Discrete and continuous models of linear elasticity: history and connections

Abstract

This paper tracks the development of lattice models that aim to describe linear elasticity of solids and the field equations of which converge asymptotically toward those of isotropic continua, thus showing the connection between discrete and continuum. In 1759, Lagrange used lattice strings/rod dynamics to show the link between the mixed differential-difference equation of a one-dimensional (1D) lattice and the partial differential equation of the associated continuum. A consistent three-dimensional (3D) generalization of this model was given much later: Poincaré and Voigt reconciled the molecular and the continuum approaches at the end of the nineteenth century, but only in 1912 Born and von Kármán presented the mixed differentialdifference equations of discrete isotropic elasticity. Their model is a 3D generalization of Lagrange's 1D lattice and considers longitudinal, diagonal and shear elastic springs among particles, so the associated continuum is characterized by three elastic constants. Born and von Kármán proved that the lattice equations converge to Navier's partial differential ones asymptotically, thus being a formulation of continuous elasticity in terms of spatial finite differences, as for Lagrange's 1D lattice. Neglecting shear springs in Born-Kármán's lattice equals to Navier's assumption of pure central forces among molecules: in the limit, the lattice behaves as a one-parameter isotropic solid ("rari-constant" theory: equal Lamé parameters, or, equivalently, Poisson's ratio υ = 1/4). Hrennikoff and McHenry revisited the lattice approach with pure central interactions using a plane truss; the equivalent Born-Kármán's lattice in plane stress in the limit tends to a continuum with Poisson's Communicated by Andreas Öchsner.

Elastic waves in periodically anisotropic heterogeneous media: bridge the gap between rigorous and phenomenological approaches

“…Despite the growing power of computer codes, analytical solutions are still of great interest. The dynamic behaviour of heterogeneous media can be described by continuum theories [6,7]. A disadvantage of these theories is that they are phenomenological in the sense that the relationship between their micro-and macroscale properties is usually not known a priori.…”