Nonlinear Mechanical Properties of Prestressed Branched Fibrous Networks

Abstract

Random fiber networks constitute the solid skeleton of many biological materials such as the cytoskeleton of cells and extracellular matrix of soft tissues. These random networks show unique mechanical properties such as nonlinear shear strain-stiffening and strain softening when subjected to preextension and precompression, respectively. In this study, we perform numerical simulations to characterize the influence of axial prestress on the nonlinear mechanical response of random network structures as a function of their micromechanical and geometrical properties. We build our numerical network models using the microstructure of disordered hexagonal lattices and quantify their nonlinear shear response as a function of uniaxial prestress strain. We consider three different material models for individual fibers and fully characterize their influence on the mechanical response of prestressed networks. Moreover, we investigate both the influence of geometric disorder keeping the network connectivity constant and the influence of the randomness in the stiffness of individual fibers keeping their mean stiffness constant. The effects of network connectivity and bending rigidity of fibers are also determined. Several important conclusions are made, including that the tensile and compressive prestress strains, respectively, increase and decrease the initial network shear stiffness but have no effect on the maximal shear modulus. We discuss the findings in terms of microstructural properties such as the local strain energy distribution.

Significance

Fibrous networks are ubiquitous and are found in a broad range of natural and manmade structures. In this work, we characterize the influence of prestress on the nonlinear mechanical response of branched random networks. The prestress is commonly seen in biological networks such as cardiovascular tissues and cytoskeleton. This work investigates the influence of material properties of individual fibers, irregularity of network microstructure, and network connectivity on prestress effects. The results are discussed in terms of microstructural details such as the local strain field distribution. This study provides additional insight into the nonlinear mechanics of random networks and could find applications in designing novel fibrous materials.

Introduction

Materials with a fibrous solid skeleton are found in a broad range of natural and manmade structures such as paper products, textiles, the cytoskeleton of cells, and collagenous tissues (1, 2, 3, 4). For example, an intricate network of protein macromolecules, known as the extracellular matrix, is responsible for the strength and structural integrity of soft tissues (3,4). Furthermore, the cytoskeleton is a network of interlinking protein fibers, e.g., actin filaments, microtubules, and intermediate filaments, that provides mechanical strength to cells (2,5). In particular, the dynamic cross-linking of actin filaments, mainly beneath the plasma membrane, forms a branched network that passively remodels and provides mechanical support to the cells during movement. Different forms of cellular motility depend on the formation of new branched actin networks. A detailed analysis of the microstructure-function relationship of random networks is required for a complete understanding of the mechanical properties of biological and natural fibrous materials.

The nonlinear strain-stiffening behavior of random fiber networks is among their unique mechanical properties that have been experimentally observed (6, 7, 8). The mechanical properties of fibrous materials and their relation to the properties of their constituting fibers can be investigated using numerical models (9,10). Previous studies have shown that the mechanical response of random networks is a function of their architecture and mechanical properties of their fibrous constituents (7, 8, 9, 10, 11, 12, 13, 14, 15). Furthermore, the specific architecture of random networks has been shown to have significant influence on certain aspects of their mechanical response (16,17). Biological networks often appear either as 1) cross-linked (z ≤ 4) networks in which fibers are interconnected by cross-linking proteins or as 2) branched (z ≤ 3) networks in which the fibers are split into other fibers at the branching points (18). The connectivity z of both types of these bionetworks is below the Maxwell isostatic threshold, i.e., their constituting fibers are weakly constrained and their nonaffine bending deformation defines their mechanics (11,12,19). The significance of the nonaffine deformation is proportional to the ratio of network connectivity/Maxwell isostatic threshold. In particular, as the connectivity of marginally stable networks becomes closer to the isostatic point, the role of stretching-dominated deformation becomes more important. Despite their significantly low average connectivity, biological networks such as collagen networks and cytoskeleton are stable even under small deformations. The finite nonzero stiffness of these structures, from the theoretical viewpoint, is due to different stabilization factors. In our study, we are particularly interested in characterizing the influence of prestress, in the form of extension and compression, on the nonlinear mechanical response of random networks subjected to large shear deformation.

The prestress, the presence of residual stresses in unloaded specimens, is commonly seen in biological networks such as cardiovascular tissues, blood clots, and cell cytoskeleton (20, 21, 22, 23). The residual stresses observed in these samples could have important effects on their mechanical response. For example, the prestress is necessary for reinforcing and homogenizing the resistance of cardiovascular tissues against external pressure (20,21). Furthermore, the prestress enhances the alignment of fibers of the extracellular matrix toward the applied loading direction (24). A prestressed hexagonal network of connective tissue fibers is responsible for the mechanical response of alveolar ducts in lung tissue (25,26). In particular, the prestress that remains after expiration is responsible for stiffening the extracellular matrix around each alveolus. Finally, the cell cytoskeleton is prestressed as the tensile force of contractile microfilaments is balanced by the compression of internal microtubules (22). These observations indicate the importance of analyzing prestress as one of the key factors in defining the mechanical properties of random networks.

