Model selection for athermal cross-linked fiber networks

Optimizing Biomimetic 3D Disordered Fibrous Network Structures for Lightweight, High-Strength Materials via Deep Reinforcement Learning

3D disordered fibrous network structures (3D-DFNS), such as cytoskeletons, collagen matrices, and spider webs, exhibit remarkable material efficiency, lightweight properties, and mechanical adaptability. Despite their widespread in nature, the integration into engineered materials is limited by the lack of study on their complex architectures. This study addresses the challenge by investigating the structure-property relationships and stability of biomimetic 3D-DFNS using large datasets generated through procedural modeling, coarse-grained molecular dynamics simulations, and machine learning. Based on these datasets, a network deep reinforcement learning (N-DRL) framework is developed to optimize its stability, effectively balancing weight reduction with the maintenance of structural integrity. The results reveal a pronounced correlation between the total fiber length in 3D-DFNS and its mechanical properties, where longer fibers enhance stress distribution and stability. Additionally, fiber orientation is also considered as a potential factor influencing stress growth values. Furthermore, the N-DRL model demonstrates superior performance compared to traditional approaches in optimizing network stability while minimizing mass and computational cost. Structural integrity is significantly improved through the addition of triple junctions and the reduction of higher-order nodes. In summary, this study leverages machine learning to optimize biomimetic 3D-DFNS, providing novel insights into the design of lightweight, high-strength materials.

Review paper: The importance of consideration of collagen cross-links in computational models of collagen-based tissues

Collagen as the main protein in Extra Cellular Matrix (ECM) is the main load-bearing component of fibrous tissues. Nanostructure and architecture of collagen fibrils play an important role in mechanical behavior of these tissues. Extensive experimental and theoretical studies have so far been performed to capture these properties, but none of the current models realistically represent the complexity of network mechanics because still less is known about the collagen's inner structure and its effect on the mechanical properties of tissues. The goal of this review article is to emphasize the significance of cross-links in computational modeling of different collagen-based tissues, and to reveal the need for continuum models to consider cross-links properties to better reflect the mechanical behavior observed in experiments. In addition, this study outlines the limitations of current investigations and provides potential suggestions for the future work.

Nonaffine Deformation of Semiflexible Polymer and Fiber Networks

Networks of semiflexible or stiff polymers such as most biopolymers are known to deform inhomogeneously when sheared. The effects of such nonaffine deformation have been shown to be much stronger than for flexible polymers. To date, our understanding of nonaffinity in such systems is limited to simulations or specific 2D models of athermal fibers. Here, we present an effective medium theory for nonaffine deformation of semiflexible polymer and fiber networks, which is general to both 2D and 3D and in both thermal and athermal limits. The predictions of this model are in good agreement with both prior computational and experimental results for linear elasticity. Moreover, the framework we introduce can be extended to address nonlinear elasticity and network dynamics.

Effect of connectivity on the elasticity of athermal network materials

Network materials with stochastic structure are ubiquitous in biology and engineering, which drives the current interest in establishing relations between their structure and mechanical behavior. In this work we focus on the effect of connectivity defined by the number of fibers emerging from a crosslink, z, and compare networks with identical (z-homogeneous) and distinct (z-heterogeneous) z at the crosslinks. We observe that the functional form of strain stiffening is z-independent, and that the central z-dependent parameter is the small strain stiffness, E₀. We confirm previous results indicating that the functional form of E₀(z) is a power function with 3 regimes and observe that this applies to a broad range of z. However, the scaling exponents are different in the z-homogeneous and z-heterogeneous cases. We confirm that increasing z across the Maxwell's central force isostatic point leads to a transition from bending to axial energy storage. However, we observe that this does not necessarily imply that deformation becomes affine in the large z limit. In fact, networks of fibers with low bending stiffness retain a relaxation mode based on the rotational degree of freedom of the crosslinks which allows E₀ in the large z limit to be smaller than the affine model prediction. We also conclude that in the z-heterogeneous case, the mean connectivity z̄ is sufficient to evaluate the effect of connectivity on E₀ and that higher moments of the distribution of z are less important.

