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

Abstract

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.

1. Introduction

Many materials of everyday use and most biological materials are made from fibers. In some cases, the material is made exclusively from fibers, with no embedding matrix, as for example in textiles, nonwovens, fiber glass and cellulose insulation, felt, and metallic wools. In some other engineering materials, the fiber network is embedded in a fluid. Engineering materials such as gels and various polymeric network in their swelled state belong to this category. Biological tissues also belong to this class. Connective tissue, such as tendons and ligaments, membranes and blood vessels are made from collagen, or a mixture of collagen and elastin fibers, embedded in a viscoelastic matrix.

In some of these materials fibers are crosslinked, while in others they interact exclusively by non-bonded interactions at contacts. All natural fibrous materials are stochastic, while some artificial ones – the notable and somewhat singular example being that of textiles – have periodic structure. In stochastic networks, fiber segments are distributed randomly in space and have random orientations. The density, ρ, is defined as the total fiber length per unit volume. It has been shown that the field ρ(x) is correlated, and the correlation length is orders of magnitude larger than the mean segments length of the network, l_c (Hatami-Marbini and Picu, 2009; Sampson, 2008). Such correlations develop during the network generation process, as for example by the flocculation of cellulose fibers in suspension during paper making (Provatas et al., 2000). Density fluctuations lead to mechanical heterogeneity on scales much larger than l_c.

Networks are intrinsically heterogeneous on smaller scales too. In 2D and 3D stochastic fibrous materials, the segment length (distance between successive crosslinks on a fiber) is Poisson distributed (Kallmes and Corte, 1960), with l_c being the mean of this distribution. This implies that a broad range of segment lengths are present in the structure. Since these are generally loaded in bending during the macroscopic loading of the network, and since the bending compliance scales with the 3^rd power of the segment length, this implies very large heterogeneity of bending rigidities at the scale of individual segments and fibers.

This discussion indicates that networks are heterogeneous on multiple scales and this is an intrinsic property of any stochastic fibrous material. The heterogeneity implies strong size effects. This implication is well documented in composites (Torquato, 2002; Ostoja-Starzewski, 2007), where the concept of the length scale of homogeneity, L_h, is used. For sample sizes L > L_h, the materials properties (e.g., Young’s and shear moduli) are independent of the sample size. Most importantly, if L > L_h the material properties are also independent of the boundary conditions applied (Hori and Nemat-Nasser, 1999; Ostoja-Starzewski, 2007). As L decreases below L_h, significant sample to sample variability is observed and evaluated properties are boundary condition-dependent. A representative volume element (RVE) of the composite must fulfill L ≥ L_h. In composites, it is generally understood, as a general guideline, that the length scale L_h is at least one order of magnitude larger than the largest characteristic length scale of the microstructure. Examples of relevant microstructural length scales are the size of inclusions, the distance between inclusions and larger scale fluctuations of the number density of inclusions.

L_h is difficult to identify in the case of networks characterized by multiscale heterogeneity. This issue was discussed in the context of 2D Mikado-type networks in (Shahsavari and Picu, 2013). It is shown that the mechanical heterogeneity of fiber networks is controlled not only by the architecture, but also by the fiber properties and the mean network density. Specifically, networks of large density and large fiber bending stiffness deform approximately affinely, despite their random structure. Since in this case the effective heterogeneity is low, size effects are expected to be weak. The opposite situation is encountered in the case of networks of low density and/or which are made from fibers of small diameter and which are soft in bending. For the cellular networks considered in this study, the structural parameter controlling this transition is $w={log}_{10}\left(\rho l_b^2\right)$, where $l_b=\sqrt{E_f{I}_{f/E_f{A}_f}}$ is a parameter with units of length, which compares the bending and axial rigidity of fibers. E_f, I_f, and A_f are the Young’s modulus of the fiber material, the fiber cross-section moment of inertia and its area. If fibers are athermal and of circular cross-section, l_b is proportional to the fiber diameter, d. An equivalent parameter may be defined for fibers of non-circular section (Deogekar and Picu, 2017).

