Experimental and numerical studies on failure modes of riveted joints under tensile load

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Experimental and numerical studies on failure modes of riveted joints under tensile load Nanjiang Chen a,b,∗ , Hongyu Luo a , Min Wan a , Jean-loup Chenot b a b

School of Mechanical Engineering and Automation, Beihang University, Beijing, 100191, China CEMEF–Mines ParisTech, BP207-06904, Sophia Antipolis Cedex, France

a r t i c l e

i n f o

Article history: Received 15 June 2013 Received in revised form 16 December 2013 Accepted 31 December 2013 Available online xxx Keywords: Failure mode Riveted joint Tensile strength Fracture FE simulation

a b s t r a c t Though engineers are not advised to utilize riveted joints in tension, rivets will inevitably bear tensile load in realities. Hence, it is interesting to investigate the failure modes of riveted joints when they are under tensile load. Following previous studies (Chen et al., 2011), in this work, the authors selected three sizes of riveted joints to conduct the riveting and tension processes experimentally and numerically. Three kinds of failure modes including pull through, shank breaking, and head breaking were observed. Simulations are able to give almost the same results as those from experiments. Furthermore, three formulas were proposed to calculate the maximum tensile strength of riveted joints. Though the values calculated from these three formulas are approximate, they have the same order of magnitude as real ones. Moreover, they could be utilized to estimate which kind of failure mode may take place when riveted joints were under tensile load. © 2014 Elsevier B.V. All rights reserved.

1. Introduction As one of the most important mechanical joints, riveted joints are extensively utilized in various configurations for a wide range of industrial products, such as aircrafts, vehicles, bridges, and many other products. In the famous Eiffel Tower in Paris, about 2.5 millions rivets were utilized to connect 15,000 iron pieces (http://corrosion-doctors.org/Landmarks/eiffel-history.htm). In modern Boeing 747 airplanes, abundance of rivets can also be found to join many parts because of their high reliability. To study the riveting process of riveted joints and their strength, engineers usually use a series of practical experiments which are time-consuming and costly. In recent years, with the development of finite element method (FEM) and computing technology, FEM softwares are able to deal with most mechanical problems, such as bulk forming, sheet forming, and many other processes. In addition, with FEM software, a better insight of the process is possible by analyzing the various mechanical parameters of the models.

∗ Corresponding author at: Corresponding author. Beihang University, 704#, Haidian District, Beijing 100191, China, Tel.: +86 10 82315095; fax: +86 10 82315095. E-mail addresses: [email protected], [email protected] (N. Chen).

There is a general piece of advice: “Never use a rivet in tension” (Slagter, 1995). However, in reality, tensile loads are inescapable. For example in airplanes, due to the upward suction explain force, the rivets that hold on the skin to the stringers and ribs undergo tensile stresses. Hence, rivets are mainly under three kinds of loads: shearing load, tensile load, and the combination of these two. When a rivet is under large shearing force, the rod of the rivet will break because of the force. However, there are several failure modes when a rivet is under tensile load. The first kind is that the rivet head is pulled through sheets. The second one is that the rivet rod breaks under tensile load which is similar to the pure extension process of a rod billet. The third mode is that the rivet head breaks. Assuming that the plate material was rigid-perfectly plastic and the tension and compression yield stresses were equal, Slagter (1995) studied the pull-through strength of flush head rivet. He deduced a very complex mathematical function by neglecting a few details and compared it with some experimental results found in literatures. The function was capable to give a nearly correct but not exact result, because a few details were neglected. During 1920s, Wilson and Oliver (1930) did much work on solid rivets. In their experimental reports, the last two kinds of failure modes in tension were both found. Langrand et al. (2001) used Arcan experimental tools to measure the tensile strength of riveted joints. The rods of rivets broke in their study and the corresponding simulations were developed in FEM software. Gurson damage criterion was utilized to depict the fracture process of the rivet rod. However, a systematic

