A General Damage Initiation and Evolution Model (DIEM) in LS-DYNA

Transcription

1 9th Euroean LS-YNA Conference 23 A General amage Initiation and Evolution Model (IEM) in LS-YNA Thomas Borrvall, Thomas Johansson and Mikael Schill, YNAmore Nordic AB Johan Jergéus, Volvo Car Cororation Kjell Mattiasson, Volvo Car Cororation and Chalmers University of Technology Paul ubois, Consulting Engineer Introduction As the automotive industry is reducing the number of hysical rototyes in favour of comuter simulations during their design rocesses, a lot of demands is ut on the accuracy of the virtual finite element models used for this urose. In this context the mathematical modelling of fracture is of or imortance and has been a field of intense research over the ast 5 some years. There are numerous fracture models imlemented in LS-YNA, but as of tradition a fracture model is statically linked to the underlying constitutive model which in ractice limits its usage to a single stage of the design rocess. Recently GISSMO [,2,3] was introduced in an attemt to remedy this shortcoming by allowing the fracture model to be searated from the constitutive model, thus facilitating results from manufacturing simulations to be transferred to subsequent crash simulations. Behind GISSMO, the amage Initiation and Evolution Model (IEM) was develoed in the same sirit and with similar caabilities, but this model has to the authors knowledge not been used extensively. The intention with this aer is to resent this latter fracture model and comare it with GISSMO and, to some extent, with CrachFEM [4] to which it has suerficial similarities. ε ε min Fig Illustration of an FL diagram and fracture along linear and nonlinear strain aths. 2 Fracture modelling 2. Plastic instability and the FL diagram Failure in sheet metals are either due to material instability with localized deformation or fracture from initiation, growth and coalescence of voids, or a combination of the two. The onset of material instabilities deends to a great extent on the material roerties including yield strength, strain hardening and rate deendency as well as geometry imerfections of the sheet. Usually lastic 23 Coyright by Aru

2 9th Euroean LS-YNA Conference 23 instability with localized necking takes lace before any other failure mechanism, and can either be mathematically redicted using for instance the Marciniak-Kuczynski method [5] or established through material tests. In either case, a common reresentation of the instability characteristics of a metal sheet is through a Forming Limit Curve (FLC) in a Forming Limit iagram (FL), as illustrated in Fig. The FLC is a function of the minor in-lane rincial (lastic) strain ε = ε * * ( ε min ) and gives the oints where instability occurs in the in-lane rincial strain sace. The safe zone is given by the inequality * ε < ε, i.e., the area below the FLC curve, and the strains should tyically reside in this region. The FLC curve is usually restricted to linear strain aths, i.e., straight lines in the FL diagram, and for nonlinear aths instability may occur either below or above the FLC curve as illustrated in Fig. ε -/3 /3 2/3 Fig 2 FLC curve reresented as critical lastic strain versus triaxiality. With the restriction to linear strain aths and von Mises associative lasticity, the FLC can be equivalently reresented in terms of critical effective lastic strain as function of either triaxiality defined as the ratio of mean stress over von Mises equivalent stress, = q, or rincial in-lane deviatoric stress ratio, s s min α =, whatever most convenient. An examle of this reresentation is shown in Fig 2 together with a linear strain ath to failure reresented by the vertical arrow and nonlinear strain aths reresented by the curved arrows. The dashed lines indicate the linear strain aths corresonding to uniaxial comression, shear, uniaxial tension and biaxial tension, and Table lists common stress states and associated triaxiality and deviatoric stress ratio values. 23 Coyright by Aru

