Geometrically Induced Cohesion

• Geometrically induced cohesion is defined as the emergence of robust, solid-like properties from the arrangement of particles or elements without adhesive forces.
• Empirical and simulation studies in granular materials, filamentous networks, and numerical meshes show that shape interlocking and topological constraints drive resistance to deformation.
• This concept underpins applications in granular metamaterials, network science, and computational fracture models, highlighting geometry’s universal role in stabilizing systems.

Geometrically induced cohesion refers to the emergence of collective or solid-like behavior resulting solely from the geometric arrangement or structure of interacting objects—particles, network nodes, data points, or mesh elements—in the absence of any explicit attractive (adhesive) forces. Manifestations span granular materials, filament packings, network communities, numerical fracture models, and geometric data analysis. The unifying feature is that geometry or topology—via mechanical interlocking, path constraints, or comparative relations—generates effective cohesion or robustness.

1. Conceptual Framework and Definitions

Geometrically induced cohesion is the ability of a system—granular packing, network subgraph, continuum mesh, or data set—to resist separation, deformation, or fragmentation, with the origin traced to the geometrical or topological properties of its constituents.

In granular assemblies, it denotes solid-like stability arising from non-convex particle shapes that interlock or entangle, without adhesive or capillary interactions. For flexible filaments or network subgraphs, it emerges from spatial configurations favoring multiple, redundant contacts or cycles. In discretized fracture, artificial mesh-induced cohesion results from forced crack path geometries (Gravish et al., 2012, Barés et al., 1 Jan 2026, Aponte et al., 2024, Rimoli et al., 2013). In data, cohesion is a function of proximity comparisons that reflect the arrangement of points (Moore, 2023).

Distinguishing Features

• Non-adhesive origin: No energetic penalty at the contact level is necessary; mechanical geometry suffices.
• Shape and arrangement dependent: Arms, branches, or cycles are essential.
• Emergent yield strength: Finite resistance appears even at very low densities.
• Reconfiguration capacity: Stability is often due to a dynamic network of contacts able to break and reform (Barés et al., 1 Jan 2026).

2. Granular Materials: Entanglement and Interlocking

Shape-induced or entanglement-based cohesion in granular matter has been extensively characterized via experiments on non-convex particles—u-shaped staples, Platonic polypods, branched metagrains—and supported by x-ray tomography and robophysical studies (Gravish et al., 2012, Barés et al., 1 Jan 2026, Aponte et al., 2024, Treers et al., 26 Jul 2025).

Mechanisms

• Interdigitation: Concave or branched shapes permit deep penetration into neighboring volumes, forming robust clusters (Barés et al., 1 Jan 2026).
• Rotational constraint: Branches with large solid angles block sliding and twisting at the contact level.
• Frictional locking: Contacts away from the centerline maximize tangential force resistance.
• Packing/entanglement competition: Too much elongation decreases number density; too little eliminates interlocks (Gravish et al., 2012).

Quantitative Descriptors

A generic local stability indicator S is developed for polypod packings,

$S = \mu\,\frac{4\pi}{n_b} \mathcal{L}$

where μ is friction, n_b is branch count, and $\mathcal{L}$ is mean interdigitation length (Barés et al., 1 Jan 2026). S grows exponentially with a concavity parameter η and linearly with n_b and μ.

Empirical and Simulation Results

• Intermediate arm lengths (l/w ≈ 0.35–0.4 for U-particles) maximize entanglement density and collapse resistance (Gravish et al., 2012).
• High k (arm number), low t/R (slenderness), and large μ enable stable columns at low solid fractions ($\phi$ < 0.2), while spheres cannot support such structures (Aponte et al., 2024).
• Tensile strength T in u-particle columns scales as $T \sim \phi^\alpha\,\sigma_c^\beta$ (packing fraction, initial compression), with α, β ≈ 1 (Treers et al., 26 Jul 2025).

Table: Geometry vs. Stability in Granular Cohesion

Parameter	Effect on Cohesion	Regime for Solid-like Behavior
Arm number k	Increases linearly	$k > 8$ for polypods (PA12, μ ≈ 0.8)
Concavity η	Exponential increase	η ≳ 0.75 required
Arm thickness t	Reduces when thick	$t/R ≲ 0.3$ optimal
Friction μ	Linear increase	μ ≳ 0.5–0.8 for high stability

3. Geometry-Driven Cohesion in Filamentous and Network Systems

Flexible filaments exhibit a geometrically controlled cohesive interaction as a function of their interfilament angle, originating from the packing multiplicity of helical contacts (Cajamarca et al., 2014). The cohesive energy is maximized at the critical skew angle $\theta_c = \pi/4$ in the sticky-tube limit, due entirely to the increase in line contact as geometry shifts from parallel to skewed configurations. In pairwise attraction models (Lennard-Jones), parallel states are always unstable, while depletion plus electrostatic repulsion can stabilize weakly parallel meta-stable alignments. These phenomena are governed by the interplay between interaction range/strength and contour geometry.

