M Oudjene
email: mourad.oudjene@insic.fr

J.-L Batoz

L Penazzi

F Mercier

A methodology for the 3D stress analysis and the design of layered sheet metal forming tools joined by screws

Keywords: FEM, Stamping process, Contact forces, Layered tooling, Rapid prototyping, Joining, Optimization

In sheet metal forming industry, several experimental trials have been carried out successfully in the last 10 years, to demonstrate the feasibility of layered forming tools, by various rapid prototyping techniques. Recently, a layered stamping punch, based on the Stratoconception ® process, was produced in a Craft European project Fastool and used successfully to produce some parts. The layered punch was joined using several screws in addition to an epoxy adhesive. Numerical simulation softwares allowing for 3D stress analysis of the layered tools are needed to evaluate their deformation and the various solutions of an assembly system. In this context, the authors proposed and developed a simplified numerical procedure, based on two steps, for the 3D stress analysis of deformable tools (layered or not). In addition, an optimization procedure, based on design of experiments and response surface method, has been established in order to optimize the screw positions, which are crucial in the aim of achieving the required high strength and life duration of the assembly technique by screws. The results show the feasibility of the developed procedure in the context of industrial applications, the potential interest of the optimization of the screw positions is also outlined.

Introduction

In the context of sheet metal forming processes, traditional manufacturing of tools is both costly and time consuming. The global cost of stamped parts is highly influenced by the cost of the tools, depending on the volume production. Therefore, several new technologies, in particular rapid prototyping techniques based on the assemblage of steel sheets in layers have been developed in the last 10 years, in order to reduce cost and manufacturing delays of tools. In the last recent years, several developments on layered tools have been undertaken. Their production and their use on pressforming to produce stamped parts have been successfully explored. Several experimental studies dealing with layered forming tools are available in the technical literature [START_REF] Nakagawa | Advanced in prototype and low volume sheet forming and tooling[END_REF][START_REF] Mueller | Rapid Prototyping Techniques to Manufacture Sheet Metal Forming Tools[END_REF][START_REF] Oudjene | Mechanical analysis of prototyping tools for sheet metal stamping[END_REF][START_REF] Muller | Rapid tooling approaches for small lot production of sheet metal parts[END_REF][START_REF] Walczyk | Design and analysis of recomfortable discrete dies for sheet metal forming[END_REF][START_REF] Walczyk | Rapid tooling for sheet metal forming using profiled edge laminations-design principles and demonstration[END_REF][START_REF] Walczyk | A comparison of rapid fabrication methods for sheet metal forming dies[END_REF][START_REF] Dickens | Principles of design for laminated tooling[END_REF][START_REF] Abel | Deux projets européens de recherche collaborative en PME (CRAFT)-Intégration du procédé de Stratoconception ® en outillage rapide direct pour l'injection, l'emboutissage et la fonderie sous pression[END_REF][START_REF] Wadman | Fluid cell forming tools produced by different rapid prototyping methods[END_REF][START_REF] Oudjene | 3D finite element modeling and analysis of the laminated tooling behaviour in sheet metal stamping process[END_REF][START_REF] Oudjene | Modélisation et optimisation du comportement mécanique des outils d'emboutissage stratifiés précontraints par vissage[END_REF][START_REF] Pepelnjak | Adaptable tooling sets for metal forming of geometrically similar products[END_REF]. None of these works, however, have examined the mechanical behaviour of the layered tools. In addition, the published papers and reports show that the research per-formed to date on layered forming tools has been only oriented towards the problems of their rapid and flexible production [START_REF] Pepelnjak | Adaptable tooling sets for metal forming of geometrically similar products[END_REF]. However, there is still a need for insight knowledge of the basic mechanisms of layered tools, in particular the joining technique.

Previous numerical analyses of forming process have concentrated exclusively on the behaviour of the workpiece (stress, strain, shape), mainly assuming rigid tools during analysis. At present a limited number of papers dealing with the elastic forming tools analysis have been published. Some numerical simulations of the deep-drawing by considering deformable tools, in relation with the wear behaviour, are available in [START_REF] Dankert | Reduction of the residual stresses in a deep-drawing cup by modifying the draw die profile[END_REF][START_REF] Damborg | Optimization of the draw die profile with regard to wear using the FEM[END_REF]. Other studies of the deformable tools, in relation with the stress analysis can be found in [START_REF] Oudjene | Modélisation et optimisation du comportement mécanique des outils d'emboutissage stratifiés précontraints par vissage[END_REF][START_REF] Burlat | Analyse Mécanique et Tribologique de l'Emboutissage: application aux Outils fabriqués par combinaison de différents matériaux[END_REF], but the works dealing with the mechanical behaviour of tools is mostly dealing with 2D situations (plane strain, axisymmetry), due to the complexity of the forming operation and computational aspects involved.

Stamping simulation taking into account the elastic deformation of tools during analysis is needed to evaluate the stresses and strains developed in the tools during the forming operation. In simulation of stamping parts with complex geometry, the incremental and dynamic explicit approach is, generally, used to deal with the contact conditions with friction. In this case, however, the mechanical behaviour of the tools cannot predicted with fairly good precision and in acceptable CPU times, if in the FE-analysis one considers the coupling between the blank forming and the linear elastic analysis of the tools [START_REF] Oudjene | Modélisation et optimisation du comportement mécanique des outils d'emboutissage stratifiés précontraints par vissage[END_REF].

In this paper, the authors proposed and developed a numerical procedure for the 3D stress analysis of the forming tools, in particular the layered ones. In the case of layered tools assembled by screws, an optimization procedure, based on the design of experiments and response surface method, has been established in order to optimize the screw positions as an attempt to improve the strength and the life duration of the joining technique. The results are presented in the context of an industrial benchmark test (layered stamping punch based on the Stratoconception ® process [21]) assembled by screws in addition to an epoxy adhesive. In the present study, the adhesive thickness was not modelled in details. In fact the presence of the adhesive leads to local stress distribution effects which are neglected here. The authors would like also to mention that the present layered punch is only foreseen for prototyping and low volume production.

Design principles of the layered punch

Within the framework of the Craft European project Fastool, an industrial layered stamping punch (Fig. 1), with dimensions: 420 mm × 220 mm × 77 mm, based on the Stratoconception ® process, was produced by stacking 7 mm-thick of 2C45 (XC48) steel sheets [21].

The main advantages of layered tools over those produced conventionally from 3D full machining of steel or cast iron block are their reduced manufacturing delays, their costs and their great adaptability (which still be greatly improved and optimized).

The rapid prototyping process

The Stratoconception ® process consists in the slicing of the part by computing, into a set of elementary layers in which plugs are inserted. The elementary layers are identified and then directly manufactured by 2.5 axis rapid milling, or by 5 axis machining from sheet material. These elementary layers are then fitted together in order to reconstruct the final product. The assembly of the layers by various techniques should be designed to support the mechanical constraints involved in working conditions [START_REF] Abel | Deux projets européens de recherche collaborative en PME (CRAFT)-Intégration du procédé de Stratoconception ® en outillage rapide direct pour l'injection, l'emboutissage et la fonderie sous pression[END_REF]21].

Machining of the layers

The different layers were directly machined with 7 mm-thick of 2C45 (XC48) steel sheets using laser cutting and micromilling (Figs. 2 and3) [START_REF] Abel | Deux projets européens de recherche collaborative en PME (CRAFT)-Intégration du procédé de Stratoconception ® en outillage rapide direct pour l'injection, l'emboutissage et la fonderie sous pression[END_REF].

The joining technique

In practice, several joining techniques (such as brazing, sticking, riveting, screwing, bolting) can be suitable for rapid manufacturing techniques. In the layered punch, shown in Fig. 1, the layers were joined by several screws with 7 mm diameter after being positioned by four plugs with 6 mm diameter. In addition, the layered punch was reinforced using an epoxy adhesive with 0.1 mm of thickness, which was accommodated between layers.

The layered punch has been successfully used on pressforming to produce some stamped parts (Fig. 4) without any visual deterioration of the punch. However, the joining technique by screws is based only on simple practical rules and experience of designers and a number of complexities are still not fully understood or not modelled to date. Accordingly, the mechanical behaviour of the layered punch could be investigated, in particular the joining technique by screws, which is vital in designing layered tools.

Elastic analysis of tools: some difficulties

To properly analyse the behaviour of forming tools, it is justified to take into account their elastic deformation during the forming operation. However, at present, little research have been carried out on the stress analysis of tools. The stresses and strains induced in the forming tools during the forming operation are seldom computed in practice if not ignored in most of the today forming simulations.

The FE ABAQUS code is used to investigate the deep-drawing of a cylindrical cup (benchmark test of Numisheet'2002) [START_REF]Proceedings of the 2nd International Conference Numerical Simulation of 3-D Sheet Metal Forming Processes-Verification of Simulation with Experiments (Numisheet[END_REF], by considering elastic deformation of the cylindrical punch during analysis. The main purpose of this study is to examine the possibility to predict the behaviour of tools by coupling the simulation of the blank forming and the linear elastic analysis of the tools. The deep drawing parameters are shown in Fig. 5 [START_REF]Proceedings of the 2nd International Conference Numerical Simulation of 3-D Sheet Metal Forming Processes-Verification of Simulation with Experiments (Numisheet[END_REF].

The material data are: isotropic elasto-plastic material with hardening given by Hollomon's model with the following parameters: n = 0.27, K = 539 MPa. Other sheet material properties are: E = 210 GPa and ν = 0.3. The punch was assumed as isotropic and elastic material with the following properties: E = 210 GPa and ν = 0.3 while the other tools (die, blankholder) were regarded as rigid. Coulomb friction model was adopted with frictional ratio µ = 0.042 as considered in Numisheet'2002 benchmark test [START_REF]Proceedings of the 2nd International Conference Numerical Simulation of 3-D Sheet Metal Forming Processes-Verification of Simulation with Experiments (Numisheet[END_REF].

Two-dimensional FE model was adopted using both static implicit and dynamic explicit approaches. All simulations have been carried out with a constant blankholder force of 70 kN and for a punch stroke of 40 mm.

Static implicit model

The blank was modelled by 4 and 270 axisymmetric elements with reduced integration (denoted CAX4R in ABAQUS) in the thickness and radial directions, respectively. The punch was modeled by 821 axisymmetric elements (denoted CAX4 in ABAQUS), see Fig. 6.

