Parameters controlling the strength of stochastic fibrous materials

Microstructural Evolution and Failure in Fibrous Network Materials: Failure Mode Transition from the Competition between Bond and Fiber

For the complex structure of fibrous network materials, it is a challenge to analyze the network strength and deformation mechanism. Here, we identify a failure mode transition within the network material comprising brittle fibers and bonds, which is related to the strength ratio of the bond to the fiber. A failure criterion for this type of fibrous network is proposed to quantitatively characterize this transition between bond damage and fiber damage. Additionally, tensile experiments on carbon and ceramic fibrous network materials were conducted, and the experimental results show that the failure modes of these network materials satisfy the theoretical prediction. The relationship between the failure mode, the relative density of network and strength of the components is established based on finite element analysis of the 3D network model. The failure mode transforms from bond damage to fiber damage as increasing of bond strength. According to the transition of the failure modes in the brittle fibrous network, it is possible to tailor the mechanical properties of fibrous network material by balancing the competition between bond and fiber properties, which is significant for optimizing material design and engineering applications.

Micromechanics of fibrous scaffolds and their stiffness sensing by cells

Mechanical properties of the tissue engineering scaffolds are known to play a crucial role in cell response. Therefore, an understanding of the cell-scaffold interactions is of high importance. Here, we have utilized discrete fiber network model to quantitatively study the micromechanics of fibrous scaffolds with different fiber arrangements and cross-linking densities. We observe that localized forces on the scaffold result in its anisotropic deformation even for isotropic fiber arrangements. We also see an exponential decay of the displacement field with distance from the location of applied force. This nature of the decay allows us to estimate the characteristic length for force transmission in fibrous scaffolds. Furthermore, we also looked at the stiffness sensing of fibrous scaffolds by individual cells and its dependence on the cellular sensing mechanism. For this, we considered two conditions- stress-controlled, and strain-controlled application of forces by a cell. With fixed strain, we find that the stiffness sensed by a cell is proportional to the scaffold's 'macroscopic' elastic modulus. However, under fixed stress application by the cell, the stiffness sensed by the cell also depends on the cell's own stiffness. In fact, the stiffness values for the same scaffold sensed by the stiff and soft cells can differ from each other by an order of magnitude. The insights from this work will help in designing tissue engineering scaffolds for applications where mechanical stimuli are a critical factor.

Quantitative evaluation of fiber network structure-property relationships in bacterial cellulose hydrogels

The present study attempts to elucidate the network structure-property relationships of bacterial cellulose (BC) hydrogels comprising cellulose nanofibrils with favorable mechanical properties. To achieve this, it is necessary to establish a method based on quantitative evaluation of nanofibril network structure, rather than a simple application of classical polymer network theory. BC hydrogels with various network structures related to their mechanical properties were prepared from seven bacterial strains. The crosslink densities of the gels were determined quantitatively by a combination of fluorescence microscopy and image analysis. The tensile tests showed that the stress-strain curves of BC hydrogels exhibited strain hardening according to the power law for strain, and the power exponent had a linear relationship with the crosslink density. This result provides insight into the structure-property relationships of BC hydrogels, which could be used to inform quality control, process optimization, and high-throughput property prediction during manufacture.

Gradient porous structures of mycelium: a quantitative structure-mechanical property analysis

Gradient porous structures (GPS) are characterized by structural variations along a specific direction, leading to enhanced mechanical and functional properties compared to homogeneous structures. This study explores the potential of mycelium, the root part of a fungus, as a biomaterial for generating GPS. During the intentional growth of mycelium, the filamentous network undergoes structural changes as the hyphae grow away from the feed substrate. Through microstructural analysis of sections obtained from the mycelium tissue, systematic variations in fiber characteristics (such as fiber radii distribution, crosslink density, network density, segment length) and pore characteristics (including pore size, number, porosity) are observed. Furthermore, the mesoscale mechanical moduli of the mycelium networks exhibit a gradual variation in local elastic modulus, with a significant change of approximately 50% across a 30 mm thick mycelium tissue. The structure-property analysis reveals a direct correlation between the local mechanical moduli and the network crosslink density of the mycelium. This study presents the potential of controlling growth conditions to generate mycelium-based GPS with desired functional properties. This approach, which is both sustainable and economically viable, expands the applications of mycelium-based GPS to include filtration membranes, bio-scaffolds, tissue regeneration platforms, and more.