While size effects in fibrous materials have been studied in 2D (Shahsavari and Picu, 2013), little is known about L_h for 3D fibrous structures. One of the goals of the present article is to provide such data. The other objective of the article is to present a method that can be used to mitigate the size effect and obtain predictions for material properties using relatively small models.

The need to mitigate size effects is obvious in most cases but becomes critical in multiscale models of fibrous materials. In the models used in (Chan et al., 2018), the macroscale problem is represented in the continuum sense and is solved using finite elements. The constitutive response is provided by subscale models in which the network is explicitly represented. One such sub-scale discrete model is used for each element of the large scale continuum mesh. To provide an adequate constitutive description, the sub-scale models must be RVEs. However, if the size effect is pronounced, very large sub-scale models are needed, which renders the multiscale problem intractable. In addition, any subscale model must be much smaller than the continuum mesh element size in order for the multiscale analysis to make sense (Feyel and Chaboche, 2000). This may not be possible if L_h is large. The boundary conditions implemented here make possible multiscale simulations of fibrous tissue at a fraction of the cost of equivalent models that use sub-scale discrete RVEs with displacement-imposed boundary conditions. We denote the discrete models of size L ≪ L_ℎ which use the generalized boundary conditions as GBC-RVE, to distinguish them from the usual RVEs of size L ≥ L_h.

2. Models and methods

Three-dimensional cellular networks of Voronoi type are considered in this work. The architecture of these structures resembles that of open cell foams (Jang et al., 2008). They are of interest here because their material behavior has been found to reproduce the experimentally observed behavior of various connective tissue and other athermal networks (Chandran and Barocas, 2006; Humphries et al., 2017; Jansen et al., 2018; Lake et al., 2012; Nachtrab et al., 2011). However, the results presented in this work are not specific to cellular networks and are expected to apply identically to other network architectures.

Networks are generated in 3D by Voronoi tessellating a random distribution of seed points defined in a cubic domain of edge length G. Fibers are defined along all edges of this tesselation. The network density, ρ, and the mean fiber length, l_c, are controlled by the number of seed points used in the Voronoi construction. The distribution of fiber lengths is truncated at l_c/100 in order to improve the solution performance for the large strain analysis which uses an explicit solver and in which the time step is controlled by the shortest element size in the structure. The generated structure is trimmed to a cubic domain with edge length L, L < G, by adding a node at all fiber trimming geometry boundaries, and by removing fiber segments outside of the trimming domain. Fibers of length less than the cutoff length are removed by merging their endpoints into a single node, whose position is chosen as an average of the original fiber node positions weighted by the connectivity z of the two nodes. The connectivity index, z, represents the average number of segments emerging from given crosslink. In an infinite 3D Voronoi structure, z = 4. Crosslinks with z ≤ 2 which are not at the model boundary are removed in an iterative fashion. This procedure ensures that the network has no internal disconnected components and no dangling ends. Alternatively, a graph traversal could enable a one step process to identify and remove the hanging nodes. Removing disconnected chains reduces the computational cost of the analysis in the explicit case and is one of the conditions required for the stiffness matrix to be positive definite in the implicit case. If excluded volume interactions (inter-fiber contacts) are not accounted for, the dangling ends have no contribution to the quasi-static response of the network.

The fiber material is considered linear elastic of Young’s modulus E_f. Fibers are straight in the undeformed configuration (no tortuosity) and randomly oriented in space. Networks with pre-aligned fibers are also considered. The alignment is quantified using the second Legendre polynomial:

$$P_2=\frac{1}{2}<3{cos}^2(\theta )-1>$$ (1)

where θ is the angle between the fiber end-to-end vector direction and the reference direction of pre-alignment, and the angular bracket represents the system average. This measure ranges from −1/2 to 1, where the two values indicate perfect alignment in orthogonal and parallel to the reference direction, respectively. The random fiber distribution corresponds to P₂ = 0.

