Nonuniversality of elastic exponents in random bond-bending networks

Schramm-Loewner Evolution in 2D Rigidity Percolation

Amorphous solids may resist external deformation such as shear or compression, while they do not present any long-range translational order or symmetry at the microscopic scale. Yet, it was recently discovered that, when they become rigid, such materials acquire a high degree of symmetry hidden in the disorder fluctuations: their microstructure becomes statistically conformally invariant. In this Letter, we exploit this finding to characterize the universality class of central-force rigidity percolation (RP), using Schramm-Loewner evolution (SLE) theory. We provide numerical evidence that the interfaces of the mechanically stable structures (rigid clusters), at the rigidification transition, are consistently described by SLE_{κ}, showing that this powerful framework can be applied to a mechanical percolation transition. Using well-known relations between different SLE observables and the universal diffusion constant κ, we obtain the estimation κ∼2.9 for central-force RP. This value is consistent, through relations coming from conformal field theory, with previously measured values for the clusters' fractal dimension D_{f} and correlation length exponent ν, providing new, nontrivial relations between critical exponents for RP. These findings open the way to a fine understanding of the microstructure in other important classes of rigidity and jamming transitions.

Evidences of Conformal Invariance in 2D Rigidity Percolation

The rigidity transition occurs when, as the density of microscopic components is increased, a disordered medium becomes able to transmit and ensure macroscopic mechanical stability, owing to the appearance of a space-spanning rigid connected component, or cluster. As a second-order phase transition it exhibits a scale invariant critical point, at which the rigid clusters are random fractals. We show, using numerical analysis, that these clusters are also conformally invariant, and we use conformal field theory to predict the form of universal finite-size effects. Furthermore, although connectivity and rigidity percolation are usually thought to be of fundamentally different natures, we provide evidence of unexpected similarities between the statistical properties of their random clusters at criticality. Our work opens a new research avenue through the application of the powerful 2D conformal field theory tools to understand the critical behavior of a wide range of physical and biological materials exhibiting such a mechanical transition.

Structural origins of cartilage shear mechanics

Articular cartilage is a remarkable material able to sustain millions of loading cycles over decades of use outperforming any synthetic substitute. Crucially, how extracellular matrix constituents alter mechanical performance, particularly in shear, remains poorly understood. Here, we present experiments and theory in support of a rigidity percolation framework that quantitatively describes the structural origins of cartilage's shear properties and how they arise from the mechanical interdependence of the collagen and aggrecan networks making up its extracellular matrix. This framework explains that near the cartilage surface, where the collagen network is sparse and close to the rigidity threshold, slight changes in either collagen or aggrecan concentrations, common in early stages of cartilage disease, create a marked weakening in modulus that can lead to tissue collapse. More broadly, this framework provides a map for understanding how changes in composition throughout the tissue alter its shear properties and ultimate in vivo function.

Nano-CT scans in the optimisation of purposeful experimental procedures: A study on metallic fibre networks

Motive Metallic fibre networks and their mechanical behaviour are only insufficiently understood. In this particular field of research, the use of nano-CT scans offers advanced opportunities for the optimised planning of experimental work and component design. Several novel applications will benefit from this research; in particular, tissue engineering applications where a controlled and reproducible mechanical stimulus on cells is required can make use of these components. MethodFor the present study, the geometry of metallic fibre network samples is measured and digitalised through the use of nano-CT scan protocols and adequate radiological post-processing steps. Fibre medial axes are transferred into finite element assemblies and are exposed to magnetic actuation models. Network displacement of input geometries is quantified by averaging of node displacement fields. Key resultsComplex 3D deformation fields with regions of tension, shear, and compression are obtained. Results from a previous study about matrix material deformation can be confirmed in this study for greater sample geometries. The strain magnitude is not uniform across the samples; several influencing parameters and deformation patterns are identified. A simple analytical model can be presented which quantifies the material deformation. ConclusionsNano-CT scans provide an efficient radiological tool in the planning of relevant experimental procedures. The present study confirms the general usability of fibre networks for the contactless creation of 3D strain fields in tissue engineering. Mechanical effects in tissue growth stimulation known from experimental work are obtained numerically for the investigated assemblies. Further work about the mechanical effects in tissue cultures appears highly worthwhile.

