Asymptotic analysis of Lennard-Jones systems beyond the nearest-neighbour setting: A one-dimensional prototypical case

Abstract

We consider a one-dimensional system of Lennard-Jones nearest-and next-to-nearest-neighbour interactions. It is known that if a monotone parameterization is assumed then the limit of such a system can be interpreted as a Griffith fracture energy with an increasing condition on the jumps. In view of possible applications to a higher-dimensional setting, where an analogous parameterization does not always seem reasonable, we remove the monotonicity assumption and describe the limit as a Griffith fracture energy where the increasing condition on the jumps is removed and is substituted by an energy that accounts for changes in orientation ('creases'). In addition, fracture may be generated by 'macroscopic' or 'microscopic' cracks.

On continuum limits of heterogeneous discrete systems modelling cracks in composite materials

“…In the last two decades progress was made in order to derive such macroscopic models from underlying microscopic (discrete) models, see e.g. [7,8,10,11,12,14,15,25,26,27]. This research started with the early contribution by Truskinovsky [29], who emphasizes that macroscopic models for fracture mechanics should contain information on the microscopic length scale of the system.…”

Asymptotic analysis of microscopic impenetrability constraints for atomistic systems

“…It has been partially addressed by Theil [23], and should be included in further investigations on the subject. The one-dimensional analysis by Braides and Solci [11] suggests that considering longer range of interactions (e.g., next-to-nearest neighbours) could be used in the place of the determinant constraint in order to eliminate non-local effects for non-opening cracks, at the expense of introducing internal and external boundary layers.…”

Elastoplastic analysis of pressure‐sensitive materials by an effective three‐dimensional mixed finite element

“…Another application field concerns the research trend on generalized continua and their applications such as, for example, reported in for gradient models; in for second gradient models; in for multi‐constituent materials, in for some hints on non‐local problems, in for pantographic structure and in for an asymptotic approach.…”

“…7 Meshes m1, m2, and m3 used in the analysis of the corner wall test and the relative displacement fields at the end of the loading phases. in [28] for some hints on non-local problems, in [2,3,34,35,38,52] for pantographic structure and in [18] for an asymptotic approach. Also the analysis of higher-order continua could make good use by effective procedures, see for example the review paper [33] which considers the dynamic case, [17] for graded materials and [4] for micropolar continua.…”

“…In the iterative scheme (18), K is the tangent stiffness matrix, whose evaluation is described below, and d = K −1 p. The global vector of the structural response is obtained by assembling the element contributions defined by:…”