🔬 materials science

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

This study evaluates novel boundary conditions for representing macroscopic cracks in periodic discrete models of concrete, finding that while Tessellation boundary conditions consistently produce well-defined localization bands, Percolation-path-aligned conditions suffer from spurious multi-band formation and displacement-jump augmented circular models sometimes fail to improve upon standard periodic results.

Original authors: Jan Raisinger, Jan Eliáš

Published 2026-06-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jan Raisinger, Jan Eliáš

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict how a block of concrete will crack when you pull it apart. To do this accurately, you can't just look at the whole block; you have to zoom in and simulate the tiny grains of sand and cement inside it. This is like looking at a forest through a microscope instead of seeing the whole tree line.

However, there's a tricky problem: How do you handle the edges of your tiny microscope view?

If you just cut a square piece out of the concrete and simulate it, the edges act like invisible walls. If a crack tries to run diagonally across your square, it hits the wall and bounces back, creating a fake, messy pattern. This makes the material look tougher and more ductile (stretchy) than it really is. It's like trying to run a marathon in a small room; you keep hitting the walls and can't run in a straight line.

The authors of this paper tested four different "rules" for how these edges should behave to see which one lets the crack run naturally. Here is what they found, using simple analogies:

1. The "Standard Wall" (Standard Periodic BCs)

The Idea: Imagine your square model is one tile in an infinite floor of identical tiles. When a crack hits the right edge, it instantly reappears on the left edge.
The Problem: If the crack wants to go diagonally, it gets confused. It has to zigzag to match the grid of the tiles. This forces the crack to take a longer, winding path, making the material absorb way more energy than it should. It's like a runner forced to run in a zigzag pattern because the track is made of square tiles.
Result: This method creates fake, overly tough behavior unless the crack goes perfectly straight up, down, or sideways.

2. The "Rotating Door" (Percolation-path-aligned BCs)

The Idea: Instead of keeping the walls fixed, you rotate the whole room so the walls are parallel to the crack.
The Problem: The authors found this creates a "tug-of-war." One side of the room is tightly held (strong constraints), while the other side is loosely held (weak constraints). The crack opens up faster on the loose side, causing the material to split in two places instead of one clean line.
Result: It often leads to multiple, messy cracks and unpredictable results. It's like trying to open a door where the hinges are loose on one side and stuck on the other; the door jams or breaks unevenly.

3. The "Sliding Puzzle" (Tessellation BCs)

The Idea: This is the paper's "hero." Imagine your square model is a puzzle piece. When a crack hits the right edge, instead of bouncing back, the model "slides" over. The piece on the right shifts slightly so the crack can continue straight into the next piece without repeating the pattern.
The Result: This allows a single, clean crack to form at any angle. The energy the material absorbs depends only on how long the crack is (which is determined by the geometry of the square), not on the artificial rules of the edge.
Why it wins: It's the most reliable. It's like a runner on a treadmill that moves with them; they can run in a straight line no matter which way they face. The authors found this method works consistently for both 2D squares and 3D cubes.

4. The "Magic Circle" (Circular Models with Displacement Jumps)

The Idea: Instead of a square, use a round model. A circle has no corners, so it seems like it shouldn't matter which way the crack goes. To let the crack cross the edge, they added a "jump" rule that allows the two halves of the circle to separate.
The Problem: It's a gamble. Whether the crack forms correctly depends entirely on where the crack starts inside the circle.

Scenario A: If the crack starts right in the middle, it hits the "jump" point, separates cleanly, and works perfectly.
Scenario B: If the crack starts slightly off-center, it avoids the "jump" point, hits the edge like a normal wall, and creates a messy, fake pattern.
Result: Since you can't always predict exactly where a crack will start in a random mix of sand and cement, this method is unreliable. It's like a magic trick that only works if the magician picks the right card by chance.

The Bottom Line

The paper concludes that if you want to simulate how concrete cracks without getting fake results:

Don't use standard square edges (they force zigzag cracks).
Don't use the "rotating" method (it causes uneven splitting).
Don't rely on the "magic circle" (it's too random).
Do use the "Sliding Puzzle" (Tessellation) method. It allows the crack to flow naturally, giving you a true picture of how the material behaves, regardless of the angle you pull it.

The authors emphasize that these findings are specific to their computer models of concrete particles. They do not claim these methods solve real-world construction problems yet, but they provide a much better tool for the scientists building those computer models.

Technical Summary: On Representation of Macroscopic Crack in Periodic Fine-Scale Discrete Mechanical Models

Problem Statement
In the multiscale modeling of heterogeneous softening materials, such as concrete, the boundary conditions (BCs) applied to the fine-scale Representative Volume Element (RVE) significantly influence the initiation and pattern of strain localization. Standard Periodic Boundary Conditions (PBCs), while suitable for elastic or hardening materials, produce artificially ductile behavior and excessive energy dissipation in softening materials when the localization band inclination does not align with the periodicity directions of the model geometry (e.g., square or cubic domains). This misalignment forces the formation of multiple, non-physical crack patterns to satisfy periodicity, distorting the macroscopic response. While spherical or circular models offer orientation-independent geometry, standard PBCs still prevent localization bands from crossing their boundaries. Although novel BCs have been proposed for continuous models, their applicability to discrete mechanical models remains unverified.

