International audienceTwo 2024 aluminum alloys are studied. Their second phase volume fraction differ as well as their mean inter-particle ....

Besson, Jacques

Bron, Frédéric

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Ductile tearing of 2024 aluminum alloy panelsFre´de´ric Bron, Jacques BessonTo cite this version:Fre´de´ric Bron, Jacques Besson. Ductile tearing of 2024 aluminum alloy panels. Internationalconference of fracture, 2005, Turin, Italy. xx, 6 p., 2005, 20-25 mars 2005. <hal-00157715>HAL Id: hal-00157715https://hal.archives-ouvertes.fr/hal-00157715Submitted on 4 Jun 2013HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s. 	 ﬀﬂﬁﬃﬂ 	"!#$! 	%	&' ()*	,+-/.10/24365 1 798 .:0<;<=>=>365 21 ?A@CBED>FHG>@JILK/@CGNMPOQ@SR>@ 2 @CBEDT@UOQBED>@CVWRT@6XZY[OQ@U\T\$@[] 0 ?^N_`]1a[b[adc>eﬂXfY[OP@C\>\1@gK<@CR>@Jh] - OPidG>BC@2 K<@CGjMQOP@SR>@CVlkminM`o@COQFpidqTh] ;<5W= kL?r]:K 5W2l=ts k 2 _du[a[aT] 0 ?%bN_`] ;fv OQIﬃBC@CR>@JhLwTeUx[x[aT] - OPidG>BC@y{zﬂ|$}6~{y6}}W6%`Tj`pmŁp1jp[$LNCECŁp[A}6nŁptJdd``J>`pmLUjnCŁpTŁH An¡`ﬃ6n6`CdŁHlmd`¡Łp:CnJ¢£NJCŁpdpﬂJ`nŁ¤¥}ZnEClNCﬂ[jJCŁp[L`4`LjC¤Tg¦¨§©}Sª/jdpS§«J[ﬃC¬­CnCmŁpWC~S¢£dC>¯®°dN1Łp`­ªg`J`ppnﬃ`®°jUjCjJ¨J°dnŁmndty*±ŁHC¡nnmd:¬m1d<Uj²1Ł¤³Łp:C´jdd`:$nCŁpp{`j¤T`L`ECŁpµjŁpJ`JC`:mŁpWJdmCﬃ$n±ﬂJU`EnU`®ŁpŁHQC:d[$C®°nPWddmJj`jC¤`¶CjmpdLj¡6nj¡®°nQ6dd&®°jCNCnCŁ`pd&}6¶C:n[$C³Łp¡®`J[&T,C¶$n±ŁpCŁp`jgNCnCŁ`r­jU`­CnCŁpECŁﬃpd¤`C´$CT°`JCŁp``rC¡C{Ł:CnJ¢£jJCŁpdp¬JjdŁp¤·L¸C¹¯º¬»ﬃ¼¡½t¾L¿{ºµ¸­¼¡¹AA2024 aluminum sheets are used to manufacture parts of the fuselage of aircrafts. Mechanicaltests are performed on the sheets using large center-cracked tension panels M(T) in order to measurethe crack initiation toughness as well as the R-curve behavior. With new improved materials,methods of analysis based on the linear fracture mechanics using plastic zone corrections can hardlybe used since valid tests, according to the standard, would require very large panels (several meters)(ASTM [1]), (Chabanet [2]). Consequently, the prediction of the resistance to ductile tearing is ofgreat interest in particular if the modeling is based on a good understanding of physical phenomenarelated to plastic deformation and ductile damage.Two materials are investigated. They are two different grades of a 2024 aluminum alloy. Onegrade corresponds to the standard material. The second one corresponds to an improved grade inwhich the Fe and Si contents have been reduced. The main difference between the materials istheir particle coarse content and particle spacing. The present study describes the application ofa continuum damage model to simulate crack propagation. The plastic anisotropy is accounted forusing a newly developed yield criterion (Bron [3]). The aim of the study is to develop a methodologyallowing the simulation of the tests on wide M(T) plates for both materials based on a parameteradjustment on much smaller samples (KA specimens) carried out on one material only.À¶Á&Âﬂº¬Ãg»¬¸CÂ¯ÄWÅ%Ât¹¯½ÆÁ&Ã{¿{ÇtÂ¯¹ﬃ¸­¿{ÂtÄº¬ÃgÅ°ºµ¸C¹¡ÈTwo grades of 2024 aluminum alloy sheets (T4 heat-treated) with a nominal thickness of 1.6 mmwere supplied by the aluminum manufacturer Pechiney. They are subsequently referred to as 2024-1and 2024-2. Variant 2024-1 is a commercial alloy and variant 2024-2 is a high purity alloy improvedfor high damage tolerance. In the following, the rolling direction is referred to as L, the longtransverse direction as T and the short transverse direction (thickness) as S. One of the major factorsregarding damage tolerance is the presence of coarse intermetallic second phase particles and voids.They are nucleation sites for damage. The volume fraction of coarse intermetallic particles is largelyinfluenced by the iron content which has been reduced in 2024-2 material. Most of those particlesare composed of either Al-Cu-Fe-Mn(-Si) or Al-Cu-Mg. Second phase particles and voids have beencharacterized by image analysis on 2D micrographs. The materials also contain a small amount ofpreexisting cavities. Corresponding volume fractions are summarized on table 1.fp (10   3) fv (10   3) ¯dp (µm)2024-1 