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cold rolling sequence: Analysis of edge cracking
Journal of Materials Processing Technology 212 (2012) 1049–1060
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Journal of Materials Processing Technology journal homepage: www.elsevier.com/locate/jmatprotec
Finite Element simulation of the edge-trimming/cold rolling sequence: Analysis of edge cracking C. Hubert a,b,∗ , L. Dubar a,b , M. Dubar a,b , A. Dubois a,b a b
Univ Lille Nord de France, F-59000 Lille, France UVHC, TEMPO EA 4542, F-59313 Valenciennes, France
a r t i c l e
i n f o
Article history: Received 6 May 2011 Received in revised form 3 November 2011 Accepted 21 December 2011 Available online 5 January 2012 PACS: 87.10.Kn 46.50.+a Keywords: Edge cracking Finite Element Method Failure criterion Damage model
a b s t r a c t The aim of the present study is to develop a strategy of numerical simulation to study the sequencing of two processes, when the modeling geometry assumptions differ. In this study, the first process is assumed continuous, leading to a plane strain model. The developed strategy of simulation takes advantages of this assumption to extrude, in the out-of-plane direction, the 2D mesh and the mechanical fields to create a 3D model corresponding to the second process. In a previous experimental study achieved by the authors, an edge-trimming/strip thickness reduction sequence was analyzed. The aim was to quantify the effect of the work hardening accumulated in the first step on the occurrence of cracks in the strips edges during the second one. The present study uses this experimental sequence to validate the proposed approach and to better understand the reasons of edge cracking, according to the plastic strain level and thickness reduction conditions. Finally, the proposed approach is used as a predictor, to quantify the effect of a major parameter in edge-trimming on the strips edges integrity, at the end of the full sequence. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The manufacturing processes of metallic parts, or their assembly, often involve multiple operations, such as machining, cutting, forming or drilling. For some processes, the residual stress state and the accumulated work hardening generated by the upstream processes can be strongly reduced or even cancelled by means, for instance, of a recrystallization annealing, where new strain-free grains nucleate and grow to replace those existing. This case has been encountered in coil coating, for which the coils are annealed before galvanizing (Dubar et al., 2006) and coating (Szczurek et al., 2009). In other cases, for processes such as self-piercing riveting, the rivet and assembled parts deformations must be considered for the mechanical strength evaluation of the whole assembly (Hoang et al., 2010). The numerical approaches, especially the Finite Element Method, are often used to simulate the manufacturing processes and to study the thermomechanical behavior of the resulting parts or assemblies. To perform an accurate simulation of multi-stages manufacturing, it is thus necessary to keep the thermomechanical material history through the different processing steps. Generally,
∗ Corresponding author at: UVHC, TEMPO EA 4542, F-59313 Valenciennes, France. Tel.: +33 3 27 51 14 54; fax: +33 3 27 51 13 17. E-mail address: [email protected] (C. Hubert). 0924-0136/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2011.12.011
this is easily achieved by creating steps within the numerical simulation, that allow to simulate several forming stages at once. Options in Finite Element solvers also allow to create backup points at a given simulation time, in order to recover the strain and stress fields to a new Finite Element model, with new boundary and simulation conditions. But these methods can only be applied if the model geometry is kept between two simulation stages, which is not always the case. The aim of this paper is to develop a modeling strategy in order to study a sequence of two steel strip working processes, by means of the Finite Element Method. The first process is modeled using the plane strain assumption to reduce computations costs (Hubert et al., 2009). It is followed by a second process, which is modeled in 3D since the regions of interest for the results analysis are not in-plane. The proposed strategy takes advantages of the continuous nature of the first process: it assumes that the part geometry is rectilinear and the material mechanical state is homogeneous along the out-of-plane direction. The 2D deformed mesh resulting from the first operation is thus extruded following the out-of-plane direction to obtain a 3D mesh, and the 2D material mechanical fields are propagated along this direction. After application of the boundary conditions related to the second process, the simulation can be run and the results are analyzed in the same way as for a full 3D simulation. This study is applied to a forming sequence, involving an industrial edge-trimming step followed by an experimental thickness
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reduction operation. The first process is needed to remove hot rolling edge defects by reducing the strips width. It is a continuous shear cutting process and, under some assumptions, it can be modeled in plane strain. The second process, named Upsetting Rolling Test (URT, Huart et al., 2004; Hubert et al., 2010), is an experimental testing stand used to reproduce the industrial cold rolling contact conditions. This sequence corresponds to the one performed in a previous study achieved by the present authors (Hubert et al., 2010), which highlighted that the strain hardening state caused by edge-trimming is of major importance in the integrity of the edges after thickness reduction. The use of such a numerical approach thus make sense. The previous experimental results will be used to validate the developed simulation strategy. The strip working processes encountered is this study involve large deformations, damage and failure of the part. A material behavior model is thus developed to consider these features. The phenomenological approach is chosen to model damage. It is based on the consideration of a Representative Elementary Volume, in which a given amount of voids is represented by a scalar or a tensor. The most common model is Lemaitre’s (Lemaitre, 1985); it has been used by Hambli (2002a) in the field of sheet blanking, or by César de Sá et al. (2006) in the case of external inversion of tubes. It allows simple implementation and identification, and to consider the load reversal. This model is chosen for this study and is implemented in the Finite Element code. To model material failure, studies in the field of shear slitting (Ghosh et al., 2005; Farzin et al., 2006) or sheet blanking (Hambli, 2002b) showed that the most used failure criteria were of very simple formulation, such as the Cockcroft–Latham (Cockroft and Latham, 1968) and the Shear Failure. The studies related to these failure criteria gave good results in terms of prediction of the sheared profile and burr resulting from the process settings. In the field of edge-trimming, validation tests have been performed by Hubert (2010), in which criteria such as the Cockcroft–Latham, the Oh et al. (1979), the Shear Failure and the Lemaitre critical damage were compared to industrially edge-trimmed profiles. The Shear Failure criterion was the most accurate one. The validation has been performed in three stages. The extent of the mechanically affected region to the edge has been compared to experimental hardness measurements, and the transition line between the sheared and the burnished regions were compared by means of the reaction forces curves. Finally, the equivalent plastic strain level predicted numerically was more realistic with the Shear Failure than with the other criteria. The Shear Failure criterion is thus adopted here. In this paper is first introduced the global methodology to create a 3D model from a 2D plane strain one, and to apply the thermomechanical fields coming from the 2D simulation results as initial state in the 3D model. Then the constitutive equations of the material behavior model and the failure criterion are presented. Secondly, the Finite Element model of the edge-trimming process and the involved boundary conditions are described. The methodology to create the 3D model is applied and the resulting model, corresponding to the edges specimens used with the URT, is detailed. The validity of the developed approach is quantified according to the experimental results of Hubert et al. (2010). Then, the experimental sensitivity study of the forward slip on edge cracking performed by Hubert et al. (2010) is reproduced. It will allow to better understand the phenomena that lead to edge cracking during plastic deformation. Finally, the developed approach is used as a predictor: one of the main edge-trimming process parameter, the clearance, is modified in order to quantify its effect on the specimens edges integrity.
Fig. 1. 2D to 3D elements extrusion in the case of reduced integration.
2. Modeling strategy The aim of this paper is to develop a modeling strategy in order to study, by means of the Finite Element Method, the effects of a metal working process on the part behavior when it is submitted to a second working operation. The sequence considered in this paper involves a 2D model followed by a 3D one, but Finite Element solvers do not allow transfer neither the geometry nor the thermomechanical material history between two models of different modeling dimensions. A methodology is thus implemented to overcome this restriction. The first point is to create a 3D mesh from the 2D one. The second point is to apply the relevant final mechanical fields to this new mesh as initial conditions. These two points are detailed in the next sections. 2.1. Mesh geometrical extrusion The 2D FE simulation on which the present modeling strategy is based on must follow the plane strain theory. It allows to assume the model geometry and the material mechanical state as continuous in the out-of-plane direction. This strategy takes advantages of these assumptions by performing, in a first time, a geometrical extrusion of the 2D model. The nodes of the deformed mesh, at the end of the 2D simulation, are projected in the third space dimension, and the initially plane elements become 3D: 8-nodes linear bricks for the initial quadrangular elements, and 6-nodes linear triangular prisms for the initially triangular plane elements. The elements length in the extrusion direction is adjusted to obtain a ratio of maximum to minimum element length between 1:1 and 2:1. The number of integration points in the 3D elements depend on those in the plane elements. Fig. 1 shows the extrusions for rectangular and triangular elements, with reduced integration. 2.2. Mechanical fields application The significant mechanical fields at the end of the 2D simulation must now be applied to the extruded mesh. In the case of builtin methods, the full tensors are applied to the new simulation as initial mechanical fields. In this modeling strategy, the procedure is simplified: this methodology is developed in the field of isotropic plasticity, and the equivalent plastic strain is sufficient to recover the equivalent stress, instead of the full tensor. Same deduction is applied for any other quantity such as damage or failure variables. This paper deals with large deformations in metal forming, a damage model coupled to the isotropic plasticity model is implemented and will be detailed in the next section. The mechanical variables to transfer to the extruded mesh are thus the isotropic p damage variable D and the equivalent plastic strain ε . These variables will be introduced in the 3D Finite Element model as initial
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Fig. 2. Mechanical field from the 2D model applied to the 3D one.