The results quality of the simulation procedure has been already evaluated in previous work [START_REF] Oudjene | Modélisation et optimisation du comportement mécanique des outils d'emboutissage stratifiés précontraints par vissage[END_REF], with regard to both the final part and the punch behaviour. The results showed good agreements with the literature ones regarding the workpiece. Fig. 7 shows the equivalent von-Mises stress distribution induced in the punch at the end of the forming stage. The maximum computed equivalent von-Mises stress is around 35 MPa.

Furthermore, the contact forces (total punch reaction force) obtained by considering elastic deformation of the punch was compared to that obtained in the case of classical simulation by considering rigid tools (Fig. 8).

It can be seen that the elastic deformation of the tools do not have significant effect on the contact forces and it can be concluded that the deep drawing process contact force results are almost the same between rigid and elastic standard steel tools.

Dynamic explicit model

For the simulation of stamping parts with complex geometries, the dynamic explicit scheme is generally used because of the contact conditions with friction. The objective of this model is to check that we obtain the same results regarding the punch behaviour as with a static implicit scheme. The simulations were run using a classical punch velocity V p = 10 m/s. The effect of the mesh quality of the punch on the results with regard to the stress distribution was studied [START_REF] Oudjene | Modélisation et optimisation du comportement mécanique des outils d'emboutissage stratifiés précontraints par vissage[END_REF], but it was found that, when one uses a classical punch velocity of 10 m/s, it is impossible to obtain a good estimation of the stress developed in the punch (Fig. 9) compared to that obtained in the case of static implicit scheme (Fig. 7). The maximum amount of von-Mises stress developed in the punch is around 150 MPa. This is an important (numerical) result.

However, with the decrease of the punch velocity from 10 to 1 m/s, it became possible to obtain fairly good estimation of the stress distribution in the punch, based on the comparison between Figs. 7 and 10. In addition, the maximum value of the computed von-Mises stress is around 37 MPa. However, the CPU time has increased by a factor of 10.

The use of a punch velocity of 1 m/s allows also to obtain good estimation of the contact forces compared to that obtained in the case of rigid tools with a punch velocity of 10 m/s (Fig. 11).From the above results, it can be concluded that general 3D stamping Based on these difficulties, it is therefore relevant to decouple the simulation of the blank forming and the linear elastic analysis of the tools. Thus, the 3D stress analysis of the tools can be performed separately by exploiting the knowledge of the distribution of the contact forces acting on the rigid tools during the forming operation.

Description of the proposed method

Principle and assumptions

The contact forces acting on the rigid tools during the forming operation have been found almost identical to those acting on the real standard steel tools (see Section 3). Hence, the stress analysis of tools can be performed using the classical stamping simulation approach by considering rigid tools and by exploiting the knowledge of the distribution of the resulting contact forces. Thus, our procedure involves two separate steps (Fig. 12): In the present study, the STAMPACK finite element code and the CATIA V5 finite element module were used to perform the steps 1 and 2, respectively.

The above procedure, in two steps, is based on the assumption that the contact forces acting on the rigid tools are almost identical to those acting on the real steel deformable ones.

Estimation of the contact forces

In the beginning of the step 1, the surface of the rigid tool (active tool parts) is divided into small sectors called "Sets" (Fig. 13). For each "Set" (i), the global nodal contact force vector, with its three cartesian components, is automatically computed in STAMPACK [START_REF] Quantech | Stampack User Guide-Version 5.6, Quantech edt., Edificio Nexus[END_REF] and can be expressed as follows:

F (i) = nnd j=1 F (i) j (1)
with

F (i) = (F (i)
x , F (i) y , F (i) z ), where nnd is the number of nodes in "Set" (i).

The third component F (i) z represents the vertical punch reaction force acting on the "Set" (i). Then, the total punch reaction force, denoted F Z , can be expressed as follows:

F Z = nset i=1 F (i) z (2)
where nset is the number of sectors (or "Sets") in the active tool parts. Fig. 12. Procedure for the 3D FE analysis and the design of deformable forming tools. Fig. 13. Schematization of elementary "Sets"/corresponding contact force vector.

Equivalent static external load

In order to simplify the transfer of the distribution of the contact forces from the rigid tool to the deformable one (layered or not), the elementary force vectors F (i) computed previously, in step 1, for each "Set", are automatically replaced (by considering their three cartesian components) by a system of equivalent static loads P (i) , distributed according to the following relation:

F (i) = s P (i) ds (i) (3)
where ds (i) is the area of the elementary "Set" number (i).

The above procedure has already been validated in [START_REF] Oudjene | Modélisation et optimisation du comportement mécanique des outils d'emboutissage stratifiés précontraints par vissage[END_REF] in the context of simple deep-drawing geometries. The obtained results showed that this procedure gives fairly good stress and strain estimations in the tools compared to that obtained in the context of deep-drawing simulation taking into account elastic deformation of tools, using the ABAQUS static implicit code.

Application to the layered tools: 3D case study

The benchmark test resulting from the Craft European project Fastool was considered as the example of the application. According to the procedure illustrated in Fig. 12, two separate steps of analysis are required to make 3D stress analysis of the tool.

Step 1: classical stamping FE-analysis

In the finite element model, a mesh of three-nodes BST shell elements was used in the full model of the blank involving 382518 elements, while the other tools (punch, die and blankholder) were assumed rigid. The blankholder force was 50 kN, punch stroke was 50 mm. The material data are: isotropic elasto-plastic behaviour with isotropic hardening given by Hollomon's model with the following material constant: n = 0.23, k = 522. [START_REF] Muller | Rapid tooling approaches for small lot production of sheet metal parts[END_REF] MPa, E = 210 GPa, ν = 0.3.

The contact formulation is based on the masterslave contact approach. The tools are considered as the master surfaces and the sheet is the slave surface. The Coulomb friction model was adopted with a frictional ratio µ = 0.1 and the interaction between the sheet and the tools is formulated using the finite sliding approach, which allows the separation of the two surfaces during sliding.

First, some results with regard to the feasibility of the blank forming (shape, strain, stress, thickness distribution) were examined [START_REF] Oudjene | Modélisation et optimisation du comportement mécanique des outils d'emboutissage stratifiés précontraints par vissage[END_REF]. The obtained workpiece final shape (Fig. 14) is in agreement with the experimental one (Fig. 4).

Second, simulations were run in order to obtain an estimation of the distribution of the contact forces acting on the rigid punch during the stamping operation.

Fig. 15 shows the contact pressure distribution on the rigid punch at the end of the forming stage. It can be seen that the contact forces are mainly located along the punch radius (active parts). This result is in good agreement with the literature [START_REF] Oudjene | Mechanical analysis of prototyping tools for sheet metal stamping[END_REF][START_REF] Oudjene | Modélisation et optimisation du comportement mécanique des outils d'emboutissage stratifiés précontraints par vissage[END_REF][START_REF] Dankert | Reduction of the residual stresses in a deep-drawing cup by modifying the draw die profile[END_REF][START_REF] Damborg | Optimization of the draw die profile with regard to wear using the FEM[END_REF][START_REF] Burlat | Analyse Mécanique et Tribologique de l'Emboutissage: application aux Outils fabriqués par combinaison de différents matériaux[END_REF].

Step 2: modelling and FE-analysis of the punch

In this second step of simulation, the layered punch was analysed separately from the step 1. As shown in Fig. 16, the punch CAD model was generated in CATIA V5 and sliced into 7 mm-thick elementary layers to be consistent with the real one.

The joining technique consists of two plugs (G 1 and G 2 ) and five screws (S 1 -S 5 ), in addition to an epoxy adhesive.

The finite element model involves 210124 four-nodes tetrahedral elements with a mesh refinement in the vicinity of the screw holes.

The 2C45 (XC48) steel layers are elastic and isotropic with the following properties: E = 210 GPa, ν = 0.3.

The adhesive with thickness of 0.1 mm was not modelled in details but each interface between the steel layers is clearly identified in the 3D FE model. As a first assumption for this study, the nodes at the interfaces belonging to the layers are tied leading in fact to a continuous medium. Attention is then paid at the post-processing stage to the maximum tensile and shear stresses which are compared with the corresponding admissible yield stresses of the adhesive. If the stress levels are satisfactory, then sliding or separation of the layers is not possible. Of course the role of the adhesive on layers assembled by screws is important as mentioned in [START_REF] Oudjene | Modélisation et optimisation du comportement mécanique des outils d'emboutissage stratifiés précontraints par vissage[END_REF][START_REF] Yoneno | The strength of joints combining adhesives with bolts (case where adherends are pipe flanges of which the interfaces are bonded partially)[END_REF] but its precise evaluation is considered beyond the scope of the present study and should be addressed in a future research paper.

The loading conditions are as follows:

(i) The initial clamping state. This stage is automatically generated in CATIA V5, by assuming minimal clamping force of 6000 N for each screw. (ii) The forming load. The distribution of the contact forces, provided by the numerical simulation, in step 1, is automat- ically imported in CATIA V5 and applied to the layered punch FE model as static load.

Results and discussions

The stress and strain distributions in the layered punch have already been investigated in [START_REF] Oudjene | Modélisation et optimisation du comportement mécanique des outils d'emboutissage stratifiés précontraints par vissage[END_REF] and it was found that the joining technique by screws requires more attention and plays an important role in the design decision of layered tools. Thus, we focussed on what happens in the screws, under the forming load.

When the distribution of the contact forces (due to the stamping) is applied to the layered punch, which is joined with an initial clamping force (F s ) i , increments (variations) F sj , in the axial screw forces, are produced. Fig. 17 shows the increments in the axial screw clamping forces as a function of the axial forming load F Z . It can be seen that the increments F sj occurring in axial screw forces increase linearly with the increase of the forming load (contact forces). This result is in agreement with the literature [START_REF] Yoneno | The strength of joints combining adhesives with bolts (case where adherends are pipe flanges of which the interfaces are bonded partially)[END_REF].

From Fig. 17, it can be seen that the highest values of increments are produced in screws S 1 and S 5 , while the increment values in the other screws are not significant. This can be explained using the theoretical relationship between the increment and the axial forming load, occurring in each screw, which can be expressed as follows:

F s = [(F s ) i -(F s ) f ] = K t K t + K cg K ′ cg K pt F Z (4)
The quantity φ = (K t /(K t + K cg ))(K ′ cg /K pt ), is called the screw load factor.

