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lc because the entire domain ends up being tiled when such large squares are used. The analysis in ref. 26 leads to the conclusion that the density map for a 2D network Soft Matter, 2011, 7, 6768–6785 | 6771

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Downloaded by RMIT Uni on 12 March 2013 Published on 24 May 2011 on http://pubs.rsc.org | doi:10.1039/C1SM05022B

F Vd ¼ C M,

Fig. 4 Auto-correlation function (ACF) of density for networks with different mean density, lc/L0, probed on different scales, d/lc. The ACF has a cut-off at |x| z L0/2.26

(Fig. 3) is rough, with a fractal dimension 2.54 (the surface embedded in 3D). The fractality of the density surface cannot be inferred from the fractality of the geometry probed at length scales smaller than lc.

3. Geometric and stiffness percolation Geometric percolation is defined as the formation of a continuous path/cluster spanning the entire problem domain. This structure may or may not be sufficiently rigid to provide nonzero elastic moduli on the global scale. Rigidity percolation refers to the critical density at which the structure acquires stiffness. Geometric and rigidity percolation are, in general, distinct events.28 Geometric percolation has been studied extensively for lattice and off-lattice models.29,30 We are concerned here with off-lattice models as they are more appropriate for random networks. For 2D structures of the type shown in Fig. 1c and d, geometric percolation takes place at a density L0rth ¼ 5.71.31 The number of cross-links per fiber, L0/lc ¼ (2/p)rL0, at percolation is also evaluated in ref. 32 and 33 as 5.42. The percolated cluster is fractal, with fractal dimension 1.8534 (to compare with D2D ¼ 1.5 evaluated by Kaye27 for networks much denser than the percolation cluster). The rigidity percolation depends on the nature of the crosslinks and on the coordination. Therefore, before discussing this issue, it is useful to analyze conditions under which a structure of fibers becomes rigid. Considering a pin-jointed frame made from F fibers and having V cross-links defined in d spatial dimensions, the total number of degrees of freedom is Vd. The Maxwell counting method35 indicates that rigidity transition takes place when F Vd ¼ 0. This relation may be written in terms of the average coordination number, hzi ¼ 2FV1. This indicates that the transition takes place when hzi ¼ 2d. Maxwell counting is not exact, as fibers may be added to the structure without constraining additional degrees of freedom. Taking this into account, one arrives at the modified Maxwell formula,36 6772 | Soft Matter, 2011, 7, 6768–6785

(3)

where C is the number of overconstraints and M is the number of mechanisms. For example, for two-dimensional frames the formula becomes F 2V ¼ 3, where the three mechanisms are two rigid body translations and one rotation in plane. In 2D (d ¼ 2), the critical coordination number is 4 and pinjointed structures such as that in Fig. 1d, for which z ¼ 4 at all cross-links, are not rigid. The structure in Fig. 1c is likewise floppy. This is equivalent to saying that central force networks with coordination z # 4 are not rigid (for these networks, the cross-links may be represented schematically as in Fig. 2a (e.g. ref. 37). These networks may be stabilized in various ways, including introducing second-nearest-neighbor links and transforming some of the cross-links in welds (Fig. 2c).34 This does not change the nature of the percolation which remains in the same universality class as that of 2D central force rigidity percolation in diluted lattices.34,38 The most common way to provide rigidity to networks of type Fig. 1d (z ¼ 4 by construction) is to account for the fiber bending stiffness. Consequently, the cross-links become similar to those in Fig. 2b or c, i.e. either transmit transverse forces which induce bending in both fibers or transmit forces and moments. If the cross-links are welds (Fig. 2c), the geometric and rigidity percolation take place at the same critical density. For freely rotating pin-jointed structures (Fig. 2b), the 2D rigidity percolation takes place at a slightly higher density, L0rth ¼ 6.7.31,38 The transition is continuous in all cases, with the elastic moduli increasing smoothly from zero. It is generally considered that both Young’s and shear moduli become non-zero simultaneously, at the same value of the density. However, this is not obvious given the fact that networks may be floppy and rigid in different deformation modes. Close to the critical point the shear modulus scales as G z (r rth)h. The exponent h was evaluated numerically for a number of models and is given in Table 1. Note that models predicting the same geometric and topological exponents (characterizing for example the fractal structure of the percolated cluster) may lead to different values of the exponent h.39 The stiffness exponent is more sensitive to the details of the model than the exponents characterizing the geometry. This indicates the expected difference between the percolation of a scalar field, as in transport problems, and that of a tensor field such as stress. An analysis of the partition of energy between axial and bending deformation modes in semi-flexible networks above but close to the percolation threshold was performed in ref. 39. The strain energy is stored mostly in bending modes. As discussed in Section 5.1.2, this situation is reversed when the density increases and/or as the coordination number increases. Table 1 Critical exponent h of stiffness percolation predicted by several models Model

Exponent h

Reference

Site percolation Central force bond percolation Random network with ‘‘welds’’ Random network with ‘‘rotating joints’’

1.35 0.06 3.57 0.3 3.96 0.04 3.0 0.2

39,162 39,162 39 31,39

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The results discussed above correspond to fully relaxed systems that carry no stress in the initial, unloaded state. The presence of residual, self-equilibrated internal stress may alter the mechanical stability of the fiber ensemble.40 Such effects are associated with the geometric non-linearity of the networks. For example, if the internal stress is large enough to produce buckling of some fibers, the response to an external load is expected to be different than that in the absence of the residual stress state. Internal stresses are also known to restrict the applicability of the Maxwell counting rule,36,41 and hence may change conclusions regarding the rigidity state of the network. Head42 has shown that the presence of an internal stress field in a random spring network changes the nature of the rigidity transition from continuous to first order (elastic moduli take finite values at the onset of rigidity). First order percolation transitions were observed in Bethe lattices43 and in some networks of infinitely rigid rods44 in the absence of residual stresses, but not in other random networks. The observation is important because it applies to systems which are not relaxed to their minimum energy state, as are most systems encountered in the physical world. Fiber waviness may affect some of the conclusions discussed in this section. Kallmes14 and Corte14,45 considered the effect of moderate fiber curvature on the geometry and moduli of paper, providing analytic results for the dependence of the mean segment length lc and other parameters on the ‘‘curl ratio,’’ a measure of the degree of waviness. They indicate a strong dependence of the geometric means on this parameter. The results were challenged recently by Yi et al.46 who suggest that the mean number of crossings per fiber is independent of the curl. They also indicate that the percolation threshold increases as the curl ratio increases.

