On Representation of Macroscopic Crack in Periodic Fine‐Scale Discrete Mechanical Models

Abstract

ABSTRACT The evolution of strain localization and the corresponding macroscopic response obtained in multiscale modeling of softening heterogeneous materials strongly depend on the boundary conditions imposed on the fine‐scale model. For rectilinear models (e.g., squares or cubes), employing standard Periodic BCs leads to overestimation of energy dissipation and artificially ductile behavior, whenever the localization band orientation deviates from the periodic directions. Recently proposed Tessellation and Percolation‐path‐aligned BCs promise to address this by aligning the periodicity frame with evolving localization bands. Alternatively, spherical/circular models provide an orientation independent response by design. Unfortunately, the standard Periodic BCs do not allow development of proper localization band crossing the spherical model's boundaries. A recently proposed modification addresses this by adding a displacement jump to the spherical periodic BCs. This study evaluates the applicability of these novel BCs to a mesoscale discrete particle model of concrete. Two‐dimensional square and circular models under uniaxial tension with different loading directions are analyzed, with selected approaches extended to three‐dimensional cube models. Percolation‐path‐aligned BCs were found to suffer from several drawbacks. Uneven straining of the weakly and strongly constrained sections can trigger localization into multiple bands, and the weakly constrained section is susceptible to spurious strain localization. Tessellation BCs, on the other hand, consistently allow the development of a single localization band of predictable length, determined only by the model geometry, enabling a straightforward correction of its influence during post‐processing. As a result, variations in dissipated energy with loading direction can be attributed solely to model geometry rather than to artifacts introduced by the boundary conditions. Periodic boundary conditions augmented with a displacement jump applied to a circular model yield inconsistent results. Some cases exhibit correct localization band development, but some produce crack patterns similar to those under the standard Periodic BCs. Which of these occurs is dictated by the internal heterogeneity of the model, which effectively determines whether strain localization is initiated in the boundary segment enhanced by the displacement jump.

The authors' abstract, as published at the source. International Journal for Numerical Methods in Engineering, 2026 · DOI ↗