Anisotropic mechanical response of layered disordered fibrous materials

Mechanically bonded fabrics account for a significant portion of nonwoven products, and serve many niche areas of nonwoven manufacturing. Such fabrics are characterized by layers of disordered fibrous webs, but we lack an understanding of how such microstructures determine bulk material response. Here we numerically determine the linear shear response of needle-punched fabrics modeled as cross-linked sheets of two-dimensional (2D) Mikado networks. We systematically vary the intra-sheet fiber density, inter-sheet separation distance, and direction of shear, and quantify the macroscopic shear modulus alongside the degree of affinity and energy partition. For shear parallel to the sheets, the response is dominated by intrasheet fibers and follows known trends for 2D Mikado networks. By contrast, shears perpendicular to the sheets induce a softer response dominated by either intrasheet or intersheet fibers depending on a quadratic relation between sheet separation and fiber density. These basic trends are reproduced and elucidated by a simple scaling argument that we provide. We discuss the implications of our findings in the context of real nonwoven fabrics.

Mechanical bone growth stimulation by magnetic fibre networks obtained through a competent finite element technique

Fibre networks combined with a matrix material in their void phase make the design of novel and smart composite materials possible. Their application is of great interest in the field of advanced paper or as bioactive tissue engineering scaffolds. In the present study, we analyse the mechanical interaction between metallic fibre networks under magnetic actuation and a matrix material. Experimentally validated FE models are combined for that purpose in one joint simulation. High performance computing facilities are used. The resulting strain in the composite’s matrix is not uniform across the sample volume. Instead we show that boundary conditions and proximity to the fibre structure strongly influence the local strain magnitude. An analytical model of local strain magnitude is derived. The strain magnitude of 0.001 which is of particular interest for bone growth stimulation is achievable by this assembly. In light of these findings, the investigated composite structure is suitable for creating and for regulating contactless a stress field which is to be imposed on the matrix material. Topics for future research will be the advanced modelling of the biological components and the potential medical utilisation.

Mechanical Cell-Cell Communication in Fibrous Networks: The Importance of Network Geometry

Cells contracting in extracellular matrix (ECM) can transmit stress over long distances, communicating their position and orientation to cells many tens of micrometres away. Such phenomena are not observed when cells are seeded on substrates with linear elastic properties, such as polyacrylamide (PA) gel. The ability for fibrous substrates to support far reaching stress and strain fields has implications for many physiological processes, while the mechanical properties of ECM are central to several pathological processes, including tumour invasion and fibrosis. Theoretical models have investigated the properties of ECM in a variety of network geometries. However, the effects of network architecture on mechanical cell-cell communication have received little attention. This work investigates the effects of geometry on network mechanics, and thus the ability for cells to communicate mechanically through different networks. Cell-derived displacement fields are quantified for various network geometries while controlling for network topology, cross-link density and micromechanical properties. We find that the heterogeneity of response, fibre alignment, and substrate displacement fields are sensitive to network choice. Further, we show that certain geometries support mechanical communication over longer distances than others. As such, we predict that the choice of network geometry is important in fundamental modelling of cell-cell interactions in fibrous substrates, as well as in experimental settings, where mechanical signalling at the cellular scale plays an important role. This work thus informs the construction of theoretical models for substrate mechanics and experimental explorations of mechanical cell-cell communication.

