Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 56, 3, pp. 781-792, Warsaw 2018 DOI: 10.15632/jtam-pl.56.3.781 NUMERICAL ANALYSIS OF THE DEEP DRAWING PROCESS INCLUDING THE HISTORY OF STRESS AND STRAIN Pawel Kaldunski Koszalin University of Technology, Koszalin, Poland e-mail: pawel.kaldunski@tu.koszalin.pl This paper discusses the results of a numerical study of circular cup drawing of steel sheets using finite elementmethod.Thedrawingprocess is consideredas a geometrical andphysical nonlinear problem with unknown boundary conditions in the contact area of the system, such as the tool and the workpiece. The updated Lagrangian description is used to charac- terize these nonlinear phenomena on a typical incremental step time. Numerical results are obtained using an explicit method in Ansys/Ls-Dyna program. The constitutive Cowper- -Symondsmaterialmodel with linear hardening strain to predictmaterial plasticity is used. The results of implementation of stresses and strains from a blanking operation flat disc of a sheet of metal for deep-drawing process are presented. After the blanking process simula- tion, an implicit springback analysis is performed. Then a numerical analysis of cup forming from this flat disc platewas carried out. The analysis results are comparedwith one another through reading of the sheet thickness in several characteristic points and the overall height of the product. Keywords: deep drawing, modeling, numerical analysis, FEM, stress history, strain history 1. Introduction At present, in metal forming industry, preparation of three-dimensional models for the drawing is very important to explain and to understand specific forming processes which occur in the drawpiece. There aremanydifferentmethods of building finite elementmodels of drawing exam- ples using 2Dnumerical and analytical models (Yao andCao, 2001). Currently, in themodelling and simulation of drawing processes, the influence on state of stress and strain prior treatments is mainly omitted (Nagasekhar et al., 2006; Trzepiecinski and Gelgele, 2011). Apart from the unknown state after sheet rolling (KimandOlver, 1998), during the blanking process, additional stress is placed on the disc periphery. The blanking process is understood as cutting out a flat disc from a flat metal sheet (Bohdal et al., 2014). As the tool, a die and a punch with cutting edgeswere used, and to prevent possiblewarping of the disc after cutting out, a blankholderwas used (Urriolagoitia-Sosa et al., 2011). Then, the drawpiece without a flange was formed from the flat disc. Thispaperpresents the results of a comprehensivemodellingof thedrawingprocess including thehistoryof stress and strain.Theblankingprocessmodellinghasbeen carried out,which takes into account a springback phenomenon that occurs after the load is removed. Such an analysis allows amoreaccurate determinationof thefinishedproductdimensions, thedrawing force in the whole process, and it permits one to determine the state of stress at any place of the drawpiece. From themechanical perspective, this process is treated as a doubly nonlinear initial-boundary value problem with movable nonlinear sources and boundaries and only partial knowledge of the boundary conditions. The following nonlinearities occur in the process: geometric, physical and boundary conditions in the area of tool-sheet contact. The geometric nonlinearity, which is thought tobeanonlineardependencebetween the strain anddisplacement, results fromachange 782 P. Kaldunski in the object geometry. The physical (i.e., material) nonlinearity is caused by nonlinearity of the mechanical properties of thematerial. Amathematical description of these nonlinear phenomena requires the use of rules regarding formulation of boundary and initial problems. 2. Finite element modelling The description of nonlinearity of the material is conducted using an incremental model that takes into account the influence of the history of strains. The object is treated as a body in which elastic strains may occur (in the scope of reversible strains) together with plastic strains (in the scope of irreversible strains) with nonlinear hardening (Gambin and Kowalczyk, 2003). For the purpose of constructing amaterial model, the following are used:Huber-Mises-Hencky’s nonlinear plasticity condition, the associated flow law and isotropic hardening (Jemiolo and Gajewski, 2014). The state of the material after the aforementioned processing is taken into account by introducing the following initial states: displacement, stresses and strains. The states of strains are described with nonlinear dependences and no linearization (Simo and Hughes, 1998). In this description, adequate measures are used for an increment of strains and for an increment of stresses (i.e., an increment of the Green-Lagrange strain tensor and an increment of the second Piola-Kirchhoff symmetric stress tensor). The incremental contact model covers contact forces, contact rigidity, contact boundary conditions and friction coefficients in this area. Themathematicalmodel is supplementedwith incremental equations ofmotionof theobject and uniqueness conditions. An incremental function of the total energy of the system is introduced. From the stationary condition of this function, it is possible to derive a variational nonlinear equation to describe motion and deformation of the object for a typical incremental step. This equation is untangled with spatial discretization using the finite element method, which results in discrete systems of equations formotion and deformation of the object in the drawing process (Zienkiewicz and Taylor, 2006). 