On Closing the Constitutive Gap Between Forming and Crash Simulation

Transcription

1 0 th International LS-DYNA Users Conerence etal Forming (3) On Closing the Constitutive Ga Between Forming and Crash Simulation F. Neukamm *,. Feucht *, A. Haue **, K. Roll * * Daimler AG, 7059 Sindelingen, Germany {rieder.neukamm; markus.eucht; karl.roll}@daimler.com ** DYNAmore GmbH, Industriestrasse, Stuttgart, Germany andre.haue@dynamore.de Abstract With increasing requirements on crashworthiness, and light-weight car body structures being a central issue in uture automotive develoment, the use o high strength steel qualities has become wide-sread in modern cars. Since these materials oten show signiicantly lower ductility than conventional steels, it is o great imortance to recisely redict ailure under crash loading conditions. Hence constitutive models in crashworthiness alications as or instance the Gurson/Johnson-Cook model which is alied widely at Daimler AG need to be initialized with correctly determined internal variables maed rom a corresonding sheet metal orming simulation. Here two rincile ways could be used theoretically: On the one hand dierent understanding o damage and ailure in crashworthiness and sheet metal orming alications may be uniied by a generalized incremental stress state deendent damage model (GISSO). This aroach can be considered as an attemt to relace the currently used FLD or the ailure descrition in orming simulations. Furthermore, an advantage would be the inherent ability to account or load-ath deendent ailure behavior. On the other hand the already alied Gurson model in crash simulations may be ed by an estimation o the internal damage value rom the orming simulation. The idea here would be to erorm the orming simulation with a state-o-the-art anisotroic material model like e.g. the Barlat model, with a simultaneously executed estimation o Gurson s damage evolution law. The resent aer will enlighten these two ossible aroaches. Furthermore it will be shown that damage rediction in metal orming rocesses and subsequently the use o the results as initial damage values in crash simulations is ossible and necessary to redict structural ailure in crashworthiness simulations. Keywords: orming to crash rocess chain, ailure and damage rediction. Introduction Crashworthiness simulations o car body structures are an imortant art o the CAE develoment chain or car design. In recent years, the requirements on assive saety o cars have grown to high standards, leading to a ermanent demand on an increase in simulation accuracy. Additionally, demands on uel eiciency and CO reduction are conronting the car body designers with the need o weight reduction to an immense eort. One way to achieve light-weight structures with good crash saety roerties is to relace conventional dee-draw steels by more sohisticated materials. Besides o using classic lightweight materials such as aluminum, magnesium or ibre reinorced lastics, new high strength steel grades are gaining more and more imortance or the construction o car body structures. Oten showing rather comlex work-hardening and racture behaviour, new methods are to be develoed to recisely redict ailure as crack develoment under crash loading. Quantitatively considering local re-damage rom oregoing orming rocesses, seems to be a necessary extension o existing ailure rediction methods or crashworthiness calculations. -

2 etal Forming (3) 0 th International LS-DYNA Users Conerence. The manuacturing rocess chain Since the use o metal orming simulations is common ractice in the automotive industry, it seems on hand to transer calculated results such as lastic strains, resultant work-hardening and damage rom orming to crash simulations. ethods have been develoed to ma data between the dierent meshes used in orming and crash simulations. Figure : The rocess chain o orming to crash simulation Since racture behaviour o high strength steels strongly deends on the loading conditions, lastic re-strain as a scalar quantity alone is not suicient to redict the remaining ductility o these materials. Due to the need o taking into account ossible changes in load ath, an additional damage arameter as an internal variable o a constitutive model is used. By emloying a cumulative damage ormulation, changes in strain ath (non-roortional loading) with variations o the resective ailure strain are considered.. aterial models along the rocess chain On the orming side o the rocess chain, the most imortant issue is to accurately describe the material yield locus. Taking into account ossible initial anisotroy o sheets, comlex ormulations o yield locus and work hardening are used. As an examle o many dierent ormulations in use, the model o Barlat&Lian (989) [] may be named. For the urose o crashworthiness simulation, sheet metal anisotroy usually is not considered. ain issues are the rediction o structural olding atterns and energy absortion roerties. Due to these dierences, the use o the same constitutive model or both simulations would oer caabilities that are not needed to both sides. Additionally, a great amount o exerience has been collected so ar by using the established material models on both sides. Furthermore, there are many roven material roerty cards already existing or the resective models. - Figure : aterial models The concet that is actually ollowed by Daimler AG Sindelingen is to add a damage model to an existing orming material model, so the resulting damage data can be transerred to a crash simulation later on. To do this, several combinations o constitutive models or orming and