The mechanical response of prestressed random fiber networks can be investigated using two-dimensional and three-dimensional network models. For example, Vahabi et al. investigated the prestress effects by considering disordered triangular lattice-based networks with a network connectivity z of ∼3–4 (27). Furthermore, Ban et al. used a three-dimensional network model to characterize the mechanics of extracellular matrices under uniaxial, biaxial, and shear loading (28). These and similar studies showed that the uniaxial tensile prestress increases the shear modulus of random networks, whereas the compressive prestress softens their shear response. Indeed, the difference between the shear stiffening or softening behavior of soft tissues and pure polymer networks in compression and tension is expected to be partly because biopolymer networks, such as extracellular matrix and cytoskeleton, are prestressed (29, 30, 31, 32). The presence of particle inclusions, e.g., cells, inside these networks has also been proposed to have a role in their strain-stiffening response (33).

The primary objective of this study is to provide a thorough investigation of how various mechanical and geometrical network parameters, such as architectural irregularity and randomness or nonlinearity of material properties of individual fibers, affect the important structural role of prestress in the nonlinear mechanics of random networks. For this purpose, we represent the microstructure of random networks as disordered and diluted hexagonal lattices with 2 < z ≤ 3. Similar branched networks have previously been used to represent numerically the mechanical response of naturally occurring bionetworks such as those that are formed by collagen and actin (1,34, 35, 36, 37, 38). The work is organized as follows: Methods describes the details of the numerical model including the geometry of branched networks. Results presents the results along with discussing the effects of prestress, network architectural disorder, and the mechanical response of constituting fibers on the nonlinear elasticity of branched fiber networks. Conclusions concludes the study with a brief summary and presentation of the main conclusions.

Methods

In this work, disordered and diluted hexagonal lattices are used to represent branched networks. A regular hexagonal lattice is a submarginal branched network with a coordination number of three at every node. Although this study does not intend to represent the mechanical response of any specific native or synthetic network of biological interest, it is interesting to note that certain types of collagen form only hexagonal networks. For example, a noticeable hexagonal-like pattern is seen in immunofluorescently labeled collagen networks of lung tissue (25). A coordination number of less than three, along with a disordered architecture, has been reported for collagen networks (25,39). Here, we perform a dilution of regular hexagonal lattices by randomly removing fiber segments with the probability of 1 − p, where p is the bond existence probability. The connectivity of diluted lattices will be below three, and the network architecture becomes disordered because of the dilution process. Fig. 1 a shows a typical diluted hexagonal lattice.

Figure 1.

(a) A portion of representative diluted hexagonal lattice network that is used in numerical network models of this study. (b) A portion of typical irregular diluted hexagonal network is shown; the irregular architecture is introduced by perturbing the position of the vertices without changing the network connectivity. (c) A representative branch of diluted hexagonal lattices composed of linear elastic fibers having different elastic moduli selected from a uniform probability distribution function is shown. (d) A schematic representation of three material models considered for the mechanical response of individual fibers is shown: model 1, linear elastic in both tension and compression; model 2, nonlinear stiffening after an initial elastic response in tension but linear elastic in compression; and model 3, nonlinear stiffening after an initial elastic response in tension and a different linear elastic response in compression.

Despite the random dilution process, the resulting disordered networks inherit certain geometric features of regular hexagonal lattices, for example, the initial orientation of fiber segments. These features may affect the numerical results. Thus, we create irregularity in the architecture of lattice-based random networks. Keeping the network connectivity constant, we introduce irregularity through random perturbation of the position of every node. In particular, each node is moved by a random distance δr along an arbitrary direction such that its new position is anywhere inside a circle of radius r_max that is centered around the initial nodal position. In this work, r_max is taken to be half of the distance between two adjacent vertices of the regular hexagonal lattice: r_max = l₀/2. A small portion of a typical irregular lattice generated by the above algorithm is shown in Fig. 1 b; the resulting networks have different distributions of initial bond angles and fiber segment lengths.

Furthermore, fibers are rigidly cross-linked at the branching points. It is noted that the nature of these cross-links could have significant influence on the mechanical response of random networks (40). The fibers are assumed to behave as beam elements with axial rigidity μ and bending rigidity κ. The dimensionless bending stiffness $\overline{\kappa }$ = κ/μl₀², where l₀ is the distance between two adjacent vertices in an undiluted hexagonal lattice, quantifies the relative strength of bending stiffness to stretching stiffness of the fibers. The commercial finite-element package ABAQUS is used to conduct the numerical simulations. All fiber segments are meshed with multiple beam elements to accurately represent their deformation. The following boundary conditions are considered to study the mechanical response of random networks under prestress.