K t [START_REF] Yoneno | The strength of joints combining adhesives with bolts (case where adherends are pipe flanges of which the interfaces are bonded partially)[END_REF] is the spring constant for a screw system and K cg [START_REF] Yoneno | The strength of joints combining adhesives with bolts (case where adherends are pipe flanges of which the interfaces are bonded partially)[END_REF] is the compressive spring constant for a layer around a screw hole. K ′ cg and K pt are the compressive spring constants defined using the mean displacements produced in the screw holes when the initial clamping force and the stamping load are respectively acting. (F s ) i and (F s ) f are respectively the initial and the final axial screw forces.

The slopes of F sj (Fig. 17) are independent of the initial clamping force (F s ) i and can be used in order to obtain the individual screw load factors.

On the other hand, it is found that the value of (F s ) f (=(F s ) i + F s ), in each screw, decreases because the value of F s is negative.

In such screwed joint, the initial pre-stressed screws should remain in tension under the loading conditions, for a proper strength of the joint. Hence two problems, related to this joining technique, can be identified:

(i) The decrease in the initial clamping force, due to the forming load, is not allowed when the final screw clamping force tends to zero (this can leads to screw loosening); (ii) The increment occurring in the axial screw clamping force can leads to large force increments in the screws, which can greatly affect the fatigue behaviour (inducing cyclic stress in the screws). The cyclic stress in each screw can be expressed as follows:

σ cycl = (F s ) i -(F s ) f 2A s = F s 2A s (5)
where A s is the cross-section of the screw.

Based on these remarks, it appears then necessary to reduce the increments occurring in axial screw clamping forces, due to the forming load, in order to improve the strength and the life duration of the joining technique. This can be achieved by optimizing the positions of the screws.

Optimization of the screw positions

Formulation of the optimization problem

Here, only the screw positions P j are considered as the design variables. The final objective is to obtain in a layered punch in which the increments occurring in the axial screw forces are as small as possible. Each screw position P j is then defined by the cartesian coordinates (x j , y j ) of the screw hole center, leading to two design variables per screw.

Cost function

The cost (or objective) function we consider in this paper is defined in terms of the increments occurring in the axial screw forces, such that:

F = 1 ns ns j=1 [ σ sj ] 2 = 1 ns ns j=1 [(σ sj ) i -(σ sj ) f ] 2 (6)
where σ sj = (F sj ) i -(F sj ) f /A sj is the pre-stress variation in screw j, expressed as a function of initial and final axial screw forces (F sj ) i and (F sj ) f , and the screw cross section A sj . ns is the screw number. A normalized form of Eq. ( 6) is defined as follows:

f cost = F -F min F max -F min (7)

Design constraints

The research domain of the design variables is bounded to satisfy the design rules of screwed joints [START_REF]Centre Technique Industriel de la Construction Métallique, Régles de calcul des constructions en acier "Régles CM66[END_REF]. For example, the distance T between screw axes, and the distance L between screw axes and outside edge of layers (Fig. 18) are closely related to the screw diameter d. These feasibility constraints can be expressed as inequality relations [START_REF]Centre Technique Industriel de la Construction Métallique, Régles de calcul des constructions en acier "Régles CM66[END_REF]:

G = 1.5d ≤ L ≤ 3.5d 3.5d ≤ T ≤ 7d ( 8
)
where d is the screw diameter.

Optimization strategy

To solve the optimization problem, computations based on the design of experiments (DoE) technique were conducted and coupled to an optimization method in Matlab. The initial optimization problem based on the DoE with five screws leading to 10 design variables involves 2 n + 2n + 1 = 1045 (n = 10 design variables) FE-analyses in CATIA V5 (if the central composite DoE with three levels for each variable is considered). However, the effort and experimental cost required for such a design could be prohibitive and unrealistic. In order to reduce the size of the problem (the number of design variables), we did adopt the following strategy:

(i) First, the displacements of the screws S 1 and S 5 in the Y direction are eliminated because of the small width of the layers (Fig. 18); (ii) Second, the screws S 2 , S 4 , and S 3 are restrained to move respectively according to the (D) and (D ′ ) axes. Thus the problem is reduced to five design variables (one variable per screw). In addition, preliminary simulations show that the optimization problem can be transformed in two separate sub-problems since it is found that there is no interaction between the positions of screws S 1 and S 5 with the other ones. Thus the initial optimization problem can be conducted in two steps:

(i) Step 1. The positions of screws S 1 and S 5 are first optimized (optimization step with two design variables). (ii) Step 2. The positions of screws S 2 -S 4 are then optimized (optimization step with three design variables).

Optimization results

As shown in Table 1, the developed optimization procedure allows to noticeably reduce the individual screw pre-stress increments, due to the forming load. In this case study, the minimization of the cost function (f cost ) reaches 27%.

Conclusion

In this paper, a simplified numerical procedure for the 3D FEanalysis of deformable stamping tools is proposed and discussed with particular attention to the analysis of layered tools joined by adhesive and screws. The stress analysis procedure is based on the decoupling of the simulation of the blank forming with the elastic analysis of the deformable tools.

The adhesive is not fully considered in the present analysis, but the stress field is particularly analyzed at the layer interfaces to check the validity of the assumption of material continuity.

The numerical approach has been applied on the industrial application of a layered punch designed and manufactured using the Stratoconception ® system. The following remarks can be mentioned:

(i) The decrease in screw pre-stresses are highly related to the stamping load and also to the spring constants of the joined layers around the screw holes. It is then advisable to reduce these decreases, as much as possible, by considering both appropriate screw positions and minimum allowable height of the layered tooling; (ii) The initial clamping force must be considered as an important parameter to guarantee minimal tensile force in axial screws when the decrease in axial screw force is large;

(iii) The decrease in axial screw forces can lead to large stress increments (or variations) in the screws, which can affect the fatigue behaviour.

In order to improve the design of the layered forming tools joined by screws, an optimization procedure has been proposed to optimize the screw positions. The results show its ability to reduce the force increments occurring in each screw, due to the forming load.

More detailed research works can be proposed such as (i) Advanced modelling of the layer interfaces to estimate more precisely the role of contact with friction and adhesive material on the compliance and the 3D stresses in the layered tooling and screws. (ii) Influence of inertia forces, vibrations and fatigue on the behaviour, integrity and life duration of the non-standard layered tools.

Fig. 1 .

 1 Fig. 1. Stamping layered punch. Source: CIRTES.

Fig. 2 .

 2 Fig. 2. Laser cutting of layers. Source: CIRTES.

Fig. 3 .

 3 Fig. 3. Micro-milling of layers. Source: CIRTES.

Fig. 4 .

 4 Fig. 4. Stamped part produced by the layered punch. Source: CIRTES.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Schematic illustration of the cylindrical cup [12,17]. R 1 = 50 mm; R 2 = 51.25 mm; R 3 = 9.5 mm; R 4 = 7.0 mm; R 0 = 105 mm; e 0 = 1 mm.

Fig. 7 .

 7 Fig. 7. Distribution of equivalent von-Mises stress in the punch at the end of the forming stage (static implicit scheme).

Fig. 8 .

 8 Fig. 8. Punch force-displacement curves: comparison between rigid and elastic tools (static implicit scheme).

Fig. 9 .

 9 Fig. 9. Distribution of equivalent von-Mises stress in the elastic punch: case where V p = 10 m/s.

Fig. 10 .

 10 Fig. 10. Distribution of equivalent von-Mises stress in the elastic punch: case where V p = 1 m/s.

Fig. 11 .

 11 Fig. 11. Punch force-displacement curves: comparison between rigid and elastic tools (dynamic explicit scheme).

  (i) Step 1. The stamping simulation is first conducted classically by considering rigid tools, allowing to obtain an estimation of the distribution of the contact forces acting on the rigid tools during the forming operation; (ii) Step 2. The stamping tool (layered or not) is analyzed separately by considering its elastic deformation and by applying the distribution of the contact forces computed previously (in step 1) and considered as static external load.

Fig. 14 .

 14 Fig. 14. Workpiece final shape (STAMPACK)/comparison with experiments.

Fig. 15 .

 15 Fig. 15. Contact pressure distribution on the rigid punch surface (STAMPACK).

Fig. 16 .

 16 Fig. 16. CAD model of the layered punch (CATIA V5).

Fig. 17 .

 17 Fig. 17. Increments (variations) in initial axial screw forces as a function of the stamping load (punch reaction force).

Fig. 18 .

 18 Fig. 18. Illustration of the design constraints.

Table 1 Initial

 1

	and final parameters
		Initial	Final
	f cost (%)	100	73
	P 1 (x 1 , y 1 ) P 2 (x 2 , y 2 ) P 3 (x 3 , y 3 ) P 4 (x 4 , y 4 )	(35, -56) (115, -55.5) (180, -52) (190, 19)	(35, -56) (110, -53.5) (178.2, -42) (186.5, 15.8)
	P 5 (x 5 , y 5 )	(148, 55.5)	(112, 55.5)

Acknowledgments

The authors would like to thank: Dr. A. Delamézière from ERMeP/GIP-InSIC for the fruitful discussions regarding the Design of Experiments and the response surface method; Dr. Laurentiu Neamtu from Quantech ATZ (Spain) for the helpful discussions regarding the use of the STAMPACK finite element code; CIRTES Company for providing informations on the Craft European Project Fastool.


Abderrahim Michrafy
email: michrafy@mines-albi.fr

Harona Diarra
email: harona.diarra@ulg.ac.be

John A Dodds
email: dodds@mines-albi.fr

Mohammed Michrafy
email: mohamed.michrafy@bem.edu

Luc Penazzi
email: penazzi@mines-albi.fr

Analysis of strain stress state in roller compaction process

Keywords:

come

Introduction

The roller compaction process of powders is a process of dry granulation agglomeration of powders. This process is employed in different industries including pharmaceutical, mineral and chemical industries. In the pharmaceutical industry, the process is employed for granulating formulations from mixtures of drug substances and excipient powders to improve flow ability for direct compression. The action of the process leads to modifications of mechanical properties of powders on the particle scale.

