Rigidity of random networks of stiff fibers in the low-density limit

RIGID GRAPH COMPRESSION: MOTIF-BASED RIGIDITY ANALYSIS FOR DISORDERED FIBER NETWORKS

Using particle-scale models to accurately describe property enhancements and phase transitions in macroscopic behavior is a major engineering challenge in composite materials science. To address some of these challenges, we use the graph theoretic property of rigidity to model mechanical reinforcement in composites with stiff rod-like particles. We develop an efficient algorithmic approach called rigid graph compression (RGC) to describe the transition from floppy to rigid in disordered fiber networks ("rod-hinge systems"), which form the reinforcing phase in many composite systems. To establish RGC on a firm theoretical foundation, we adapt rigidity matroid theory to identify primitive topological network motifs that serve as rules for composing interacting rigid particles into larger rigid components. This approach is computationally efficient and stable, because RGC requires only topological information about rod interactions (encoded by a sparse unweighted network) rather than geometrical details such as rod locations or pairwise distances (as required in rigidity matroid theory). We conduct numerical experiments on simulated two-dimensional rod-hinge systems to demonstrate that RGC closely approximates the rigidity percolation threshold for such systems, through comparison with the pebble game algorithm (which is exact in two dimensions). Importantly, whereas the pebble game is derived from Laman's condition and is only valid in two dimensions, the RGC approach naturally extends to higher dimensions.

Effect of crosslink torsional stiffness on elastic behavior of semiflexible polymer networks

Networks of semiflexible filaments are building blocks of different biological and structural materials such as cytoskeleton and extracellular matrix. The mechanical response of these systems when subjected to an applied strain at zero temperature is often investigated numerically using networks composed of filaments, which are either rigidly welded or pinned together at their crosslinks. In the latter, filaments during deformation are free to rotate about their crosslinks while the relative angles between filaments remain constant in the former. The behavior of crosslinks in actual semiflexible networks is different than these idealized models and there exists only partial constraint on torques at crosslinks. The present work develops a numerical model in which two intersecting filaments are connected to each other by torsional springs with arbitrary stiffness. We show that fiber networks composed of rigid and freely rotating crosslinks are the limiting case of the present model. Furthermore, we characterize the effects of stiffness of crosslinks on effective Young's modulus of semiflexible networks as a function of filament flexibility and crosslink density. The effective Young's modulus is determined as a function of the mechanical properties of crosslinks and is found to vanish for networks composed of very weak torsional springs. Independent of the stiffness of crosslinks, it is found that the effective Young's modulus is a function of fiber flexibility and crosslink density. In low density networks, filaments primarily bend and the effective Young's modulus is much lower than the affine estimate. With increasing filament bending stiffness and/or crosslink density, the mechanical behavior of the networks becomes more affine and the stretching of filaments depicts itself as the dominant mode of deformation. The torsional stiffness of the crosslinks significantly affects the effective Young's modulus of the semiflexible random fiber networks.

Simulation of the mechanical behavior of random fiber networks with different microstructure

Filamentous protein networks are broadly encountered in biological systems such as cytoskeleton and extracellular matrix. Many numerical studies have been conducted to better understand the fundamental mechanisms behind the striking mechanical properties of these networks. In most of these previous numerical models, the Mikado algorithm has been used to represent the network microstructure. Here, a different algorithm is used to create random fiber networks in order to investigate possible roles of architecture on the elastic behavior of filamentous networks. In particular, random fibrous structures are generated from the growth of individual fibers from random nucleation points. We use computer simulations to determine the mechanical behavior of these networks in terms of their model parameters. The findings are presented and discussed along with the response of Mikado fiber networks. We demonstrate that these alternative networks and Mikado networks show a qualitatively similar response. Nevertheless, the overall elasticity of Mikado networks is stiffer compared to that of the networks created using the alternative algorithm. We describe the effective elasticity of both network types as a function of their line density and of the material properties of the filaments. We also characterize the ratio of bending and axial energy and discuss the behavior of these networks in terms of their fiber density distribution and coordination number.