The uniaxial tensile or compressive prestress strain ε_p is applied by increasing or decreasing the vertical position of all nodes on the top edge of diluted lattices, i.e., u_y = ε_p L, u_x = 0, where L is the network dimension in both directions and u_x and u_y are displacements of nodes in the x and y directions, respectively. The origin of the coordinate system is located at the bottom left corner of the simulation domain. The nodes on the bottom edge, y = 0, are fully constrained horizontally and vertically but are set free to rotate. A periodic boundary condition is imposed on side edges x = 0 and x = L. The shear response of the prestressed networks is obtained by applying incremental shear strain δγ. To apply the shear strain, we fix the nodes at the lower edge y = 0 in the horizontal and vertical directions, translate horizontally (u_x = δγ L and u_y = 0) those at the upper edge y = L, and consider a periodic boundary condition for the nodes on the vertical edges. From the finite-element simulation results, we calculate the shear stress by dividing the sum of forces in the fibers intersecting the upper edge by L (41). The differential shear modulus K of networks, also referred to as stiffness herein, at each increment is defined as the slope of the stress-strain response, i.e., K = dτ/dγ, where τ is the shear stress in units of μ/l₀. The initial shear modulus of the networks at γ = 1% is denoted by K₀. Furthermore, we compute the stretching energy U^a = $\sum _{fibers}\int \frac{\mu }{2}$(∂u_i/∂s)²ds and bending energy U^b = $\sum _{fibers}\int \frac{\kappa }{2}$(∂ψ_i/∂s)²ds, where ∂u_i/∂s is the axial strain and ∂ψ_i/∂s is the curvature change of a beam element about its neutral surface.

In this work, we also investigate the influence of randomness in the elastic modulus of fibers on the mechanical response of prestressed networks under shear. To this end, we assign random stiffness to all fiber segments by multiplying the elastic modulus E with a random variable having a uniform distribution between (1 − λ) and (1 + λ) where |λ| < 1 (Fig. 1 c). Furthermore, we consider fibers with different mechanical behavior in tension and compression as well as with the nonlinear tensile response (Fig. 1 d). In particular, in addition to networks composed of linear elastic fibers (referred to as model 1), we consider fibers with strain-stiffening in tension (42),

$${\sigma }_f=\{\begin{array}{l}\rho E{\epsilon }_f{\epsilon }_f\le 0\hfill \\ E{\epsilon }_{f}0\le {\epsilon }_f\le {\epsilon }_y\hfill \\ E{\epsilon }_y+a\left(e^{b\left({\epsilon }_f-{\epsilon }_y\right)}-1\right){\epsilon }_y\le {\epsilon }_f\hfill \end{array},$$ (1)

where E is the linear elastic modulus; σ_f is the stress in fibers; ε_f is the strain in fibers; ρ, a, and b are constants; and ε_y is the strain at which fibers become nonlinear. Without loss of generality, we consider b = 10 and a = 10 E in this work. It is noted that the linear elastic behavior is only considered for the mechanical response of fibers in compression. However, following what has previously been suggested (43, 44, 45, 46), we consider fiber microbuckling as a loss of stiffness in compression (ρ = 0.5). Similar overall results are expected for any other ratio 0 < ρ ≪ 1 (43). In the following, the cases with ρ = 1 and ρ = 0.5 are referred to as strain-stiffening response without microbuckling (model 2) and strain-stiffening with microbuckling response (model 3), respectively.

Results

Effects of uniaxial prestress on nonlinear mechanics of fiber networks

Fig. 2 a shows the differential shear modulus versus shear strain when no uniaxial prestress is applied. In agreement with previous results in the literature (18), if the applied shear strain is less than γ₀ (where γ₀ denotes the onset of nonlinear stiffening and is shown by star symbols in Fig. 2 a), there is a negligible variation in the shear modulus of networks at constant $\overline{\kappa }$. In this regime, the initial shear modulus, K₀, increases with $\overline{\kappa }$ (results not shown explicitly but can be inferred from Fig. 2 a). In this study, γ₀ (for networks with $\overline{\kappa }$ < 10⁻³) is the strain at which the minimum of the function K/σ vs. σ occurs (47) (Fig. 2 b). Simpler methods also exist to determine γ₀. For example, it can be taken as the strain at which the shear modulus becomes three times the initial linear modulus (18). As the applied strain increases (γ > γ₀), the shear modulus increases nonlinearly. Similar to the mechanical behavior of diluted triangular networks (48), the nonlinear shear stiffening beyond γ₀ can be divided into two distinct regimes. After the significantly nonlinear growth up to the critical strain γ_c (shown by the cross symbols in Fig. 2 a), the shear modulus shows an almost linear variation with the shear strain. This transition from the nonlinear to the linear variation of shear modulus occurs because the network deformation switches from bending dominated to stretching dominated, in which central force interactions and stretching of fibers become the primary mechanisms resisting the external deformation (see Fig. 3, a and b for ε_p = 0%). There are several ways to determine γ_c (49,50). Here, we determine γ_c as the strain for which the maximal value of dlog(K)/dlogγ versus the shear strain is obtained. For strains much larger than γ_c, our numerical simulations (results not shown here) show that the shear modulus approaches a constant value for all $\overline{\kappa }$-values. This elusive observation is due to the drawbacks of numerical simulations, including the assumption that highly stretched fibers behave linear elastically without showing nonlinear response or failing (42). Experimental measurements suggest that localized failure or total collapse of the network structure is expected at these very high strains (48).

