Mixed Atomistic-Continuum Models of Material Behavior: The Art of Transcending Atomistics and Informing Continua
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Mixed Atomistic–
Continuum Models of Material Behavior: The Art of Transcending Atomistics and Informing Continua M. Ortiz, A.M. Cuitiño, J. Knap, and M. Koslowski
Introduction The recent development of microscopes that allow for the examination of defects at the atomic scale has made possible a more direct connection between the defects and the macroscopic response they engender (see, e.g., the December 1999 issue of MRS Bulletin1). Techniques ranging from highresolution electron microscopy, which makes possible the determination of the atomic-level structure of dislocation cores and grain boundaries, to the atomic force microscopes that enhance our understanding of nanoindentation phenomena, all pose deep challenges for the modeling of the mechanics of materials. Each of these experiments calls for renewed efforts to strengthen the connection between defect mechanics and macroscopic constitutive descriptions. However, the link between the defects themselves and the observed macroscopic behavior is often a difficult one to forge theoretically and remains an active area of research. Many of the fundamental mechanisms underlying the inelastic behavior of materials are mediated by crystal-lattice defects and are, therefore, accessible to direct atomistic simulation, either by means of empiri-
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cal potentials or through ab initio quantum mechanical calculations. However, the relevance of atomistic calculations to the study of the macroscopic behavior of materials is often overstated. To be sure, there are macroscopic phenomena that can be directly elucidated at the atomic scale. A notable example is furnished by firstprinciples calculations of the equation of state and elastic moduli of bcc metals up to high pressures and temperatures.2–6 However, atomic-scale mechanisms are in general separated from the macroscopic behavior they engender by a vast array of intervening continuum scales. These mesoscopic scales both filter (average) and modulate (set the boundary conditions or driving forces for) the atomic-scale phenomena and are an essential part of the constitution of materials. Conversely, continuum theories rest on the assumption that the relevant fields that describe the state of the material vary slowly on the atomic scale. Therefore, continuum theories a fortiori break down in the vicinity of lattice defects or any other entity possessing structure on the atomic scale. Continuum theories can be “enriched”
in an attempt to incorporate additional atomistic information and avert this breakdown. The notorious core cutoff radius of the elastic theory of dislocations is a case in point. Another notable example is furnished by Mura’s theory of eigendistortions,7 which allows an otherwise linear-elastic material or a harmonic lattice to undergo crystallographic slip in discrete Burgers vector quanta, thus substantially extending the scope of linear elasticity. Ultimately, however, a complete understanding of these phenomena, as well as the computation of the relevant material constants, requires atomistic model
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