3 0 th International LS-DYNA Users Conerence etal Forming (3) crash simulation are ossible. As damage values are usually not to be easily converted rom one damage model to another, a sensible combination should involve the same damage model on both sides o the rocess chain. Two ossible combinations o models are considered herein: - The Gurson model, used in combination with an anisotroic material model (e.g. Barlat89) only or damage accumulation in the orming simulation, and as stand-alone constitutive model with damage or the crash simulation - A generalized incremental, stress-state deendent model (GISSO), combined with an anisotroic material model or orming simulation, and with a von ises material model or crash simulation Both combinations lead to the act that a damage model has to run in the background o a orming simulation, without any interaction o the damage model on the orming constitutive model. 3. The Gurson-model in combination with orming material models The Gurson-model with extension by Tvergaard and Needleman [8] is based on a micromechanical model describing growth and nucleation o sheroid voids in rigid-erectly lastic material. It oers a comlete descrition o ductile material behaviour, including sotening and ailure. When combined with a orming simulation, the calculated void volume raction can be maed as a re-damage arameter to the crash simulation later on. Figure 3: Combination o Gurson and Barlat models The model oers a yield unction deendent on hydrostatic ressure and the eective void volume raction : Φ q * 3q + q cosh σ σ ( q * ) 0 () With σ : actual low stress in matrix material : hydrostatic ressure q: equivalent (von ises) stress * : eective void volume raction Damage evolution is deined in a cumulative way: -3

4 etal Forming (3) 0 th International LS-DYNA Users Conerence with A s l l Δ ( ) Δε + AΔε () N N e π void growth l ε ε N sn π void nucleation As can be seen rom equation (), damage evolution consists o void growth due to volumetric lastic straining, and the nucleation o voids due to deviatoric lastic straining. Usually, void growth is considered the dominating mechanism o material deterioration under tensile loading. l This imlies the volumetric art o the lastic strain rate ε being dierent rom zero as long as the void volume raction and thereore the damage is growing. This will haen under arbitrary loading conditions o tensile nature, i.e. ositive mean stress. Although based on the von ises lastic otential, the Gurson model violates by its deinition the assumtion o isochoric lastic low, which is common in classical lasticity theory. In terms o ractical use, this is shown by a lastic Poisson s ratio ν being dierent rom 0.5. ν ε ε, yy (3), xx The rise in volume is caused by a growing void volume raction. 3. The model o Barlat&Lian 989 The model o Barlat and Lian [] on the other hand is based on the assumtion o isochoric l lastic behaviour, thus by deinition yielding a volumetric strain rate ε equal to zero. For the lane stress case (imlemented in LS-DYNA as at_036), the yield unction is deined as + K + a K K + c K Φ a K σ (4) Y with K σ x + hσ y K σ x hσ y + τ xy Here, Y σ is the actual yield stress; a, c, h and are anisotroy arameters usually calculated rom lanar r-values. -4