Figure 2.

(a) The effect of different of $\overline{\kappa }$ on the differential shear modulus variation of diluted hexagonal lattices with average network connectivity z = 2.6 when no prestress strain is applied. The star and cross symbols denote the onset of the nonlinear stiffening and the transition from the bending-dominated to the stretching-dominated response, respectively. (b) The variation of K/σ as a function of shear stress σ is shown; the minimum of these curves gives the critical strain γ₀. The variation of normalized initial shear modulus as a function of normalized prestress stress σ_p is also shown. The dashed lines are linear fits to the numerical results. The effect of axial prestress strain ε_p on the normalized differential shear modulus of networks with (c) $\overline{\kappa }$ = 0.0001 and (d) $\overline{\kappa }$ = 0.01 is shown, where K₀ is the initial shear modulus of networks with ε_p = 0%.

Figure 3.

The relative energies (a) U^b/U^t and (b) U^a/U^t, where U^a, U^b, and U^t are the stretching, bending, and total energy, respectively. (c) The effect of axial prestress strain ε_p on the relative energy U^b/U^t and U^a/U^t (left y axis) and the variation of the applied shear strain at which the bending and axial total energies are equal, U^b = U^a (right y axis), are given. (d) The fiber angle distribution for branched networks subjected to 20% uniaxial prestress strain is shown. The results shown are for networks with $\overline{\kappa }$ = 0.0001.

The possible effects of tensile or compressive prestress on the shear modulus variation versus shear strain are shown in Fig. 2, c and d for networks composed of linear elastic fibers with $\overline{\kappa }$ = 0.0001 and 0.01, respectively. Here, the shear modulus is normalized by the initial shear modulus K₀ of unloaded (no prestress) networks. In general, and in agreement with what has recently been reported (27,28), tensile prestress increases the shear modulus, whereas compressive prestress makes the networks softer. The numerical simulations of our study also agree with recent rheological experiments showing that network storage modulus has a linear variation with the axial prestress stress (28), i.e., the amount of stiffening is linearly proportional to the magnitude of the tensile prestress; this dependence is more significant for small $\overline{\kappa }$ (Fig. 2 c). As the networks are prestressed in tension, fibers rotate in the loading direction. The amount of this rotation increases when fibers are flexible (small $\overline{\kappa }$) and/or the prestress level is large. Less rotation in the loading direction is seen in networks with stiff fibers. On the other hand, networks soften under the initial compression; a similar behavior is observed experimentally for collagen fiber samples (30). This softening behavior could be due to bending and buckling of fibers when networks are prestressed by compressive axial strain; note that buckling and bending of fibers are energetically more favorable than their pure compression.

Fig. 3, a and b show the relative energy contribution of bending and stretching deformation for networks with $\overline{\kappa }$ = 0.0001, respectively. When the axial prestress strain is zero, the bending energy contribution decreases (and the stretching energy contribution increases) with increasing the applied shear strain. As discussed earlier, bending of fibers dominates the network behavior throughout the nonlinear stiffening regime until the critical strain γ_c, at which the bending dominated behavior becomes stretching dominated. In other words, bending of fibers is not only important in the small strain elastic regime but also significant throughout the nonlinear stiffening regime (48,51).

The application of axial prestress strain affects the relative importance of stretching and bending energy contribution in the network mechanical behavior. In particular, with increasing the initial extension, γ_c shifts toward smaller strains, and the nonlinear strain-stiffening occurs only over a limited range of shear strains. The compressive prestress plays an opposite role, i.e., it shifts the critical strain toward larger strains and makes the fiber networks softer. This is because bending or buckling of fibers, caused by the initial uniaxial compression, creates excess floppiness in the networks. Fig. 3, a and b suggest that the sharp transition from bending- to stretching-dominated deformation disappears for large applied prestress extension (ε_p ≥ 20%). Fig. 3 c shows the effect of axial prestress strain on the relative contribution of bending energy U_b and stretching energy U_a at γ = 1%. Furthermore, this figure shows the variation of the applied shear strain at which U_b = U_a as a function of the axial prestress strain. The curves shown in this figure confirm the above discussion for the role of prestress on the nonlinear mechanics of fiber networks. It is expected that for sufficiently large extension, at which stretching of fibers is the primary mode of deformation, affine models could give an accurate prediction of the network shear response at any applied shear strain. It is noted that the ratio of bending/axial rigidity of the fibers has a direct influence on the prestress effects. For example, 20% axial extension increases the normalized linear shear modulus by ∼25 and 2.5 when $\overline{\kappa }$ = 0.0001 and 0.01, respectively. This is because it is energetically more costly to build excess lengths in the fibers with large $\overline{\kappa }$. Fibers of non-prestressed networks are in the 0, 60, and 120° directions when angles are measured counterclockwise from the x axis to fibers. Fig. 3 d shows the influence of the axial prestress of +20% on the orientation of fibers before the application of the shear load. It is seen that applying the axial strain significantly alters the initial uniform distribution of network fibers and reorients a portion of fibers in 45°, i.e., the shear loading direction. Fibers with 45° orientation are expected to reduce the initial floppiness of networks by resisting the applied shear deformation. Thus, these prestressed networks have larger initial shear modulus in comparison with those without prestress (Fig. 2). Furthermore, the reoriented fibers in 45° carry the applied shear load primarily in tension, which explains the observed dependence of the ratio of the bending energy U_b and stretching energy U_a on the axial prestress (Fig. 3, a–c).

