Huetink, J.

Wisselink, H.H.

English

Damage and fracture are important criteria in the design of products and processes. Traditional assessments of the formability of sheet metal materials are based on the appearance of geometrical instabilities, the necking of the sheet. A commonly used method to predict geometrical instabilities in sheet metal forming processes is the Forming Limit Curve (FLC). However, the use of an FLC has some limitations in case of small radius over thickness ratios, non proportional strain paths and non-traditional forming materials with less formability, such as aluminium and high-strength steels. These materials often fail due to physical instabilities, even before necking starts. For crash simulations the FLC is too conservative as also during necking energy is absorbed. Ductile damage models can be used complementary to the FLC approach to improve the failure predictions

Huétink, J.

University of Twente Research Information

IX International Conference on Computational PlasticityCOMPLAS IXE. Oñate and D.R.J. Owen (Eds)c©CIMNE, Barcelona, 2007APPLICATION OF NONLOCAL DAMAGE MODELS TOSHEET FORMING APPLICATIONSH. H. Wisselink⋆, J. Huétink†⋆Netherlands Institute for Metals Research (NIMR)P.O. Box 5008, 2600GA Delft, The Netherlandse-mail: h.h.wisselink@utwente.nl, web page: www.nimr.nl†University of Twente (UT)Faculty of EngineeringP.O. Box 217, 7500AE Enschede, The Netherlandse-mail: j.huetink@utwente.nl, web page: www.dieka.orgKey words: Damage, Nonlocal, Sheet Forming1 INTRODUCTIONDamage and fracture are important criteria in the design of products and processes. Tra-ditional assessments of the formability of sheet metal materials are based on the appearanceof geometrical instabilities, the necking of the sheet. A commonly used method to predictgeometrical instabilities in sheet metal forming processes is the Forming Limit Curve (FLC).However, the use of an FLC has some limitations in case of small radius over thickness ratios,non proportional strain paths and non-traditional forming materials with less formability, suchas aluminium and high-strength steels. These materials often fail due to physical instabilities,even before necking starts. For crash simulations the FLC is too conservative as also duringnecking energy is absorbed.Ductile damage models can be used complementary to the FLC approach to improve thefailure predictions. A nonlocal damage model will be presented in Section 2 and some resultswill be shown in Section 3. The extension of this model to shells is discussed in Section 4.2 NONLOCAL DAMAGE MODELNonlocal damage models are used to avoid the mesh dependency problems of local damagemodels. The nonlocal damage model presented here is based on [1, 2]. The local damage drivingvariable z is a function of the local stress and strain history as defined in Equation 1. Thetriaxiality σhσeqin this equation proves to be important factor [3].z =∫εp〈1 + Aσhσeq〉εBp dεp (1)The nonlocal damage driving variable z̄ is obtained using a Helmholtz partial differentialequation with a Neumann boundary condition (Equation 2). l is the internal length scale, whichcontrols the width of the localisation bands.1H.H. Wisselink and J. Huétinkz̄ − l2∇2z̄ = z; ∇z̄ · n = 0 on Γ (2)The evolution of history parameter κ is according the Kuhn-Tucker loading-unloading con-ditions.κ̇ ≥ 0; z̄ − κ ≤ 0; κ̇(z̄ − κ) = 0 (3)The degradation of the material properties ω is calculated from the history parameter κusing a damage evolution law. Here a linear law is used, the degradation initiates at κi and thematerial fails completely at κu.ω = 0 for κ < κi; ω =κ − κiκu − κiκi ≤ κ ≤ κu; ω = 1 for κ > κu (4)The yield stress is calculated using a strain hardening function h, which competes with thesoftening due to damage.σy = (1 − ω)h(εp) (5)The nonlocal damage model is implemented using user subroutines into a commercial explicitFinite Element code [4] as well as a private implicit code [5] using an operator split approach.This method is relatively easy to implement into existing codes. The local and nonlocal damagedriving variables z and z̄ are updated only at the end of an increment. This means that thedegradation ω is kept constant during an increment. Therefore the results become stepsizedependent, but will converge for decreasing stepsize. The increment size used in an explicitsimulation is normally much smaller than used in an implicit simulation. Therefore the stepsizedependency is negligible for explicit codes.The discretisation of Equation 2 results in a symmetric linear system of equations, whichhas to be solved every increment to calculate the nonlocal variable z̄ . The calculation timeof the nonlocal values is less than the time needed for one increment in an implicit simulation.Contrary to this, for an explicit simulation the calculation of the nonlocal values takes moretime than an increment (depending on the problem size). Therefore, in order to reduce thetotal calculation times using an explicit code, the nonlocal values are calculated only every nincrements (typically 100).Cracks initiate as the degradation (ω) is equal to one and there is no stiffness left. Crackgrowth is often modelled by element erosion, i.e. failed elements are removed from the problemand the simulation is continued on this updated geometry. Element erosion results in massloss and a faceted crack surface. This may have a strong influence on the local stress and straindistributions and may lead to to numerical instabilities. This may result in convergence problemsusing implicit integration, but is attractive for explicit codes due to its simplicity, compared toalternative computational techniques to describe evolving cracks, such as remeshing [2].The nonlocal model has a number of extra material parameters (l, κi, κu,A,B) besides thenormal ones, which have to be determined experimentally. Microhardness measurements orinverse modelling can be used for this purpose [3, 6], but no standard