2.1. Basic relationships Components of theGreen-Lagrange strain tensor increment for a typical time step ∆tand for a non-linear isotropic material with mixed hardening are calculated from the formula (Bohdal and Kukielka, 2014; Bohdal, 2015) ∆εij = 1 1− S̃∗∗ ( D (E) ijkl ∆σkl− 2 3 σY (·)ĖT∆ε̇eS̃ij S̃ijC (E) ijkl S̃kl+ 2 3 σ2Y (·) ( C̃(·)+ 2 3 ET ) ) (2.1) Components of the Piola-Kirchhoff stress tensor increment for a typical time step ∆t and for a non-linear material with mixed hardening are calculated using the formula ∆σij = C (E) ijkl ( ∆εkl−ψ S̃kl ( S̃ijC (E) ijkl ∆εkl− 2 3 σY (·)Ė (VP) T ∆ε̇ (VP) z ) S̃ijC (E) ijkl S̃kl+ 2 3 σ2 Y (·) ( C̃(·)+ 2 3 ET ) ) (2.2) where ψ is the load factor and is ψ = 1 for loading and ψ = 0 for unloading processes, S̃∗∗ = S̃∗ijC (E) ijmnS̃mn is a positive scalar variable, S̃ij = Sij − αij (i,j = 1,2,3) are the stress deviator components, αij are the translation tensor components, D̃σ and D (E) ijkl are compo- nents of the tensor D(E) = C(E) −1 in time t, C (E) ijkl are the elastic constitutive tensor compo- nentsC(E), C̃(·)= C̃(ε (VP) e , ε̇ (VP) e ) is the temporary translation hardening parameter in time t. σY (·)= σY (ε (VP) e , ε̇ (VP) e ) is the accumulated material yield stress which depends on the history Numerical analysis of the deep drawing process including... 783 of viscoplastic strain and strain rate, ε (VP) e and ε̇ (VP) e are the cumulative effective viscoplastic strain and strain rates, respectively, ET is the strain hardening modulus at time t, ĖT is the strain hardeningmodulus rate at time t (Bohdal, 2015). In the analysis of the deep drawing process, the bilinear model is used. Thismaterial model has a linear hardening relationship between stress and strain describedwith the empiricalmodel: σY = σY0 + Etanε (P) eq , where σY0 is the initial yield stress, Etan = (ETE)/(E − ET) is the material parameter dependenton themodulusof plastic hardeningET = dσY/dε (P) eq andYoung’s elasticmodulusE, ε (P) eq is the equivalent plastic strain.TheCowper-Symonds elastic/viscoplastic material model is used for computer simulation of the cutting process. According to the strain criterion, material separation occurs when the strain value of the leading node is greater than or equal to a limiting value. The limiting strain used is εf = 1. When an element of the matrixmaterial reaches the limiting strain value, the corresponding elementwill be deleted. The Cowper-Symonds model allows for linear isotropic (β = 1) assumed in simulations, kinematic (β = 0) or mixed (0 < β < 1) plastic strain hardening, and the effect of the plastic strain velocity is given by the following power relation σp = ( 1+ ϕ̇ (p) i C )m (Re+βEpϕ (p) i ) (2.3) where β is the plastic strain hardening parameter, Re [MPa] is the initial static yield point, ϕ̇ (p) i [s −1] is the plastic strain rate, C [s−1] is the material parameter defining the effects of the plastic strain rate, m =1/P is the material constant defining the sensitivity of the material to the plastic strain rate, ϕ (p) i [–] is the plastic strain intensity, and Ep = (ETE)/(E −ET) is a material parameter dependent both on the plastic strain hardening modulus, ET = ∂σp/∂ϕ (p) i , and Young’s modulus E. 3. Numerical model and results Theblankinganddrawing tools are treatedasnon-deformablebodies, i.e.,E →∞.The following material parameters are assumed forDC01deep-drawing steel (Kaldunski, 2009):Young’smodu- lus E =210GPa, Poisson’s ratio ν =0.29, density ρ =8000kg/m3, yield stress Re =200MPa, tangent modulus Etan =1050MPa. In this study, the objects aremeshed with an 8-node SOLID 164 element type with reduced integration and hourglass control. The sheet has been divided into cubic elements to eliminate the influence of non-uniform mesh on the final results. The contact between ideally rigid tools and the deformable sheet metal is described using Coulomb’s friction model according to the formula µc = FD +(FS −FD)e −DC|vrel| (3.1) where FD is the dynamic friction coefficient, FS – static friction coefficient, DC – exponential decay coefficient, e – Euler’s number, vrel – relative velocity of the surfaces in contact. The following coefficients of friction in the contact zone between the stamp and sheet have been adopted: static friction coefficient µs =0.2, dynamic friction coefficient µd =0.1. The following coefficients of friction in the contact zone between the die block and sheet have been adopted: static friction coefficient µs =0.1, dynamic friction coefficient µd =0.01. The DC exponential decay coefficient is 10. The dependence of the coefficients of friction between the sheet and the stamp on the relative velocity of the surfaces in contact is presented with the solid line on the graph (Fig. 1), while the dotted line shows the dependence of the friction coefficients between the sheet and the die block. 