13.0 1.2 192024-2 3.7 1.6 35Table 1: Image analysis on 2024-1 and 2024-2 materials. f p: particle volume fraction, fv: voidvolume fraction, ¯dp: particle mean spacing.641218r101810606025 10r  3  7530 601360(L),1230(T)7602531160(L),1030(T)TB NUr NV KA M(T)Figure 1: Specimens for mechanical tests (all dimensions in mm) – For TB specimen, longitudinaland transverse deformations are measured. For NUr (r  0  5, 1 or 2 mm) and NV, left and rightopening displacements are measured; the mean value is used. For KA and M(T), the crack openingdisplacement and the crack length are measured. The notch radius of KA and NV is less than 60 µm.The length of the M(T) specimen slightly differs in directions L and T because of the limited widthof the as-received sheets.Seven types of samples are used (Fig. 1). The TB sample is a conventional smooth tensile barused to determine the elastic-plastic behavior. The differently notched samples NU05, NU1, NU2and NV are used to characterize the behavior under various stress triaxiality ratios and to evaluatedamage properties. Center-cracked tension panels M(T) are used to obtain a stable crack propagationover more than 60 mm at each side of the initial crack. This standard R-curve test (ASTM [1]) isexpensive and could be replaced by tests on small sized Kahn specimens (KA) which also allowa stable crack propagation over more than 20 mm (ASTM [4]). The original crack length of largeM(T) specimens is 253 mm after fatigue precracking. To prevent buckling two rigid face plates areaffixed to the central portion of the specimen. KA specimens are not precracked. The radius of theV-notch is less than 60 µm. For notched specimens, opening displacement is measured on both sides.TB and NUr tests are used to adjust the model parameters describing the anisotropic plastic behavior(Bron [3]). KA specimens are used to adjust the model parameters describing ductile failure. FinallyM(T) samples are used to check the validity of the model as they are simulated using the parametersobtained on small samples.The fracture mechanisms were described by the authors in a previous publication (Bron [5]). Theappearance of the macroscopic fracture surface depends on the notch severity. M(T), KA and NVexhibit the same aspect: the crack initiates with a flat triangular shape perpendicular to the loadingdirection (Fig. 2). The remaining fracture surface is slanted with a 45  angle with respect to theloading direction. When notch severity decreases (NU05) the triangle diminishes in size and finally,for NU1, NU2 and smooth specimens, there is no triangle at all and the whole surface is slanted.Voids are first initiated at intermetallic particles in both cases. In the flat triangular region, largevoids grow from the particles up to coalescence by “internal necking” (Thomason [6]). In slantedregions, coalescence occurs by a “void sheet mechanism” (Garrison [7]) which is associated withthe creation of smaller dimples in the inter-void ligaments.intermetallic particlesinternal neckingdispersoidsband localization20 µmFigure 2: Two failure mechanisms: internal necking (left) or localization of the deformation (right). ¶Á9¼¡½tÃgÄ4¸C¹tÈThe constitutive model for porous materials proposed in (Rousselier [8]) and extended in (Tanguy[9]) is used. Damage is represented by a single scalar variable, the porosity f . The plastic flowpotential, φ, is written as:φ  σ  R  p  (1)where R  p  represents the hardening behavior of the undamaged material. p is a scalar measure ofthe plastic strain. The effective stress, σ  , is defined by:ψ  σa 1  f  σ 23 f Dexp3q2σm 1  f  σ 	  1def. σ 0 (2)where σa is an equivalent stress and σm the mean stress D and q are material parameters adjustedfrom experiments. σa does not correspond to the usual von Mises stress but is defined according to(Bron [3]) in order to account for plastic anisotropy. The plastic flow is obtained assuming normalityso that the plastic strain rate tensor is given by:ε˙p~~ 1  f  p˙ ∂φ∂σ~~ 1  f  p˙∂σ ∂σ (3)where σ is the stress tensor. Noting that σ  is an homogeneous function