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and comprehensive investigation of the failure modes of riveted joints in tension is very scarce if any in the literature. Hence, in the current article, the main work is to develop the experimental and numerical models to analyze these three failure modes. 2. Finite element method For a more complete description of the scientific approach of FE approach of metal-forming process, the reader may refer to Wagoner and Chenot (2001).



and the mass conservation equation can be written as:

−

˙ ∗ dV = 0 (div(v) + p)p

where  is the compressibility coefficient of the material. The most widely used time integration method remains the simple one-step Euler scheme. In this simple approach, the total time is decomposed into small increments t and the displacement field is proportional to the velocity field at the beginning of the increment: u = t v

2.1. Constitutive law For cold-forming processes, an elastic plastic law must be chosen. A widely used approximation, which reveals accurate enough in most applications, is based on an additive decomposition of the strain rate tensor ε˙ into an elastic part ε˙ e and a plastic one ε˙ p : ε˙ = ε˙ e + ε˙ p

(8)

In the same way, the stress increments are introduced, so that the Eqs. (9) and (10) are rewritten as:





(  +   ) : ε˙ ∗ dV −



(p + p)div(v∗)dV −

˝

˝

 v ∗ dS = 0 ∂˝c

(9)

(1)

  



In our case, where material rotations remain relatively small, the use of a non-objective derivative of the stress tensor is acceptable. The hypo-elastic law is written as:

−

d = e trace(ε˙ e )I + 2e ε˙ e dt

2.4. Finite element approximation

e

(2)

e

where and are the Lamé coefficients. The plastic component of the strain rate tensor obeys the flow rule:  =

(7)

˝

2 ε˙ p 0 (¯ε) 3 ε¯˙

(3)

where   is the deviatoric stress tensor,  0 is the yield stress, ε¯˙ is the equivalent strain rate, and ε¯ is the equivalent strain. Different laws can be used to describe work-hardening. To represent our materials accurately, the following form is chosen: √ 0 (¯ε) = 3k(1 + a¯εn ) (4)

t

˝

div(u) + p p ∗ dV = 0

(10)

Many different finite element formulations were proposed, and developed at the laboratory level, but it is now realized that the discretization scheme must be compatible with other numerical and computational constraints, among which the following can be quoted: • Remeshing and adaptive remeshing. • Unilateral contact analysis, iterative solving of nonlinear and linear systems. • Domain decomposition and parallel computing. • Easy transfer of physical internal parameters, when multi-physic coupling must be taken into account.

where a, n, and k are the material parameters. To obtain the exact values of these three parameters of each kind of material, uniaxial tension test was conducted to get the load-displacement curve of this process and the curve was transferred to the true stress strain curve to obtain the initial material parameters. The initial material parameters were improved by iterative computation with Forge2009® .

To-day, a satisfactory compromise is based on a mixed displacement and pressure formulation using tetrahedral elements, and a bubble function for the velocity in order to stabilize the solution for incompressible or quasi-incompressible materials. The displacement field is discretized with shape functions Nn , in terms of nodal displacement vectors Un :

2.2. Friction

u =



Un Nn ( )

(11)

n

At the interface between part and tool, the friction shear stress can be modeled by a Coulomb law:  = −˛f |n |

v 0 with || ≤ √ |v| 3

(5)

where ˛f is the friction coefficient,  n is the normal stress, and v is the tangential velocity. 2.3. Virtual work principle For an incompressible or quasi-incompressible flow, it is desirable to utilize a mixed formulation. In the domain of the part, for any virtual velocity and pressure fields, v*, p*, this formulation can be written as:





  : ε˙ ∗ dV − ˝



pdiv(v∗)dV − ˝

 v ∗ dS = 0 ∂˝c

(6)

The pressure field is generally expressed in terms of different shape functions: p=



pm Mm

(12)

m

Using isoparametric elements, the mapping between the physical space with coordinates x and the reference space with coordinate is: x=



Xn Nn ( )

(13)

n

The strain increment tensor can be computed with the help of the conventional B discretized linear operator: ε =



Un Bn

(14)

n

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Fig. 1. Configurations of riveted joints, the left one is for case a and b, while the right one is for case c as shown in Table 1.

