2.)W.Horton ]r.,D.S.Powars,G.S.Gohn,Eds.,U.S.Geol. 17.D.S.Powars,U.S.GeoL Surv.Prof.Pap.1622 (2000) 33.The International Continental Scientific Drilling Program, Surv.Prof.Pap.1688 (2005). 18.C.W.Poag.The Chesopeake Invoder (Princeton Univ. the U.S.Geological Survey,and the NASA Science Mission 3.C.Koeberl,C.W.Poag,W.U.Reimold,D.Brandt, Press,Princeton,N].1999). Directorate provided funding for the drilling project. Science271,1263(1996). 19.E.R.McFarland,T.S.Bruce,U.S.Geol.Surv.Prof.Pop NSF and the Austrian Science Foundation provided 4.A.Deutsch.C.Koeberl.Meteorit.Plonet.Sai.41.689 (2006). 1688-K(2005). supplementary funding for the drill-site operations 5.G.S.Collins,K.Winnemann,Geology 33,925 (2005). 20.W.E.Sanford et ol.,Eos 85,369 (2004). DOSECC Inc.conducted the administrative and 6.G.S.Gohn et al.,Eos 87,349 (2006). 21.W.E.Sanford,J.Geochem.Explor.78-79,243(2003). operational management of the deep drilling project. 7.)W.Horton Jr.et al.,Geol.Soc.Am.Abstr.Prog.39. 22.W.E.Sanford,Geofluids 5,185 (2005). We thank the Buyrn family for use of their land as a 451,abstract167-5(2007). 23.See supporting material on Science Online. drilling site,the scientific and technical staff of the 8.)W.Horton Jr.,)N.Aleinikoff,M )Kunk,C.W.Naeser, 24.L Radford,L B.Cobb,R.L McCoy,U.S.DOE Rep. Chesapeake Bay Impact Structure Drilling Project for N.D.Naeser,U.S.GeoL Surv.Prof.Pap.1688 (2005). D0E/ET28373-1(1980). their many contributions,A.Gronstal (Open University. 9.P.B.Robertson,R.A.F.Grieve,in Impact and Explosion 25.F.T.Manheim,M.K.Horn,Southeast.Geol 9,215 UK)for the microbe enumeration data,L Edwards Crotering:Planetary and Terrestrial Implicotions, (1968). (USGS)for discussions of the post-impact geology,and D.)Roddy,R O.Pepin,R.B.Merrill,Eds.(Pergamon, 26.A.L.Gronstal et al.,Geol.Soc.Am.Abstr.Prog.39,316 G.Collins(Imperial College,UK)and K.Wunnemann New York,1977,pp.687-702. abstract116-22(2007). (Humboldt-University Berlin)for making available the 10.A.Wittmann,T.Kenkmann,L.Hecht,D.Stoffler, 27.R.]Parkes et al.,Noture 371,410 (1994). results of their numerical modeting. GeoL Soc.Am.Bull.119,1151 (2007). 28.5.D'Hondt et al.,Science 306,2216 (2004) 11.D.Stoffler et ol.,Meteorit.Planet.Sci.39,1035(2004). 29.R.]Parkes,B.A.Cragg,P.Wellsbury.Hydrogeol J.8, 12.T.Hayden et al.,Geology 36,327 (2008). 11(2000). Supporting Online Material 13.D.S.Powars,T.S.Bruce,U.5.Geol.Surv.Prof.Pap.1612 30.W.D.Grant,Philos.Trans.R.Soc.London Ser.B 359 www.sciencemag.org/cgi/content/full320/5884/1740/DC1 1249(2004). Materials and Methods 1999) 14.K.G.Miller,G.S.Mountain,Leg 150 Shipboard Party. 31.M.L.Malinconico,W.E.Sanford,]W.Horton Jr.. Figs.S1 to S10 Members of the New Jersey Coastal Plain Drilling Project, abstract for Conference on Large Meteorite Impacts and Table S1 Science271,1092(1996). References Planetary Evolution IV,Vredefort Dome,South Africa, 15.K.G.Miller et al.,Science 310,1293 (2005). abstract3068(2008). 4 April 2008:accepted 28 May 2008 16.]V.Browning et al,GeoL Soc.Am.Bull.118,657 (2006). 32.K.Kashefi,D.R.Lovley,Science 301,934 (2003). 10.1126/science.1158708 REPORT location glide paths and the investigation of vol- Dislocation Mean Free Paths umes that are representative of the bulk material (4).One can then derive a continuum formulation and Strain Hardening of Crystals on the basis of physically justified mechanisms and parameters,and this formulation is further integrated at the scale of a bulk crystal.Face- B.Devincre,1 T.Hoc,2 L.Kubin1* centered cubic (foc)crystals are taken as bench- mark materials because of their well-documented. Predicting the strain hardening properties of crystals constitutes a long-standing challenge for dislocation but rather complicated,stress/strain response. theory.The main difficulty resides in the integration of dislocation processes through a wide range of We started by considering the critical stresst time and length scales,up to macroscopic dimensions.In the present multiscale approach,dislocation for the activation of slip system i as a function of dynamics simulations are used to establish a dislocation-based continuum model incorporating discrete the