5 0 th International LS-DYNA Users Conerence etal Forming (3) 3. Adating the Gurson model The dierence in volumetric lastic straining by itsel is the reason or the act that the Gurson model cannot be couled to an isochoric material model by simly transerring the calculated stress and strain tensors. To calculate the corresonding ore volume raction rom an isochoric constitutive model, the volumetric strain rate o the Gurson model has to be estimated rom the existing strain rate tensor. For this urose, the comatibility equation and the low rule o the Gurson model are used: The associated low rule Φ Δ ε ij Δλ (5) σ ij is searated into a volumetric and deviatoric art Δε kk Δε eq Φ Δε Δλ Φ Φ Δε + Δε q 0 Φ q Δε q Δλ q Φ Δε Δε q Φ (6) q Emloying the resective derivatives o the low rule Φ ( σ ij ), and aroximating using a Taylor series exansion, leads to the ollowing relation or the volumetric strain increment as a unction o the deviatoric strain increment: Δε Δε q 3q sinh σ 3 * qqσ q (7) By using this relation, the adjacent volumetric strain increment o the Gurson model can be estimated rom an isochoric model like Barlat. The dierences in mechanical behaviour between the two models are yet not cured. Since the Gurson model would lead to a material change in volume, which the Barlat model does not, dierent strains will be calculated. This leads to incorrect values o damage when comared to a ure Gurson model, getting worse the higher the void volume raction, and thereore the change in volume is. This is a rincial roblem o the two material models, which can be considered undamentally incomatible. A simulation using the Gurson model, simly leads to dierent results in terms o strains comared to e.g. the Barlat model. To solve this roblem, a correction term to the Gurson damage evolution is considered. Based on the known relation o two rincial lastic strains, or incomressible models like Barlat in -5

6 etal Forming (3) 0 th International LS-DYNA Users Conerence uniaxial tension ( ε ε ), and the relation or arbitrary Poisson s ratio ( νε ε ), a correction term was derived: with 3q qσ Δ c ( ) Δε q q * 3q η sinh + AΔε q (8) σ 4 c * q q η (9) The relation derived as equation (8) associates isochoric strain increments o the Barlat model to an increment o void volume raction o the Gurson model. For the uniaxial tension case, the correction term is exact or the known aearance o the strain rate tensors o both models. For dierent load cases such as equibiaxial tension, this relation has to be set u searately, as no closed ormulation o the correction term or arbitrary values o triaxiality η can be ound. As a workaround, a correction actor S was introduced based on henomenological indings. The correction term now reads as ollows: -6 4 (0) 4 + q q S c * Using simle numerical tests o characteristic load cases, a table o correction actors S can be deined, to get a satisactory it o damage evolution or arbitrary values o triaxiality η. Further inormations about this issue can be ound in Schmeing et al. [7]. 3.3 Extension o the Gurson model to shear-dominated ailure Secial imortance or an exerimental roo o concet comes to the use o secimen which actually show reroducible crack ormation undergoing a orming rocess. One such secimen, which is used in the orming methods develoment deartment at Daimler, is the Cross-die (see igure 7). Intended or ormability tests, it shows a wide range o triaxialities rom equibiaxial tension ( η / 3) and ure shear ( η 0) to negative values, corresonding to comressive stress states. In these tests, certain grades o high strength steels show ailure in areas undergoing shear-dominated deormation. The Gurson model is not suited to describe this behaviour (see also Feucht et al. [5]). Possible extensions o the Gurson model to shear dominated ailure have been recently roosed by Nahshon and Hutchinson [6], and Xue [0]. A successul alication o the Gurson model couled to orming simulations, will thereore make it necessary to add such an extension to the Gurson model, to allow or the descrition o ailure in a wide range o alications. 4. The generalized incremental stress-state deendent damage model (GISSO) ost o the roblems addressed above, resulting rom couling the Gurson model to an isochoric orming material model, can be avoided by the use o a less comlicated damage model. Widely in use is the damage model o Johnson and Cook [4].