values in the material data, and are directly applied to the integration points by the FE code at the first calculation increment. Fig. 2 shows an illustration of an arbitrary mechanical field coming from the 2D deformed mesh transferred to the 3D model. The developed mesh extrusion algorithm has been integrated into ABAQUS via its graphical user interface. The principle of this algorithm is outlined in Fig. 3. 3. Local integration 3.1. Elastoplasticity and damage The processes being modeled in this study lead to large deformations until failure of small regions of the worked part or even the entire part. First, it is necessary to take into account the elastoplastic behavior of the material, which can be defined by a Ludwik law, p
y = 0 + Kε m,
(1)
where 0 and y are the initial and the current yield stress, K and p m are material parameters and ε is the equivalent plastic strain. The identification of the material parameters K and m, performed on tensile specimens, showed a strong dependency of the material to damage (Hubert et al., 2009). It has been decided to develop a user material subroutine to take into account the elastoplastic behavior coupled to a damage law. The Lemaitre one has been chosen, mainly for its capability to predict damage increase in compression (Li et al., 2010). It is defined by D˙ =
−Y s S
p ε˙ ,
(2)
where D˙ is the isotropic damage rate, S and s are material paramp
eters and ε˙ is the equivalent plastic strain rate. −Y is the damage strain energy release rate, defined by −Y =
q2 R
,
(3)
2 (1 + ) + 3(1 − 2)2 , 3
(4)
2E(1 − D)2
with R =
where q is the von Mises equivalent stress, E is the material Young modulus, is its Poisson ratio and is the stress triaxiality ratio ( = p/q, p being the hydrostatic stress). D is the damage variable, defined such that 0 ≤ D ≤ Dc ≤ 1, with Dc the damage threshold at which a macro crack occurs. In the initial Lemaitre damage model (Lemaitre, 1985), the evolution of the damage variable does not depend on the sign of the stress triaxiality ratio. However, when loading a damaged specimen in compression, there is a voids closure effect leading to a slowed damage evolution. This phenomenon has been highlighted, among other authors, by Andrade Pires et al. (2004) and César de Sá et al. (2006). As outlined in introduction of this paper, the second process will apply a thickness reduction of the steel strip, and the voids
closure effect will occur. It is thus taken into account in a simplified, conditional form: −Y ∗ = −˛Y
(5)
with
˛=1
>0
(tensile)
˛=h
≤0
(compression),
where −Y* is the damage strain energy release rate depending on the sign of the stress triaxiality ratio, through the variable h. The value of h is taken from the literature (César de Sá et al., 2006) and is h = 0.2: in the case of a compressive loading, the damage rate will be slowed by 80%. The above damage law is fully coupled to the elastoplastic behavior of the material, and the governing system of equations is solved by means of a Newton–Raphson minimization method (Hubert, 2010). This material model is implemented in a Vectorized User MATerial (VUMAT) for ABAQUS/Explicit. The material plastic and damage properties, i.e. the hardening coefficient K, the hardening exponent m and the damage parameter S, have been determined by means of inverse methodology, on the force–displacement response of tensile specimens (Hubert, 2010). The second damage parameter is fixed at s = 1 (Lemaitre and Sermage, 1997). The final values for all the parameters are given in Table 1. 3.2. Failure In order to model the material separation caused by the first process of the working sequence, the edge-trimming, a failure criterion must be implemented. In the study of Hubert (2010), the capability of five criteria to predict failure in edge-trimming has been quantified. The evaluation of these criteria has been performed according to the extent of the hardened region to the cut edge and on the equivalent plastic strain value on the cut edge. Two of these failure criteria were the one related to the Lemaitre damage model and the Shear Failure criterion. For the criteria based on maximum damage, the obtained cut profiles were unrealistic, with large elements distortion. It was also strongly penalizing for the application of the strategy developed in the previous section. Thus, it has been concluded that this failure criterion was not appropriate to model material separation for this process, that involve an inclined knife and large clearance. The most appropriate failure criterion was the Shear Failure running in parallel with the damage model to keep
Table 1 Elastoplastic and damage material parameters. Parameter
K
m
S
s
Value
387 MPa
0.498
2.7 MPa
1
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Fig. 3. Extrusion algorithm flow chart for the 3D model creation.
material softening. This criterion is implemented in the VUMAT, it is defined by p
Csf =
ε0 +
p
εf
,
(6)
p
where ε0 is the initial equivalent plastic strain, is the equivap lent plastic strain increment and εf is the equivalent plastic strain at p
failure. The identification of εf is achieved using the Finite Element model of the tensile specimen and the critical value is picked up from the FEM result once the experimental elongation is reached. p From this result, the criterion is activated when εf = 1.17, correp
sponding to Csf = 1. It is important to notice that this value of εf is directly related to the specimen mesh size. Thus, to keep the consistency of this value all along the simulation stages, the same elements size will be applied to the edge-trimming model (in the sheared region), as well as to the strip thickness reduction model in the next section. The material failure is modeled using the element deletion technique, that removes the elements from the mesh once all their integration points reach Csf = 1. As explained above, the failure criterion and the damage model are running together; it means that the damage value D increases as well as Csf , and failure occur when Csf = 1. 4. Reproduction of the experimental sequence The experimental sequence performed in the authors previous study (Hubert et al., 2010) involves an industrial edge-trimming process, followed by a thickness reduction step applied by means of an experimental testing device, the Upsetting Rolling Test, designed to reproduce the cold rolling contact conditions. The aim of this previous study was to quantify the effect of the rolling contact conditions on the edge-trimmed specimens edges integrity. After thickness reduction, all configurations gave cracks on the specimens edges. But the analysis showed a strong dependency of the cracks width on the forward slip: the higher its value was, the thinner the cracks were. Moreover, the influence of the material state after edge-trimming has been analyzed, by means of a softening annealing stage in between the two manufacturing processes. It
led to a decrease of −20% of the hardness and no crack appeared after thickness reduction, leaving prospects upon improvements concerning the industrial edge-trimming process. In this section, this experimental sequence is reproduced. The first process is introduced with the assumptions on which it is based on. Once the results are obtained, the extrusion algorithm is applied to the final mesh to create the 3D FE model of the strip edge and the running conditions are detailed. Both models are supplied by the material behavior subroutine developed in the previous section.
4.1. Edge-trimming process simulation As described in introduction of this paper, the edge-trimming process is a continuous shear cutting operation, performed in order to remove possible defects created by the upstream process, the hot rolling. Considering only one strip edge, the edge-trimming process consists in two circular knives and three polyurethane rolls that allow to clamp the strip during cutting, driven by the strip motion. Fig. 4 shows a schema of the process. In Fig. 4(a) are depicted two angles, ˛1 and ˛2 : the cant angle and the rake angle, respectively. ˛2 is recalled onto Fig. 4(b), with two other process parameters, the clearance and the overlap, corresponding to the horizontal and vertical gaps between the upper and the lower knives. Due to angles ˛1 and ˛2 , the path of a given point on the upper knife edge evolves in 3D and thus, the edge-trimming process should be modeled in three dimensions. In a previous study, Hubert et al. (2009) showed that modeling the edge-trimming process in 3D gives roughly the same results as a 2D plane strain computation, especially in terms of final equivalent plastic strain on the cut edges. Moreover, the computation time for a 30 mm long strip is several days on a SGI Altix XE1200 with 16 dual core processors. For these reasons, the assumption of a plane strain configuration is adopted. The industrial values of the clearance, the overlap, the cant and the rake angles are applied to the Finite Element model, and the upper knife path is projected in the modeling plane. The Finite Element model of edge-trimming is illustrated in Fig. 5. Fig. 5 shows a 2.5 mm × 40 mm strip, the two knives and the polyurethane roll no. 2 (Fig. 4), used to clamp the strip during
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Fig. 4. Schema of the edge-trimming process and the involved parameters. (a) 3D view of the process. (b) 2D view of the process parameters.
cutting. This polyurethane roll is modeled as a linear elastic material, its viscous behavior being not considered as dominating. Its Young modulus is given by the manufacturer to E = 345 MPa. The knives are considered as rigid bodies, and the strip properties are given in Table 1. The strip mesh involves four nodes quadrangular plane strain elements with reduced integration and hourglass modes control. The element size in the coarse region is a1 = 5 mm and is a2 = 62.5 m in the fine region, to obtain a better accuracy of the results in the cut zone. This mesh size in this zone is consistent with the mesh size used for the failure criterion calibration. This precaution allows to ensure the accuracy of elements deletion onset. After its initiation, the crack propagation maybe inaccurate because no mesh regularization has been used. This choice also comes from
the fact that the propagation technique is the element deletion, that leads to material removal and thus inaccuracies. The transition between the coarse and fine regions is ensured by triangular elements. Finally, the model is running with the industrial speed, V = 1 m s−1 , applied onto the knives. The simulation of this model gives the profile illustrated in Fig. 6. Two characteristic regions are denoted on this figure, the sheared and the burnished regions, that will be used in the following of the study. 4.2. Thickness reduction simulation The developed extrusion algorithm is applied to the deformed mesh of the edge-trimming model (Fig. 6), to create the geometry of
Fig. 5. Schema of the Finite Element model of edge-trimming process, boundary conditions and magnification of the finely meshed region.