4. Single fiber mechanics The mechanical behavior of a single filament is an essential input to any fiber network model. This discussion is divided into two parts addressing thermal and athermal fibers. The deformation of athermal fibers, such as those used in nonwovens, is typically described using continuum mechanics. Energy is stored in bending, axial, torsion and shear deformation modes. In fact, in most models only the axial and bending modes are considered. If the constitutive law of the fiber is linear elastic, these modes are independent. Specifically, the strain energy in a fiber is expressed in terms of transverse, u(s), and axial displacements, l(s), as 2 ð ð 2 1 2 1 dlðsÞ U¼ EI V uðsÞ ds þ EA ds: (4) 2 2 ds Here, E is Young’s modulus, A is the cross-sectional area of the beam, I its moment of inertia and the integration is performed along the respective fiber. The quantity lb ¼ EI/EA was used to quantify the relative stiffness of bending and axial deformation. Note that lb has units of length and, for cylindrical fibers, it is equal to half of the cylinder radius. Expression (4) for the strain energy is based on the classical, Euler–Bernoulli beam theory. It is valid for long, slender beams in which the shear stresses developing due to the applied transverse forces are small compared to stresses induced by This journal is ª The Royal Society of Chemistry 2011

bending. The Timoshenko beam theory includes a correction due to the shear deformation mode and hence is more appropriate for short, stubby beams. The stiffness matrix of Timoshenko and Euler–Bernoulli beams can be found in many texts, e.g. in ref. 47. The slightly more complicated formulation of the Timoshenko beam is rarely used in random fiber network models. It should be observed that given the broad distribution of fiber segment lengths in such systems (eqn (2)), a large number of segments are short enough (compared with the characteristic dimension of the cross-section) for the Euler– Bernoulli theory to become inaccurate. This suggests that the use of the Timoshenko theory for the entire network is desirable. An observation which may provide additional insight useful in the selection of an appropriate model for athermal beams is that most strain energy is stored in soft deformation modes of the network. This is discussed in detail in Section 5.1.2. At high network densities, the energy is stored primarily in the axial deformation mode and hence the Euler–Bernoulli and Timoshenko models are expected to provide similar predictions. The situation should be different when the network density is low and beam bending is the dominant deformation mode. In such conditions the overall response of the network should depend significantly on the model employed. In a small number of publications athermal fibers are not modeled with the beam theory, rather are represented using modified bead spring models,48,49 which are commonly employed as coarse grained representations of polymeric chains in polymer physics (e.g. ref. 50). Alternatively, the volume of each fiber is discretized with finite elements and hence continuum mechanics is used on the scale of individual fibers.51 The advantage of these models is that the excluded volume constraint (non-crossing condition) is enforced at all times during deformation. However, they are much more computationally expensive than models representing fibers as beams of trusses. Finally, ad hoc models for the fiber constitutive behavior have also been used; for example, inref. 52 and 53 the axial force is chosen as an exponential function of the Green strain in the direction of the segment. The mechanics of molecular filaments has a strong entropic component. The classical example of such system is rubber. This material is a network of cross-linked polymeric chains (polyisoprene) and is above the glass transition temperature in ambient conditions. The elastic response of a single chain is described by the entropic spring model.54 This is fundamentally different from the elasticity of all other solids which is primarily enthalpic. In the entropic spring model, the variation of the free energy of the chain induced by a change of its end-to-end distance is associated with the variation of the chain entropy. Specifically, the number of conformations available to the respective molecule decreases as the chain is stretched, and hence the entropy decreases. The thermodynamic force (acting axially) is given by f ¼ TvS/vl, where S is the entropy of the filament. The force depends on temperature and this leads to effects which are not observed in enthalpic systems, such as adiabatic cooling upon contraction and the development of a tensile force when the temperature is increased at constant stretch. Several models have been developed for this type of filaments. The freely rotating chain model assumes that the filament is composed of n segments/links and neighboring segments may Soft Matter, 2011, 7, 6768–6785 | 6773

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take any orientation and may overlap. The shape of the resulting filament is similar to an unconstrained random walk in 3D. The probability for the end-to-end vector to have length between R and R + dR is 3=2 3 3R2 4pR2 pðRÞdR ¼ 4pR2 dR; (5) exp 2pna2 2na2 where a is the length of a link. A link is a Kuhn segment or a segment of the filament equal in length to the persistence length. The mean square end-to-end distance is given by hR2i ¼ na2. The entropy of the filament is computed as S ¼ kB log U, where U is the number of conformations a chain of given endto-end vector and number of links may take. This may be written in terms of p(R) and the axial force results by taking the derivative of the free energy with respect to R, f ¼

3kB T R: na2

(6)

The effective axial modulus is proportional to the temperature (unlike the variation with temperature of the modulus of enthalpic materials). Real chains cannot overlap and cannot cross. When embedded in good solvents, the excluded volume constraint leads to chain swelling which, in turn, leads to an entropy loss. Taking these considerations into account, Flory proposed corrections to the formulation of the ideal chain. In this theory, the mean square end-to-end distance of the filament scales as hR2i z n2n , where nF is Flory’s exponent. The force required to stretch the filament becomes: 1=ð1nF Þ R : (7) f zkB T annF F

