Random Fiber Networks With Superior Properties Through Network Topology Control

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.

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.

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.

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.

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.