identification methodexists. The element size in the damaged areas should be smaller than the length scale tocapture the gradients properly in the nonlocal model. Ideally this length scale is related tothe microstructure. However, a small length scale leads to a large number of elements. Inpractice the length scale might be chosen larger in order to decrease the problem size. Adaptiveremeshing of the damaged area might solve this problem [2].2H.H. Wisselink and J. Huétink3 BENDING APPLICATIONThe nonlocal model has been applied to a three point sheet bending case with a punch radiusof 0.2 mm and a die radius of 5 mm. The sheet of 1 mm thickness has been modelled with40 plane strain elements in thickness direction. An elasto-plastic VonMises material model iscombined with the nonlocal damage model. The used material properties are κi = 0.05, κu =0.5, A = 3, B = 0, h = 983(εp + 4.1e−2)0.256. The length scale l has been varied from 0.0 (localdamage) to 0.2 mm. 0 5 10 15 20 25 30 0  1  2  3  4  5  6  7Force (kN)Displ. (mm)local (l=0.0)nonlocal l=0.05mmnonlocal l=0.2mmFigure 1: Results of bending simulations. Left: ω for l = 0.05mm. Right: force-displacement diagramfor various l.Figure 1 shows the resulting crack shape. During bending shearbands develop at the outerfibers. This results in a crack inclined to the outer surface, which grows along the symmetry lineuntil separation of the parts. These shear bands disappear for larger values of the length scale(l = 0.2mm) and the crack follows the symmetry line completely. From the force-displacementgraph can be seen that an increasing length scale leads to a more ductile response. Thesedetailed 2D simulations of bending (and combinations with tension) can serve as a reference forsimulations with shell elements.4 SHELL ELEMENTSFor large scale simulations of forming or crash of sheet products shell elements are the mostefficient method. Shell elements have a limited description of the kinematics, as normals remainstraight during the deformation. Therefore these types of elements are not capable of describingfailure with through thickness shear bands as shown in Figure 1. However, often such localdetails are not of interest and only the correct moment of failure and the energy absorption isimportant. Furthermore the result should be independent of the used meshsize.The extension of the nonlocal damage model from solid elements to shell elements is notstraight forward. The size of shells elements is normally larger than the sheet thickness. There-fore it is not possible anymore to use the same length scale as in simulations with solid elements.Often the nonlocal model is applied independently for every layer of integration points of theshell element.Even without damage the necking in simulations with shell elements is mesh size dependent3H.H. Wisselink and J. Huétinkas illustrated in Figure 2. The necking of a tensile bar with an imperfection has been modelledwith 2D solids and shells with the same plane strain boundary condition. In the 2D case theneck has a fixed width where for the shells the neck has a width of one element, because thethickness of two neighbouring elements is not coupled. The described nonlocal damage modelis not capable of solving this type of mesh dependency completely. 0 100 200 300 400 500 600 700 800 0  1  2  3  4  5Force (N)Displ. (mm)2D solids: 240x32 elmsShells: 240 elms 0.4 0.5 0.6 0.7 0.8 0.9 1 0  2  4  6  8  10  12  14  16  18thickness (mm)x-coord. (mm)2D solids: 240x32 elmsShells: 240 elmsFigure 2: Results of tensile test simulation of a strip of 15x1 mm. Left: force-displacement diagram.Right: thickness distribution at displacement of 3 mm.5 CONCLUSIONSA nonlocal damage model has been presented, which gives mesh independent results for solidelements. Some issues concerning the extension of this nonlocal damage models to shell elementshave been discussed. A solution for the observed problems with thin shells might be the use ofsolid shells or cohesive zones [7].ACKNOWLEDGEMENTSThis research was carried out under project number MC1.05205 in the framework of theStrategic Research programme of the Netherlands Institute for Metals Research.REFERENCES[1] R. A. B. Engelen, M.G.D. Geers, and F.P.T. Baaijens. Nonlocal implicit gradient-enhanced elasto-plasticity for the modelling of softening behavior. Int. J. Plasticity., 19:403–433, 2003.[2] J. Mediavilla, R.H.J. Peerlings, and M.G.D. Geers. An integrated continuous-discontinuous approachtowards damage engineering in sheet metal forming processes. Eng. Frac. Mech., 73(7):895–916, 2006.[3] A.M. Goijaerts, L.E. Govaert, and F.P.T. Baaijens. Evaluation of ductile fracture models for differentmetals in blanking. J. Mat. Proc. Tech., 110:312–323, 2001.[4] LTSC. LS-DYNA keyword user’s manual, 971 edition, 2007.[5] DiekA development group. DiekA manual, 8.1 edition, 2006.[6] A. Mkaddem, F. Gassara, and R. Hambli. A new procedure using the microhardness technique forsheet material damage characterisation. J. Mat. Proc. Tech., 178:111–118, 2006.[7] F. Cirak, M. Ortiz, and A. Pandolfi. A cohesive approach to thin-shell fracture and fragmentation.Computer Methods in Applied Mechanics and Engineering, 194:2604–2618, 2005.4

Application of nonlocal damage models to sheet forming applications

Huetink, Han

NARCIS 

A cohesive approach to thin-shell fracture and fragmentation.

A new procedure using the microhardness technique for sheet material damage characterisation.

An integrated continuous-discontinuous approach towards damage engineering in sheet metal forming processes.

DiekA development group.

Evaluation of ductile fracture models for diﬀerent metals in blanking.

LS-DYNA keyword user’s manual, 971 edition,

Nonlocal implicit gradient-enhanced elastoplasticity for the modelling of softening behavior.

http://purl.utwente.nl/publications/59682

Application of nonlocal damage models to sheet forming applications

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