Dynamics and quasicontinuum analogs of a lattice with simple microstructures
Résumé
The dynamic behaviour of, even simple, discrete structures is actually already very complex in the linear regime because of the presence of multiple internal scales of times and lengths. For discrete elastic structures comparable to atomic systems, this complexity is incorporated into phonon dispersion relations that are highly non-trivial, but well known for some of them. To describe the behaviour of such structures on a macroscopic scale, many numerical approaches naively rely on the classical continuum (CC) theory that is non-dispersive and gives only a very limited description of the singular behaviour of the discrete microscopic structure. As a result, the CC theory misses many important physical effects that are not relevant to the structures of macroscopic size, but actually become dominant at the scale of the considered microstructures. The non-trivial dynamic behaviour of a simple linearly elastic monatomic chain is provided as a useful theoretical illustration for the development of generalized models and numerical coupling methods involving multi-scales and, or multi-physical models. This academic model has some dispersive properties or dynamics (violating the Einstein's interpretation of causality in the case of a domain unbounded) that are poorly known in the literature. Their accurate analysis and proper use lead in particular to two kinds of exact (quasi-) continuum models: the Eringen's [1] and Kunin's [2] spatially-nonlocal, but temporally-local, models; the Charlotte and Truskinovsky's temporally-nonlocal model [3], which is spatially-local for the considered discrete model. In terms of numerical coupling methodologies, the first continuum models preserve the existence of energy potentials, while the second one the generalized force-based or virtual power. Both continualisations involve length or, and time scales that are non-arbitrary but are related to the dispersive and attenuation structural properties of the atomic interactions. A comparison with the CC theory of elasticity shows that the CC prediction can be artificially improved by taking into account the simultaneity of the atomic response, associated with the violation of Einstein's causality. This non-trivial behavior can be interpreted in a continuous model by the presence of inertial and pseudo-dissipative post-Newtonian forces that are ignored by most multi-scale numerical coupling methods. References: 1 A.C. Eringen, "Nonlocal field theories", In A.C. Eringen, (Editor), "Continuum Physics", Academic press, New York, 4, 205-267, 1976. 2 I. Kunin,"Elastic media with microstructure, v.I (One dimensional models)", Springer, Berlin,: 1982. 3 M. Charlotte, L. Truskinovsky, "Lattice dynamics from a continuum viewpoint", J. Mechanics and Physics of Solids, PII, S0022-5096(12)00058-0, 2012. doi:10.1016/j.jmps.2012.03.004