The material properties of individual fibers

In this section, we examine the effects of mechanical behavior of fibers on the macroscopic behavior of networks particularly under axial prestress. As stated before, this work considers different constitutive laws for the mechanical response of fibers: linear elastic in tension and compression (model 1), strain-stiffening in tension and linear elastic in compression (model 2), and strain-stiffening in tension and buckling in compression (model 3); see Eq. 1. The mechanical properties of networks without any prestress are initially discussed. Fig. 4 a shows the effect of material properties of individual fibers on the shear modulus of branched networks with $\overline{\kappa }$ = 0.0001 and 0.01. Comparing the shear modulus versus shear strain curves obtained for networks having similar geometric microstructure but composed of fibers with different mechanical response, we observe that the network shear modulus depends on the constitutive response of individual fibers. For example, the maximal stiffness of networks with model 2 fibers is almost sevenfold larger than that of networks with model 1 fibers. Moreover, buckling of individual fibers, modeled by ρ = 0.5, decreases the initial shear modulus of networks with model 1 fibers by about 30%; however, it is noted that this constitutive model does not significantly affect the shear modulus at large shear strain. The conclusion that buckling of the fibers primarily affects network elasticity at the small far-field strains agrees with previous studies (44,45). When loaded in shear, a number of force chains oriented at about 45° angle with respect to the horizontal axis emerge (Fig. 4 b); the fibers belonging to these force chains carry positive axial force. Other force chains composed of fibers in compression are also formed at an almost 135° angle with respect to the imposed shear strain direction (Fig. 4 b). Decreasing the elastic modulus of fibers in compression reduces the ability of compressive force chains to resist external forces and subsequently leads to a decrease in the macroscopic shear modulus of networks. The fibers at 135° direction inhibit the bending of other fibers and are primarily important when bending modes govern the mechanics of networks. When the applied shear strain becomes very large, the reorientation of fibers in the loading direction is complete, and compressive force chains have a negligible role in resisting external loads. This is why reducing the elastic modulus of fibers does not have any influence on the mechanical response of networks at large shear strains (Fig. 4 a).

Figure 4.

(a) The effect of material properties of fibers on the variation of differential shear moduli of branched networks $\overline{\kappa }$ = 0.01 and 0.0001; no uniaxial prestress strain is applied, and average network connectivity z = 2.6. (b) The fibers carrying effective positive and negative axial force are shown. The effect of prestress strain on the normalized initial shear modulus at γ = 1% of prestressed networks with (c) $\overline{\kappa }$ = 0.01 and (d) $\overline{\kappa }$ = 0.0001 is shown when different constitutive models are used to represent the mechanical response of individual fibers. The shear modulus K₀ of similar networks without uniaxial prestress strain is used as a normalization factor. (e) The normalized maximal strain inside random networks as a function of the applied shear strain for networks composed of model 1 and model 3 fibers is shown. The normalization factor is the maximal tensile strain at γ = 100% for networks of linear elastic (model 1) fibers without uniaxial prestress strain.

Fig. 4, c and d plot the normalized initial shear stiffness, calculated at γ = 1%, for branched networks with $\overline{\kappa }$ = 0.01 and 0.0001 at various prestress levels, respectively. The network stiffness at each prestress level is normalized by the stiffness of the same networks when no prestress is applied. It is seen that prestress has the same effect on the shear modulus of networks independent of the material properties of their constituting fibers. The networks stiffen when the uniaxial strain is positive and soften when it is negative. The actual amount of stiffening and softening depends on 1) the specific material model used to represent the mechanical response of fibers, 2) the magnitude of the prestress strain, and 3) the value of the dimensionless bending stiffness $\overline{\kappa }$. At any constant $\overline{\kappa }$, networks composed of fibers represented by model 3 exhibit the most stiffness alternation because of the prestress. For these networks, fibers reorient more easily in the direction of the axial load and they subsequently get stretched when the networks are sheared. This means that less excess length can be built up in these fibers compared with those having other constitutive models. Thus, the stretching modes become more significant than the soft bending deformation modes. The bending rigidity of fibers is an important factor in the amount of fiber rotation in the prestress stage. At 20% uniaxial prestress extension, the normalized initial moduli of networks with model 3 fibers are ∼3.5 and 35 when $\overline{\kappa }$ = 0.01 and 0.0001, respectively. The significantly larger value of the initial shear modulus of networks with $\overline{\kappa }$ = 0.0001 is because their constituting fibers easily get oriented in the direction of the applied uniaxial extension, and they could subsequently resist against the shear strain in the stretching mode.