The process is based on the transport of powders using a screw feeder which is required for poor flow ability powder mixtures. The powder is then drawn between two counter-rotating rolls in the compaction zone where a high pressure is applied for forming a strip of compacted powder. The formed strip is then milled or stored. Beyond the benefits of this process (continuous, economical and easy to use); it is far from being completely controlled. The difficulty comes mainly from the strong dependence between the process parameters and the properties of powders.

The process of drawing and densification by deforming the powders between rollers involves various complex mechanisms. First, the screw feeding which must achieve a uniform and continuous flow should be synchronous with the speed of the rolls to avoid the sub-feeding or the over-feeding problems [START_REF] Guigon | Roll press-influence of force feed systems on compaction[END_REF]. A correlation between the screw and roll velocities is needed to determine the area of the good compaction rate of the press.

To predict the powder behavior between rolls, Johanson [START_REF] Johanson | A rolling theory for granular solids[END_REF] introduced a continuous model. The granular medium was considered compressible, cohesive and could be characterized with the Jenike-Shield yield function [START_REF] Jenike | On the plastic flow of Coulomb solids beyond original failure[END_REF]. To describe the process by which the granular medium is drawn and compacted between rollers, the Johanson model introduced the concept of the nip angle which defines two contiguous zones. Before the nip angle, the granular medium is transported by relative sliding on the roll, and once the nip angle is achieved, the powder is assumed to be drawn by sticking to the roll surface and to be densified by deforming under the high roll pressure. Following these assumptions, two equations were used for the determination of the nip angle and the pressure distribution in the compaction area. In this model, the granular material is described by the internal friction angle, the wall friction angle and the compressibility coefficient. The merit of the Johanson model is to propose a semi-analytic description of the basic properties of a granular material and the rolling process variables. However, this model has limitations [START_REF] Sommer | Flow and compression properties of feed solids for roll-type presses and extrusion presses[END_REF] and shows discrepancies between calculated and measured values for very compressible materials. The basic description of the powder behavior and the "a priori" definition of two zones with two types of properties of contact (slip and stick) to determine the nip angle are the most critical hypothesis. Various modeling approaches used for sheet and plate metal roll forming were adapted for the prediction of the roller compaction of a granular material. A review of some of them was published in [START_REF] Dec | Comparison of various modeling methods for analysis of powder compaction in roller press[END_REF].

The aim of this work is to study the drawing and densification process of powdered microcrystalline cellulose by roller press using a 2D finite element modeling. The modified Drucker-Prager Cap model is used as material behavior. The distributions of the process variables in contact surface between the powder and the roll are analyzed to gain better understanding of the slip-stick zones and the effect of material parameters on the maximum pressure, the nip angle and the maximum density. A comparison between the predicted and measured variables is discussed.

Experimental methods

Roll press

Experiments were carried out with a Komarek® B050PH laboratory press with a horizontal feed screw. This press was fitted with 100 mm diameter and 38 mm wide smooth rolls. The roll speed can be varied in the range of 0-7.5 rpm, the feed screw varied in the range of 0-137 rpm and the hydraulic pressure varied up to 50 kN (13.2 kN/ cm).

The upper roll was instrumented with three strain gauges to measure the normal force along the width of the roll. The strain gauges were positioned at 10 mm from the roll surface and are aligned in the direction of the axis (one gauge at the centre and the two others at 5 mm from both sides of the roll). Thus, there is no direct contact of the sensor with the powder. The signals are transmitted to the computer using a slip ring assembly fixed to the press.

Powder

Microcrystalline cellulose (MCC) known commercially as Avicel PH102, with a mean particle size (d 50 ) of 90 μm, and apparent and true densities of 270 and 1600 kg/m 3 , was used in the experiments. The low cohesion measured using an annular shear tester was 0.5 kPa and corresponded approximately to the density of 500 kg/cm 3 .

Characterization of the compaction

Both the roll speed and the roll gap affect the compacted strip density. However, in the roll compaction with horizontal feeding, the screw speed is a dominant parameter that controls the process [START_REF] Guigon | Roll press-influence of force feed systems on compaction[END_REF][START_REF] Th | Dry granulation of organic powders-dependence of pressure 2D distribution on different process parameters[END_REF]7]. It is then important to determine correlations between the screw and the roll speeds for obtaining compacts with good mechanical properties. In this study, the gap was fixed to 2 mm and a high hydraulic pressure of 34 kN was employed to avoid fluctuations in the thickness of the compacted strip.

To determine the speed conditions used to produce compacted strips and hence their resulting densities, three speed zones were identified which are associated with the under-feeding region where the powder material is not compacted, to the over-feeding zone where the machine becomes blocked, and to the good compaction area. This characterization will not be employed in the simulation presented here. The feeding system will be represented with a feeding pressure and only the roll speed will be considered. However, this characterization gives the operating conditions for the production of a satisfactory strip with the horizontal feeding system.

The process parameters such as roll pressure, nip angle and density of compacted strips are sensitive to the fluctuations of the screw and the roll speeds. To moderate this effect, the analyzed strips were produced by maintaining substantially constant the ratio of the screw-roll speeds in the good compaction area. By varying the roll speed and keeping the ratio of the screw-roll speed approximately constant, small cubs were cut from the manufactured strips and their densities (mass/volume) were characterized. The result of the averaged density for different strips was plotted in Fig. 1 according to the roll speed. The density of strips was in the range of 1000-1200 kg/cm 3 and shows an increase with the roll speed up to 6 rpm and decreases after. This decrease could be attributed to the time under the maximum pressure. However, the measured bulk density of the strips could not reflect the gradient density in the depth. As in die compaction, the transmitted pressure to the powder is higher in the areas in the contact with the rolls and decreases from the rolls to the middle of the strip. It results in a density gradient in the depth with higher values in the contact zone as it can be observed in the micrograph in Fig. 2 obtained using a numerical microscope from Keyence France.

Two-dimensional finite element modeling

Finite element modeling is a method often used in the optimization and the control of forming processes such as die compaction [START_REF] Brewin | Modelling of powder die compaction[END_REF] and roller powders compaction [START_REF] Dec | Comparison of various modeling methods for analysis of powder compaction in roller press[END_REF]. A comparison of various modeling methods applied to roller compaction was published in [START_REF] Dec | Comparison of various modeling methods for analysis of powder compaction in roller press[END_REF] and it concluded about the advantages of the finite element method over the other methods describing the roller compaction process. Roll speed, Vr (rpm) In the context of this study, the goal is to develop a 2D finite element modeling (plan strain assumption) of roller compaction to examine the development of stress, strain, velocity and density at the contact surface between the roll and the powder in order to gain more understanding of the mechanisms of drawing and densification by deforming the powder between rolls. Also, the effect of the material parameters on the distribution of the process variables is investigated.

Powder behavior

The analysis of roller compaction was conducted by assuming that the powder behaves as a single phase porous medium (without air) described by the modified Drucker-Prager Cap model implemented in Abaqus® [START_REF] Hibbit | [END_REF]. In the space (p, q) "p: hydrostatic pressure, q: equivalent stress", the yield surface is essentially represented by the shear surface (shear failure line) and the cap yield surface. The constitutive equations of the model are not listed here and can be found in the references cited above.

The model parameters that have to be identified from the experiments, such as instrumented die compaction, simple compression and diametrical compression, are: elastic properties E and ν, powder cohesion d, internal friction β, the eccentricity of the cap R which corresponds to the curvature of the cap and P b , the evolution of the cap surface (hardening function) according to the volumetric inelastic strain. For the MCC powder, these parameters depend on the material density. In the present study, the simulations were conducted using the constant parameters listed in Table 1 and with the hardening function P b plotted in Fig. 3. The Drucker-Prager Cap surface evolution resulting from the above parameters was plotted in Fig. 4. To incorporate the effect of the variation of material parameters on the results, two parameters considered as potentially influential were examined: Young's modulus E and the cap eccentricity R. Different simulations were conducted to evaluate their impacts on the maximum roll pressure, the nip angle and the maximum density.

Wall friction

In sheet and plate metal forming, the friction between the piece and the roll is essential for the forming process. For metal powder rolling, Chekmarev et al. [START_REF] Chekmarev | Investigation of specific pressure, specific friction and the coefficient of friction during metal powder rolling[END_REF] measured the shear stress along the arc of contact between the powder and the roll and concluded that the friction could be represented using the Coulomb friction model. In our roll press, no instruments were available to measure the tangential force on the contact interface. To progress in study, the Coulomb friction was assumed between the rolls and the powder as above and the friction coefficient was therefore characterized in cylindrical geometry from the die compaction experiment.

During die compaction, the tangential force to the contact area (die wall) F t = (σ uσ l )πr 2 is proportional to the normal (radial) force F r = 2πrhσ r . This is also the case in metal powder rolling [START_REF] Chekmarev | Investigation of specific pressure, specific friction and the coefficient of friction during metal powder rolling[END_REF]. The normal pressure is considered uniform along the compact height h.

The friction coefficient μ(σ r ) was then estimated as:

μ σ r ð Þ = σ u -σ l ð Þ πr 2 2πrhσ r ð1Þ
where the height h is obtained from the initial height of the powder bed (10 mm) and the measured punch displacement u u and r = δ 2 (δ = 11.28 mm) is the internal diameter of the die. Fig. 5 shows the evolution of the coefficient of wall friction with the normal (radial) pressure in un-lubricated conditions. As is seen in Fig. 5, the friction coefficient decreases down to 0.24 for low radial pressure (b5 MPa) and increases progressively up to 0.35. This limit value was considered in the simulation of the steady state analysis of the rolling compaction. The friction coefficient along the arc between the roll and the powder depends on the pressure and shear stress distributions. Its predicted distribution between rolls will be discussed in comparison with the measured friction coefficient from the work of Chekmarev [START_REF] Chekmarev | Investigation of specific pressure, specific friction and the coefficient of friction during metal powder rolling[END_REF].

Numerical procedure

The numerical simulation was performed using commercial software (Abaqus® explicit version 6.7.4). The behavior is assumed to be symmetrical between rollers and only half of the domain was considered. A roll of 100 mm in diameter and 38 mm in width was assumed to be a rigid body and rotates with a speed of 5.6 rpm (0.058 m/s). The domain occupied by the powder was meshed with continuum elements CPE4R with reduced integration for plane strain conditions. The gap was adjusted to 2.0 mm and the entry height of the material corresponds to approximately 20°roll angle. The initial material density was fixed at 0.48 • 10 3 kg/m 3 (corresponding to 0.3 of the relative density).