The Flory exponent is given as nF ¼ 3/(d + 2), where d is the dimensionality of the space in which the chain is embedded. In order to incorporate the finite extensibility of the filament, the Langevin correction is required. For a purely entropic filament (the enthalpic contribution associated with stretching is neglected), the force must diverge as thermal fluctuations are straightened out. The relation between the mean end-to-end vector length and the applied force becomes R/Rmax ¼ L(fa/kBT), where L(x) ¼ coth(x) 1/x is the Langevin function and Rmax ¼ na is the contour length of the filament. The Langevin function is linear for small values of the argument, L(x) z x/3, x / 0 and asymptotes to 1 for large values, L(x) z 1 1/x, x / N. Consequently, the axial force diverges for extensions R / na as fz

kB T Rmax : a Rmax R

(9)

Fig. 5 shows the constitutive response of freely rotating and worm-like chains (limit Rmax [ lp), along with their small deformation asymptote (the Gaussian chain approximation). Accounting for both the enthalpic and entropic contributions to the constitutive response of a filament is more involving.58–63 Thermal fluctuations producing transverse vibrations of the filament act against the filament bending stiffness. If the energy equipartition condition requiring that the same amount of energy is associated with all vibration modes is imposed, one may compute the amplitudes of all modes and hence estimate the set of shapes the chain may take. This derivation was performed in ref. 62 and 63 where the PDF of the end-to-end vector length for such a semi-flexible chain was derived as: " # N ðpiÞ2 lp 2lp X R 2 iþ1 pðRÞ ¼ 2 1 : (10) ðpiÞ ð1Þ exp Rmax Rmax i¼1 Rmax A 2D version of this PDF is available in ref. 63–65. The formula is approximate and properly normalized for large lp/Rmax values. It was verified against numerical results in ref. 66. The relation between the applied force and the end-to-end vector length is given in the implicit form: Rð f Þ Rð0Þ ¼

N R2max X f =f0 ; 2 p lp i¼1 i2 i2 þ f =f0

(11)

with f0 ¼ p2EI/R2max (note the resemblance of the normalization constant f0 to the critical force for Euler buckling; the similarity is fortuitous). It should be observed that in the context of semiflexible filaments for which bending is important, the persistence length may be evaluated as lp ¼ EI/kBT.67 A static model of a wavy filament being loaded along its endto-end vector was studied in ref. 68. The filament is assumed to be stress free in its wavy state and the work done during pulling is stored primarily as bending strain energy. The model was compared with the thermal version of the inextensible semiflexible chain described above62,63 in ref. 68, and it was concluded that despite quantitative differences, the general behavior is qualitatively similar. For example, both models predict the same divergence of the force at large extensions, R / Rmax,

(8)

The worm-like chain is a version of the freely rotating chain in which the angles made by successive bonds are close to p. This is considered a good model for relatively stiff polymers with large persistence length, lp.55 The mean square end-to-end vector becomes hR2i z 2lpna, much larger than that for the freely rotating chain of same n and a. After incorporating the Langevin correction in the worm-like chain, one may compute the axial entropic force in the two limits (e.g. ref. 56 and 57): for small extensions and for contour lengths much larger than lp, the model gives the same prediction as the freely rotating chain, eqn (6), while for large extensions R / Rmax ¼ na, 6774 | Soft Matter, 2011, 7, 6768–6785

2 kB T Rmax fz : 2a Rmax R

Fig. 5 Force acting along the end-to-end vector of filaments modeled using the worm-like chain model, the freely rotating chain model and the Gaussian approximation for the freely rotating chain model.


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f z 1/(Rmax R)2,

(12)

(i.e. identical to the expression for the worm-like chain). An experimental and numerical analysis of the force required to pull a wavy athermal filament is presented in ref. 69. These authors show that even though the force–displacement curve of a filament with shape described by a single frequency is convex at small loads and becomes concave at large loads in the R / Rmax limit, the respective curve corresponding to filaments of more complex shape described by a superposition of wavelengths is always concave. This has implications for the discussion on the nature of stiffening in fiber networks subjected to large deformations.62,66,68 It is interesting to compare filaments with Rmax [ lp and semiflexible filaments with Rmax z lp in the two limits of large and small extensions. The response at large extensions is general to inextensible thermal filaments, independent of the relationship between contour length and persistence length (eqn (12)). However, in the linear regime (small extensions), filaments with contour length much larger than the persistence length respond as described in eqn (6) (Fig. 5), while semi-flexible filaments with lp comparable with Rmax behave as described in eqn (11). ðEIÞ2 ðRð f Þ Rð0ÞÞ Eqn (11) becomes in the linear range f z kB TR4max (see also ref. 70). Note the important difference between this expression and eqn (6) with respect to the dependence of the force on temperature. In order to account for the enthalpic stretching of filaments once the waviness is entirely eliminated during stretching, an ad hoc relation was proposed:58,62,66 # " N f R2 X f ð1 þ f =EAÞ=f0 Rð0Þ þ max : Rð f Þ ¼ 1 þ EA p2 lp i¼1 i2 i2 þ f ð1 þ f =EAÞ=f0 (13)

5. Mechanics of the network The mechanics of the network is entirely determined by the constitutive behavior of fibers and cross-links, the spatial distribution of fibers and the non-bonded, excluded volume interactions between fibers. Fiber–fiber friction and the interaction of fibers with an embedding medium may be important in some applications. These are usually neglected in cross-linked networks, but become important in entangled networks. The following discussion is divided into two parts addressing separately networks with and without cross-links. Networks made from flexible and semi-flexible fibers are discussed separately.