3 9th Euroean LS-YNA Conference 23 Table Triaxiality and deviatoric stress ratio for some common stress states. Load case α Biaxial tension 2/3 Plane strain tension / 3 Uniaxial tension /3 -/2 Shear - Uniaxial comression -/3-2 Plane strain comression -/ 3 - Biaxial comression -2/3 The effective lastic strain evolves with the rincial comonents of the lastic strain rate tensor as & ε = & ε min + & ε & + ε 3 min & ε and the safe zone is for lastic strains smaller than the critical lastic strain, ε < ε, i.e., the area under the FLC reresented in Fig 2. Obviously nonlinear strain aths is a concern, and one way to account for this is to introduce an instability indicator ω that somehow evolves with lastic strain and deends on the stress state, i.e., & ω = & ω (& ε, ) = & ω (& ε, α ), and instability is said to occur when ω =. This instability model has to be accomanied with the restriction that it reconciles with the FLC reresentation for linear strain aths, which mathematically can be exressed as ε & ω & ε d ε = for constant or α. This suggests the following obvious evolution law, & ε & ω =, () ε although there is freedom to choose any other within reason. 2.2 Post-instability and damage modelling Indeendent of the model of choice to redict instability, there is the delicate matter of what haens after instability has occurred, i.e., the ost-instability stage. To this end it is imortant to note that the reason for having an instability model in the first lace is that the actual material instability henomenon cannot be catured in the finite element model itself due to coarseness of the numerical discretization. This means that numerical results are in effect only reliable u to the instability oint, and to model what really haens after that and u to the oint of fracture borders to the art of 23 Coyright by Aru

4 9th Euroean LS-YNA Conference 23 guesswork. The simlest way out would be to reconcile a fracture model with the assumtion that fracture in ractice occurs very soon after instability onset and whence simly erode the material when instability has been detected. If this aroach leads to results that are inconsistent with the real-life behaviour then one may need to actually model the ost-instability stage. A damage model can be seen as a homogenization model of the micromechanical comosition of material and void that accordingly augments the stress from the constitutive model for the formation of internal forces. A simle henomenological aroach is taken by introducing a damage arameter, <, that is assumed to reduce the effective cross-sectional area in the direction of loading. This is to say that the homogenized stress, i.e., the stress used to model the global resonse, can be exressed in terms of the stress used in the constitutive law as σ ( ) σ~ =, (2) i.e., the material is isotroically degraded with damage. This is the Continuum amage Mechanics concet and was introduced by Lemaitre [6]. The hysical interretation of this model is usually that the area is reduced due to the formation of cavities that develo in the material during lastic loading, i.e., the material is damaged in the true sense of the word, but other interretations may aly in the event that it just serves as a means to model, or induce, the ost-instability resonse of the material. As the instability arameter, the damage arameter is assumed to evolve with lastic strain and deend on the stress state, but in this case we introduce two further deendencies for the following reasons. u* u f u Fig 3 Hyothetical lastic dislacement to failure function and the effect in a uniaxial forcedislacement resonse. u* 23 Coyright by Aru

5 9th Euroean LS-YNA Conference 23 An imortant asect is that the resulting strain softening behavior that is induced by a damage model leads to results that deend highly on the numerical discretization, the finite element mesh imoses the size of the obtained localization areas and a coarse mesh is generally less rone to strain localization when comared to a fine mesh. To attenuate this effect a regularization is imerative and this is here done by that the damage arameter is assumed to deend on the mesh size h. Furthermore, in order to accurately model the shae of different engineering stress-strain resonses as shown in Fig 3, a deendency on the damage arameter itself is introduced, resulting in a damage evolution law on the form & = & ( ε&,,, h). This is further simlified by assuming that the damage evolves with the lastic dislacement defined by & = (3) u hε& and introduce the lastic strain at failure function u f = u f (, ) (4) and the resulting damage evolution as = u& u f (, ) &. (5) The interretation of the lastic strain at failure function is illustrated in a uniaxial tension case (=/3) in Fig 3, where the value at = governs the elongation of a test secimen between instability and fracture while the aearance governs the tail shae of the force dislacement curve. 3 The IEM model The IEM model is thoroughly treated in the LS-YNA keyword manual on *MAT_A_EROSION [7], and should be read in arallel to the following. 3. Combining criteria The IEM model allows for an arbitrary number of initiation and evolution criteria to be defined and combined. Assuming that n initiation/evolution tyes have been defined, damage initiation and evolution history variables, and,, i=, n, are introduced for each integration oint. These are initially set to zero and then evolve with the deformation of the elements according to rules associated with the secific damage initiation and evolution tye chosen. The damage initiation variables do not influence the results but just serve as an indicator for the onset of damage. The damage evolution variables govern the damage in the material and are used to form the global damage,. When multile criteria are active, n>, each individual criterion can be of maximum,, or multilicative,, tye. The global damage variable is defined as where =max(, ) =max = ( ) 23 Coyright by Aru