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Sfwd = 2.15%. Using these conditions, two material mechanical states are considered, as for the experiments:
Fig. 6. Deformed mesh, superimposed contour plot of the equivalent plastic strain and characteristic regions.
the experimental edge-trimmed strip. The extruded mesh is illustrated in Fig. 7. This model represents a 5 mm wide region from the cut edge; a symmetry condition is applied on the edge opposite to the cut one, in the yz-plane, to match the experimental strip width (Hubert et al., 2010). This new Finite Element model is 35 mm long, with two regions coarsely meshed in order to save on computation time: the initial contact region and the end of the strip, in which the results will be scattered anyway. The stationary regime region is 10 mm long. The strip is clamped in the initial contact region (z = 0). The mechanical history coming from the edge-trimming model p is then applied, it corresponds to the equivalent plastic strain ε and the damage level D, introduced as initial values in the material properties of the new model, in the region denoted in Fig. 7. The steel strip is deformed in between a slab of an industrial work roll, with radius R = 27.98 mm, and a steel plate. In the model, both tools are assumed as rigid bodies. The steel plate is fully clamped; the work roll is driven in velocity in the extrusion way (+zaxis) with a linear speed Vs , corresponding to the strip exit speed, and its rotation is applied by an angular speed ω around the +x-axis. The two parameters to apply to match the experimental tests are the reduction ratio p, and the forward slip Sfwd , defined by p=
t1 − t0 t0
(7)
and Sfwd =
Vs − ωR , ωR
(8)
where t0 and t1 are the initial and final strip thicknesses. The computations use the experimental running conditions (Hubert et al., 2010), in terms of measured reduction ratio and forward slip, for the first reduction pass. Two studies will be performed. In a first time, a validation step is performed to quantify the accuracy of the developed methodology, with p =− 28.30 % and
1. the model is called “Simply Trimmed” (ST), since the states of equivalent plastic strain and of damage are those obtained directly from the edge-trimming model, using the Shear Failure criterion; 2. the model is called “Trimmed and Heat treated” (TH), and corresponds to specimens that have been heat treated before thickness reduction using the URT. The experiments showed that the decrease of the hardness level by heat treatment allows to preserve the specimens edges integrity during reduction. For this last model, TH, the heat treatment applied on the experimental specimens allowed to decrease the hardness level in the strip edge by 20%. In that case, it is assumed that this treatment, performed at 550 ◦ C during 35 min, is not able to remove the voids created by damage and thus, the damage level remains unchanged. To match the experiments, the TH model has been generated by applying a coefficient of 0.8 to the equivalent plastic strain value. In a second time, the effect of the forward slip on the strips edges behavior is analyzed, with the same reduction ratio as for the validation step, and forward slips Sfwd = 2.15% and Sfwd = 11.57%. The results of both studies are given in the next section. As for the edge-trimming model, the Finite Element computations are performed using the explicit solver of ABAQUS. In that case, in order to decrease the computation time, and since the inertia effects are not taken into account in this study, it is chosen to increase the experimental strip exit speed from Vs = 90 mm min−1 to Vs = 8.33 m s−1 . This new value for Vs corresponds to a common exit speed used on an industrial cold mill. Using an explicit solver, it is actually possible to decrease the computation time by increasing the loading speeds, as long as the material does not have any viscous property, and if, a posteriori, the model inertia is not preponderant in the whole simulation energy balance. In the case of the considered model, the kinetic energy Ec is very small compared to the internal energy of the model Ei : for the whole simulation, the ratio Er = Ec /Ei × 100 remains lower than 7%, and is 0.07% when the work roll reaches the finely meshed region. It can be concluded that the explicit solver can be used for the following of the study. A second point must checked: the stable time lapse in the simulation. It is determined thanks to the reaction forces obtained on the work roll during the simulation. The longitudinal and tangential forces are plotted in the graph of Fig. 8. The graph of Fig. 8 shows that the stable region of the model is located in between t = 1.6 ms and t = 3.46 ms; the analysis of the following results will be focused in this time interval. The modeling of failure is still managed by the element deletion technique, and the failure criterion used for the rest of the
Fig. 7. Finite Element model of the cut edge specimen, boundary conditions and region affected by the edge-trimming operation.
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central bursting. The critical value of Lemaitre’s failure criterion has also been identified numerically; it corresponds to Dc = 0.625.
5. Results 5.1. Validation of the methodology
Fig. 8. Normal and tangential forces applied on the work roll for the URT model with Sfwd = 2.15% (ST model).
computations is Lemaitre one. This choice is justified by the use of variable ˛ (Eq. (5)), that allows to slow down the failure initiation in compression, and by the need to reproduce a discontinuous defect, as explained in the study of Saanouni et al. (2004), on
In the previous study, the results showed a significant difference between the ST and TH specimens: the heat treatment allowed to decrease the hardening created by the edge-trimming process by 20%, and the effect on the specimens edges integrity was clear, there was definitely no edge cracking due to thickness reduction from 28.30% up to 66%. In the case of the numerical model, the same observation is made: using the ST model, with the full equivalent plastic strain field, failed elements appeared, as shown in the magnifying glass in Fig. 9(a). The regions with both failed ( = 0) and stress carrying ( > 0) elements denote regions that are potentially affected by a crack.
Fig. 9. Results comparison for the models deformed directly after edge-trimming (ST) and after decrease of the equivalent plastic strain (TH). (a) With full material history (ST model). (b) With equivalent plastic strain decreased by 20% (TH model).
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Fig. 10. Results comparison for the ST models, deformed with Sfwd = 2.15% and Sfwd = 11.57%. (a) ST model, Sfwd = 2.15%. (b) ST model, Sfwd = 11.57%.
In the case of the model with decreased hardening (Fig. 9(b)), such a behavior is not noticed, and means that the specimen edge is free of crack. Actually, the decrease of the equivalent plastic strain allows to slow down the evolution of the damage variable, that does not reach the critical value Dc = 0.625. As for the experimental study, the probability for cracks occurrence is very low with annealed specimens. It can be concluded that the assumptions on which the developed methodology is based on is sensitive enough to model the edges behavior encountered experimentally. In the next section, this approach will be used to understand the causes of edge cracking according to the forward slip value. 5.2. Effect of forward slip on edge cracking The above model, with the full edge-trimming material history, is now adapted to represent a high forward slip value, about ∗ Sfwd = 10
. Actually, even if this case is never used in the industry, it highlighted an interesting effect on the edges specimens integrity after thickness reduction: the averaged cracks width measured on the specimens was decreased about 36% after the first thickness reduction pass, using a high forward slip. In that case, the averaged forward slip value was Sfwd = 11.57%, and the averaged reduction ratio was p =− 30.01 %. But in order to perform a relevant study, the applied reduction ratio for this second model is the same as for the first one, i.e. p =− 28.30 %. The deformed mesh obtained using this configuration is illustrated in Fig. 10(b), and the one obtained with a small forward slip, presented in Fig. 9(a), is recalled in Fig. 10(a). A first remark on these two deformed meshes is related to the regions affected by fracture: both models show failed elements, but in the case of a small forward slip, both the sheared and the burnished regions are affected, while in the case of a high forward slip, only the sheared region is cracked (Fig. 10(b), the specific regions have been illustrated in Fig. 6). It can also be noticed that the number of failed elements is roughly the same for both forward slip
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Fig. 13. Tracking of the stress triaxiality ratio, the equivalent plastic strain and the damage variable according to the time, in the sheared region (element set no. 1).
Fig. 11. Locations of mechanical variables tracking.
cases: 447 failed elements in the case of Sfwd = 2.15% and 437 in the case of Sfwd = 11.57%. In these Finite Element models, the total number of elements is 184,500 in the finely meshed region. It can be concluded that in both cases, an equilibrium is reached which is related to the strip stiffness, with a different distribution of failed elements in the potentially crack sensitive regions. In order to understand which mechanical factors lead to the modification of the strip edge behavior with the forward slip, two sets of seven elements on each specimen edge are chosen (Fig. 11), in the sheared and the burnished regions. The mechanical response of the integration points is averaged for each element set and will allow to track the mechanical variables involved in the Lemaitre damage law during rolling. The Lemaitre damage law, described in Section 3, depends on p the equivalent plastic strain ε , the stress triaxiality ratio and the material parameters E, , S, s. The last parameters being constants, the parameters contributing to damage evolution are the equivalent plastic strain and the stress triaxiality ratio. In incremental form, the damage law becomes Dn+1 (, ) = Dn +
1 − Dn+1 (, )
−Y
n+1 (, )
S
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s
,
(9)
where is the equivalent plastic strain increment. Eq. (9) is plotted on a 3D graph (Fig. 12) to better highlight the effect of the stress p triaxiality ratio and of the equivalent plastic strain ε . The tracking of the first element set, in the sheared region (Fig. 11), which is plotted on the graph of Fig. 13, shows the evolup tion of D, and ε according to the simulation time, for both forward slips. In the case of the small forward slip value, the stress triaxiality ratio is positive right before the contact entry. Once in the contact,
it is negative until t = 1.92 ms and it stays positive until t = 2.05 ms. will remain negative till the end of the simulation. This means that the damage energy release rate −Y is affected by ˛ (Eq. (5)) and then, the damage evolution is slowed down. In the case of a high forward slip value, the trend for is similar to the one observed for a small forward slip, aside from its amplitude. Before the contact entry, the values of the stress triaxiality ratio are roughly the same as for Sfwd = 2.15%. Once the element set enters the contact region, remains in the positive part of the graph until the end of the contact. This means that the partial closure parameter ˛ equals 1, and thus damage increases more quickly. It can actually p be observed on the graph of Fig. 12, that shows, for ε = 0.5, the fast increase of the damage value for positive stress triaxiality ratios. The tracking of the second element set, located in the burnished region of the edge-trimmed profile (Fig. 11), shows that whatever the forward slip value, the stress triaxiality ratio is positive in almost the whole contact area (Fig. 14). According to Fig. 12, it is favorable to the increase of damage. Moreover, the initial value of the damage is also higher in the burnished region (D0 = 0.11) than in the sheared one (D0 = 0.043). The equivalent plastic strain, in the case of the burnished region, is also smaller than for the sheared region, that limits the increase of damage. The curves plotted on the graphs of Figs. 13 and 14 allow to explain the failure of elements only in the sheared region, for a high forward slip value. In this case, the damage state is high in the sheared region, leading the equivalent stress to soften. The elements stress carrying capacity is thus reduced and they relieve the burnished region, in which no failed elements are observed. On the other hand, with a small forward slip value, the sheared region is
Fig. 12. Damage surface, plotted according to the stress triaxiality ratio and to the equivalent plastic strain.