Finally, the discretized mixed integral formulation for the mechanical problem is:





RnU =





˝

∂˝c

=

1 b( , u ) = 2

(p + p) trace(Bn ) dV

h

˝

˛f |n |

+

P Rm



( +  ) : Bn dV −

  

(15)



div(u) + p Mm dV = 0

t

˝

v

  Nn dS = 0 v

always possible in the reality. Hence, the symmetric formulation was proposed in which the integration happens in both surfaces:

(16)



h

aC



 ha h(uha ,

uhb )ds +

C

hb h(uhb ,

uha )ds

(19)

b

However, if the discretizations of the two surfaces are similar then this method will introduce too many contact conditions and may artificially stiffen the contact interface. Consequently, in the quasi-symmetric formulation, hb is ˜ h that is calculated from h only: replaced by an approximation  1 2

b



˜ h h(uh , uh )ds  a b b

a

Remark: At each time increment, Eq. (14) is first written without the Jaumann correction on the stress increment and the system of Eqs. (14) and (15) is solved. Then, the stress tensor is transformed using the rotation matrix corresponding to the spin increment.

b(h , uh ) =

2.5. Contact formulation

where bh (ha ) is the orthogonal projection of ha on aC (Fourment, 2008).

The same approach of Forge2009® is utilized which can deal with multi-body and multi-material structures. A nodal incremental form of the penalty technique is used for contact analysis. At each time step, the nodes that are potentially going to penetrate the opposite surface are searched. Based on the in-depth penetration measurement, a penalty contribution is added to the functional for these nodes. The contact terms arising from contact between different bodies are computed through a coupled approach based on a master-slave algorithm.

 b(, u) =

C

h(ua , ub )ds

(17)

where C is the contact surface,  is the contact normal stress, and h(ua , ub ) is the distance function between the two contact surfaces. However, after finite element discretization, the two surfaces are not similar, therefore, it is necessary to define on which surface the right part of Eq. (16) should be integrated. In the usual master/slave formulations, b(, u) is only integrated on the slave body surface:

 b(ha , uh ) =

aC

ha h(uha , uhb )ds

(18)

where ha is the normal stress on the slave body surface, while

aC is the slave body surface. To obtain an accurate integration result, the slave body should be discretized more finely than the master body, but that is not

aC

ha h(uha , uhb )ds +





˜ h = h (h ) with  a b b

C

,

b

(20)

2.6. Numerical issues The nonlinear equations resulting from the mechanical behavior are linearized with the help of the Newton-Raphson method. Now, the resulting linear systems are often solved with iterative methods, which appear faster and require much less CPU memory than the direct ones. Prediction of possible formation of folding defects during forming processes is based on the analysis of the contact of the part with itself, therefore providing a problem similar to the coupling with tools. Automatic dynamic remeshing during the simulation of the whole forming process is almost always necessary, as elements undergo very high strain which could produce degeneracy. Before this catastrophic event, decrease in element quality must be evaluated and a remeshing module must be launched periodically to recover a satisfactory element quality. The global mesh can be completely regenerated, using a Delaunay or any front tracing method, but the method of iterative improvement of the mesh, with a possible local change of element structure and connectivity, seems to be much more effective. For more details about the remeshing technique, the reader may refer to Wagoner and Chenot (2001). For industrial complicated applications, with short delays, the computing time can be decreased dramatically using several or several tens of processors. This requires to use an iterative solver and to define a partition of the domain, each sub-domain being associated with a processor. But the parallelization is made more complex due to remeshing and the remeshing process itself must be parallelized.

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Fig. 2. Hydraulic press machine in riveting (left) and the rivet joint assembly (right).

