Probing Soft Fibrous Materials by Indentation

Abstract

Indentation is often used to measure the stiffness of soft materials whose main structural component is a network of filaments, such as the cellular cytoskeleton, connective tissue, gels, and the extracellular matrix. For elastic materials, the typical procedure requires fitting the experimental force-displacement curve with the Hertz model, which predicts that f = kδ^1.5 and k is proportional to the reduced modulus of the indented material, E/(1 − v²). Here we show using explicit models of fiber networks that the Hertz model applies to indentation in network materials provided the indenter radius is larger than approximately 12l_c, where l_c is the mean segment length of the network. Using smaller indenters leads to a relation between force and indentation displacement of the form f = kδ^q, where q is observed to increase with decreasing indenter radius. Using the Hertz model to interpret results of indentations in network materials using small indenters leads to an inferred modulus smaller than the real modulus of the material. The origin of this departure from the classical Hertz model is investigated. A compacted, stiff network region develops under the indenter, effectively increasing the indenter size and modifying its shape. This modification is marginal when large indenters are used. However, when the indenter radius is small, the effect of the compacted layer is pronounced as it changes the indenter profile from spherical towards conical. This entails an increase of exponent q above the value of 1.5 corresponding to spherical indenters.

Graphical Abstract

1. Introduction

Indentation has been used by many authors to determine the material behavior of biological network materials such as collagen [1–3], the cytoskeleton [4–7], cartilage [8–11], gels [12–14]. Biological network materials are those whose main structural component is a network of protein filaments. Indentation experiments are relatively easy to perform and can be done on multiple scales from standard large scale indentation [15–17] to nano-indentation [18–20], to indentation with the tip of an atomic force microscope [21,22].

An indentation experiment produces an indentation force vs. indentation displacement curve, F(δ). The response of soft biological materials is non-linear elastic, with zero or negligible residual deformation upon unloading. The force-displacement curve is typically interpreted using the Hertz model for a spherical indenter [23] which predicts:

$$f=\frac{4E^{*}}{3R}{a}^3,$$ (1)

where the reduced modulus E* for in infinitely rigid indenter is E* = E/(1 − v²), R is the indenter radius and a is the radius of the contact spot. E and ν are the Young’s modulus and Poisson ratio of the indented material. The model also predicts $a=\sqrt{R\delta }$ which, when inserted in Eq. (1) leads to

$$f=\frac{4}{3}{E}^{*}{R}^{1/2}{\delta }^{3/2}=k{\delta }^{3/2}.$$ (2)

In the case of elastic-plastic materials, the Oliver-Pharr approach must be used, which identifies the modulus, E*, from the tangent of the unloading branch of the force-displacement curve [20], The networks described here are non-linear elastic and fully reversible (no dissipation), see Fig. S1. Hence, the loading curve can be directly used for the identification of E* [1,24,25]. Since the material is nonlinear elastic, it is not directly apparent that the Hertz model is a reasonable choice for determining the small strain elastic modulus. Here, we show that the Hertz model can be used for data interpretation to accurately recover the small strain modulus when sufficiently large indenters are used (see figure 4). Since biological network materials are embedded in a volume preserving fluid, the Poisson ratio is typically assumed to be 0.5, which allows inferring the Young’s modulus, E. The Hertz model is based on the following key assumptions: the material is isotropic and homogeneous, strains are small (a ≪ R), the material is linear elastic, and the effect of adhesion between the tip and surface is negligible [23].

Figure 4.

Data from Fig. 2 replotted as normalized instantaneous modulus, $E_H^{*}/E_0^{*}$, function of the indentation depth, for (a) w = −3.86 and (b) w = −5.2. A value of 1 indicates the applicability of Hertz’s model. Data obtained from indentation tests performed with indenters with R/l_c < 12.5 lead to predicted stiffness values smaller than the true stiffness of the network. The legend shown in (a) applies to both panels.

If adhesion is important, the Johnson-Kendall-Roberts (JKR) [26] or the Derjaguin-Muller- Toporov (DMT) [27] models can be used. These models are accurate at the limits of soft materials with large intenters, and stiff materials with small indenters, respectively. Alternatively, the Maugis-Dugdale model, which uses a square-well approximation of the adhesive interactions, encompasses both the JKR and DMT limits [28]. In practice the analytical expressions of the Maugis-Dugdale model are hard to use, so empirical fits have been developed [29,30]. Here we limit attention to situations in which adhesion between tip and substrate is not important.