(Langevin formulation). The excluded volume interactions between chains are thought to have no effect on the mechanics of the network, despite the fact that the material is dense and essentially incompressible. This approximation is based on the phantom chain concept commonly used in polymeric melts. Hence, the elasticity is entirely due to the behavior of the molecules viewed as entropic springs. Since classical rubber elasticity theory is presented in many texts,71 we limit to reviewing few key concepts. The first models assumed Gaussian chain behavior. It was conjectured that the response of the network is similar to that of three chains connected at one end and oriented in the three directions of the coordinate axes.72 If the network contains a total of N strands, one considers N/3 sets of such chain triads loaded by the far-field. Non-Gaussian statistics was incorporated in this ‘‘3-chain’’ model.73,74 A ‘‘4-chain’’model was proposed as an alternative.75,76 Somewhat later, Treloar77 and Treloar and Riding78 developed a network theory in which chains have full spatial distribution of end-to-end vector orientations and the global stress is obtained by averaging over this distribution. The chain deformation is considered affine, i.e. each end-to-end vector stretches and rotates as dictated by the far field. A simple formula is derived for the shear modulus of the network: G ¼ rckBT,

where, as in Section 2, rc is the chain number density. This theory was further developed by many authors (e.g. ref. 79 and 80) and a review is presented in ref. 81. The affine assumption is a major approximation in these theories. It is equivalent to assuming that each chain is tied to an underlying homogeneous continuum. In reality, network nodes are free to fluctuate and move non-affinely in order to reduce the overall free energy. These fluctuations were incorporated in the phantom network model.82,83 To introduce the central concept, let us consider that all filaments have n links and the coordination is z at all cross-links. Consider a set of z 1 filaments linked to the surface of the model and meeting at a cross-link which belongs to the ‘‘first layer of cross-links’’ below the (affinely deforming) surface. This cross-link, and in fact all cross-links in this ‘‘first layer,’’ may be considered connected with the surface through a single filament of n/(z 1) links and with a filament of n links with the interior of the material. This procedure may be applied successively to homogenize the connectivity between successive ‘‘layers of cross-links’’ to infer the stiffness of the entire network. The shear modulus of the network results G ¼ rc kB T

5.1.

Cross-linked networks

5.1.1 Flexible filaments. Networks with flexible filaments are those in which no bending moments are transmitted along individual filaments. Cross-links can only be pin joints (Fig. 2a) and the stability condition discussed in Section 3 requires that the average coordination per cross-link must be larger than 4. Such networks are representative for rubber and other crosslinked molecular structures. In rubbers, each molecule is viewed as an entropic spring in 3D which may be linear (Gaussian approximation of the freely rotating chain) or non-linear This journal is ª The Royal Society of Chemistry 2011

(14)

z2 : z

(15)

This value is always smaller than that of the affine network (eqn (14)), but the difference decreases as the coordination z increases. This dependence of the non-affinity of the deformation field on the coordination number is an important result of the phantom network theory. Despite the fact that the underlying physics of deformation is expected to be different, entropic network theories have been used to model the constitutive behavior of amorphous glassy polymers. The intense strain hardening observed in such systems was interpreted as being due to stretching of chain segments Soft Matter, 2011, 7, 6768–6785 | 6775


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between entanglements and subsequent chain alignment. A version of the original 3-chain model84 and a 8-chain model (8 chains with a common cross-link and connected to the corners of a cube which deforms affinely)85,86 were used to reproduce the constitutive response of various polymers below the glass transition temperature. In all these models, the network strands are considered entropic springs. Clearly, one may use any other chain stiffness within the same formalism, e.g. by replacing the entropic stiffness of the Gaussian approximation, 3kBT/Na2, with another elastic constant representing an enthalpic spring. It is important to observe that the entropic springs (both linear and non-linear) are in tension at all times. The axial force vanishes only when R / 0, which suggests that the unloaded network should collapse. The missing component is the excluded volume interactions which, in the classical theory, oppose the reduction of volume balancing the entropic forces. However, as mentioned above, these interactions are thought not to contribute to the deviatoric stress and hence are neglected in constitutive formulations. A significantly different view is provided by the ‘‘intrinsic stress formulation’’ in which the excluded volume interactions are given the central role.87–90 In dense systems, such as rubber, non-bonded enthalpic interactions take place just as in monatomic fluids and, for example, in crystalline metals. One may compute the atomic scale stress by taking into account all non-bonded interactions between a representative atom or group of atoms (e.g. a link in a freely rotating chain) and its neighbors. This stress is hydrostatic if the average is performed in a fixed, global coordinate system. However, if the average is performed in a coordinate system tied to the principal directions of the representative molecule, the ‘‘intrinsic frame’’ (e.g. with one of the axes aligned with the filament direction), the resulting ‘‘intrinsic stress’’ has a deviatoric component. This asymmetry is due to the non-uniform distribution of neighbors around the representative atom induced by steric shielding. The physical picture emerging from this analysis is that each filament segment carries an elementary deviatoric stress. The value of this intrinsic stress is largely independent of the deformation. When these intrinsic frames are random, e.g. in thermal equilibrium and in the absence of imposed deformation, the global stress obtained by summing up contributions of intrinsic frames results hydrostatic. A global deviatoric stress results upon preferential alignment of the intrinsic frames. This physical picture is scale independent: remains unchanged if one selects a short or a long segment of the filament to attach an intrinsic frame to it.91 It is also observed that the stress is enthalpic, just like in all other materials. However, due to the fact that the intrinsic (enthalpic) stress increases with increasing the temperature, the global stress inherits this apparently entropic characteristic.92 In the intrinsic stress formulation there is no need to make supplementary assumptions to prevent chain collapse and, given the fact that enthalpic interactions dominate, the framework applies equally to the rubbery and glassy states without supplementary assumptions being needed. 5.1.2 Semi-flexible filaments. Networks of semi-flexible filaments may be enthalpic or may include an entropic component. In both cases, filament bending is an important deformation 6776 | Soft Matter, 2011, 7, 6768–6785