784 P. Kaldunski Fig. 1. Friction coefficient as a function of relative velocity of the surfaces in contact An additional blankholder is applied from the bottom, which eliminates the bulge out effect of the blanked element.The type of thematerial used for this calculation is sheetmetal stamping DC01 with thickness g =2mm. Diameter of the blanked disc D =70mm. Figure 2 shows the initial mesh of the blanking process. The cutting areas of the die and punch are divided on a denser mesh to increase the calculation accuracy. Fig. 2. Initial mesh of the blanking process Figure 3 shows the equivalent stress distribution in the blanked disc. It may be observed that the maximum stress values are located on the disc periphery in the cutting area. Fig. 3. Equivalent stress distribution in the blanked disc Figure 4 shows the equivalent stress distribution in the disc after unloading. This analysis is necessary in a multistage process. Otherwise, in further simulations, it does not receive any correct results. It can be observed that the values of the equivalent stress at the periphery of the disc after removing the elastic strain is almost 3 times reduced. Numerical analysis of the deep drawing process including... 785 Fig. 4. Equivalent stress distribution in the blanked disc after springback 3.1. Deep drawing without the history of stress and strain Figure 5 shows a cross-sectional view of a discrete model of the drawing process (Kaldunski andKukielka, 2014).Thediewhich isusedhas roundingradiusrd =16mmand internaldiameter d = 40mm. The punch diameter has rounding radius rp = 4mm and its diameter is equal to dp =35mm. The clearance die is therefore 2.5mm. An acceptable reduction of the diameter in the drawing process determined by the m1 coefficient for the ratio D/g = 35 equals m1 = 0.5 (Table 1). This means that the value m1 = d/D =0.57 is acceptable. Fig. 5. A discrete model of the drawing process Table 1. Acceptable reduction of the diameter in the drawing process for the ratio D/g (Mar- ciniak, 1998) D/g m1 = d/D 630 0.62 400 0.60 250 0.58 160 0.56 100 0.54 63 0.52 40 0.50 25 0.48 786 P. Kaldunski At first, analysis of the numerical drawing process without taking into account the history of stress and strain from the blanking has been performed. Thus, the initial values of stress and strain in all elements and nodes were 0. What was left was only the mesh deformation of the disc as a result of blanking. Figure 6a shows a map of the equivalent stresses in the finished drawpiece. It can be observed that the values are rising from the bottom transition area through the side wall to the periphery. The maximum values of stresses during the drawing process occur at the periphery as a result of circumferential compressive stresses. Unevenness on the edge results from sheet discretization. After the drawpiece forming simulation, a springback type analysis has been performed. It is analysis performed with the implicit integration method. It enables removal of all the nodes and elements of elastic strains, and it simulates the release of the product from the die. As a result of the removal of elastic strains, the product undergoes expansion, and its external diameter increases from 40mm (Fig. 6a) to 40.13mm (Fig. 6b). Also, the stresses occurring at the periphery of the drawpiece are reduced by approximately 25%. Moreover, their location change. They occur in around half of the product height. Fig. 6. Maps of the equivalent stress in the drawpiece without history: (a) before springback, (b) after springback 3.2. Deep drawing with the history of stress and strain To simulate the drawing process including the history of stress and strain, the disc after blanking and after springback analysis has been imported.To simulate the drawing process in- cluding the history of stress and strain, in the disc after blanking and after springback, analysis was performed taking into account the stress and strain values.All the conditions for theprocess, that is: diameters and radii remained unchanged. Figure 7a showsmaps of the equivalent stress in the finished product. It can be seen that also in this case, the maximum stresses are located on the periphery of the product. However, the values are about 66% higher than in the case without including the history of stress and strain. In addition, the process of shaping the product proceeded in the samemanner as in the previous case. The obtained diameter of the drawpiece corresponds to the internal die diameter, i.e. 40mm. At thenext stage, the springbackanalysis hasbeenperformed(Fig. 7b).Theproductbehaved in a similarmanner as in the case without including the history. Themaximum stress value was migrated from the periphery to about half of the drawpiece height and decreased by 45%. The diameter of the product was increased to 40.11mm, so that was almost the same as without including the stress and strain history. 