Methodology
This study evaluates the applicability of three novel boundary condition approaches to a mesoscale discrete particle model of concrete (specifically, the Lattice-Discrete Particle Model or LDPM framework). The models consist of interconnected rigid particles with constitutive behavior defined at inter-particle contacts. The investigation covers two-dimensional square and circular models, as well as selected three-dimensional cube models, subjected to uniaxial tension at various loading angles.

The study compares the following boundary condition strategies:

Standard Periodic BCs: The baseline approach where boundary nodes are constrained to move relative to each other based on the macroscopic strain.
Minimal Kinematic BCs: Constraints that enforce zero average fine-scale strain but allow rigid body motion.
Percolation-path-aligned BCs (Aligned BCs): These rotate the periodicity frame to align with the evolving localization band. They enforce strong periodicity on boundary sections perpendicular to the crack and weak (integral) periodicity on sections where normals do not match.
Tessellation BCs: These shift the periodicity frame (by an offset $\Delta x$ ) to allow the localization band to continue into adjacent domains without repeating the crack pattern, effectively tessellating space with deformed models.
Embedded-crack BCs (for circular models): A modification to PBCs that introduces an unknown displacement jump at a specific "opening node pair" to allow the crack to separate the circular domain.

The models utilize both "smooth" surface geometries (with auxiliary boundary nodes) and "rough" surface geometries (where the tessellation is statistically identical to the bulk). Simulations were performed using the Open Academic Solver (OAS) with 20 realizations for 2D square models and 10 for circular and 3D cube models.

Key Results

Standard Periodic BCs: Confirmed to produce orientation-dependent softening. For loading angles other than $0, \pi/4, \pi/2$ (in 2D) or specific orthogonal directions (in 3D), the energy dissipation increases significantly (up to 2.5 times the reference) due to the formation of multiple, artificially long cracks.
Minimal Kinematic BCs: Found unsuitable for softening models. They lead to spurious strain localization at the boundaries where detachment of a few nodes is energetically favorable, resulting in unrealistically low energy dissipation and peak stresses.
Percolation-path-aligned BCs: Exhibited major shortcomings. The distinction between strongly and weakly constrained boundary sections caused uneven straining. This often led to the formation of multiple localization bands (secondary cracks) or spurious localization in the weakly constrained sections, particularly for loading angles between $\pi/8$ and $3\pi/8$ . The results showed a strong dependence on the internal heterogeneity of the model.
Tessellation BCs: Consistently yielded a single, well-defined localization band for all loading directions. The length of the band was determined solely by the model geometry and the loading angle. Consequently, the observed dependence of dissipated energy on the loading direction was attributed to geometric factors (crack length) rather than boundary condition artifacts. In 2D, the energy dissipation matched a theoretical geometric factor ( $1/h$ ) closely; in 3D, the correlation was reasonable but weaker due to discretization effects.
Embedded-crack BCs (Circular Models): Produced inconsistent results. While they allowed for a full localization band when the crack initiated at the specific "opening node pair," the outcome was dictated by the internal heterogeneity. If the crack initiated elsewhere, the model behaved similarly to one under standard PBCs, often resulting in multiple cracks and excessive energy dissipation.

Significance and Conclusions
The paper concludes that Tessellation BCs offer the most significant improvement over standard approaches for discrete models of softening materials. They provide a consistent macroscopic response where the energy dissipation is governed by the geometry of the localization band rather than artificial boundary constraints. This allows for straightforward post-processing corrections based on the calculated crack length.

In contrast, Percolation-path-aligned BCs are deemed unreliable for discrete models due to their tendency to induce multiple cracks or spurious localization. Embedded-crack BCs for circular models are limited by their reliance on the specific initiation site of the crack, making the results stochastic and dependent on the random internal structure.

The authors note that these conclusions apply specifically to discrete models with homogeneous material parameters under monotonic uniaxial tension. They acknowledge that the applicability of these methods to more complex loading scenarios or continuum models with heterogeneous material distributions remains an open question. Furthermore, the practical implementation of these novel BCs requires a robust technique to determine the crack normal direction at the onset of localization, which is not always trivial in iterative solutions.

1. The "Standard Wall" (Standard Periodic BCs)

2. The "Rotating Door" (Percolation-path-aligned BCs)

3. The "Sliding Puzzle" (Tessellation BCs)

4. The "Magic Circle" (Circular Models with Displacement Jumps)

The Bottom Line

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