DUCTILE FRACTURE PREDICTION IN THIN METALLIC PLATES SUBJECTED TO EXPLOSIONS

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1 DUCTLE FRACTURE PREDCTON N TN METALLC PLATES SUBJECTED TO EXPLOSONS Łukasz Pyrzowski 1, Paweł Kłosowski, Krzyszto Woznica 3, Olivier Pennetier 4 1 Gansk University o Technology, Narutowicza str. 11/1, Gańsk, Polan. University o Orleans, nstitut PRSME, Léonar e Vinci str. 8, 4507 Orléans, France. lpyrzow@pg.ga.pl Gansk University o Technology, Narutowicza str. 11/1, Gańsk, Polan. klosow@ pg.ga.pl 3 Bourges igher National School o Engineering, nstitut PRSME, Lahitolle blv. 88, 1800 Bourges, France. krzyszto.woznica@ensi-bourges.r 4 University o Orleans, UT Bourges, nstitut PRSME, Lattre e Tassigny ave. 63, 1800 Bourges, France. opennetier@bourges.univ-orleans.r Abstract. Numerical simulations o uctile amage an racture preiction in metallic structures are an important challenge. n recent years this branch o soli mechanics has mae a great evelopment, however very rarely authors veriy their moels with experiments more complicate then uniaxial tensile or compression tests o ierent specimen shapes. This paper presents an experimental stuy on amage evelopment in metallic plates subjecte to shockwave impulses explosions. The experiments were perorme using a special tube appropriate or making insie etonations. During the tests ierent initial pressures o the explosive mixture were use, allowing observation o ierent stages o the plate s amage process. These experiments are the great base or veriication o uctile amage an racture preiction moelling, which is the main part o the presente paper. Keywors: uctile racture, triaxiality stress, plane stress state, experiment, ientiication, numerical simulation, aluminium, explosion. ntrouction The main goal o this paper is to present an compare ierent types o numerical simulations o uctile racture preiction. The paper is ivie into three main parts. The irst two chapters contain the escription o preparation, carrying out an results o experiments perorme or amage evelopment analysis on thin metallic plates subjecte to explosions an uniaxial tensile tests realize or material parameters ientiication. n the next three chapters the theoretical basis are presente: the constitutive uation an racture criterions plus ientiication o the material parameters. The last part consists o the escription o numerical simulations, veriication o ientiication results, racture criterion calibration, results presentation an comparison. All these experiments an numerical analysis are perorme or better unerstaning an improving the knowlege level in this kin o phenomenon moelling. The practical example o such problem is assuring better saety o metallic pressurize cylinrical vessels, which are wiely use to transport an store o lammable luis in inustry an army or analysis o all other cases o ailures, where explosion energy acts on metallic constructions. Experiments on plates subjecte to explosions The main experiments or analyse problem are the tests on aluminium alloy plates subjecte to gas mixture explosions. The experimental evices assembly iagram, prepare or those tests, is presente in Fig 1. The main part o the research stan is the tube shown in Fig. The tube is appropriate or making etonations insie. ts imension are: 196 mm o inner iameter, an 1300 mm o total length. The tube is compose o two separate parts between which a specimen a plate, can be ixe. The tube is uippe with two pressure sensors, installe beore an behin the specimen. The sensors are connecte to ampliiers an arther to an oscilloscope, which recors pressure time histories. 1000

2 The pressure time histories, or ierent initial pressures levels o the explosive gas mixture, registere uring the tests, are presente in Fig 3. The teste plates are shown in Fig 4. The results o ater experiments plates geometry measurement are presente in Fig 5. Fig 1. Experimental evices connection iagram Fig. Research stan Fig 4. Damage plates The proceure o the single experiment consists o the ollowing actions: irstly the air rom both tubes is pumpe out until reaching vacuum, then the irst tube is ille with the explosive gas mixture o an appropriate pressure, the secon one, in the same time, is ille with the air o the same pressure. Ater closing all valves the etonation is initiate by