Steady-State Cracks in Viscoelastic Lattice Models

TL;DR

This paper investigates the steady-state behavior of mode III cracks in viscoelastic lattice models, revealing limitations of continuum theories in capturing lattice-specific phenomena like trapping and velocity selection influenced by viscosity.

Contribution

It introduces a detailed analysis combining numerical and analytical methods to compare lattice and continuum models, highlighting the effects of viscoelasticity and the limitations of continuum approximations.

Findings

Lattice models exhibit trapping phenomena not captured by continuum theories.

Viscosity introduces new effects in crack velocity selection.

Continuum approaches fail to predict certain lattice-dependent behaviors at large N.

Abstract

We study the steady-state motion of mode III cracks propagating on a lattice exhibiting viscoelastic dynamics. The introduction of a Kelvin viscosity $\eta$ allows for a direct comparison between lattice results and continuum treatments. Utilizing both numerical and analytical (Wiener-Hopf) techniques, we explore this comparison as a function of the driving displacement $\Delta$ and the number of transverse sites $N$. At any $N$, the continuum theory misses the lattice-trapping phenomenon; this is well-known, but the introduction of $\eta$ introduces some new twists. More importantly, for large $N$ even at large $\Delta$, the standard two-dimensional elastodynamics approach completely misses the $\eta$-dependent velocity selection, as this selection disappears completely in the leading order naive continuum limit of the lattice problem.