To generate a model with pre-aligned fibers, we begin by generating in a large domain the reference network of specified density and with random orientations of fibers, after which the crosslinks are moved affinely by a mapping similar to that of a constant volume uniaxial deformation. The deformation proceeds until the desired value of P₂ is reached. The model is then trimmed to the final cubic shape of desired edge length, L. The pre-aligned networks considered in this work have P₂ = 0.224.

Fibers are discretized using Timoshenko beam elements. The element aspect ratio is kept close to 5. This value results from studies of the effect of h-refinement on the simulation output and based on our experience with models of fibrous networks. We have found that this provides an acceptable representation of the analytical strain energy in simple beam bending simulations.

The networks are deformed by applying boundary conditions as defined in section 3 and uniaxial loading conditions are considered in most cases. Young’s modulus, E₀, is evaluated directly, while the reported shear modulus is computed in the small strain limit using the expression from linear isotropic elasticity, $G_0=E_0/2(1+v)$. This is adequate since all networks considered, without pre-aligned fibers, are nominally isotropic (unless stated otherwise). The Poisson ratio, v, is evaluated along with E₀ in each simulation.

The solution is obtained with the general purpose finite element solver Abaqus, version 2018. The implicit solver is used for small deformations, while damped dynamic explicit simulations are performed to response under large deformations and rotations. We monitor the kinetic and damping energies, requiring these to remain well below 5% of strain energy. Additionally, we monitor the difference in the internal and external forces at each node normalized by the fiber area and the fiber elastic modulus requiring that they remain below 10⁻³. The out of balance force measure was used extensively in the dynamic relaxation literature because it gives a point-wise measure of how far the current system state is from static equilibrium (Kadkhodayan et al., 2008; Namadchi and Alamatian, 2016; Papadrakakis, 1981; Underwood, 1986; Zhang et al., 1994). These conditions are not fulfilled during the initial part of the deformation extending up to ~3% strain due to dynamic effects associated with the start-up. Therefore, the explicit solution is not accurate at small strains and the implicit solution is mandatory in order to obtain accurate predictions of the small strain modulus. Further, the formations of contacts between fibers during deformation is not accounted for. Such events are rare when the network is loaded in tension, as discussed in (Islam and Picu, 2018).

3. Results

3.1. Size effect in three-dimensional random networks

Size effects affecting the small strains mechanical behavior of 2D Mikado-type random networks were studied in (Shahsavari and Picu, 2013). It was shown that networks with w corresponding to the affine deformation regime (large density and/or large l_b) exhibit weak size effects. In this limit, a model size of approximately L ≥ 10l_c provides an adequate value of the network stiffness. This prediction is also independent of the boundary conditions applied, which is an indication that. L_h ≅10l_c. The situation is quite different when w takes values in the non-affine deformation regime (low density and/or small l_b), case in which the predictions are not independent of boundary conditions even for L = 20l_c. No such study is available in the literature for 3D networks, although model predictions for a variety of network parameters and architectures abound.

Figure 1a shows the bounds of Young’s modulus, E₀, of affinely deforming Voronoi networks with w = −1, function of the model size, L. Figure 1b shows similar data for a non-affinely deforming network with w = −4. The two networks have the same density and same geometry, but their l_b differs by 2.5 orders of magnitude. The upper bound is obtained by applying displacement boundary conditions (Voigt bound), while the lower bound results by using traction boundary conditions (Reuss bound). The figure indicates that the size effect is present in both cases, but it is significantly more pronounced in the non-affine, w = −4, case. Note that the vertical axis in Fig. 1b is logarithmic. The two bounds do not converge even for model sizes as large as L = 40l_c and hence L_h is not reached even with the largest models considered here.

Figure 1.

Effective Young’s modulus of (a) affine, w = −1, and (b) non-affine networks, w = −4, with the model size, L. The vertical axis is normalized by the fiber modulus, E_f, while the horizontal axis is normalized by the mean segment length, l_c. The curves represent the upper (Voigt) and lower (Reuss) bounds obtained by applying displacement and traction boundary conditions, respectively. The size effect disappears when the two bounds meet. This condition is not reached even with the largest models considered here. The The bars represent standard deviation based on $160/\left(L/l_c\right)$ realizations.