Strain-driven criticality underlies nonlinear mechanics of fibrous networks

Networks with only central force interactions are floppy when their average connectivity is below an isostatic threshold. Although such networks are mechanically unstable, they can become rigid when strained. It was recently shown that the transition from floppy to rigid states as a function of simple shear strain is continuous, with hallmark signatures of criticality [Sharma et al., Nature Phys. 12, 584 (2016)1745-247310.1038/nphys3628]. The nonlinear mechanical response of collagen networks was shown to be quantitatively described within the framework of such mechanical critical phenomenon. Here, we provide a more quantitative characterization of critical behavior in subisostatic networks. Using finite-size scaling we demonstrate the divergence of strain fluctuations in the network at well-defined critical strain. We show that the characteristic strain corresponding to the onset of strain stiffening is distinct from but related to this critical strain in a way that depends on critical exponents. We confirm this prediction experimentally for collagen networks. Moreover, we find that the apparent critical exponents are largely independent of the spatial dimensionality. With subisostaticity as the only required condition, strain-driven criticality is expected to be a general feature of biologically relevant fibrous networks.

Elastic regimes of subisostatic athermal fiber networks

Athermal models of disordered fibrous networks are highly useful for studying the mechanics of elastic networks composed of stiff biopolymers. The underlying network architecture is a key aspect that can affect the elastic properties of these systems, which include rich linear and nonlinear elasticity. Existing computational approaches have focused on both lattice-based and off-lattice networks obtained from the random placement of rods. It is not obvious, a priori, whether the two architectures have fundamentally similar or different mechanics. If they are different, it is not clear which of these represents a better model for biological networks. Here, we show that both approaches are essentially equivalent for the same network connectivity, provided the networks are subisostatic with respect to central force interactions. Moreover, for a given subisostatic connectivity, we even find that lattice-based networks in both two and three dimensions exhibit nearly identical nonlinear elastic response. We provide a description of the linear mechanics for both architectures in terms of a scaling function. We also show that the nonlinear regime is dominated by fiber bending and that stiffening originates from the stabilization of subisostatic networks by stress. We propose a generalized relation for this regime in terms of the self-generated normal stresses that develop under deformation. Different network architectures have different susceptibilities to the normal stress but essentially exhibit the same nonlinear mechanics. Such a stiffening mechanism has been shown to successfully capture the nonlinear mechanics of collagen networks.

Active Entanglement-Tracking Microrheology Directly Couples Macromolecular Deformations to Nonlinear Microscale Force Response of Entangled Actin

We track the deformation of discrete entangled actin segments while simultaneously measuring the resistive force the deformed filaments exert in response to an optically driven microsphere. We precisely map the network deformation field to show that local microscale stresses can induce filament deformations that propagate beyond mesoscopic length scales (60 μm, >3 persistence lengths l_p). We show that the filament persistence length controls the critical length scale at which distinct entanglement deformations become driven by collective network mechanics. Mesoscale propagation beyond l_p is coupled with nonlinear local stresses arising from steric entanglements mimicking cross-links.

The Elastic Behaviour of Sintered Metallic Fibre Networks: A Finite Element Study by Beam Theory

Background

The finite element method has complimented research in the field of network mechanics in the past years in numerous studies about various materials. Numerical predictions and the planning efficiency of experimental procedures are two of the motivational aspects for these numerical studies. The widespread availability of high performance computing facilities has been the enabler for the simulation of sufficiently large systems.

Objectives and Motivation

In the present study, finite element models were built for sintered, metallic fibre networks and validated by previously published experimental stiffness measurements. The validated models were the basis for predictions about so far unknown properties.

Materials and Methods

The finite element models were built by transferring previously published skeletons of fibre networks into finite element models. Beam theory was applied as simplification method.

Results and Conclusions

The obtained material stiffness isn’t a constant but rather a function of variables such as sample size and boundary conditions. Beam theory offers an efficient finite element method for the simulated fibre networks. The experimental results can be approximated by the simulated systems. Two worthwhile aspects for future work will be the influence of size and shape and the mechanical interaction with matrix materials.

