On the Cauchy—Born Rule:
TL;DR: The Cauchy-Born rule as mentioned in this paper has been widely used in the literature to relate continuum theories of elastic phenomena to typical atomic or molecular theories, and it has been used to correlate changes in positions of the entities in the latter with descriptions of deformation in the former.
Abstract: To relate continuum theories of elastic phenomena to typical atomic or molecular theories, one needs some way of correlating changes in positions of the entities in the latter with descriptions of deformation used in the former. For crystals, the most commonly used bridge used for this is the Cauchy—Born rule. In recent years, there has been a flurry of activity aimed at better understanding it and in trying to generalize it. I will describe and comment on important ideas that have been generated, point out some successes and failures of the rule and indicate kinds of applications that have motivated such work.
Citations
More filters
Book•
24 Nov 2011TL;DR: In this paper, the authors present the complete fundamentals of multiscale modeling for graduate students and researchers in physics, materials science, chemistry, and engineering, with examples drawn from modern research on the thermodynamic properties of crystalline solids.
Abstract: Material properties emerge from phenomena on scales ranging from Angstroms to millimeters, and only a multiscale treatment can provide a complete understanding. Materials researchers must therefore understand fundamental concepts and techniques from different fields, and these are presented in a comprehensive and integrated fashion for the first time in this book. Incorporating continuum mechanics, quantum mechanics, statistical mechanics, atomistic simulations and multiscale techniques, the book explains many of the key theoretical ideas behind multiscale modeling. Classical topics are blended with new techniques to demonstrate the connections between different fields and highlight current research trends. Example applications drawn from modern research on the thermo-mechanical properties of crystalline solids are used as a unifying focus throughout the text. Together with its companion book, Continuum Mechanics and Thermodynamics (Cambridge University Press, 2011), this work presents the complete fundamentals of materials modeling for graduate students and researchers in physics, materials science, chemistry and engineering.
381 citations
TL;DR: This review presents the state of the art in strain and ripple-induced effects on the electronic and optical properties of graphene by providing the crystallographic description of mechanical deformations, as well as the diffraction pattern for different kinds of representative deformation fields.
Abstract: This review presents the state of the art in strain and ripple-induced effects on the electronic and optical properties of graphene. It starts by providing the crystallographic description of mechanical deformations, as well as the diffraction pattern for different kinds of representative deformation fields. Then, the focus turns to the unique elastic properties of graphene, and to how strain is produced. Thereafter, various theoretical approaches used to study the electronic properties of strained graphene are examined, discussing the advantages of each. These approaches provide a platform to describe exotic properties, such as a fractal spectrum related with quasicrystals, a mixed Dirac-Schrodinger behavior, emergent gravity, topological insulator states, in molecular graphene and other 2D discrete lattices. The physical consequences of strain on the optical properties are reviewed next, with a focus on the Raman spectrum. At the same time, recent advances to tune the optical conductivity of graphene by strain engineering are given, which open new paths in device applications. Finally, a brief review of strain effects in multilayered graphene and other promising 2D materials like silicene and materials based on other group-IV elements, phosphorene, dichalcogenide- and monochalcogenide-monolayers is presented, with a brief discussion of interplays among strain, thermal effects, and illumination in the latter material family.
358 citations
TL;DR: In this article, a review of the state of the art in strain and ripple-induced effects on the electronic and optical properties of graphene is presented, with a focus on the Raman spectrum.