an electric ilament. During the explosion the pressure time histories are registere on an oscilloscope. The gas mixture use or experiments is the mixture o propane an oxygen, the initial pressures use are: 0.3, 0.4, 0.5, 0.6, 0.7 an 0.8 bars. The specimens are the square 330x330 mm the plates are mae rom Al017 aluminum alloy 1 mm thick sheets. Fig 5. Plates geometry measurement results Fig 3. Registere pressure time histories n all presente above results, or each initial pressure, only one example has been chosen. The results o all perorme experiments are presente in Fig 6, where the maximum elections or the amage plates without holes an the hole uivalent iameters or the plates with holes are reerre to the pressure impulses. The pressure impulses have been calculate by integrating the 1001

3 pressure signal with respect to time, or the time interval rom 0 to 0.9 ms, which is the time o main impact. Fig 8. Results o tension tests with constant strain rates Fig 6. Delections an hole iameters vs. impulses pressure Uniaxial experiments All uniaxial experiments have been realize using Zwick/Roell Z00 strength machine uippe in an extensometer presente in Fig 7. The specimens have been prepare as bars mm, cut rom the sheets use or the main experiments. Fig 9. Results o loa-unloa tension tests Constitutive an amage moel The willingness o the racture preiction FEM moelling ruires making a ecision about moels selection. For this phenomenon moelling the Chaboche viscoplastic constitutive moel has been chosen (Lemaitre an Chaboche 1990). The inelastic strain rate in this moel is expresse by the ollowing ormula Fig 7. Test machine The specimens have been cut in both irections: along an across sheet s rolling irection. The machine s clamps spacing has been set to 75 mm, the gauge length has been 40 mm. All tests have been perorme in room temperature. Two kin o experiments have been realize: the tensile tests with constant strain rates an loa-unloa cyclic tension tests. The irst have been perorme or three ierent strain rates: /s, /s an /s. The examples o recore stress vs. strain plots are presente in Fig 8. The cyclic tests have been carrie with the constant strain rates /s uring loa phases an controlle by orce ecreasing own to zero value uring unloa phases with the same velocity. The successive cycles have been procee ater o the total strain increment. The results are presente in Fig 9 as the plots o uniaxial orce versus isplacement, what is useul or urther material parameters ientiication. 3 & = p& ' X ', J ( ' X ') (1) where an X are the eviatoric parts o stress an kinematic harening tensors, J(.) is the secon invariant an p is the accumulate plastic strain. The accumulate plastic strain rate p& is eine by the uation p& = ( J ( ' X ') / (1 D)) R k K n, () where k, K an n are the material parameters, R is the isotropic harening scalar unction an D is the amage parameter. The kinematic harening tensor X an the isotropic harening R are escribe as 100 & = a & cxp &, X 3 R& = b( R1 R) p&, (3)

4 where a, c, b, an R 1 are the harening material parameters. For amage moelling the Lemaitre moel has been chosen (Lemaitre 1985). The amage parameter evolution D is expresse in this moel by the ollowing uation s Y β + 1 D& = ( 1 D) p&, (4) S where s, S an β are the amage material parameters, Y is the amage strain energy release rate. The unction Y is etermine by (1 ) 3(1 ) Y = + ν + ν, (1 D) E 3 where an are the hyrostatic stress an von Misses uivalent stress, E is the elastic moulus an ν is the Poisson ratio. Fracture criterions The selection o the appropriate racture initiation criterion is the most important an the most iicult problem in the attempt o racture preiction FEM moelling. The most important actor, that controls initiation o uctile racture, o course besies the strain intensity, is the stress triaxiality /. t has been prove in the laboratory tests by (Bonora 1997) or (Bao an Wierzbicki 004). n the analyse problem the range o expecte stress triaxiality is between uniaxial an biaxial tension (1/3 / /3). Among many criterions, which can be oun in literature, our base on ierent theoretical assumptions have been chosen. The irst criterion assumes that racture occurs in a material when the uivalent plastic strain reaches a critical value (uber 1904) (5) =. (6) For an incompressible