6 9th Euroean LS-YNA Conference 23 Now to the evolution of the individual damage initiation and evolution history variables, and for the sake of clarity we ski the suerscrit i from now on. 3.2 Initiation criteria 3.2. uctile criterion For the ductile initiation otion a function = (, ) reresents the lastic strain at onset of damage. Otionally this function can deend on the effective lastic strain rate. The damage initiation history variable evolves according to Shear criterion =. For the shear initiation otion a function = (, ) reresents the lastic strain at onset of damage. This is a function of a shear stress function defined as =(+ )/ with being the ressure, q the von Mises equivalent stress and τ the maximum shear stress defined as a function of the rincial stress values = /2. Introduced here is also the ressure influence arameter k s. Otionally this function can deend on the effective lastic strain rate. The damage initiation history variable evolves according to MSFL and FL criteria =. The MSFL and FL initiation otions are restricted to lane stress shell elements, and only the midsurface is considered in an attemt to characterize the cross section as a whole. For this a function = (, ) reresents the lastic strain at the onset of damage. Otionally this can function can also deend on the effective lastic strain rate, which must be ositive for the initiation variable to evolve. For the MSFL criterion, the lastic strain used in this failure criterion is a modified effective lastic strain that only evolves when the ressure is negative, i.e., the material is not affected in comression. The damage initiation history variable evolves according to =max, which should be interreted as the maximum value u to this oint in time. In effect, this means that damage starts evolving as soon as the (modified) lastic strain reaches the critical value. The FL otion differs from the MSFL otion in that the damage initiation history variable evolves just as the ductile and shear criteria, i.e., =. and by that the lastic strain here is not modified by the sign of ressure. 23 Coyright by Aru

7 9th Euroean LS-YNA Conference 23.5 Alha Fig 4 MSFL inut rincial stress ratio vs. triaxiality. The MSFL (and FL) instability curve is with resect to the ratio of rincial deviatoric stress comonents in the lane, α, and as a reference we resent here the conversion formulae between this quantity and the triaxiality. This relation is obtained via the following two relations, assuming >-/3, and arametrized in β>- 2β + α = 2 + β β = β + β and is shown in Fig 4. Table lists some of these relations. 3.3 Evolution criteria For the evolution of the associated damage variable we introduce the lastic dislacement which evolves according to = < h with h being a characteristic length of the element introduced to suress the mesh deendence often encountered in finite element damage models. Note that this quantity starts evolving after the corresonding damage initiation variable reaches unity and also that each criterion has its unique lastic dislacement variable Linear damage evolution Triaxiality The damage variable evolves linearly with the lastic dislacement according to = with being the lastic dislacement at failure function. The lastic dislacement at failure can be constant or deend on the triaxiality and damage, which in the latter case means that = (,). This can be used to characterize and differentiate the ost-instability stage by designing this function according to the illustration in Fig Coyright by Aru