Abstract: This review presents the state of the art in strain and ripple-induced effects on the electronic and optical properties of graphene. It starts by providing the crystallographic description of mechanical deformations, as well as the diffraction pattern for different kinds of representative deformation fields. Then, the focus turns to the unique elastic properties of graphene, and to how strain is produced. Thereafter, various theoretical approaches used to study the electronic properties of strained graphene are examined, discussing the advantages of each. These approaches provide a platform to describe exotic properties, such as a fractal spectrum related with quasicrystals, a mixed Dirac-Schrodinger behavior, emergent gravity, topological insulator states, in molecular graphene and other 2D discrete lattices. The physical consequences of strain on the optical properties are reviewed next, with a focus on the Raman spectrum. At the same time, recent advances to tune the optical conductivity of graphene by strain engineering are given, which open new paths in device applications. Finally, a brief review of strain effects in multilayered graphene and other promising 2D materials like silicene and materials based on other group-IV elements, phosphorene, dichalcogenide- and monochalcogenide-monolayers is presented, with a brief discussion of interplays among strain, thermal effects, and illumination in the latter material family.
271 citations
138 citations
Cites background or methods from "On the Cauchy—Born Rule:"
...Here, we give one example of 4-tap Daubechies’ wavelet and interested reader should find more in [16]....
[...]
...and the first-principle-based molecular dynamics, initiated in the 1980’s by Car and Parrinello [16]....
[...]
...246 6.1.1 The Car-Parrinello molecular dynamics . . . . . . . . . . . . . . . 246 6.1.2 The quasi-continuum method . . . . . . . . . . . . . . . . . . . . 248 6.1.3 The kinetic scheme . . . . . . . . . . . . . . . . . . . . . . . . . ....
[...]
...Two of the best known examples are the QM-MM (quantum mechanics–molecular mechanics) approach in chemistry and the Car-Parrinello molecular dynamics....
[...]
...The importance of concurrent coupling was emphasized in the work of Car and Parrinello [16], as well as the earlier work of Warshel and Levitt [48] on QMMM methods (see also [47])....
[...]
TL;DR: In this article, the authors propose a method based on rectifying the singular points in the parameter space by using a blow-up argument and then asymptotically matching the approximations around such points with the regular approximation away from them.
Abstract: Our starting point is a parameterized family of functionals (a ‘theory’) for which we are interested in approximating the global minima of the energy when one of these parameters goes to zero. The goal is to develop a set of increasingly accurate asymptotic variational models allowing one to deal with the cases when this parameter is ‘small’ but finite. Since Γ-convergence may be non-uniform within the ‘theory’, we pose a problem of finding a uniform approximation. To achieve this goal we propose a method based on rectifying the singular points in the parameter space by using a blow-up argument and then asymptotically matching the approximations around such points with the regular approximation away from them. We illustrate the main ideas with physically meaningful examples covering a broad set of subjects from homogenization and dimension reduction to fracture and phase transitions. In particular, we give considerable attention to the problem of transition from discrete to continuum when the internal and external scales are not well separated, and one has to deal with the so-called ‘size’ or ‘scale’ effects.
129 citations
References
More filters
TL;DR: In this article, the authors focus on the distinction between the strain due to complete twinning and the macroscopic strain caused by formation of a thin twin lamella, and the subdivision of the orientation diagram according to the twinning system which can be operated by axial loading is a more complicated one than customarily indicated.
1,282 citations
"On the Cauchy—Born Rule:" refers background in this paper
...Christian and Mahajan present some sketches of possible shuffles....
[...]
...Research on these was reviewed by Christian and Mahajan [28]....
[...]
TL;DR: In this paper, the authors make predictions based on an analysis of a new nonlinear theory of martensitic transformations introduced by the authors, where the crystal is modelled as a nonlinear elastic material, with a free-energy function that is invariant with respect to both rigid-body rotations and the appropriate crystallographic symmetries.