Stiffness Percolation in Stochastically Fragmented Continua

We study the mechanical behavior of three-dimensional, randomly microcracked continua for crack densities up to and above the transport percolation threshold. We show the existence of a fully fragmented material state in which stiffness is preserved due to topological interlocking of fragments. In this regime, the mechanical behavior is controlled by the contacts between fragments and becomes nonlinear. The upper limit of crack densities for which this behavior is observed, the stiffness percolation threshold, is identified. The variation of the effective material stiffness for crack densities ranging from 0 to the stiffness percolation threshold is reported.

Exceptional stiffening in composite fiber networks

We study the small strain elastic behavior of composite athermal fiber networks constructed by adding stiffer fibers to a cross-linked base network. We observe that if the base network is in the affine deformation regime, the composite behaves similar to a fiber-reinforced continuum. When the base network is in the nonaffine deformation regime, the stiffness of the composite increases by orders of magnitude upon the addition of a small fraction of stiff fibers. The increase is not gradual, but rather occurs in two steps. Of these, one is associated with the stiffness percolation of the network of added fibers. The other, which occurs at very small fractions of stiff fibers, is due to the percolation of perturbation zones, or "interphases," induced in the base network by the stiff fibers, regions where the energy is stored mostly in the axial deformation mode. Their size controls the stiffening transition and depends on base network parameters and the length of added fibers. It is also shown that the perturbation field introduced in the base network by the presence of a stiff fiber is much longer ranged than in the case when the fiber is tied to a continuum of same modulus with the base network.

Wrinkling of random and regular semiflexible polymer networks

We investigate wrinkling of two-dimensional random and triangular semiflexible polymer networks under shear. Both types of semiflexible networks exhibit wrinkling above a small critical shear angle, which scales with an exponent of the bending modulus between 1.9 and 2.0. Random networks exhibit hysteresis at the wrinkling threshold. Wrinkling lowers the total elastic energy by up to 20% and strongly affects the elastic properties of all semiflexible networks such as the crossover between bending and stretching dominated behavior. In random networks, we also find evidence for metastable wrinkled configurations. While the disordered microstructure of random networks affects the scaling behavior of wrinkle amplitudes, it has little effect on wrinkle wavelength. Therefore, wrinkles represent a robust, microstructure-independent assay of shear strain or elastic properties.

Effective-medium theory of a filamentous triangular lattice

We present an effective-medium theory that includes bending as well as stretching forces, and we use it to calculate the mechanical response of a diluted filamentous triangular lattice. In this lattice, bonds are central-force springs, and there are bending forces between neighboring bonds on the same filament. We investigate the diluted lattice in which each bond is present with a probability p. We find a rigidity threshold p(b) which has the same value for all positive bending rigidity and a crossover characterizing bending, stretching, and bend-stretch coupled elastic regimes controlled by the central-force rigidity percolation point at p(CF)=/~2/3 of the lattice when fiber bending rigidity vanishes.

Elasticity of a filamentous kagome lattice

The diluted kagome lattice, in which bonds are randomly removed with probability 1-p, consists of straight lines that intersect at points with a maximum coordination number of 4. If lines are treated as semiflexible polymers and crossing points are treated as cross-links, this lattice provides a simple model for two-dimensional filamentous networks. Lattice-based effective-medium theories and numerical simulations for filaments modeled as elastic rods, with stretching modulus μ and bending modulus κ, are used to study the elasticity of this lattice as functions of p and κ. At p=1, elastic response is purely affine, and the macroscopic elastic modulus G is independent of κ. When κ=0, the lattice undergoes a first-order rigidity-percolation transition at p=1. When κ>0, G decreases continuously as p decreases below one, reaching zero at a continuous rigidity-percolation transition at p=p(b)≈0.605 that is the same for all nonzero values of κ. The effective-medium theories predict scaling forms for G, which exhibit crossover from bending-dominated response at small κ/μ to stretching-dominated response at large κ/μ near both p=1 and p(b), that match simulations with no adjustable parameters near p=1. The affine response as p→1 is identified with the approach to a state with sample-crossing straight filaments treated as elastic rods.

