JOINING AND MECHANICAL STRENGTH OF SELF-PIERCING RIVETED
STRUCTURE – NUMERICAL MODELING AND EXPERIMENTAL VALIDATION
S. Fayolle, P.O. Bouchard and K. Mocellin
CEMEF – Ecole des Mines de Paris
BP 207, 06904 Sophia-Antipolis Cedex, France
[email protected]
ABSTRACT
In this paper simulations of the self-piercing riveting process as well as the mechanical strength of riveted specimens are
performed and compared with experimental results. To get accurate simulations of such a process, we have to deal with large
plastic deformation, multimaterial contact, damage and fracture. We use here a damaged elastic-plastic material. Ductile
damage is computed using the Lemaitre damage model. The material parameters of the sheets and the rivet, i.e. hardening
and damage parameters are identified using an inverse method. The joining process is simulated on a 2D axisymmetric
configuration using the finite element software Forge2005. The simulations are compared to the experiments on the basis of
geometrical cuts and load-displacement curves. Good agreements are observed. After the generation of a 3D specimen from
the final 2D simulation of riveting, we compare the mechanical strength of the virtual connection with results obtained using an
ARCAN type device. We show that the use of a coupled damage approach is essential to get predictive results.
Introduction
To decrease the weight of new cars, aluminium alloys are progressively replacing steels in parts of the “body-in-white”.
However, the use of aluminium alloys requires sometimes new joining techniques to replace classical welding-points. Indeed
the welding of aluminium alloy sheets is not easy due to the high thermal conductivity, low melting range and the natural
surface oxide layer. In the same way, it is difficult to join steel and aluminium alloy sheets together due to the great difference
of melting point. To overcome this problem new joining techniques need to be used. Self-piercing riveting (SPR) or clinching
are two of relatively new joining techniques in which joining comes from the materials plastic deformation. The present study
focuses on the self-piercing riveting: its numerical modelling and experimental validation.
A literature review shows that the SPR process is getting more and more studied. These studies deal with the experimental
testing of riveted specimens [1], or with the numerical simulation of riveting process [2-3]. The originality of our approach,
initially presented in [4], is the use of damage in the modelling of material behaviors sheets. Throughout this paper, we will
show the importance of taking damage into account in order to get more accurate results.
In the present work the possibility to simulate the riveting process and the mechanical strength of the joint using the finite
element software Forge2005 is presented. All the methodology from the material parameters characterization to the
mechanical tests of the riveted specimens is described.
Self-Piercing Riveting Process
During the SPR process, a semi-tubular rivet is pressed by a punch into two or more sheets which are maintained between a
blank-holder and a lower die. The rivet pierces (through the thickness of) the upper sheet and flares into the bottom sheet. The
process is a combination of the following four steps (Fig. 1). First the blank holder presses the two sheets against the die (a);
then the punch presses the rivet so that the rivet pierces the upper sheet (b). Next the rivet flares into the lower sheet due to
the lower die geometry (c). To finish the punch and the blank-holder release the joining point (d).
(a)
(b)
(c)
(d)
Figure 1: Self-piercing riveting process
The mechanical strength of the final joint depends on the die and rivet geometries, pre-clamping pressure of the blank holder
and on rivet setting pressure.
Damage Elastic-Plastic Model
Materials for the rivet and the metal sheets used in this study are aluminum alloy and steel. The behavior of these materials is
damaged elastic-plastic. Hardening and damage are assumed to be isotropic. To model the ductile damage, we have opted for
the Lemaitre damage model [5]. Damage is represented by a new internal variable D . The value of D is ranging from 0
(sound material) to 1 (fracture). This variable allows define the effective stress tensor ~ which represents the stress tensor in
the equivalent sound material:
σ
σ
σ
where
σ
~=
(1)
(1 − D )
is the stress tensor for the damaged material given by the Hooke’s law:
= (1 − D )E :
σ
e
(2)
ε
The yield function used is the von Mises yield criterion:
J2 ( )
− σ (ε ) ≤ 0
(1 − D ) 0
(3)
) is the second invariant, J 2 ( ) =
3 2 s : s in which s is the deviator stress tensor.
σ
f ( , ε , D) =
σ
where σ 0 is the isotropic hardening law. J 2 (
σ
σ
The evolution laws of the plastic strain tensor and the damage are respectively defined by:
ε
& p = λ&
∂f
λ& 3 s
=
∂
1− D 2 J2 (
σ
D& =
σ
Y