mode and the persistence length, lp, is large compared to the mean segment length, lc. The athermal models are adequate representations of paper and various types of nonwovens. A large literature addressing these systems exists and its overview is beyond the goal of the present review. The reader is directed to comprehensive reviews of the older literature.15,93 Most of these works employed the affine approximation to predict the motion and deformation of fibers. One starts with the distribution function of fiber orientations (ODF) before deformation and assumes that each fiber moves as dictated by the far field strain. If small strains are applied, the ODF is assumed to be essentially deformation-independent. The total strain energy stored in the network and the effective stress may be computed based on these assumptions (e.g. ref. 94–97). Under large deformations, the ODF must be updated as deformation proceeds. With the affine deformation assumption still in place, the constitutive response for large deformations may be predicted for (at least) simple loads such as uniaxial and biaxial tension/compression.80,95 The agreement of these predictions with experimental data is not always good. Generally, these models perform poorly when applied to networks of low density98 and/or networks subjected to complex deformation paths. It is now well documented that the deformation of random fiber networks is non-affine (NA)31,33,53,99,100 i.e., the local strain field is not homogeneous and is distinct from the far-field imposed strain. As discussed below, the degree of non-affinity depends on the network density and fiber properties. This is actually expected considering the large heterogeneity of the network discussed in Section 2 (Fig. 3).101 The energy level at which the actual NA deformation occurs is lower than the energy corresponding to the affine deformation. Hence, the material is more compliant in the presence of NA. Chandran and Barocas53 indicate that the actual system-scale stress in random fiber networks subjected to displacement boundary conditions is about three times lower than the stress predicted using the affine deformation assumption. They indicate that there is no correlation between the strain in a fiber and its orientation. The NA strain was measured on the micrometre scale by Liu et al.102 in a F-actin network subjected to shear in which tracer beads were embedded. They observe the expected increase of the NA strain with decreasing network density. Various measures of non-affinity have been used. In ref. 33 and 68 the NA is quantified with the measure h(q qaff)2ir, where q is the change of the angle between the position vectors of two crosslinks separated by a distance r, change due to a far field imposed strain, and qaff is the corresponding affine value. Evaluating this quantity for different distances r leads to the conclusion that the NA decreases with increasing r and the trend is more pronounced for small lb (fibers with low bending stiffness). A strain-based measure of NA was introduced in ref. 100. The strain is probed on scale r by selecting triads of cross-links forming approximately equilateral triangles of area r2. The local strain is computed based on the relative displacements of the corners of the triangle—a procedure similar to that used in tensometry. The NA is computed as H(r) ¼ h(X Xaff)2ir, where X ¼ {311,322,312, u12} is a vector containing the non-zero strain and rotation components. Xaff is the affine equivalent vector, i.e. the far field strain imposed along the model boundaries. Neither X nor H has This journal is ª The Royal Society of Chemistry 2011


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tensorial properties. Fig. 6a shows the four entries of H for a network with rc ¼ 150 (lc/L0 ¼ 0.046) and lb ¼ 5 105(lb/L0 ¼ 104) subjected to uniaxial tension in direction 1, i.e. 311aff s 0. All NA strains decay with r following a power-law over the entire range of probed length scales. Interestingly, there is no difference between the various strain components and the rotation, despite the fact that the far field is pure tension and 3ij ¼ 0 for i,j s 1. The local rotation is also non-zero, which is in agreement with the observation of rotatory displacement fields in granular media subjected to simple strains (e.g. shear) in the far field.103 Fig. 6b shows the dependence of one of the components of H on parameter lb. As the bending stiffness of filaments increases relative to the axial stiffness, the NA decreases. A similar decrease is observed when the network density is increased.33,100 This suggests that denser networks with filaments relatively stiff in bending deform more affinely and hence the affine models described above should provide accurate predictions of the elastic moduli. This issue is discussed further below. However, it should be observed that this can only be an approximation since the deformation of the random structure is never exactly affine. As discussed in ref. 104 and 105, the central parameter controlling the NA is the local density (or fluctuations in density). Flocculated fiber networks are expected to have higher NA levels (and therefore lower effective moduli at given mean

fiber density) than unflocculated networks even at high nominal density. A similar effect may be induced by introducing spatial variations of the cross-link coordination number.11 The NA was studied theoretically and numerically in a related class of random structures in ref. 106. Considering du ¼ u uaff, the NA displacement of cross-links, this field exhibits spatial correlations as hu(x)u(0)i z log|x| in 2D and hu(x)u(0)i z |x|1 in 3D. The coefficient of proportionality scales with the square of the applied far field strain and the variance of the elastic moduli. This emphasizes the importance of the local density fluctuations in defining the NA level. Small deformations. The modulus of cross-linked networks of semi-flexible fibers of finite length, L0, subjected to small deformations was studied both experimentally and numerically (e.g. ref. 31,33,104,105, and 107). These works revealed nontrivial behavior which was not reported in the previous literature on nonwovens and paper. Fig. 7 shows numerical results for networks of the type shown in Fig. 1d having various densities, r, and ratios lb/L0. The axes are normalized as Enet/(r rth)l2b and (r rth)5.7l1.8 b , which leads to data collapse. The various data points correspond to densities r ˛ [50, 400] and lb/L0 ˛ [2 106, 2 101]. The curve shows two significantly different regimes. At large densities and large lb values, a plateau results, i.e. G z (r rth)l2b z l2b/lc,

(16)

where rth is the density at stiffness percolation. Since l2b ¼ EI/EA, the ratio of the bending and axial stiffness of the fiber, it results that in this regime the modulus of the network scales linearly with the density. The linear scaling of the modulus with the density is well-known in the literature108 and has been predicted by affine/effective medium models for the random network (e.g. ref. 109). Hence, if fibers are relatively stiff in bending and the density is high, the network behaves similarly to a homogeneous body which deforms affinely. The effective elastic constants of this effective medium can be predicted based on the

Fig. 6 (a) The variation of the strain-based non-affinity measure H with the probing length scale. (b) Dependence of the non-affinity on lb.100


Fig. 7 Dependence of Young’s modulus of a 2D cross-linked athermal network on the density, r, and lb. The exponents of the horizontal axis variable are adjusted to obtain data collapse. The vertical axis is normalized with the prediction of the affine deformation model.