For the small strain implicit solutions, the time it takes to solve a model increases nonlinearly with the number of linear equations. We note that the computational time required to solve a model of size L = 40l_c is two orders of magnitude larger than that required for the L = 10l_c model at same compute power.

The transition between the affine and non-affine deformation regimes is usually represented in the form of a ‘master plot’ in which the network modulus, E₀, is shown versus w (Head et al., 2003; Picu, 2020; Shahsavari and Picu, 2013). E₀ is normalized by the affine prediction, which is also the absolute upper bound. The affine limit corresponds to moving the boundary and all crosslinks of the model affinely with the macroscale deformation. However, it is also reached by increasing the density of the network and/or increasing l_b, while applying displacement boundary conditions in the normal way.

Figure 2 shows the master plot for Voronoi networks computed in the two limits (Voigt and Reuss) and using models of size L = 20l_c. The Voigt limit is obtained by imposing affine displacements at all nodes on the model surface, while the interior degrees of freedom are allowed to relax. As expected, the curves approximately overlap in the affine limit, but large differences are observed at smaller w values. The scaling of E₀ with the model parameters in the low w non-affine regime is $E~E_f{I}_f{\rho }^2$, while in the affine regime (large w) it is $E~E_f{A}_f\rho$. These scalings are not affected by the size effect. The importance of this data is that it indicates that the 𝑤 value marking the transition between the affine and non-affine regimes is a function of the boundary conditions used.

Figure 2.

Master plot showing how the normalized small strain modulus, E₀, varies with the structural parameter w. Three curves are shown corresponding to the Voigt and Reuss estimates of E₀, and to the estimate obtained with GBC-RVEs with the optimal set size (p = 3), as discussed in section 3.3. The size of the models used is L = 20l_c. Since L < L_h (Fig. 1), the size effect is present and modifies the master plot.

Comparing the results presented here with those for 2D Mikado networks in (Shahsavari and Picu, 2013), it results that the size effect is more pronounced in 3D than in 2D. This poses serious problems when attempting to obtain quantitative predictions using numerical models. A solution to this problem is presented in section 3.3.

3.2. Generalized boundary conditions: definition and implementation

To mitigate the size effect described in section 3.1, we implement in the context of fibrous materials the generalized boundary conditions described in (Glüge, 2013). Let be the outer surface of the model and n be the number of nodes (fiber terminations) on A. Following the procedure proposed in (Glüge, 2013), the model boundary is divided in i = 1…k surface subdomains, Aⁱ, each containing p nodes (n = kp). The total surface area of the boundary is $\text{A}={\displaystyle {\displaystyle {\sum }_{i=1}^k}{A}^i}$. Each subdomain area Aⁱ is made up from the areas corresponding to the p nodes in the set i.e., $A^i={\displaystyle {\displaystyle {\sum }_{j=1}^p}{A}_j^i}$. We maintain the superscript i representing the subdomain as a reminder that summation is performed only over the nodes which are contained in the i^th set. Figure 3 shows a network realization of size $L/l_c=12$ and the partition of the model surface in subdomains. The partition is shown only for one of the model faces, but it is applied over its entire surface A. The areas corresponding to each surface node (termination of a fiber) is shown. The ensemble of areas shown with the same color form a subdomain and the p corresponding nodes form a set.

Figure 3.

Network of size $\text{ e }L/l_c=12$ shown together with the partition of one face of the 3D problem domain in subdomains. Each Voronoi cell represents an $A_j^i$. The cells are colored by set, so the union of all cells of a given color has area Aⁱ. Black points show the location of the fiber terminations on the surface.

We seek to apply to the model a prescribed macroscopic strain state described by the displacement gradient, H^appl. Since the generalized boundary condition meets the Hill-Mandel condition (Glüge, 2013), the mean displacement gradient over the network volume, $\overline{\mathbf{\text{H}}}$, is equal to the applied displacement gradient on the boundary i.e., $H^{\text{appl}}=\overline{H}$.

