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

Abstract

International audienceIt 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

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

“…Regardless of the considered method, these kinematic-based boundary conditions are always limited to load lower than β max /(1 + 1/2n) in such a case. A similar result was obtained in [13] where the local kinematic boundary conditions have shown some inconsistencies, especially close to the peak load in a lattice DDM system.…”

“…In contrast, as highlighted in Figure 5, the softening process is highly controlled by the number of cells n: the response becomes more brittle for a sufficiently large number of cells, whereas a quasi-brittle response is obtained for a structural system composed of few damage cells. Wood's paradox is found again for an infinite number of cells, with the elastic unloading beyond the peak load as observed in other lattice DDM system [13].…”

“…Equation (74) is equivalent to the implicit gradient damage model already stated by Peerlings et al [35]. A difference with the model considered in [35] is that the elasticity in the present model is affected by the nonlocal terms, as also shown by Challamel et al [13] from a damage lattice model. In absence of nonlocal effects in the effective stress law, the present model coincides with the one of Peerlings et al [35] using this nonlocal strain measure or, formally, with the model of Pijaudier-Cabot and Bazant [1] if the nonlocal strain measure is associated with some other phenomenological nonlocal kernels.…”

“…To our knowledge, very few results have been presented in the literature for the calibration of nonlocal CDM models from DDM systems, at least for axial systems. It is only recently that the calibration of a nonlocal CDM bending system has been characterized from a DDM bending lattice [13]. This paper extends the methodology to axial DDM lattices.…”

Using a penalty term to deal with spurious oscillations in second gradient finite elements

“…A non constant internal length is also introduced in 7 , 8 and 9 . Finally, the link between a discrete and a nonlocal continuum has been recently studied in 10 .…”