In networks, the concept is formalized via the triangle-based cohesion metric (Friggeri et al., 2011):

$$C(S) = \frac{T_{\mathrm{in}}(S)}{\binom{|S|}{3}} \cdot \frac{T_{\mathrm{in}}(S)}{T_{\mathrm{in}}(S)+T_{\mathrm{out}}(S)}$$

where $T_{\mathrm{in}}(S)$ is the number of triangles fully inside S and $T_{\mathrm{out}}(S)$ the number of outbound triangles. Internal triangles indicate high subgraph density, while outbound triangles penalize leakage of strong ties, connecting directly to classical sociological notions (triadic closure, weak/strong tie theory). Experimental validation on social graphs finds cohesion the strongest available predictor of subjective "community-ness" (Friggeri et al., 2011).

4. Numerical and Computational Manifestations: Mesh-Induced Cohesion

In cohesive zone modeling and fracture mechanics, discretization of a continuum induces artificial cohesive effects: mesh-induced anisotropy and mesh-induced toughness. These artifacts are geometric in origin—the crack path can only propagate along mesh edges or faces, and the minimum graph distance between nodes generally exceeds the Euclidean length (Rimoli et al., 2013).

The path-deviation ratio $\eta(\theta)$,

$\eta(\theta) = \frac{L_g(\theta)}{L_e}$

quantifies the mesh-induced cohesive penalty in any direction θ. Regular grids exhibit strong anisotropy (e.g., $η=1$ at 0°, $η\approx 1.414$ at 45° for squares); isotropic K-means and conjugate-direction meshes minimize both mean and standard deviation of $\eta(\theta)$. Adoption of such advanced mesh strategies is essential to minimize artificial geometrically induced toughness in numerical fracture simulations.

5. Data Cohesion: Ordinal Geometry and Cluster Formation

The notion of cohesion in data analysis, inspired by social interaction models, is realized as a function of geometric proximity triplets rather than distance per se (Moore, 2023). For a finite metric space $(\mathcal{X}, d, p)$, the cohesion $C_{\mathcal{X}}(x, w)$ is defined as a vote across all y ∈ $\mathcal{X}$ evaluating whether x or w is closer to y considering only outlier relationships,

$$C_{\mathcal{X}}(x, w) = \sum_{y\in \mathcal{X}} \frac{1_{\{d(x,w) < \min(d(x,y), d(w,y))\}}}{U_{\mathcal{X}}(x,y)}\, p_y$$

where $U_{\mathcal{X}}(x, y)$ is the local mass in the (x,y)-vicinity.

The key geometric implications are:

• Transformation-invariance within clusters: Shrinking intra-cluster distances or expanding inter-cluster distances beyond a separation threshold leaves the cohesion structure unchanged.
• Point-like aggregates: Compact clusters act as single weighted points under cohesion.
• Local density adaptation: Cohesion "renormalizes" by local point mass without explicit bandwidth selection or parametrization.

This framework uniquely satisfies two axioms: average value equals one-half and outlier-influence proportional to mass.

6. Continuum Fracture and Phase-Field Models

Geometrically nonlinear cohesive fracture models and their phase-field approximations describe the energetics of material separation in the presence of large deformations (Conti et al., 2022, Reinoso et al., 2015). The limiting functional involves three terms: an elastic energy in the bulk, a geometric jump energy concentrated on fracture surfaces, and a Cantor part for diffuse damage,

$$F_0(u, 1) = \int_\Omega W(\nabla u(x))\,dx + \int_{J_u} \varphi([u](x), \nu_u(x))\,d\mathcal{H}^{n-1}(x) + \int_\Omega \ell(d D^c u)$$

Cohesive surface density $\varphi(z, \nu)$ depends on the local crack opening and orientation, and is constructed via cell problems that encode local geometric opening. For frame-indifferent models (e.g., $\Psi(\xi) = \mathrm{dist}^2(\xi, SO(n))$), the cohesive law is independent of orientation; in anisotropic settings, geometric orientation of the crack front modifies cohesive response.

Under finite deformation, cohesive interface elements must account for large kinematic rotations, with geometric terms in the stiffness modifying both softening behavior and apparent toughness—neglecting these terms greatly mispredicts energy release, especially in mode-I or peeling scenarios (Reinoso et al., 2015).

7. Applications and Emergent Phenomena

Geometrically induced cohesion is a central mechanism in diverse contexts:

• Robophysical and robotic manipulation: Excavation and construction methods exploiting entanglement properties of granular substrates demand explicit measurement and control of shape-induced tensile strength (Treers et al., 26 Jul 2025).
• Architected materials: Platonic polypod assemblies and optimized non-convex grains form the basis of new, reconfigurable solid-like granular metamaterials, where the geometric design sets the mechanical response (Aponte et al., 2024).
• Additive manufacturing: Geometry of triply-periodic minimal surface (TPMS) structures determines powder retention and jamming by interacting with adhesion and gravity-induced flow, affecting depowderability (Gupta et al., 25 Nov 2025).
• Network science and data: Quantification of community structure or data clusters via geometric cohesion metrics enables identification of robust groups or point-like sets, supplementing traditional spectral or density-based measures (Friggeri et al., 2011, Moore, 2023).

In all cases, the core principle is that geometry—via interlocking, spatial proximity, or combinatorial organization—can be harnessed to produce emergent cohesion, with predictive mathematical frameworks and practical design implications across material, algorithmic, and social systems.