Scale effects in the static response of a one-dimensional quasi-brittle damage lattice

Abstract

In this paper, we investigate the failure of a discrete elastic-damage axial system using both a discrete and an equivalent continuum approach. The microstructured damage chain consists of a one-dimensional damage lattice with direct nearest–neighbour interactions, which is composed of a series of periodic elastic-damage springs (axial lattice system treated in terms of discrete damage mechanics). We show that the damage lattice equations are equivalent to the centred finite difference formulation of a continuum damage mechanics (CDM) evolution problem. Such a discrete damage system reveals some scale effects on both the structural strength and stiffness. The nonlocal CDM in the hardening branch and cohesive damage model in the softening branch considered here are built using a continualisation procedure applied to the nonlinear difference equations of the lattice system. With this procedure, the difference equations to be solved are approximated by higher order differential equations. Using a rational asymptotic method, the continualised model appears to be equivalent to a nonlocal CDM model in the damage propagation zone. A finite length cohesive model is obtained in the softening range. A comparison of the discrete and the continuous problems for damage chains brings out the effectiveness of the new micromechanics-based nonlocal and cohesive continuum damage model, especially for capturing scale effects.