To address the mesh distortion in the roller compaction problem, the arbitrary Lagrangian-Eulerian (ALE) procedure with adaptive meshing was employed with explicit integration using the Abaqus® Explicit code. The powder feed system was approximated by a uniform feed load (0.1 N which corresponds to a uniform pressure of 0.046 MPa). Eulerian inflow and outflow boundary conditions were employed to ensure the continuously delivered powder to the entry and the free exit of the material. For the computational efficiency of the quasi-static problem, a mass-scaling factor was used and optimized based on the examination of the level of the kinetic energy that must be without oscillations and be small compared to the internal energy. The Coulomb friction was assumed for contact behavior between the roll and the powder. The friction coefficient was μ = 0.35.

A rigid surface was created in the feed zone to represent the feeder barrel with a frictionless condition. The presence of this surface avoids the severe distortion of meshes during the feeding.

Results and discussions

Predicted results

For simulations conducted with material parameters listed in Table 1 and the data of the hardening function plotted in Fig. 3, the distribution of process variables such as pressure, shear stress, velocity, shear stress-pressure ratio and density at roll-powder contact surface were plotted in Figs. 678910.

The shape of the predicted pressure plotted in Fig. 6 shows a gradual increase and reaches its maximum at the angle (-1.7°) just before the gap (angle 0°). The corresponding shear stress changes its sign at the angle -1.32°and achieves its maximum at -2.70°and its minimum at -0.01°(before the gap position). The shapes of the shear stress and the pressure are supported by the measurements obtained in [START_REF] Schönert | Shear stresses and material slip in high-pressure roller mills[END_REF] and [START_REF] Chekmarev | Investigation of specific pressure, specific friction and the coefficient of friction during metal powder rolling[END_REF] and those obtained by the simulations in [START_REF] Dec | Comparison of various modeling methods for analysis of powder compaction in roller press[END_REF]. For the determination of the slip-stick zones and the nip angle, the ratio of shear stress over pressure and the linear velocity at the roll-powder contact surface are considered.

Figs. 7 and8 show the velocity of the powder at the roll surface and the shear stress-pressure ratio.

In the context of the Coulomb friction, the shear stress τ and the normal stress σ n (known as roll pressure) should verify τ ≤ μ σ n . Fig. 8 shows the slip region τ b μ σ n between -20°and -9°and the stick region τ = μ σ n between -9°and 0°. The angle -9°characterizes here the transition between the slip and the stick regions as in the Johanson modeling and corresponds to the nip angle. The powder is first drawn by the slip mechanism in the first region and is densified by deforming in the second time in the stick zone. Just after the 0°a ngle, the shear stress-pressure ratio reaches again the friction coefficient 0.35. This distribution is in agreement with the friction coefficient distribution along the metal/roll contact arc in [START_REF] Chekmarev | Investigation of specific pressure, specific friction and the coefficient of friction during metal powder rolling[END_REF].

The above zones could be also determined from the profile of the powder velocity in the rolling direction (Fig. 7). In the feed zone, the powder velocity is as 0.03 m/s and is uniform in the strip depth because no friction was assumed between the powder and the feeder barrel. From the contact angle with the roll to the nip angle, the Fig. 6. Distribution of roll pressure and shear stress in the compaction area. velocity increases up to the roll speed and presents a vertical gradient that participates in the reorganization of the particles during the flow between rolls and contributes probably to the final density distribution. In the stick region, the powder in contact with the roll flows with the same speed as the roll and the vertical gradient in the strip decreases down to zero at the gap angle. In the ejection phase, the strip is expulsed with a velocity faster than the roll.

Fig. 9 shows the density distribution of the meshes in contact with the roll and in the meshes of the centreline. In the slip zone, the density of meshes in contact with the roll is higher than the density at the centreline. This is due to the low pressure applied on the middle line at this region. The maximum of the density is reached before the 0°angle. At this position where the pressure reaches its maximum, the predicted density on the contact zone and on the centreline is identical. This prediction does not reflect the observation of the internal structure shown in micrograph 2. This is a limitation of the model which does not take into account the effect of the air pressure in the powder.

Contour plots of the density in the rolling direction, were represented in Fig. 10. The initial material density was at 448 kg/m 3 (relative density = 0.3). The powder is continuously fed, drawn and densified between the rolls. In the feed zone, the density is constant and is equal to the initial density except in the upper small zone just before the contact with the roller. In this zone a lower pressure is observed and seems to have the same numerical origin as in the fluid flow simulations in the presence of a stair. At the gap position (angle 0°), the predicted density in the strip depth was in the range of 1050-1110 kg/m 3 which presents a good agreement with the measured bulk density (1000-1200 kg/m 3 ) of compacts prepared using 2 mm of the gap (Fig. 1).

Predicted process parameters such as pressure, shear stress, nip angle and powder velocity showed profiles in agreement with the measurements in [START_REF] Schönert | Shear stresses and material slip in high-pressure roller mills[END_REF][START_REF] Chekmarev | Investigation of specific pressure, specific friction and the coefficient of friction during metal powder rolling[END_REF] and with the numerical simulations in [START_REF] Dec | Comparison of various modeling methods for analysis of powder compaction in roller press[END_REF]. They give an acceptable image of the drawing and the densification of powders in the roller compaction process as it was proposed in the model of Johanson [START_REF] Johanson | A rolling theory for granular solids[END_REF]. However, these predictions depend on the material parameters and feed conditions. In the work of Dec [START_REF] Dec | Comparison of various modeling methods for analysis of powder compaction in roller press[END_REF], the effects of the feed pressure and friction coefficient on the process parameters were investigated. Additionally to these, we focus here on the effects of the material rigidity and the eccentricity of the yield surface cap on the process variables.

Effect of Young's modulus and eccentricity on process variables

Simulations by varying material parameters such as Young's modulus E and cap eccentricity R were conducted to examine their effects on the process variables.

The effect of material rigidity on the results was investigated by varying Young's modulus in the range of 0.5-5 GPa and with keeping the other parameters constant (ν = 0.3, d = 0, β = 65°, R = 0.6 and P b plotted in Fig. 5).

In a similar manner, the impact of the eccentricity R on the process variable was examined by varying R in the range of 0.4-1 as shown in Table 2. The other parameters were given as E = 5 GPa, ν = 0.3, d = 0, β = 65°and P b plotted in Fig. 3.

Fig. 11 shows the material rigidity on the roll pressure. The major changes introduced by varying Young's modulus concern the maximum roll pressure. The evolution of the maximum pressure according to the rigidity is plotted in the same figure. The maximum pressure evolves as a logarithmic function of the material rigidity. However, no significant changes of the pressure shape were observed. This invariability character of the pressure distribution was demonstrated in the work of Chekmarev [START_REF] Chekmarev | Investigation of specific pressure, specific friction and the coefficient of friction during metal powder rolling[END_REF] where the experimental data obtained by rolling metal powders showed that the pressure distribution does not depend on the type of the metal powder being rolled or the strip width and thickness.

The second material parameter investigated was the eccentricity of the cap yield surface. The cap yield surface has an elliptical shape and the eccentricity of the cap is defined as its curvature. This curvature depends on the material parameters such as the cohesion, the internal friction angle and the hardening function P b . To simplify the analysis, the cohesion was considered null and the variation of the eccentricity was made by keeping the same hardening function P b as it is shown in Fig. 12.

By varying the eccentricity in the range of 0.4-1, effects were observed on the maximum pressure, the nip angle and the maximum density. In Table 2 was summarized the predicted values according to the eccentricity. As it can be seen in Fig. 13, the increase of the eccentricity is accompanied by the decrease of the nip angle, the decrease of the maximum pressure and the decrease of the maximum density. These results are consistent and supported by the general observations where the increase of the nip angle (by decreasing the eccentricity or by increasing the wall friction, for example) increases the maximum pressure and the density of the strip. The effects introduced by the variation of the eccentricity demonstrate the importance of accurate material data in the modeling.

Conclusion

In this work, it was demonstrated that the mechanisms of drawing and densification by deforming the powder with rolls could be described using modeling based on the finite element method. This approach, which is similar to the modeling used for sheet and plate metal roll forming and where the behavior and feeding process of a powdered material were specified, showed the capabilities to predict the process variables such as roll pressure distribution, shear stress, and nip angle. The predictions are in agreement with the measurements and numerical simulations published in the literature. The prediction of the density in the gap zone was in accord with the measured bulk density of strips. However, the behavior of powder is complex and could not be completely described with this modeling, particularly the density gradient in the depth of the strip.

The effects of material parameters such as rigidity and eccentricity of the cap of yield surface on the predictions were analyzed. It could be concluded that the need to obtain accurate material parameters with gradual dependencies with material density and their incorporation in the modeling render this approach more predictive and attractive for the optimization and the control of the process. A perspective of this approach will concern its extension to the 3D in order to examine the observed heterogeneity of properties over the strip width resulting from the effect of the screw feeder. It will be then required to model the feeding system, which is responsible for the heterogeneous behavior of the strip.

Nomenclature

Fig. 1 .

 1 Fig. 1. Bulk density measurements of strips versus roll speed (gap = 2 mm).

Fig. 2 .

 2 Fig. 2. Microstructure of the MCC strip depth (micrograph obtained with numerical microscope -Keyence France).

YoungFig. 3 .

 3 Fig. 3. Hardening function P b versus volumetric inelastic strain.

Fig. 4 .

 4 Fig.4. Evolution of yield surfaces with relative density (friction angle = 65°, initial cohesion = 0, eccentricity = 0.6, and P b in Fig.5).

Fig. 5 .

 5 Fig. 5. Wall friction coefficient versus normal pressure (from die compaction experiments).

Fig. 7 .

 7 Fig. 7. Velocity in rolling direction versus rolling angle (on the contact surface and on the centreline).

Fig. 8 .

 8 Fig. 8. Shear stress-pressure ratio along the arc of contact between powder and roll.

Fig. 9 .

 9 Fig. 9. Density distribution at contact surface and in the centreline.

Fig. 10 .