The objective of this work is to examine the applicability of the Hertz model to network materials probed with small tips. The drive to use tips with small radii is motivated by the need to probe small samples (e.g., cells) and by the hope that using small tips when probing a heterogeneous material produces estimates closer to the values of the stiffness at the site of indentation. The difficulty encountered in such experiments emerges from the fact that networks are not continua, have stochastic structure and are intrinsically mechanically heterogeneous. Networks are composed from filaments connected at crosslinks and the length of segments bounded by two successive crosslinks along a filament is Poisson distributed, which implies a broad distribution of filament stiffnesses [31]. The mean segment length, l_c (or the ‘mesh size’) is the only parameter of this distribution and is taken as the characteristic length of the network.

The mechanical behavior of networks depends on the nature and behavior of filaments, crosslink properties and network architecture. Filaments small enough to be subjected to thermal fluctuations (as in gels, cytoskeleton and molecular networks) are denoted as thermal, while those insensitive to thermal fluctuations are referred to as athermal. In this work we consider athermal networks which may represent collagen-based biological structures. Athermal filaments may be represented by beams with axial and bending rigidities proportional to E_fA_f and E_fI_f, where E_f is Young’s modulus of the filament material, while A_f and I_f are the filament cross-section area and moment of inertia, respectively. Assuming filaments to be cylindrical, the only relevant parameter of the cross-section is the diameter, d. In collagen networks, crosslinks are formed by the exchange of fibrils from filament to filament (filaments are actually bundles of fibrils [32,33]) and transmit both forces and moments between and along filaments. The network architecture is defined here in terms of a set of parameters, including the network density, ρ, which represents the total length of filament connected to the network per unit network volume (excluding dangling ends and free filaments), the number density of crosslinks, ρ_b, and the mean connectivity number, z, representing the mean number of filament segments merging into a crosslink. Geometric considerations mandate that these parameters are related as ρ = zl_cρ_b/2 (this relation emerges by considering that the fiber length corresponding to one crosslink of connectivity z is zl_c/2 are there are ρ_b such crosslinks per unit volume). In many biological materials, such as collagen, the connectivity number typically ranges from 3 to 4 [34]. In this work, we use a Voronoi network architecture which has connectivity number of 4. Preferential filament orientation introduces anisotropy and additional parameters must be considered to describe it. Here, we consider networks with random orientation of filaments and isotropic behavior.

The stress-strain behavior of a network subjected to uniaxial tension (or shear) is non-linear and exhibits 3 regimes. At strains smaller than few percent, the response is linear and is defined by the small-strain Young’s modulus, E₀. Exponential stiffening is observed at larger strains, followed by a power law regime in which the stress is proportional to the strain to a power between 1 and 2 [35,36]. Such behavior was observed in many experiments with biological materials and is reproduced closely by the models used here and described in section 2 [37–40]. In compression, deformation localizes in bands oriented approximately orthogonal to the loading direction and this reflects in the stress-strain curve which, after the short initial linear regime, exhibits softening followed by a slowly rising plateau [41]. The bands appear stochastically and may not percolate across the sample due to the intrinsic stochasticity of the network. Therefore, the stress-strain curve is strongly tension-compression asymmetric, with the compressive branch being much softer than the tensile branch [41,42].

Athermal networks of high density (large ρ) and/or filaments stiff in bending (large d) exhibit an approximately affine deformation, while those of low density and/or small d, deform in a non-affine way [43–46]. Hence, for given network architecture, it is possible to change the behavior from affine to non-affine by decreasing the filament diameter and hence allowing filaments to deform in the softer bending mode. This transition is described by the non-dimensional parameter w = log₁₀ρl_b²; large and small values of w correspond to affine and non-affine behaviors, respectively, with the transition taking place at a w value which depends on the network architecture [45,46]. Biological network materials belong to the non-affine category.

Many attempts have been made to extend Hertzian indentation for use with materials with nonlinear constitutive parameters. Horkay et al. made use of an empirical measure to relate the mean stress and strain to the indenter contact radius, indenter radius, and indentation force to derive an estimate for the force-displacement response for many nonlinear constitutive models, including the Mooney-Rivlin [25,47,48]. Giannakopoulos and Triantafyllou employed second order elasticity to show that the force-displacement equation for a Mooney-Rivlin material is the same as that of incompressible linear elasticity [49]. Guo et al. provided an improved estimate for the contact radius in nonlinear materials which compares favorably against deep indentations with neo-Hookean materials [50].

Much interest has also been paid to the source of errors in indentation of biological materials. For example, Crick and Yin use simulated force-deflection curves to show how a nonlinear constitutive response can cause misidentification of the indenter contact point [51]. Wu et al. have shown using polydimethylsiloxane (PDMS) that the force-displacement relation can be affected by boundary effects and, if the sample has a thickness of more than 20 times the indenter radius, the Hertz solution is valid only up to δ/R = 0.66 [24].

The progress made towards accurately approximating deep indentations in nonlinear materials has undoubtedly improved experimental measurements of biological materials. However, these traditional continuum models do not contain an internal length scale, so they are unable to capture size effects associated with the size of the indenter.

