A percolation-based theoretical model reveals the structural origin of strain-stiffening in semiflexible fibrous networks

Abstract

Semiflexible fibrous networks are extensively found in biological tissues and engineered materials, exhibiting distinct mechanical properties situated between purely flexible and rigid networks. Understanding the underlying mechanisms governing their nonlinear strain-stiffening behavior remains challenging due to limitations in current theoretical frameworks, which predominantly focus on fiber stiffness, crosslinking density, and fiber interactions, without fully addressing changes in the network topology during deformation. Here, we propose a percolation-based theoretical model to elucidate the mechanical response and strain-stiffening behavior of semiflexible fibrous networks under tensile loading. By explicitly defining percolation parameters such as node connectivity, fiber connection probability, percolation threshold, and the rigidity percolation giant component (RPGC), we quantitatively correlate the microscopic fiber rigidity transitions with macroscopic network mechanics. Our molecular dynamics simulations combined with graph theory analysis demonstrate the central role of RPGC formation in the transition from non-affine to affine deformation regimes. Numerical results confirm that our percolation model captures the nonlinear stress-strain trend observed in simulation, particularly the onset of strain-stiffening. Additionally, we identify a three-stage deformation behavior: initially non-affine, transitioning to entropy-driven affine deformation, and ultimately dominated by enthalpic affine deformation, which provides detailed insights into the microstructural evolution of fibrous networks under strain. This percolation-based framework offers a comprehensive mechanistic understanding of semiflexible fiber networks, which may inform the rational design and optimization of biomimetic materials and engineered network structures.