A discrete fiber network finite element model of arterial elastin network considering inter-fiber crosslinking property and density

Inter-fiber crosslinks within the extracellular matrix (ECM) play important roles in determining the mechanical properties of the fibrous network. Discrete fiber network (DFN) models have been used to study fibrous biological material, however the contribution of inter-fiber crosslinks to the mechanics of the ECM network is not well understood. In this study, a DFN model of arterial elastin network was developed based on measured structural features to study the contribution of inter-fiber crosslinking properties and density to the mechanics and fiber kinematics of the network. The DFN was generated by randomly placing line segments into a given domain following a fiber orientation distribution function obtained from multiphoton microscopy until a desired fiber areal fraction was reached. Intersections between the line segments were treated as crosslinks. The generated DFN model was then incorporated into an ABAQUS finite element model to simulate the network under equi- and nonequi-biaxial deformation. The inter-fiber crosslinks were modeled using connector elements with either zero (pin joint) or infinite (weld joint) rotational stiffness. Furthermore, inter-fiber crosslinking density was systematically reduced and its effect on both network- and fiber-level mechanics was studied. The DFN model showed good fitting and predicting capabilities of the stress-strain behavior of the elastin network. While the pin and weld joints do not seem to have noticeable effect on the network stress-strain behavior, the crosslinking properties can affect the local fiber mechanics and kinematics. Overall, our study suggests that inter-fiber crosslinking properties are important to the multiscale mechanics and fiber kinematics of the ECM network.

Parameters controlling the strength of stochastic fibrous materials

Many materials of everyday use are fibrous and their strength is important in most applications. In this work we study the dependence of the strength of random fiber networks on structural parameters such as the network density, cross-link density, fiber tortuosity, and the strength of the inter-fiber cross-links. Athermal networks of cellular and fibrous type are considered. We conclude that the network strength scales linearly with the cross-link number density and with the cross-link strength for a broad range of network parameters, and for both types of networks considered. Network strength is independent of fiber material properties and of fiber tortuosity. This information can be used to design fiber networks for specified strength and, generally, to understand the mechanical behavior of fibrous materials.

Random fiber networks with inclusions: The mechanism of reinforcement

The mechanical behavior of athermal random fiber networks embedding particulate inclusions is studied in this work. Composites in which the filler size is comparable with the mean segment length of the network are considered. Inclusions are randomly distributed in the network at various volume fractions, and cases in which fibers are rigidly bonded to fillers and in which no such bonding is imposed are studied separately. In the presence of inclusions, the small strain modulus increases, while the ability of the network to strain stiffen decreases relative to the unfilled network case. The reinforcement induced by fillers is most pronounced in sparse networks of floppier filaments that deform in the bending-dominated mode in the unfilled state. As the unfilled network density or the bending stiffness of fibers increases, the effect of filling diminishes rapidly. Fillers lead to a transition from the soft, bending-dominated, to the stiffer, stretching-dominated, deformation mode of the network, a transition which is primarily responsible for the observed overall reinforcement. The confinement, i.e., the restriction on network kinematics imposed by fillers, causes this transition. These results provide a justification for the observed difference in reinforcement obtained in sparsely versus densely cross-linked networks at a given filling fraction and provide guidance for the further development of network-based materials.

Multiscale model of fatigue of collagen gels

Fatigue as a mode of failure becomes increasingly relevant with age in tissues that experience repeated fluctuations in loading. While there has been a growing focus on the mechanics of networks of collagen fibers, which are recognized as the predominant mechanical components of soft tissues, the network's fatigue behavior has received less attention. Specifically, it must be asked (1) how the fatigue of networks differs from that of its component fibers, and (2) whether this difference in fatigue behaviors is affected by changes in the network's architecture. In the present study, we simulated cyclic uniaxial loading of Voronoi networks to model fatigue experiments performed on reconstituted collagen gels. Collagen gels were cast into dog-bone shape molds and were tested on a uniaxial machine under a tension-tension cyclic loading protocol. Simulations were performed on networks modeled as trusses of, on average, 600 nonlinear elastic fibers connected at freely rotating pin-joints. We also simulated fatigue failure of Delaunay, and Erdős-Rényi networks, in addition to Voronoi networks, to compare fatigue behavior among different architectures. The uneven distribution of stresses within the fibers of the unstructured networks resulted in all three network geometries being more endurant than a single fiber or a regular lattice under cyclic loading. Among the different network geometries, for low to moderate external loads, the Delaunay networks showed the best fatigue behavior, while at higher loads, the Voronoi networks performed better.