Since indentation is performed with a tip of certain radius, R, the probing method introduces a length scale. It is natural to inquire whether the interaction of this length scale with the network-intrinsic length scale, l_c, leads to any special effects, not accounted for in the Hertz model. This investigation is the scope of the present work. To this end, we perform indentations in model networks using spherical indenters of size comparable with the network mesh size, 2 ≤ R/l_c ≤ 14 and explore the dependence of the force-displacement curve on this parameter. We conclude that the Hertz model applies only for R values one order of magnitude larger than l_c, while for smaller R/l_c, the exponent of the force-displacement relation increases above the 1.5 value predicted by the Hertz model, Eq. (2). We also observe that this scaling remains valid for deep indentations, beyond the range of applicability of the Hertz model. Further, we investigate the origin of the departure of the exponent from the value predicted by Hertz model and associate this effect with the development of a compacted network region under the indenter, which has the net effect of changing the indenter size and shape. The shape change leads to the observed variation of the exponent of the force-displacement function. The present results also imply that using Hertz model to infer material stiffness when using indenters smaller than approximately 12l_c does not lead to accurate results.

2. Models and methods

Two types of models are used in this work: a model in which all fibers are explicitly represented, which we call the ‘network model,’ and a continuum, homogeneous model of the indentation problem denoted here as the ‘continuum model.’

Networks of cellular, Voronoi type are used here. A similar network model was used previously to represent the mechanics of collagen networks with [52–55] and without [44,56] cellular embeddings. Networks are constructed using a custom Python script by performing a Voronoi tessellation of space based on a random distribution of seed points. A cube of edge length L = 200l_c, where l_c is the mean segment length of the network to be obtained, is defined and is used as initial problem domain. 300,000 seed points with randomly selected positions are defined within the cube. The seed points are used to tessellate the domain and network filaments are defined along the edges of each polyhedron of the Voronoi tessellation. This procedure produces a network of straight filaments with z = 4. The density, ρ, is defined by the seed number density and, for the networks considered here, $\rho =1.1/l_c^2$. The network is then clipped to a semi-sphere with a radius R_N = 50l_c by placing nodes at the intersection of fibers with the boundary of the clipping manifold. Any disconnected regions are identified and removed. Nodes that are not located on the boundary and with z = 1, and the associated fibers, are removed iteratively until all nodes are either on a boundary or have z > 1. Fibers of length smaller than l_c/25 are removed by merging the nodes at the end of the fiber. The location of the merged node is determined by taking the average of the fiber node locations weighted by the nodal connectivity numbers.

The semi-spherical model is used for indentation. The boundary conditions are shown schematically in Fig. 1a: zero displacements are imposed on the spherical part of the boundary, while the flat upper part of the model is traction free. The rotations of the boundary nodes are not constrained. Indentation is performed with a spherical indenter of radius R (R/l_c is the relevant variable), which is represented as a rigid surface whose vertical position is controlled.

Figure 1.

Schematic representations of (a) network and (b) continuum models. Boundary conditions are indicated in both cases. The grid in (b) indicates schematically the grading of the mesh.

Fibers are discretized with B32 Timoshenko beam finite elements and the solution is obtained with the commercial software Abaqus/Explicit 2018. Contact is defined between the rigid indenter surface and fibers using the general contact algorithm. Contacts may also form between fibers and the effect of this process on the force-displacement curve is evaluated in dedicated simulations. Interfibrillar friction and friction between the fibers and the indenter were not considered. To reproduce quasi-static conditions, the kinetic energy of the model is kept below 5% of the total energy in all cases. An algorithmic damping is introduced to reduce the degree of oscillatory motion at contacts.

A continuum model is also developed for the indentation problem. This model does not represent the network explicitly, rather it uses a constitutive relation of hyperfoam type to describe the effective behavior of the network. We use the Storåkers’[57] representation of the hyperfoam strain energy function given as:

$$U=\sum _{i=1}^N\frac{2{\mu }_i}{{\alpha }_i^2}\left[{\lambda }_1^{{\alpha }_i}+{\lambda }_2^{{\alpha }_i}+{\lambda }_3^{{\alpha }_i}-3+\frac{1}{{\beta }_i}\left(J^{-{\alpha }_i{\beta }_i}-1\right)\right].$$

Here, λ_i, represent the principal stretches with i ranging from 1 to 3, J = λ₁λ₂λ₃ is the Jacobian of the deformation, and β_i = v_i/(1 − 2v_i). We consider all v_i to be equal to the effective small strain Poisson ratio. The μ_i parameters are related to the initial shear modulus as ${\mu }_0={\sum }_{i=1}^N{\mu }_i$ and the initial bulk modulus as $K_0={\sum }_{i=1}^{N}2{\mu }_i\left(1/3+{\beta }_i\right)$. Under uniaxial deformation λ₁ = λ and λ₂ = λ₃. These constitutive parameters are controlled as described in section 3.2 to explore the effect of tension-compression asymmetry on the indentation force-displacement relation. This model was chosen over other more popular non-linear models, such as hyperelastic models, due to its ability to better represent the compression behavior of the network [58].

