During the past decade, computer simulations based on a quantum-mechanical description of the interactions between electrons and between electrons and atomic nuclei have developed an increasingly important impact on solid-state physics and chemistry and on materials science—promoting not only a deeper understanding, but also the possibility to contribute significantly to materials design for future technologies. This development is based on two important columns: (i) The improved description of electronic many-body effects within density-functional theory (DFT) and the upcoming post-DFT methods. (ii) The implementation of the new functionals and many-body techniques within highly efficient, stable, and versatile computer codes, which allow to exploit the potential of modern computer architectures. In this review, I discuss the implementation of various DFT functionals [local-density approximation (LDA), generalized gradient approximation (GGA), meta-GGA, hybrid functional mixing DFT, and exact (Hartree-Fock) exchange] and post-DFT approaches [DFT + U for strong electronic correlations in narrow bands, many-body perturbation theory (GW) for quasiparticle spectra, dynamical correlation effects via the adiabatic-connection fluctuation-dissipation theorem (AC-FDT)] in the Vienna ab initio simulation package VASP. VASP is a plane-wave all-electron code using the projector-augmented wave method to describe the electron-core interaction. The code uses fast iterative techniques for the diagonalization of the DFT Hamiltonian and allows to perform total-energy calculations and structural optimizations for systems with thousands of atoms and ab initio molecular dynamics simulations for ensembles with a few hundred atoms extending over several tens of ps. Applications in many different areas (structure and phase stability, mechanical and dynamical properties, liquids, glasses and quasicrystals, magnetism and magnetic nanostructures, semiconductors and insulators, surfaces, interfaces and thin films, chemical reactions, and catalysis) are reviewed. © 2008 Wiley Periodicals, Inc. J Comput Chem, 2008

Ab-initio simulations of materials using VASP: Density-functional theory and beyond.

Graphene-based $s{p}^{2}$-carbon nanostructures such as carbon nanotubes and nanofibers can fail near their ideal strengths due to their exceedingly small dimensions. We have calculated the phonon spectra of graphene as a function of uniaxial tension by density functional perturbation theory to assess the first occurrence of phonon instability on the strain path, which controls the strength of a defect-free crystal at $0\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. Uniaxial tensile strain is applied in the $x$ (nearest-neighbor) and $y$ (second nearest-neighbor) directions, related to tensile deformation of zigzag and armchair nanotubes, respectively. The Young's modulus $E=1050\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$ and Poisson's ratio $\ensuremath{\nu}=0.186$ from our small-strain results are in good agreement with previous calculations. We find that in both $x$ and $y$ uniaxial tensions, phonon instabilities occur near the center of the Brillouin zone, at (${\ensuremath{\epsilon}}_{xx}=0.194$, ${\ensuremath{\sigma}}_{xx}=110\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$, ${\ensuremath{\epsilon}}_{yy}=\ensuremath{-}0.016$) and (${\ensuremath{\epsilon}}_{yy}=0.266$, ${\ensuremath{\sigma}}_{yy}=121\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$, ${\ensuremath{\epsilon}}_{xx}=\ensuremath{-}0.027$), respectively. Both soft phonons are longitudinal elastic waves in the pulling direction, suggesting that brittle cleavage fracture may be an inherent behavior of graphene and carbon nanotubes at low temperatures. We also predict that a phonon band gap will appear in highly stretched graphene, which could be a useful spectroscopic signature for highly stressed carbon nanotubes.

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Ab initio calculation of ideal strength and phonon instability of graphene under tension

A brief introduction to the ABINIT software package is given. Available under a free software license, it allows to compute directly a large set of properties useful for solid state studies, including structural and elastic properties, prediction of phase (meta)stability or instability, specific heat and free energy, spectroscopic and vibrational properties. These are described, and corresponding applications are presented. The emphasis is also laid on its ease of use and extensive documentation, allowing newcomers to quickly step in.

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A brief introduction to the ABINIT software package

We describe a group of alloys that exhibit “super” properties, such as ultralow elastic modulus, ultrahigh strength, super elasticity, and super plasticity, at room temperature and that show Elinvar and Invar behavior. These “super” properties are attributable to a dislocation-free plastic deformation mechanism. In cold-worked alloys, this mechanism forms elastic strain fields of hierarchical structure that range in size from the nanometer scale to several tens of micrometers. The resultant elastic strain energy leads to a number of enhanced material properties.

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Multifunctional Alloys Obtained via a Dislocation-Free Plastic Deformation Mechanism.

High entropy alloys: A focused review of mechanical properties and deformation mechanisms

The ideal shear strength is the minimum stress needed to plastically deform an infinite dislocation-free crystal and is an upper bound to the strength of a real crystal. We calculate the ideal shear strengths of Al and Cu at zero temperature using pseudopotential density functional theory within the local density approximation. These calculations allow for structural relaxation of all five strain components other than the imposed shear strain and result in strengths on ${111}$ planes of 1.85 and 2.65 GPa for Al and Cu, respectively ( $8%$-- $9%$ of the shear moduli). In both Al and Cu, the structural relaxations reduce the ideal shear strengths by $35%$ to $45%$, but the directions of relaxation strain in each are qualitatively different.