7 0 th International LS-DYNA Users Conerence etal Forming (3) Figure 4: Combination o Johnson-Cook criterion and Barlat model In its original ormulation, this model shows a linear accumulation o damage D deending on the ratio o ailure strain to the actual equivalent lastic strain increment. dε e D ε < with ε ( ( )) e ε + + d d ex d3η d4 ln () ε 0 The ailure strain is hereby deined as a monotonically alling unction o triaxiality η. Contrary to this, recent ublications by Bao and Wierzbicki [], Barsoum and Faleskog [3] and others are ointing to a deendence o ailure strains not only on triaxiality, but also on the Lode angle, which is reresenting a third invariant o stress tensor. Thus making it ossible to distinguish between stress states o axisymmetric nature and lane strain conditions. 4. Needed extensions o the Johnson-Cook criterion For the lane stress case, the stress tensor exressed in rincial stresses er deinition yields one rincial stress equal to zero. Due to this, the stress tensor can be uniquely described by two invariants, which imlies that a descrition o ailure strain deending on triaxiality alone is suicient as long as lane stress conditions revail. In automotive structures, mostly consisting o thin sheets, lane stress as an assumtion is also maniested in simulations by the use o shell element discretisation. Due to this, stress states rom calculations are lane er deinition, excet or out-o-lane shear stresses that result rom some shell ormulations. 4.. Failure strain in lane stress conditions Nevertheless, a ailure surace or arbitrary states o stress, deined in the coordinates o triaxiality and a Lode-deendent arameter can lead to a more comlex relationshi between triaxiality and ailure strain or the lane stress case also. Generally, a minimum in ailure strain can be exected or deormation under lane strain conditions. Keeing to this, the shae o the curve o ailure strain vs. triaxiality is susected to look dierent rom the monotonically decreasing Johnson-Cook curve. Local minima in ailure strains can be exected or triaxiality values o η 0 (shear), and η / 3 0, 58, since these stress states ulil lane stress as well as lane strain conditions. () -7

8 etal Forming (3) 0 th International LS-DYNA Users Conerence Figure 5: Failure strain vs. triaxiality (lane stress) The ractical imlementation o this concet at Daimler is to deine a curve o ailure strains as a unction o triaxiality η, deined on certain characteristic oints that can be determined by sheet metal couon tests. Data inut is made lexible by deining a load curve o ailure strain vs. triaxiality. 4.. Damage accumulation Another asect, which is comletely dierent in the Gurson and Johnson-Cook model, is the way o damage accumulation. Being linear in the Johnson-Cook model, it is quite similar to the well known iner rule used or atigue calculations. The micromechanical Gurson model on the other hand, shows a strongly nonlinear relation between the damage arameter and increasing lastic strain ε, even or load cases o constant triaxiality. Resulting rom the theory o void growth behaviour, simulation results mostly show good accordance to ractical observations in tensile load cases. Observations on model materials, e.g. by Weck et al. [9], also show a rather exonential growth o damage or void volume with increasing lastic strain. Looking at the desired use o a damage model to estimate the re-damage induced to sheet metal arts during orming oerations, it seems very imortant to realistically describe the accumulation o damage, since in orming oerations the material usually will not be elongated to strains close to ailure. Considering the accumulation o damage ollowing a load ath o varying triaxiality, it seems obvious that an incremental ormulation deending on the actual value o damage has to be ound. This leads to an ordinary dierential equation o Damage D: ( D,η) D (3) As a simle solution satisying this requirement, a ower law unction can be used: D n ε or ε ε const. only! (4) -8

9 0 th International LS-DYNA Users Conerence etal Forming (3) Figure 6: Normalized damage accumulation By dierentiating, one gets to an incremental ormulation o non-linear damage evolution: D n ( ) n D ε (5) ε By choosing an exonent n, (4) is simliied to the linear Johnson-Cook criterion. This ormulation was also roosed by Xue [], motivated by considerations on low cycle atigue. By imlementing the extensions described above, the damage model has lost its similarity to the Johnson-Cook model, and will be called GISSO (generalized incremental stress-state deendent damage model) rom here on. 5. Simulation o a demonstrator art As an examle or the ractical use o a orming simulation couled with a damage model, the orming simulation o a Cross-die was used. The simulation was done with LS-DYNA, using at_036 (Barlat89) couled with the GISSO damage model running in background, as described above. The arameters used are or DP600 dual hase steel. As inut to the damage model, a curve o ailure strains vs. triaxiality similar to the one dislayed in igure 5 was used. 5. Dierences in distribution o strain and damage One observation that is quite obvious rom the results, is that the distribution o equivalent lastic strain, and the calculated damage distribution can dier undamentally. For this art, a maximum in equivalent lastic strain can be ound at the lower hal o the ront side (let icture in igure 7). In these sots, the strain state is o comressive nature, combined with shear. High ailure strains can be exected or this strain state or ductile materials like DP600. Consequently, the calculated damage values are not reaching the critical level o in these areas (right icture in igure 7). Crack initiation is redicted at the ront edge o the art, where the equivalent lastic strain does not reach as high values as it does below. The redicted sot o crack initiation its to exerimental results quite well, as well as the redicted drawing deth. -9