Table 1 Parameters of riveted joints used in the study. Case

d (mm)

L (mm)

dh (mm)

L1 (mm)

L2 (mm)

h (mm)

a b c

14.9 9.8 5.8

36 30.4 15.4

16 10.25 6.45

7.8 7.8 2.8

5.8 5.8 2.8

3

Table 2 Materials used in each case. Case

Rivet

Upper sheet

Lower sheet

a b c

S235JR FR5 S700MC

HLES600MC HLES600MC HLES355MC

HLES355MC HLES355MC HLES355MC

Force control method was applied in riveting processes and end forces were set as 280 kN, 140 kN, and 90 kN for these three cases, respectively. Fig. 2 shows the hydraulic press machine and the rivet joint assembly before riveting process. The velocity of the lower die was set as 6 mm/s during the riveting process. The Arcan test procedure (Arcan et al., 1987) was utilized to conduct the tensile processes of riveted joints for case a and case b (see Fig. 3). A special tool was designed to finish the tensile process of riveted joint for case c (see Fig. 4). The velocity of the tool was set as 6 mm/s and the press machine stopped when one kind of failure happened. 4. Simulations

To avoid the necessity for the user to perform several computations, with different meshes to check the accuracy, error estimation can be developed using, for example, the generalization of the method proposed by Zienckiewicz and Zhu (1987). Then, if the rate of convergence of the approximation is known, the local mesh refinement necessary to achieve a prescribed tolerance can be computed, and the meshing modules are improved to be able to respect the refinement when generating the new mesh. 3. Experimental approach To illustrate these three failure modes under tensile load, three sizes of riveted joints were selected. Configurations and sizes were shown in Fig. 1 and Table 1, respectively. Materials used in tests were illustrated in Table 2. In riveting processes, upper dies were fixed and lower dies moved upward to form the driven heads.

An implicit finite element code Forge3 designed for the simulations of three-dimensional metal forming was selected as FEM tool to simulate the riveting and tensile processes of the riveted joints. It includes chaining module which can chain several simulations. With the help of this function, the code is able to load the previous final deformed shapes, residual stress distribution, local yield stresses due to work hardening and dies corresponding to the next deformation stage and then start automatically the tension process of riveted joint after riveting process. All deformed parts including rivets and sheets were meshed with tetrahedral elements while surfaces of dies were considered as rigid and meshed with triangular elements. To avoid distortion of the meshes, decrease in element quality was evaluated and a remeshing module was launched periodically to recover a satisfactory element quality. Half symmetry was utilized during the simulations with symmetry boundary conditions (see Fig. 5).

Fig. 3. Arcan test procedure for case a and case b. The left is the riveted joint and the right is the tensile assembly.

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Fig. 4. Tensile assembly for case c. The left is the riveted joint and the right part is the tensile assembly. Table 3 Material parameters of the rivets and sheets used in the study.

Fig. 5. FEM models in Forge2009.

Uniaxial tension tests and upsetting tests were utilized to obtain the constitutive parameters of sheets and rivets, respectively. Several hardening laws were compared with the experimental behavior of the rivet material and Eq. (4) was found to be the most accurate one. Table 3 shows the material parameters of the rivets and the sheets used in the simulations. In riveting processes, the rise of temperature is mainly because of strain energy of the deformed materials. To estimate the rise of temperature, adiabatic heating was assumed during the riveting processes. According to energy equilibrium, following equation is obtained.





ε

ε

√ 1 dε = 3k(1 + aεn )dε c 0 0 √ √ (n + 1) 3k + 3kaεn+1 = (n + 1)c

1 T = c

(21)

Name

k (MPa)

A

N

E (GPa)

