Rectification of stress by fiber networks: Manifestation of non-linear screening through self-organized buckling

TL;DR

The paper addresses how force transmission in malleable fiber networks under active forcing yields nontrivial large-scale responses. They simulate a minimal nonlinear triangular lattice with mid-point buckling subjected to a force dipole of strength $2|F|$ and analyze the far-field response via the stress dipole $\hat{\mathscr{D}}$ and total stress $\hat{\mathscr{S}}$, revealing three distinct mechanical regimes (L,N,R). A key finding is that buckling organizes into domain patterns that map onto Kagome soft modes, driving a strong, nonlinear renormalization of the Poisson ratio $\nu(|F|)$ from roughly $1/3$ in the linear regime to near $1$ in the rectified regime, consistent with a nonlinear mechanical screening described by Vector Charge Theory (VCT). This screening mechanism explains how the network can completely mask, or even invert, the imposed force at large distances, and provides a framework for understanding active remodeling in cytoskeletal-like networks.

Abstract

Force transmission at large length scales is crucial for such biological functions as cell motility and morphogenesis. The networks that transmit these forces are malleable, patterned by active forces generated at the microscale by biological motors. In this paper we explore a simple model of a non-linear fiber network which has only two modes of deformation, but exhibits diverse mechanical phases with distinct large-scale response, tuned by the strength of a microscopic force dipole. We demonstrate, via numerical simulations, that the network is remodeled by organized patterns of buckling, which lead to a renormalization of the Poisson ratio. Finally, we show that the emergent behavior at large length scales can be ascribed to "mechanical screening" of the force dipole, analogous to dielectric screening of charges in electrostatics.

Figures (7)

• Figure 1: (A) (Top Row) $\mathscr{S}$, and $\mathscr{D}$, visualized using the Voigt notation (see text). A cube represents the three components:$\lbrace \mathscr{D}_{xx},\mathscr{D}_{yy},\mathscr{D}_{xy}\rbrace$), and the mid-plane represents:$\lbrace \mathscr{S}_{xx},\mathscr{S}_{yy},\mathscr{S}_{xy}=0\rbrace$). Arrows pointing outward (inward) represent positive (negative) stress. (Second Row) Annular pressure (Blue curve:) ($Tr(\mathscr{D}$)), and Poisson ratio (Red curve), $\nu$, exhibit three distinct mechanical phases: - linear (L), non-linear (N) and rectified (R) . (B) Buckling pattern is shown at three consecutive values of applied force ($F=14, 15, 16$). Ordered domain patterns signal the discontinous change at $|F|=15$-there are no domains in the network preceding or following these phase changes. Similar trend is seen at $|F|=25$ (not shown).
• Figure 2: A schematic of the simulated network of non-linear springs. Each spring has two junctions (orange circles) where it is connected to other springs and a midpoint (blue circles) where it can buckle. Each spring can deform either by stretching and contracting or by buckling at its midpoint. Large black arrows indicate the applied force dipole. The network contains 2900 junctions (nodes) and 8700 springs.
• Figure 3: (A) Schematic of a triangular network drawn in green. The circles denote junctions of the springs (orange) and midpoint of each spring (cyan). Connecting each midpoint with its four nearest-midpoint-neighbors gives rise to the Kagome lattice shown in teal. (B) under deformation, a pair of adjacent up and down triangles may rotate in two opposite directions forming a twisted unit:colored blue or red according to their rotation direction. Non-twisted units are not colored. (C) Spatial distribution of twisted units at three consecutive force values $F=25$ (left), $F=26$ (center), and $F=27$ (right). The center panel depicts the organized pattern at the transition from N to R.
• Figure 4: Top Panel: Spatial pattern of $\sigma_{yy}$ as we move from phase $L \rightarrow$ phase $N$ (left) and phase $N \rightarrow$ phase $R$ (right). Bottom Panel: Corresponding patterns of $\sigma_{\theta \theta}.$ Fig \ref{['first_image']}B and fig \ref{['kagome']}C show the changes in the kagome patterns at these force values.
• Figure 5: $\sigma_{\theta \theta}$ calculated from simulation at (A) F=10 and (B) F=30. Predictions of $\sigma_{\theta \theta}$ from VCTG where the value of $\nu$ is obtained by fitting the data to theory: (C) $\nu$ = 0.34 and (D) $\nu$=1.0 .