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.

Mechanics of a two-fiber model with one nested fiber network, as applied to the collagen-fibrin system

The mechanical behavior of collagen-fibrin (col-fib) co-gels is both scientifically interesting and clinically relevant. Collagen-fibrin networks are a staple of tissue engineering research, but the mechanical consequences of changes in co-gel composition have remained difficult to predict or even explain. We previously observed fundamental differences in failure behavior between collagen-rich and fibrin-rich co-gels, suggesting an essential change in how the two components interact as the co-gel's composition changes. In this work, we explored the hypothesis that the co-gel behavior is due to a lack of percolation by the dilute component. We generated a series of computational models based on interpenetrating fiber networks. In these models, the major network component percolated the model space but the minor component did not, instead occupying a small island embedded within the larger network. Each component was assigned properties based on a fit of single-component gel data. Island size was varied to match the relative concentrations of the two components. The model predicted that networks rich in collagen, the stiffer component, would roughly match pure-collagen gel behavior with little additional stress due to the fibrin, as seen experimentally. For fibrin-rich gels, however, the model predicted a smooth increase in the overall network strength with added collagen, as seen experimentally but not consistent with an additive parallel model. We thus conclude that incomplete percolation by the low-concentration component of a co-gel is a major determinant of its macroscopic properties, especially if the low-concentration component is the stiffer component.

STATEMENT OF SIGNIFICANCE

Models for the behavior of fibrous networks have useful applications in many different fields, including polymer science, textiles, and tissue engineering. In addition to being important structural components in soft tissues and blood clots, these protein networks can serve as scaffolds for bioartificial tissues. Thus, their mechanical behavior, especially in co-gels, is both interesting from a materials science standpoint and significant with regard to tissue engineering.

Poisson's Contraction and Fiber Kinematics in Tissue: Insight From Collagen Network Simulations

Connective tissue mechanics is highly nonlinear, exhibits a strong Poisson's effect, and is associated with significant collagen fiber re-arrangement. Although the general features of the stress-strain behavior have been discussed extensively, the Poisson's effect received less attention. In general, the relationship between the microscopic fiber network mechanics and the macroscopic experimental observations remains poorly defined. The objective of the present work is to provide additional insight into this relationship. To this end, results from models of random collagen networks are compared with experimental data on reconstructed collagen gels, mouse skin dermis, and the human amnion. Attention is devoted to the mechanism leading to the large Poisson's effect observed in experiments. The results indicate that the incremental Poisson's contraction is directly related to preferential collagen orientation. The experimentally observed downturn of the incremental Poisson's ratio at larger strains is associated with the confining effect of fibers transverse to the loading direction and contributing little to load bearing. The rate of collagen orientation increases at small strains, reaches a maximum, and decreases at larger strains. The peak in this curve is associated with the transition of the network deformation from bending dominated, at small strains, to axially dominated, at larger strains. The effect of fiber tortuosity on network mechanics is also discussed, and a comparison of biaxial and uniaxial loading responses is performed.

Hindrances to precise recovery of cellular forces in fibrous biopolymer networks

How cells move through the three-dimensional extracellular matrix (ECM) is of increasing interest in attempts to understand important biological processes such as cancer metastasis. Just as in motion on flat surfaces, it is expected that experimental measurements of cell-generated forces will provide valuable information for uncovering the mechanisms of cell migration. However, the recovery of forces in fibrous biopolymer networks may suffer from large errors. Here, within the framework of lattice-based models, we explore possible issues in force recovery by solving the inverse problem: how can one determine the forces cells exert to their surroundings from the deformation of the ECM? Our results indicate that irregular cell traction patterns, the uncertainty of local fiber stiffness, the non-affine nature of ECM deformations and inadequate knowledge of network topology will all prevent the precise force determination. At the end, we discuss possible ways of overcoming these difficulties.

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.

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.

