We introduce the density functional theory (DFT) local quasicontinuum method: a first principles multiscale material model that embeds DFT unit cells at the subgrid level of a finite element computation. The method can predict the onset of dislocation nucleation in both single crystals and those with inclusions, although extension to lattice defects awaits new methods. We show that the use of DFT versus embedded-atom method empirical potentials results in different predictions of dislocation nucleation in nanoindented face-centered-cubic aluminum. In this paper we establish the feasibility of embedding density functional theory (DFT) into large-scale macroscopic finite-element calculations. The embedding takes place at the subgrid level and exploits the Cauchy-Born hypothesis, 1 whereby the local deformation computed at a quadrature point of the finite-element grid is applied to an infinite crystal lattice, whose energy and state of stress is then evaluated with DFT. We call our method DFT-LQC in reference to the local quasicontinuum (LQC) or Cauchy-Born approach, 2 in anticipation of a future nonlocal extension of the method.The impetus for multiscale descriptions of materials such as DFT-LQC derives from two main sources. From a topdown viewpoint, empirical constitutive models which remain reliable under extreme conditions of pressure, deformation, and deformation rate are often not available for use in largescale engineering simulations. Alternately, it is often desirable to extend the applicability of fundamental theories to engineering spatial and temporal scales. Methods such as DFT-LQC meet these demands by supplying a fundamental description of the material and embedding this description into large-scale engineering simulations.The LQC method, including extensions dealing with complex lattices, has been extensively used by Tadmor et al.,3,4 who used the method to study the nanoindentation of silicon and to analyze the process of polarization switching in PbTiO 3 . Tadmor et al. based their subgrid atomistic calculations on empirical potentials or effective Hamiltonians fitted to first principles calculations.The chief contribution of the present work is to establish the feasibility of using DFT, as opposed to empirical or effective interatomic potentials, as the fundamental description of materials such as aluminum in LQC calculations. Since DFT is a first-principles theory (see Ref. 5, and references therein for background), it may be expected to result in increased fidelity of the calculations, especially when the material is subject to an environment where the empirical potentials were not calibrated. A case in point is provided by nanoindentation, which induces high pressures and deformations under the indenter, even in the elastic range. Indeed, we show that the use of DFT versus embedded-atom method (EAM) empirical potentials results in vastly different predictions of dislocation nucleation in nanoindented facecentered-cubic (fcc) aluminum.The general DFT-LQC method consists of embedding DFT calcul...
We present a practical algorithm for partially relaxing multiwell energy densities such as pertain to materials undergoing martensitic phase transitions. The algorithm is based on sequential lamination, but the evolution of the microstructure during a deformation process is required to satisfy a continuity constraint, in the sense that the new microstructure should be reachable from the preceding one by a combination of branching and pruning operations. All microstructures generated by the algorithm are in static and configurational equilibrium. Owing to the continuity constrained imposed upon the microstructural evolution, the predicted material behavior may be path-dependent and exhibit hysteresis. In cases in which there is a strict separation of micro and macrostructural lengthscales, the proposed relaxation algorithm may effectively be integrated into macroscopic finite-element calculations at the subgrid level. We demonstrate this aspect of the algorithm by means of a numerical example concerned with the indentation of an Cu-Al-Ni shape memory alloy by a spherical indenter.
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