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Fig. 14. Tracking of the stress triaxiality ratio, the equivalent plastic strain and the damage variable according to the time, in the burnished region (element set no. 2).
Fig. 15. Cut profiles for the two clearances: (a) initial and (b) 80% of the initial value.
less damaged than for a high forward slip. The burnished region, in which the initial damage state is higher than in the sheared region, reaches more quickly the critical damage threshold Dc , and failed elements are observed. 5.3. Effect of clearance on the edges integrity Hilditch and Hodgson (2005), or formerly Taupin et al. (1996), performed studies in the field of metal cutting, especially on the cutting parameters. The aim of these studies was to decrease the probability of burr formation after cutting, and they showed that decreasing the clearance (E, the horizontal gap between the upper and the lower knives, Fig. 4(b)) limits this defect. Golovashchenko (2006) get the same trend: a higher sheared region with less defects,
like burrs or material fines after cutting. The author postulated that decreasing the clearance increases the hydrostatic pressure in the strip edge, leading to the improvement of the trimming quality. In this section, the developed approach is used to quantify the effect of the clearance on the occurrence of cracks during plastic deformation. The clearance used in Section 4.1 of this paper is decreased by 20%, leading the knives path to be slightly modified. The other edge-trimming parameters remain unchanged. Fig. 15 shows the new cut profile, obtained with this new clearance value, compared to the initial one. The equivalent plastic strain fields and the extent of the affected regions are superimposed on the deformed profiles.
Fig. 16. Quantification of the extent to the edge of the region affected mechanically by edge-trimming.
C. Hubert et al. / Journal of Materials Processing Technology 212 (2012) 1049–1060
Fig. 17. Deformed mesh in the case of a 20% reduction of the clearance during edgetrimming.
Based on the equivalent plastic strain contour lines, Fig. 15 shows that the extend of the region affected by the edge-trimming is decreased by 15%. A more accurate comparison is shown on the graph of Fig. 16. It shows the extent to the edge of the region affected by cutting, in terms of equivalent plastic strain profile. This graph shows that the extent to the edge of the affected region is roughly the same for the two clearance values, but the gradient is more pronounced in the case of the small clearance. The clearance reduction allows to locate the high values of the material hardening near the edge: the equivalent plastic strain is decreased by 30% at 0.2 mm to the edge. According to the study of Golovashchenko (2006), this should allow to decrease the probability of crack occurrence during rolling. To check this hypothesis, the deformed mesh illustrated in Fig. 15 is submitted to the extrusion algorithm presented in Section 2. Same boundary conditions are applied to this new model as for the ST model with Sfwd = 2.15% (Section 5.2). The final mesh is illustrated in Fig. 17, after deformation. On one hand, compared to the deformed mesh of Fig. 10(a), the number of failed elements is increased by 27%. On the other hand, the failed elements are all located on the first elements layer of the edge (which is almost 70 m wide). The modification of the equivalent plastic strain gradient (Fig. 16) thus has an important role in edge cracking: even if more cracks are noticed on the edge, the propagation length is decreased by the fast recovering of the initial material state in the edge. 6. Conclusion This paper focuses on the development of a methodology to study a process sequence when the modeling geometrical assumptions differ from a process to another. This study is based on the Finite Element Method. The first sequence process is modeled assuming the plane strain condition. The second one is modeled in 3D because of the location of the regions of interest. The procedure is based on the continuity assumption of the geometry and of the mechanical fields in the out-of-plane direction to create a 3D mesh from the 2D one, and to transfer the thermomechanical fields to the new model. This methodology is applied to an experimental study achieved by the present authors, in the field of steel strips working. It involves an industrial edge-trimming step followed by an experimental thickness reduction test. At the end of this sequence, the specimens edges were cracked owing to the work hardening accumulated during the edge-trimming operation. The accuracy of the developed strategy is qualitatively compared to the previous experimental results. The model sensitivity to the hardening state is in good agreement with the experimental observations: a decrease of 20% of the hardening state is sufficient to completely remove any crack initiation during thickness reduction. This leads to the conclusion that the developed strategy, as well as the material model, are accurate enough.
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The effect of the forward slip on cracks occurrence is then analyzed, based on the previous experimental study. Two extreme forward slip values are considered: the first value is commonly used in the industry, while the second one is very high. The results show that the forward slip strongly modifies the stress triaxiality ratio, which is directly linked to the damage law, along the contact length. The density of failed elements, that represent the cracked regions, is roughly the same for both forward slip values, but their distribution is located in the top of the specimen edge for the high forward slip value. It can suppose that since the failed elements are all located on one place of the edge, the cracks should be deeper. This could correlate the experimental observations, but the selected method to model cracks is not accurate enough to validate this hypothesis. Finally, the methodology is used to study the effect of an edge-trimming parameter, the clearance, on the specimens edges integrity after thickness reduction. Decreasing the clearance by 20% allowed to locate the plastic strain near the edge, leading to less deep cracks after thickness reduction comparing to the industrial configuration. These results follow the trends observed experimentally but the results accuracy is affected by the element deletion technique, leading to material volume loss in the modeled part. This needs further investigations, for instance to enhance the modeling of cracks initiation and propagation applying the eXtended Finite Element Method (XFEM), which could give more accurate and realistic sheared or cracked profiles. Also, the material model introduced in this paper is rate and temperature independent, while the strip temperature may reach 120 ◦ C in the cold mill roll bite, and strain rates may highly increase with thickness reduction. These limitations may be investigated in further work. Acknowledgements The present research work has been supported by International Campus on Safety and Intermodality in Transportation, the Nord-Pas-de-Calais Region, the European Community, the Regional Delegation for Research and Technology, the Ministry of Higher Education and Research, and the National Center for Scientific Research. The authors gratefully acknowledge the support of these institutions. References Andrade Pires, F.M., de Souza Neto, E., Owen, D., 2004. On the finite element prediction of damage growth and fracture initiation in finitely deforming ductile materials. Computer Methods in Applied Mechanics and Engineering 193, 5223–5256. César de Sá, J.M.A., Areias, P.M.A., Zheng, C., 2006. Damage modelling in metal forming problems using an implicit non-local gradient model. Computer Methods in Applied Mechanics and Engineering 195, 6646–6660. Cockroft, M., Latham, D., 1968. Ductility and workability of metals. Journal Institute of Metals 96, 33–39. Dubar, M., Huart, S., Deltombe, R., Dubois, A., Dubar, L., 2006. Cold rolling of galvanized strips: A new approach to the evaluation of zinc fines formation. Journal of Materials Processing Technology 177, 505–508. Farzin, M., Javani, H., Mashayekhi, M., Hambli, R., 2006. Analysis of blanking process using various damage criteria. Journal of Materials Processing Technology 177, 287–290. Ghosh, S., Li, M., Khadke, A., 2005. 3D modeling of shear-slitting process for aluminum alloys. Journal of Materials Processing Technology, 91–102. Golovashchenko, S.F., 2006. A study on trimming of aluminum autobody sheet and development of a new robust process eliminating burrs and slivers. International Journal of Mechanical Sciences 48, 1384–1400. Hambli, R., 2002a. Inverse technique identification of material damage law with experimental validation. The International Journal of Advanced Manufacturing Technology 20, 223–229. Hambli, R., 2002b. Prediction of burr height formation in blanking processes using neural network. International Journal of Mechanical Sciences 44, 2089–2102. Hilditch, T., Hodgson, P., 2005. Development of the sheared edge in the trimming of steel and light metal sheet. Part 1. Experimental observations. Journal of Materials Processing Technology 169, 184–191. Hoang, N., Porcaro, R., Langseth, M., Hanssen, A.G., 2010. Self-piercing riveting connections using aluminium rivets. International Journal of Solids and Structures 47, 427–439.
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Huart, S., Dubar, M., Deltombe, R., Dubois, A., Dubar, L., 2004. Asperity deformation, lubricant trapping and iron fines formation mechanism in cold rolling processes. Wear 257, 471–480. Hubert, C., 2010. Stratégie de modélisation de la séquence cisaillage-laminage en mise en forme à froid des aciers. Ph.D. Thesis. Université de Valenciennes et du Hainaut-Cambrésis. Hubert, C., Dubar, L., Dubar, M., Dubois, A., 2010. Experimental simulation of strip edge cracking in steel rolling sequences. Journal of Materials Processing Technology 210, 1587–1597. Hubert, C., Dubar, M., Dubois, A., Dubar, L., 2009. 3D modeling of edge trimming process. International Journal of Material Forming 2, 837–840. Lemaitre, J., 1985. A continuous damage mechanics model for ductile fracture. Journal of Engineering Materials and Technology 107, 83. Lemaitre, J., Sermage, J.P., 1997. One damage law for different mechanisms. Computational Mechanics 20, 84–88.
