4.1 Effect of hole diameter on rivet driven head dimension
The relationship between rivet driven head dimension and hole diameter under different squeeze forces is illustrated in Fig. 6. It can be seen from Fig. 6a that the influence of hole diameter on rivet head diameter is very small under the conditions of 15 KN and 23 KN squeeze force. The height of the driven head decreases along with the increase of hole diameter, especially under the condition of 23 KN squeeze force, but the decreasing amplitude is very small, see Fig. 6b. It can be concluded that the influence of the hole diameter on the rivet driven head dimension is very small under the same squeeze force. Therefore, there are limitations in using rivet driven head dimensions to evaluate riveting quality, which is consistent with the conclusion of Lei  and Skorupa .
4.2 Influence of hole diameter on interference distribution and material flow characteristics
Interference is another parameter to evaluate the riveting quality, which has an important influence on the mechanical properties of riveted joints. In the process of riveting, the interference can be expressed as absolute interference and relative interference. The absolute interference is the ratio of twice the radial displacement of the hole wall and the hole diameter. However, the relative interference is the ratio of the difference between the rivet shank diameter before and after riveting and the rivet shank diameter before riveting. The absolute interference and relative interference are shown in Eq. (1) and (2) respectively.
Where ΔH is the radial displacement of the hole wall after riveting, dH is the diameter of the rivet hole before riveting, d is the diameter of the rivet shank after riveting, d0 is the diameter of rivet shank before riveting.
Fig. 7 shows the degree of hole expansion in different hole diameters ranges from 15 KN to 23 KN squeeze force. The left side of the vertical line represents the outer sheet, and the right side represents the inner sheet. It can be seen from the figure that the expansion degree of the inner sheet is greater than that of the outer sheet under each hole diameter, and the difference of hole wall expansion between inner and outer sheets increases with the increase of the squeeze force. There is little difference in the expansion level of the outer sheet along the thickness direction under different hole diameters. However, the difference of hole expansion level of inner sheet increases, especially near the rivet driven head side. The expansion degree of the hole wall on the side of the rivet driven head is greater than that on the side of the rivet manufactured head, whether it is 15 KN or 23 KN squeeze force. The above phenomena are related to the material flow characteristics during riveting.
Generally, the rivet and hole belong to clearance fit in the initial state, and the diameter of the hole is larger than the diameter of the rivet shank. In the early stage of the riveting process, the whole rivet shank belongs to the free upsetting stage when the rivet shank does not contact the hole wall, and the material mainly flows in the axial direction. The radial expansion of the rivet shank is limited by the hole wall when the rivet shank contacts with the hole wall and the axial material flow of the rivet shank tend to be saturated. Only a small amount of material will flow into the hole under the action of squeeze force, the remaining material in the rivet shank will form the rivet driven head. The axial path of rivet shank at different radial positions is shown in Fig. 3. The change of radial displacement under different paths is illustrated in Fig. 8. In the figure, the left side of the vertical line represents inside the hole and the right side represents outside the hole. As the figure shows, the radial displacement of the rivet shank inside the hole has little change, and the increase of radial displacement mainly occurs outside the hole. The radial displacement of the rivet near the surface outside the hole first increases and then decreases, which is caused by the uneven distribution of friction in the forming process of rivet driven head. The radial displacement of 23 KN squeeze force is greater than that of 15 KN squeeze force under the same path condition. The radial displacement values of different paths are significantly different, and the radial displacement value of path 1 is the largest, and the radial displacement value of path 4 is the smallest. The mutation points of radial displacement in paths 2, 3, and 4 are earlier compared with path 1. Therefore, it is more appropriate to describe the radial flow with the increase of the radial displacement of the material inside the rivet shank, which is consistent with the previous research conclusion . Careful observation shows that the radial deformation of rivet shank increases with the increase of hole diameter, especially when the hole diameter is 5.00 mm and 5.10 mm (see Fig. 8g ~ Fig. 8j). The main reason is that the reaction force of rivet by hole wall is reduced when the clearance between rivet and hole is increased, and the rivet shank is easier to be upset.