To better understand the role of prestress in the mechanical properties of networks and its relation to the material properties of individual fibers, we compute the maximal tensile strain in branched networks with $\overline{\kappa }$ = 0.0001 as a function of the applied shear strain in different material models and at different uniaxial extension, Fig. 4 e. We use the maximal tensile strain at γ = 100% in networks with linear elastic (model 1) fibers to normalize the results. In all cases, the maximal tensile strain is almost constant with increasing shear strain to a certain point and then rises sharply after a certain shear strain, referred to as γ₀ in this work. This plot clearly shows that the initial uniaxial extension generates initial tensile strain, slightly lower than the macroscopic applied strain, inside the networks when γ = 0. With increasing the applied strain, the role of material model of the fibers becomes significant. Overall, networks with stiff-buckling material models (models 2 and 3) have lower maximal local tensile strain in their fibers at γ = 100%. This is because the strain-stiffening response of fibers makes their extension energetically less favorable as the applied strain increases. The change in the local strain distribution due to the material models of fibers should be important in the failure analysis of polymer networks (52).

Effects of irregularity in network architecture

The nonlinear mechanical response of submarginal disordered and diluted lattices and randomly connected networks agrees with each other; therefore, both models are suitable for investigating the mechanical response of biological networks (12,19,50,51,53, 54, 55, 56, 57). The microstructure of disordered lattice-based models is obtained from removing bonds from regular lattices. Removing bonds from regular lattices, in addition to creating geometric disorder, has a direct effect on the network connectivity.

In this section, we investigate how the introduction of geometric disorder while keeping the network connectivity constant may affect the mechanical response of disordered hexagonal lattice networks. This is implemented by introducing randomness in the nodal positions of diluted lattices as described in Methods. The perturbation of nodal positions alters the initial length and orientation of fibers. The diluted lattices, without geometric disorder, consist of fibers oriented at angles 0, 60, and 120° with respect to the horizontal axis. Depending on the parameters chosen for the perturbation algorithm, different angle distribution is obtained. However, we leave such detailed discussion for a future study. Here, we only note that the irregular networks have different fiber angle distributions and almost an unaltered average fiber segment length.

Fig. 5 a differentially plots the shear moduli of regular and irregular diluted networks, composed of fibers with linear elastic behavior (model 1), versus shear strain when the axial prestress strain is zero. Independent of the bending rigidity of fibers, only a slight reduction of the shear modulus is observed because of the geometric irregularity (R/l₀ = 0.5). Networks with irregular microstructure are more compliant because they are composed of fibers cross-linked together in a manner favoring bending deformation. Fig. 5 b investigates the role of microstructure irregularity on the initial shear modulus, at γ = 1%, of prestressed networks. In this plot, the shear modulus of networks with no prestress is used as the normalization parameter, and the normalized shear modulus is plotted as a function of the applied uniaxial tensile strain. It is seen that the effect of geometric disorder on the shear modulus increases with increasing the axial extension and decreasing the bending rigidity of the fibers. In general, the irregular morphology induces a smaller rate of stiffening under axial prestress in comparison with the regular architecture. For example, when the axial prestress strain is 20%, geometric disorder decreases the normalized shear modulus by ∼15 and 70% for networks with $\overline{\kappa }$ = 0.01 and $\overline{\kappa }$ = 0.0001, respectively. Fig. 5 b also shows that the shear modulus increases linearly up to 20% axial strain for large $\overline{\kappa }$; however, it initially increases linearly and then exponentially for small $\overline{\kappa }$.

Figure 5.

(a) The effect of microstructure irregularity on the shear modulus of networks with z = 2.6 that are not prestressed for $\overline{\kappa }$ = 0.01 and 0.0001. (b) The initial shear modulus at γ = 1% is shown as a function of the uniaxial prestress strain for regular and irregular networks with $\overline{\kappa }$ = 0.01 and 0.0001. (c) The force chain distribution in regular and irregular branched networks when $\overline{\kappa }$ = 0.0001 and ε_p = +20% is shown.

It is instructive to note that the microstructural irregularity is an effective tool in avoiding excessive stiffening in natural biological networks. The excessive stiffening due to prestress may result in very high levels of stress in fiber segments and ultimately their failure. Although prestress is commonly seen in biological materials, these structures are often active structures and may create geometric disorder in their microstructure to adjust the required level of stiffening and prevent failure. Fig. 5 c depicts the influence of disordered network architecture on the force distribution when the axial prestress strain is 20%. Force chains with relatively straight paths from the top to bottom edges are seen in diluted regular networks. The force chains become more uniform as irregularity is introduced in the network structure and more fibers contribute to resisting the external load. In addition to homogenous stress distribution, with more fibers sharing the applied load, the magnitude of the force that they carry becomes significantly low, causing them to become softer.