10 etal Forming (3) 0 th International LS-DYNA Users Conerence Figure 7: Cross-die; Contours o equivalent lastic strain (let) and damage D (right) This shows, that an estimation o re-damage rom orming oerations by simly considering the equivalent lastic strain values at the end o the rocess, may not be suicient or materials that show a rather comlex correlation between strain state and the resective ailure strain. 5. Eects o non-linear damage accumulation on damage distribution To show the dierences resulting rom a modiied damage evolution, identical models o the Cross-die were used. Figure 8: Cross-die; contours o damage D Figure 8 shows the dierences in damage distribution resulting rom dierent exonents n in the evolution law, at the moment o crack initiation (same drawing deth). Damage values o indicate ailure, which is redicted at the same sots on the edge o the art or both exonents. The dierences resulting rom dierent exonents in the damage evolution law result in lower damage values in regions that are not close to ailure. Assuming the correctness o the investigations mentioned above, local re-damage would thereore be overestimated by using a linear damage evolution law. -30

11 0 th International LS-DYNA Users Conerence etal Forming (3) 6. Conclusions The described ossibilities or determination and transer o local re-damage data rom orming to crash simulations, are romising otential to make crack rediction in crash simulations more accurate in the uture. Both otions roosed, or the combination o a material model or orming simulations (like Barlat89), with a crash damage model, can be imroved by imlementing the described extensions to the damage models. As some unintended, but very welcome side-eect, the damage models also show a romising results in redicting ductile ailure in orming simulations. The use o these damage models could thereore also lead to an imroved ailure rediction in orming simulations. 7. Acknowledgement Thanks to Dr. Tobias Erhart or atiently imlementing the numerous extensions in LS-DYNA. Part o this work was suorted by unds o the roject WING by the German Federal inistry o Education and Research (BBF). Reerences [] Barlat F., Lian J.: Plastic Behaviour and Stretchability o Sheet etals. Part I: A Yield Function or Orthotroic Sheets Under Plane Stress Conditions. Int. J. o Plasticity 5,. 5-66, 989 [] Bao Y., Wierzbicki T.: On Fracture Locus in the Equivalent Strain and Stress Triaxiality Sace. Int. J. o echanical Sciences 46,. 8-98, 004. [3] Barsoum I., Faleskog J.: Ruture in Combined Tension and Shear: Exeriments. Int. J. o Solids A/Structures 44, , 007. [4] Johnson G. R., Cook W. H.: Fracture Characteristics o Three etals Subjected to Various Strains, Strain Rates, Temeratures and Pressures. Eng. Fracture echanics,. 3-48, 985 [5] Feucht., Sun D.-Z., Erhart T., Frank Th.: Recent Develoment and Alications o the Gurson odel. 5 th German LS-DYNA Forum D-II-3, 006 [6] Nahshon K., Hutchinson J.W.: odiication o the Gurson odel or Shear Failure. Eur. J. o echanics A/Solids 7,. -7, 008 [7] Schmeing F., Haue A., Feucht.: Forming and Crash Induced Damage Evolution and Failure Prediction. Part I: Extension o the Gurson odel to Forming Simulations. 6 th German LS-DYNA Forum H-I-0, 007 [8] Tvergaard V., Needleman A.: Analysis o the Cu-Cone Fracture in a Round Tensile Bar. Archive o echanics 3, , 984 [9] Weck A., Wilkinson D.S., Toda H., aire E.: D and 3D Visualization o Ductile Fracture. Adv. Eng. aterials 8 (6), [0] Xue, L.: Ductile Fracture odeling Theory, Exerimental Investigation and Numerical Veriication. Ph.D. Thesis, assachusetts Institute o Technology, 007 [] Xue, L: Damage Accumulation and Fracture Initiation in Uncracked Ductile Solids Subject to Triaxial Loading. Int. J. o Solids and Structures 44, , 007-3

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