 10 Fig. 10. Contour plots of the density in the powder bed.

Fig. 11 .

 11 Fig. 11. Effect of the material rigidity on the roll pressure distribution.

Fig. 13 .

 13 Fig. 13. Effects of eccentricity on the nip angle (left), and on the maximum roll pressure (right).

Table 1

 1 Material parameters used in the 2D simulation.

Table 2

 2 Effects of eccentricity on the maximum pressure, nip angle and maximum relative density.

	Eccentricity	Maximum roll pressure	Nip angle	Maximum relative
	(R)	(MPa)	(degree)	density
	0.4	120.2	12.39	0.81
	0.5	78.79	9.55	0.79
	0.6	47.2	9.56	0.76
	0.7	31.9	8.99	0.74
	0.8	23.3	8.43	0.7
	0.9	18.5	7.88	0.68
	1	16.17	7.35	0.65

  σ = (σ ij ) Stress tensor ε = (ε ij ) Strain σ r

		Radial stress
	p	Hydrostatic pressure
	q	Equivalent stress
	σ n ,τ	Normal stress and shear stress
	F t ,F r	Tangential and radial forces (die compaction)
	d,β	Cohesion and friction angle of powder
	E,ν	Young's modulus and Poisson ratio
	μ	Wall friction coefficient
	ρ	Density
	R	Cap eccentricity
	p b	Hydrostatic yield stress
	δ	Diameter of the die
	H	Tablet height (die compaction)


M Oudjene
email: mourad.oudjene@insic.fr

L Penazzi

J.-L Batoz

Towards the three-dimensional FE analysis of rapid prototyping tools for sheet metal stamping process

Keywords: FEM, Stamping process, Contact forces, Layered tooling, Rapid prototyping, Joining

Towards the three-dimensional FE analysis of rapid prototyping tools for sheet metal stamping process M. Oudjene, Luc Penazzi, Jean-Louis

Introduction

Sheet metal stamping process is among the oldest and most widely used industrial manufacturing process. The production of high quality parts with low cost is the ultimate goal for the manufacturing industry. On the other hand, the tools play a crucial role and the global cost of a stamped part is highly influenced by the cost of the tools, depending on the production volume. Therefore, new technologies, based on rapid prototyping techniques, have been continuously developed and evaluated in order to reduce cost and manufacturing delays of tools. In the last decade, several developments on rapid stamping tools, in particular those based on the assemblage of steel sheets and joined by screws (or bolts), have been undertaken. Their production and their use on pressforming, to produce stamped parts, have been successfully explored [START_REF] Nakagawa | Advanced in prototype and low volume sheet forming and tooling[END_REF][START_REF] Mueller | Rapid Prototyping Techniques to Manufacture Sheet Metal Forming Tools[END_REF][START_REF] Oudjene | Mechanical analysis of prototyping tools for sheet metal stamping[END_REF][START_REF] Muller | Rapid tooling approaches for small lot production of sheet metal parts[END_REF][START_REF] Walczyk | Design and analysis of reconfortable discrete dies for sheet metal forming[END_REF][START_REF] Walczyk | Rapid tooling for sheet metal forming using profiled edge laminations-design principles and demonstration[END_REF][START_REF] Walczyk | A comparison of rapid fabrication methods for sheet metal forming dies[END_REF][START_REF] Dickens | Principles of design for laminated tooling[END_REF][START_REF] Abel | Deux projets européens de recherche collaborative en PME (CRAFT)-intégration du procédé de Stratoconception en outillage rapide direct pour l'injection, l'emboutissage et la fonderie sous pression[END_REF]. None of these works, however, have examined the mechanical behaviour of the tools, which is vital in the aim of achieving high strength of tools, in particular the joining technique. In addition, the published papers and reports showed that the research performed to date on layered tools has been oriented only towards the problem of their rapid and flexible production [START_REF] Walczyk | Rapid tooling for sheet metal forming using profiled edge laminations-design principles and demonstration[END_REF]. Thus, prototyping tools have been designed and manufactured based only on guess and simple practical rules of thumb. At present, a number of complexities related in particular to the joining by screws are still not fully understood or not modelled. The mechanical strength of layered tools is closely related to the strength of the joining technique used. It is then of primary importance to understand the relation between the applied loads (stamping forces) and the given behaviour of screws.

The evaluation of the strength of layered tools needs an evaluation of the stresses (and strains) induced in the tools during the stamping operation. However, those stresses are seldom computed in practice if not ignored in most of the to date stamping simulations. For accurate stress analysis of stamping tools, it would be necessary to take into account their elastic deformation during analysis within a coupled stamping simulation. However, previous work [START_REF] Wadman | Fluid cell forming tools produced by different rapid prototyping methods[END_REF][START_REF] Oudjene | Modélisation et optimisation du comportement mécanique des outils d'emboutissage stratifiés précontraints par vissage[END_REF] showed that the effort and computational times required for such a coupled simulation, in particular in a general three-dimensional (3D) case, could be prohibitive and unacceptable (strong practical difficulties are involved).

FE analysis of stamping tools taking into account their elastic deformation during analysis have been conducted in relation with the wear behaviour [START_REF] Oudjene | 3D finite element modeling and analysis of the laminated tooling behaviour in sheet metal stamping process[END_REF][START_REF] Oudjene | Modélisation et optimisation du comportement mécanique des outils d'emboutissage stratifiés précontraints par vissage[END_REF][START_REF] Dankert | Reduction of the residual stresses in a deep-drawing cup by modifying the draw die profile[END_REF][START_REF] Damborg | Optimization of the draw die profile with regard to wear using the FEM[END_REF] or stress analysis [START_REF] Wadman | Fluid cell forming tools produced by different rapid prototyping methods[END_REF][START_REF] Oudjene | Modélisation et optimisation du comportement mécanique des outils d'emboutissage stratifiés précontraints par vissage[END_REF][START_REF] Pepelnjak | Adaptable tooling sets for metal forming of geometrically similar products[END_REF][START_REF] Dankert | Reduction of the residual stresses in a deep-drawing cup by modifying the draw die profile[END_REF][START_REF] Damborg | Optimization of the draw die profile with regard to wear using the FEM[END_REF][START_REF] Burlat | Analyse Mécanique et Tribologique de l'emboutissage: application aux outils fabriqués par combinaison de différents matériaux[END_REF]. However, the limited number of works dealing with the behaviour of tools is mostly dealing with 2D situations (plane strain, axisymmetry).

In this paper, the authors proposed and developed a numerical simplified procedure, in two steps, for the 3D stress analysis of sheet metal stamping tools (layered or not). That procedure has the advantage to avoid the strong practical difficulties involved in a coupled stamping simulation by considering the elasticity of tools. It is based on the assumption that the contact forces acting on the rigid tools during stamping operation are almost identical to those acting on the elastic tools (standard steel tools). Thus, it exploits the knowledge of the distribution of the contact forces acting on the rigid tools to study, separately, the behaviour of elastic tools within a linear elastic analysis, by assuming the distribution of the contact forces as static external load.

First, the paper presents a series of stamping simulations taking into account elastic deformation of tools in order to show the validity of the assumption made on the contact forces, i.e. the stamping process contact forces results are the same between rigid and elastic tools. After that, the developed procedure is applied to study the 3D mechanical behaviour of a layered stamping punch joined by screws and adhesive. Both stress and strain developed in the layered punch have been investigated in previous work [START_REF] Oudjene | Modélisation et optimisation du comportement mécanique des outils d'emboutissage stratifiés précontraints par vissage[END_REF] and it was found that the maximum amount of both equivalent von-Mises stress and elastic deformation developed in the punch are still much lower than the yield point for the steel material used for the punch. Thus, we focussed in this study on the screws behaviour, which is of extreme importance for the strength and life duration of the layered punch. The results show several issues with the layered punch related to the joining by screws which need to be investigated more and mastered before it should be considered as suitable for production. Finally, some answers are established and some important considerations to improve the design of layered tools joined by screws are outlined.

Coupling stamping FE analysis: adopted strategy

The numerical simulation of the stamping process is extensively used for the analysis and the design of industrial parts to avoid the long and expensive experimental tryout procedures. Thus, the previous stamping simulations have been concentrated exclusively on the workpiece behaviour mainly assuming rigid tools.

Two major numerical approaches are often used for the simulation of such problems: dynamic explicit and static implicit approaches. However, the implicit scheme is often limited by convergence problems, due to the contact conditions with friction.

Several numerical tests have been performed by the authors on some practical tools-part system where the behaviour of the tools has been evaluated regarding the intensity and the distribution of the stresses induced in the elastic tools, as well as the contact forces between the workpiece and the tools (rigid or elastic tools). The present section, deals with the results in the case of an axisymmetrical deep-drawing test. Both implicit and explicit codes are used, showing the limitations of explicit codes to deal with the full coupling between the blank forming and the stress analysis of elastic tools.

Case study description

The deep drawing of a cylindrical cup of Numisheet'2002 benchmark test [START_REF]Proceedings of the Second International Conference on Numerical Simulation of 3-D Sheet Metal Forming Processes-Verification of Simulation with Experiments (Numisheet'2002)[END_REF], by considering the elasticity of the punch, was considered as the example of the application. In order to calibrate the numerical model with respect to the literature results, the simulation is, first, conducted classically by assuming a 3D situation with rigid tools as considered in [START_REF]Proceedings of the Second International Conference on Numerical Simulation of 3-D Sheet Metal Forming Processes-Verification of Simulation with Experiments (Numisheet'2002)[END_REF], using ABAQUS/explicit finite element code. After that, coupling deep-drawing simulations were run by considering elastic deformation of the punch using ABAQUS/implicit (2D situation) and ABAQUS/explicit (3D situation). The quality of the simulation procedure was evaluated with regard to both the workpiece and the behaviour of the punch.

The deep-drawing parameters are shown in Fig. 1. All simulations have been carried out with a constant blankholder force of 70 kN and punch stroke of 40 mm. Only one quarter of the model was considered as in Numisheet'2002. The blank was modelled as an isotropic elastoplastic material with isotropic hardening characteristics given by Hollomon's model with the following material constants: K = 539 MPa, n = 0.27, E = 210 GPa and = 0.3. Coulomb frictional model was assumed with = 0.04 as considered in Numisheet'2002 [START_REF]Proceedings of the Second International Conference on Numerical Simulation of 3-D Sheet Metal Forming Processes-Verification of Simulation with Experiments (Numisheet'2002)[END_REF].