Li, H., Fu, M., Lu, J., Yang, H., 2010. Ductile fracture: experiments and computations. International Journal of Plasticity. Oh, S.I., Chen, C.C., Kobayashi, S., 1979. Ductile fracture in axisymmetric extrusion and drawing. Journal of Engineering for Industry 101, 36–44. Saanouni, K., Mariage, J., Cherouat, A., Lestriez, P., 2004. Numerical prediction of discontinuous central bursting in axisymmetric forward extrusion by continuum damage mechanics. Computers and Structures 82, 2309–2332. Szczurek, E., Dubar, M., Deltombe, R., Dubois, A., Dubar, L., 2009. New approach to the evaluation of the free surface position in roll coating. Journal of Materials Processing Technology 209, 3187–3197. Taupin, E., Breitling, J., Wu, W., Altan, T., 1996. Material fracture and burr formation in blanking results of fem simulations and comparison with experiments. Journal of Materials Processing Technology 59, 68–78.
Contents lists available at SciVerse ScienceDirect
Journal of Materials Processing Technology journal homepage: www.elsevier.com/locate/jmatprotec
Finite Element simulation of the edge-trimming/cold rolling sequence: Analysis of edge cracking C. Hubert a,b,∗ , L. Dubar a,b , M. Dubar a,b , A. Dubois a,b a b
Univ Lille Nord de France, F-59000 Lille, France UVHC, TEMPO EA 4542, F-59313 Valenciennes, France
a r t i c l e
i n f o
Article history: Received 6 May 2011 Received in revised form 3 November 2011 Accepted 21 December 2011 Available online 5 January 2012 PACS: 87.10.Kn 46.50.+a Keywords: Edge cracking Finite Element Method Failure criterion Damage model
a b s t r a c t The aim of the present study is to develop a strategy of numerical simulation to study the sequencing of two processes, when the modeling geometry assumptions differ. In this study, the first process is assumed continuous, leading to a plane strain model. The developed strategy of simulation takes advantages of this assumption to extrude, in the out-of-plane direction, the 2D mesh and the mechanical fields to create a 3D model corresponding to the second process. In a previous experimental study achieved by the authors, an edge-trimming/strip thickness reduction sequence was analyzed. The aim was to quantify the effect of the work hardening accumulated in the first step on the occurrence of cracks in the strips edges during the second one. The present study uses this experimental sequence to validate the proposed approach and to better understand the reasons of edge cracking, according to the plastic strain level and thickness reduction conditions. Finally, the proposed approach is used as a predictor, to quantify the effect of a major parameter in edge-trimming on the strips edges integrity, at the end of the full sequence. © 2011 Elsevier B.V. All rights reserved.
1. Introduction The manufacturing processes of metallic parts, or their assembly, often involve multiple operations, such as machining, cutting, forming or drilling. For some processes, the residual stress state and the accumulated work hardening generated by the upstream processes can be strongly reduced or even cancelled by means, for instance, of a recrystallization annealing, where new strain-free grains nucleate and grow to replace those existing. This case has been encountered in coil coating, for which the coils are annealed before galvanizing (Dubar et al., 2006) and coating (Szczurek et al., 2009). In other cases, for processes such as self-piercing riveting, the rivet and assembled parts deformations must be considered for the mechanical strength evaluation of the whole assembly (Hoang et al., 2010). The numerical approaches, especially the Finite Element Method, are often used to simulate the manufacturing processes and to study the thermomechanical behavior of the resulting parts or assemblies. To perform an accurate simulation of multi-stages manufacturing, it is thus necessary to keep the thermomechanical material history through the different processing steps. Generally,
∗ Corresponding author at: UVHC, TEMPO EA 4542, F-59313 Valenciennes, France. Tel.: +33 3 27 51 14 54; fax: +33 3 27 51 13 17. E-mail address: [email protected] (C. Hubert). 0924-0136/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2011.12.011
this is easily achieved by creating steps within the numerical simulation, that allow to simulate several forming stages at once. Options in Finite Element solvers also allow to create backup points at a given simulation time, in order to recover the strain and stress fields to a new Finite Element model, with new boundary and simulation conditions. But these methods can only be applied if the model geometry is kept between two simulation stages, which is not always the case. The aim of this paper is to develop a modeling strategy in order to study a sequence of two steel strip working processes, by means of the Finite Element Method. The first process is modeled using the plane strain assumption to reduce computations costs (Hubert et al., 2009). It is followed by a second process, which is modeled in 3D since the regions of interest for the results analysis are not in-plane. The proposed strategy takes advantages of the continuous nature of the first process: it assumes that the part geometry is rectilinear and the material mechanical state is homogeneous along the out-of-plane direction. The 2D deformed mesh resulting from the first operation is thus extruded following the out-of-plane direction to obtain a 3D mesh, and the 2D material mechanical fields are propagated along this direction. After application of the boundary conditions related to the second process, the simulation can be run and the results are analyzed in the same way as for a full 3D simulation. This study is applied to a forming sequence, involving an industrial edge-trimming step followed by an experimental thickness
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reduction operation. The first process is needed to remove hot rolling edge defects by reducing the strips width. It is a continuous shear cutting process and, under some assumptions, it can be modeled in plane strain. The second process, named Upsetting Rolling Test (URT, Huart et al., 2004; Hubert et al., 2010), is an experimental testing stand used to reproduce the industrial cold rolling contact conditions. This sequence corresponds to the one performed in a previous study achieved by the present authors (Hubert et al., 2010), which highlighted that the strain hardening state caused by edge-trimming is of major importance in the integrity of the edges after thickness reduction. The use of such a numerical approach thus make sense. The previous experimental results will be used to validate the developed simulation strategy. The strip working processes encountered is this study involve large deformations, damage and failure of the part. A material behavior model is thus developed to consider these features. The phenomenological approach is chosen to model damage. It is based on the consideration of a Representative Elementary Volume, in which a given amount of voids is represented by a scalar or a tensor. The most common model is Lemaitre’s (Lemaitre, 1985); it has been used by Hambli (2002a) in the field of sheet blanking, or by César de Sá et al. (2006) in the case of external inversion of tubes. It allows simple implementation and identification, and to consider the load reversal. This model is chosen for this study and is implemented in the Finite Element code. To model material failure, studies in the field of shear slitting (Ghosh et al., 2005; Farzin et al., 2006) or sheet blanking (Hambli, 2002b) showed that the most used failure criteria were of very simple formulation, such as the Cockcroft–Latham (Cockroft and Latham, 1968) and the Shear Failure. The studies related to these failure criteria gave good results in terms of prediction of the sheared profile and burr resulting from the process settings. In the field of edge-trimming, validation tests have been performed by Hubert (2010), in which criteria such as the Cockcroft–Latham, the Oh et al. (1979), the Shear Failure and the Lemaitre critical damage were compared to industrially edge-trimmed profiles. The Shear Failure criterion was the most accurate one. The validation has been performed in three stages. The extent of the mechanically affected region to the edge has been compared to experimental hardness measurements, and the transition line between the sheared and the burnished regions were compared by means of the reaction forces curves. Finally, the equivalent plastic strain level predicted numerically was more realistic with the Shear Failure than with the other criteria. The Shear Failure criterion is thus adopted here. In this paper is first introduced the global methodology to create a 3D model from a 2D plane strain one, and to apply the thermomechanical fields coming from the 2D simulation results as initial state in the 3D model. Then the constitutive equations of the material behavior model and the failure criterion are presented. Secondly, the Finite Element model of the edge-trimming process and the involved boundary conditions are described. The methodology to create the 3D model is applied and the resulting model, corresponding to the edges specimens used with the URT, is detailed. The validity of the developed approach is quantified according to the experimental results of Hubert et al. (2010). Then, the experimental sensitivity study of the forward slip on edge cracking performed by Hubert et al. (2010) is reproduced. It will allow to better understand the phenomena that lead to edge cracking during plastic deformation. Finally, the developed approach is used as a predictor: one of the main edge-trimming process parameter, the clearance, is modified in order to quantify its effect on the specimens edges integrity.
Fig. 1. 2D to 3D elements extrusion in the case of reduced integration.