Elastic response of filamentous networks with compliant crosslinks

Experiments have shown that elasticity of disordered filamentous networks with compliant crosslinks is very different from networks with rigid crosslinks. Here, we model and analyze filamentous networks as a collection of randomly oriented rigid filaments connected to each other by flexible crosslinks that are modeled as wormlike chains. For relatively large extensions we allow for enthalpic stretching of crosslink backbones. We show that for sufficiently high crosslink density, the network linear elastic response is affine on the scale of the filaments' length. The nonlinear regime can become highly nonaffine and is characterized by a divergence of the elastic modulus at finite strain. In contrast to the prior predictions, we do not find an asymptotic regime in which the differential elastic modulus scales linearly with the stress, although an approximate linear dependence can be seen in a transition from entropic to enthalpic regimes. We discuss our results in light of recent experiments.

Viscoelasticity of cross-linked actin networks: experimental tests, mechanical modeling and finite-element analysis

Filamentous actin is one of the main constituents of the eukaryotic cytoskeleton. The actin cortex, a densely cross-linked network, resides underneath the lipid bilayer. In the present work we propose a continuum mechanical formulation for describing the viscoelastic properties of in vitro actin networks, which serve as model systems for the cortex, by including the microstructure, i.e. the behavior of a single filament and its spatial arrangement. The modeling of the viscoelastic response in terms of physically interpretable parameters is conducted using a multiscale approach consisting of two steps: modeling of the single filament response of F-actin by a worm-like chain model including the extensibility of the filament, and assembling the three-dimensional biopolymer network by using the microsphere model which accounts for filaments equally distributed in space. The viscoelastic effects of the network are taken into account using a generalized Maxwell model. The Cauchy stress and elasticity tensors are obtained within a continuum mechanics framework and implemented into a finite-element program. The model is validated on the network level using large strain experiments on reconstituted actin gels. Comparisons of the proposed model with rheological experiments recover reasonable values for the material parameters. Finite-element simulations of the indentation of a sphere on a network slab and the aspiration of a droplet in a micropipette allow for further insights of the viscoelastic behavior of actin networks.

Nonaffine displacements in crystalline solids in the harmonic limit

A systematic coarse graining of microscopic atomic displacements generates a local elastic deformation tensor D as well as a positive definite scalar χ measuring nonaffinity, i.e., the extent to which the displacements are not representable as affine deformations of a reference crystal. We perform an exact calculation of the statistics of χ and D and their spatial correlations for solids at low temperatures, within a harmonic approximation and in one and two dimensions. We obtain the joint distribution P(χ,D) and the two-point spatial correlation functions for χ and D. We show that nonaffine and affine deformations are coupled even in a harmonic solid, with a strength that depends on the size of the coarse-graining volume Ω and dimensionality. As a corollary to our work, we identify the field h(χ) conjugate to χ and show that this field may be tuned to produce a transition to a state where the ensemble average <χ> and the correlation length of χ diverge. Our work should be useful as a template for understanding nonaffine displacements in realistic systems with or without disorder and as a means for developing computational tools for studying the effects of nonaffine displacements in melting, plastic flow, and the glass transition.

Stiffness transition in anisotropic fiber nets

We demonstrate the existence of a percolationlike stiffness transition in fiber networks with a bidisperse orientation distribution and with fiber densities clearly above the geometrical and the ordinary stiffness transition. The fibers are oriented parallel and perpendicular to a strain direction and they have a large fiber aspect ratio. The stiffness K of the fiber nets can be described by a scaling relation, K [proportionally] τ(α) g[(ε - ε(c))/τ(-β)], where τ is the fraction of fibers parallel to strain. g is a scaling function that is roughly described by a power law g(x) [proportionally ] x(γ) for stiffness above the transition and by a constant below the transition. The transition point is characterized by qualitative changes in the distribution of the elastic deformation energy of the fibers, the deformation mode of the fibers, the effective Poisson ratio of the nets, the distribution of elastic energy on fibers and cross links, and the ratio of elastic and viscous dissipation energy. This transition opens the possibility of extreme stiffness variations with minimal mesh manipulations in the vicinity of the transition (i.e., a stiffness gate). It is possible that this transition affects the mechanical behavior of the cytoskeleton in cells.

