Statistical Mechanics of Elasticity

Abstract

This volume contains 21 chapters on various topics in applied mathematics. The chapters have been written by 20 individuals, each of whom has been involved with the applications of mathematics, mostly in the physical sciences and engineering. The following topics are discussed in one or more chapters: basic analysis, vectors, tensors, complex variables, ordinary differential equations, partial differential equations, special functions, integral equations, transform methods, asymptotic methods, perturbation methods, linear algebra, functional approximation, numerical analysis, optimization techniques, probability, and statistics. Also included are chapters on several particular areas of application, namely, oscillations, wave propagation, and formulation of mathematical models. The volume is not a handbook in the usual sense, that is, it is not a collection of tables and formulas. Instead, each chapter represents the author's summary description of the chapter topic, with emphasis on the ideas and techniques of potential value in problem solving. Motivation is provided from the applied sciences for most of the topics discussed and the presentation is generally descriptive, rather than formal. Based on a close look at several of the chapters in areas most familiar to me, the treatment is quite complete. The chapters have been prepared independently, without cross-referencing, so that some topics are covered more than once, usually from different points of view. The volume does include an extensive index of 32 pages, so that the place in which any particular topic is considered can easily be determined. For example, the Wiener-Hopf method can be found in the chapter on transform methods, variational methods are discussed mainly in the chapters on ordinary differential equations and on optimization, and the finite element method is briefly introduced in the chapter on partial differential equations of second and higher order. The coverage appears to be quite complete within the branch of mathematics commonly termed analysis. The rationale for including two chapters on specific applications, namely, wave propagation and oscillations (nonlinear

Finite crystal elasticity of carbon nanotubes based on the exponential Cauchy-Born rule

“…There are two equivalent approaches for extracting the elastic moduli from the atomistic description, 31,32 namely the method of the homogeneous deformations, [33][34][35] also called the Cauchy-Born hypothesis, 36 and the method of the long waves within lattice-dynamical theories. 31 For examples of the application of these classical methods to the in-plane response of graphene, see Refs.…”

“…[33][34][35]42,46 The optical modes are the analog of the inner displacements in lattice dynamical theories. 32 Let denote the inner displacements field, which following Ref. 42, is defined in the undeformed body, previous to the ''macroscopic'' deformation ⌽.…”

“…Consequently, as the crystalline solid deforms, lattice vectors undergo a linear transformation. 32,33,46 This approach is often abstracted through the Cauchy-Born rule: 36,40 aϭFA, ͑1͒…”

Atomistic long-term simulation of heat and mass transport

“…These kinetic equations, when inserted into local energy-balance and mass-balance equations, define an evolution of the system. In subsequent derivations, we follow and adapt the notation of Weiner (2002).…”

Simple optimized Brenner potential for thermodynamic properties of diamond

“…Arroyo and Belytschko [6], Zhang et al [7][8][9][10] and Jiang et al [11] have proposed nanoscale continuum theories for carbon nanotubes based on interatomic potentials for carbon. Based on the local harmonic approximation [12], Jiang et al [13] established a Enite-temperature continuum theory directly from the interatomic potential. The interatomic potential is incorporated into the continuum analysis via the constitutive model and the effect of Enite temperature is also taken into account.…”