2. Modeling strategy The aim of this paper is to develop a modeling strategy in order to study, by means of the Finite Element Method, the effects of a metal working process on the part behavior when it is submitted to a second working operation. The sequence considered in this paper involves a 2D model followed by a 3D one, but Finite Element solvers do not allow transfer neither the geometry nor the thermomechanical material history between two models of different modeling dimensions. A methodology is thus implemented to overcome this restriction. The first point is to create a 3D mesh from the 2D one. The second point is to apply the relevant final mechanical fields to this new mesh as initial conditions. These two points are detailed in the next sections. 2.1. Mesh geometrical extrusion The 2D FE simulation on which the present modeling strategy is based on must follow the plane strain theory. It allows to assume the model geometry and the material mechanical state as continuous in the out-of-plane direction. This strategy takes advantages of these assumptions by performing, in a first time, a geometrical extrusion of the 2D model. The nodes of the deformed mesh, at the end of the 2D simulation, are projected in the third space dimension, and the initially plane elements become 3D: 8-nodes linear bricks for the initial quadrangular elements, and 6-nodes linear triangular prisms for the initially triangular plane elements. The elements length in the extrusion direction is adjusted to obtain a ratio of maximum to minimum element length between 1:1 and 2:1. The number of integration points in the 3D elements depend on those in the plane elements. Fig. 1 shows the extrusions for rectangular and triangular elements, with reduced integration. 2.2. Mechanical fields application The significant mechanical fields at the end of the 2D simulation must now be applied to the extruded mesh. In the case of builtin methods, the full tensors are applied to the new simulation as initial mechanical fields. In this modeling strategy, the procedure is simplified: this methodology is developed in the field of isotropic plasticity, and the equivalent plastic strain is sufficient to recover the equivalent stress, instead of the full tensor. Same deduction is applied for any other quantity such as damage or failure variables. This paper deals with large deformations in metal forming, a damage model coupled to the isotropic plasticity model is implemented and will be detailed in the next section. The mechanical variables to transfer to the extruded mesh are thus the isotropic p damage variable D and the equivalent plastic strain ε . These variables will be introduced in the 3D Finite Element model as initial
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Fig. 2. Mechanical field from the 2D model applied to the 3D one.
values in the material data, and are directly applied to the integration points by the FE code at the first calculation increment. Fig. 2 shows an illustration of an arbitrary mechanical field coming from the 2D deformed mesh transferred to the 3D model. The developed mesh extrusion algorithm has been integrated into ABAQUS via its graphical user interface. The principle of this algorithm is outlined in Fig. 3. 3. Local integration 3.1. Elastoplasticity and damage The processes being modeled in this study lead to large deformations until failure of small regions of the worked part or even the entire part. First, it is necessary to take into account the elastoplastic behavior of the material, which can be defined by a Ludwik law, p
y = 0 + Kε m,
(1)
where 0 and y are the initial and the current yield stress, K and p m are material parameters and ε is the equivalent plastic strain. The identification of the material parameters K and m, performed on tensile specimens, showed a strong dependency of the material to damage (Hubert et al., 2009). It has been decided to develop a user material subroutine to take into account the elastoplastic behavior coupled to a damage law. The Lemaitre one has been chosen, mainly for its capability to predict damage increase in compression (Li et al., 2010). It is defined by D˙ =
−Y s S
p ε˙ ,
(2)
where D˙ is the isotropic damage rate, S and s are material paramp
eters and ε˙ is the equivalent plastic strain rate. −Y is the damage strain energy release rate, defined by −Y =
q2 R
,
(3)
2 (1 + ) + 3(1 − 2)2 , 3
(4)
2E(1 − D)2
with R =
where q is the von Mises equivalent stress, E is the material Young modulus, is its Poisson ratio and is the stress triaxiality ratio ( = p/q, p being the hydrostatic stress). D is the damage variable, defined such that 0 ≤ D ≤ Dc ≤ 1, with Dc the damage threshold at which a macro crack occurs. In the initial Lemaitre damage model (Lemaitre, 1985), the evolution of the damage variable does not depend on the sign of the stress triaxiality ratio. However, when loading a damaged specimen in compression, there is a voids closure effect leading to a slowed damage evolution. This phenomenon has been highlighted, among other authors, by Andrade Pires et al. (2004) and César de Sá et al. (2006). As outlined in introduction of this paper, the second process will apply a thickness reduction of the steel strip, and the voids
closure effect will occur. It is thus taken into account in a simplified, conditional form: −Y ∗ = −˛Y
(5)
with
˛=1
>0
(tensile)
˛=h
≤0
(compression),
where −Y* is the damage strain energy release rate depending on the sign of the stress triaxiality ratio, through the variable h. The value of h is taken from the literature (César de Sá et al., 2006) and is h = 0.2: in the case of a compressive loading, the damage rate will be slowed by 80%. The above damage law is fully coupled to the elastoplastic behavior of the material, and the governing system of equations is solved by means of a Newton–Raphson minimization method (Hubert, 2010). This material model is implemented in a Vectorized User MATerial (VUMAT) for ABAQUS/Explicit. The material plastic and damage properties, i.e. the hardening coefficient K, the hardening exponent m and the damage parameter S, have been determined by means of inverse methodology, on the force–displacement response of tensile specimens (Hubert, 2010). The second damage parameter is fixed at s = 1 (Lemaitre and Sermage, 1997). The final values for all the parameters are given in Table 1. 3.2. Failure In order to model the material separation caused by the first process of the working sequence, the edge-trimming, a failure criterion must be implemented. In the study of Hubert (2010), the capability of five criteria to predict failure in edge-trimming has been quantified. The evaluation of these criteria has been performed according to the extent of the hardened region to the cut edge and on the equivalent plastic strain value on the cut edge. Two of these failure criteria were the one related to the Lemaitre damage model and the Shear Failure criterion. For the criteria based on maximum damage, the obtained cut profiles were unrealistic, with large elements distortion. It was also strongly penalizing for the application of the strategy developed in the previous section. Thus, it has been concluded that this failure criterion was not appropriate to model material separation for this process, that involve an inclined knife and large clearance. The most appropriate failure criterion was the Shear Failure running in parallel with the damage model to keep
Table 1 Elastoplastic and damage material parameters. Parameter
K
m
S
s
Value
387 MPa
0.498
2.7 MPa
1
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Fig. 3. Extrusion algorithm flow chart for the 3D model creation.
material softening. This criterion is implemented in the VUMAT, it is defined by p
Csf =
ε0 +
p
εf
,
(6)
p
where ε0 is the initial equivalent plastic strain, is the equivap lent plastic strain increment and εf is the equivalent plastic strain at p
failure. The identification of εf is achieved using the Finite Element model of the tensile specimen and the critical value is picked up from the FEM result once the experimental elongation is reached. p From this result, the criterion is activated when εf = 1.17, correp
sponding to Csf = 1. It is important to notice that this value of εf is directly related to the specimen mesh size. Thus, to keep the consistency of this value all along the simulation stages, the same elements size will be applied to the edge-trimming model (in the sheared region), as well as to the strip thickness reduction model in the next section. The material failure is modeled using the element deletion technique, that removes the elements from the mesh once all their integration points reach Csf = 1. As explained above, the failure criterion and the damage model are running together; it means that the damage value D increases as well as Csf , and failure occur when Csf = 1. 4. Reproduction of the experimental sequence The experimental sequence performed in the authors previous study (Hubert et al., 2010) involves an industrial edge-trimming process, followed by a thickness reduction step applied by means of an experimental testing device, the Upsetting Rolling Test, designed to reproduce the cold rolling contact conditions. The aim of this previous study was to quantify the effect of the rolling contact conditions on the edge-trimmed specimens edges integrity. After thickness reduction, all configurations gave cracks on the specimens edges. But the analysis showed a strong dependency of the cracks width on the forward slip: the higher its value was, the thinner the cracks were. Moreover, the influence of the material state after edge-trimming has been analyzed, by means of a softening annealing stage in between the two manufacturing processes. It
led to a decrease of −20% of the hardness and no crack appeared after thickness reduction, leaving prospects upon improvements concerning the industrial edge-trimming process. In this section, this experimental sequence is reproduced. The first process is introduced with the assumptions on which it is based on. Once the results are obtained, the extrusion algorithm is applied to the final mesh to create the 3D FE model of the strip edge and the running conditions are detailed. Both models are supplied by the material behavior subroutine developed in the previous section.
4.1. Edge-trimming process simulation As described in introduction of this paper, the edge-trimming process is a continuous shear cutting operation, performed in order to remove possible defects created by the upstream process, the hot rolling. Considering only one strip edge, the edge-trimming process consists in two circular knives and three polyurethane rolls that allow to clamp the strip during cutting, driven by the strip motion. Fig. 4 shows a schema of the process. In Fig. 4(a) are depicted two angles, ˛1 and ˛2 : the cant angle and the rake angle, respectively. ˛2 is recalled onto Fig. 4(b), with two other process parameters, the clearance and the overlap, corresponding to the horizontal and vertical gaps between the upper and the lower knives. Due to angles ˛1 and ˛2 , the path of a given point on the upper knife edge evolves in 3D and thus, the edge-trimming process should be modeled in three dimensions. In a previous study, Hubert et al. (2009) showed that modeling the edge-trimming process in 3D gives roughly the same results as a 2D plane strain computation, especially in terms of final equivalent plastic strain on the cut edges. Moreover, the computation time for a 30 mm long strip is several days on a SGI Altix XE1200 with 16 dual core processors. For these reasons, the assumption of a plane strain configuration is adopted. The industrial values of the clearance, the overlap, the cant and the rake angles are applied to the Finite Element model, and the upper knife path is projected in the modeling plane. The Finite Element model of edge-trimming is illustrated in Fig. 5. Fig. 5 shows a 2.5 mm × 40 mm strip, the two knives and the polyurethane roll no. 2 (Fig. 4), used to clamp the strip during
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Fig. 4. Schema of the edge-trimming process and the involved parameters. (a) 3D view of the process. (b) 2D view of the process parameters.
cutting. This polyurethane roll is modeled as a linear elastic material, its viscous behavior being not considered as dominating. Its Young modulus is given by the manufacturer to E = 345 MPa. The knives are considered as rigid bodies, and the strip properties are given in Table 1. The strip mesh involves four nodes quadrangular plane strain elements with reduced integration and hourglass modes control. The element size in the coarse region is a1 = 5 mm and is a2 = 62.5 m in the fine region, to obtain a better accuracy of the results in the cut zone. This mesh size in this zone is consistent with the mesh size used for the failure criterion calibration. This precaution allows to ensure the accuracy of elements deletion onset. After its initiation, the crack propagation maybe inaccurate because no mesh regularization has been used. This choice also comes from
the fact that the propagation technique is the element deletion, that leads to material removal and thus inaccuracies. The transition between the coarse and fine regions is ensured by triangular elements. Finally, the model is running with the industrial speed, V = 1 m s−1 , applied onto the knives. The simulation of this model gives the profile illustrated in Fig. 6. Two characteristic regions are denoted on this figure, the sheared and the burnished regions, that will be used in the following of the study. 4.2. Thickness reduction simulation The developed extrusion algorithm is applied to the deformed mesh of the edge-trimming model (Fig. 6), to create the geometry of
Fig. 5. Schema of the Finite Element model of edge-trimming process, boundary conditions and magnification of the finely meshed region.
