Evaluation of the parallel coupling constitutive model for biomaterials using a fully coupled network-matrix model.

In this article we discuss the effective properties of composites containing a crosslinked athermal fiber network embedded in a continuum elastic matrix, which are representative for a broad range of biological materials. The goal is to evaluate the accuracy of the widely used biomechanics parallel coupling model in which the tissue response is defined as the additive superposition of the network and matrix contributions, and the interaction of the two components is neglected. To this end, explicit, fully coupled models are used to evaluate the linear and non-linear response of the composite. It is observed that in the small strain, linear regime the parallel model leads to errors when the ratio of the individual stiffnesses of the two components is in the range 0.1-10, and the error increases as the matrix approaches the incompressible limit. The data presented can be used to correct the parallel model to improve the accuracy of the overall stiffness prediction. In the non-linear large deformation regime linear superposition does not apply. The data shows that the matrix reduces the stiffening rate of the network, and the response is softer than that predicted by the parallel model. The correction proposed for the linear regime mitigates to a large extent the error in the non-linear regime as well, provided the matrix Poisson ratio is not close to 0.5. The special case in which the matrix is rendered auxetic is also evaluated and it is seen that the auxeticity of the matrix may compensate the stiffening introduced by the network, leading to a composite with linear elastic response over a broad range of strains.

Emergence of an apparent yield phenomenon in the mechanics of stochastic networks with inter-fiber cohesion.

In this work we investigate the contribution of inter-fiber cohesion to defining the mechanical behavior of stochastic crosslinked fiber networks. Fibers are athermal and store energy primarily in their bending and axial deformation modes. Cohesion between fibers is defined by an interaction potential. These structures are in equilibrium with the inter-fiber cohesive forces before external load is applied and their mechanical behavior is probed in uniaxial tension. Two types of configurations are considered: a state with high initial free volume in which contacts between fibers are scarce, and a state with low free volume and large number of fiber contacts. While in the absence of cohesion the response is hyperelastic, we observe that a yield point-like phenomenon develops as the strength of cohesion increases in both network types considered; we refer to this as an 'unlocking phenomenon'. The small strain stiffness increases as cohesion becomes more pronounced. The stiffness and unlocking stress are expressed in terms of network parameters and cohesion strength through a product of two functions, one dependent on network parameters only, and the other is a function of the cohesion strength. While the small strain response is controlled by cohesion, the large strain behavior is shown to be largely controlled by the network. Therefore, varying the strength of cohesion has no effect on strain stiffening. These observations provide a physical basis for the unlocking observed in both athermal and thermal network materials and are expected to facilitate the design of soft materials with novel properties.

Stiffening mechanisms in stochastic athermal fiber networks.

Stochastic athermal networks composed of fibers that deform axially and in bending strain stiffen much faster than thermal networks of axial elements, such as elastomers. Here we investigate the physical origin of stiffening in athermal network materials. To this end, we use models of stochastic networks subjected to uniaxial deformation and identify the emergence of two subnetworks, the stress path subnetwork (SPSN) and the bending support subnetwork (BSSN), which carry most of the axial and bending energies, respectively. The BSSN controls lateral contraction and modulates the organization of the SPSN during deformation. The SPSN is preferentially oriented in the loading direction, while the BSSN's preferential orientation is orthogonal to the SPSN. In nonaffine networks stiffening is exponential, while in close-to-affine networks it is quadratic. The difference is due to a much more modest lateral contraction in the approximately affine case and to a stiffer BSSN. Exponential stiffening emerges from the interplay of the axial and bending deformation modes at the scale of individual or small groups of fibers undergoing large deformations and being subjected to the constraint of rigid cross-links, and it is not necessarily a result of complex interactions involving many connected fibers. An apparent third regime of quadratic stiffening may be evidenced in nonaffinely deforming networks provided the nominal stress is observed. This occurs at large stretches, when the BSSN contribution of stiffening vanishes. However, this regime is not present if the Cauchy stress is used, in which case stiffening is exponential throughout the entire deformation. These results shed light on the physical nature of stiffening in a broad class of materials including connective tissue, the extracellular matrix, nonwovens, felt, and other athermal network materials.

Probing soft fibrous materials by indentation.