The mean displacement gradient corresponding to the i^th boundary subdomain, ${\overline{H}}^i$, is written as:

$${\overline{H}}^{\text{i}}={\displaystyle {\sum }_{\text{j}=1}^{\text{p}}{u}_j^i}\otimes n_{0j}^i{A}_{0j}^i,$$ (2)

which can be re-arranged as:

$${\overline{H}}^{\text{i}}=\overline{H}\cdot {\displaystyle {\sum }_{\text{j}=1}^{\text{p}}{\mathit{x}}_{0j}^i}\otimes {\mathit{n}}_{0j}^i{A}_{0j}^i,$$ (3)

where $u_j^i,x_{0j}^i$ and $n_{0j}^i$ are the displacement, position vector and outward unit normal at the j^th node on the i^th boundary subdomain, and $A_{0j}^i$ is the area corresponding the same node. A zero subscript indicates the reference, undeformed configuration. If all components of $\overline{H}$ are defined, Eq. (3) can be used to evaluate the mean displacement gradient for the i^th boundary subdomain, ${\overline{H}}^i$. Then, Eq. (2) is added to the system of equations resulting from the finite element variational problem as a weak constraint on the displacements of the boundary nodes in set i. If $\overline{H}$ has unspecified components, these appear as additional degrees of freedom in the respective system of equations and are solved for in a manner similar to that used to evaluate nodal displacements. Note that Eq. (2) is reminiscent of the equation relating $\overline{H}$ to the boundary displacements using the Gauss theorem, which for a continuum of volume V surrounded by the closed surface A takes the form $\overline{H}=(1/V){\displaystyle {\int }_{\mathit{A}}u}\otimes ndA$.

When the number of sets is equal to the number of nodes, i.e., n = k and p = 1, the generalized boundary condition reduces to the usual displacement-imposed boundary condition, and the model boundary moves affinely. It is shown in (Glüge, 2013; Miehe, 2003) for the continuum that when the number of sets is equal to 1, k = 1 and p = n, the generalized boundary conditions become the usual homogenous stress boundary condition.

The major advantages of this type of boundary conditions, and the reason for which they are used in the present context, is that they allow interpolating continuously between the two limits of imposed displacements and imposed tractions, by simply varying the set size, p

Since the boundary conditions applied correspond to uniaxial tension, the only specified non-zero component of H^appl is $H_{33}^{appl}$ (loading in the x₃ direction). The off diagonal components are set to zero, which forces the system to undergo no shear strain, in average. $H_{11}^{appl}$ and $H_{22}^{appl}$ are not specified and are solved for as independent variables, such to represent traction free boundary conditions in the respective directions.

Convergence of the discrete form of the generalized boundary condition discussed here to the continuous form in (Glüge, 2013) is only guaranteed when the surface nodal density goes to infinity, or conversely the surface area associated with each node goes to zero. We do not directly address this issue, however we show that the generalized boundary conditions are effective under a range of network densities which suggests that the surface nodal densities observed in Voronoi networks are high enough to approximate well the continuum equations.

The areas $A_j^i$ shown in Fig. 3 are defined by performing a Voronoi tesselation of each outer surface of the model. Further, the surface nodes need to be assigned to the nodal sets. There are multiple ways in which this can be done. Here we choose to select the nodes assigned to each set randomly from the population of n surface nodes. The way in which the nodes are assigned to sets introduces a negligible degree of variability in the model response, as analyzed in the next section and also discussed in (Glüge, 2013).

3.3. Optimal boundary conditions to mitigate size effects

3.3.1. Small deformations

We examine first the application of the generalized boundary conditions to evaluate the small strain elasticity of fiber networks. Figure 4 shows plots similar to those in Fig. 1, with the modulus predicted by using various p values. Figures 4a and 4b show results for the affine and non-affine cases, with w = −1 and w = −4, respectively. The vertical axis represents either Young’s modulus (shown by continuous lines) or shear modulus (shown by dashed lines), while the horizontal axis represents the model size in units of l_c.

Figure 4.