plastic material the uivalent plastic strain is eine as where 1,, 3 = + +, (7) 3 are the principal plastic strains. The 1 3 origin o this criterion goes back to the beginning o 0 th century, but still is very popular in contemporary commercial coes. The secon criterion takes into account both the stress an the strain values, it assumes that the strain energy ensity is constant w( ) = 0 ijij = Wcr (8) 1003 This criterion has been use by (Dauonet 006). The thir criterion is base on the relation between the applie stress in rupture an a critical value o amage epening upon the material an the loaing (Lemaitre 1996). n the uniaxial loaing conitions this relation can be written as 1 D1 cr =, where D 1cr is the amage critical value in uniaxial loaing conitions. n three imensions the critical value o the amage is given by ollowing ormula u 1 cr * (9) Dcr = D 1, (10) where u is the ultimate stress an * is the eective amage uivalent stress 1 * (1 ) 3(1 ) = + +. ( 1 D) 3 ν ν (11) The ourth criterion assumes that racture initiates at the critical point o structure, when the accumulate uivalent plastic strain with a suitable triaxiality weighing unctions reaches a critical value (Bao an Wierzbicki 004) = Dc. (1) 0 n the range o expecte stress triaxiality this ormula can be reuce to = Dc. (13) 0 entiication o constitutive an amage moel material parameters The irst material parameters, which shoul be ientiie, are the elastic moulus E an conventional plastic limit R 0.. The estimation has been one using the results o the tension tests with constant strain rates. The Chaboche moel assumes that material is isotropic. The parameters have been ientiie separately rom the results o tests on the specimens cut along an across sheet s rolling an later the average values have been calculate. The inal value o elastic moulus E is 70.0 GPa an conventional plastic limit R 0. is 8.3 MPa. The secon step o ientiication is estimating the values o the viscoplastic parameters o the moel. The ientiication has been mae on the basis o proceure given in (Kłosowski an Woznica 007). The material in room temperature almost oes not exhibit viscous properties, the plastic limit in zero strain rate parameter k has been assume as ual to R 0., the strain-rate epenency parameters n an K have been assume by values n = 10 an K = 5 MPa. Estimation o other parameters ruires writing the Chaboche moel uations (1)-(3) in uniaxial loaing conitions. This operation an reversing o ua-

5 tion (3b) gives the ollowing ormulas, which are the base o the ientiication 1/ X( ) R( ) k K, n = + ν + ν + ν & (14) a (1 exp( X = c )), (15) 3 c R = R (1 exp( b )). (16) 1 Using the results o laboratory tests irectly, cause problems in the ientiication, smoothing o the unctions is necessary. Thereore the last square polynomial approximation metho has been applie. The stress unction has been approximate by the polynomial egree o 10, the strain unction by the polynomial egree o 1. The example result is presente in Fig 10. The examples o inal unctions, ater proceure o the approximation, use in the ientiication are shown in Fig 11. Fig 10. Approximation o laboratory test result Fig 11. Functions use in ientiication All material parameters have been estimate using the Marquart-Levenberg last square metho variant. The results o the Chaboche moel material parameters ientiication are presente in Table 1. results o uniaxial loa-unloa cyclic tension tests have been use. The values o the amage may be evaluate rom the ollowing uation (Lemaitre 1996) E D E0 D = 1 (17) where E D an E 0 are the elastic moulus o the amage an virgin material. The ientiication o elastic moulus or each loaunloa cycle o the tension test in this metho is necessary. Accoring to (Celentano an Chaboche 007) the uniaxial moulus corresponing to an elastic path o the i cycle can be obtaine with Pu Pl Al Ei =, (18) Lu Ll L where P is the current axial loa, A is the current transversal area o the sample, L is the current extensometer length an sub inexes u an l, respectively, reer to the upper an lower points locate in the elastic path o the stress-strain curve. Unortunately uring the tests the current transversal area o the sample has not been measure. n ientiication the constant value o the transversal area an initial value o the sample s length have been assume. The example o the amage parameter calculation is presente in Table. Full results or the specimens cut along sheet s rolling (squares) an across (triangles) are shown in Fig 1. Table. Damage parameter calculation True strain E i [GPa] D [-] nitial Cycle Cycle Cycle Cycle Cycle Cycle Cycle Cycle Cycle l Table 1. Material parameters o the Chaboche moel E [GPa] k [MPa] b [-] R 1 [MPa] a [MPa] c [-] n [-] K [MPa] The ollowing step o ientiication is estimating o the amage parameter evolution. The easiest in practical use is the nonirect metho, using the inluence o amage on elasticity. For perorming this measurement the 1004 Fig 1. Results o the amage measurement