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Sfwd = 2.15%. Using these conditions, two material mechanical states are considered, as for the experiments:
Fig. 6. Deformed mesh, superimposed contour plot of the equivalent plastic strain and characteristic regions.
the experimental edge-trimmed strip. The extruded mesh is illustrated in Fig. 7. This model represents a 5 mm wide region from the cut edge; a symmetry condition is applied on the edge opposite to the cut one, in the yz-plane, to match the experimental strip width (Hubert et al., 2010). This new Finite Element model is 35 mm long, with two regions coarsely meshed in order to save on computation time: the initial contact region and the end of the strip, in which the results will be scattered anyway. The stationary regime region is 10 mm long. The strip is clamped in the initial contact region (z = 0). The mechanical history coming from the edge-trimming model p is then applied, it corresponds to the equivalent plastic strain ε and the damage level D, introduced as initial values in the material properties of the new model, in the region denoted in Fig. 7. The steel strip is deformed in between a slab of an industrial work roll, with radius R = 27.98 mm, and a steel plate. In the model, both tools are assumed as rigid bodies. The steel plate is fully clamped; the work roll is driven in velocity in the extrusion way (+zaxis) with a linear speed Vs , corresponding to the strip exit speed, and its rotation is applied by an angular speed ω around the +x-axis. The two parameters to apply to match the experimental tests are the reduction ratio p, and the forward slip Sfwd , defined by p=
t1 − t0 t0
(7)
and Sfwd =
Vs − ωR , ωR
(8)
where t0 and t1 are the initial and final strip thicknesses. The computations use the experimental running conditions (Hubert et al., 2010), in terms of measured reduction ratio and forward slip, for the first reduction pass. Two studies will be performed. In a first time, a validation step is performed to quantify the accuracy of the developed methodology, with p =− 28.30 % and
1. the model is called “Simply Trimmed” (ST), since the states of equivalent plastic strain and of damage are those obtained directly from the edge-trimming model, using the Shear Failure criterion; 2. the model is called “Trimmed and Heat treated” (TH), and corresponds to specimens that have been heat treated before thickness reduction using the URT. The experiments showed that the decrease of the hardness level by heat treatment allows to preserve the specimens edges integrity during reduction. For this last model, TH, the heat treatment applied on the experimental specimens allowed to decrease the hardness level in the strip edge by 20%. In that case, it is assumed that this treatment, performed at 550 ◦ C during 35 min, is not able to remove the voids created by damage and thus, the damage level remains unchanged. To match the experiments, the TH model has been generated by applying a coefficient of 0.8 to the equivalent plastic strain value. In a second time, the effect of the forward slip on the strips edges behavior is analyzed, with the same reduction ratio as for the validation step, and forward slips Sfwd = 2.15% and Sfwd = 11.57%. The results of both studies are given in the next section. As for the edge-trimming model, the Finite Element computations are performed using the explicit solver of ABAQUS. In that case, in order to decrease the computation time, and since the inertia effects are not taken into account in this study, it is chosen to increase the experimental strip exit speed from Vs = 90 mm min−1 to Vs = 8.33 m s−1 . This new value for Vs corresponds to a common exit speed used on an industrial cold mill. Using an explicit solver, it is actually possible to decrease the computation time by increasing the loading speeds, as long as the material does not have any viscous property, and if, a posteriori, the model inertia is not preponderant in the whole simulation energy balance. In the case of the considered model, the kinetic energy Ec is very small compared to the internal energy of the model Ei : for the whole simulation, the ratio Er = Ec /Ei × 100 remains lower than 7%, and is 0.07% when the work roll reaches the finely meshed region. It can be concluded that the explicit solver can be used for the following of the study. A second point must checked: the stable time lapse in the simulation. It is determined thanks to the reaction forces obtained on the work roll during the simulation. The longitudinal and tangential forces are plotted in the graph of Fig. 8. The graph of Fig. 8 shows that the stable region of the model is located in between t = 1.6 ms and t = 3.46 ms; the analysis of the following results will be focused in this time interval. The modeling of failure is still managed by the element deletion technique, and the failure criterion used for the rest of the
Fig. 7. Finite Element model of the cut edge specimen, boundary conditions and region affected by the edge-trimming operation.
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central bursting. The critical value of Lemaitre’s failure criterion has also been identified numerically; it corresponds to Dc = 0.625.
5. Results 5.1. Validation of the methodology
Fig. 8. Normal and tangential forces applied on the work roll for the URT model with Sfwd = 2.15% (ST model).
computations is Lemaitre one. This choice is justified by the use of variable ˛ (Eq. (5)), that allows to slow down the failure initiation in compression, and by the need to reproduce a discontinuous defect, as explained in the study of Saanouni et al. (2004), on
In the previous study, the results showed a significant difference between the ST and TH specimens: the heat treatment allowed to decrease the hardening created by the edge-trimming process by 20%, and the effect on the specimens edges integrity was clear, there was definitely no edge cracking due to thickness reduction from 28.30% up to 66%. In the case of the numerical model, the same observation is made: using the ST model, with the full equivalent plastic strain field, failed elements appeared, as shown in the magnifying glass in Fig. 9(a). The regions with both failed ( = 0) and stress carrying ( > 0) elements denote regions that are potentially affected by a crack.
Fig. 9. Results comparison for the models deformed directly after edge-trimming (ST) and after decrease of the equivalent plastic strain (TH). (a) With full material history (ST model). (b) With equivalent plastic strain decreased by 20% (TH model).
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Fig. 10. Results comparison for the ST models, deformed with Sfwd = 2.15% and Sfwd = 11.57%. (a) ST model, Sfwd = 2.15%. (b) ST model, Sfwd = 11.57%.
In the case of the model with decreased hardening (Fig. 9(b)), such a behavior is not noticed, and means that the specimen edge is free of crack. Actually, the decrease of the equivalent plastic strain allows to slow down the evolution of the damage variable, that does not reach the critical value Dc = 0.625. As for the experimental study, the probability for cracks occurrence is very low with annealed specimens. It can be concluded that the assumptions on which the developed methodology is based on is sensitive enough to model the edges behavior encountered experimentally. In the next section, this approach will be used to understand the causes of edge cracking according to the forward slip value. 5.2. Effect of forward slip on edge cracking The above model, with the full edge-trimming material history, is now adapted to represent a high forward slip value, about ∗ Sfwd = 10
. Actually, even if this case is never used in the industry, it highlighted an interesting effect on the edges specimens integrity after thickness reduction: the averaged cracks width measured on the specimens was decreased about 36% after the first thickness reduction pass, using a high forward slip. In that case, the averaged forward slip value was Sfwd = 11.57%, and the averaged reduction ratio was p =− 30.01 %. But in order to perform a relevant study, the applied reduction ratio for this second model is the same as for the first one, i.e. p =− 28.30 %. The deformed mesh obtained using this configuration is illustrated in Fig. 10(b), and the one obtained with a small forward slip, presented in Fig. 9(a), is recalled in Fig. 10(a). A first remark on these two deformed meshes is related to the regions affected by fracture: both models show failed elements, but in the case of a small forward slip, both the sheared and the burnished regions are affected, while in the case of a high forward slip, only the sheared region is cracked (Fig. 10(b), the specific regions have been illustrated in Fig. 6). It can also be noticed that the number of failed elements is roughly the same for both forward slip
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Fig. 13. Tracking of the stress triaxiality ratio, the equivalent plastic strain and the damage variable according to the time, in the sheared region (element set no. 1).
