On representation of macroscopic crack in periodic fine-scale discrete mechanical models

Abstract

In multiscale modeling of heterogeneous softening materials, boundary conditions (BC) in the fine-scale model strongly influence the strain localization pattern and the macroscopic response. For rectilinear models (e.g., squares or cubes), standard Periodic BCs produce artificially ductile behavior with excessive energy dissipation when the localization band inclination does not match the periodicity directions. Recently proposed Tessellation and Percolation-path-aligned BCs promise to address this by adapting the periodicity frame to align with the 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 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 the selected approaches extended to three-dimensional cube models. Results show that Percolation-path-aligned BCs exhibit major shortcomings: they can lead to multiple localization bands due to uneven straining of the two boundary sections and their weakly constrained section can be prone to spurious strain localization. In contrast, Tessellation BCs consistently yield a well-defined localization band, whose length is determined solely by the model geometry, making it straightforward to account for in post-processing. Periodic boundary conditions augmented with a displacement jump applied to a circular model sometimes incorrect produce crack patterns similar to those under the standard Periodic BCs.