(1 − D )  S 0
λ&



(4)
)
b
(5)
where S 0 , b are parameters and Y is the damage strain energy release rate given by:
2

J 2 ( )  2
(1 + ν ) + 3(1 − 2ν ) p

2 E (1 − D ) 3
 J2 (

σ
Y=
σ
where p is the hydrostatic stress,


) 
2




p = (1 3) tr( ) , E is the Young modulus andν is the Poisson ratio.
σ
(6)
λ& is the plastic multiplier which is subjected to the Kuhn-Tucker conditions:
λ& ≥ 0,
f ≤ 0, λ&f = 0
(7)
To complete the damage model, we need to introduce the strain threshold ε d from which the damage starts and the critical
damage parameter Dc . Dc is the trigger value from which fracture occurs. This damaged elastic-plastic model was
implemented in the finite element software Forge2005®.
Identification of the Material Parameters
To get predictive results, it is important to be able to identify accurately materials parameters. This is done through an inverse
analysis technique by comparing numerical and experimental results. The experimental data are provided from different
mechanical tests. For metal sheets we have performed tensile tests on plate specimens as well as indentation tests (Fig. 2).
Tensile specimens were cut along the grain flow direction. An extensometer was mounted in the gage section to determine the
tensile strain. Tensile tests were performed on a servo-hydraulic machine with a controlled speed of 0.5 mm/s. Figure 2a gives
an example of typical load-displacement curves obtained during a tensile test of a 5754-O aluminum alloy. Indentation tests
were performed on the same device at a speed of 0.1 mm/s. The displacement was measured thanks to an LVDT sensor. An
example of the response obtained during the test is shown on Fig 2b. The objective of this second test is to identify sheets
mechanical behavior using a test which is more representative with respect to the SPR process.
0,25
0,4
0,35
0,2
0,15
Load (T)
Load (T)
0,3
0,1
0,25
0,2
0,15
0,1
Experimental
0,05
0,05
Numerical
0
0
0
(a)
2
4
6
8
10
0
(b)
Displacement (m m)
0,2
0,4
0,6
0,8
1
1,2
1,4
Displacem ent (m m)
Figure 2: AA 5754-O characterization: (a) tensile test, (b) indentation test
0,5
1,8
0,45
1,6
0,4
1,4
0,35
1,2
0,3
Load (T)
Load (T)
The rivet, provided by Böllhoff, is made of boron high-strength steel. Its material properties are not easy to identify. Two
different tests have been performed for the rivet. The first one is a flare test of the whole rivet whereas the second is a
compression test of a cylinder cut in the shank of the rivet like in [3]. The rivet used in these tests is a K 5 x 8 rivet. The
displacement is measured thanks to an LVDT sensor and the prescribed speed is 0.1 mm/s. Figure 3 presents typical loaddisplacement curves obtained for these two tests.
0,25
0,2
0,6
0,15
Experimental
0,1
0,05
0,4
0,2
Numerical
0
0
(a)
1
0,8
0
0,5
1
Displacem ent (mm )
1,5
2
(b)
0
1
2
Displacem ent (m m )
Figure3: Rivet characterization: (a) compression test, (b) flare test
3
4
To determine the parameters of the isotropic hardening and of the damage law, we use an inverse analysis method. The
methodology is based on an evolutionary algorithm [6]. Due to the great number of unknown parameters, we perform the
identification in two steps. First the isotropic hardening law is identified on the first part of the load displacement curve, i.e.
before the softening occurs. Then damage parameters are determined on the rest of the curve. For the hardening law, we use
a Krupkowski power law:
σ 0 = 3K (ε + ε 0 )n
(8)
where K is the consistency, n is the isotropic hardening exponent, ε 0 allows us to take into account the yield stress. Table 1
summarizes the identified parameters of the hardening law.
Table 1: Material Parameters
Material
Young modulus, E (GPa)
K
AA 5754-O
XSG steel
Rivet-
64
210
190
250
350
1250
(MPa)
Initially, to define the damage law, we need to determine four parameters:
n
ε 0 (%)
0.3
0.282
0.01
0.45
1.22
ε d , Dc , S 0 and b . In a first approximation, we can
identify ε d and Dc thanks to the stress-strain curves [5]. Indeed, it can be assumed that the damage starts when the plastic
strain reaches the values corresponding to the ultimate stress σ U . Moreover the critical damage parameter is given by:
Dc = 1 −
σR
σU
(9)
where σ R is the stress value at rupture. Table 2 summarizes the identified parameters of the damage law.
Table 2: Damage Parameters
Material
εd
S 0 (MPa)
b
Dc
AA 5754-O
XSG steel
0.23
0.28
0.8
3
1
2
0.18