8 9th Euroean LS-YNA Conference 23 4 Examles Fig 5 Secimens for calibrating the IEM model (A,R4,R,S and S3). 4. Crude calibration of the IEM model Fig 5 shows some secimens (labeled as in the cation) develoed at Fraunhofer IWM, Freiburg, Germany, under the suervision of r. ong-zhi Sun. When elongated in the vertical direction these are subjected to a reasonably homogenous stress state in the deformation region with triaxiality values as listed in Table 2. The urose of this examle is to design damage initiation and evolution arameters to get the failure characteristics of the aluminium alloy EN-AW 682 in temer T6 sulied by SAPA. We use the FL initiation criterion in Section and want to find the critical lastic strains for the triaxiality values in Table 2, and thereafter use a damage evolution law as in Section 3.3. to find the table giving critical lastic dislacement as function of triaxiality and damage. Although damage and lastic deformation are two distinct dissiative rocesses they influence each other, and furthermore the stress state is not homogeneous in the secimens, which makes the calibration of the model non-trivial and an otimization tool such as LS-OPT is generally recommended. The estimation of the IEM arameters was done manually and intuitively, with results reflecting this fact. Table 2 Rough triaxiality value for the different secimens shown in Fig 5. Secimen Triaxiality A.33 R4.45 R.66 S. S3.2 In Fig 6, the results from the arameter calibrations for GISSMO and IEM are shown, of which the former is done with LS-OPT. One immediate conclusion can be drawn, being that a erfect fit to tests for all the secimens is simly out of reach. This can be at least artly attributed to the comlex mechanisms occurring in the ost-instability region of the different tests, i.e., the couling and transition between different stress states. A more comlex model and better understanding of micromechanical effects are robably needed to obtain a better fit. It can also be seen that the otimization of the GISSMO model leads to a result seemingly different from (and erhas better than) the IEM results, which is robably exlained by the difference in aroach as the latter is obtained through manual tweeking of arameters with lack of systemacy. 23 Coyright by Aru

9 9th Euroean LS-YNA Conference 23 Normalized force A Test No amage GISSMO IEM Normalized dislacement Normalized force R Test No amage GISSMO IEM Normalized dislacement Normalized force R4 Test No amage GISSMO IEM Normalized dislacement 23 Coyright by Aru

10 9th Euroean LS-YNA Conference 23 Normalized force S Test No amage GISSMO IEM Normalized dislacement Normalized force S3 Test No amage GISSMO IEM Fig 6 Results from model calibration for GISSMO and IEM. It is here aroriate to discuss the results from a comarative ersective and briefly summarize the differences between GISSMO and IEM. The GISSMO damage evolution model is of the form (5), but where the lastic dislacement and lastic dislacement at failure are defined by and -.2 & = r( h) u ε& Normalized dislacement u f γ = r ) ε ( ) ϕ( & ε ), (6) ( h ref f resectively. The introduction of a regularization function r and a strain rate scaling function φ makes the model more general, while the actual damage evolution law is a γ-arameterized restriction of the more general exression (4). Furthermore, the Lemaitre-tye couling to the stress as in (2) is generalized in GISSMO by introducing an exonent δ, σ ( δ ) σ~ =, which together with the exonent γ can be used to determine the ost-instability behaviour of the material. The damage initiation model in GISSMO is analogous to that of its evolution model defined 23 Coyright by Aru

11 9th Euroean LS-YNA Conference 23 by (6) excet for no element size deendency. The initiation model utilizes the same exonent γ and strain rate scaling function φ as in the evolution model. This should be comared to the IEM corresondence in (), which is a simlified variant of the GISSMO counterart excet for a more general strain rate deendence. For linear strain aths they more or less amount to the same thing, but GISSMO allows for more flexibility when it comes to fitting the instability onset for nonlinear strain aths. All in all the two models are similar and regardless of reference, the erformance hinges mostly uon how well the damage initation and evolution arameters are fit to reflect the failure characteristics of the material of interest. 4.2 IEM and CrachFEM Some single element examles with different strain aths will be used to illustrate the IEM model and comare it to two alternative failure rediction models: CrachFEM by Matfem [4] and the standard LS- yna model MAT9 [7], an anisotroic elastic-lastic material model with a strain based failure criterion. The material used is a relatively brittle high strength steel. Hence failure can be assumed to occur close to instability and therefore only damage initiation is considered. The FL criterion is used and although the uctile and Shear criteria are also included in the model, they are never critical Linear strain ath In Fig 7, constant uniaxial tension is alied until failure. We can see that all three models behave as * * exected with failure at the FLC, both in the ε ε ( ) ε = ε ε diagrams. = and the ( min ) ε. ε..5 IEM FL CrachFEM MAT_ Fig 7 Linear strain ath examle, uniaxial tension Non-linear strain aths When first alying uniaxial tension and then switching to lane-strain loading, see Fig 8, we see that the behaviour looks different in the two different variable saces. We must then remember that in * * IEM, the FL is given as ε = ε ( ) and in MAT9 it is given as ε = ε ( ε min ). The translation between the two is only straight forward as long as the strain ath is linear. The situation is similar in Fig 9, where the load is first biaxial tension and then lane-strain. In IEM, the damage initiation (and evolution, not considered here) variables are stored as extra history variables in 3PLOT and ELOUT. More or less corresonding variables are available in CrachFEM and MAT9. A comarison for the last examle is shown in Fig. The staggered look of the CrachFEM FL curve is due to fact that the rather costly Crach algorithm is only evaluated with a certain lastic strain interval, in this case 2%..5 ε min 23 Coyright by Aru