Young’s (continuous lines) and shear (dashed lines) moduli function of the model size, L/l_c. Curves for various values of p are shown; those with small p converge from above, while those with large p converge from below. The curve for p = 3 is approximately flat and hence predicts the asymptote value. The vertical axis is normalized by the value of this prediction, E_∞ and G_∞ for the continuous and dashed lines, respectively. (a) shows results for the affine networks with w = −1, and (b) shows data for the non-affine case with w = −4. The bars represent standard deviation based on 160/(L/l_c) realizations.

The figure shows that increasing p leads to smooth interpolation between the upper bound (which is reached when p = 1) and the lower bound (reached by imposing p = n). The actual, model size-independent value of the modulus corresponds to the asymptote of these curves, which are supposed to merge once L = L_h. The curves corresponding to small p values converge to this asymptote from above. Those corresponding to large p converge from below. A certain p value is identified for which the curve is flat, i.e., its curvature vanishes. Therefore, this represents the asymptote itself. The observation is of central importance here since it allows identifying the asymptote using small models, provided the optimal p values is used.

The data indicates that for cellular networks the optimal p is 3. The vertical axes in Fig. 4 are normalized by the value of the modulus (either Young’s or shear) predicted using p = 3. This leads to the collapse of the Young and shear moduli curves, which indicates that their convergence to the asymptote as L increases is identical. The value p = 3 leads to model size-independent stiffness predictions in both w = −1 and w = −4 cases. Furthermore, the master curve obtained with p = 3 is shown in Fig. 2 along with the Voigt and Reuss bounds which correspond to p = 1 and p = n, respectively.

To support the conclusion that the optimal set size for this network architecture is p = 3, we also consider networks with preferential fiber alignment in the x₃ direction. In all cases considered, the alignment in the undeformed configuration is P₂ = 0.224. These networks are transversely isotropic, with the axis of transverse isotropy being the axis of initial fiber alignment, x₃. These are probed in uniaxial tension in the x₃ (parallel) and, in separate simulations, in the x₁ (perpendicular) direction. The effective stiffness evaluated is $E_{eff}={\sigma }_{mm}/{\epsilon }_{mm}$ for m = 3 and m = 1, respectively, which represents the inverse of the compliance in the respective directions. The values of E_eff are reported in Fig. 5 function of the model size for multiple p values and for the two cases, w = −1 and w = −4 considered above (Figs. 5a and 5b, respectively). The vertical axis is normalized by the model prediction of the horizontal asymptote obtained with p = 3. The conclusions are similar to those emerging from Fig. 4. The optimal set size is p = 3, for which model size-independent predictions are obtained for any $L/l_c$. The convergence rate of the moduli of the aligned networks towards the respective asymptote is similar to that of the non-aligned networks. The non-affine case with w = −4 exhibits a more pronounced size effect than its affine counterpart.

Figure 5.

The effective stiffness of networks with pre-aligned fibers (P₂ = 0.224) function of the model size L/l_c obtained in uniaxial tension performed in the parallel (continuous lines) and perpendicular directions (dashed lines) relative to the initial fiber alignment. The affine case with w = −1 is shown in (a) and the non-affine case with w = −4 is shown in (b). Multiple values of p are considered. The bars represent standard deviation based on 160/(L/l_c) realizations.

The bars in Figs. 4 and 5 represent variability of the moduli (standard deviation) evaluated over 160/(L/l_c) realizations (32 realizations are used for the data points at the left end of the curves and 4 realization of the largest models with $L/l_c=40$ are considered). The surface nodes are assigned to sets randomly for each realization. Therefore, the bars represent the combined effect of variability introduced by the replicas (different network geometries) and by the definition of the nodal sets. The variability decreases as the model size increases, as expected. Further, at given $L/l_c$ and hence given number of replicas, the variability decreases as p increases. This trend is due to numerical error associated with the solution of the boundary value problem in which the boundary conditions are imposed in the form of constraint equations of the type of Eq. (2). The larger the number of sets (smaller p), the larger this algorithmic error. On the other hand, the bars for p = n are essentially not affected by the algorithmic error since only one nodal set exists in this limit. We conclude that the variability introduced by the random assignment of nodes to sets is negligible.