6 The last material constants, which shoul be ientiie, are the amage parameters s, S an β. This estimation has been mae on the basis o the proceure given in (Kłosowski an Pyrzowski 008). Taking the amage unction (4) an the amage strain energy unction (5) in uniaxial loaing conitions an substituting the secon to the irst, the ollowing ormula is obtaine s β 1 + ( D) D( ) & D = = 1. (19) t E(1 D) S t The unction o the experimental values D( ) shoul be approximate using the last square metho, the ollowing unction has been chosen ( ) D( ) = a 1 exp( b ). (0) By comparison o the erivate unction (1) with inelastic strain, which a an b parameters are known, with ormula (19) with reuce t D( ) = abexp( b ), = (1) E(1 D) S the approximation o amage material parameters is possible. The results o this estimation are presente in Table 3. Table 3. Damage material parameters S [GPa] s [-] β [-] s two integration points, ivie into ive layers have been applie. A relatively ine mesh has been assume (elem. size o 1. mm). The plate s bounary conitions are presente in Fig 13, they correspon to those which have been encountere in experiments. The astening o the plates has not assure ull clamping, the plates uring experiments have been sli insie with the average value o mm. n the moel also mm gap has been assume in plate s ixing. The rouning o ixing elements in the tube has been moelle by a rigi contact curve. The elements in which the racture criterion is ulille have been eactivate. The pressure in particular calculations is applie accoring to the registere pressure time histories. Fig 13. Bounary conitions in moeling Veriication o material parameters ientiication The veriication o ientiication is perorme by comparing the experimental results with the results o uniaxial tension test numerical simulation. The simulation o the test with constant strain rate /s, have been perorme. The results are presente in Fig 14 an Fig 15. A goo correlation is observe, the correlation coeicients R or both comparisons are over Numerical moel escription n the numerical analysis the non-linear initeelement MSC.Marc system has been use. t is a multipurpose coe or avance engineering simulations, which great avantage is possibility o user-eine subroutines application. The viscoplastic Chaboche moel with amage has been applie into the program using the UVSCPL subroutine. The algorithm use in this subroutine can be oun in (Kłosowski an Pyrzowski 008). The program has been use to simulate numerically the process o uniaxial tension test an to calculate the main problem plates subjecte to explosions. Dynamic, geometrically non-linear analysis using the oubolt integration algorithm has been perorme. n calculations o the uniaxial tension tests the ournoe thin-shell elements, with our integration points, ivie into ive layers have been use. n the specimen moel (bar) the isplacement an rotations o noes corresponing to the clampe ege have been ixe. The loaing has been controlle by isplacement o the noes corresponing to the crosshea. n numerical calculations o the plates subjecte to explosions the axisymmetric moel has been assume. The three-noe thin-shell axisymmetric elements, with Fig 14. Comparison o strain vs. stress plots Fig 15. Comparison o strain vs. amage plots 1005

