Critical wave lengths and instabilities in gradient-enriched continuum theories

Abstract

Continuum material models can be enriched with additional gradients in order to model phenomena that are driven by processes at lower levels of observation. For instance, higher-order gradients of strains or inertia can be included in the equations of elasticity in order to model wave dispersion that occurs in heterogeneous media. These higher-order gradients may be obtained by homogenisation or continualisation procedures applied to heterogeneous micro-structured media. For a systematic comparison of various gradient-enriched continua, dispersion analysis may be used. In this contribution, we will explore the occurence of critical wave lengths and its implications for the material stability. In particular, we will present a unifying theorem that permits to assess the stability of elastic, hardening and softening gradient-enriched continua by means of a critical wave length analysis, whereby the upper or lower bound nature of the critical wave length indicates whether the model is stable or unstable.

On the Kachanov-Rabotnov continuum damage model

“…Dispersion is the phenomenon that harmonic waves with different wave lengths or frequencies propagate with different velocities. The ability to transform the shape of waves is a necessary though not sufficient condition for continua to capture localisation phenomena [25]. In a classical strain-softening continua, the waves are not dispersive, which means that the continuum is not able to transform propagating waves into stationary localisation waves [24].…”

Nano-scale wave dispersion beyond the First Brillouin Zone simulated with inertia gradient continua

“…However, for the strain gradient models certain truncations (second-order, sixth-order, etc.) lead to imaginary frequencies for the larger wave numbers, which in turn result in artificial model instabilities 17 . The other truncations (fourth-order, eighth-order, etc.)…”