In this paper, the radial displacement value of path 2 is selected for research, and the radial displacement changes with the axial position under different hole diameters, as shown in Fig. 9. As can be seen, the position of radial displacement mutation will be later with the increase of squeeze force. The mutation point is located in the axial position of 5mm when the squeeze force is 15KN, as shown in Fig. 9a. However, when the squeeze force is 23KN, the mutation point is located in the axial position of 6mm, as shown in Fig. 9b. In other words, there will be more materials flow into the hole and the greater the expansion of the hole wall when the squeeze force increase, this is also consistent with the results of interference analysis. With the increase of the hole diameter, the material flowing into the hole increases slightly.
4.3 Effect of hole diameter on shear properties of riveted lap joints
The shear test results are shown in Fig. 10. The peak load of each specimen with different hole diameters can be obtained from the corresponding load-displacement curve. As the figure showed that the variation trend of the load-displacement curve is almost the same at different hole diameters. In the stage of loading, there is a parallel straight line segment of 0.1 mm, which is caused by the clamping slip of the joint and has no effect on the final shear force. Then, it enters the elastic deformation stage. In this stage, the rivet begins to deform with the increase of the tensile force until it finally breaks. This is mainly because the interference between the rivet and the hole becomes smaller when the squeeze force decreases, the rivet is easy to tilt in the hole under the action of tensile force, which will produce greater displacement when the rivet shank breaks. On the contrary, the diameter of the rivet shank increases with the increase of squeeze force, the rivet is not easy to tilt in the hole because the hole wall reacts on the rivet shank. In this case, the contact area between the rivet shank and hole wall will increases, so the bearing capacity is stronger.
To further illustrate the influence of hole diameter on the bearing capacity of riveted lap joints, the average value of shear force under the same parameters is taken as the graph, and the histogram of the influence of hole diameter on the shear force of riveted lap joints under different squeeze force is obtained, as shown in Fig. 11. The figure shows that the influence of the hole diameter on the shear force under different squeeze forces is consistent, the shear strength of the rivet increases with the increase of the hole diameter. It is noted that the deformation degree of the hole wall decreases with the increase of the hole diameter (see Fig. 7 and Fig. 9). At the same time, the rivet shank on the outer side of the sheet has more material flowing into the hole, and a larger rivet shank diameter can be obtained. The bearing strength of the riveted lap joints is usually determined by the diameter of the rivet shank. The larger the rivet shank diameter is, the larger the tensile force is needed to achieve the failure of the specimen, which means that the shear strength of the riveted lap joints increases with the increase of the hole diameter.
On the other hand, as Fig. 11 reveals that the shear strength of the specimen increases with the increase of squeeze force. Apparently, increasing the hole diameter can effectively improve the bearing strength of the riveted lap joints compared with increasing the squeeze force, which is contrary to the conclusion of Zeng . The main reason is that the fatigue process is significantly different from the static load process, and the influencing factors of the fatigue process are more complex. Besides, the hole diameter in this paper is larger than the rivet diameter. However, three kinds of fit states were used in Zeng's research, including interference fit, transition fit, and clearance fit. Only from the point of view of improving the static strength, increasing the hole diameter is beneficial to improve the shear strength of the specimen. However, the fretting effect of fretting on the fatigue life of riveted lap joints can not be ignored in the process of service while pursuing high shear strength. It has been pointed out that the deformation of the hole wall decreases and the corresponding interference decreases with the increase of the hole diameter under the same squeeze force condition. Under cyclic load, cracks are easy to initiate from the hole edge for the sample with small interference, which eventually leads to the premature failure of the riveted lap joints . However, some literature  shows that the effect of interference fit on fatigue life gain is weakened and even the fatigue life is reduced when the interference is too large. Therefore, choose the appropriate hole diameter can not only improve the bearing strength of the riveted lap joints, but also improve the fatigue life. It is crucial to achieving the maximum benefit of the riveted lap joints.