dislocation densities p'stored (ie.,tempo- and intermittent aspects of plastic flow.This is performed through the modeling of a key quantity, rarily or permanently immobilized)in all slip sys- the mean free path of dislocations.The model is then integrated at the scale of bulk crystals,which temsj.This critical stress is given by a generalized allows for the detailed reproduction of the complex deformation curves of face-centered cubic crystals. Because of its predictive ability,the proposed framework has a large potential for further applications. Taylor relation ()of th omb where u is the shear modulus and b the modu- islocations are complex defects of crystal- translation of the crystal lattice.During plastic flow, lus of the Burgers vector.In fec crystals,the line materials,which have been investigated dislocations multiply and their mutual interactions symmetric tensor a contains six independent for more than 70 years.Their fundamental hinder their motion.As a consequence,a shear dimensionless coefficients,which account for the and economical importance arises from the number stress increase dt has to be imposed to produce a average strength of pair interactions between slip of properties they govem,such as the high strength shear strain increase dy.By definition,the ratio systems that result from short-and long-range of nanostructured materials,the reliability of semi- dildy is the strain hardening rate.Although dis- interactions.Their values were recently deter- conductor devices,the processing and service life location theory has successfully explained many as- mined from DD simulations(5). of structural materials,or the rheological proper- pects of the strength of crystalline solids,predicting For determining strain hardening,the key quan- ties of tectonic events in Earth's crust. strain hardening is "the most difficult remaining tity is the rate at which the critical stress evolves The irreversible,or plastic,deformation of crys- problem"(3).The present dislocation-based models with strain or,equivalently,the rate at which dis- tals results from the motion on crystallographic for strain hardening still have difficulties integrating locations accumulate under strain.For this purpose, planes of linear defects,the dislocations (/2). elementary dislocation properties into a continuum it is useful to define a dislocation mean free path L These defects carry an elementary amount of shear description of bulk crystals or polycrystals.As a which is the distance traveled by a dislocation seg- (the Burgers vector)that is usually the smallest consequence,current approaches cannot avoid ment of length before it is stored by interaction making use of extensive parameter fitting. with the microstructure.When the line moves by a Laboratoire d'Etude des Microstructures,Unite Mixte de Re- The present work takes advantage of three- distance d,it sweeps an area ldr and produces a cherche (UMR)104 CNRS,CNRS-Office National d'Etudes et dimensional dislocation dynamics (DD)simula- shear strain dy=bldd/V,where Vis the volume of de Recherches Aerospatiales (ONERA),20 Avenue de la Division tions (4-8)for averaging dislocation properties at the crystal.The stored density has then statistically Lederc,BP 72,92322 Chatillon Cedex,France.Laboratoire MSSMat,UMR 8579 CNRS,Ecole Centrale Paris,Grande Voie the intermediate scale of slip systems,which are increased by dp =(dL)V and the incremental des Vignes,92295 Chatenay-Malabry Cedex,France. ensembles of dislocations having the same Burgers storage rate is dp/dy=1/bL.This definition is only *To whom correspondence should be addressed.E-mail: vector and slip plane.The use of periodic boundary valid in differential form.as dislocation lines mul- ladislas.kubin@onera.fr conditions allows for the tayloring of large dis- tiply when they move.Following Kocks et al.and www.sciencemag.org SCIENCE VOL 320 27 JUNE 2008 17452. J. W. Horton Jr., D. S. Powars, G. S. Gohn, Eds., U.S. Geol. Surv. Prof. Pap. 1688 (2005). 3. C. Koeberl, C. W. Poag, W. U. Reimold, D. Brandt, Science 271, 1263 (1996). 4. A. Deutsch, C. Koeberl, Meteorit. Planet. Sci. 41, 689 (2006). 