The central observation from this study is that a value of p exists for which the predicted small strain stiffness is model size-independent. For the cellular network architecture considered here, this optimal value is p = 3. We see that this value holds for approximately affine and strongly non-affine networks, with randomly oriented and with preferentially oriented fibers. However, it is possible that for different network architectures, the optimal p is different. Nevertheless, the method proposed here can be used to predict the model size-independent moduli even if the optimal p is not known, while using samples of small size. For example, one may analyze models of two sizes, e.g., $L/l_c=5$ and 10, for several p values and evaluate the trends of the predicted moduli. This provides the left side segment of the curves in Fig. 4. These segments defined by just two data points are sufficient to predict the optimal p based on the change of their slopes from positive (at large p) to negative (at small p). Such analysis is much cheaper computationally than using the large models required to directly evaluate the asymptotic values of the moduli in the L > L_h limit.

3.3.2. Large deformations

It is of interest to investigate how the size effect discussed in the previous sections for the small deformations regime affects the large strains response of the network. Crosslinked networks without an embedding matrix (or embedded in a viscoelastic fluid) are hyperelastic in the absence of damage and fiber plasticity. Collagen-based connective tissues, blood vessel walls and various membranes such as the basal membrane, the amnion and the liver capsule, exhibit hyperelastic behavior. In the normal physiological range, their deformation can be described in terms of an initial regime (regime I), which is linear elastic, and a second regime (regime II) characterized by exponential stiffening. Gels and some nonwovens (hydroentangled and needle-punched nonwovens) also exhibit these regimes before the onset of damage.

This behavior is reproduced by the networks used in this study. Figure 6a shows the normalized nominal stress (first Piola-Kirchoff), $S/E_f$, versus stretch for networks with 𝑤 = −1 obtained with a model of size L = 20l_c. Curves corresponding to the upper bound (p = 1), lower bound (p = n) and the model size-independent curve obtained with the GBC-RVE for set size p = 3 are shown. Two other curves are shown, which are obtained with boundary conditions usually applied when modeling uniaxial tension of networks (not GBC): Case (A) is a mixed boundary condition in which the nodes on the face of normal x₃ have specified displacements in the x₃ direction and are free in the other two directions, while the model boundaries parallel to the loading direction are left free; Case (B) is similar to Case (A) except that the model boundaries parallel to the loading direction are forced to remain planar, but are free to move in the direction of their normal (zero tractions are applied in the average sense). It is observed that cases (A) and (B) are close to each other and are close to the optimal GBC-RVE case with p = 3.

Figure 6.

(a) Nominal stress-stretch curves for affine models with w = −1 and model size L = 20l_c obtained with p = 1 (upper bound), p = n (lower bound), and p = 3 (optimal GBC-RVE case), along with cases (A) and (B) defined in text. The vertical axis is normalized by the fiber material modulus. (b) Data in (a) replotted as tangent stiffness versus nominal stress. The vertical axis is normalized by the small strain stiffness of the respective network, E₀, while the horizontal axis is normalized by the nominal stress at the transition from regime I to II, S₀.

To clarify the model size effect in the non-linear regime, it is useful to replot the data in Fig. 6a as tangent stiffness versus stress. The tangent stiffness is computed based on the nominal stress as $E_t=dS/d\lambda$. In this representation, the two regimes appear well defined as a horizontal lines corresponding to regime I, and a line of slope 1/2, which indicates that $E_t=dS/d\lambda ~S^{1/2}$ and hence the stress-stretch curve is quadratic in regime II, $S~\alpha {(\lambda -1)}^2$. This representation is shown in Fig. 6b. The vertical axis is normalized by the small strain modulus (shown in Fig. 4a), while the horizontal axis is normalized by the stress at the transition between regimes I and II, S₀.