Indentation is often used to measure the stiffness of soft materials whose main structural component is a network of filaments, such as the cellular cytoskeleton, connective tissue, gels, and the extracellular matrix. For elastic materials, the typical procedure requires fitting the experimental force-displacement curve with the Hertz model, which predicts that f=kδ1.5 and k is proportional to the reduced modulus of the indented material, E/(1-ν2). Here we show using explicit models of fiber networks that the Hertz model applies to indentation in network materials provided the indenter radius is larger than approximately 12lc, where lc is the mean segment length of the network. Using smaller indenters leads to a relation between force and indentation displacement of the form f=kδq, where q is observed to increase with decreasing indenter radius. Using the Hertz model to interpret results of indentations in network materials using small indenters leads to an inferred modulus smaller than the real modulus of the material. The origin of this departure from the classical Hertz model is investigated. A compacted, stiff network region develops under the indenter, effectively increasing the indenter size and modifying its shape. This modification is marginal when large indenters are used. However, when the indenter radius is small, the effect of the compacted layer is pronounced as it changes the indenter profile from spherical towards conical. This entails an increase of exponent q above the value of 1.5 corresponding to spherical indenters. STATEMENT OF SIGNIFICANCE: The article presents a study of indentation in network biomaterials and demonstrates a size effect which precludes the use of the Hertz model to infer the elastic constants of the material. The size effect occurs once the indenter radius is smaller than approximately 12 times the mean segment length of the network. This result provides guidelines for the selection of indentation conditions that guarantee the applicability of the Hertz model. At the same time, the finding may be used to infer the mean segment length of the network based on indentations with indenters of various sizes. Hence, the method can be used to evaluate this structural parameter which is not easily accessible in experiments.

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.

Random Fiber Network Loaded by a Point Force.

This article presents the displacement field produced by a point force acting on an athermal random fiber network (the Green function for the network). The problem is defined within the limits of linear elasticity, and the field is obtained numerically for nonaffine networks characterized by various parameter sets. The classical Green function solution applies at distances from the point force larger than a threshold which is independent of the network parameters in the range studied. At smaller distances, the nonlocal nature of fiber interactions modifies the solution.

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.

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.

Tensile behavior of non-crosslinked networks of athermal fibers in the presence of entanglements and friction.

Many biological and soft artificial materials contain a random network of non-crosslinked fibers as their main structural component. The excluded volume interactions (contact forces) at fiber contacts control the mechanical behavior of these systems. This physics has been studied extensively in compression, but little is known about the relation between network structure and its mechanical response in tension. In particular, although occasionally used conjecturally, the notion of fiber entanglements in athermal networks is not well defined, nor is it clear what role entanglements play in athermal network mechanics. The primary contribution of this work is the introduction of a measure of the degree of entanglement of a system of random athermal fibers, and the definition of its relationship with the mechanical behavior of the network. Entanglements confine the fibers during tensile loading, reduce the auxetic effect in mat-like networks, and maintain the inter-fiber contact density. In the absence of this contribution, reduction of the contact density during tensile loading due to auxeticity results in stress reduction. Entanglements stabilize the network via a tensegrity mechanism similar to that operating in woven materials and lead to network stiffening. The relation between the proposed measure of entanglements and the fiber volume fraction is defined. The effect of inter-fiber friction on the mechanics of entangled mat-like non-crosslinked fiber networks is also evaluated.

Size Effects in Random Fiber Networks Controlled by the Use of Generalized Boundary Conditions.

Materials with a stochastic fiber network as the main structural constituent are broadly encountered in engineering and in biology. These materials are characterized by multiscale heterogeneity and hence their properties evaluated numerically or experimentally are generally dependent on the size of the sample considered. In this work we evaluate the size effect on the linear and non-linear mechanical response of three-dimensional stochastic fiber networks and determine its dependence on material parameters and on the degree of affinity of network deformation. The size effect is more pronounced in non-affine than in affine networks and decreases slowly when the model size increases. In order to eliminate this effect, models lager than can be effectively solved with current computers have to be considered. To address this issue, we propose a method that allows using relatively small models, while accurately predicting the small and large strain behaviors of the network. The method is based on the generalized boundary conditions introduced in (Glüge 2013, Computational Materials Science 79, 408-416), which are being adapted here to the requirements imposed by fibrous 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.

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.

Mechanical behavior of nonwoven non-crosslinked fibrous mats with adhesion and friction.