Inferring spatial variations of microstructural properties from macroscopic mechanical response

Disease alters tissue microstructure, which in turn affects the macroscopic mechanical properties of tissue. In elasticity imaging, the macroscopic response is measured and is used to infer the spatial distribution of the elastic constitutive parameters. When an empirical constitutive model is used, these parameters cannot be linked to the microstructure. However, when the constitutive model is derived from a microstructural representation of the material, it allows for the possibility of inferring the local averages of the spatial distribution of the microstructural parameters. This idea forms the basis of this study. In particular, we first derive a constitutive model by homogenizing the mechanical response of a network of elastic, tortuous fibers. Thereafter, we use this model in an inverse problem to determine the spatial distribution of the microstructural parameters. We solve the inverse problem as a constrained minimization problem and develop efficient methods for solving it. We apply these methods to displacement fields obtained by deforming gelatin-agar co-gels and determine the spatial distribution of agar concentration and fiber tortuosity, thereby demonstrating that it is possible to image local averages of microstructural parameters from macroscopic measurements of deformation.

Effect of Fiber Crimp on the Elasticity of Random Fiber Networks With and Without Embedding Matrices

Fiber networks are assemblies of one-dimensional elements representative of materials with fibrous microstructures such as collagen networks and synthetic nonwovens. The mechanics of random fiber networks has been the focus of numerous studies. However, fiber crimp has been explicitly represented only in few cases. In the present work, the mechanics of cross-linked networks with crimped athermal fibers, with and without an embedding elastic matrix, is studied. The dependence of the effective network stiffness on the fraction of nonstraight fibers and the relative crimp amplitude (or tortuosity) is studied using finite element simulations of networks with sinusoidally curved fibers. A semi-analytic model is developed to predict the dependence of network modulus on the crimp amplitude and the bounds of the stiffness reduction associated with the presence of crimp. The transition from the linear to the nonlinear elastic response of the network is rendered more gradual by the presence of crimp, and the effect of crimp on the network tangent stiffness decreases as strain increases. If the network is embedded in an elastic matrix, the effect of crimp becomes negligible even for very small, biologically relevant matrix stiffness values. However, the distribution of the maximum principal stress in the matrix becomes broader in the presence of crimp relative to the similar system with straight fibers, which indicates an increased probability of matrix failure.

Softening in Random Networks of Non-Identical Beams

Random fiber networks are assemblies of elastic elements connected in random configurations. They are used as models for a broad range of fibrous materials including biopolymer gels and synthetic nonwovens. Although the mechanics of networks made from the same type of fibers has been studied extensively, the behavior of composite systems of fibers with different properties has received less attention. In this work we numerically and theoretically study random networks of beams and springs of different mechanical properties. We observe that the overall network stiffness decreases on average as the variability of fiber stiffness increases, at constant mean fiber stiffness. Numerical results and analytical arguments show that for small variabilities in fiber stiffness the amount of network softening scales linearly with the variance of the fiber stiffness distribution. This result holds for any beam structure and is expected to apply to a broad range of materials including cellular solids.

Foam-like compression behavior of fibrin networks

The rheological properties of fibrin networks have been of long-standing interest. As such there is a wealth of studies of their shear and tensile responses, but their compressive behavior remains unexplored. Here, by characterization of the network structure with synchronous measurement of the fibrin storage and loss moduli at increasing degrees of compression, we show that the compressive behavior of fibrin networks is similar to that of cellular solids. A nonlinear stress-strain response of fibrin consists of three regimes: (1) an initial linear regime, in which most fibers are straight, (2) a plateau regime, in which more and more fibers buckle and collapse, and (3) a markedly nonlinear regime, in which network densification occurs by bending of buckled fibers and inter-fiber contacts. Importantly, the spatially non-uniform network deformation included formation of a moving "compression front" along the axis of strain, which segregated the fibrin network into compartments with different fiber densities and structure. The Young's modulus of the linear phase depends quadratically on the fibrin volume fraction while that in the densified phase depends cubically on it. The viscoelastic plateau regime corresponds to a mixture of these two phases in which the fractions of the two phases change during compression. We model this regime using a continuum theory of phase transitions and analytically predict the storage and loss moduli which are in good agreement with the experimental data. Our work shows that fibrin networks are a member of a broad class of natural cellular materials which includes cancellous bone, wood and cork.

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.

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.