3.3. Comparision of two examples The next step is a detailed comparison of the drawpiece obtained without including the history of stress and strain with the drawpiece in which that history is included. Figure 8 shows Numerical analysis of the deep drawing process including... 787 Fig. 7. Maps of the equivalent stress in the drawpiece with history: (a) before springback, (b) after springback the graph of the drawing force as a function of punch displacement for these two cases. The solid line represents the dependence with including the history. The dotted line represents the dependence without including the history. It can be observed that the two graphs are similar in characteristics. The differences are only in terms of the value, and there is a slight shifting. The fluctuations at the first stage of the graph are probably a result of discontinuous contact at the beginning of the drawing process. Themaximum drawing force in the case with the history included is equal to 65kN, without including the history it is less by about 7kN. The difference in the value due to the fact that the disc has hardening at the periphery after blanking. This requires a greater force to deform it further. In both cases, the achievement of the maximum drawing force occurred at the punch displacement being equal to 25mm. This is the moment where there is the greatest peripheral drawpiece flange compression. Fig. 8. Dependence of the drawing force from punch displacement Figure 9 shows the deformation of the product for the maximum drawing force without including the history of stress and strain. After passing the maximum value of the drawing force, its value decreases until the periphery of the drawpiece is located directly between the die and the punch. At this point, the drawing force increases again, in the case with including 788 P. Kaldunski the history up to 46kN, and for the case without the history up to 41kN. This increase results directly from the clearance between the die and the punch. In this case, this is adopted less intentionally to observe the effect it would have on the formation of the product. This moment is illustrated in Fig. 10. The final step of drawpiece forming can be observed, which consists in compressionof thedrawpieceperipheral to the size of the clearance equal to2.5mm.An improper selection of the clearancemay cause at this point, during the experiment on the hydraulic press, jamming and damage of the tools. Fig. 9. Deformation of the product for the maximum drawing force without the history of stress and strain Fig. 10. Final drawpiece formation without the history of stress and strain Numerical analysis of the deep drawing process including... 789 Apart from the drawing force, it is also possible to compare products in terms of their geometry and shape. Namely, the two drawpieces sheet thicknesses in characteristic locations and their total heights are compared. The reading is performed between the same nodes in each case. This proceeding allows us to eliminate errors of the mesh deformation, or other reading inaccuracies. Figure 11a shows a graph of the sheet thickness changes at the drawpiece periphery as a function of the punch displacement. It can be observed that the drawpiece, almost from the beginning of the drawing process with the history included, has a greater thickness at its periphery. This is due to the fact that the compressive circumferential stress during the process is cumulative with the stress after blanking. The compression of such a tensioned material is difficult, therefore, some of the material moves inwards or outwards and causes enlargement of the sheet thickness. Themaximumdifference in the thickness of the sheetmetal at the periphery occurs before the punch and reaches a 40mmdisplacement. For the drawpiece with the history included, the thickness is equal to 2.77mm, whereas without the history is equal to 2.68mm. This situation occurs immediately before the moment the periphery is compressed between the punchand the die, as shown inFig. 10.Then, after forming thedrawpiecewalls to the cylindrical form, the edge thickness is nearly identical for the two cases and it is 2.54mm for the case with and2.53mmwithout the history.Thevalue at end in both cases that is larger than the clearance value equal to 2.5mm is due to friction. The nodes betweenwhich the reading ismade are not in a perfectly horizontal position. As a result of sliding friction on the die and static on the punch, its minor displacement occurs. Another dimension that is compared in both cases is the sheet thickness measured at the center of the drawpiece bottom. The reading has been conducted in the samemanner as before. What ismeasuredduring thewhole process is the distance between nodes, at the outer and inner surface, in themiddle of the disc. The reading results are presented in the graph inFig. 