The continuum model is axisymmetric and the boundary conditions are shown in Fig. 1b. The model size is L_x = L_y ≈ 31R in both x and y direction, much larger than the size of the network model due to the much lower computational effort required in this case. The continuum model has no internal length scale, and model dimensions are sufficiently large such that the solution is guaranteed to be independent of the indenter radius. Quadratic axisymmetric quadrilateral elements (CAX8) are used to discretize the problem domain and the mesh is graded linearly along both axes, from the point of initial contact. The ratio of the largest and smallest element sizes is 70. The indenter is modeled as a rigid quarter circle (axisymmetric). The solution is obtained using the implicit scheme in the finite element software Abaqus/Standard (2018).

3. Results

3.1. Network model results

Quasi-static indentation is performed with indenters of R/l_c = 2, 3, 5, 8, 12.5, and 14. In each case, 4 replicas of the model (different realizations of the network) are considered. Further, networks with w = −5.2 and w = −3.86 are considered such to evaluate the effect of this structural parameter on indentation behavior. Both w values correspond to non-affine networks, but w = −3.86 is closer to the non-affine-to-affine transition, which for this type of network takes place in the vicinity of w = −2.2 [59,60]. As indicated in section 2, all networks considered have the same density and the different w values are obtained by adjusting the fiber diameter, d. The diameters corresponding to w = −5.2 and w = −3.86 are d = 0.01l_c and d = 0.06l_c, respectively. These network parameters result in a network small strain stiffness of E₀/E_f = 5.11 * 10⁻⁶ for w = −3.86 and E₀/E_f = 3.14 * 10⁻⁹ for the w = −5.20 case. The Poisson’s ratio of the network in the small strain regime is v ≈ 0.326 and has limited variation with w. The loading and unloading branches of an indentation simulation with a network of w = −3.86 and an indenter with R/l_c = 12.5 is shown in Fig. S1 of the Supplementary material.

In reconstructed collagen networks the mean segment length, l_c, is reported in the range 1–3 μm for concentrations between 1 and 4 mg/ml [34,61,62]. The diameter of collagen fibers (fibril bundles) depends on pH and the temperature at which the network is constructed, with the smaller diameter of ~150 nm being reported for collagen gels reconstructed at the body temperature of 37C [61]. This corresponds to a ratio d/l_c in the range 0.05 to 0.15. The range of lengths and diameters of collagen fibers in the extracellular matrix network are reported as 20 – 200 μm and 200 – 400 nm, respectively [63,64], which corresponds to a broader range of d/l_c of 0.001 to 0.02.

To check the applicability of the Hertz model, we evaluate first the exponent q of the indentation force-displacement function f = kδ^q; the Hertz model predicts q = 1.5 (Eq. 2). Figure 2 shows this relation in double logarithmic coordinates for all indenter sizes and the two w values considered. The slope of these curves represents q. Each curve is an average over 4 realizations of the network. Data points represent raw data, while lines represent a moving median with a kernel length of 9. The continuous line corresponds to indentation depths smaller than the indenter radius (δ < R), while the dashed line corresponds to deeper indentations. Deep indentations (to depths way beyond the range of applicability of Hertz’s theory) are usually performed in experiments, particularly when using sharp tips.

Figure 2.

Force-displacement curves averaged over 4 realizations of the network for (a) w = −3.86 and (b) w = −5.2. Symbols show the data and lines correspond to moving median with a kernel size of 9. Continuous and dashed lines indicate regimes in which δ < R and δ > R, respectively. The slope of 1.5 corresponding to the Hertz model is shown.

It is observed that, beyond an initial set-in range (δ/l_c ≤ 2/3), the curves are approximated closely by the power function f = kδ^q, and exponent q increases as R/l_c decreases. This trend is shown in the plot of Fig. 3, where q is represented as a function of R/l_c. The figure indicates that Hertz scaling (q = 1.5) is recovered for sufficiently large indenters, i.e. R/l_c > 8. This observation aligns with the general rule of thumb that homogenized behavior results when the probing domain (size of the sample in a uniaxial test, indenter size, size of the volume element considered in simulations of heterogeneous materials) is at least one order of magnitude larger than the largest material length scale.

Figure 3.

Variation of exponent q with the indenter radius R/l_c for networks with the two w values considered. The continuous line is added to indicate the trend of the data.