v

S235JR FR5 S700MC HLES600MC HLES355MC

242.5 242.2 242.7 346.2 225.0

0.9542 0.5025 0.5740 1.072 1.249

0.4796 0.6298 0.1300 0.5635 0.6660

210 210 210 210 210

0.3 0.3 0.3 0.3 0.3

in Table 4. Hence, in adiabatic condition, T for S235JR, FR5, and S700MC in riveting processes are 188 ◦ C, 149 ◦ C, and 172 ◦ C, respectively. In reality, it is air condition but not adiabatic condition, so the rise of temperature must be smaller. In conclusion, the heat effects can be neglected during riveting processes. In the simulation, Coulomb’s law is selected as the friction law between the contacted parts and the coefficient is set as 0.15. Further study in the simulation to be published will show that the friction coefficient has little impact on the final result in riveting process. To simulate potential fracture phenomenon of rivets in tensile process, uncoupled damage law normalized (Cockcroft and Latham, 1968) was applied to compute damage values in rivets and “killelement” technique (Bouchard et al., 2008) was utilized to delete those completely damaged elements. Iterative simulations of uniaxial tensile processes of the rivet materials were utilized to obtain the critical values of these three materials. They are 1.2, 1.7, and 0.6 for S235JR, FR5, and S700MC, respectively.



0

ε¯ f

1 d¯ε = C ¯

(22)

where ε¯ f is the critical equivalent strain,  1 is the first principle stress, ¯ is the equivalent stress, ε¯ is the equivalent strain, and C is the critical damage value. 5. Results 5.1. Riveting process

where T is the rise of temperature, c is the specific heat capacity,  is the density, and the other parameters are defined in Eq. (4). In riveting processes, the maximum equivalent strain is about 1. As an approximate approach, c is set as 464 J/(kg·◦ C) and  is set as 7900 kg/m3 . The other three coefficients k, a, and n are shown

For riveting process, final configurations and load-displacement curves are shown in Figs. 6–8. Through these three figures, it is obvious that the results from simulations are almost the same as those from experiments. Fig. 9 shows the equivalent strain distribution during the riveting process of case c. From this figure, it can

Table 4 Tensile strengths calculated from the three formulas, simulations, and experiments. Case

Pull through (P1 )

Shank breaking (P2 )

Head breaking (P3 )

Experiment

Simulation

a b c

67 60 24

101 33 19.6

143 44 19.4

48.7 39.2 17.9

49 42.6 16.5

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Fig. 6. Comparisons of final configurations of riveted joint (left) and load-displacement curves (right) between experiment and simulation of case a.

Fig. 7. Comparisons of final configurations of riveted joint (left) and load-displacement curves (right) between experiment and simulation of case b.

be noticed that the area of maximum equivalent strain changes with the movement of lower die. In the beginning, it locates at the area where the rivet contacts with the lower die (see Fig. 9(b)). Afterward, the centre part of the driven head has the maximum equivalent strain (see Fig. 9(c)). In the end, the area where rivet contacts with the lower sheet has the maximum equivalent strain (see Fig. 9(d)). 5.2. Tensile process The pull-through failure mode of riveted joint under tensile load is shown in Fig. 10. In this figure, the left part is the

specimen in experiment while the right part is the situation in simulation. The load-displacement curve of this case is illustrated in Fig. 11. This curve can be divided into three parts: OA, AB, and BC. The lower sheet starts to yield at point A and becomes a cone from point A to point B. After point B, the inner diameter of the lower sheet is as big as the diameter of the driven head of the rivet and the load decreases from this point (Fig. 12). Fig. 13 illustrates the shank-breaking mode of riveted joint under tensile load. Because the sheets in case b have only small deformation, the tension process of this riveted joint is similar to a uniaxial tension process of a rod. Hence, the load-displacement

Fig. 8. Comparisons of final configurations of riveted joint (left) and load-displacement curves (right) between experiment and simulation of case c.

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Fig. 9. Equivalent strain distribution of riveted joint during riveting process of case c.

Fig. 10. Pull-through failure mode of riveted joint under tensile load of case a.

Fig. 11. Load-displacement curves of riveted joint under tensile load of case a.

curves in Fig. 14 are also similar to those curves from uniaxial tension processes of rods (Fig. 15). The head-breaking mode of riveted joint under tensile load is shown in Fig. 16. Fractures occur at the edges between the head and the shank. The load-displacement curves are shown in Fig. 17. In this case, the deformation of the sheets is relatively large. In conclusions, simulations are able to represent the tension processes of riveted joints, no matter which kind of failure mode occurs. In this work, theoretical analysis of these three kinds of failure modes are also conducted and compared with the experimental and numerical results. For the pull-through failure mode, analyzed in Slagter’s (1995) article, it is assumed that the rivet was rigid and the sheets were ideal plastic. Based on the limit analysis theory, a pull-through strength prediction model was developed. The governing equation

Fig. 12. Equivalent strain distribution and VonMises stress distribution of case a.

