Article Mathematics and Mechanics of Solids 1–23 Ó The Author(s) 2019 Article reuse guidelines: sagepub.com/journals-permissions DOI: 10.1177/1081286519881668 journals.sagepub.com/home/mms Nonlocality of one-dimensional bilinear hardening–softening elastoplastic axial lattices Vincent Picandet and Noe¨l Challamel Universite´ de Bretagne-Sud, Institut de Recherche Dupuy de Loˆme, UMR CNRS Lorient, France Received 30 January 2019; accepted 20 September 2019 Abstract The static behaviour of an elastoplastic axial lattice is studied in this paper through both discrete and nonlocal continuum analyses. The elastoplastic lattice system is composed of piecewise linear hardening–softening elastoplastic springs con- nected between each other via nodes, loaded by concentrated tension forces. This inelastic lattice evolution problem is ruled by some difference equations, which are shown to be equivalent to the finite difference formulation of a continuous elastoplastic bar problem under distributed tension load. Exact solutions of this inelastic discrete problem are obtained from the resolution of this piecewise linear difference equations system. Localization of plastic strain in the first elasto- plastic spring, connected to the fixed end, is observed in the softening range. A continuous nonlocal elastoplastic theory is then built from the lattice difference equations using a continualization process, based on a rational asymptotic expan- sion of the associated pseudo-differential operators. The continualized lattice-based model is equivalent to a distributed nonlocal continuous elastoplastic theory coupled to a cohesive elastoplastic model, which is shown to capture efficiently the scale effects of the reference axial lattice. The hardening–softening localization process of the nonlocal elastoplastic continuous model strongly depends on the lattice spacing, which controls the size of the nonlocal length scales. An analogy with the one-dimensional lattice system in bending is also shown. The considered microstructured elasto- plastic beam is a Hencky bar-chain connected by elastoplastic rotational springs. It is shown that the length scale calibra- tion of the nonlocal model strongly depends on the degree of the difference equations of each lattice model (namely axial or bending lattice). These preliminary results valid for one-dimensional systems allow possible future developments of new nonlocal elastoplastic models, including two- or even three-dimensional elastoplastic interactions. Keywords Microstructures, plastic collapse, constitutive behaviour, elastic-plastic material, analytic functions, finite differences, non- local mechanics 1. Introduction Establishing a rigorous foundation of continuum field theories from atomistic interactions is an old fun- damental topic in physics and mechanics, which probably dates, in the case of linear elasticity, from the 19th century, with the pioneering works of Navier [1], Cauchy [2,3] and Piola [4,5] devoted to molecular and continuum linear elasticity. This problem is mathematically related to the asymptotic justification of partial differential equations valid for the equivalent continuous medium from difference equations, which govern the discrete medium (or the atomistic one). Much effort has been made during the 20th Corresponding author: Vincent Picandet, Universite´ Bretagne-Sud, Institut de Recherche Dupuy de Luˆme, UMR CNRS 6027, Lorient, 56100 France Email: vincent.picandet@univ-ubs.fr