On the role of the filament length distribution in the mechanics of semiflexible networks

This paper explores the effects of filament length polydispersity on the mechanical properties of semiflexible crosslinked polymer networks. Extending previous studies on monodisperse networks, we compute numerically the response of crosslinked networks of elastic filaments of bimodal and exponential length distributions. These polydisperse networks are subject to the same affine to nonaffine (A/NA) transition observed previously for monodisperse networks, wherein the decreases in either crosslink density or bending stiffness lead to a shift from affine, stretching-dominated deformations to nonaffine, bending-dominated deformations. We find that the onset of this transition is generally more sensitive to changes in the density of longer filaments than shorter filaments, meaning that longer filaments have greater mechanical efficiency. Moreover, in polydisperse networks, mixtures of long and short filaments interact cooperatively to generally produce a nonaffine mechanical response closer to the affine prediction than comparable monodisperse networks of either long or short filaments. Accordingly, the mechanical affinity of polydisperse networks is dependent on the filament length composition. Overall, length polydispersity has the effect of sharpening and shifting the A/NA transition to lower network densities. We discuss the implications of these results on experimental observation of the A/NA transition, and on the design of advanced materials.

Exploring the elastic features of spherically shaped biological assemblies and soft matter systems

Using a numerical simulation, we study the elastic features of biological assemblies (e.g. viruses and bacteria) and soft matter systems (e.g. colloidosomes and nanoparticle covered droplets) that possess a spherical shape in which the proteins (particles) on the colloidosomes or virus shells are mechanically linked to form a stress-bearing spherical structure that may dramatically enhance the surface rigidity. The dependence of the rigidity enhancement upon the density of the cross-linked proteins situated on the surface of the virus is explored. We determine the percolation threshold P(ce) by considering bond percolation on the spherical elastic networks involving nearest neighbor forces. The percolation threshold of such networks is very different from that of a two-dimensional triangular lattice due to the topological effect. We find that the threshold probability for the spherical elastic network is considerably smaller than for an unwrapped network, which reveals that the spherical topology induces more rigidity to the network.

Internal stresses, normal modes, and nonaffinity in three-dimensional biopolymer networks

We numerically investigate deformations and modes of networks of semiflexible biopolymers as a function of crosslink coordination number z and strength of bending and stretching energies. In equilibrium filaments are under internal stress, and the networks exhibit shear rigidity below the Maxwell isostatic point. In contrast to two-dimensional networks, ours exhibit nonaffine bending-dominated response in all rigid states, including those near the maximum of z=4 when bending energies are less than stretching ones.

Coherent potential approximation of random nearly isostatic kagome lattice

The kagome lattice has coordination number 4, and it is mechanically isostatic when nearest-neighbor sites are connected by central-force springs. A lattice of N sites has O(√N) zero-frequency floppy modes that convert to finite-frequency anomalous modes when next-nearest-neighbor (NNN) springs are added. We use the coherent potential approximation to study the mode structure and mechanical properties of the kagome lattice in which NNN springs with spring constant κ are added with probability P=Δz/4, where Δz=z-4 and z is the average coordination number. The effective medium static NNN spring constant κ(m) scales as P(2) for P≪κ and as P for P≫κ, yielding a frequency scale ω*~Δz and a length scale l*~(Δz)(-1). To a very good approximation at small nonzero frequency, κ(m)(P,ω)/κ(m)(P,0) is a scaling function of ω/ω*. The Ioffe-Regel limit beyond which plane-wave states become ill-defined is reached at a frequency of order ω*.