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elasticity of individual fibers (e.g. ref. 33 and 97). Interestingly, the strain energy is stored predominantly in the axial deformation mode. In fact, if one eliminates bending and considers only the axial stiffness, a simple affine model leads immediately to the linear scaling of the shear modulus of the network with r.33 In the other extreme, when fibers are relatively soft in bending (small lb/L0) and/or at small densities, the modulus changes rapidly with lb and with density. In this regime,


Enet zðr rth Þm1 lbm2 ;

(17)

with both exponents being quite large, m1 ¼ 6.7 and m2 ¼ 3.8. A similar scaling relationship was determined in ref. 31, where m1 ¼ 6.67 and m2 ¼ 4. The relationship determined in ref. 33 is also amenable to this type of scaling with relatively similar exponents. The behavior in the range of system parameters in which the moduli depend so strongly on density cannot be predicted with affine models. In this case, the strain energy is stored predominantly in the bending mode. This may be compared with the situation close to percolation (Section 3), when energy storage in bending modes and a power law scaling of the percolating cluster modulus with the density are also observed.110 Fig. 8 shows experimental data for Young’s modulus of paper versus the density15,111 exhibiting trends similar to those obtained in softer network-based materials. At low densities the modulus increases rapidly with the density and eventually converges to the affine model prediction (modulus proportional to the density). Increasing the fiber length decreases the percolation threshold and leads to faster convergence to the affine prediction. This is also supported by the numerical study in ref. 112. These results are obtained for fibers cross-linked at all crossing points (Fig. 1d). However, one may consider situations in which cross-links are placed more sparsely. Then, the effective density

entering the above equations is lower than the actual fiber number density (reff z 1/lc, with lc controlled by the density of cross-links). Taking advantage of the large value of the exponents, one may vary the modulus over several orders of magnitude just by changing the cross-link density, without modifying the actual fiber number density.107 It must be observed that in networks of flexible filaments, varying the cross-link density at constant fiber density has a much weaker effect on the stiffness.113 Let us observe again that as the bending stiffness of the filaments decreases, the strain energy is stored predominantly in bending modes and the dependence of the network modulus on fiber density and stiffness cannot be predicted with affine models. When the trend is reversed and the axial mode becomes softer, the strain energy is stored predominantly in this mode and the network deformation is similar to that of a homogeneous body. This indicates that energy is stored always in the softer mode, i.e. as if the two modes were ‘‘springs connected in series.’’ Then, one expects a scaling of the network modulus of the form: 1/Enet z (EI/l3c)1 + (EA/lc)1, where the two terms may be weighted by some constants. This is, of course, just a heuristic approximation of the results discussed above. This reasoning has been extended in ref. 104 to incorporate the contribution of the entropic axial stiffness. To clarify under what conditions more strain energy is stored in the soft regions of the network, let us consider first the two composites sketched in Fig. 9a and b. The black regions are stiffer than the white ones. Both models are loaded in displacement control in the vertical direction. It is clear that in (a) the hard and soft regions are in parallel and more energy will be stored in the hard domains. In situation (b), the two are in series and more energy is stored in the soft domains. If the hard regions

Fig. 8 Variation of Young’s modulus of paper with the density (adapted from ref. 15). The two datasets correspond to networks with different fiber length. As the density increases, the modulus converges to the value predicted by the affine model.

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When discussing the results presented above, in particular the numerical data, it is necessary to remember that the moduli inferred from tests using stress-imposed and displacementimposed boundary conditions are not necessarily identical.117,118 If stress is imposed, one measures the strains and evaluates the compliance, Sd (the index indicates that the size of the patch or observation scale considered is d). Under displacement boundary conditions, one measures the stress and evaluates the stiffness tensor, Cd. In general, Cd is not an inverse of Sd for finite d, rather 119 S1 This is a type of ‘‘size effect’’ since in the limit d / N d # Cd. the two tensors coincide. So, in the context of statistically homogeneous networks one may write: hSd/2i1 # hSdi1 # C # hCdi # hCd/2i

Fig. 9 Composite continua with stiff (black) and compliant (white) domains arranged (a) in parallel, (b) in series, and (c) randomly. (d) Prediction of the model dividing networks in systems in which the strain energy is stored predominantly in compliant regions and systems in which the energy is stored predominantly in stiffer regions.

are randomly distributed as in Fig. 9c, energy will be stored in the soft domains as long as a path of soft regions percolates across the field. The site percolation threshold for the square lattice is pc ¼ 0.592, i.e. the condition is fulfilled when the percentage of soft regions is larger than 59.2%. Let us consider now a random network which is probed on scale d as in Fig. 3. The stiffness of each patch of area d2 depends on the local density (Fig. 7). Let us assume that the patch density distribution, p(rd), is lognormal with mean r0 and variance s and let us define a threshold density rth d which separates ‘‘soft’’ and ‘‘hard’’ domains. In order to decide whether the strain energy will be stored in ‘‘soft’’ regions of density rd < rth d , one needs to compute the fraction of ‘‘soft’’ sites in the lattice (the cumulative probability) and compare it with pc. This leads to a relationship between rth d , s and the network mean density r0 giving a threshold below which the energy is stored predominantly in the ‘‘hard’’ domains and above which the strain energy is stored in ‘‘soft’’ domains. This relation is shown in Fig. 9d. Similar results are obtained if a different distribution p(rd) is used. Hence, given the large heterogeneity of the network (large s), the likelihood of observing that the energy is stored predominantly in the more compliant regions is rather high. The effective Poisson ratio of the network results is very close to n ¼ 0.5 for a broad range of fiber densities (e.g. ref. 33 and 114). This is also the affine prediction for 2D systems. The 3D affine model predicts n ¼ 0.3.115 Note that n ¼ 0.5 does not imply incompressibility in 2D since the relationship between bulk and Young’s modulus in 2D is K ¼ E/[2(1 n)]. In 2D n may take values as high as 1, while in the 3D case the Poisson ratio must be smaller than 0.5. For more insight into the relationship between 3D and 2D elasticity, see ref. 116. This journal is ª The Royal Society of Chemistry 2011