Biomechanics, Energetics, and Structural Basis of Rupture of Fibrin Networks

Fibrin provides the main structural integrity and mechanical strength to blood clots. Failure of fibrin clots can result in life-threating complications, such as stroke or pulmonary embolism. The dependence of rupture resistance of fibrin networks (uncracked and cracked) on fibrin(ogen) concentrations in the (patho)physiological 1-5 g L-1 range is explored by performing the ultrastructural studies and theoretical analysis of the experimental stress-strain profiles available from mechanical tensile loading assays. Fibrin fibers in the uncracked network stretched evenly, whereas, in the cracked network, fibers around the crack tip showed greater deformation. Unlike fibrin fibers in cracked networks formed at the lower 1-2.7 g L-1 fibrinogen concentrations, fibers formed at the higher 2.7-5 g L-1 concentrations align and stretch simultaneously. Cracked fibrin networks formed in higher fibrinogen solutions are tougher yet less extensible. Statistical modeling revealed that the characteristic strain for fiber alignment, crack size, and fracture toughness of fibrin networks control their rupture resistance. The results obtained provide a structural and biomechanical basis to quantitatively understand the material properties of blood plasma clots and to illuminate the mechanisms of their rupture.

Toughness of Network Materials: Structural Parameters Controlling Damage Accumulation

Many materials have a network of fibers as their main structural component and are referred to as network materials. Their strength and toughness are important in both engineering and biology. In this work we consider stochastic model fiber networks without pre-existing cracks and study their rupture mechanism. These materials soften as the crosslinks or fibers fail and exhibit either brittle failure immediately after the peak stress, or a more gradual, ductile rupture in the post peak regime. We observe that ductile failure takes place at constant energy release rate defined in the absence of pre-existing cracks as the strain derivative of the specific energy released. The network parameters controlling the energy release rate are identified and discussed in relation to the Lake-Thomas theory which applies to crack growth situations. We also observe a ductile to brittle failure transition as the network becomes more affine and relate the embrittlement to the reduction of mechanical heterogeneity of the network. Further, we confirm previous reports that the network strength scales linearly with the bond strength and with the crosslink density. The present results extend the Lake-Thomas theory to networks without pre-existing cracks which fail by the gradual accumulation of distributed damage and contribute to the development of a physical picture of failure in stochastic network materials.

Stretchy and disordered: Toward understanding fracture in soft network materials via mesoscopic computer simulations

Soft network materials exist in numerous forms ranging from polymer networks, such as elastomers, to fiber networks, such as collagen. In addition, in colloidal gels, an underlying network structure can be identified, and several metamaterials and textiles can be considered network materials as well. Many of these materials share a highly disordered microstructure and can undergo large deformations before damage becomes visible at the macroscopic level. Despite their widespread presence, we still lack a clear picture of how the network structure controls the fracture processes of these soft materials. In this Perspective, we will focus on progress and open questions concerning fracture at the mesoscopic scale, in which the network architecture is clearly resolved, but neither the material-specific atomistic features nor the macroscopic sample geometries are considered. We will describe concepts regarding the network elastic response that have been established in recent years and turn out to be pre-requisites to understand the fracture response. We will mostly consider simulation studies, where the influence of specific network features on the material mechanics can be cleanly assessed. Rather than focusing on specific systems, we will discuss future challenges that should be addressed to gain new fundamental insights that would be relevant across several examples of soft network materials.

Humidity dependence of fracture toughness of cellulose fibrous networks

Cellulose-based materials are increasingly finding applications in technology due to their sustainability and biodegradability. The sensitivity of cellulose fiber networks to environmental conditions such as temperature and humidity is well known. Yet, there is an incomplete understanding of the dependence of the fracture toughness of cellulose networks on environmental conditions. In the current study, we assess the effect of moisture content on the out-of-plane (i.e., z-dir.) fracture toughness of a particular cellulose network, specifically Whatman cellulose filter paper. Experimental measurements are performed at 16% RH along the desorption isotherm and 23, 37, 50, 75% RH along the adsorption isotherm using out-of-plane tensile tests and double cantilever beam (DCB) tests. Cohesive zone modeling and finite element simulations are used to extract quantitative properties that describe the crack growth behavior. Overall, the fracture toughness of filter paper decreased with increasing humidity. Additionally, a novel model is developed to capture the high peak and sudden drop in the experimental force measurement caused by the existence of an initiation region. This model is found to be in good agreement with experimental data. The relative effect of each independent cohesive parameter is explored to better understand the cohesive zone-based humidity dependence model. The methods described here may be applied to study rupture of other fiber networks with weak bonds.