of degree 1 of σ Euler’stheorem applies so that:σ : ε˙p 1  f  σ  p˙ (4)This relation is interpreted as the equality between the macroscopic plastic work (left hand-side) andthe microscopic plastic work (right hand-side). The evolution of the porosity f is given by massconservation modified to account for strain controlled void nucleation (Chu [10]):˙f   1  f  traceε˙p  An p˙  1  f  2 ∂σ ∂σ : 1 Anp˙  (5)Nucleation correspond to cracking/debonding of second phase particles.KAOpening displacement (mm)∆a(mm)2420161284043210F   S0(MPa)120100806040200M(T)Opening displacement (mm)∆a(mm)605040302010043210F   S0(MPa)300250200150100500Figure 3: Simulations (lines) on KA and M(T) samples for material 2024-2 compared to experiments(symbols); loading is in the T direction. Empty symbols are used for the load F  S0 and blacksymbols for the crack advance ∆a.¸C¹ﬃ¸Cº¬Ã ÃÄlÃÁ&Ãg¹¯ºÅ°¸CÁ&¾¯ÄlÂﬂºµ¸­¼¡¹Calculations were done using quadratic elements with reduced integration. The region wherethe crack propagates was meshed with 20 node 3D elements (8 integration points) and far from thisregion, plane stress elements are used. Besides, due to the symmetries, only an eighth of the samplesis meshed (a fourth for KA samples). A post-increment procedure removes broken elements. In thecase of 3D bricks with twenty nodes and eight Gauss points, it is performed when four Gauss pointsare broken. Details on the simulations procedures can be found in (Besson [11])When modeling crack propagation using continuum damage mechanics, the crack is a thinvolume which height is half the element height in the case of quadratic elements. Thus crack growthis very much affected by the height of the elements  in the crack region which should be consideredas a material parameter (Rivalin [12]), (Xia [13]), (Skallerud [14]). The two other mesh dimensionsonly influence the accuracy of the results as a classical mesh refinement. Indeed the crack growth isalso controlled by the damage parameters D and q. The mesh size only affects the behavior in thecrack region but the damage parameters modify the behavior in the whole sample. Thus an optimalmesh size should allow the use of the same set of damage parameters for all stress/strain conditions.Previous publications show that the order of magnitude of the mesh size should be the inter-particle spacing (Rousselier [8]), (Brocks [15]), (Steglich [16]), (Gullerud [17]). The mean secondphase particle spacing ¯d is estimated from the surface particle density (table. 1). In the present case,¯d is so small that a mesh corresponding to this size would be too large to allow to run the simulations.simulationstestsNU2NU1NU05NVOpening displacement (mm)1.21.00.80.60.40.200.80.60.40.200.60.40.200.40.20F S0(MPa)4003002001000Figure 4: Simulations on notched samples for material 2024-2; loading is in the T direction.In this work the mesh dimension,  , was fixed a priori for material 2024-2:  2024-2  175µm. Themesh size for the second grade (2024-1) is defined so that:  2024-2   2024-1  ¯d2024-2  ¯d2024-1 so thatthe mesh size is scaled according to the microstructural characteristic length. The initial porosityis taken from the image analysis as well as the particle volume fraction which can nucleate newcavities. Based on observations which indicate that all particles are broken in a tensile test at theonset of necking the nucleation rate is given by: An   fp  p0  if p  p0 and An  0 otherwise. p0is the plastic strain at the onset of necking (p0  0  16 for both materials). The damage parameters,D  4  6 and q  1  0, were adjusted to fit results on KA samples for material 2024-2 only. Material2024-1 is simulated using the same values for D and q and values for the initial porosity and thenucleation rate corresponding to image analysis results. The adjustment of the parameters used torepresent plastic anisotropy is described in (Bron [3]).