0.14
Simulation and Validation of Self-Piercing Process
Forge2005® is used to perform simulation of the self-piercing riveting process. This software enables to perform 2D or 3D
simulations. Moreover the key phenomena, namely multi-body contact [7], large deformations, damage and simulation of
fracture, which appear during the process, are incorporated into the finite element solver. An automatic remesher enables to
preserve a good mesh quality during the process.
To simulate the self-piercing riveting process, a 2D axisymmetric configuration was carried out and represented on Figure 4.
The blank-holder, the punch and the lower die were assumed as rigid bodies.
Punch
Blank-holder
Rivet
Top sheet
Bottom sheet
Lower die
Figure 4: 2D axisymmetric representation of the SPR process
Two configurations are listed in table 3. In every configuration, the pre-clamping pressure is 594N and the punch velocity is 20
mm/s. The phenomenon of upper sheet rupture is modeled using a kill-element technique. When damage reaches a critical
value inside an element (typically Dc ), the element mechanical contribution to the mechanical problem is set to 0 and the
element is deleted from the mesh.
Table 3: Definition of two different configurations
Configuration
Material
Sheet 1
Sheet 2
1
AA 5754-O
AA 5754-O
2
AA 5754-O
XSG steel
Sheet 1
1
1
Thickness
Sheet 2
2
2
Rivet
Die
C5x5
C5x5
DZ0902000
DZ0902000
In order to validate our simulations, an extensive experimental program was performed to obtain experimental data. The
joining process stage was achieved on a Böllhoff device. For each configuration, we have recorded the evolution of the punch
load versus displacement during the process and we have performed geometrical cuts. In Figure 5, the final computed
geometry is superimposed with a cut of the experimental specimen.
Numerical
geometries
Figure 5: Numerical –experimental comparison of geometrical cuts. Left: configuration 1. Right: configuration 2.
The comparison between numerical and experimental geometries is very good. Indeed, the shape of the sheets as well as the
flare of the rivet is well predicted. In Figure 6, experimental and numerical load-displacement curves are compared for
configuration 2. Once again a good agreement is observed. However we can notice a slight difference at the end of the curve.
This may come from the deformation of the experimental device due to the high level of load reached at the end of the riveting
process.
4,5
4
Experimental
3,5
Load (T)
3
Numerical
2,5
2
1,5
1
0,5
0
0
1
2
3
4
5
6
Displacem ent (m m )
Figure 6: Comparison of numerical and experimental load – displacement curves
Figure 7a represents the influence of damage on the load – displacement curve. Damage seems to modify the riveting load
only at the end of the process. Indeed when the rivet pierces the top sheet, damage localizes in a narrow band near the tip of
rivet. This damaged area is not wide enough to change the riveting load. On the contrary when the rivet flares into the bottom
sheet, high strain level appears and damage plays an important role (Fig. 7b).
4,5
4
3,5
Experimental
Load (T)
3
Numerical with damage
Numerical without damage
2,5
2
1,5
1
0,5
0
(a)
0
1
2
3
4
5
6
(b)
Displacem ent (m m )
Figure 7: Influence of damage on the numerical load – displacement curve and damage map at the end of process
Tests and Simulations of the Mechanical Strength of the Joint
The second aim of this paper is to present the ability of our model to simulate the mechanical strength of the joined specimen
under static loading (tensile, shear and mixed solicitations). The experimental data are obtained thanks to an ARCAN test (Fig.
8a) as in [8]. This specific device enables to mix and control tensile and shear loadings on a riveted cross-shaped specimen
(Fig. 8b). For each configuration we obtain the load-displacement curves for three different angular positions (0° pure traction,
45° and 90° pure shear) as well as the failure mode of the specimens.
(a)
(b)
Figure 8: ARCAN device and a riveted cross-shaped specimen
Simulations of the mechanical strength of the riveted specimen are achieved using Forge2005®. Mechanical fields are
exported from a 2D axisymmetric mesh to a 3D mesh. The creation of the 3D sample is performed in many steps using the
final geometries of the SPR process. First, a 3D geometry is extrapolated from the 2D geometry using a simple revolution.
