig. 10. The EMR riveted specimen.
Fig. 11. The comparison between experimental and theoretical results of interference-fit values.
end to the other end. The upsetting deformation preferentially occurs in the position close the loading, and the position preferentially contact with the hole wall. Consequently, the interference fit is also gradually formed from the loading end to the other end. The partial energies in the preferential positions are transferred to the hole wall, leading to gradual energy loss from the loading end to the other end. It can be seen that the interference-fit values of the riveted joints gradually increase with the axial distance (from manufactured head to driven head). The variation trend of interference-fit values is consistent with the previous studies [14,15]. In general, the theoretical values agree well with experimental values. Therefore, the analytical model is accurate and can commend-
0
𝜎𝑝𝜃 𝜎𝑠
( ) 𝑟 1 = ln + 𝑟𝑝 2 𝑅
(67) 0 →∞
3.3. The effect of process parameters on interference-fit values The interference-fit model (Eq. (64)) shows that the process parameters affecting this model include riveting forces, material properties and structure dimensions of rivets, riveted sheets and punch. The material 269
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Fig. 12. The size of plastic deformation areas for riveted sheets: (a) the schematic diagram of deformation areas, (b) the radius values of plastic area.
Fig. 13. The residual stress distribution around hole walls of riveted sheets: (a) radial stress, (b) circumferential stress. Fig. 14. The relationship between parameters and interference-fit values: (a) the rivet shaft effect, (b) discharge energy effect.
properties and structure dimensions of the punch have no choice if the existing EMR equipment is used. In addition, the material properties of rivets and riveted sheets are also given before the riveting. Consequently, the length of rivet shaft and riveting forces can usually be adjusted according to the joined structures. Fig. 14 shows the relationship between the two parameters and relative interference-fit values. In this paper, the rivets with the diamater of 5 mm are used as an example. In order to insure the dismension of driven head, the rivet shaft must ex-
ceed 1.2 × 5 mm than the total thickness of all riveted sheets before the riveting pocess. Consequently, the length of rivet shaft increases as total thickness of all riveted sheets increases. It can be seen in Fig. 14(a) that the whole relative interference-fit values decrease with the increase of length of rivet shaft. The rivet shaft with a larger length-diameter ratio can be regarded as a slender rod. For the slender rod, stress wave effect is very signifanctly and can be considered as one-dimension stress. Due to transverse dispersion of stress wave, plastic deformations maily
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concentrate in the part closing the loading end of rivet. Therefore, The rivet shafts with the larger length have the smaller relative interferencefit values. Only riveting forces can be adjusted to achieve the thicksheet structure with enough interference-fit values. The EMR process with higer discharge energy has larger riveting forces. The precise control of discharge energy is one advantage of EMR. Fig. 14(b) demonstrates increasing the discharge energies can remarkably improve the interference-fit values. Moveover, EMR riveted sheets prepared under high discharge energy have relatively poorer uniform interference-fit. However, the increase of rivet shaft can improve the uniform, which can just compensate for this shortage.
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4. Conclusion This paper investigated the interference fit in electromagnetic riveted joint using the theoretical methods. The analytical model of interference fit is derived based on the stress wave and thick wall cylinder theory. The accuracy of the model is well verified by EMR experiments. The residual stress distribution law of riveted sheets is obtained by this model. And the relationship between relatve interference-fit values and two critical parameters (riveting forces and rivet shaft length) is also estibulished. This can guide the interference-fit design of EMR joining structures according to the existing equipments and materials. In additon, this model include many factors (not only the material properties and size of rivet, sheets and punch, but also the riveting force). Consequently, the residual stress distribution law and interference-fit values can be predicted to reduce the waste during repeated tests. Interferencefit value is most impantant factor affacting riveted structures. Taking the interference-fit value as the ultimate optimization objective, this model can provide scientific basis for improving material properties and size of rivet and sheet, and designing riveting setup. Conflict of interest None. Acknowledgements This project is supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (51621004); the Key Research and Development Program of Hunan Province (2017GK2090). References [1] Skorupa A, Skorupa M. Riveted lap joints in aircraft fuselage: design, analysis and properties. Dordrecht, Heidelberg, New York, London: Springer; 2012. [2] Hu JS, Zhang KF, Yang QD, Cheng H, Liu P, Yang Y. An experimental study on mechanical response of single-lap bolted CFRP composite interference-fit joints. Compos Struct 2018;196:76–88. [3] Binnur GK. Effect of the clearance and interference-fit on failure of the pin-loaded composites. Mater Des 2010;31:85–93. [4] Skorupa M, Skorupa A, Machniewicz T, Korbel A. Effect of production variables on the fatigue behaviour of riveted lap joints. Inter J Fatigue 2010;32:996–1003.
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