Fig. 12 shows the effect of different hole diameters on the bearing capacity of the specimen. As the figure depicts the shear force of the specimen with a diameter of 4.90 mm, 4.94 mm, 5.00 mm, and 5.10 mm increases by 3.51%, 6.52%, 9.99%, and 17.77% respectively, compared with the specimen with a diameter of 4.82 mm under the condition of 15 KN squeeze force. When the squeeze force increases to 23 KN, the shear force of the specimen with a diameter of 4.90 mm, 4.94 mm, 5.00 mm, and 5.10 mm increases by 4.47%, 7.04%, 10.59%, and 17.24% respectively, compared with the specimen with a diameter of 4.82 mm. Except for the specimen with a diameter of 5.10 mm, the increased amplitude of shear force under the condition of 23 KN squeeze force is greater than that under the condition of 15 KN squeeze force.
4.4 Shear fracture analysis
In order to study the failure modes of riveted lap joints with different hole diameters, the macro characterization of the tested specimens was carried out. Fig. 13 shows the shear failure specimens of partially riveted lap joints under different combinations of hole diameter and squeeze force. For all the riveted specimens, the fracture occurs at the rivet shank of the interface of the laminate. There is no obvious necking phenomenon in all specimens, which indicates the brittle fracture of rivet under tensile force. It is worth mentioning that there is an obvious gap between the cross-section of the rivet shank and the hole wall, which is the result of the plastic deformation of the hole wall caused by the extrusion of the rivet shank. The plastic deformation zone can not recover after the fracture of the rivet shank, and the circular rivet hole is elongated to form an elliptical hole. Except for the deformation around the hole, the sheet has no obvious deformation. From the radial expansion diagram of the hole wall (see Fig. 7), a large interference step at the junction of the upper and lower sheets can be seen. Meanwhile, the single lap riveted joint is easy to produce secondary bending under the action of tensile force, and the out-of-plane deflection constraint is small, so the specimen breaks from the middle of the rivet shank. What's more interesting is that the rivet shank at rivet manufactured head-end stays in the outer sheet under the condition of 15 KN squeeze force, except that the rivet manufactured head of some specimens with a hole diameter of 5.10 mm fall off from the outer sheet, as shown in Fig. 13a. Different from the 15 KN squeeze force specimens, the rivet manufactured head of all specimens did not fall off from the outer sheet under 23 KN squeeze force, this is significantly different from specimens under the condition of 15 KN squeeze force. However, the rivet shank at the rivet driven head end of all shear specimens ejected from the hole under the condition of 15 KN squeeze force, regardless of the hole diameter. The above phenomenon was not observed under the condition of 23 KN squeeze force. The reason is that the interference close to rivet driven head under the condition of 23 KN squeeze force is larger than that 15 KN squeeze force. When the riveted lap joints are subjected to tensile force, the rivet shank squeezes the hole wall and makes the hole plastic deformation, forming a gap between the rivet shank and the hole. The specimens with small interference are not enough to offset the effect of the gap at the moment of fracture, so the rivet head falls off from the hole. The rivet manufactured head falls off from the outer sheet for the sample with a hole diameter of 5.10 mm, which is mainly because the extrusion of the rivet shank on the outer side sheet becomes smaller when the hole diameter increases to 5.10 mm. It is estimated that if the hole diameter continues to increase, all rivet manufactured heads will fall off from the outer sheet.