Rapid Quantification of 3D Collagen Fiber Alignment and Fiber Intersection Correlations with High Sensitivity

Metastatic cancers aggressively reorganize collagen in their microenvironment. For example, radially orientated collagen fibers have been observed surrounding tumor cell clusters in vivo. The degree of fiber alignment, as a consequence of this remodeling, has often been difficult to quantify. In this paper, we present an easy to implement algorithm for accurate detection of collagen fiber orientation in a rapid pixel-wise manner. This algorithm quantifies the alignment of both computer generated and actual collagen fiber networks of varying degrees of alignment within 5°°. We also present an alternative easy method to calculate the alignment index directly from the standard deviation of fiber orientation. Using this quantitative method for determining collagen alignment, we demonstrate that the number of collagen fiber intersections has a negative correlation with the degree of fiber alignment. This decrease in intersections of aligned fibers could explain why cells move more rapidly along aligned fibers than unaligned fibers, as previously reported. Overall, our paper provides an easier, more quantitative and quicker way to quantify fiber orientation and alignment, and presents a platform in studying effects of matrix and cellular properties on fiber alignment in complex 3D environments.

Remodeling of fibrous extracellular matrices by contractile cells: predictions from discrete fiber network simulations

Contractile forces exerted on the surrounding extracellular matrix (ECM) lead to the alignment and stretching of constituent fibers within the vicinity of cells. As a consequence, the matrix reorganizes to form thick bundles of aligned fibers that enable force transmission over distances larger than the size of the cells. Contractile force-mediated remodeling of ECM fibers has bearing on a number of physiologic and pathophysiologic phenomena. In this work, we present a computational model to capture cell-mediated remodeling within fibrous matrices using finite element-based discrete fiber network simulations. The model is shown to accurately capture collagen alignment, heterogeneous deformations, and long-range force transmission observed experimentally. The zone of mechanical influence surrounding a single contractile cell and the interaction between two cells are predicted from the strain-induced alignment of fibers. Through parametric studies, the effect of cell contractility and cell shape anisotropy on matrix remodeling and force transmission are quantified and summarized in a phase diagram. For highly contractile and elongated cells, we find a sensing distance that is ten times the cell size, in agreement with experimental observations.

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.

Predictive maps for stochastic nonaffine stiffening and damage in fibrous networks

The macroscopic responses of synthetic and natural filamentous networks are determined by a combination of microstructure and filament properties. Biofilament networks such as those of actin and fibrin have become vehicles for studying important concepts in mechanics such as rigidity percolation, linearity and nonlinearity, isotropy and anisotropy, affinity and nonaffinity, hardening and softening, bending and stretching transitions, etc. In this work, we consider generic fibrous network architectures to map out their mechanical responses over a wide range of filament properties. Using the finite element method, we perform two-dimensional simulations of discrete networks subjected to shear deformation. These simulations encompass stochastic effects arising from network topology (filament arrangement, orientation, and length distribution) and the thermally activated crosslink scission. We study the mechanics of these random networks up to a strain of 10%, including damage that is induced by crosslink scission. The response is nonlinear and the initial elastic modulus alone is not sufficient to give an understanding about the overall response. We show that the nonlinear elastic response of the network can be captured using a few parameters that depend on some well known length scales in network mechanics. For networks with filament density above the rigidity percolation threshold, by increasing filament density and bending stiffness, we observe a crossover from the bending dominated elastically compliant stiffening regime to a stretching dominated rigid nonstiffening regime. We show that in the bending dominated regime there are large deviations from the predictions of affine continuum theories. We also give a simple qualitative model for describing the contours of the incubation strain which marks the onset of damage in networks.

Cross-Linked Fiber Network Embedded in Elastic Matrix

The mechanical behavior of a three-dimensional cross-linked fiber network embedded in matrix is studied in this work. The network is composed from linear elastic fibers which store energy only in the axial deformation mode, while the matrix is also isotropic and linear elastic. Such systems are encountered in a broad range of applications, from tissue to consumer products. As the matrix modulus increases, the network is constrained to deform more affinely. This leads to internal forces acting between the network and the matrix, which produce strong stress concentration at the network cross-links. This interaction increases the apparent modulus of the network and decreases the apparent modulus of the matrix. A model is developed to predict the effective modulus of the composite and its predictions are compared with numerical data for a variety of networks.