³»¯ÃgÅ°¾tÄ°ºﬃÅÂt¹¯½ ¿¬¼¡¹L¿{Ä¥¾¡Å¸­¼¡¹tÅThe present simulation cannot reproduce slant fracture and crack advance remains flat throughoutthe calculation. It should however be outlined that, in the literature, the flat to slant crack path changehas not yet been satisfactorily simulated. It is believed that a very fine mesh (mesh size of about theinter-particle distance) is required to perform the calculation of the flat to slant transition. Suchcalculations require a computational power that is not yet available. The present calculation is ableto represent both the crack growth rate and the load level.Fig. 3 compares the experimental and simulated load vs. opening and crack advance (∆a) vs.opening curves for both KA and M(T) specimens for material 2024-2. The crack length is slightlyunderestimated. This is possibly due to the inability of the model to capture the transition betweenflat and slanted fracture. Is the case of the M(T) specimen the load is overestimated; this is due to thepartial buckling of the large plate during testing which is not fully prevented using an anti-bucklingsetup during the test (Roychowdhury [18]).Fig. 4 compares experimental and simulated loads for NV and NUr specimens. A goodagreement is obtained for the deeply notched specimen (NV). For other specimens the ductilityis underestimated. This discrepancy is related to the fact that the mesh size was fixed to be ablerun calculations and not adjusted to be representative of the material rupture characteristic length.This leads to a high value of the adjusted D which is suitable to simulate crack propagation with thechosen mesh size but leads to early rupture in notched specimens.Fig. 5 compares the experimental and simulated load vs. opening and crack advance curves formaterial 2024-1. A good agreement is obtained considering that the damage parameters D and qwere not fitted. This also shows that considering that the interparticle mean distance is the rupturecharacteristic length is consistent. A better agreement could have been obtained considering themean distance between particles and voids considered as a single defect population as in that casethe ¯d2024-1  ¯d2024-2 would have been slightly larger.KAOpening displacement (mm)∆a(mm)2420161284043210F   S0(MPa)120100806040200M(T)Opening displacement (mm)∆a(mm)605040302010043210F   S0(MPa)300250200150100500Figure 5: Simulations on KA and M(T) samples for material 2024-1; loading is in the T direction.Empty symbols are used for the load F  S0 and black symbols for the crack advance ∆a.REFERENCES1 ASTM (1999) Annual Book of ASTM Standards, 03.01, 509–521.2 Chabanet, O., Steglich, D., Besson, J., Heitmann, V., Hellmann, D. and Brocks, W. (2003)Comput. Mater. Sci., 26, 1–12.3 Bron, F. and Besson, J. (2004) Int. J. Plasticity, 20 (4–5), 937–963.4 ASTM (2001) Annual Book of ASTM Standards, 02.02.5 Bron, F., Besson, J. and Pineau, A. (accepted) Mater. Sci. Engng A.6 Thomason, P. F. (1985) Acta Metall., 33 (6), 1079–1085.7 Garrison, W. M. and Moody, N. R. (1987) J. Phys. Chem. Solids, 48 (11), 1035–1074.8 Rousselier, G. (1987) Nuclear Engng Design, 105, 97–111.9 Tanguy, B. and Besson, J. (2002) Int. J. Fract., 116 (1), 81–101.10 Chu, C. C. and Needleman, A. (1980) J. Engng Mater. Tech., 102 (3), 249–256.11 Besson, J., Steglich, D. and Brocks, W. (2001) Int. J. Solids Struct., 38 (46–47), 8259–8284.12 Rivalin, F., Besson, J., Di Fant, M. and Pineau, A. (2001) Engng Fract. Mech., 68, 347–364.13 Xia, L., Shih, C. F. and Hutchinson, J. W. (1995) J. Mech. Phys. Solids, 43 (3), 389–413.14 Skallerud, B. and Zhang, Z. L. (1999). In: Fatigue and fracture mechanics: 29th vol. ASTM STP1332, Panontin, T. L. and Sheppard, S. D. (Eds.), pp. 201–214.15 Brocks, W., Hao, S. and Steglich, D. (1996) J. Phys. IV , 6 (6), 43–52.16 Steglich, D. and Brocks, W. (1998) Fatigue Fract. Engng Mater. Struct., 21 (10), 1175–1188.17 Gullerud, A., Gao, X., Dodds Jr, R. and Haj-Ali, R. (2000) Engng Fract. Mech., 66 (1), 65–92.18 Roychowdhury, S., Roy, Y. and Dodds Jr, R. (2002) Engng Fract. Mech., 69 (8), 983–1002.

Ductile tearing of 2024 aluminum alloy panels

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Ductile tearing of 2024 aluminum alloy panels

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