Stress relaxation in network materials: the contribution of the network

Stress relaxation in network materials with permanent crosslinks is due to the transport of fluid within the network (poroelasticity), the viscoelasticity of the matrix and the viscoelasticity of the network. While relaxation associated with the matrix was studied extensively, the contribution of the network remains unexplored. In this work we consider two and three-dimensional stochastic fiber networks with viscoelastic fibers and explore the dependence of stress relaxation on network structure. We observe that relaxation has two regimes - an initial exponential regime, followed by a stretched exponential regime - similar to the situation in other disordered materials. The stretch exponent is a function of density, fiber diameter and the network structure, and has a minimum at the transition between the affine and non-affine regimes of network behavior. The relaxation time constant of the first, exponential regime is similar to the relaxation time constant of individual fibers and is independent of network density and fiber diameter. The relaxation time constant of the second, stretched exponential regime is a weak function of network parameters. The stretched exponential emerges from the heterogeneity of relaxation dynamics on scales comparable with the mesh size, with higher heterogeneity leading to smaller stretch exponents. In composite networks of fibers whose relaxation time constant is selected from a distribution with set mean, the stretch exponent decreases with increasing the coefficient of variation of the fiber time constant distribution. As opposed to thermal glass formers and colloids, in these athermal systems the dynamic heterogeneity is introduced by the network structure and does not evolve during relaxation. While in thermal systems the control parameter is the temperature, in this athermal case the control parameter is a non-dimensional structural parameter which describes the degree of non-affinity of the network.

Sharing the Load: Stress Redistribution Governs Fracture of Polymer Double Networks

The stress response of polymer double networks depends not only on the properties of the constituent networks but also on the interactions arising between them. Here, we demonstrate, via coarse-grained simulations, that both their global stress response and their microscopic fracture mechanics are governed by load sharing through these internetwork interactions. By comparing our results with affine predictions, where stress redistribution is by definition homogeneous, we show that stress redistribution is highly inhomogeneous. In particular, the affine prediction overestimates the fraction of broken chains by almost an order of magnitude. Furthermore, homogeneous stress distribution predicts a single fracture process, while in our simulations, fracture of sacrificial chains takes place in two steps governed by load sharing within a network and between networks, respectively. Our results thus provide a detailed microscopic picture of how inhomogeneous stress redistribution after rupture of chains governs the fracture of polymer double networks.

Strength and deformability of fibrin clots: Biomechanics, thermodynamics, and mechanisms of rupture

Fibrin is the major determinant of the mechanical stability and integrity of blood clots and thrombi. To explore the rupture of blood clots, emulating thrombus breakage, we stretched fibrin gels with single-edge cracks of varying size. Ultrastructural alterations of the fibrin network correlated with three regimes of stress vs. strain profiles: the weakly non-linear regime due to alignment of fibrin fibers; linear regime owing to further alignment and stretching of fibers; and the rupture regime for large deformations reaching the critical strain and stress, at which irreversible breakage of fibers ahead of the crack tip occurs. To interpret the stress-strain curves, we developed a new Fluctuating Spring model, which maps the fibrin alignment at the characteristic strain, network stretching with the Young modulus, and simultaneous cooperative rupture of coupled fibrin fibers into a theoretical framework to obtain the closed-form expressions for the strain-dependent stress profiles. Cracks render network rupture stochastic, and the free energy change for fiber deformation and rupture decreases with the crack length, making network rupture more spontaneous. By contrast, mechanical cooperativity due to the presence of inter-fiber contacts strengthens fibrin networks. The results obtained provide a fundamental understanding of blood clot breakage that underlies thrombotic embolization. STATEMENT OF SIGNIFICANCE: Fibrin, a naturally occurring biomaterial, is the major determinant of mechanical stability and integrity of blood clots and obstructive thrombi. We tested mechanically fibrin gels with single-edge cracks and followed ultrastructural alterations of the fibrin network. Rupture of fibrin gel involves initial alignment and elastic stretching of fibers followed by their eventual rupture for deformations reaching the critical level. To interpret the stress-strain curves, we developed Fluctuating Spring model, which showed that cracks render rupture of fibrin networks more spontaneous; yet, coupled fibrin fibers reinforce cracked fibrin networks. The results obtained provide fundamental understanding of blood clot breakage that underlies thrombotic embolization. Fluctuating Spring model can be applied to other protein networks with cracks and to interpret the stress-strain profiles.

