Forming kinematics analysis of countersinking process

I. Jallouli*, H. Mosbah**, A. Krichen*** *Faculty of Sciences Gafsa, Campus Universitaire Sidi Ahmed Zarrouk, Gafsa University, Tunisia, E-mail: brahimjallouli@yahoo.fr **Faculty of Sciences Sfax, University of Sfax, B.P. 1199, 3031 Sfax, Tunisia, E-mail: hassen.mosbah@yahoo.fr ***Laboratoire de Génie de Production Mécanique et Matériaux, National Engineering School of Sfax, University of Sfax, B.P. 1188, 3032 Sfax, Tunisia, E-mail: Abdelkader.krichen@yahoo.fr


Introduction
Countersinking is a process used in sheet metal forming.This process consists of forming a truncated hole.The obtained internal shape is called countersink and it is used to locate a rivet or screw head.When placed in conical hole, the head of these components must sit flush with the surface with the surrounding material.In practice, the sheet is rigidly clamped around its periphery by a blankholder, and then a truncated punch concentrates deformation to enlarge the side of existing hole Fig. 1.Workpiece performed by countersinking are known by their high mechanical properties and their low cost beside the low material waste and the high productivity.
Due to the conservation of metal volume in plastic domain, countersinking process induces a complex forming kinematics in sheet.Therefore, expansion and retraction of the workpiece can occur.Industrial analyses show that these phenomena affect widely the final shape of the obtained countersink.In addition, some problems can be observed when countersinking process was performed by progressive die.The problems become serious when pilots and punched holes were designed for proper location of sheet.For countersinking process, researches were mainly focused on surface characteristics [1], fatigue life [2][3][4][5], the cracking behavior [6] and the characteristics of the riveted link [7].Up to now, there is a very scarce of literature relating firmly to the countersinking process.In the preliminary study [8] performed by the authors; a FE model has been built and validated experimentally.Jallouli et al., 2011 improved the methodology can be used in determining the parameters of the countersinking process.
In this paper, a finite element (FE) model has been developed to understand forming kinematics.In particular, efforts were concentrated to quantify expansion and retraction phenomena in countersinking process by analyzing the evolution of hole diameter, sheet thickness near countersink and diameter of workpiece.Results were also discussed as a function of blank-holding way.

FE modelling and experiments
In view of the large number of interacting factors and the complexities of the countersinking process (plasticity, bending, stretching and friction), our attention is focused on developing FE models to lead numerical analysis.The standard version of the FE code ABAQUS has been used.Some experiments were performed to verify the efficiency of the FE models Since the stress field in the workpiece during the countersinking process is mainly in a compression state, the material behavior was characterised under compression test.A special sandwich sample had been made of four disks of sheet which were put together [10].They have a diameter of 10.5 mm so that no buckling effect was observed [8].The tests were carried out under displacement control at a constant rate of 5 mm per minute.To obtain the stress-strain curve shown in Fig. 2, the conventional analysis assumes a homogeneous state of stresses and strains in the specimen.
For the FE analysis, the metal sheet was assumed to have elastic-plastic behavior with an isotropic hardening rule.The material was assumed to be isotropic and to obey the von Mises yield criterion given by: J2 denotes the second stress invariant that is defined as follows: where S is the deviatoric part of the Cauchy stress tensor  .R denotes the isotropic hardening function that is chosen with form of Swift type: where ε0, σ0, K and m are material parameters for the Swift model.

Workpiece Die
Blank-holder The evolution of the plastic strain rate is obtained by using the normality rule: where n gives the direction of the plastic strain increment and d  , called the plastic multiplier, determines the magnitude of the plastic strain increment.
The values of the model parameters (ε0 = 278 MPa, K = 580 MPa, m = 0.125 and ε0 = 0.00276) were obtained by the best fit to the experimental stressstrain curve in the rolling direction as shown in Fig. 2.

Countersinking parameters
To succeed countersinking process, some parameters have to be controlled, especially the tool geometry parameters.A schematic diagram of the countersinking process and the final screw hole is shown in Fig. 3.
The workpiece is a commercial steel sheet with a thickness e = 2.7 mm.It is a washer with an external diameter dr = 30 mm and an initial hole diameter di = 6 mm.The cone semi-angle Øp of the truncated punch was set to 90°.To analyze the effect of blank-holding way, three cases were performed: without blank-holder (C1), with fixed blank-holder (C2) and with applying a constant displacement (δbh = 0.01 mm) on the blank-holder (C3).Countersinking tests were carried out with an universal traction-compression testing machine.The tests were performed using displacement control at a constant low rate of the punch equal to 5 mm per minute.The outputs of displacement gauge, and load cell were continuously stored with data acquisition.