Classical stamping FE analysis with rigid tools

A four-node shell element with reduced integration (element S4R of ABAQUS) is used to discrete a quarter of the initial blank involving in 1400 elements, while the other tools (punch, die, blankholder) were regarded as rigid (meshed with the rigid four-node element R3D4 of ABAQUS). A constant punch speed of 10 m/s was adopted.

The thickness distribution along the radial direction is presented in Fig. 2, from which it is observed that the result obtained with ABAQUS/explicit is in agreement (maximum thickness deviation 0.01 mm) with the literature one, but the thickness picks are not detected at the same distance from the centre (at 69 mm for the reference solution and at 76 mm for ABAQUS).

Furthermore, the punch reaction force is presented in Fig. 3 and it can be seen that the result is in good agreement with the literature one.

Stamping FE analysis with elastic punch

To our best knowledge we do not have reference results for the punch behaviour. All the results available in the literature deal only with the stamped sheet. In order to compute an accurate solution (reference solution) regarding the punch behaviour, one wishes to deal with a 2D deep-drawing simulation using static implicit scheme (ABAQUS/implicit). Furthermore, we make a 3D deep-drawing simulation using dynamic explicit scheme to check that we obtain the same results regarding the punch behaviour as with a static implicit scheme.

2D static implicit model

The blank was modelled by 4 and 270 axisymmetric elements (denoted CAX4R in ABAQUS) in the thickness and radial directions, respectively.

The punch was modelled as an elastic and isotropic material with the following properties: E = 210 GPa and = 0.3. A four-node element (CAX4 element of ABAQUS) was used to discrete the punch (Fig. 4), involving 821 elements.

By analysing the results, with regard to the workpiece final shape, it is found that the elastic deformations induced in the punch do not seem to have significant effects on both thickness and strains distributions.

The contact forces between the workpiece and the punch (rigid punch and elastic punch) have been evaluated regarding both the intensity and the distribution. Fig. 5 shows the contact forces acting on both the rigid and the elastic punch. It can be seen that the contact forces are almost identical, i.e. the elastic deformation induced in the punch are infinitesimal, so there is no effect on the contact forces.

In order to examine the distribution of the contact forces acting on the tools (rigid or elastic) during the stamping operation, see Fig. 6 which shows the distribution of the contact pressure as reaction forces applied to the deformed sheet. For the punch, it can be seen that the contact forces are mainly located along the punch radius and present two picks with a maximum amount value of 54 MPa. These two picks correspond to the beginning and the end of the contact between the deformed sheet and the punch radius. The result is qualitatively in good agreement with the literature [START_REF] Oudjene | Mechanical analysis of prototyping tools for sheet metal stamping[END_REF][START_REF] Oudjene | Modélisation et optimisation du comportement mécanique des outils d'emboutissage stratifiés précontraints par vissage[END_REF][START_REF] Dankert | Reduction of the residual stresses in a deep-drawing cup by modifying the draw die profile[END_REF][START_REF] Damborg | Optimization of the draw die profile with regard to wear using the FEM[END_REF][START_REF] Burlat | Analyse Mécanique et Tribologique de l'emboutissage: application aux outils fabriqués par combinaison de différents matériaux[END_REF]. By taking into account the elasticity of the punch, during the deep-drawing operation, it was also possible to obtain the distribution of the equivalent von-Mises stress (or elastic strain) developed in the punch (Fig. 7). The maximum amount of equivalent stress computed at the end of forming stage is around 35 MPa.

From the above results, it can be concluded that the elastic deformation of standard steel tools are infinitesimal and do not have significant influence on both the workpiece shape (strain, thickness distribution) and the contact forces.

3D dynamic explicit model

The FE-simulation based on dynamic explicit scheme is standard in stamping simulation of parts with complex geometries. The objective of this model is to check that we obtain the same results regarding the punch behaviour as with a static implicit scheme.

The same mesh as in Section 2.2 was used for a quarter of the blank sheet (Fig. 8) while the quarter of the punch was meshed with 16 469 eight-node brick elements with reduced integration (C3D8R element of ABAQUS).

The simulation was made by moving the punch download from its upper surface using a punch speed of 10 m/s.

The effect of the mesh quality in the punch on the results with regard to the stress distribution was already studied in [START_REF] Oudjene | Modélisation et optimisation du comportement mécanique des outils d'emboutissage stratifiés précontraints par vissage[END_REF]. Despite that, it was impossible to obtain a good estimation of the stress developed in the punch (Fig. 9), with a classical punch velocity of 10 m/s, compared to that obtained in the case of a static implicit scheme (Fig. 7). Moreover, the maximum amount of von-Mises stress developed in the punch is around 92 MPa. This is an important numerical result. In addition, the computed contact forces (punch reaction force curves) are strongly irregular. However, with the decrease of the punch velocity from 10 to 1 m/s, it became possible to obtain a fairly good estimation of the stress distribution in the punch, based on the comparison between Figs. 7 and10. The maximum value of the computed von-Mises stress is now around 37 MPa. However, the CPU time has noticeably increased (Table 1). It can be concluded that for an accurate solution, it would be necessary to noticeably reduce the punch speed because of the inertia effects and the propagation of the elastic waves in the elastic punch.

With the use of a punch velocity of 1 m/s, the contact forces acting on the elastic punch are found almost identical to those acting on the rigid one (Fig. 11), as in the static implicit case. Furthermore, the CPU times of the different simulations, carried out using dynamic explicit scheme, are summarized in Table 1 (Pentium VI, 2.8 GHz, 2 Go RAM).

The above results show that the full coupling is not necessary regarding the distribution and the intensity of the contact forces. Moreover, if explicit codes are used (which is standard in stamping simulations) very small time steps are required to obtain good stresses in the elastic tools, leading to large and unacceptable CPU times. It can be concluded that in a general 3D stamping simulation, the full coupling between the elasticity of tools and part, involves strong practical difficulties, if explicit code is used:

(i) it would be necessary to use hundreds of thousands or even more elements in the FE model; (ii) in order to obtain an accurate solution, it is necessary to noticeably reduce the punch speed involving massive and unacceptable computational times.

Based on these difficulties, it is therefore relevant to decouple the simulation of the blank forming and the linear elastic analysis of the tools. Thus, the 3D stress analysis of the tools can be performed separately by exploiting the knowledge of the distribution of the contact forces acting on the rigid tools during the forming operation.

Description of the proposed method

Principle and assumptions

The contact forces acting on the rigid tools during the forming operation have been found almost identical to those acting on the real standard steel tools (see Section 2). Hence, the stress analysis of tools can be performed using the classical stamping simulation approach by considering rigid tools and by exploiting the knowledge of the distribution of the resulting contact forces. Thus, our procedure involves two separate steps (Fig. 12):

(i) step 1, the stamping simulation is first conducted classically by considering rigid tools, allowing to obtain an estimation of the distribution of the contact forces acting on the rigid tools during the forming operation; (ii) step 2, the stamping tool (layered or not) is analysed separately by considering its elastic deformation and by applying the distribution of the contact forces computed previously (in step 1) and considered as a static external load.

In the present study, the STAMPACK finite element code [START_REF] Quantech | Stampack user guide-version 5.6, Quantech edition, Edificio Nexus[END_REF] and the CATIA V5 finite element module [START_REF][END_REF] were used to perform the step 1 and step 2, respectively. The reason behind this choice is that STAMPACK and CATIA V5 offer some facilities and advantages to deal with the above procedure in two steps [START_REF] Oudjene | Modélisation et optimisation du comportement mécanique des outils d'emboutissage stratifiés précontraints par vissage[END_REF].

The above procedure, in two steps, is based on the assumption that the contact forces acting on the rigid tools are almost identical to those acting on the real steel deformable ones.

Estimation of the contact forces

At the beginning of step 1, the surface of the rigid tool (active tool parts) is divided into small sectors called "Sets" (Fig. 13). For each "Set" (i), the global nodal contact force vector, with its three cartesian components, is automatically computed in STAMPACK and can be expressed as follows:

F (i) = nnd j =1 F (i) j (1)
with

F (i) =(F (i) x , F (i) y , F (i) z )
, where nnd is the number of nodes in "Set" (i).

The third component F (i) z represents the vertical punch reaction force acting on the "Set" (i). Then, the total punch reaction force, denoted F Z , can be expressed as follows:

F Z = nset i=1 F (i) z (2)
where nset is the number of sectors (or "Sets") in the active tool parts.

Equivalent static external load

In order to simplify the transfer of the distribution of the contact forces from the rigid tool to the deformable one (layered or not), the elementary force vectors F (i) computed previously, in step 1, for each "Set", are automatically replaced (by considering their three cartesian components) by a system of equivalent static loads P (i) , distributed according to the following relation:

F (i) = s P (i) ds (i) , ( 3
)
where ds (i) is the area of the elementary "Set" number (i).

The above procedure has already been validated in [START_REF] Oudjene | Modélisation et optimisation du comportement mécanique des outils d'emboutissage stratifiés précontraints par vissage[END_REF] in the context of simple deep-drawing geometries. The obtained results showed that this procedure gives fairly good stress and strain estimations in the tools compared to that obtained in the context of deep-drawing simulation taking into account elastic deformation of tools, using the ABAQUS static implicit code.

Application to the layered tools: 3D case study

Description of the case study

A layered stamping punch (Fig. 14) with dimensions: 420 mm × 220 mm × 77 mm, based on the Stratoconception ᭨ process, was considered as the example of the application. The layered punch was built by stacking 7 mm thick of EN DIN 2C45 (NF EN XC48) steel sheets [START_REF] Abel | Deux projets européens de recherche collaborative en PME (CRAFT)-intégration du procédé de Stratoconception en outillage rapide direct pour l'injection, l'emboutissage et la fonderie sous pression[END_REF], which were directly machined by laser cutting and micro-milling. Several screws were used for the joining of the layers in addition to an epoxy adhesive with thickness of 0.1 mm, which was inserted between layers.

According to the procedure illustrated in Fig. 12, two separate steps of analysis are required to make 3D stress analysis of the tool.