Percolation of collagen stress in a random network model of the alveolar wall

Fibrotic diseases are characterized by progressive and often irreversible scarring of connective tissue in various organs, leading to substantial changes in tissue mechanics largely as a result of alterations in collagen structure. This is particularly important in the lung because its bulk modulus is so critical to the volume changes that take place during breathing. Nevertheless, it remains unclear how fibrotic abnormalities in the mechanical properties of pulmonary connective tissue can be linked to the stiffening of its individual collagen fibers. To address this question, we developed a network model of randomly oriented collagen and elastin fibers to represent pulmonary alveolar wall tissue. We show that the stress-strain behavior of this model arises via the interactions of collagen and elastin fiber networks and is critically dependent on the relative fiber stiffnesses of the individual collagen and elastin fibers themselves. We also show that the progression from linear to nonlinear stress-strain behavior of the model is associated with the percolation of stress across the collagen fiber network, but that the location of the percolation threshold is influenced by the waviness of collagen fibers.

Strength of stochastic fibrous materials under multiaxial loading

Many biological and engineering materials are made from fibers organized in the form of a stochastic crosslinked network, and the mechanics of the network controls the behavior of the material. In this work we investigate the strength of stochastic networks without pre-existing damage which fail due to crosslink rupture. Athermal networks ranging from approximately affine to strongly non-affine are subjected to multiaxial loading and the strength is evaluated using numerical models. It is observed that once the stress is normalized by the strength measured in uniaxial tension, the failure surface becomes approximately independent of network parameters. This extends the relation between strength and network parameters previously established in (S. Deogekar, M. R. Islam, R. C. Picu, Parameters controlling the strength of stochastic fibrous materials, Int. J. Solids Struct., 2019, 168, 194-202) to the multiaxial case. The failure surface depends on both first two invariants of the stress. Strongly non-affine networks behave somewhat different from the affine networks under loadings close to the hydrostatic and pure shear loading modes, while the difference disappears in the first quadrant of the principal stress space. The results are compared with experimental data from the literature.

The role of temperature in the rigidity-controlled fracture of elastic networks

We study the influence of thermal fluctuations on the fracture of elastic networks, via simulations of the uniaxial extension of central-force spring networks with varying rigidity. Studying their failure response, both at the macroscopic and microscopic level, we find that an increase in temperature corresponds to a more homogeneous stress (re)distribution and induces thermally activated failure of springs. As a consequence, the material strength decreases upon increasing temperature, the microscopic damage spreads over a larger area and a more ductile fracture process is observed. These effects are modulated by network rigidity and can therefore be tuned via the network connectivity and the rupture threshold of the springs. Knowledge of the interplay between temperature and rigidity improves our understanding of the fracture of elastic network materials, such as (biological) polymer networks, and can help to refine design principles for tough soft materials.

Random Fiber Networks With Superior Properties Through Network Topology Control

In this work, we study the effect of network architecture on the nonlinear elastic behavior and strength of athermal random fiber networks of cellular type. We introduce a topology modification of Poisson-Voronoi (PV) networks with convex cells, leading to networks with stochastic nonconvex cells. Geometric measures are developed to characterize this new class of nonconvex Voronoi (NCV) networks. These are softer than the reference PV networks at the same nominal network parameters such as density, cross-link density, fiber diameter, and connectivity number. Their response is linear elastic over a broad range of strains, unlike PV networks that exhibit a gradual increase of the tangent stiffness starting from small strains. NCV networks exhibit much smaller Poisson contraction than any network of same nominal parameters. Interestingly, the strength of NCV networks increases continuously with an increasing degree of nonconvexity of the cells. These exceptional properties render this class of networks of interest in a variety of applications, such as tissue scaffolds, nonwovens, and protective clothing.