Ideal Shear Strengths of fcc Aluminum and Copper

Using pseudopotential density functional theory within the local-density approximation, we calculate the ideal shear strengths of W for slip on {110}, {112} and {123} planes allowing for complete structural relaxation orthogonal to the applied shear. The strengths in the weak directions on all planes are found to be very nearly equal (about 18GPa, or 11% of the shear modulus G). Moreover, the shear instability occurs at approximately the same applied shear strain (17–18%). This unusual isotropy is explained in terms of the atomic configurations of high-energy saddle points reached during shear. Analysis of these saddle points may also offer a simple explanation for the prevalence of the pencil glide of dislocations on planes containing a (111) direction in bcc metals. Finally, we calculate the ideal cleavage strengths of W on {100} and compare our calculated ideal shear and cleavage strengths with experimental nanoindentation and whisker measurements. All these results can be rather simply unders...

The ideal strength of tungsten

We have calculated the phonon spectra of aluminum as a function of strain using density functional perturbation theory for $⟨110⟩$, $⟨100⟩$, and $⟨111⟩$ uniaxial tension, as well as relaxed $⟨112⟩{111}$ shear. In all four cases, phonon instabilities occur at points away from the center of the Brillouin zone and intrude before the material becomes unstable according to elastic stability criteria. This is the first time the ideal strength of a metal has been shown to be dictated by instabilities in the acoustic phonon spectra. We go on to describe the crystallography of the unstable modes, all of which are shear in character. This work further suggests that shear failure is an inherent property of aluminum even in an initially dislocation-free perfect crystal.

Phonon instabilities and the ideal strength of aluminum.

This paper investigates the conditions of elastic stability that set the upper limits of mechanical strength Following Gibbs, we determine the conditions that ensure stability against reconfigurations that leave the boundary of the system unchanged The results hold independent of the nature of properties of the loading mechanisms but are identical with those derived previously for a solid in contact with a reservoir that maintains the Cauchy stress Mechanisms that control the stress in some other way may add further conditions of stability; nonetheless, the conditions of internal stability must always be obeyed and can be consistently used to define the ultimate strength The conditions of stability are contained in the requirement that λ ijkl δe ij δe kl ≥ 0 for all infinitesimal strains, where λ ijkl = 1/2(B ijkl + B klij ), and B is the tensor that governs the change in the Cauchy stress t during incremental strain from a stressed state τ: t ij = T ij + B ijkl δe kl  Since λ has full Voigt symmetry, it can be written as the 6 x 6 matrix λ ij with eigenvalues λ α  Stability is lost when the least of these vanishes The conditions of stability are exhibited for cubic (hydrostatic), tetragonal (tensile) and monoclinic (shear) distortions of a cubic crystal and some of their implications are discussed Elastic stability and the limits of strength are now being explored through first-principles calculations that increment uniaxial stretch or shear to find the maximum stress We discuss the nature of this limiting stress and the steps that may be taken to identify orthogonal instabilities that might intrude before it is reached

The internal stability of an elastic solid

The ideal structural steel combines high strength with excellent fracture toughness. In this paper we consider the limits of strength and toughness from two perspectives. The first perspective is theoretical. It has recently become possible to compute the ideal shear and tensile strengths of defect-free crystals. While the ferromagnetism of bcc Fe makes it a particularly difficult problem, we can estimate its limiting properties from those of similar materials. The expected behavior at the limit of strength contains many familiar features, including cleavage on {100}, slip on multiple planes, "conditionally" brittle behavior at low temperature and a trend away from brittle behavior on alloying with Ni. The behavior of fcc materials at the limit of strength suggests that true cleavage will not happen in austenitic steels. The results predict an ideal cleavage stress near 10.5 GPa, and a shear strength near 6.5 GPa. The second perspective is practical: how to maximize the toughness of high-strength steel. Our discussion here is limited to the subtopic that has been the focus of research in our own group: the use of thermal treatments to inhibit transgranular brittle fracture in lath martensitic steels. The central purpose of the heat treatments described here is grain refinement, and the objective of grain refinement is to limit the crystallographic coherence length for transgranular crack propagation. There are two important sources of transgranular embrittlement: thermal (or, more properly, mechanical) embrittlement at the ductile–brittle transition, and hydrogen embrittlement from improper heat treatment or environmental attack. As we shall discuss, these embrittling mechanisms use different crack paths in lath martensitic steels and, therefore, call for somewhat different remedies.

C. R. Krenn

Papers

Ideal Shear Strengths of fcc Aluminum and Copper

The ideal strength of tungsten

Phonon instabilities and the ideal strength of aluminum.

The internal stability of an elastic solid

The Limits of Strength and Toughness in Steel