Anisotropic linear elastic properties of fractal-like composites

In this work, the anisotropic linear elastic properties of two-phase composite materials, made up of square inclusions embedded in a matrix, are investigated. The inclusions present a fractal hierarchical distribution and are supposed to have the same Poisson's ratio as the matrix but a different Young's modulus. The effective elastic moduli of the medium are computed at each fractal iteration by coupling a position-space renormalization-group technique with a finite element analysis. The study allows to obtain and generalize some fundamental properties of fractal composite materials.

Energy distribution in disordered elastic networks

Disordered networks are found in many natural and artificial materials, from gels or cytoskeletal structures to metallic foams or bones. Here, the energy distribution in this type of networks is modeled, taking into account the orientation of the struts. A correlation between the orientation and the energy per unit volume is found and described as a function of the connectivity in the network and the relative bending stiffness of the struts. If one or both parameters have relatively large values, the struts aligned in the loading direction present the highest values of energy. On the contrary, if these have relatively small values, the highest values of energy can be reached in the struts oriented transversally. This result allows explaining in a simple way remodeling processes in biological materials, for example, the remodeling of trabecular bone and the reorganization in the cytoskeleton. Additionally, the correlation between the orientation, the affinity, and the bending-stretching ratio in the network is discussed.

Hierarchical models of rigidity percolation

We introduce models of generic rigidity percolation in two dimensions on hierarchical networks and solve them exactly by means of a renormalization transformation. We then study how the possibility for the network to self organize in order to avoid stressed bonds may change the phase diagram. In contrast to what happens on random graphs and in some recent numerical studies at zero temperature, we do not find a true intermediate phase separating the usual rigid and floppy ones.

Wrinkling and strain softening in single-wall carbon nanotube membranes

The nonlinear elasticity of thin supported membranes assembled from length purified single-wall carbon nanotubes is analyzed through the wrinkling instability that develops under uniaxial compression. In contrast with thin polymer films, pristine nanotube membranes exhibit strong softening under finite strain associated with bond slip and network fracture. We model the response as a shift in percolation threshold generated by strain-induced nanotube alignment in accordance with theoretical predictions.

Algorithms for three-dimensional rigidity analysis and a first-order percolation transition

A fast computer algorithm, the pebble game, has been used successfully to analyze the rigidity of two-dimensional (2D) elastic networks, as well as of a special class of 3D networks, the bond-bending networks, and enabled significant progress in studies of rigidity percolation on such networks. Application of the pebble game approach to general 3D networks has been hindered by the fact that the underlying mathematical theory is, strictly speaking, invalid in this case. We construct an approximate pebble game algorithm for general 3D networks, as well as a slower but exact algorithm, the relaxation algorithm, that we use for testing the new pebble game. Based on the results of these tests and additional considerations, we argue that in the particular case of randomly diluted central-force networks on bcc and fcc lattices, the pebble game is essentially exact. Using the pebble game, we observe an extremely sharp jump in the largest rigid cluster size in bond-diluted central-force networks in 3D, with the percolating cluster appearing and taking up most of the network after a single bond addition. This strongly suggests a first-order rigidity percolation transition, which is in contrast to the second-order transitions found previously for the 2D central-force and 3D bond-bending networks. While a first order rigidity transition has been observed previously for Bethe lattices and networks with "chemical order," here it is in a regular randomly diluted network. In the case of site dilution, the transition is also first order for bcc lattices, but results for fcc lattices suggest a second-order transition. Even in bond-diluted lattices, while the transition appears massively first order in the order parameter (the percolating cluster size), it is continuous in the elastic moduli. This, and the apparent nonuniversality, make this phase transition highly unusual.