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Fig. 13. Shank-breaking failure mode of riveted joint under tensile load of case b.

for the rivet pull-through problem was derived using the theorem of minimum total potential energy. The total potential energy (denoted by Utotal in Eq. (22)) consisted of the potential energy of the external load (denoted by W in Eq. (22)) and the energy dissipation in plastic deformation. The latter could be divided into two parts: the internal energy dissipation in the plate due to the circumferential bending moment and circumferential tensile stress (denoted by Up in Eq. (22)); the internal energy dissipation on the circular hinge line r3 (denoted by Ub in Eq. (22)). By applying zero, the partial derivative of the total potential energy with respect to the deflection ω, ¯ the rivet pull-through load is able to be obtained (Fig. 18). Utotal = W + Up + Ub

(23)

Fig. 14. Load-displacement curves of riveted joint under tensile load of case b.

W = −P

r3 − r2 ω ¯ r3 − r1

(24)

Fig. 15. Equivalent strain distribution and VonMises stress distribution of case b.

Fig. 16. Head-breaking failure mode of riveted joint under tensile load of case c.

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Fig. 17. Load-displacement curves of riveted joint under tensile load of case c.



2



r3

Up =

(M  + N ε )r drd

(25)

r2

0

Fig. 19. Illustration of pull-through failure mode of riveted joint under tensile load.





2

Ub =

Mp 0

∂ω ∂r

r3 d

(26)

where P is the pull-through load, r1 , r2 , and r3 are defined in Fig. 19, M is the circumferential bending moment,  is the circumferential curvature, N is the circumferential force, ε is the circumferential strain, Mp is the yielding bending moment, and ω is the deflection. A leaner displacement function is assumed, that is, the deflection ω is a leaner function of radius r. ω(r) =

r3 − r ω ¯ r3 − r1

(27)

For more details about this model, readers can refer Slagter (1995). In his model, a lot of details were neglected or simplified, such as the rivet was assumed as rigid, the sheets were assumed as ideal rigid-plastic, the deflection was assumed as a leaner function of radius r, etc. Therefore, it was impossible to obtain very precise result through this formula. An illustration of this model is shown in Fig. 19. P=



2r3 − r2  2r2 εmax (r3 − r2 ) y t 2 + 2 r3 − r1 t2

(28)

where  y is the yield stress of the sheet, t is the thickness of the sheet, r3 is the radius of the area over which lateral displacement takes place, r2 is the radius of critical value enabling rivet pull through, r1 is the radius of rivet, and εmax is the strain at failure of sheet material.

For shank-breaking failure mode, it can be simplified as a uniaxial tension process of a rod specimen. Then, the maximum tensile strength of riveted joint is: P = m · r1 2

(29)

where  m is the maximum tensile stress and r1 is the radius of rivet. For head-breaking failure mode, it can be simplified as a pure shearing process. Then, the maximum tensile strength of riveted joint is: P = m · 2r1 · h

(30)

where  m is the maximum shearing stress, r1 is the radius of rivet, and h is the thickness of rivet head. Because the three formulas above are all simplified, only approximate results can be obtained from them. For the three kinds of riveted joints used in this study, Table 4 shows maximum tensile strengths calculated from the three formulas above and obtained from experiments and simulations. For case a, P1 is smaller than P2 and P3 , so the pull-through failure mode occurs when the riveted joint is under tensile load. However, this value is about 37% bigger than that obtained from the experiment. For case b, P2 is the smallest among these three values, so the shank-breaking failure mode takes place when the riveted joint is under tensile force. However, this value is about 19% smaller than that from the experiment.

Fig. 18. Equivalent strain distribution and VonMises stress distribution of case c.