We present a study of the mechanical behavior of planar fibrous mats stabilized by inter-fiber adhesion. Fibers of various degrees of tortuosity and of infinite and finite length are considered in separate models. Fibers are randomly distributed, are not cross-linked, and interact through adhesion and friction. The variation of structural parameters such as the mat thickness and the mean segment length between contacts along given fibers with the strength of adhesion is determined. These systems are largely dissipative in that most of the work performed during deformation is dissipated frictionally and only a small fraction is stored as strain energy. The response of the mats to tensile loading has three regimes: a short elastic regime in which no sliding at contacts is observed, a well-defined sliding regime characterized by strain hardening, and a rapid stiffening regime at larger strains. The third regime is due to the formation of stress paths after the fiber tortuosity is pulled out and is absent in mats of finite length fibers. Networks of finite length fibers lose stability during the second regime of deformation. The scaling of the yield stress, which characterizes the transition between the first and the second regimes, and of the second regime's strain hardening modulus, with system parameters such as the strength of adhesion and friction and the degree of fiber tortuosity are determined. The strength of mats of finite length fibers is also determined as a function of network parameters. These results are expected to become useful in the design of electrospun mats and other planar fibrous non-cross-linked networks.

Strength of filament bundles - The role of bundle structure stochasticity.

Most biological fibrous materials are hierarchical, in the sense that fibers of finite length assemble in bundles, which then form networks with structural role. Examples include collagen, silk, fibrin and microtubules. Some artificial fiber-based materials share this characteristic, examples including carbon nanotube (CNT) yarns and unidirectional composites. Here we study bundles in which filaments do not break, while bundle rupture happens by the failure of inter-filament crosslinks, followed by pull-out. In all cases, the crosslinks are randomly distributed along interfaces. The strength of such bundles depends on material parameters of the filaments and crosslinks, such as their stiffness and strength, and on the cross-link density. We focus on the dependence of the bundle strength on two parameters: filament waviness and filament staggering. Bundles with regular staggering are stronger than those with stochastic staggering. We identify the optimal regular staggering that maximizes the strength. Filament waviness increases the strength of stochastically staggered bundles at constant crosslink density but decreases the strength of regularly staggered bundles. Results for bundles with permanent crosslinks, which never reform once they break, as well as transient crosslinks capable of reforming during deformation are presented, and it is shown that the general trends are independent of the nature of the crosslinks. The results are discussed in the context of collagen and carbon nanotube bundles.

Filamentary structures that self-organize due to adhesion.

We study the self-organization of random collections of elastic filaments that interact adhesively. The evolution from an initial fully random quasi-two-dimensional state is controlled by filament elasticity, adhesion and interfilament friction, and excluded volume. Three outcomes are possible: the system may remain locked in the initial state, may organize into isolated fiber bundles, or may form a stable, connected network of bundles. The range of system parameters leading to each of these states is identified. The network of bundles is subisostatic and is stabilized by prestressed triangular features forming at bundle-to-bundle nodes, similar to the situation in foams. Interfiber friction promotes locking and expands the parametric range of nonevolving systems.

Bone toughening through stress-induced non-collagenous protein denaturation.

Bone toughness emerges from the interaction of several multiscale toughening mechanisms. Recently, the formation of nanoscale dilatational bands and hence the accumulation of submicron diffuse damage were suggested as an important energy dissipation processes in bone. However, a detailed mechanistic understanding of the effect of this submicron toughening mechanism across multiple scales is lacking. Here, we propose a new three-dimensional ultrastructure volume element model showing the formation of nanoscale dilatational bands based on stress-induced non-collagenous protein denaturation and quantify the total energy released through this mechanism in the vicinity of a propagating crack. Under tensile deformation, large hydrostatic stress develops at the nanoscale as a result of local confinement. This tensile hydrostatic stress supports the denaturation of non-collagenous proteins at organic-inorganic interfaces, which leads to energy dissipation. Our model provides new fundamental understanding of the mechanism of dilatational bands formation and its contribution to bone toughness.

Structural evolution and stability of non-crosslinked fiber networks with inter-fiber adhesion.

Adhesion plays an important role in the mechanics of nanoscale fibers such as various biological filaments, carbon nanotubes and artificial polymeric nanofibers. In this work we study assemblies of non-crosslinked filaments and characterize their adhesion-driven structural evolution and their final stable structure. The key parameters of the problem are the network density, the fiber length, the bending stiffness of fibers and the strength of adhesion. The system of fibers self-organizes in one of three types of structures: locked networks, in which fibers remain in the as-deposited state, cellular networks, in which fibers form bundles and these organize into a larger scale network, and disintegrated networks, in which the network of bundles becomes disconnected. We determine the parametric space corresponding to each of these structures. Further, we identify a triangular structure of bundles, similar to the Plateau triangle occurring in foams, which stabilizes the network of bundles and study in detail the stabilization mechanism. The analysis provides design guidelines and a physical picture of the stability and structure of random fiber networks with adhesion.

Publisher Correction: Morphology and mechanics of fungal mycelium.

A correction to this article has been published and is linked from the HTML and PDF versions of this paper. The error has been fixed in the paper.

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.