(18)

for any finite d, where the angular brackets indicate that the respective quantity is computed as an ensemble average over replicas of the system. An interesting aspect of the nature of heterogeneity in random networks is the presence of spatial correlations of the elastic moduli.26,120 These are induced by spatial correlations of density. Specifically, let us probe the network on scale d (Fig. 3) and evaluate the density in each patch, rd(x). The value in patch i is defined by the number density of fibers having their center in the respective patch, and by fibers with centers in other patches j and crossing patch i, |xi xj| < L0/2. In 2D this induces spatial correlations of type hrd(|x|)rd(0)i z |x|1 (first order) with cut-off at |x| ¼ L0/2. Note that the cut-off is significantly larger than the segment length, lc. If the fibers are ‘‘infinite,’’ the cut-off is defined by the persistence length, lp. This rather trivial correlation of the density implies a correlation in moduli. Fig. 10 shows the spatial correlations of Young’s modulus computed for 2D networks of various densities and with fibers of same length (corresponding to the density ACF shown in Fig. 4). This leads to correlations in mechanical fields, as discussed in ref. 106. The implication of these correlations for the larger scale mechanics of the network is not trivial, as discussed in ref. 120. Specifically, this observation limits the scale at which homogenization can be performed to values larger than the correlation cut-off.

Fig. 10 Auto-correlation function (ACF) of local Young’s moduli for networks with different density, lc/L0, probed on different scales, d/lc.26

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The effect of wavy fibers on the moduli was studied in a number of works (e.g. ref. 121–123). For example, Perez121 accounted for fiber waviness by assuming that segments remain passive until straightened. In recent works,68,124 athermal networks with prescribed levels of fiber waviness were studied numerically. Both bending and axial stiffness of fibers were considered. In ref. 124 it is observed that as the degree of waviness increases and fibers are relatively soft in bending, a new scaling regime develops, in which G z (r rth)m1, with m1 decreasing from 6.67 for small waviness, to 3 for larger waviness. The response of random networks to localized loads has been studied.125 From a continuum mechanics point of view, this problem is complex due to the extreme heterogeneity of the elastic moduli, presence of spatial correlations and high field gradients expected close to the point of application of the force. Using discrete numerical models, Head et al.125 concluded that the deformation in the vicinity of the force is quite different from the elastic field in a homogenized continuum, but in the far field it converges to the affine predictions. In this limit, the energy decays as |x|2 with |x| being the distance to the point where the force is applied, i.e. as predicted by elasticity. Furthermore, close to the force, in the NA region, energy is stored primarily in bending modes, while away from it energy is stored primarily in axial modes. These conclusions are obtained only after significant averaging over statistically similar replicas. Large deformations. The large deformation behavior of fiber networks is more important from the point of view of applications than the small deformation behavior. Strain stiffening is generally observed in all networks subjected to large strains. However, stiffening is more pronounced in networks of semiflexible fibers. Their moduli may increase by more than one order of magnitude for relatively small strains.126 The study of strain stiffening by experimental means started a few decades ago (e.g. ref. 127–136), and led to theoretical and numerical investigations continuing today.31,33,99,106,137 The effect is due either or both to the axial pulling out of undulations of segments between cross-links, and to the preferential fiber alignment induced by the deformation. The filament waviness can be due to thermal fluctuations or, in the case of athermal filaments, to the way these are deposited during network fabrication (crimp). Before the ‘‘slack’’ is eliminated by the axial deformation, the work done is primarily used to reduce the entropy in the case of thermal filaments, and to ‘‘unbend the waviness’’ in the case of athermal fibers. Once the fiber is straightened, the stiffness increases dramatically (see Section 4) and energy is stored in the axial mode. Storm et al.62 developed an affine-network model which includes the entropic behavior of fibers with axial and bending stiffness. Onck et al.68 developed a 2D discrete model (including NA) in which fibers are athermal and wavy in the undeformed state. The two models are discussed and compared in ref. 66. The NA discrete model predicts a significantly smaller stiffness than the affine model in all conditions, as expected. The difference is more pronounced at small and intermediate fiber densities. However, the stiffening behavior at large strains is qualitatively similar. Stiffening is due both to the non-linear constitutive behavior of individual fibers (whether of entropic or enthalpic origin), and to filament re-orientation induced by the 6780 | Soft Matter, 2011, 7, 6768–6785