Please cite this article in press as: Chen, N., et al., Experimental and numerical studies on failure modes of riveted joints under tensile load. J. Mater. Process. Tech. (2014), http://dx.doi.org/10.1016/j.jmatprotec.2013.12.023

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For case c, P3 is smaller than P1 and P2 , so the head-breaking failure mode happens when the rivet joint is under tensile load. Though P2 and P3 are very close in this case, the edge between the head and the shank is bearing a bending moment because of the deformation of the upper sheet (see Fig. 16). The fracture of this riveted joint is the combination of the shearing force and this bending moment. Hence, in experiments and simulations, head-breaking failure mode took place but not shank-breaking mode. In conclusions, although the three formulas are not able to give very precise tensile strengths, the values calculated from them are at the same order of magnitude as those from experiments and simulations. Furthermore, they are qualified as the basis to judge the failure mode when riveted joints are under tensile loads. 6. Conclusions To investigate the failure modes of riveted joints when they are under tensile loads, three sizes of riveted joints were utilized to conduct riveting and tension processes. Three failure modes including pull through, shank breaking, and head breaking were observed in experiments. Meanwhile, Forge2009 was acted as FEM tool to simulate the riveting and tension processes of the riveted joints. The results from the simulations, including configurations, loaddisplacement curves, and fractures were almost the same as those from the experiments both in riveting and tension processes. For the three failure modes, corresponding formulas of tensile strengths were proposed and compared with the experimental results. Though the tensile strengths calculated from them are not very precise, they are at the same order of magnitude of real values from experiments. Furthermore, they are able to be utilized to judge the failure mode of riveted joint when they are under tensile load.

Acknowledgement The authors would like to acknowledge the MonaLisa project, which is financed by Cetim with a close cooperation between Cetim, Cemef, and Transvalor. The authors would like to thank Alain Le Floc’h, Francis Fournier, and Gilbert Fiorucci for their dynamic technical assistance. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.jmatprotec. 2013.12.023. References Arcan, L., Arcan, M., Daniel, I.M., 1987. SEM fractography of pure and mixed mode interlaminar fracture in graphite/epoxy composites. ASTM Special Technical Publications 948, 41–47. Bouchard, P.O., Laurent, T., Tollier, L., 2008. Numerical modeling of self-pierce riveting—from riveting process modeling down to structural analysis. J. Mater. Process. Tech. 202, 290–300. Chen, N., Ducloux, R., Pecquet, C., Malrieux, J., Thonnerieux, M., Wan, M., Chenot, J.L., 2011. Numerical and experimental studies of the riveting process. Int. J. Mater. Forming. 4 (1), 45–54. Cockcroft, M.G., Latham, D.J., 1968. Ductility and the workability of metals. J. Inst. Met. 96, 33–35. Fourment, L., 2008. A quasi-symmetric formulation for contact between deformable bodies. Eur. J. Comput. Mech. 17, 907–918. Langrand, B., Deletombe, E., Markiewicz, E., Drazetic, P., 2001. Riveted joint modeling for numerical analysis of airframe crashworthiness. Finite. Elem. Anal. Des. 38, 21–44. Slagter, S.J., 1995. On the tensile strength of rivets in thin sheet materials and fibre metal laminates. Thin. Wall. Struct. 21, 121–145. Wagoner, R.H., Chenot, J.L., 2001. Metal Forming Analysis. Cambridge University Press, Oxford, UK, pp. 25–84. Wilson, W.M., Oliver, W.A., 1930. Tension tests of rivets. University of Illinois, Engineering Experiment Station, Bulletin No. 210, Urbana. Zienckiewicz, O.C., Zhu, J.Z., 1987. A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Numer. Meth. Eng. 24, 337–357 (Available at: http://corrosion-doctors.org/Landmarks/eiffel-history.htm. 201303-14).

Please cite this article in press as: Chen, N., et al., Experimental and numerical studies on failure modes of riveted joints under tensile load. J. Mater. Process. Tech. (2014), http://dx.doi.org/10.1016/j.jmatprotec.2013.12.023