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.

Affine-nonaffine transition in networks of nematically ordered semiflexible polymers

We study the mechanics of nematically ordered semiflexible networks showing that they, like isotropic networks, undergo an affine to nonaffine crossover controlled by the ratio of the filament length to the nonaffinity length. Deep in the nonaffine regime, however, these anisotropic networks exhibit a much more complex mechanical response characterized by a vanishing linear-response regime for highly ordered networks and a dependence of the shear modulus on shear direction at both small (linear) and finite (nonlinear) strains that is different from the affine prediction of orthotropic continuum linear elasticity. We show that these features can be understood in terms of a generalized floppy modes analysis of the nonaffine mechanics and a type of cooperative Euler buckling.

Faster strain fluctuation methods through partial volume updates

Elastic systems that are spatially heterogeneous in their mechanical response pose special challenges for molecular simulations. Standard methods for sampling thermal fluctuations of a system's size and shape proceed through a series of homogeneous deformations, whose magnitudes can be severely restricted by its stiffest parts. Here we present a Monte Carlo algorithm designed to circumvent this difficulty, which can be prohibitive in many systems of modern interest. By deforming randomly selected subvolumes alone, it naturally distributes the amplitude of spontaneous elastic fluctuations according to intrinsic heterogeneity. We describe in detail implementations of such "slice moves" that are consistent with detailed balance. Their practical application is illustrated for crystals of two-dimensional hard disks and random networks of cross-linked polymers.

An algorithm for extracting the network geometry of three-dimensional collagen gels

The geometric structure of a biopolymer network impacts its mechanical and biological properties. In this paper, we develop an algorithm for extracting the network architecture of three-dimensional (3d) fluorescently labeled collagen gels, building on the initial work of Wu et al., (2003). Using artificially generated images, the network extraction algorithm is then validated for its ability to reconstruct the correct bulk properties of the network, including fiber length, persistence length, cross-link density, and shear modulus.

Micromechanical model for elasticity of the cell cytoskeleton

Semiflexible polymer networks, such as cell cytoskeleton, differ significantly from their flexible counterparts in their deformation energy storage mechanism. As a result, the network elasticity is governed by both enthalpic and entropic variations. In addition, the enthalpic effect shows two distinct regimes of energy storage mechanism, the affine and nonaffine regimes. In the past, computation-based modeling on random networks, such as the Mikado model, was used to demonstrate the physical mechanism of mechanical deformation of semiflexible networks. These models are computationally intensive and hence are difficult to apply to studying whole cells. In this paper, we develop a micromechanical model to predict the average macroscopic elastic properties of a random, semiflexible, biopolymer network. The model employs a unit cell consisting of four semiflexible chains and four equivalent axial-bending springs. The proposed unit-cell-based micromechanical model represents a statistically average realization of the actual network and gives the average mechanical properties, such as the shear modulus. Comparisons between the model predictions and Mikado model results confirm that this micromechanical model captures the essential deformation physics revealed from previous studies on the actual network and is capable of predicting the transition between nonaffine and affine deformations. This model can be used to develop efficient continuum constitutive models of the cytoskeleton in the future.

Acoustic probing of the jamming transition in an unconsolidated granular medium

Experimentally determined dispersion relations for acoustic waves guided along the mechanically free surface of an unconsolidated granular packed structure provide information on the elasticity of granular media at very low pressures that are naturally controlled by the gravitational acceleration and the depth beneath the surface. The experiments confirm recent theoretical predictions that relaxation of the disordered granular packing through nonaffine motion leads to a peculiar scaling of shear rigidity with pressure near the jamming transition corresponding to zero pressure.

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.

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.