7 Calibration o racture criterions parameters The calibrations o all racture criterion parameters have been one using the results o uniaxial laboratory tests an results o their numerical simulations. The irst racture criterion ruires calibrating the critical value o uivalent plastic strain. The average value o total strain to racture, calibrate on the basis o laboratory tests results, is = The uivalent plastic strain has been estimate using the uniaxial numerical simulation, its value in the moment o racture is = The secon criterion ruires calculating the total strain energy ensity by integrating the stress unction with respect to the total strain. The critical total strain energy ensity has been calculate on the basis o the uniaxial numerical simulation, its value is W cr = MJ m -3. The thir criterion, base on the critical value o amage, ruires estimating two parameters: the amage critical value in uniaxial loaing conitions an the ultimate stress. The secon parameter has been calculate as the average value o maximum stress reache in laboratory tests: u = 490 MPa. Taking the irst parameter irectly rom the laboratory tests amage measurement has been impossible because the cyclic experiments have not reache the average value o strain to racture got in tension tests. The uniaxial critical amage has been estimate rom numerical simulation, its value is D 1cr = The last criterion is base on irect epenency o the racture initiation to the stress triaxiality. During the uniaxial tension tests, which are use or calibration, the stress triaxiality is changing in a relatively narrow range an is always positive. By introucing the concept o the average stress triaxiality uring the eormation = av 1, () the racture criterion (13) can be rewritten as av 0 = D. c (3) The average value o the stress triaxiality uring the uniaxial tension test is ( / ) av = Taking the uivalent plastic strain to racture rom the irst criterion = the critical value D c = The example result o plate subjecte to explosion simulation with initial pressure 0.3 bars is presente in Fig 16 an Fig 17. The maximum election in the mile point o this plate measure ater experiment is 5. mm, the election obtaine in calculations is 6.8 mm, the is 6.3 %. The next example o the plate subjecte with initial pressure 0.4 bars is presente in Fig 18 an Fig 19. The maximum election measure is 38.5 mm, obtaine in calculations is 33.4 mm, the is 13. %. Fig 16. Delections in the mile o the plate, test 0.3 bar Fig 17. Cross-sections o the plate, test 0.3 bars Fig 18. Delections in the mile o the plate, test 0.4 bars Fig 19. Cross-sections o the plate, test 0.4 bars Numerical simulations o plates without racture The irst presente numerical simulations o the plates subjecte to explosions are or the lowest values o initial pressure 0.3 an 0.4 bars. As it has been shown in bounary conitions presentation, the plates ixing conitions are quite complicate an iicult in moeling. The results o this simulations are goo veriication o the moel s behaviour The comparisons o all results obtaine in numerical calculations with the results o experiments are presente in Table 4. The numerical results correspon well with experiments. n the simulations with initial pressure 0.3 bars, the average ierence in obtaine elections is 5.6 %. n the test with initial pressure o 0.4 bars the is higher, however this ierence occurs only in the mile part o the plate.

8 Table 4. Numerical an experimental elections o plates nit. pressure [bar] Test number Exp. el Num. el Error Numerical simulations o plates with racture The numerical simulations o the plates subjecte to explosions, in which the racture has been occurre, are presente or two highest initial pressures 0.6 an 0.7 bars. The calculations have been compute until one o the elements, in which the racture criterion has been ulille, has been eactivate. This moment has been assume as the racture initiation. The example is presente in Fig 0. n each racture criterion three ierent approaches have been analyse. The elements have been eactivate when the critical value occurre in all integration points o single layer, in at least three layers an in all layers over the thickness. The etaile results o numerical simulations or particular experiments presenting the calculate hole s iameters an the relative s or each racture criterion an each approach are shown in Tables The empty cells, which occur or the irst criterion, mean that clear crack localisation is impossible, the elements eactivation have starte rom the centre o the plate. Table 7. Numerical simulation results, pressure 0.6 bars, test bars 1/5 layer 3/5 layers 5/5 layers test mm Crit Crit Crit Crit Table 8. Numerical simulation results, pressure 0.6 bars, test bars 1/5 layer 