FE modeling
The simulations were performed by the standard version of the FE software ABAQUS.The workpiece was modeled as a deformable body.Die, punch and blankholder were modeled as rigid surfaces.Owing to the revolution symmetry of the geometry and loading, countersinking process were considered as an axisymmetric problem.A typical FE mesh is shown in Fig. 5.A 3-nodes linear axisymmetric triangle hybrid constant pressure element (CAX3H) was used.An automatic mesh generator was used to generate the FE mesh.The elements in the contact area were appropriately refined.Due to the large strain level in the countersinking process, an updated formulation of the mesh was used during the FE calculation.Coulomb's friction law was chosen to solve the frictional effect of the tool-workpiece interfaces.To describe reasonable contact conditions between the tools and the workpiece, a friction coefficient of 0.2 was assumed in the simulation.

Experimental validation
The described FE model is validated by comparison to experimental results in terms of relationship between the punch load and the punch travel (Fig. 6).The FE results can also be used to estimate the countersink shape for a given punch load.Fig. 7 shows a comparison between experimental and simulated countersink.It is worth to note that the results are in good agreement not only in the contact zone but also in the compressed zone under the punch.

Results and discussion
The investigations were first focused on the pre-diction of the load-displacement relationship.Efforts are secondly concentrated on the forming kinematics during countersinking process.

Punch load-punch displacement
Punch load versus punch displacement was plotted in Fig. 8 for the three cases (C1, C2 and C3).For each blank-holding case, non-linear relationship was observed.Several parameters cause this observed nonlinearity (nonlinearity of the stress-strain behavior, complexity of the boundary conditions, and nonlinearity of the contact).It was found that the change in blank-holding does not affect the punch load-punch displacement relationship at the beginning of the process.But at the final stage of loading, the punch load increases.Stronger the holding, higher the increase of punch load.Nevertheless, the small change in punch load does not describe the change in forming kinematics.
Fig. 8 Punch load versus punch displacement

Forming kinematics
When countersinking was performed without blank-holder (C1), the outer edge of the workpiece moves up to a certain level (δM).Thus, the final shape reveals a loss of the flatness in the outer edge of the finished product.To accomplish the countersink in the others cases (C2 and 3), deformed shape of workpiece shows that the punch pushed the material towards the free regions as shown in Fig. 9. Fig. 9 Deformed shape of workpiece That is to say, material flows to the regions where no contact area was observed (neither with truncated surface nor with blank-holder).The motion of material toward the center induces retraction of the initial hole.The motion in opposite direction of punch displacement induces axial expansion near the obtained countersink.Finally, sliding under blank-holder produces a radial expansion of the workpiece.
It is clear that analysis of deformed shape gives a good idea about forming kinematics.But it is also important to quantify the observed retraction and expansion and illustrate their evolution as a function of blank-holding conditions.In Figs. 10 and 11 were showed the minimum hole diameter (dint), the maximum sheet thickness (ef) and the increase (Δdext) of the workpiece diameter (dext) during countersinking process respectively.The decrease of (dint) indicated that the retraction of the initial hole was started immediately with the beginning of punch displacement.On the other hand, evolution of (ef) illustrates that axial expansion was started progressively.As regards (Δdext) evolution, it illustrates that the radial expansion was radial expansion of workpiece is appeared; both retraction and axial expansion were softened.

Conclusion
This study shows that four geometrical changes in the shape of workpiece were observed when countersinking process was performed.Consequently, a complex forming kinematics, which is widely linked to the blankholding conditions, was found.This leads to a difficulty in the prediction of the final shape and proves that the FE model can be an appropriate tool for production and design of sheet parts involving countersink.
The comparison between FE simulations and experiments shows that the developed FE model can be an appropriate tool to predict the punch load, the forming kinematics and the final shape with an acceptable agreement.Finally, it seems interesting to study the effect of the initial anisotropy of the sheet on the FE results.This will be done in future works.

I. Jallouli, H. Mosbah, A. Krichen FORMING KINEMATICS ANALYSIS OF COUNTERSINKING PROCESS S u m m a r y
This paper deals with the countersinking process which permits to modify the entrance of an existing drilled hole and to obtain the screw hole countersink.Nevertheless forming kinematics during countersinking process was studied as a function of blank-holding conditions.The effect of blank-holding on the countersinking process was investigated using numerical and experimental analysis.An axisymmetric elasto-plastic finite element model was developed.Four geometrical changes were identified in the

Fig. 6
Fig. 6 Punch travel curves, experimental and FE analysis