Elasticity of Poissonian fiber networks

Normal Stresses, Contraction, and Stiffening in Sheared Elastic Networks

When elastic solids are sheared, a nonlinear effect named after Poynting gives rise to normal stresses or changes in volume. We provide a novel relation between the Poynting effect and the microscopic Grüneisen parameter, which quantifies how stretching shifts vibrational modes. By applying this relation to random spring networks, a minimal model for, e.g., biopolymer gels and solid foams, we find that networks contract or develop tension because they vibrate faster when stretched. The amplitude of the Poynting effect is sensitive to the network's linear elastic moduli, which can be tuned via its preparation protocol and connectivity. Finally, we show that the Poynting effect can be used to predict the finite strain scale where the material stiffens under shear.

Topology effects on nonaffine behavior of semiflexible fiber networks

Filamentous semiflexible networks define the mechanical and physical properties of many materials such as cytoskeleton. In the absence of a distinct unit cell, the Mikado fiber network model is commonly used algorithm for representing the microstructure of these networks in numerical models. Nevertheless, certain types of filamentous structures such as collagenous tissues, at early stages of their development, are assembled by growth of individual fibers from random nucleation sites. In this work, we develop a computational model to investigate the mechanical response of such networks by characterizing their nonaffine behavior. We show that the deformation of these networks is nonaffine at all length scales. Furthermore, similar to Mikado networks, the degree of nonaffinity in these structures decreases with increasing the probing length scale, the network fiber density, and/or the bending stiffness of constituting filaments. Nevertheless, despite the lower coordination number of these networks, their deformation field is more affine than that of the Mikado networks with the same fiber density and fiber mechanical properties.

Comparing the microstructure and mechanical properties of Bombyx mori and Antheraea pernyi cocoon composites

Silkworm cocoon material is a natural composite consisting of silk fibres and sericin glues. Both domestic and wild silkworms produce cocoons but with different functionality - one selected by man for textile manufacture whereas the other selected by Nature to provide damage-tolerant housing. To understand the structure--property relationship of cocoons, we evaluated and compared the microstructure and mechanical properties of two representative cocoon walls. It appears that a "brittle and weak" composite is produced by domestic Bombyx mori (B. mori) while a "tough and strong" composite is made by wild Antheraea pernyi (A. pernyi). The superior mechanical performance of A. pernyi cocoons can be attributed to both the material properties and the fibre network microstructures. Failure mechanisms and different failure modes for cocoon fibre composites were also proposed. A finite element model revealed qualitatively the effect of fibre properties and inter-fibre bonding strength on the mechanical properties of the fibre network. It emerged that both good mechanical properties of fibres and robust inter-fibre bonding were required for tough and strong fibre composites. The new insights could inspire new designs of synthetic fibre composites with enhanced mechanical properties.

STATEMENT OF SIGNIFICANCE

Natural cocoons are an important group of natural fibre composites with versatile functionalities. Previous studies have focused on the diversity of cocoon species and different morphological and mechanical features. It was suggested that the cocoon network structure determined the final mechanical properties of the cocoon composite. Nevertheless, the full structure-propertyfunction relationships for the cocoon composite are not understood. By studying two distinct cocoon species with specific functionalities, we prove that the mechanical properties of two cocoons are determined by both network properties and fibre properties. A robust fibre network is the prerequisite, within which the good mechanical properties of the fibres can play a part. The finding will inspire new designs of synthetic composites with desirable and predictable mechanical properties.

On the origins of the elasticity of cellulose nanofiber nanocomposites and nanopapers: a micromechanical approach

The elastic properties of cellulose nanofibril (NFC) nanocomposites and nanopapers are predicted by a multiscale network model that shows that the deformation mechanisms are governed by the bonds between rigid NFC segments and in the kinked regions.

Contact formation in random networks of elongated objects

The effect of steric hindrance is an important aspect of granular packings as it gives rise to, e.g., limitations on the densities of ordered and disordered packings, both of which are essentially defined by the geometry of the constituents. Here we focus on the random packing of rods via deposition and their distributions of contact number and segment length. Such statistical properties are relevant for mechanical properties of the structures, but the (quite large) steric effects on them have not been addressed in previous studies. We therefore develop a theory that describes the statistical properties of rod packings, while taking into account that the deposited rods cannot overlap and thus induce steric hindrances. The distributions derived from the theory are compared with experimental results and numerical simulations of networks constructed via deposition. The results explain the non-Poisson statistics observed in the experiments and show that the induced steric range of the rods can be large compared to their diameter and decreases with compactification of the pile, implying local orientational ordering of the structure.

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.

Silkworm cocoons inspire models for random fiber and particulate composites

The bioengineering design principles evolved in silkworm cocoons make them ideal natural prototypes and models for structural composites. Cocoons depend for their stiffness and strength on the connectivity of bonding between their constituent materials of silk fibers and sericin binder. Strain-activated mechanisms for loss of bonding connectivity in cocoons can be translated directly into a surprisingly simple yet universal set of physically realistic as well as predictive quantitative structure-property relations for a wide range of technologically important fiber and particulate composite materials.

Semiflexible filamentous composites

Inspired by the ubiquity of composite filamentous networks in nature, we investigate models of biopolymer networks that consist of interconnected floppy and stiff filaments. Numerical simulations carried out in three dimensions allow us to explore the microscopic partitioning of stresses and strains between the stiff and floppy fractions cs and cf and reveal a nontrivial relationship between the mechanical behavior and the relative fraction of stiff polymer: when there are few stiff polymers, nonpercolated stiff "inclusions" are protected from large deformations by an encompassing floppy matrix, while at higher fractions of stiff material the stiff network is independently percolated and dominates the mechanical response.

