LS-DYNA Simulation of Hemispherical-punch Stamping Process Using an Efficient Algorithm for Continuum Damage Based Elastoplastic Constitutive Equation

Transcription

1 LS-YNA Simulatio of Hemispherical-puch Stampig Process Usig a Efficiet Algorithm for Cotiuum amage Based Elastoplastic Costitutive Equatio Nima Salajegheh*, Nader Abedrabbo ad Farhag Pourboghrat epartmet of Mechaical Egieerig, Michiga State Uiversit, East Lasig, MI Abstract. A efficiet itegratio algorithm for cotiuum damage based elastoplastic costitutive equatios is implemeted i LS-YNA. The isotropic damage parameter is defied as the ratio of the damaged surface area over the total cross sectio area of the represetative volume elemet. This parameter is icorporated ito the itegratio algorithm as a iteral variable. The developed damage model is the implemeted i the FEM code LS-YNA as user material subroutie (UMAT). Pure stretch experimets of a hemispherical puch are carried out for copper sheets ad the results are compared agaist the predictios of the implemeted damage model. Evaluatio of damage parameters is carried out ad the optimized values that correctl predicted the failure i the sheet are reported. Predictio of failure i the umerical aalsis is performed through elemet deletio usig the critical damage value. The set of failure parameters which accuratel predict the failure behavior i copper sheets compared to experimetal data is reported as well. INTROUCTION Applicatio of fracture mechaics to characterize ductile fracture has led to the itroductio of differet ductile fracture criteria, which otabl iclude J- itegral. These criteria offer a tool for predictig the crack propagatio at the macroscale but the fail to take ito accout the cotiuous deterioratio of material properties as the effect of microcrack ucleatio ad accumulatio. Moreover, fracture mechaics deals with the aalsis of existig cracks which might ot be preset or exactl located. Formig limit diagrams, o the other had, have proved to be reliable for predicatig the iitiatios of material discotiuities, however, FL s also eglect the material softeig behavior resultig from cotiuous material deterioratio. Hece, the cocept of cotiuum damage mechaics has gaied credit for takig ito accout the cotiuous deterioratio of material ad predictig failure. The idea was itroduced whe i 1958 L. M. Kachaov [1] published a simple model of material damage for creep aalsis. Ever sice, the cocept has bee expaded through excellet work of researchers such as. Krajciovic et al [, ], F. A. Leckie et al [4], J. L. Chaboche et al [5], S. Murakami et al [6], C. L. Chow et al [7-11], J. Lemaitre et al [1-17] ad other fie researchers to the poit of applicatio to real-life problems. amage Parameter, Effective Stress, ad the Equivalece Priciple amage ma be i the form of creatio of discotiuous surfaces, breakig of atomic bods, or growth of microcavities. Followig the work of L. M. Kachaov [1], at the mesoscale, damage ma be approximated i a plae b the area of the itersectios of all the flaws with that plae. I order to work with a dimesioless quatit, this area is scaled b the size of the represetative volume elemet (RVE). For a RVE orieted alog the directio, we ca defieδ S as the area of the sectio of the RVE ad δ S as the area of the itersectios of all microcracks or microcavities which lie i δ S as see i Figure