A notable feature of the results in Fig. 2 is that some of the data points are significantly above the median lines. Such spikes of the force occur occasionally during indentation, particularly in networks with smaller w. The spikes are a manifestation of instabilities which are a characteristic feature of the deformation of stochastic networks.

The contact area is also compared with the prediction of the Hertz model for models with w = −3.86 and R/l_c = 3 and 14. To evaluate the contact area, the network crosslinks in contact with the indenter at any given time are identified and projected in the plane orthogonal to the indentation direction. The resulting set of co-planar points is tessellated and the area of the enclosed polyhedron is determined using the convex hull algorithm. The contact area is similar to that predicted by Hertz model in the R/l_c = 14 case but it is smaller than this prediction for R/l_c = 3.

In view of the results in Figs. 2 and 3 indicating non-Hertz scaling of the force with indentation depth, it is useful to evaluate the error made when interpreting experimental data with the Hertz model in a regime in which the Hertz solution does not apply. To this end, we compute the instantaneous modulus as $E_H^{*}=3f/(4\delta a)$ where the contact radius,

$$\text{a}=\{\begin{array}{c}\sqrt{\delta R},\text{when}\delta \le R,\\ R,\text{when}\delta >R,\end{array}$$

and report it in Fig. 4 after normalization with the reduced small strain modulus of the network, $E_0^{*}$. If the indentation test is an accurate method to evaluate the network stiffness, the ratio $E_H^{*}/E_0^{*}$ should be 1. The network modulus was determined separately, by applying uniaxial deformation to cubic subdomains extracted from the network models used for indentation. The size of models considered for this purpose was L/l_c = 31.6, which provides a reasonable compromise between mitigating size effects while remaining within the available computational power limits. The size effect affecting network stiffness and ways to mitigate it are discussed in [60].

Figure 4 indicates that, for both w values, $E_H^{*}/E_0^{*}$ is essentially 1 for all δ/l_c, provided R/l_c is larger than approximately 12. Interestingly, the ratio remains 1 even at indentation depths larger than the limit of applicability of the Hertz model (note that large indentation depths entail large strains, which go beyond the applicability of linear elasticity). Therefore, for indenter sizes above the R/l_c ≈ 12 threshold, the indentation test interpreted with the Hertz model provides an accurate estimate of the network stiffness. Smaller indenters lead to estimates significantly lower than the true network modulus. This trend continues at low indentation depths, and the inferred, apparent instantaneous modulus monotonically decreases with decreasing indenter radius. Other authors noted this trend when using sharp pyramidal indenters and attributed the effect to the fact that the strain gradients increase as indentation depth increases [65]. For a spherical indenter, the imposed strain gradient does not increase with increasing indentation depth, however it is proportional to 1/R, so it increases as R decreases.

Note that exponent q becomes equal to 1.5 once R/l_c > 8 (Fig. 3), but in the interval 12 > R/l_c > 8 ratio $E_H^{*}/E_0^{*}$ is smaller than 1 (Fig. 4) and hence Hertz model does not apply in this range.

In the case of reconstructed collagen for which l_c ≈ 1 – 3 μm, as discussed above, indenter tips larger than ~12 – 37 μm should be used to ensure applicability of Hertz’s model, while much larger indenters are required when probing biological tissue with larger l_c.

3.2. Origin of the departure from Hertz model predictions

The departure from Hertz solution shown by the data in Fig. 2 is not necessarily surprising, given that the network is not a continuum and the effect is observed when the probing length scale becomes comparable with the network mesh size. However, this observation does not provide a physical explanation. In this section we explore several potential mechanisms that may lead to the observed effect. Specifically, we explore the role of inter-filament contacts, the effect of the tension-compression asymmetry, discuss the potential contribution of non-locality, and of strain localization within the network below the indenter.

3.2.1. Role of contacts

Contact formation between filaments has a strong effect on material behavior in uniaxial compression [42,66]. It is useful to ask whether this mechanism modifies the indentation force-displacement curve. Exploring this issue is straightforward when using discrete network models since simulations may be performed with and without contact detection enabled. If contacts are not allowed to form, filaments are free to cross each other (phantom fibers), while in the opposite case, contacts form dynamically preventing filament crossing. A test of this type was performed with networks with w = −3.86 and using indenters of various radii. Figure 5 shows force-displacement curves similar to those in Fig. 2a obtained with and without contacts. The figure indicates that contacts have no effect on the force-displacement curve for all R/l_c values considered. This also eliminates the possibility that contacts are responsible for the departure from the Hertz model discussed in section 3.1.

Figure 5.

Force-displacement curves for networks with w = −3.86 and indenters of different radii. The formation of contacts between fibers is allowed and prevented in separate simulations (shown by dotted and continuous lines, respectively). The curves corresponding to the two situations and for given indenter size differ by less than the symbol size.