Predicting polymer nanofiber interactions via molecular simulations

Physical and functional properties of nonwoven textiles and other fiberlike materials depend strongly on the number and type of fiber-fiber interactions. For nanoscale polymeric fibers in particular, these interactions are governed by the surfaces of and contacts between fibers. We employ both molecular dynamics (MD) simulations at a temperature below the glass transition temperature T(g) of the polymer bulk, and molecular statics (MS), or energy minimization, to study the interfiber interactions between prototypical polymeric fibers of 4.6 nm diameter, comprising multiple macromolecular chains each of 100 carbon atoms per chain (C100). Our MD simulations show that fibers aligned parallel and within 9 nm of one another experience a significant force of attraction. These fibers tend toward coalescence on a very short time scale, even below T(g). In contrast, our MS calculations suggest an interfiber interaction that transitions from an attractive to a repulsive force at a separation distance of 6 nm. The results of either approach can be used to obtain a quantitative, closed-form relation describing fiber-fiber interaction energies U(s). However, the predicted form of interaction is quite different for the two approaches, and can be understood in terms of differences in the extent of molecular mobility within and between fibers for these different modeling perspectives. The results of these molecular-scale calculations of U(s) are used to interpret experimental observations for electrospun polymer nanofiber mats. These findings highlight the role of temperature and kinetically accessible molecular configurations in predicting interface-dominated interactions at polymer fiber surfaces, and prompt further experiments and simulations to confirm these effects in the properties of nonwoven mats comprising such nanoscale fibers.

Heterogeneous long-range correlated deformation of semiflexible random fiber networks

The deformation of dense random fiber networks is important in a variety of applications including biological and nonliving systems. In this paper it is shown that semiflexible fiber networks exhibit long-range power-law spatial correlations of the density and elastic properties. Hence, the stress and strain fields measured over finite patches of the network are characterized by similar spatial correlations. The scaling is observed over a range of scales bounded by a lower limit proportional to the segment length and an upper limit on the order of the fiber length. If the fiber bending stiffness is reduced below a threshold, correlations are lost. The issue of solving boundary value problems defined on large domains of random fiber networks is also addressed. Since the direct simulation of such systems is impractical, the network is mapped into an equivalent continuum with long-range correlated elastic moduli. A technique based on the stochastic finite element method is used to solve the resulting stochastic continuum problem. The method provides the moments of the distribution function of the solution (e.g., of the displacement field). It performs a large dimensionality reduction which is based on the scaling properties of the underlying elasticity of the material. Two examples are discussed in closure.

A microstructurally informed model for the mechanical response of three-dimensional actin networks

We propose a class of microstructurally informed models for the linear elastic mechanical behaviour of cross-linked polymer networks such as the actin cytoskeleton. Salient features of the models include the possibility to represent anisotropic mechanical behaviour resulting from anisotropic filament distributions, and a power law scaling of the mechanical properties with the filament density. Mechanical models within the class are parameterized by seven different constants. We demonstrate a procedure for determining these constants using finite element models of three-dimensional actin networks. Actin filaments and cross-links were modelled as elastic rods, and the networks were constructed at physiological volume fractions and at the scale of an image voxel. We show the performance of the model in estimating the mechanical behaviour of the networks over a wide range of filament densities and degrees of anisotropy.

Strain hardening, avalanches, and strain softening in dense cross-linked actin networks

Actin filament networks enable the cytoskeleton to adjust to internal and external forcing. These dynamic networks can adapt to changes by dynamically adjusting their cross-links. Here, we model actin filaments as cross-linked elastic fibers of finite dimensions, with the cross-links being approximately 1 mum apart, and employ a full three-dimensional model to study their elastic properties by computer simulations. The results show compelling evidence that dense actin networks are characterized by (a) strain hardening without entropic elasticity, (b) avalanches of cross-link slippage leading to strain softening in the case of breakable cross-links, and (c) spontaneous formation of stress fibers in the case of dynamic cross-link formation and destruction.

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.

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.

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.

Carbon nanotube mats and fibers with irradiation-improved mechanical characteristics: a theoretical model

We employ a theoretical model to calculate mechanical characteristics of macroscopic mats and fibers of single-walled carbon nanotubes. We further investigate irradiation-induced covalent bonds between nanotubes and their effects on the tensile strength of nanotube mats and fibers. We show that the stiffness and strength of the mats can be increased at least by an order of magnitude, and thus small-dose irradiation with energetic particles is a promising tool for making macroscopic nanotube materials with excellent mechanical characteristics.

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.

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.

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

Rigidity percolation is analyzed in two-dimensional random networks of stiff fibers. As fibers are randomly added to the system there exists a density threshold q=q(min) above which a rigid stress-bearing percolation cluster appears. This threshold is found to be above the connectivity percolation threshold q=q(c) such that q(min)=(1.1698+/-0.0004)q(c). The transition is found to be continuous, and in the universality class of the two-dimensional central-force rigidity percolation on lattices. At percolation threshold the rigid backbone of the percolating cluster was found to break into rigid clusters, whose number diverges in the limit of infinite system size, when a critical bond is removed. The scaling with system size of the average size of these clusters was found to give a new scaling exponent delta=1.61+/-0.04.