Fig. 11. Locations of mechanical variables tracking.
cases: 447 failed elements in the case of Sfwd = 2.15% and 437 in the case of Sfwd = 11.57%. In these Finite Element models, the total number of elements is 184,500 in the finely meshed region. It can be concluded that in both cases, an equilibrium is reached which is related to the strip stiffness, with a different distribution of failed elements in the potentially crack sensitive regions. In order to understand which mechanical factors lead to the modification of the strip edge behavior with the forward slip, two sets of seven elements on each specimen edge are chosen (Fig. 11), in the sheared and the burnished regions. The mechanical response of the integration points is averaged for each element set and will allow to track the mechanical variables involved in the Lemaitre damage law during rolling. The Lemaitre damage law, described in Section 3, depends on p the equivalent plastic strain ε , the stress triaxiality ratio and the material parameters E, , S, s. The last parameters being constants, the parameters contributing to damage evolution are the equivalent plastic strain and the stress triaxiality ratio. In incremental form, the damage law becomes Dn+1 (, ) = Dn +
1 − Dn+1 (, )
−Y
n+1 (, )
S
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s
,
(9)
where is the equivalent plastic strain increment. Eq. (9) is plotted on a 3D graph (Fig. 12) to better highlight the effect of the stress p triaxiality ratio and of the equivalent plastic strain ε . The tracking of the first element set, in the sheared region (Fig. 11), which is plotted on the graph of Fig. 13, shows the evolup tion of D, and ε according to the simulation time, for both forward slips. In the case of the small forward slip value, the stress triaxiality ratio is positive right before the contact entry. Once in the contact,
it is negative until t = 1.92 ms and it stays positive until t = 2.05 ms. will remain negative till the end of the simulation. This means that the damage energy release rate −Y is affected by ˛ (Eq. (5)) and then, the damage evolution is slowed down. In the case of a high forward slip value, the trend for is similar to the one observed for a small forward slip, aside from its amplitude. Before the contact entry, the values of the stress triaxiality ratio are roughly the same as for Sfwd = 2.15%. Once the element set enters the contact region, remains in the positive part of the graph until the end of the contact. This means that the partial closure parameter ˛ equals 1, and thus damage increases more quickly. It can actually p be observed on the graph of Fig. 12, that shows, for ε = 0.5, the fast increase of the damage value for positive stress triaxiality ratios. The tracking of the second element set, located in the burnished region of the edge-trimmed profile (Fig. 11), shows that whatever the forward slip value, the stress triaxiality ratio is positive in almost the whole contact area (Fig. 14). According to Fig. 12, it is favorable to the increase of damage. Moreover, the initial value of the damage is also higher in the burnished region (D0 = 0.11) than in the sheared one (D0 = 0.043). The equivalent plastic strain, in the case of the burnished region, is also smaller than for the sheared region, that limits the increase of damage. The curves plotted on the graphs of Figs. 13 and 14 allow to explain the failure of elements only in the sheared region, for a high forward slip value. In this case, the damage state is high in the sheared region, leading the equivalent stress to soften. The elements stress carrying capacity is thus reduced and they relieve the burnished region, in which no failed elements are observed. On the other hand, with a small forward slip value, the sheared region is
Fig. 12. Damage surface, plotted according to the stress triaxiality ratio and to the equivalent plastic strain.
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Fig. 14. Tracking of the stress triaxiality ratio, the equivalent plastic strain and the damage variable according to the time, in the burnished region (element set no. 2).
Fig. 15. Cut profiles for the two clearances: (a) initial and (b) 80% of the initial value.
less damaged than for a high forward slip. The burnished region, in which the initial damage state is higher than in the sheared region, reaches more quickly the critical damage threshold Dc , and failed elements are observed. 5.3. Effect of clearance on the edges integrity Hilditch and Hodgson (2005), or formerly Taupin et al. (1996), performed studies in the field of metal cutting, especially on the cutting parameters. The aim of these studies was to decrease the probability of burr formation after cutting, and they showed that decreasing the clearance (E, the horizontal gap between the upper and the lower knives, Fig. 4(b)) limits this defect. Golovashchenko (2006) get the same trend: a higher sheared region with less defects,
like burrs or material fines after cutting. The author postulated that decreasing the clearance increases the hydrostatic pressure in the strip edge, leading to the improvement of the trimming quality. In this section, the developed approach is used to quantify the effect of the clearance on the occurrence of cracks during plastic deformation. The clearance used in Section 4.1 of this paper is decreased by 20%, leading the knives path to be slightly modified. The other edge-trimming parameters remain unchanged. Fig. 15 shows the new cut profile, obtained with this new clearance value, compared to the initial one. The equivalent plastic strain fields and the extent of the affected regions are superimposed on the deformed profiles.
Fig. 16. Quantification of the extent to the edge of the region affected mechanically by edge-trimming.
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Fig. 17. Deformed mesh in the case of a 20% reduction of the clearance during edgetrimming.
Based on the equivalent plastic strain contour lines, Fig. 15 shows that the extend of the region affected by the edge-trimming is decreased by 15%. A more accurate comparison is shown on the graph of Fig. 16. It shows the extent to the edge of the region affected by cutting, in terms of equivalent plastic strain profile. This graph shows that the extent to the edge of the affected region is roughly the same for the two clearance values, but the gradient is more pronounced in the case of the small clearance. The clearance reduction allows to locate the high values of the material hardening near the edge: the equivalent plastic strain is decreased by 30% at 0.2 mm to the edge. According to the study of Golovashchenko (2006), this should allow to decrease the probability of crack occurrence during rolling. To check this hypothesis, the deformed mesh illustrated in Fig. 15 is submitted to the extrusion algorithm presented in Section 2. Same boundary conditions are applied to this new model as for the ST model with Sfwd = 2.15% (Section 5.2). The final mesh is illustrated in Fig. 17, after deformation. On one hand, compared to the deformed mesh of Fig. 10(a), the number of failed elements is increased by 27%. On the other hand, the failed elements are all located on the first elements layer of the edge (which is almost 70 m wide). The modification of the equivalent plastic strain gradient (Fig. 16) thus has an important role in edge cracking: even if more cracks are noticed on the edge, the propagation length is decreased by the fast recovering of the initial material state in the edge. 6. Conclusion This paper focuses on the development of a methodology to study a process sequence when the modeling geometrical assumptions differ from a process to another. This study is based on the Finite Element Method. The first sequence process is modeled assuming the plane strain condition. The second one is modeled in 3D because of the location of the regions of interest. The procedure is based on the continuity assumption of the geometry and of the mechanical fields in the out-of-plane direction to create a 3D mesh from the 2D one, and to transfer the thermomechanical fields to the new model. This methodology is applied to an experimental study achieved by the present authors, in the field of steel strips working. It involves an industrial edge-trimming step followed by an experimental thickness reduction test. At the end of this sequence, the specimens edges were cracked owing to the work hardening accumulated during the edge-trimming operation. The accuracy of the developed strategy is qualitatively compared to the previous experimental results. The model sensitivity to the hardening state is in good agreement with the experimental observations: a decrease of 20% of the hardening state is sufficient to completely remove any crack initiation during thickness reduction. This leads to the conclusion that the developed strategy, as well as the material model, are accurate enough.
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The effect of the forward slip on cracks occurrence is then analyzed, based on the previous experimental study. Two extreme forward slip values are considered: the first value is commonly used in the industry, while the second one is very high. The results show that the forward slip strongly modifies the stress triaxiality ratio, which is directly linked to the damage law, along the contact length. The density of failed elements, that represent the cracked regions, is roughly the same for both forward slip values, but their distribution is located in the top of the specimen edge for the high forward slip value. It can suppose that since the failed elements are all located on one place of the edge, the cracks should be deeper. This could correlate the experimental observations, but the selected method to model cracks is not accurate enough to validate this hypothesis. Finally, the methodology is used to study the effect of an edge-trimming parameter, the clearance, on the specimens edges integrity after thickness reduction. Decreasing the clearance by 20% allowed to locate the plastic strain near the edge, leading to less deep cracks after thickness reduction comparing to the industrial configuration. These results follow the trends observed experimentally but the results accuracy is affected by the element deletion technique, leading to material volume loss in the modeled part. This needs further investigations, for instance to enhance the modeling of cracks initiation and propagation applying the eXtended Finite Element Method (XFEM), which could give more accurate and realistic sheared or cracked profiles. Also, the material model introduced in this paper is rate and temperature independent, while the strip temperature may reach 120 ◦ C in the cold mill roll bite, and strain rates may highly increase with thickness reduction. These limitations may be investigated in further work. Acknowledgements The present research work has been supported by International Campus on Safety and Intermodality in Transportation, the Nord-Pas-de-Calais Region, the European Community, the Regional Delegation for Research and Technology, the Ministry of Higher Education and Research, and the National Center for Scientific Research. The authors gratefully acknowledge the support of these institutions. References Andrade Pires, F.M., de Souza Neto, E., Owen, D., 2004. On the finite element prediction of damage growth and fracture initiation in finitely deforming ductile materials. Computer Methods in Applied Mechanics and Engineering 193, 5223–5256. César de Sá, J.M.A., Areias, P.M.A., Zheng, C., 2006. Damage modelling in metal forming problems using an implicit non-local gradient model. Computer Methods in Applied Mechanics and Engineering 195, 6646–6660. Cockroft, M., Latham, D., 1968. Ductility and workability of metals. Journal Institute of Metals 96, 33–39. Dubar, M., Huart, S., Deltombe, R., Dubois, A., Dubar, L., 2006. Cold rolling of galvanized strips: A new approach to the evaluation of zinc fines formation. Journal of Materials Processing Technology 177, 505–508. Farzin, M., Javani, H., Mashayekhi, M., Hambli, R., 2006. Analysis of blanking process using various damage criteria. Journal of Materials Processing Technology 177, 287–290. Ghosh, S., Li, M., Khadke, A., 2005. 3D modeling of shear-slitting process for aluminum alloys. Journal of Materials Processing Technology, 91–102. Golovashchenko, S.F., 2006. A study on trimming of aluminum autobody sheet and development of a new robust process eliminating burrs and slivers. International Journal of Mechanical Sciences 48, 1384–1400. Hambli, R., 2002a. Inverse technique identification of material damage law with experimental validation. The International Journal of Advanced Manufacturing Technology 20, 223–229. Hambli, R., 2002b. Prediction of burr height formation in blanking processes using neural network. International Journal of Mechanical Sciences 44, 2089–2102. Hilditch, T., Hodgson, P., 2005. Development of the sheared edge in the trimming of steel and light metal sheet. Part 1. Experimental observations. Journal of Materials Processing Technology 169, 184–191. Hoang, N., Porcaro, R., Langseth, M., Hanssen, A.G., 2010. Self-piercing riveting connections using aluminium rivets. International Journal of Solids and Structures 47, 427–439.
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