Effect of network connectivity

In this section, we investigate the effect of network connectivity on the nonlinear mechanics of prestressed branched networks. The initial shear modulus of networks composed of linear elastic fibers with network connectivity 2 < z < 3 is found as they are subjected to uniaxial prestress strain of 10 and 20%. Fig. 6 a shows the effect of the tensile prestress on the normalized network stiffness. The data shown in this plot suggest that the role of network connectivity in the elasticity of prestressed networks is primarily controlled by $\overline{\kappa }$. The normalized shear modulus at $\overline{\kappa }$ = 0.0001 increases with increasing the network connectivity; the increase of normalized shear modulus is gradual when 2.3 < z < 2.5 and rapid for z > 2.5. By contrast, networks with larger $\overline{\kappa }$ ≥ 0.01 exhibit less sensitivity with respect to the network connectivity variation (Fig. 6 a). With increasing the network connectivity, more elements contribute to the load bearing, and the network stiffness increases. This effect is more important when the bending rigidity of fibers is small because increasing the network connectivity moves their mechanical behavior to the affine limit (stretching-dominated regime); however, the stretching deformation mode already governs the behavior of networks composed of fibers with large $\overline{\kappa }$ (12,19).

Figure 6.

(a) The influence of network connectivity on the initial normalized shear modulus of random fiber networks with $\overline{\kappa }$ = 0.01 and 0.0001 as they are subjected to uniaxial prestress strain of 10 and 20%. K₀ is the initial shear modulus of respective networks when ε_p = 0. At least three replicas are considered for each network connectivity; the error bars are about the same size as the symbols used to represent the average. The variation of the normal stress is shown as a function of uniaxial prestress and network connectivity for networks with (b) $\overline{\kappa }$ = 0.0001 and (c) $\overline{\kappa }$ = 0.01.

Another interesting quantity in prestressed fiber networks is the variation of the normal stress as a function of the applied extension. Unlike the expected behavior of linear elastic materials under shear stress, polymer networks contract in the direction normal to the shearing direction, causing the appearance of negative normal stresses. This phenomenon occurs because of nonaffine deformations in athermal networks and the asymmetric extension-compression response of their fibers (27,50,58, 59, 60, 61). Fig. 6, b and c plot the variation of the normalized normal stress as a function of the uniaxial strain ε_p during the initial linear elastic behavior of networks. The results are normalized with respect to the linear shear modulus of the networks at ε_p = 0. It is noted that the normal stress is negative when networks are stretched. In other words, the negative normal stress means that networks have a tendency to collapse when the shear strain is applied. We see an increase in the absolute value of the stress with increasing the axial strain (Fig. 6, b and c). The network connectivity and the bending rigidity of the fibers have paramount effects on the dependence of the normal stress on the axial prestress. With increasing the network connectivity, the amount of asymmetry in the extensional response of constituting fibers increases when $\overline{\kappa }$ is small. This is because increasing z creates more constraint on the flexible fibers and more energy is required for bending them. The increased resistance of fibers to bending means an increase in their asymmetric extension response. However, the resulting asymmetry is less significant in networks composed of fibers with large $\overline{\kappa }$ (Fig. 6 c).

Randomness in individual fiber stiffness

The mechanical response of individual fibers is different in natural networks. For example, several studies have reported variation in the mechanical properties of collagen fibrils (62, 63, 64). This variation could be due to differences in the diameter of fibrils during the formation of collagen bundles (65,66). There are few studies in the literature that investigated the mechanical properties of networks composed of both stiff and soft fibers (67, 68, 69). It was found that significant network stiffening occurs because of adding stiff fibers to an otherwise homogeneous network. Furthermore, it was shown that increasing the variance in fiber stiffness, keeping the mean constant, decreases the overall network stiffness (70). However, these previous studies primarily focused on the network response under small deformation. In this section, we investigate the influence of randomness in individual fiber stiffness on the nonlinear mechanical response of prestressed branched networks under large deformation. For this purpose, the stiffness of fiber segment between each two cross-links is randomly, and according to a uniform distribution, assigned a number from (λ)E to (2 − λ)E, where E is the average elastic modulus of fibers. Fig. 7 a shows the variation of differential shear modulus as a function of axial prestress strain for networks with λ = 1.0, 0.25, and 0.1. It is seen that the randomness in the stiffness of individual fibers affects the macroscopic response of both prestressed and not prestressed networks. This figure is plotted for $\overline{\kappa }$ = 0.0001 and z = 2.6, but a similar trend is found for networks composed of fibers with different dimensionless bending rigidity and network connectivity.

Figure 7.

