On the emergence of a universal crack-bridging law in fiber-reinforced geopolymer composites: a variational and scale-invariant formulation

On the emergence of a universal crack-bridging law in fiber-reinforced geopolymer composites: a variational and scale-invariant formulation

Authors:

Kinga Korniejenko

Cracow University of Technology, Faculty of Materials Engineering and Physics, Kraków, Poland

Zholaman Shashpan, Lyazzat Aruova

L.N. Gumilyov Eurasian National University, Department of Technology of Industrial and Civil Engineering, Astana, Kazakhstan

Corresponding author:

Kinga Korniejenko kinga.korniejenko@pk.edu.pl

Introduction

Applications of fiber‑reinforced geopolymer composites (FRGC) span structural and infrastructure domains where fracture control and sustainability are critical. Typical use‑cases include precast beams, panels and slabs subjected to flexure and opening‑mode cracking; thin architectural elements and façade panels where low shrinkage and crack bridging enhance serviceability; repair and retrofit overlays; thermally exposed components; marine and chemically aggressive environments; and digitally manufactured (3D‑printed) elements [Abbas et al., 2022; Ranjbar & Zhang 2020; Shilar & Shilar, 2026].

Fiber-reinforced geopolymer composites integrate principles of sustainability with enhanced resistance to fracturing. The fracture dynamics of these materials are influenced by quasi-brittle crack propagation, which is stabilized through the mechanism of fiber pull-out. Notwithstanding extensive scholarly investigations, cohesive traction-separation laws are predominantly approached as empirical functions that are individually calibrated for each specific composition [Oliwa et al., 2026; Ranjbar & Zhang 2020].

This engenders a pertinent structural inquiry: Is the phenomenon of crack-bridging solely reliant on the composition, or does it demonstrate universal characteristics when articulated in suitable dimensionless parameters?

This research endeavors to elucidate that inquiry through an amalgamation of variational fracture mechanics and micromechanical analysis under certain conditions: straight polymer fibers (PP, PVA) and selected straight steel fibers; approximately constant interfacial shear; nearly isotropic fiber orientation; low‑to‑moderate fiber volume fractions; and loading scenarios governed by classical single‑fiber pull‑out. These conditions underpin the dimensionless collapse and invariance results are demonstrated later in the manuscript [Wang et al., 2021; Sadrmomtazi & Rad, 2024].

Such a model can be especially applied to:

The present scaling is physically valid under classical single‑fiber pull‑out with approximately constant interfacial shear, straight slender fibers, nearly isotropic orientation, and low‑to‑moderate fiber volume fractions; deviations (e.g., hooked/twisted fibers, evolving adhesion, strong anisotropy, high V_f, or rate‑dependent matrices) may require extensions discussed later in the paper [Mandal et al., 2020; Zheng et al., 2022; Feng et al., 2022].

Variational Formulation of Crack Bridging

Consider a cracked body Ω ⊂ R3 with displacement field u. The total potential energy is:

where:

Bridging traction is defined as:

Equilibrium follows from stationarity:

The fracture energy associated with bridging is:

Equation (4) guarantees energetic consistency.

Micromechanical Derivation

Single-Fiber Pull-Out

For a fiber of diameter df​, embedded length le​, and interfacial shear stress τint, force equilibrium gives:

Integrating:

The macroscopic traction from randomly oriented fibers:

Fiber density per crack area:

Substituting (6)–(8):

Equation (9) reveals natural scaling variables.

Dimensionless Scaling and Invariance

Define:

Dimensionless fracture energy:

Substituting (11) into (4):

If

then scale invariance holds.

Bridging Modulus

Let tp be peak traction. Define:

If dimensionless curves collapse:

The Bridging Modulus measures fiber engagement efficiency independent of fiber content.

Discussion (Extended Analytical Depth)

Energetic Interpretation

From (4) and (9):

Thus fracture energy scales linearly with fiber volume fraction and fiber diameter, explaining invariance under normalization.

Operator Perspective

Define operator:

Under scaling transformation:

The operator becomes approximately invariant.

Implications

The proposed variational–micromechanical framework provides a compact route to define traction–separation relations for fiber‑reinforced geopolymer composites (FRGC) without per‑composition empirical fitting. By normalizing crack opening and bridging traction with fiber diameter d_f, fiber volume fraction V_f, and interfacial shear strength τ_int, traction–separation curves from different mixes collapse to an almost universal dimensionless response. This enables: (i) transfer of cohesive parameters across FRGC formulations; (ii) reduced experimental campaigns for model calibration; and (iii) direct embedding of scale‑invariant cohesive laws in finite element cohesive zone models and digital‑twin pipelines. The “Bridging Modulus,” defined from the peak dimensionless traction, further offers an invariant descriptor of fiber engagement efficiency that is independent of V_f [Abbas et al., 2022; Sadrmomtazi & Rad, 2024; Samal & Blanco, 2021].

Summarizing the main benefits from model are:

The collapse of normalized curves indicates that crack-bridging behavior possesses intrinsic structural organization governed by interfacial shear transfer.

Where and Why Results May Fail

The presented model has some limitations and a boundary of validity. While the formulation is mathematically consistent, its physical validity relies on specific micromechanical assumptions used to derive the single-fiber pull-out and the crack-bridging density. The model is appropriate when these assumptions hold, and it should be extended or recalibrated when they do not.

Assumptions under which the scaling is expected to hold

Situations where the current model may be inaccurate and require extension

Practical guidance

For applications outside the baseline assumptions, we recommend augmenting the dimensionless law with: (i) an orientation tensor in the areal density, (ii) snubbing/bending factors in the single-fiber equilibrium, and (iii) an interface damage/softening variable to capture evolving int. These additions preserve the variational structure while expanding the envelope of validity.

Mini-Dialogue with the Reader

Use of Artificial Intelligence

AI tools were used exclusively for language refinement and structural editing. All mathematical derivations, theoretical constructs, and interpretations were developed by the author.

Appendix A. Extended Micromechanical Derivation

Starting from interfacial equilibrium:

Integrating:

Macroscopic traction:

Substitute x=df x^

Hence:

which confirms the dimensionless scaling relation.

List of references