Given the critical importance of contacts in uniaxial compression, it is surprising that their role in indentation is minimal. To clarify this issue, we show in Fig. 6 a map of the ratio of the current density to the density of the unloaded network (i.e., the inverse of the Jacobian of the deformation) in the region below the indenter for the case R/l_c = 12.5 and at an indentation depth δ/l_c = 5. The density is computed by partitioning the deformed network domain in volume elements defined in a spherical coordinate system centered at the center of the indenter. The density is averaged in the azimuthal direction and is shown in Fig. 6 projected in the x − y plane of Fig. 1a. Within this plane, averaging regions have a length 0.5l_c in the radial direction and a span of π/30 radians in the polar direction.

Figure 6.

Ratio of the current density to the density of the undeformed material (the inverse of the Jacobian of the deformation field) in the region under an indenter of radius R/l_c = 12.5 and at an indentation depth δ/l_c = 5 obtained with a network model with w = −3.86. This field is obtained by averaging in the azimuthal direction about the indenter axis, and over 4 replicas of the network. The white semi-circular lines and labels show the distance from the first contact site in units of l_c.

Figure 6 indicates the formation of a highly compacted region of thickness approximately 2l_c immediately under the indenter. The volume variation in the region outside this band is much smaller. This behavior is similar to that observed in uniaxial compression [41,42], when highly compacted localization bands form at random locations in the material, which is otherwise only weakly compressed. Many contacts form in the compacted region leading to significant local stiffening. Due to its localization right under the indenter, the net effect of the compacted region is to modify the size and shape of the indenter, the effect of which is discussed further in section 3.2.4. The density of inter-filament contacts outside this band is close to zero and hence the indentation force-displacement curve remains essentially insensitive to the formation of contacts.

A separate simulation was run while disabling the formation of contacts between fibers and maintaining contacts between fibers and indenter. The resulting density map is indistinguishable from that shown in Fig. 6, which indicates that the compacted region is due to the direct action of the indenter on the network. This result is due to the small fiber diameter relative to the mesh size. As indicated in section 3.1, d/l_c = 0.01 and 0.06 in networks with w = −5.2 and w = −3.86, respectively. Since the volume fraction of fibers is small, their excluded volume contribution is largely inconsequential.

3.2.2. Role of the tension-compression asymmetry and non-linearity

The mechanical behavior of network materials is highly non-linear and asymmetric. Strong stiffening is observed in tension and softening takes place in compression. In this section we use the continuum model to evaluate the effect these features have on the force-displacement curve. The network behavior is non-local and non-linear and exhibits tension-compression asymmetry [67–70]. The continuum model uses a local formulation, while the asymmetry and non-linearity may be modified to reproduce those of the network. Therefore, the procedure used here allows separating the effect of non-locality from that of asymmetry and non-linearity.

Figure 7a shows the uniaxial nominal stress-stretch curve obtained with a network with w = −3.86, the same as is used in indentation tests described in section 3.1. The tension branch of the curve is also shown in Fig. S2 of the Supplementary material. Since this response is obtained in a uniaxial test with no imposed strain gradients, the non-local effects are absent. Beyond a short linear regime observed at strains below ~3% in both tension and compression (regime characterized by the small strain modulus, E₀), the curve stiffens exponentially in tension, while in compression it reaches a softening point, followed by a quasi-plateau. This data is fitted with the hyperfoam model of Eq. 3 with N = 3 using the built-in least square fitting procedure in Abaqus (2018). The constitutive parameters resulting from this fit are as follows: μ₁ = 2.95, μ₂ = −3, μ₃ = 0.55, α₁ = 10.73, α₂ = 10.71, and α₃ = 10.19. To enhance the accuracy, additional data corresponding to loading the same network in simple shear is added to the dataset used for fitting, see Fig. S2 of Supplementary material. The fitted constitutive description is then used in the continuum model to simulate indentation with a spherical indenter. We note that, the hyperfoam model used here does not replicate all aspects of the network mechanical response. For instance, the model assumes a strain-independent Poisson’s ratio (v = 0), which is inconsistent with the network behavior [42]. Consequently, the volumetric response of the fitted model does not compare well to that of the network. However, the uniaxial and simple shear test data used to calibrate the model is reproduced with reasonable accuracy (overall RMSD = 20.36% in ±50% nominal strain range), as shown in Fig. 7a.

Figure 7.

(a) Uniaxial nominal stress-stretch behavior of a fiber network (red circles) and the fit with the hyperfoam model of Eq. 3 (continuous red line labeled ‘Ref. case’). Two other curves are constructed by scaling the tensile branch and shear response (not shown) by ½ and 2, respectively, to create additional datasets with softer (Case I) and stiffer (Case II) tensile responses. All curves are normalized by the small strain stiffness of the reference network, $E_0^{Ref.}$. The straight line shows the linear elastic approximation of the curves. Diamond markers indicate maximum (tensile) and minimum (compressive) principal stretches over the entire problem domain in the indentation simulations leading to the data in (b). (b) Force-displacement curves corresponding to the three constitutive behaviors in (a). The force is normalized by $4E_0^{*}{R}^2/3$ corresponding to the Hertz solution. The exponent q obtained for each force-displacement curve is shown in the legend.

