From discrete to nonlocal continuum damage mechanics: Analysis of a lattice system in bending using a continualized approach

It is shown herein that the bending problem of a discrete damage system, also called microstructured damage system or lattice damage system, can be rigorously handled by a nonlocal continuum damage mechanics approach. It has been already shown that Eringen's nonlocal elasticity was able to capture the scale effects induced by the discreteness of a microstructured system. This paper generalizes such results for inelastic materials and first presents some results for engineering problems modelled within continuum damage mechanics. The microstructured model is composed of rigid periodic elements connected by rotational elastic damage springs (discrete damage mechanics). Such a discrete damage system can be associated with the finite difference formulation of a continuum damage mechanics problem, i.e. the Euler-Bernoulli damage beam problem. Starting from the discrete equations of this structural problem, a continualization method leads to the formulation of an Eringen's type nonlocal model with full coupling between nonlocal elasticity and nonlocal continuum damage mechanics. Indeed, the nonlocality appears in this continualized approach both in the constitutive law and in the damage loading function. A comparison of the discrete and the continuous problems for the cantilever shows the efficiency of the new micromechanics-based nonlocal continuum damage modelling for capturing scale effects. The length scale of the nonlocal continuum damage mechanics model is rigorously calibrated from the size of the cell of the discrete repetitive damage system. The new micromechanics-based nonlocal damage mechanics model is also analysed with respect to available nonlocal damage mechanics models.