Unfolding cross-linkers as rheology regulators in F-actin networks

We report on the nonlinear mechanical properties of a statistically homogeneous, isotropic semiflexible network cross-linked by polymers containing numerous small unfolding domains, such as the ubiquitous F-actin cross-linker filamin. We show that the inclusion of such proteins has a dramatic effect on the large strain behavior of the network. Beyond a strain threshold, which depends on network density, the unfolding of protein domains leads to bulk shear softening. Past this critical strain, the network spontaneously organizes itself so that an appreciable fraction of the filamin cross-linkers are at the threshold of domain unfolding. We discuss via a simple mean-field model the cause of this network organization and suggest that it may be the source of power-law relaxation observed in in vitro and in intracellular microrheology experiments. We present data which fully justify our model for a simplified network architecture.

Filamin cross-linked semiflexible networks: fragility under strain

The semiflexible F-actin network of the cytoskeleton is cross-linked by a variety of proteins including filamin, which contains Ig domains that unfold under applied tension. We examine a simple filament network model cross-linked by such unfolding linkers that captures the main mechanical features of F-actin networks cross-linked by filamin proteins and show that, under sufficient strain, the network spontaneously self-organizes so that an appreciable fraction of the filamin cross-linkers are at the threshold of domain unfolding. We propose and test a mean-field model to account for this effect. We also suggest a qualitative experimental signature of this type of network reorganization under applied strain that may be observable in intracellular microrheology experiments of Crocker et al.

Nonaffine correlations in random elastic media

Materials characterized by spatially homogeneous elastic moduli undergo affine distortions when subjected to external stress at their boundaries, i.e., their displacements from a uniform reference state grow linearly with position , and their strains are spatially constant. Many materials, including all macroscopically isotropic amorphous ones, have elastic moduli that vary randomly with position, and they necessarily undergo nonaffine distortions in response to external stress. We study general aspects of nonaffine response and correlation using analytic calculations and numerical simulations. We define nonaffine displacements as the difference between and affine displacements, and we investigate the nonaffinity correlation function and related functions. We introduce four model random systems with random elastic moduli induced by locally random spring constants (none of which are infinite), by random coordination number, by random stress, or by any combination of these. We show analytically and numerically that scales as where the amplitude is proportional to the variance of local elastic moduli regardless of the origin of their randomness. We show that the driving force for nonaffine displacements is a spatial derivative of the random elastic constant tensor times the constant affine strain. Random stress by itself does not drive nonaffine response, though the randomness in elastic moduli it may generate does. We study models with both short- and long-range correlations in random elastic moduli.

Mechanical response of semiflexible networks to localized perturbations

Previous research on semiflexible polymers including cytoskeletal networks in cells has suggested the existence of distinct regimes of elastic response, in which the strain field is either uniform (affine) or nonuniform (nonaffine) under external stress. Associated with these regimes, it has been further suggested that a mesoscopic length scale emerges, which characterizes the scale for the crossover from nonaffine to affine deformations. Here, we extend these studies by probing the response to localized forces and force dipoles. We show that the previously identified nonaffinity length [D. A. Head, Phys. Rev. E 68, 061907 (2003)] controls the mesoscopic response to point forces and the crossover to continuum elastic behavior at large distances.

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.

Viscoelasticity of single wall carbon nanotube suspensions

We investigate the viscoelastic properties of an associating rigid rod network: aqueous suspensions of surfactant stabilized single wall carbon nanotubes (SWNTs). The SWNT suspensions exhibit a rigidity percolation transition with an onset of solidlike elasticity at a volume fraction of 0.0026; the percolation exponent is 2.3+/-0.1. At large strain, the solidlike samples show volume fraction dependent yielding. We develop a simple model to understand these rheological responses and show that the shear dependent stresses can be scaled onto a single master curve to obtain an internanotube interaction energy per bond approximately 40k(B)T. Our experimental observations suggest SWNTs in suspension form interconnected networks with bonds that freely rotate and resist stretching. Suspension elasticity originates from bonds between SWNTs rather than from the stiffness or stretching of individual SWNTs.

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.