In indentation, principal strains are both positive and negative throughout the problem domain. Therefore, it is of interest to determine the effect of the tension-compression asymmetry of the constitutive behavior of the network on the force-displacement curve. To this end, the tensile branch of the stress-strain curve is modified as shown in Fig. 7a, such to create softer (Case I) and stiffer (Case II) responses in tension relative to the reference model fitted to the network response. The straight line in Fig. 7a (curve L) corresponds to the linear elastic behavior with Young’s modulus E₀. The asymmetry may be quantified by computing the parameter $\chi ={\int }_{1-{\lambda }^{*}}^{1+{\lambda }^{*}}\left|S_{11}/E_o^{ref}\right|d\lambda /2{\int }_0^{1+{\lambda }^{*}}{S}_{11}/E_o^{ref}d\lambda$, where λ* is taken here 0.20. χ is zero for a symmetric stress-stretch curve and takes the maximum values of 0.5 when the compressive stress vanishes identically. χ = 0.4, 0.43 and 0.46 correspond to case I, the reference case, and case II, respectively.

Figure 7b shows the indentation force-displacement curves obtained with the continuum model having constitutive behavior defined by the three curves in Fig. 7a. The curve corresponding to the linear elastic model is added for reference (labeled L).

To demonstrate the range of strains experienced by the continuum during indentation, we show in Fig. 7a by filled diamond symbols the maximum (tensile) and minimum (compressive) stretches in the indented model with δ/R = 0.9, i.e., corresponding to the state of maximum indentation depth of the curves in Fig. 7b. Under these conditions, the minimum (compressive) stretch is approximately 0.2, while the maximum (tensile) stretch is 1.15 and approximately the same in all 3 cases of Fig. 7a. This is to indicate to what degree the tension-compression asymmetry of the network is sampled during these indentations.

The curves in Fig. 7b are fitted over their entire range with a power function of the form F = kδ^q, and q is reported in the caption to Fig. 7b. The power function provides a good fit to the numerical force-displacement curves over the entire δ/l_c range considered. The exponent decreases as the degree of tension-compression asymmetry increases: case L with χ = 0 has q = 1.5, case I with χ = 0.4 has q = 1.4, the reference case with χ = 0.43 has q = 1.32 and case II with χ = 0.46 has q = 1.29. The effect of asymmetry cannot be tested with the network model because the asymmetry of the network constitutive response cannot be adjusted without changing the network structure. We conclude that, since the network predicts q values equal to or larger than 1.5, the observed increase of q with decreasing indenter radius (Fig. 3) cannot be caused by the tension-compression asymmetry of the network mechanical response.

3.2.3. Role of non-locality

It is of interest to discuss the effect of non-locality on the response of the network to indentation. Interactions in networks are non-local by definition and fibers support both forces and moments. Non-local effects are present in network models but are not captured by the local continuum model used here. However, results for the indentation of non-local elastic half spaces with spherical indenters are available in the literature [71,72]. Dhaliwal and Khan [72] derived the solution for the indentation problem in a micropolar elastic half space with an axisymmetric indenter of arbitrary profile. Their results indicate that the exponent and functional form of the indentation force-displacement relation remain identical to those of the Hertz model. A similar conclusion is obtained by Gourgiotis et al. [71] who study the indentation problem in couple-stress elasticity considering a spherical indenter. While an analytic expression is not provided, the results reported indicate that the exponent of the force-displacement relation is insensitive to the internal length scale of the couple-stress model. Based on these results, we infer that the increase of q with decreasing R obtained here with network models is not due to the intrinsic non-locality of network elasticity.

3.2.4. Role of network densification under indenter

We turn attention now to the effect of the compacted network developing under the indenter. Figure 6 shows that the network density increases by a factor of 2 or more in a layer of thickness approximately 2l_c under the indenter. Densification is larger in the vicinity of the indenter axis. The stiffness of the densified region is much larger than that of the network of density close to the nominal value. Therefore, it may be envisioned that the densified sub-domain becomes part of the indenter and hence, the effective indenter has size and shape different from those of the nominal indenter.