12 9th Euroean LS-YNA Conference ε. ε..5 IEM FL CrachFEM.5 MAT_ ε min..2 Fig 8 Non-linear strain ath examle, uniaxial tension + lane strain ε. ε..5 IEM FL CrachFEM.5 MAT_ ε min..2 Fig 9 Non-linear strain ath examle, biaxial tension + lane strain..8.6 IEM FL IEM uctile IEM Shear.8.6 CF FL CF uctile CF Shear M9 FL Ψ Ψ ε ε Fig amage initiation (or failure risk) variables in non-linear strain ath examle (CF = CrachFEM). 4.3 Comarison between MSFL and FL in IEM As described in Section above, two different instability criteria are available in IEM. In order to illustrate the difference between them, we will study an examle where we start near ure shear loading and then go to uniaxial tension. However, as we start either slightly into the comressive or the tensile region, we will see comletely different behaviour of IEM MSFL in Fig. This is due to the effect of ressure sign in the MSFL criterion that is not resent in the FL criterion. 23 Coyright by Aru

13 9th Euroean LS-YNA Conference ε.2 ε.2. IEM FL. IEM FL CrachFEM CrachFEM IEM MSFL IEM MSFL Fig LHS starting withα =., RHS starting with α = Summary The keyword MAT_A_EROSION in LS-YNA is today equied with two damage initiation and evolution models, GISSMO and IEM, and this aer has been exloiting the features of the latter. A comarison can conclude that the two models are very similar in terms of functionality with a few excetions. One fundamental difference is that the IEM model can be used to redict instabilities in forming alications with shell elements using an FL criterion that attemts to consider the characteristics of the cross section and not each integration oint individually. Excet for this, the differences lie mostly in the mathematical exressions of the instability and evolution criteria, and either model could be referable as it comes down to how well test data can be fitted within the realm of arametrization. One asect in terms of referring one over the other is the matter of taste and comfort, and from this asect the users of CrachFEM could find IEM attractive as it is aarently similar in that several criteria can be combined. With a ositive mindset, IEM can be seen as CrachFEM light. Two examles have been resented to highlight these conclusions. 6 Acknowledgments The authors are grateful to SAPA Technology and Fraunhofer IWM for allowing data to be used for comleting this document. 7 References [] Neukamm F., Feucht M. and Haufe A., Considering amage History in Crashworthiness Simulations, Proc. 7 th Eur. LS-YNA Users Conf., Salzburg, Austria, 29. [2] Haufe A., Neukamm F., Feucht M. and Borrvall T., A Comarison of Recent amage and Failure Models for Steel Materials in Crashworthiness Alication in LS-YNA, Proc. th Int. LS-YNA Users Conf., earborn, MI, USA, 2. [3] Effelsberg J., Haufe A., Feucht M., Neukamm F. and ubois P., On Parameter Identification for the GISSMO amage Model, Proc. 2 th Int. LS-YNA Users Conf., earborn, MI, USA, 22. [4] CrachFEM, [5] Marciniak Z. and Kuckzynski K., Limit Strains in the Process of Stretch-Forming Sheet Metal, Int. J. Mech. Sci., 9, , 967. [6] Lemaitre J., A Continuous amage Mechanics Model for uctile Fracture, J. Mat. Tech., 7, , 985. [7] LS-YNA Keyword User s Manual, R7.., Volume II Material Models, Livermore Software Technology Cororation (LSTC), Coyright by Aru