A comparative discrete-dislocation/nonlocal crystal-plasticity

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typeset2:/sco3/jobs1/ELSEVIER/msa/week.17/Pmsa15088y.001 Wed May 16 07:53:37 2001 Page Wed 16 D. Columbus, M. Grujicic / Materials Science and Engineering A000 (2001) 000–000 2. localization of plastic deformation in the form of deformation bands emanating from the crack tip is substantially more pronounced in the discrete-dislocation case (e.g. Fig. 4c) than in the crystal-plasticity case (e.g. Fig. 8c). This difference can be readily explained. In the crystal-plasticity analysis, while plastic deformation takes place on specific crystallographic slip systems, the systems are distributed in a continuous manner throughout the material. Consequently, plastic deformation varies continuously throughout the material. In sharp contrast, in the discrete-dislocation analysis, the same crystallographic slip systems are represented in a discrete fashion. Consequently, plastic deformation is the result of the nucleation and motion of discrete dislocations on discrete slip planes; 3. while the overall (stress and strain) fields which resulted from the discrete-dislocation and the nonlocal crystal-plasticity analyses are qualitatively (and to a good extent quantitatively) similar, the global (stress intensity versus crack extension) responses show significant differences. Primarily, the response obtained using the crystal-plasticity analysis is monotonic, while the one resulting from the discrete-dislocation analysis has a step-wise character. In addition, the overall slopes of the K I/K I0 versus a curves differ in the two cases by a factor of three to four which can give rise to a substantial difference in the stress intensity factor at the crack extensions significantly larger than the ones analyzed in the present work. These differences can be attributed to the fact that within the discrete-dislocation approach, the dislocation structure is undergoing a continuous evolution. Consequently, various dislocation configurations with greatly different net stress and displacement fields are being generated. Some of these configurations impede crack extension resulting in a steep K I/K I0 versus a response. Others, on the other hand interfere with plastic deformation promoting crack extension. In the case of crystal plasticity, due to the continuum nature of crystallographic slip, the fields and the overall material response evolve only in a continuous manner. 4. Conclusions Based on the results obtained in the present work, the following main conclusions can be drawn: (1) Mode I fracture problem in the presence of crystallographic sip can be analyzed in a consistent manner using both UNCORRECTED PROOF discrete-dislocation and continuum crystal-plasticity formulations. Within both formulations, the cohesive zone approach can be used to represent the force-displacement crack constitutive behavior. (2) When com- pared at comparable levels of the applied stress intensity factor, the dislocation/strain and stress fields, as well as distorted meshes and crack profiles are quite comparable for the two approaches. The observed differences in these fields can be readily attributed to the discrete nature of dislocations and to the discrete nature of slip systems in the discrete-dislocation approach. (3) The main difference in the results obtained using the two formulations pertains to the global stress intensity versus crack extension response. The continuum-nature of the crystal-plasticity analysis yields a monotonic K I/K I0 versus a response. In sharp contrast, the discrete nature of dislocations and their sources, as well as the interaction between dislocations and the advancing crack tip, gives rise to a step-like K I/K I0 versus a behavior. 5. Uncited references [26]. Acknowledgements The material presented here is based on work supported by the National Science Foundation, Grant Numbers DMR-9906268 and CMS-9531930 and by the US Army Grant Number DAAH04-96-1-0197. The authors are indebted to Drs Bruce A. MacDonald and Daniel C. Davis of NSF and Dr David M. Stepp of ARO for the continuing interest in the present work. The help of Dr Erik van der Giessen in providing the preprints of his work and in clarifying some aspects of the computational procedure is greatly appreciated. The authors also acknowledge the support of the Office of High Performance Computing Facilities at Clemson University. Appendix A. Derivation of an evolution equation for the density of geometrically necessary dislocations. If dX represents an infinitesimal vector connecting two material points in the reference configuration, then its image in the intermediate configuration obtained after plastic deformation is given as: dY=F p dX (A1) Then, for an infinitesimal surface S in the reference configuration which has a unit normal r 0 and is enclosed counterclockwise by a circuit C, the total Burgers vector content of all the dislocations piercing the area S can be evaluated using the standard right-hand Burgers circuit. The image of C in the intermediate configuration obtained after plastic shearing of the