5. G. S. Collins, K. Wünnemann, Geology 33, 925 (2005). 6. G. S. Gohn et al., Eos 87, 349 (2006). 7. J. W. Horton Jr. et al., Geol. Soc. Am. Abstr. Prog. 39, 451, abstract 167-5 (2007). 8. J. W. Horton Jr., J. N. Aleinikoff, M. J. Kunk, C. W. Naeser, N. D. Naeser, U.S. Geol. Surv. Prof. Pap. 1688 (2005). 9. P. B. Robertson, R. A. F. Grieve, in Impact and Explosion Cratering: Planetary and Terrestrial Implications, D. J. Roddy, R. O. Pepin, R. B. Merrill, Eds. (Pergamon, New York, 1977), pp. 687–702. 10. A. Wittmann, T. Kenkmann, L. Hecht, D. Stöffler, Geol. Soc. Am. Bull. 119, 1151 (2007). 11. D. Stöffler et al., Meteorit. Planet. Sci. 39, 1035 (2004). 12. T. Hayden et al., Geology 36, 327 (2008). 13. D. S. Powars, T. S. Bruce, U.S. Geol. Surv. Prof. Pap. 1612 (1999). 14. K. G. Miller, G. S. Mountain, Leg 150 Shipboard Party, Members of the New Jersey Coastal Plain Drilling Project, Science 271, 1092 (1996). 15. K. G. Miller et al., Science 310, 1293 (2005). 16. J. V. Browning et al., Geol. Soc. Am. Bull. 118, 657 (2006). 17. D. S. Powars, U.S. Geol. Surv. Prof. Pap. 1622 (2000). 18. C. W. Poag, The Chesapeake Invader (Princeton Univ. Press, Princeton, NJ, 1999). 19. E. R. McFarland, T. S. Bruce, U.S. Geol. Surv. Prof. Pap. 1688-K (2005). 20. W. E. Sanford et al., Eos 85, 369 (2004). 21. W. E. Sanford, J. Geochem. Explor. 78–79, 243 (2003). 22. W. E. Sanford, Geofluids 5, 185 (2005). 23. See supporting material on Science Online. 24. L. Radford, L. B. Cobb, R. L. McCoy, U.S. DOE Rep. DOE/ET/28373-1 (1980). 25. F. T. Manheim, M. K. Horn, Southeast. Geol. 9, 215 (1968). 26. A. L. Gronstal et al., Geol. Soc. Am. Abstr. Prog. 39, 316, abstract 116-22 (2007). 27. R. J. Parkes et al., Nature 371, 410 (1994). 28. S. D’Hondt et al., Science 306, 2216 (2004). 29. R. J. Parkes, B. A. Cragg, P. Wellsbury, Hydrogeol. J. 8, 11 (2000). 30. W. D. Grant, Philos. Trans. R. Soc. London Ser. B 359, 1249 (2004). 31. M. L. Malinconico, W. E. Sanford, J. W. Horton Jr., abstract for Conference on Large Meteorite Impacts and Planetary Evolution IV, Vredefort Dome, South Africa, abstract 3068 (2008). 32. K. Kashefi, D. R. Lovley, Science 301, 934 (2003). 33. The International Continental Scientific Drilling Program, the U.S. Geological Survey, and the NASA Science Mission Directorate provided funding for the drilling project. NSF and the Austrian Science Foundation provided supplementary funding for the drill-site operations. DOSECC Inc. conducted the administrative and operational management of the deep drilling project. We thank the Buyrn family for use of their land as a drilling site, the scientific and technical staff of the Chesapeake Bay Impact Structure Drilling Project for their many contributions, A. Gronstal (Open University, UK) for the microbe enumeration data, L. Edwards (USGS) for discussions of the post-impact geology, and G. Collins (Imperial College, UK) and K. Wünnemann (Humboldt-University Berlin) for making available the results of their numerical modeling. Supporting Online Material www.sciencemag.org/cgi/content/full/320/5884/1740/DC1 Materials and Methods Figs. S1 to S10 Table S1 References 4 April 2008; accepted 28 May 2008 10.1126/science.1158708 REPORTS Dislocation Mean Free Paths and Strain Hardening of Crystals B. Devincre,1 T. Hoc,2 L. Kubin1 * Predicting the strain hardening properties of crystals constitutes a long-standing challenge for dislocation theory. The main difficulty resides in the integration of dislocation processes through a wide range of time and length scales, up to macroscopic dimensions. In the present multiscale approach, dislocation dynamics simulations are used to establish a dislocation-based continuum model incorporating discrete and intermittent aspects of plastic flow. This is performed through the modeling of a key quantity, the mean free path of dislocations. The model is then integrated at the scale of bulk crystals, which allows for the detailed reproduction of the complex deformation curves of face-centered cubic crystals. Because of its predictive ability, the proposed framework has a large potential for further applications. Dislocations are complex defects