Step 1: classical stamping FE analysis

In the finite element model, a mesh of three-node BST (Belytschko Shell Triangular) shell element was used in the full model of the blank involving 382 518 elements, while the other tools (punch, die and blankholder) were assumed rigid. The blankholder force was 50 kN, punch stroke was 50 mm. The material data are: isotropic elasto-plastic behaviour with isotropic hardening given by Hollomon's model with the following material constant: k=522.4 MPa, n=0.23, E=210 GPa, = 0.3. The contact formulation is based on the master-slave contact approach [START_REF] Quantech | Stampack user guide-version 5.6, Quantech edition, Edificio Nexus[END_REF]. The tools are considered as the master surfaces and the sheet is the slave surface. The Coulomb friction model was adopted with a frictional ratio = 0.1 and the interaction between the sheet and the tools is formulated using the finite sliding approach, which allows the separation of the two surfaces during sliding.

First, some results with regard to the feasibility of the blank forming (shape, strain, stress, thickness distribution) were examined [START_REF] Oudjene | Modélisation et optimisation du comportement mécanique des outils d'emboutissage stratifiés précontraints par vissage[END_REF]. The obtained workpiece (Fig. 15) is qualitatively in agreement with the experimental one, regarding the part shape.

Second, simulations were run in order to obtain an estimation of the distribution of the contact forces acting on the rigid punch during the stamping operation.

Fig. 16 shows the contact pressure distribution on the rigid punch at the end of the forming stage. It can be seen that the contact forces are mainly located along the punch radius (active parts). This result is in good agreement with the literature [START_REF] Oudjene | Mechanical analysis of prototyping tools for sheet metal stamping[END_REF][START_REF] Oudjene | Modélisation et optimisation du comportement mécanique des outils d'emboutissage stratifiés précontraints par vissage[END_REF][START_REF] Dankert | Reduction of the residual stresses in a deep-drawing cup by modifying the draw die profile[END_REF][START_REF] Damborg | Optimization of the draw die profile with regard to wear using the FEM[END_REF][START_REF] Burlat | Analyse Mécanique et Tribologique de l'emboutissage: application aux outils fabriqués par combinaison de différents matériaux[END_REF]. The CPU time was around 10 h (Pentium VI, 2.8 GHz, 2 Go RAM).

Step 2: modelling and FE analysis of the punch

In this second step of simulation, the layered punch was analysed separately from step 1. As shown in Fig. 17, the punch CAD model was generated in CATIA V5 and sliced into 7 mm thick elementary layers to be consistent with the real one.

The joining technique consists in 2 plugs (G 1 and G 2 ) and 5 screws (S 1 to S 5 ), in addition to an epoxy adhesive. The finite element model involves 210 124 four-nodes tetrahedral elements with a mesh refinement in the vicinity of the screw holes.

The EN DIN 2C45 (NF EN XC48) steel layers are elastic and isotropic with the following properties: E = 210 GPa, = 0.3.

The adhesive with thickness of 0.1 mm was not modelled in detail but each interface between the steel layers is clearly identified in the 3D FE model. As a first assumption for this study, the nodes at the interfaces belonging to the layers are tied leading in fact to a continuous medium. Attention is then paid to the post-processing stage to the maximum tensile and shear stresses which are compared to the corresponding admissible yield stresses of the adhesive. If the stress levels are satisfactory, then sliding or separation of the layers is not possible. Of course the role of the adhesive on layers assembled by screws is important as mentioned in [START_REF] Oudjene | Modélisation et optimisation du comportement mécanique des outils d'emboutissage stratifiés précontraints par vissage[END_REF][START_REF] Yoneno | The strength of joints combining adhesives with blolts (case where adherends are pipe flanges of which the interfaces are bonded partially)[END_REF] but its precise evaluation is considered beyond the scope of the present study and should be addressed in a future research paper.

The loading conditions are as follows:

(i) The initial clamping state: this stage is automatically generated in CATIA V5, by assuming minimal clamping force of 6000 N for each screw. (ii) The forming load: the distribution of the contact forces, provided by the numerical simulation, in step 1, is automatically imported in CATIA V5 and applied to the layered punch FE model as a static load.

Results and discussions

The stress and strain distributions in the layered punch have already been investigated in [START_REF] Oudjene | Modélisation et optimisation du comportement mécanique des outils d'emboutissage stratifiés précontraints par vissage[END_REF] and it was found that the joining technique by screws requires more attention and plays an important role in the design decision of layered tools. Thus, we focussed on what happens in the screws, under the forming load.

When the distribution of the contact forces (due to the stamping) is applied to the layered punch, which is joined with an initial clamping force (F S ) i , increments (variations) F sj , in the axial screw forces, are produced. Fig. 18 shows the increments in the axial screw clamping forces as a function of the axial forming load F Z . It can be seen that the increments F sj occurring in axial screw forces increase linearly with the increase of the forming load (contact forces). This result is in agreement with the literature [START_REF] Yoneno | The strength of joints combining adhesives with blolts (case where adherends are pipe flanges of which the interfaces are bonded partially)[END_REF].

From Fig. 18, it can be seen that the highest values of increments are produced in screws S 1 and S 5 (Fig. 17), while the increment values in the other screws are not significant. This can be explained using the theoretical relationship between the increment and the axial forming load, occurring in each screw, which can be expressed as follows [START_REF] Yoneno | The strength of joints combining adhesives with blolts (case where adherends are pipe flanges of which the interfaces are bonded partially)[END_REF]:

F S = [(F S ) i -(F S ) f ] = K t K t + K cg K ′ cg K pt .F Z . (4)
The quantity = (K t /(K t + K cg ))(K ′ cg /K pt ), is called the screw load factor.

K t is the spring constant for a screw system and K cg is the compressive spring constant for layers around a screw hole. K ′ cg and K pt are the compressive spring constants defined using the mean displacements produced in the screw holes when the initial clamping force and the stamping load are, respectively, acting. (F S ) i and (F S ) f are, respectively, the initial and the final axial screw forces.

The slopes of F Sj (Fig. 18) are independent of the initial clamping force (F S ) i and can be used in order to obtain the individual screw load factors.

On the other hand, it is found that the value of (F S ) f (=(F S ) i + F S ), in each screw, decreases because the value of F S is negative.

In such a screwed joint, the initial prestressed screws should remain in tension under the loading conditions, for a proper strength of the joint. Hence two problems, related to this joining technique, can be identified:

(i) The decrease in the initial clamping force, due to the forming load, is not allowed when the final screw clamping force tends to zero (this can lead to screw loosening). (ii) The increment occurring in the axial screw clamping force can lead to large force increments in the screws, which can greatly affect the fatigue behaviour (inducing cyclic stress in the screws). The cyclic stress in each screw can be expressed as follows:

cycl = [(F S ) i -(F S ) f ] 2.A S = F S 2.A S , (5)
where A S is the cross section of the screw.

Based on these remarks, it is therefore relevant to reduce the increments occurring in axial screw clamping forces, due to the forming load, in order to improve the strength and the life duration of the joining technique.

Conclusion

In this paper, a simplified numerical procedure for the 3D FE analysis of elastic sheet metal stamping tools, in particular those based on rapid prototyping techniques, is proposed and discussed with particular attention to the analysis of layered tools joined by adhesive and screws. The stress analysis procedure is based on the decoupling of the simulation of the blank forming with the elastic analysis of the deformable tools.

The adhesive is not fully considered in the present analysis, but the stress field is particularly analysed at the layer interfaces to check the validity of the assumption of material continuity.

The numerical approach has been applied to the industrial application of a layered punch designed and manufactured using the Stratoconception ᭨ system. The following remarks can be mentioned:

(i) The decrease in screw prestresses are highly related to the stamping load and also to the spring constants of the joined layers around the screw holes. It is then advisable to reduce these decreases, as much as possible. (ii) The initial clamping force must be considered as an important parameter to guarantee minimal tensile force in axial screws when the decrease in axial screw force is large. (iii) The decrease in axial screw forces can lead to large stress increments (or variations) in the screws, which can greatly affect the fatigue behaviour of screws.

More detailed research work is now in progress towards the improvement of the strength and the life duration of the joining technique by screws. Important optimization issues can be proposed, such as the optimization of the screw positions which should be addressed in a future work.

Fig. 1 .

 1 Fig. 1. Schematic illustration of the deep-drawing test [16].

Fig. 2 .

 2 Fig. 2. Thickness distribution along the radial direction.

Fig. 3 .

 3 Fig. 3. Punch force-displacement curves.

Fig. 4 .

 4 Fig. 4. 2D axisymmetric FE model with elastic punch.

Fig. 5 .

 5 Fig. 5. Punch reaction force: comparison between rigid and elastic tools.

Fig. 6 .

 6 Fig. 6. Contact pressure distribution between the workpiece and the tools.

Fig. 7 .

 7 Fig. 7. Distribution of the von-Mises stress in the punch (reference solution).

Fig. 8 .

 8 Fig. 8. 3D axisymmetric FE model with elastic punch.

Fig. 9 .

 9 Fig. 9. Distribution of equivalent von-Mises stress in the elastic punch: case where V p = 10 m/s.

Fig. 10 .

 10 Fig. 10. Distribution of equivalent von-Mises stress in the elastic punch: case where V p = 1 m/s.

Fig. 12 .

 12 Fig. 12. The flowchart of the proposed procedure in two steps.

SetFig. 13 .

 13 Fig. 13. Schematization of elementary "Sets"/corresponding contact force vector.

Fig. 14 .

 14 Fig. 14. The studied stamping layered punch (source: CIRTES).

Fig. 15 .

 15 Fig. 15. Workpiece final shape: comparison with experiments.

Fig. 16 .

 16 Fig. 16. Contact pressure (MPa) distribution on the rigid punch (STAMPACK).

Fig. 17 .

 17 Fig. 17. CAD model of the layered punch (CATIA V5).

Fig. 18 .

 18 Fig. 18. Increments (variations) in initial axial screw forces as a function of the stamping load (punch reaction force).

Table 1

 1 CPU times involved in the different simulations based on dynamic explicit scheme

	3D models (explicit)	CPU times
	V With rigid tools 0 h 44 min	-
	With elastic punch	3 h 26 min	33 h 4 min
	Fig. 11. Punch force-displacement curves: comparison between rigid and
	elastic tools (dynamic explicit scheme).

p = 10 m/s V p = 1 m/s