11b.The reading starts from the initial value of the sheet thickness of 2.01mm.Then, during the process, small thinning follows in the bottom sheet. In the case where the history of stress and strain is included, the thickness in the final product is 1.96mm, whereas without history it is 1.97mm. A reduction in the sheet thickness in this area is due to radial extention of the bottom during the drawing process. This is often so slight thinning that besides the reading of the simulation result it is very difficult to observe. Also in this case, the difference on the level of 0.01mm is only a general value, that the history of stress and strain causes amarginal effect on the change in the thickness of the bottom drawpiece. Fig. 11. Dependence of: (a) edge thickness from punch displacement, (b) bottom thickness from punch displacement Much more important is the reading of the sheet metal thickness on the curved edge of the drawpiece, where the bottom passes into the cylindrical wall. An improper selection of the clearance or a too large diameter of the output disc in relation to the internal die diameter may 790 P. Kaldunski cause bottombreak offduring the drawingprocess.Thebottombreak off occursmost frequently in the area where the bottom passes into the cylindrical wall, because there occurs the greatest thickness reduction of the sheet. The dependence of the sheet thickness changes as a function of the punch displacement is plotted in Fig. 12a. The reading starts from the initial value of the sheet thickness of 2.01mm. Larger sheet thinning occurred when including the history of stress and strain. The sheet thickness decreases to 1.74mm. This is due to significant hardening of the drawpiece periphery, which makes entry into the die difficult, and this, in turn, promotes thinning of the sheet metal at its rounding. In the case whithout the history included, the sheet thickness in this area is 1.76mm. Similarly, as for measuring of the bottom thickness, these differences are difficult to observe and are measured in a traditional experiment. The last element which is compared is the drawpiece height. The dependence, as a function of the drawpiece height from the punch displacement, is shown in the graph in Fig. 12b. Both characteristics are almost identical to the punch displacement of 40mm. The last stage of the drawingprocess is from thedisplacement 40mmwhereanoticable differencebetween theheights are plotted in Fig. 12c. The drawpiece height shaped with the history included is equal to 26.21mm, whereas without the history it is 25.96mm. Marginally larger height of the product with the history included may result directly from the hardening of the drawpiece periphery. This hardening, in turn, influences substantial thickening of the periphery. This, in turn, at the final stage makes the transition difficult between the die and the punch. This affects the greater flow on the sidewall, bottom and the area where the bottom passes into the cylindrical wall. Similarly as in the reading of the sheet thickness at the periphery of drawpiece, in the bottom and rounding, the differences are minor and difficult to measure in a traditional experiment. Numerical analysis allows one to detect these slight size differences. Fig. 12. Dependence of: (a) rounding thickness from punch displacement, (b) drawpiece height from punch displacement, (c) drawpiece height from punch displacement (zoom) Numerical analysis of the deep drawing process including... 791 4. Conclusions In this paper, the results of numerical complexmodelling of the blanking and drawing processes have been compared. After blanking simulation, a springback analysis has been conducted. Then, two analyses of drawing were carried out and the results were compared. One with the history included of stress and strain after the blanking process, and the other one without that history. The comparative elements between the analyses were: the drawing force as a function of punch displacement, changing of the sheet thickness at the bottom, at the periphery and the rounding, and also the final drawpiece height. In addition, the equivalent stress distributions in the drawpieces were compared. It was observed that including the history in the drawing process caused an increase of the maximum drawing force by 12%. There were also marginal differences in the drawpiece dimensions, whichwere impossible tomeasure with traditional methods during the experiment. The sheet thickness on the periphery, until the whole drawpiece had been formed, was higher by 3.4%. The drawpiece bottom thickness decreased by 0.5%. Also, the rounding thickness decreased by 1.1%. The product with the history included was 1% higher than without the history. The most significant difference was noted in the stress values. The drawpiece with the history included had a 66% higher stress values in the last step of the analysis. After the springback analysis, the stress values was higher by 20.7% than in the analysis without the history. References 1. Bohdal L., 2015,Application of FEMand vision-basedmethods to analysis of shearing processes in the aspect of scrap reduction,Annual Set the Environment Protection, 17, 90-103 2. Bohdal L., Kukielka L., 2014, Application of variational and FEM methods to the modelling and numerical analysis of guillotining process for geometrical and physical nonlinearity,Mechanics, 20, 2, 197-204 3. 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