typeset2:/sco3/jobs1/ELSEVIER/msa/week.17/Pmsa15088y.001 Wed May 16 07:53:37 2001 Page Wed D. Columbus, M. Grujicic / Materials Science and Engineering A000 (2001) 000–000 17 crystal lattice, C 1 , is generally an open contour, and hence, the line integral C 1dY defines the closure failure. Since the total Burgers vector B completes the circuit C 1 , it can be defined as: B=− dY=− FpdX (A2) C 1 C Eq. (A2) can be transformed into: B=− (×F pT ) Tr0 dS= r0 dS (A3) S using a generalized Stokes theorem. in Eq. (A2) is the so-called Nye’s tensor [26], and is defined as the tensorial curl of F p as: −(×F pT ) T (A4) The material derivative of the Nye’s tensor can be defined as: −(×F pT ) T (A5) which through the use of Eq. (19), becomes: =−[×(LpFp ) T ] T =−× m pnTT 0n0 F (A6) Next, the portion of associated with activity of slip system is defined as: −{×[( m 0 n0 )F p ] T } T S =−m 0 [×( F pT n0 )] (A7) Thus = (A8) Next, a phenomenological equation for is defined as follows. is assumed to scale with the accumulation rate of three sets of geometrically necessary dislocations all with Burgers vectors in the m0 direction: (a) a set of screw dislocations with tangent vector in the m0-direc tion; (b) a set of edge dislocations with tangent vector in the x3-direction; and (c) a set of edge dislocations with tangent vector in the n0-direction. Hence, for a cubic lattice with a common Burgers vector magnitude, b, can be written as: =m0b( G m0+ G p0+ G n0) (A9) (s ) (e )1 (e )2 where G , G , and G are the accumulation (s ) (e )1 (e )2 rates of density of the three aforementioned sets of geometrically necessary dislocations, respectively. A comparison of the second Eq. (A7) and Eq. (A9) yields: UNCORRECTED PROOF −×( F pT n 0 )=b( G (s)m 0 + G (e)1p 0 + G (e)2n 0 ) (A10) It can be easily shown that for in-plane-strain deformation, (F P 13=F P 23=F P 31=F P 32=0, F P 33=1), Eq. (A10) yields: G (s)=0, G (e)2=0 and G (e )1 = 1 d [ P P (F 11n 0(1) +F21n 0(2) )] bp 0(3) dx2 − d [ dx1 P P (F 12n 0(1) +F22n 0(2) )] (A11) Eq. (A11) is an evolution equation for the density of geometrically necessary dislocations and is used in Section 2.2 to evaluate the non-local contribution to the slip resistance. References [1] G.R. Irwin, Fracture, in: S. Flugge (Ed.), Encyclopedia of Physics, vol. 6, Springer New York, New York, 1958, pp. 551–590. [2] L.B. Freund, The mechanics of dislocations in strained-layer semiconductor-materials, Adv. Appl. Mech. 30 (1994) 1–66. [3] V. Tvergaard, J.W. Hutchinson, The relation between crack growth resistance and fracture process parameters in elastic-plastic solids, J. Mech. Phys. Solids 40 (1992) 1377–1397. [4] I.-H. Lin, R. Thomson, Cleavage, dislocation emission, and shielding for cracks under general loading, Acta Metall. 34 (1986) 187–206. [5] V. Shastry, P.M. Anderson, R. Thomson, Nonlocal effects of existing dislocations on crack-tip emission and cleavage, J. Mater. Res. 9 (1994) 1–16. [6] A. Needleman, A continuum model for void nucleation by inclusion debonding, J. Appl. Mech. 54 (1987) 525–531. [7] N.A. Fleck, J.W. Hutchinson, Strain gradient plasticity, Adv. Appl. Mech. 33 (1997) 295–361. [8] N.A. Fleck, G.M. Muller, M.F. Ashby, J.W. Hutchinson, Strain gradient plasticity: theory and experiment, Acta Metall. Mater. 42 (1994) 475–487. [9] A. Acharya, J.L. Bassani, Incompatibility and crystal plasticity, J. Mech. Phys. Solids (1998) (in press). [10] H. Dao, Parks, Geometrically necessary dislocation density in continuum plasticity theory, FEM implementation and application, PhD Thesis, MIT, August 1997. [11] Z. Suo, C.F. Shih, A.G. Varias, A theory for cleavage cracking in the presence of plastic flow, Acta Metall. Mater. 41 (1993) 1551–1557. [12] H.H. M. Cleveringa, E. van der Giessen, A. Needleman, A discrete dislocation analysis of mode I crack growth, J. Mech. Phys. Solids (in press) [13] H.H. M. Cleveringa, E. van der Giessen, A. Needleman, A discrete dislocation analysis of rate effects in mode I crack growth, Mater. Sci. Eng. (in print). [14] ABAQUS Theory Manual, Version 5.8, Habbilt, Karlsson, and Soreassen, Providence, RI, 1999. [15] J.H. Rose, J. Ferrante, J. Smith, Universal binding energy curves for metals and bimetallic interfaces, Phys. Rev. Lett. 47 (1981) 675–678. [16] M. Grujicic, D. Columbus, Discrete-dislocation analysis of ductile-to-brittle transition in BCC-like metallic materials, J. Mater. Sci. (2000) (in print). [17] A.S. Krausz, H.E. Eyring, Deformation Kinetics, Wiley, New York, 1975. [18] V.F. Kocks, A.S. Argon, M.F. Ashby, Thermodynamics and Kinetics of Slip, Pergamon, London, 1975 in Progress in Materials Science.
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A comparative discrete-dislocation/nonlocal crystal-plasticity

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