Force distributions and force chains in random stiff fiber networks

We study the elasticity of random stiff fiber networks. The elastic response of the fibers is characterized by a central force stretching stiffness as well as a bending stiffness that acts transverse to the fiber contour. Previous studies have shown that this model displays an anomalous elastic regime where the stretching mode is fully frozen out and the elastic energy is completely dominated by the bending mode. We demonstrate by simulations and scaling arguments that, in contrast to the bending dominated elastic energy, the equally important elastic forces are to a large extent stretching dominated. By characterizing these forces on microscopic, mesoscopic and macroscopic scales we find two mechanisms of how forces are transmitted in the network. While forces smaller than a threshold Fc are effectively balanced by a homogeneous background medium, forces larger than Fc are found to be heterogeneously distributed throughout the sample, giving rise to highly localized force chains known from granular media.

Nonaffine rubber elasticity for stiff polymer networks

We present a theory for the elasticity of cross-linked stiff polymer networks. Stiff polymers, unlike their flexible counterparts, are highly anisotropic elastic objects. Similar to mechanical beams, stiff polymers easily deform in bending, while they are much stiffer with respect to tensile forces ("stretching"). Unlike in previous approaches, where network elasticity is derived from the stretching mode, our theory properly accounts for the soft bending response. A self-consistent effective medium approach is used to calculate the macroscopic elastic moduli starting from a microscopic characterization of the deformation field in terms of "floppy modes"-low-energy bending excitations that retain a high degree of nonaffinity. The length scale characterizing the emergent nonaffinity is given by the "fiber length" lf, defined as the scale over which the polymers remain straight. The calculated scaling properties for the shear modulus are in excellent agreement with the results of recent simulations obtained in two-dimensional model networks. Furthermore, our theory can be applied to rationalize bulk rheological data in reconstituted actin networks.

Effective medium theory of semiflexible filamentous networks

We develop an effective medium approach to the mechanics of disordered, semiflexible polymer networks and study the response of such networks to uniform and nonuniform strain. We identify distinct elastic regimes in which the contributions of either filament bending or stretching to the macroscopic modulus vanish. We also show that our effective medium theory predicts a crossover between affine and nonaffine strain, consistent with both prior numerical studies and scaling theory.

Role of architecture in the elastic response of semiflexible polymer and fiber networks

We study the elasticity of cross-linked networks of thermally fluctuating stiff polymers. As compared to their purely mechanical counterparts, it is shown that these thermal networks have a qualitatively different elastic response. By accounting for the entropic origin of the single-polymer elasticity, the networks acquire a strong susceptibility to polydispersity and structural randomness that is completely absent in athermal models. In extensive numerical studies we systematically vary the architecture of the networks and identify a wealth of phenomena that clearly show the strong dependence of the emergent macroscopic moduli on the underlying mesoscopic network structure. In particular, we highlight the importance of the polymer length, which to a large extent controls the elastic response of the network, surprisingly, even in parameter regions where it does not enter the macroscopic moduli explicitly. Understanding these subtle effects is only possible by going beyond the conventional approach that considers the response of typical polymer segments only. Instead, we propose to describe the elasticity in terms of a typical polymer filament and the spatial distribution of cross-links along its backbone. We provide theoretical scaling arguments to relate the observed macroscopic elasticity to the physical mechanisms on the microscopic and mesoscopic scales.

Floppy modes and nonaffine deformations in random fiber networks

We study the elasticity of random fiber networks. Starting from a microscopic picture of the nonaffine deformation fields, we calculate the macroscopic elastic moduli both in a scaling theory and a self-consistent effective medium theory. By relating nonaffinity to the low-energy excitations of the network ("floppy modes"), we achieve a detailed characterization of the nonaffine deformations present in fibrous networks.

Stiff polymers, foams, and fiber networks

We study the elasticity of fibrous materials composed of generalized stiff polymers. It is shown that, in contrast to cellular foam-like structures, affine strain fields are generically unstable. Instead, a subtle interplay between the architecture of the network and the elastic properties of its building blocks leads to intriguing mechanical properties with intermediate asymptotic scaling regimes. We present exhaustive numerical studies based on a finite element method complemented by scaling arguments.