Then a new mesh is generated and the mechanical fields (residual stresses, damage, …) are transported using an
interpolation technique. To finish, the 3D circular meshes are cut to have a rectangular numerical specimen closest to the
experimental geometry (Fig. 9).
Figure 9: Creation of the 3D numerical sample
Figure 10 represents experimental and numerical curves for configuration 1. Experimental and numerical curves exhibit the
same global behavior. For the pure tensile loading, we can notice that the numerical load does not reach the same maximum
value than in experiment. Numerical damage is higher than in reality. A better investigation of the damage parameters might
0,2
0,18
0,16
0,16
0,14
0,14
0,12
0,12
0,1
0,08
Experimental
0,06
0,3
Numerical
0,1
0,08
0,06
Numerical
0,04
0,4
0,35
Experimental
Load (T)
0,2
0,18
Load (T)
Load (T)
be performed in further computations. For the mixed solicitation, the numerical curve reproduces the global behavior. The
computed curve obtains for the pure shear loading is underneath the experimental one.
4
Experimental
0,1
0,05
0,02
0
2
0,2
0,15
0,04
0,02
0
0,25
6
8
Numerical
0
0
0
5
Displacement (m m)
10
0
15
1
2
(a)
3
4
5
6
Displacem ents (mm )
Displacem ent (m m)
(b)
(c)
Figure 10: Numerical – experimental comparison of load – displacement curves: (a) 0°, (b) 45°, (c) 90°
Figure 11 shows a comparison between numerical and experimental failure modes. The failure mode is reproduced with a
good accuracy for pure traction condition (Fig. 11a) even if the numerical fracture is wider than the experimental one. This is
another evidence that the damage parameters are not accurate enough. For mixed solicitation (Fig. 11b), the failure mode is
well reproduced. For pure shear loading (Fig. 11c), the rivet rotates during the numerical simulation and is pulled out from the
bottom sheet. Inversely, in the rivet remains in the bottom sheet during the experiments whereas the upper sheet is pulled out.
(b)
(a)
(c)
Figure 11: Comparison between numerical and experimental failure modes: (a) 0°, (b) 45°, (c) 90°
As for the self-piercing riveting process, simulations were performed to characterize the influence of damage on loaddisplacement curve. Figure 12 summarizes these results on a pure tensile loading. Damage obviously modifies the whole
behavior of the assembly. Indeed, damage induces softening in the material behaviors until fracture. Then the computation
without damage is more resistant than the computation with damage. Moreover, without damage, it is not possible to predict
the failure mode. Damage plays an important role on the mechanical strength of the final joint.
0,25
Experimental
Numerial with damage
Load (T)
0,2
Numerical without damage
0,15
0,1
0,05
0
0
0,5
1
1,5
2
2,5
3
Displacement (mm )
Figure 12: Influence of damage on load – displacement curves
Conclusion
The self-piercing riveting process as well as the mechanical strength of riveted specimens were simulated and compared with
experimental results. We use a damaged elastic-plastic material using the Lemaitre ductile damage model. After identifications
of material parameters, we have performed 2D axisymmetric computations of the riveting process and 3D computations of
mechanical strength of the joining point. Simulations were compared to experiments and good agreements were observed. For
the riveting process and the static loading computations, we have shown the importance of damage in order to have good
accuracy between computations and experiments.
Some improvements might be done in order to enhance the numerical model. Indeed, we have pointed out the fact that a
better investigation of the damage parameters might be performed. Moreover the use of coupled damage generates a wellknown numerical issue: the damage localization phenomena, i.e. numerical results become dependant on the mesh size and
orientation. A sensitivity study might be performed. From the material behavior point of view, we need to investigate the
dependency of the material to the strain rate especially for steels.
In the future, such a tool can be used to improve SPR mechanical strength by optimizing rivets or SPR tools geometries.
Acknowledgments
We acknowledge the support of CETIM Technical Center, P.S.A and Böllhoff for providing experimental data and financial
support for this study.
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