To quantify this effect, we consider networks with w = −3.86 and identify the network regions in the vicinity of the indenter of density larger than 2.5 times the mean density. Further, we merge these regions with the volume of the nominal spherical indenter to infer the profile of the effective indenter. The factor of 2.5 selected to identify the compacted network domain below the indenter is somewhat arbitrary. This threshold must be chosen high enough to eliminate the noise, while still being low enough to capture the compaction band. We check that reducing the threshold to 2 leaves the effective indenter profiles unchanged. The highest density under the indenter is approximately 5 times the mean (nominal) density of the network and hence, increasing the threshold to 5 largely eliminates the compacted region. The spatial distribution of high-density regions and the effective indenter profiles resulting upon the adoption of thresholds of value 2, 2.5 and 5 times the mean network density, and for an indenter of R/l_c = 12.5 are shown in Fig. S3.

Indenter profiles for the network with w = −3.86, a density threshold of 2.5 and all R/l_c considered are shown in Fig. 8a. In the figure, both coordinates are normalized with the respective indenter radius, R. Two effects are observed: (i) the shape and size of indenters of large R/l_c is not significantly modified since the thickness of the densified layer is of the order of l_c and its contribution to the effective indenter radius decreases as R/l_c increases. However, the size of the small indenters increases considerably; (ii) the indenter shape changes as R decreases. Specifically, the effective indenter profile becomes ‘sharper’ with decreasing R.

Fig. 8.

(a) Profile of the effective indenters after the compacted network region is merged with the indenter volume. The profiles are normalized by the respective nominal indenter radius, R, to demonstrate the relative contribution of the compacted region to the size of the effective indenter. All curves are obtained for the same indentation depth, δ = 5l_c. The dashed circle shows the nominal indenter. The profiles in (a) are fit with a power function to obtain q_predicted as described in the text. q_predicted is shown in (b) against R/l_c; the trends match that of Fig. 3. The continuous line is added to indicate the trend of the data.

Indentation with a generalized axisymmetric indenter was first analyzed by Steuermann [73] (see also [23,74]) who determined that for an indenter profile described by a function y = βxⁿ, the force-displacement relation is of the form f~δ^(n+1)/n. For a parabolic profile (spherical indenter), n = 2 and f~δ^1.5, while for a conical indenter with n = 1, f~δ². The profiles in Fig. 8a can be fitted with power functions of the form y = βxⁿ corresponding to exponents q_predicted = (n + 1)/n of the indentation force-displacement curve. The result of this analysis is shown in Fig. 8b Values of q_predicted agree with the corresponding data obtained directly from the force-displacement curves of Fig. 2a and shown in Fig. 3. While within the present accuracy the numerical prediction of q based on the estimate of the indenter profile is somewhat smaller than the values reported in Fig. 3, the physical basis of the effect discussed and the trend of q are well-defined.

The observations made here bear similarities with those reported for foams subjected to indentation. Gibson and Ashby [75] discuss that in an elastic-plastic foam, the indentation force-displacement curve has a peak, and that this peak scales inversely with the indenter size when indenters are sufficiently small. A similar size effect is identified numerically for the indentation of 2D periodic cellular structures with a flat punch [76]. The peak indentation stress (evaluated as the indentation force divided by the indenter width) increases with decreasing indenter width, with the effect being prominent for indenters smaller than 5 times the cell size. This result is similar to that obtained for elastic-plastic hexagonal honeycombs subjected to indentation [77] and is in agreement with experimental data for periodic cellular materials [78]. Densification under the indenter was observed in cellular materials by direct speckle interferometry and X-ray tomography measurements, and in simulations [79–81]. In view of these findings, the observation that densification takes place in fibrous materials subjected to indentation is not surprising. However, the result that the exponent of the indentation force-displacement relation changes as the indenter size decreases relative to the mean segment length, and its origin associated with the variation of the effective indenter shape due to network densification under the indenter are novel and important for the interpretation of experimental data obtained from indentation tests in network materials.

Conclusions

In this work we demonstrate the emergence of a size effect in indentation of network materials which leads to the modification of the force-displacement function when the indenter radius becomes smaller than about 12 times the mean segment length of the network. The exponent of the force-displacement power function is observed to increase as the indenter size decreases. This implies that the Hertz model, which is often used to infer the material stiffness from measured force-indentation curves, is not valid when using indenters with small radius. We show that the size effect is due to the formation of a densified network sub-domain under the indenter which effectively modifies the indenter shape. The modification is similar in the case of indenters with large and small radius, but its effect is more pronounced in the case of small indenters whose effective shape and size change significantly during indentation.

Statement of significance.

The article presents a study of indentation in network biomaterials and demonstrates a size effect which precludes the use of the Hertz model to infer the elastic constants of the material. The size effect occurs once the indenter radius is smaller than approximately 12 times the mean segment length of the network. This result provides guidelines for the selection of indentation conditions that guarantee the applicability of the Hertz model. At the same time, the finding may be used to infer the mean segment length of the network based on indentations with indenters of various sizes. Hence, the method can be used to evaluate this structural parameter which is not easily accessible in experiments.