Fig. 14 shows the macroscopic fracture morphology of the shear specimen under different combinations of hole diameter and squeeze force. As the figure shows, the shear failure position and fracture morphology are similar under different hole diameters and squeeze forces, which are the mixed mode of brittle and plastic fracture. The shear fracture of all specimens can be divided into zone 1 and zone 2. In the initial stage of shearing, there will be displacement between the upper sheet and the lower sheet under the action of shear force. In this case, the local stress concentration is easy to form in the rivet shank located at the interface of lamination. When the shear stress is much higher than the yield strength and fracture strength of the material, the crack at the fracture of zone 1 is generated and extended rapidly under the high stress, forming a shear band similar to brittle fracture. After that, the local stress concentration began to weaken, and the rivet began to change into the overall stress state. With the decrease of initial stress and enough time for crack propagation, the fracture mode changes from shear band to dimple, which is characterized by plastic fracture macroscopically. The area of zone 1 decreases with the increase of squeeze force under the same hole diameter, which means that the smaller the shear displacement when the rivet breaks, which can also be seen from the shear load-displacement curve.
Detailed micro fracture characteristics are shown in Fig. 15. Region 1 is a shear slip region, which is smooth and has obvious shear texture, and its direction is the same as the loading direction. Region 2 is relatively rough, goose down, and dark in color, which is an obvious plastic fracture zone. The aluminum alloy used in this paper belongs to the face-centered cubic structure. When the shear stress exceeds the critical shear stress, the crystal will slip along the direction of <1`1 0> and a slip shear band will be formed, as shown in Fig. 15c. With the increase of shear force, it will enter the transition zone from shear band fracture to ductile fracture. There is an obvious boundary in the transition zone that can be seen. The boundary is marked with a yellow dotted line, as shown in Fig. 15b. It is obvious that there are many elongated dimples along the direction of shear stress under the boundary of the slip zone, which is the result of plastic fracture of rivet shank. When the shear force exceeds the strength limit of the rivet shank, the bearing capacity of the rivet shank will be greatly reduced, and the rivet shank will be broken instantly. The area of region 1 is much smaller than that of region 2, which is different from the result of Li . In the present research, the rivet material is aluminum alloy 2117-T4, while the rivet material is Q235 carbon steel in the experiment of Li. The tensile stress of Q235 carbon steel is larger than that of aluminum alloy 2117-T4. There is no doubt that Q235 has stronger resistance to tensile force deformation, so the area of region 1 is larger.
4.5 Shear failure mechanism
Fig. 16 shows the shear load-displacement curve of the typical shear specimen and the explanation of shear failure mechanism. According to the changing trend of the curve, the shear load-displacement curve can be divided into four stages, as shown in Fig. 16a. The first stage is called the adjustment stage. In this stage, the gap between the fixture and the sample is adjusted under the action of tensile force. Meanwhile, the shear displacement and shear forces are very small, and the rivet and sheets are not deformed. The second stage is called the stationary stage. After the adjustment, the riveted lap joints enter the elastic stage. Due to the interference between the rivet and the sheets, the rivet is enough to resist the influence of the tensile force. The shear force and displacement increase linearly with the increase of tensile force, and the riveted lap joints have good stiffness characteristics at this stage. The third stage is called the degradation stage. With the further increase of the tensile force, the riveted lap joints enter the stage of plastic deformation, and the joint stiffness degrades continuously under the tensile force. In this stage, the lap sheets first yield and produce plastic deformation when the yield strength of the material around the hole is lower than the shear strength of the rivet, resulting in a gap at the interface between the rivet and the hole wall. Under the action of tensile force, the inner sheet and the outer sheet exert the same shear force and the opposite direction on the rivet shank, and finally the rivet tilts, which can be seen from Fig. 16b. As the tensile force continues to increase, the yield strength increases with the increase of the deformation degree of the sheets. The sheets stop deformation when the yield strength of the material around the hole is higher than the shear strength of the rivet, and the rivet is obviously damaged under the action of shear load, resulting in shear slip, as indicated in Fig. 15c. With the appearance of shear slip, the bearing capacity of the rivet shank is greatly reduced under the shear load, and the load drops sharply, resulting in fracture failure of the rivet shank. The shear force reaches the maximum at this stage.