2 material parameter ragig from zero to oe which could be determied experimetall. To iclude the damage parameter ito the costitutive equatios, differet criteria have bee postulated [10]. I this work, the Strai Equivalece Priciple itroduced b Lemaitre [1] has bee utilized. This priciple states that a strai costitutive equatio for a damaged material ma be derived i the same wa as for a virgi material except that the usual stress is replaced b the effective stress: FIGURE 1. Isotropic defiitio of damage parameter [1]. The value of the damage at poit M i the directio is defied as: ε (, ) = ε(,0) = 1 (4) δ S M (, ) = δ S (1) The scalar (isotropic) damage parameter at poit M is the maximum value of M (, ) for all possible orietatios of. It follows from this defiitio that the value of the scalar variable is bouded b 0 ad 1 as follows: = 0 Udamaged material = 1 Full broke material () I fact, the failure occurs for <1 through a process of istabilit. Sice o force is carried b the broke area represeted b S, a effective stress,, ca be itroduced which is based o the surface that effectivel resists the load, amel (S-S ): F F = = = S S S 1 S 1 S () I compressio, some defects close ad the surface that effectivel resists the load becomes larger tha (S-S ). I particular, if all the defects close, the effective stress i compressio is equal to the usual stress. Moreover, the damage evolutio is almost zero ad the damage parameter remais costat. This effect is called the crack closure effect ad is take ito accout, i this work, b itroducig a factor i the umerical implemetatio of damage evolutio equatio. Whe the material is i compressio, egative hdrostatic pressure, the factor is set equal to zero ad is set equal to oe for positive values of hdrostatic stress. However, sice the damage evolutio i compressio ma ot be egligible for some materials ad temperatures, oe ca itroduce a amage Evolutio Equatio ad the Computatioal Algorithm Assumig a isotropic power law hardeig rule, Lemaitre postulated the damage evolutio equatio to be [1]: ε c = PR ε P = K( ε 0 + p) (1 ) (1 ) H + ν + ν ( ε0 + p ) eq (5) (6) P Where ε P is the plastic strai below which the damage evolutio is egligible. ε PR is the plastic strai at rupture. c is the damage parameter at rupture called the critical value of damage. H is the hdrostatic stress. eq is the vo Mises equivalet stress. is the ield stress. ε 0 is the strai value at are isotropic hardeig coefficiets. ield. K ad p is the equivalet plastic strai which is writte as: p p p = ε : ε (7) Applig the strai equivalece priciple, ieldig occurs whe: = = = K( ε 0 + p) (8) 1 40

3 Thus, the vo Mises ield fuctio ma be writte i the followig form: f ( ) = : (1 ) 0 (9) Where is the deviatoric stress tesor. The radius of ield fuctio ad the ormal uit vector are: R = ( 1 ) (10) N = = = = : R 1 ( ) (11) Assumig that the total strai rate is the summatio e p of the elastic ad plastic parts, ε = ε + ε, ad p applig the flow rule to the plastic part, ε = γ N, the equivalet plastic strai rate ca be writte as: p p p = ε : ε = γ ) (1) Other equatios used i the implemetatio of the computatioal algorithm are as follows: Stress rate: e ( )( ( ) = 1 λtrace ε I + µε e (1) Equivalet vo Mises stress: Isotropic hardeig law: eq = : (14) ( ) h = K + p ε 0 1 (15) The computatioal algorithm aims at calculatig the stress, strai, ad damage parameters at the ed of a small time (strai) icremet give their values at the begiig of the icremet ad the compoets of total strai tesor applied over the icremet. First, the icremet is assumed to be totall elastic ad the elastic predictor is calculated as the stress for the icremet: If the elastic predictor,, is withi the curret ield surface, the elastic predictor is the actual value of stress for the curret icremet: = (17) If ot, a plastic correctio should be made as followig: ( ) ( ) p = µ ε 1 = old µ γn 1 old (18) We seek to fid γ. Sice the icremet is o loger totall elastic, the value of total equivalet plastic strai, p, eeds to be updated: p = + γ p old (19) Ad the ield stress cosiderig the plastic hardeig is calculated as: If p ε = + h p = p 1 ( ε 0 ) + K + p γ (0), the damage parameter has to be updated usig the Lemaitre s equatio: The, = + old ( ε p) c = 0 + εpr εp γ ( 1 ) ( 1 H + ν + ν) eq (1) γ is determied cosiderig costat durig oe step which is justified i the explicit calculatio because of ver small icremets. At the ed of icremet: = + h p = 1 ( ε 0 ) + K + p γ () Trial = + (1 )( λtrace( ε) I + eµ ε ) (16) old old 40

4 = R N= ( 1 ) N µ γ( 1 old ) N= ( 1 old ) + h γ N = + γ + µ γ ( 1 ) h old N = + h+ µ γ ( 1 old ) N old : = ( 1 old ) + h+ µ γ : ( 1 ) γ = h µ 1+ ( 1 ) µ old () old Hemispherical-Puch Stampig Experimet The experimets were performed usig a 1.6 mm thick copper 99.9% sheet cut ito a 179 mm square blak. The blak was the clamped betwee the die ad the holder; the puch the moves upward ad deforms the sheet ito a hemispherical-shape cup. Figure shows a schematic illustratio of the setup. For material characterizatio, several uiaxial tesile tests were performed based o ASTM E8M stadard ad the results were averaged to obtai the mechaical properties of the material as reported i table 1. TABLE 1. Material properties Propert Magitude Youg s modulus GPa Poiso s ratio 0.4 Yield stress 90.0 MPa K (Hardeig coefficiet) 491. MPa (Hardeig expoet) Figure. Begiig ad the ed of the stampig process Numerical Results The computatioal algorithm is implemeted ito the oliear fiite elemet code, LS-YNA through the user material subroutie optio, UMAT. The material model is the used i the simulatio of a hemispherical-puch stampig process. The umerical model used i the simulatio is show i Figure. Figure. Numerical model used i FEM simulatio. 404

