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An adaptive finite element approach to atomic-scale mechanics. – The quasicontinuum method. (English) Zbl 0982.74071
From the summary: The paper gives a description of the quasicontinuum method, with special reference to the ways in which the method may be used to model crystals with more than a single grain. The formulation is validated in terms of a series of calculations on grain boundary structure and energetics. The method is then illustrated on the motion of a stepped twin boundary where a critical stress for the boundary motion is calculated, and on the nanoindentation into a solid containing a subsurface grain boundary to study the interaction of dislocations with grain boundaries.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74M25 Micromechanics of solids
82D25 Statistical mechanical studies of crystals
74E15 Crystalline structure
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[1] Ackland, G.J., Tichy, G., Vitek, V., Finnis, M.W., 1987. Simple N-body potentials for the noble-metals and nickel. Philosophical Magazine A56, 735-756.
[2] Barenblatt, G.I., 1962. The mathematical theory of equilibrium cracks in brittle fracture. Advances in Applied Mechanics, 7, 55-129.
[3] Belytschko, T., Tabbara, M., 1993. H-adaptive finite-element methods for dynamic problems, with emphasis on localization. International Journal of Numerical Methods in Engineering 36, 4245-4265. · Zbl 0794.73071
[4] Chadwick, P., 1976. Continuum Mechanics. John Wiley and Sons, New York.
[5] Christian, J. W. (1983) Some surprising features of the plastic deformation of body-centered cubic metals and alloys. Metallurgical Transactions A14, 1237-1256.
[6] Dahmen, U., Hetherington, C.J., Okeefe, M.A., Westmacott, K.H., Mills, M.J., Daw, M.J., Vitek, V., 1990. Atomic-structure of a sigma-99 grain boundary in aluminum—a comparison between atomic-resolution observation and pair-potential and embedded-atom simulations. Philosophical magazine Letters 62, 327-335.
[7] Daw, M.S., Baskes, M.I., 1983. Semiempirical, quantum-mechanical calculations of hydrogen embrittlement in metals. Physical Review Letters 50, 1285-1288.
[8] Dennis, J.E. Jr., Schnabel, R.B., 1983. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs. · Zbl 0579.65058
[9] Ercolessi, F., Adams, J., 1993. Interatomic potentials from 1st-principles calculations—the force-matching method. Europhysics Letters 26, 583-588.
[10] Ericksen, J.L., 1984. The Cauchy-Born hypothesis for crystals. In Phase Transformations and Material Instabilities in Solids, ed. M. Gurtin, pp. 61-77. Academic Press.
[11] Gerberich, W.W., Nelson, J.C., Lilleodden, E.T., Anderson, P., Wyrobek, J.T., 1966. Indentation induced dislocation—the initial yield-point. Acta Materiala 44, 3585-3598.
[12] Hirth, J.P., Lothe, J., 1968. Theory of Dislocations. McGraw-Hill, New York. · Zbl 1365.82001
[13] Hull, D., Bacon, D.J., 1992. Introduction to Dislocations. Pergamon Press, Oxford.
[14] King, A.H., Smith, D A., 1980. The effects on grain boundary processes of steps in the boundary plane associated with the core of grain boundary dislocations. Acta Crystallographica A36, 335-343.
[15] Kohlhoff, S., Gumbsch, P., Fischmeister, H.F., 1991. Crack-propagation in bcc crystal studied with a combined finite-element and atomistic model. Philosophical Magazine A64, 851-878.
[16] Okabe, A., Boots, B., Sugihara, K., 1992. Spatial Tessellations. Wiley and Sons, New York.
[17] Papadrakakis, M., Ghionis, P., 1986. Conjugate-gradient algorithms in nonlinear structural-analysis problems. Computer Methods in Applied Mechanics Engineering 59, 11-27. · Zbl 0595.73097
[18] Peierls, R.E., 1940. The size of a dislocation. Proceedings of the Physical Society of London 52, 34-37.
[19] Shenoy, V.B., Phillips, R., 1997. Finite-sized atomistic simulations of screw dislocations. Philosophical Magazine A76, 367-385.
[20] Sloan, S.W., 1993. A fast algorithm for generating constrained Delaunay triangulations. Computers and Structures 47, 441-450. · Zbl 0808.65149
[21] Tadmor, E.B., 1996. The quasicontinuum method. Ph.D. Thesis, Brown University.
[22] Tadmor, E.B., Ortiz, M., Phillips, R., 1996. Quasicontinuum analysis of defects in solids. Philosophical Magazine A73, 1529-1563.
[23] Tadmor, E.B., Miller, R., Phillips, R., Ortiz, M., 1997. Quasicontinuum analysis of nanoindentation. To be submitted to Acta Materiala.
[24] Thomson, R., Zhou, S.J., Carlsson, A.E., Tewary, V.K., 1992. Lattice imperfections studied by use of lattice green-functions. Physical Review B, 46, 10,613-10,622.
[25] Xu, X.-P., Argon, A.S., Ortiz, M., 1995. Nucleation of dislocations from crack tips under mixed-modes of loading—implications for brittle against ductile behavior of crystals. Philosophical Magazine A72, 415-451.
[26] Zienkiewicz, O.C., Zhu, J.Z., 1987. A simple error estimator and adaptive procedure for practical engineering analysis. International Journal of Numerical Methods in Engineering 24, 337-357. · Zbl 0602.73063
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