Mean-field description of jamming in noncohesive frictionless particulate systems

A theory for kinetic arrest in isotropic systems of repulsive, radially interacting particles is presented that predicts exponents for the scaling of various macroscopic quantities near the rigidity transition that are in agreement with simulations, including the nontrivial shear exponent. Both statics and dynamics are treated in a simplified, one-particle level description and coupled via the assumption that kinetic arrest occurs on the boundary between mechanically stable and unstable regions of the static parameter diagram. This suggests that the arrested states observed in simulations are at (or near) an elastic buckling transition. Some additional numerical evidence to confirm the scaling of microscopic quantities is also provided.

Distinct regimes of elastic response and deformation modes of cross-linked cytoskeletal and semiflexible polymer networks

Semiflexible polymers such as filamentous actin (F-actin) play a vital role in the mechanical behavior of cells, yet the basic properties of cross-linked F-actin networks remain poorly understood. To address this issue, we have performed numerical studies of the linear response of homogeneous and isotropic two-dimensional networks subject to an applied strain at zero temperature. The elastic moduli are found to vanish for network densities at a rigidity percolation threshold. For higher densities, two regimes are observed: one in which the deformation is predominately affine and the filaments stretch and compress; and a second in which bending modes dominate. We identify a dimensionless scalar quantity, being a combination of the material length scales, that specifies to which regime a given network belongs. A scaling argument is presented that approximately agrees with this crossover variable. By a direct geometric measure, we also confirm that the degree of affinity under strain correlates with the distinct elastic regimes. We discuss the implications of our findings and suggest possible directions for future investigations.

Deformation of cross-linked semiflexible polymer networks

Networks of filamentous proteins play a crucial role in cell mechanics. These cytoskeletal networks, together with various cross-linking and other associated proteins largely determine the (visco)elastic response of cells. In this Letter we study a model system of cross-linked, stiff filaments in order to explore the connection between the microstructure under strain and the macroscopic response of cytoskeletal networks. We find two distinct regimes as a function primarily of cross-link density and filament rigidity: one characterized by affine deformation and one by nonaffine deformation. We characterize the crossover between these two.

Elasticity of stiff polymer networks

We study the elasticity of a two-dimensional random network of rigid rods ("Mikado model"). The essential features incorporated into the model are the anisotropic elasticity of the rods and the random geometry of the network. We show that there are three distinct scaling regimes, characterized by two distinct length scales on the elastic backbone. In addition to a critical rigidity percolation region and a homogeneously elastic regime we find a novel intermediate scaling regime, where the elasticity is dominated by bending deformations.

Nonuniversality of elastic exponents in random bond-bending networks

We numerically investigate the rigidity percolation transition in two-dimensional flexible, random rod networks with freely rotating cross links. Near the transition, networks are dominated by bending modes and the elastic modulii vanish with an exponent f=3.0+/-0.2, in contrast with central force percolation which shares the same geometric exponents. This indicates that universality for geometric quantities does not imply universality for elastic ones. The implications of this result for actin-fiber networks is discussed.

Dynamic rigidity transition

An inflated closed loop (or membrane) is used to demonstrate a dynamic rigidity transition that occurs when impact energy is added to the loop in static equilibrium at zero temperature. The only relevant parameter in this transition is the ratio of the energy needed to collapse the loop and the impact energy. When this ratio is below a threshold value close to unity, the loop collapses into a high-entropy floppy state, and it does not return to the rigid state unless the impact energy can escape. The internal oscillations are in the floppy state dominated by 1/f(2) noise. When the ratio is above the threshold, the loop does not collapse, and the internal oscillations resulting from the impact are dominated by the eigenfrequencies of the stretched membrane. In this state, the loop can bounce for a long time. It is still an open question whether bouncing will eventually vanish or whether a stationary bouncing state will be reached. The dynamic transition between the floppy and the rigid state is discontinuous.