It is observed that the curves in Fig. 6a overlap when re-plotted as in Fig. 6b. Hence the normalization of the two axes with E₀ and S₀ eliminates the size effect. Note that, since regime I is linear, S₀ = E₀(λ₀ −1), where λ₀ is the stretch at the transition from regime I to II. We verified that λ₀ is model size independent. Therefore, the large difference seen in Fig. 6a is entirely associated with the size effect on the small strain modulus, E₀. It results that the functional form of regime II stiffening is independent of the model size. The implication of this finding is that once the size effect on the small strain stiffness is controlled (e.g., by the use of GBC-RVE with p = 3), the non-linear response becomes also size effect free.

Figure 7a shows a plot similar to that in Fig. 6a corresponding to networks with w = −4 and of size L = 20l_c. The difference between the Voigt (p = 1) and Reuss (p = 𝑛) bounds is very large in this case. The Case (A) and Case (B) curves are close to each other, but are not that close to the optimal GBC-RVE case with p = 3.

Figure 7.

(a) Nominal stress-stretch curves for non-affine models with w = −4 and model size L = 20l_c obtained with p = 1 (upper bound), p = n (lower bound), and p = 3 (optimal GBC-RVE case), along with cases (A) and (B) defined in text. The vertical axis is normalized by the fiber material modulus. (b) Data in (a) replotted as tangent stiffness versus nominal stress. The vertical axis is normalized by the small strain stiffness of the respective network, E₀, while the horizontal axis is normalized by the stress at the regime I to II transition, S₀.

Figure 7b shows the data in Fig. 7a plotted as tangent stiffness vs. stress. The results are similar to those in Fig. 6b, except for the fact that the slope of the curves in the non-linear regime II is 1. This indicates that $E_t=dS/d\lambda ~S$ and hence the stress-stretch curve has exponential functional form. As in the affine case, we observe that the functional form of strain stiffening is not affected by the size effect, which influences only the small strain stiffness. We note that slopes of ½ and 1 are usually reported in the literature for affine and non-affine networks in their nonlinear regime, in representations similar to Figs. 6b and 7b, respectively (Licup et al., 2015; Picu, 2020).

Similar conclusions have been reached with respect to the effect of various other structural parameters on the stress-stretch curves. Fiber tortuosity (Ban et al., 2016a), the variability of fiber material properties or of the fiber diameter from fiber to fiber (Ban et al., 2016b), rendering the fiber cross-section non-circular (Deogekar and Picu, 2017), all these parameters modify the small strain modulus, but leave the functional form of strain stiffening unchanged. Consequently, the stress-stretch curves corresponding to various values of these parameters appear quite different, but they collapse once plotted as normalized tangent stiffness versus stress.

4. Conclusion

Size effects in 3D stochastic fiber networks are pronounced and the system size beyond which material parameters (e.g., small strain stiffness) become size-independent, and independent of the boundary conditions applied, is larger than can be easily simulated with today’s computers. The size effect is weaker in the limit of affine network behavior, but increases in magnitude as the degree of non-affinity increases. This situation poses serious problems when attempting to evaluate material parameters using numerical models, and in experiments using small samples or based on indentation with small tips.

We implement a formulation of generalized boundary conditions developed for heterogeneous continua in (Glüge, 2013) which allows gaining control over the size effect. We identify the optimal nodal set size (a key parameter of the generalized boundary conditions) which allows predicting the infinite model size behavior using models of any size. The same set size can be used for affine and non-affine networks, with and without preferential fiber alignment.

The size effect affecting the non-linear regime of the deformation has similar origins with that affecting the small strain modulus. We observe that once the size effect affecting the small strain modulus is eliminated by using generalized boundary conditions with the optimal set size, the non-linear regime also becomes size effect free. This applies to both the affine and non-affine networks.

The contributions of this article are twofold. We evaluate the magnitude of size effects in 3D cellular stochastic networks function of the structural parameter w. While the size effect has been studied in 2D fiber networks, this is the first analysis of this effect in 3D. More importantly, we propose a method to eliminate the size effect, which applies equally to affine and non-affine networks deforming in the linear and non-linear regimes.