3/5 layers 5/5 layers test mm Crit Crit Crit Crit Table 9. Numerical simulation results, pressure 0.7 bars, test bars 1/5 layer 3/5 layers 5/5 layers test mm Crit Crit Crit Crit Table 10. Numerical simulation results, pressure 0.7 bars, test Fig 0. Example o numerical simulation result Table 5. Numerical simulation results, pressure 0.6 bar, test bars 1/5 layer 3/5 layers 5/5 layers test mm Crit Crit Crit Crit Table 6. Numerical simulation results, pressure 0.6 bars, test 0.6 bars 1/5 layer 3/5 layers 5/5 layers test 45.mm Crit Crit Crit Crit bars test 39.3mm 1/5 layer 3/5 layers 5/5 layers Crit Crit Crit Crit Table 11. Numerical simulation results, pressure 0.7 bar, test bar 1/5 layer 3/5 layers 5/5 layers test mm Crit , Crit Crit Crit For one chosen experimental result (initial pressure 0.6 bar, test ), an the approach when in at least three layers the racture criterion must be ulille, the comparative cross-sections plots are presente in Fig

9 This analysis maybe oes not give the evient answer which criterion or approach is the best an the only right, but gives the iel to iscussion an the irection to the urther analysis in the uctile racture preiction. Acknowlegments Calculations presente in the paper were perorme at the Acaemic Computer Centre in Gansk (TASK). Researches were supporte by The French Government Scientiic Scholarship grante in 009 by The French Embassy in Polan. Reerences Fig 1. Cross-sections o the plate, test 0.4 bar Discussion an conclusions The comparative analysis o the experimental results with numerical simulations in the plates subjecte to impulse loaings explosions is presente. The material parameters ientiication is veriie by the uniaxial numerical simulations, the bounary conitions by the numerical simulations o plates without racture, all this results agree well with experiments. The racture preiction has been analyze using our ierent criterions an three ierent approaches. The irst criterion in many examples oes not give the clear crack localization, the obtaine hole iameters are too small. The secon criterion gives the lowest average. The best results are or the approach, where the elements have been eactivate in at least three layers over the thickness, the obtaine average is 1.8 %. The thir criterion gives a bit worse results then the secon. ere the lowest is reache or the thir approach, where the elements have been eactivate when the critical value occurre in all integration points in all layers, the average is 15.8 %. Though, a consieration is cause by the act, that this thir approach results are not much iversiie rom 35 to 40. mm in numerical simulations, where in experiments rom 34.5 to 58.7 mm. Thereore in spite o a bit higher 18 %, the secon approach seems to be better. The ourth criterion gives the highest s, the obtaine hole s iameters are too large. Bao, Y.; Wierzbicki, T On racture locus in the uivalent strain an stress triaxiality space, nternational Journal o Mechanical Sciences 46: oi: /j.ijmecsci Bonora, N On the eect o triaxial state o stress on uctility uning nonlinear CDM moel, nternational Journal o Fracture 88: oi:10.103/a: Celentano, D. J.; Chaboche, J. L Experimental an numerical characterization o amage evolution in steels, nternational Journal o Plasticity 3: oi: /j.ijplas Dauonet, B Etue numeriqe e la rapture elements e structures constituant es reservoirs sous chargement ynamique [Numerical researches o the structural elements rupture uner ynamic loaings]. PhD Thesis, Univer. Orleans. uber, M. T Przyczynek o postaw teoryi wytrzymałości [Contribution to the ounation o the strength o the material], Czasopismo Techniczne, Lwów : 81. Kłosowski, P.; Pyrzowski, Ł sotropic amage in viscoplastic low conitions, FEM applications with practical examples, Journal o KONES: Powertrain an Transportation 15(1): Kłosowski, P.; Woznica, K Nieliniowe lepkoplastyczne prawa konstytutywne w wybranych zastosowaniach analizy konstrukcji [Elasto-viscoplastic non-linear constitutive moels in selecte applications o structures analysis]. Gańsk: Wyawnictwo Politechniki Gańskiej. SBN Lemaitre, J.; Chaboche, J. L Mechanics o soli materials. Cambrige: Cambrige University Press. SBN Lemaitre, J A continuous amage mechanics, Journal o Engineering Materials an Technology 107: oi: / Lemaitre, J A course on amage mechanics. Berlin: Springer. SBN