Abstract: Predictions are made based on an analysis of a new nonlinear theory of martensitic transformations introduced by the authors The crystal is modelled as a nonlinear elastic material, with a free-energy function that is invariant with respect to both rigid-body rotations and the appropriate crystallographic symmetries The predictions concern primarily the two-well problem, that of determining all possible energy-minimizing deformations that can be obtained with two coherent and macroscopically unstressed variants of martensite The set of possible macroscopic deformations obtained is completely determined by the lattice parameters of the material For certain boundary conditions the total free energy does not attain a minimum, and the finer and finer oscillations of minimizing sequences are interpreted as corresponding to microstructure The predictions are amenable to experimental tests The proposed tests involve the comparison of the theoretical predictions with the mechanical response of properly oriented plates subject to simple shear Additional crystallographic background is given for the model, and the theory is compared with the `linearized' model of Khachaturyan, Roitburd and Shatalov There are some similarities in the predictions of the two theories, but also some major discrepancies
694 citations
"On the Cauchy—Born Rule:" refers background in this paper
...There is a good review of work on Martensitic transformations in shape-memory alloys by James and Hane [27], with over 100 references....
[...]
...I thank Richard James, Pradeep Sharma, Lev Truskinovsky and Giovanni Zanzotto for useful suggestions and for helping me collect references....
[...]
...As an alternative, he mentions developing a “shell theory” in which the basic kinematics uses exactly those variables, mentioning that this was done by Falk and James [50], in their analysis of an interesting instability in the bacteriophage T4 tail sheath....
[...]
...In a study of special kinds of configurations occurring in many kinds of atomic structures, James [49] explores a different kind of idea....
[...]
...According to Yang and E [40], Friesecke and James [41] were among the first to consider generalizations of CBR for some such structures, rods and thin films, so I will describe their idea related to this....
[...]
445 citations
TL;DR: It is shown that the most stable crystalline phases with rocksalt-like structures are characterized by large vacancy concentrations and local distortions, which helps to design novel phase-change materials as evidenced by new experimental data.
Abstract: Phase-change materials are of tremendous technological importance ranging from optical data storage to electronic memories. Despite this interest, many fundamental properties of phase-change materials, such as the role of vacancies, remain poorly understood. ‘GeSbTe’-based phase-change materials contain vacancy concentrations around 10% in their metastable crystalline structure. By using density-functional theory, the origin of these vacancies has been clarified and we show that the most stable crystalline phases with rocksalt-like structures are characterized by large vacancy concentrations and local distortions. The ease by which vacancies are formed is explained by the need to annihilate energetically unfavourable antibonding Ge–Te and Sb–Te interactions in the highest occupied bands. Understanding how the interplay between vacancies and local distortions lowers the total energy helps to design novel phase-change materials as evidenced by new experimental data.
416 citations
TL;DR: In this paper, a finite deformation continuum theory is derived from interatomic potentials for the analysis of the mechanics of carbon nanotubes, and the expression for the flexural stiffness of graphene sheets, which cannot be obtained from standard crystal elasticity, is derived.
Abstract: A finite deformation continuum theory is derived from interatomic potentials for the analysis of the mechanics of carbon nanotubes. This nonlinear elastic theory is based on an extension of the Cauchy-Born rule called the exponential Cauchy-Born rule. The continuum object replacing the graphene sheet is a surface without thickness. The method systematically addresses both the characterization of the small strain elasticity of nanotubes and the simulation at large strains. Elastic moduli are explicitly expressed in terms of the functional form of the interatomic potential. The expression for the flexural stiffness of graphene sheets, which cannot be obtained from standard crystal elasticity, is derived. We also show that simulations with the continuum model combined with the finite element method agree very well with zero temperature atomistic calculations involving severe deformations.
398 citations
"On the Cauchy—Born Rule:" refers background in this paper
...Then, there are the different ideas proposed a bit later by Arroyo and Belytschko [44, 45] on films that I will describe, for the case of one involving a single layer of atoms....
[...]
...They find that their approach gives a slightly different continuum theory from that used by Arroyo and Belytschko....
[...]
...To ease comparison with the ideas of Arroyo and Belytschko, I will discuss their idea for films of the Bravais lattice kind....
[...]
...Very briefly, he describes how he would construct a shell theory somewhat like that sketched above, used by Arroyo and Belytschko....
[...]