5 The puch, die, ad blak holder were meshed with rigid elemets hexagoal quadratic solid elemets were used to mesh the sheet part. Two cases were studied which differed i the material damage parameters. I case 1, these parameters were take from the literature [14]; ad i case, the were optimized ad suggested b the authors after a series of sesitivit aalsis. The values are listed i table. TABLE. Material damage parameters Parameter Case 1 Case ε P ε PR c Figure 5. Experimetal result of formig of 99.9% copper sheet. Moitorig the puch force variatios is a good wa to see the softeig behavior of material as the damage grows. Figure 4 shows a compariso of puch force vs. puch displacemet betwee both cases of stud compared agaist experimetal results Puch Force (kn) Experimetal Lametre Paper Optimized Values Puch isplacemet (mm) Figure 6. Compariso of failure betwee umerical ad experimetal fidigs Figure 4. Compariso of puch force vs. its displacemet betwee umerical ad experimetal fidigs I the aalsis, failure was predicted usig the critical value of damage ad demostrated b elemet deletio method which was implemeted ito the UMAT. Figure 5 shows a picture of the experimetal result of formig of the copper sheet. Figure 6 shows a picture of the umerical aalsis usig the optimized damage values as reported i Table. As see from Figure 4, the damage parameters reported i literature [14] for the pure copper used i this experimet were uable to predict the behavior see experimetall. The optimized values of damage, o the other had, were capable of predictig both failure locatios (Figure 5 ad Figure 6) ad the puch force (Figure 4). Coclusios The softeig behavior of material ad its effect o the puch force was satisfactoril captured b usig a cotiuum damage based material model with optimized damage parameters. The material model proved to be particularl fuctioal i predictig failure locatio compared to experimetal results. Nevertheless, mesh depedec ad the assumptio of zero damage growth i compressio are two limitatios which should be carefull dealt with whe the material model is to be used for the simulatio of other formig processes with differet loadig coditios. 405

6 ACKNOWLEGMENTS The first author would like to ackowledge ad thak r. Sag-Wook Lee, assistat professor i the departmet of Mechaical Egieerig at Soochuhag Uiversit i Korea, for his cotributio for the derivatio of the umerical algorithm. REFERENCES 1. Kachaov, L.M., Itroductio to cotiuum damage mechaics, ordrecht, Bosto: Martius Nijhoff Publishers, Krajciovic,., ad Foseka, G.U., Joural of Applied Mechaics, 48, (1981).. Krajciovic,., Joural of Applied Mechaics, 5, (1985). 4. Oat, E.T., ad Leckie, F.A., Joural of Applied Mechaics, 55, 1-10 (1988). 5. Chaboche, J.L., Nuclear Egieerig ad esig, 79, (1984). 6. Murakami, S., Joural of Applied Mechaics, 55, (1988). 7. Chow, C.L., ad Wag, J., Iteratioal Joural of Fracture,, -16 (1987). 8. Chow, C.L., ad Wag, J., Egieerig Fracture Mechaics, 7, (1987). 9. Chow, C.L., ad Lu, T.J., Egieerig Fracture Mechaics, 4, (1989). 10. Chow, C.L., ad Wag, J., Egieerig Fracture Mechaics, 0, (1988). 11. Chow, C.L., ad Wag, J., Iteratioal Joural of Fracture, 8, 8-10 (1988). 1. Lemaitre, J., Joural of Egieerig Materials ad Techolog, 107, 8-89 (1985). 1. Lemaitre, J., A Course o amage Mechaics, New York: Spriger, Lemaitre, J., ad Chaboche, J.L., Mechaics of Solid Materials, Cambridge: Cambridge Uiversit Press, Lemaitre, J., ad oghri, I., Computer methods i applied mechaics ad egieerig, 115, 197- (1994). 16. Lemaitre, J., ad ufaill, J., Egieerig Fracture Mechaics, 8, (1987). 17. Lemaitre, J., ad esmorat, R., ad Sauza, M., Europea Joural of Mechaics - A/Solids, 19, (000). 406