of crystal￾line materials, which have been investigated for more than 70 years. Their fundamental and economical importance arises from the number of properties they govern, such as the high strength of nanostructured materials, the reliability of semi￾conductor devices, the processing and service life of structural materials, or the rheological proper￾ties of tectonic events in Earth’s crust. The irreversible, or plastic, deformation of crys￾tals results from the motion on crystallographic planes of linear defects, the dislocations (1, 2). These defects carry an elementary amount of shear (the Burgers vector) that is usually the smallest translation of the crystal lattice. During plastic flow, dislocations multiply and their mutual interactions hinder their motion. As a consequence, a shear stress increase dt has to be imposed to produce a shear strain increase dg. By definition, the ratio dt/dg is the strain hardening rate. Although dis￾location theory has successfully explained many as￾pects of the strength of crystalline solids, predicting strain hardening is “the most difficult remaining problem” (3). The present dislocation-based models for strain hardening still have difficulties integrating elementary dislocation properties into a continuum description of bulk crystals or polycrystals. As a consequence, current approaches cannot avoid making use of extensive parameter fitting. The present work takes advantage of three￾dimensional dislocation dynamics (DD) simula￾tions (4–8) for averaging dislocation properties at the intermediate scale of slip systems, which are ensembles of dislocations having the same Burgers vector and slip plane. The use of periodic boundary conditions allows for the tayloring of large dis￾location glide paths and the investigation of vol￾umes that are representative of the bulk material (4). One can then derive a continuum formulation on the basis of physically justified mechanisms and parameters, and this formulation is further integrated at the scale of a bulk crystal. Face￾centered cubic (fcc) crystals are taken as bench￾mark materials because of their well-documented, but rather complicated, stress/strain response. We started by considering the critical stress ti c for the activation of slip system i as a function of the dislocation densities r j stored (i.e., tempo￾rarily or permanently immobilized) in all slip sys￾tems j. This critical stress is given by a generalized Taylor relation (9) of the form, ti c ¼ mb ffiffiffiffiffiffiffiffiffiffiffiffi ∑ j aijr j q , where m is the shear modulus and b the modu￾lus of the Burgers vector. In fcc crystals, the symmetric tensor aij contains six independent dimensionless coefficients, which account for the average strength of pair interactions between slip systems that result from short- and long-range interactions. Their values were recently deter￾mined from DD simulations (5). For determining strain hardening, the key quan￾tity is the rate at which the critical stress evolves with strain or, equivalently, the rate at which dis￾locations accumulate under strain. For this purpose, it is useful to define a dislocation mean free path L, which is the distance traveled by a dislocation seg￾ment of length l before it is stored by interaction with the microstructure. When the line moves by a distance dx, it sweeps an area ldx and produces a shear strain dg = bldx/V, where V is the volume of the crystal. The stored density has then statistically increased by dr = (dx/L)l/V, and the incremental storage rate is dr/dg = 1/bL. This definition is only valid in differential form, as dislocation lines mul￾tiply when they move. Following Kocks et al. and 1 Laboratoire d'Etude des Microstructures, Unité Mixte de Re￾cherche (UMR) 104 CNRS, CNRS–Office National d'Etudes et de Recherches Aérospatiales (ONERA), 20 Avenue de la Division Leclerc, BP 72, 92322 Chatillon Cedex, France. 2 Laboratoire MSSMat, UMR 8579 CNRS, Ecole Centrale Paris, Grande Voie des Vignes, 92295 Chatenay-Malabry Cedex, France. *To whom correspondence should be addressed. E-mail: ladislas.kubin@onera.fr www.sciencemag.org SCIENCE VOL 320 27 JUNE 2008 1745