On the failure of a discrete axial chain using a continualized nonlocal Continuum Damage Mechanics approach

The failure of a discrete elastic-damage axial system is investigated using both a discrete and an equivalent continuum approach. The Discrete Damage Mechanics approach is based on a microstructured model composed of a series of periodic elastic-damage springs (axial Discrete Damage Mechanics lattice system). Such a discrete damage system can be associated with the finite difference formulation of a Continuum Damage Mechanics evolution problem. Several analytical and numerical results are presented for the tensile failure of this axial damage chain under its own weight. The nonlocal Continuum Damage Mechanics models examined in this paper are mainly built from a continualization procedure applied to centered or uncentered finite difference schemes. The asymptotic expansion of the first-order upward difference equations leads to a first-order nonlocal model, whereas the asymptotic expansion of the centered finite difference equations leads to a second-order nonlocal Eringen's approach. To complete this study, a phenomenological nonlocal gradient approach is also examined and compared with the first continualization methods. A comparison of the discrete and the continuous problems for the chains shows the effectiveness of the new micromechanics-based nonlocal Continuum Damage modeling, especially for capturing scale effects. For both continualized approaches, the length scale of the nonlocal models depends only on the cell size, while for the so-called phenomenological approach, the length scale may depend on the loading parameter. This apparent load-dependent length scale, already discussed in the literature with numerical arguments, is found to be sensitive to the postulated structure of the nonlocal model calibrated according to a lattice approach. Copyright (c) 2015 John Wiley & Sons, Ltd.