(a) The influence of randomness in the elastic modulus of fibers on the shear modulus of networks with $\overline{\kappa }$ = 0.0001 as they are subjected to uniaxial prestress strain ε_p. The influence of intensity of randomness in the elastic modulus of fibers, as measured by (b) the parameter λ and (c) the standard deviation of the Young’s modulus probability distribution, on the initial shear modulus K₀ and maximal shear modulus K_max of networks at axial prestress strain ε_p = 0, 10, and 20% is shown. (d) The percent of elements with different ratios R = ε_rand/ε_unif, where ε_rand is the strain of fiber segments of networks with random fiber stiffness and ε_unif is the strain of same fiber segments in networks with constant fiber stiffness, is given. It is seen that R for a large number of fiber segments is less than 2.

Fig. 7, b and c show the influence of intensity of randomness in the elastic modulus of fibers on the initial shear modulus K₀ and maximal shear modulus K_max at axial prestress strain ε_p = 0, 10, and 20%. In these plots, the values of λ are taken as 1.0, 0.5, 0.25, 0.1, 0.01, and 0.001, meaning that although the average elastic modulus of the fibers is the same, the standard deviation of Young’s modulus probability distribution is different in separate simulations and equal to 0, 0.29E, 0.43E, 0.52E, 0.57E, and 0.58E, respectively. It is seen that the initial and maximal shear moduli of networks decrease as the range of variation of E increases (λ ≥ 0.01); however, they become insensitive to the variation of E for λ ≤ 0.01. These observations are in general agreement with those reported by Ban et al. (70), whose numerical and analytical arguments show that variabilities in fiber stiffness result in network softening proportional to the variance of the fiber stiffness distribution. The insensitivity of network modulus for λ ≤ 0.01 seen in Fig. 7 b is because the standard deviation of distribution function considered for the Young’s modulus of the fibers becomes independent of λ. Our results suggest that the network softening has a quadratic dependence on the standard deviation of fiber stiffness, which is in agreement with the study by Ban et al. (70) (Fig. 7 c; the dashed lines in this figure are polynomial fits of order 2). However, unlike this previous study, a similar softening effect due to the stiffness variability exists for the maximal shear stiffness. Although we first note that the nonlinear response of random networks was not the primary focus of the study by Ban et al. (70), this disagreement could be because here we used a different distribution function and network microstructure.

The variation of network nonlinear mechanical response because of variation in fiber stiffness is not significantly important when λ ≥ 0.5. This insignificant variation is possibly because the microstructure of these networks is already random and the randomness in stiffness causes the strain within the networks to fluctuate slightly. To capture this fluctuation, the local strain ε_rand of beam elements of networks with random fiber stiffness, λ = 0.5, is normalized by the strain ε_unif in corresponding beam elements of networks with constant fiber stiffness (λ = 1.0), Fig. 7 d. This normalization is performed for all beam elements after the uniaxial prestress strain of +20% is applied. The ratio R = ε_rand/ε_unif is less than 2 for ∼95% of beam elements, which is consistent with the observation that the stiffness has an insignificant effect on the shear modulus of networks. The axial prestress strain has a slight effect on the overall strain softening of random network because of variation in their fiber stiffness (Fig. 7, b and c). Also, we observe that increasing the network connectivity (results not shown) makes the network nonlinear response more dependent on the amount of randomness in fiber stiffness, as measured by the parameter λ.

Conclusions

We conduct a comprehensive numerical study of the mechanical properties of diluted hexagonal lattices under prestress to better understand the mechanical response of biological branched networks such as actin and collagen networks. The numerical simulations of this study successfully capture the stiffening or softening shear behavior of random biopolymer networks because of the axial tension or compression prestress. The important role of prestress is because it causes the fibers to reorient in or from the loading direction. We discuss the findings in terms of the microstructural features including the dominant mode of deformation determined from calculating the contribution of stretching and bending energy to the total elastic energy of networks. We model buckling of the fibers as a reduction in their Young’s modulus in compression and observe that networks with such fibers are softer than those composed of linear elastic fibers. We also consider fibers with nonlinear tension strain-stiffening and show that their networks become stiffer compared to networks of linear elastic fibers. We explain these interesting findings in terms of the local strain distribution inside the networks. We also consider networks composed of fibers with different mechanical properties and networks with irregular microstructure. We show that microstructural irregularity could be an effective tool for fibrous materials in avoiding excessive stiffening, which is important in preventing failure in fiber networks. Moreover, with increasing the randomness in the stiffness of individual fibers while keeping their mean stiffness constant, network softening proportional to the variance of the fiber stiffness distribution is found at both small and large shear deformation. Finally, we discuss the roles of the bending rigidity of fibers and the network connectivity on the nonlinear mechanics of random networks. The numerical simulations of our study provide additional insight into the mechanical properties of random fiber networks and could find important applications in designing novel biomimetic fibrous materials.

Author Contributions

H.H.-M. and M.R. designed research, performed research, analyzed data, and wrote the manuscript.

Editor: Christopher Yip.