deformation. The second mechanism dominates at large strains. These trends are observed also in the experimental study presented in ref. 107, where F-actin networks cross-linked with the actin-binding protein scruin are tested at different actin concentrations and different cross-link densities. As the fiber density increases, the stress at which non-linear behavior of the network is observed increases. When the filament density is kept constant and the cross-link density increases, the stress at the onset of non-linearity also increases. The degree of non-affinity of the deformation is observed to decrease with strain68 and is larger in 3D than in 2D models.112 The dependence of NA on strain is attributed to the preferential alignment of filaments. In the limit, when all the filaments are aligned in the loading direction and span the entire problem domain, the NA vanishes. The effect of fiber orientation on NA was studied in detail in ref. 138. The component of the NA strain associated with the macroscopic strain leading to fiber orientation decreases (e.g. X1 produced by an applied far field strain 311). The other NA strain components remain as large as in the random network or increase. Furthermore, if a preferentially oriented network is probed in the direction perpendicular to the preferential fiber direction, the NA in all strain components is larger than when probing is performed in the direction of fiber alignment. Modifying the cross-link properties has a significant influence on the network moduli. The effect of using rotating joints (Fig. 2b) or welded joints (Fig. 2c) as cross-links in random networks such as that in Fig. 1d was studied in ref. 104. At high network densities, if the ratio lp/lc > 1, the network normalized shear modulus, Gl3c/EI, is independent of the type of cross-links used. However, for lp/lc < 1, ‘‘welding’’ the cross-links leads to significantly larger values of Gl3c/EI relative to the case in which rotating joints are used. Motivated by the mechanical behavior of the cytoskeleton,9,133,139 special constitutive relations for the cross-links have been considered, e.g. cross-links with a zigzag-type constitutive response modeling unfolding of specific actin-binding proteins.10 This leads to features in the network mechanics which are too specific to be discussed here. Allowing the cross-links to break140 leads to avalanches of cross-link failure and strain softening as well as to the formation of ‘‘stress fibers.’’ 5.1.3 Multiscale and continuum models. Solving boundary value problems defined over large network domains is desirable in a number of applications. For example, if one would be able to model the entire complexity of the cytoskeleton mechanics, faster progress towards understanding chemo-mechanical transduction could be made. Likewise, modeling fiber-based consumer products on the product scale would greatly facilitate product design and improvement. Developing discrete models on this scale is prohibited by the extremely large number of fibers/segments present in the problem. This provided the motivation for the development of multiscale models and of continuum representations of fiber networks. Effective medium representations were the first type of models developed for these systems; some of these have been discussed above. One of the most influential is Cox’s model141 which has been used extensively to describe paper properties. This uses the ‘‘shear-lag’’ idea according to which fibers are considered This journal is ª The Royal Society of Chemistry 2011


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attached to a homogeneous background (which implicitly deforms affinely). The representative fiber is loaded axially and the stress varies along the contour according to a function determined from a balance equation taking into account the load transfer from the background. The shear lag model has been both criticized and improved in various ways (e.g. ref. 109,142, and 143). The idea is broadly used in composites (fibers embedded in a matrix), where the shear lag model is considered accurate. However, the success of the model in composites is due to the fact that the matrix is homogeneous and the fiber bending stiffness can be usually neglected relative to the axial stiffness. Hence, in these systems each fiber experiences a load transfer and stress distribution similar to those described by the shear lag model. However, in fibrous systems, the effective medium can be defined only on average. If the axial force in a large number of fibers is evaluated and averaged, a stress distribution along the representative fiber similar to that predicted by the shear lag model develops.144 However, as discussed in Section 5.1.2, the large heterogeneity of the structure leads to pronounced local non-affine deformation which cannot be neglected. The model is expected to be less accurate in low density networks and in networks of semi-flexible fibers where bending is important. The models discussed in Section 5.1.2 avoid focusing on a single ‘‘representative’’ fiber or segment embedded in an effective medium. The entire fiber assembly is considered and the model can be solved only numerically. The central concept is to use large enough models for which reliable effective constitutive descriptions can be derived. If one is interested in solving a very large boundary value problem, the network is replaced by a homogeneous continuum with ‘‘effective’’ constitutive equations. It is useful to ask what are the limitations of this homogenization approach. This question has two parts. One should inquire how large should the model used for the calibration of constitutive equations be. Second, assuming that a constitutive equation has been reliably fitted to the overall response of a large fiber mass, one may ask whether this equation is applicable in the presence of large field gradients. Partial answers to these questions can be found in the existing literature on random networks and on random composites. Homogenization can be performed for a heterogeneous body if scale decoupling exists. This implies that a scale can be found which is significantly larger than the largest characteristic length scale of the microstructure. This insures that the model used for homogenization is ‘‘representative.’’ Furthermore, if the heterogeneity exhibits spatial correlations, the scale at which homogenization is performed must be larger than the cut-off or the characteristic length scale of these spatial correlations. As discussed in the previous sections, fiber networks do exhibit density fluctuations extending over a broad range of scales and spatial correlations of the density field with a cut-off proportional to the fiber length. Hence, the homogenization scale for these systems must be larger than the fiber length. Let us assume that homogenization can be performed and let us replace the network with a continuum. Can one use the homogenized constitutive model for any boundary conditions applied to this effective continuum? The answer is negative. If the boundary conditions induce field gradients with characteristic length scale (e.g. the inverse of the strain gradient) smaller than This journal is ª The Royal Society of Chemistry 2011

the homogenization length scale, the continuum model is expected to be a poor representation of the actual discrete body deformation. An example of this type is a point force applied on a 2D network.125 In this simulation the continuum representation was adequate only on average and at a distance from the point force at which the effective field gradient becomes small enough. Considering these limitations one may wonder if homogenization techniques may be used at all for such systems. A definite answer for this question remains to be found. In closing this discussion, two attempts to address the multiscale nature of the deformation in fiber networks are presented. A two-scale sequential multiscale technique for fiber networks was developed by Barocas and collaborators.51,53,145 These models are intended to represent the mechanics of connective tissue. A continuum model is used on the coarse scale, where the problem domain is divided into finite elements. The constitutive behavior of the material is defined by a set of ‘‘representative volume elements’’ (RVEs) each of them being a discrete model of a small network patch. An RVE, representing the fine scale, is associated with each integration point of each element of the coarse scale. The deformation of the RVEs is imposed by the coarse mesh and the resulting stress or strain energy is returned to the coarse model where an iteration is set-up to fulfill the balance equations. The method is computationally expensive, but avoids the use of effective constitutive equations. However, it does not provide a good solution for any of the issues discussed in the preceding paragraphs. The RVEs are loaded with a uniform field and hence the response of the real network to field gradients is not captured. Furthermore, although in principle RVEs can be as large as necessary such that their output can be trusted as a real ‘‘homogenized’’ response of the network to the respective perturbation, in practice, using large RVEs is too expensive. A formulation which avoids the use of RVEs and addresses the issue of the homogenization scale was presented in ref. 120 and 146. In this method, the network domain is partitioned into subdomains of size d and each of these is replaced with a continuum with elasticity fitted to the local material response (tangent stiffness). The resulting continuum model has the elastic heterogeneity, including spatial correlations, of the actual network on scales larger than d. The heterogeneity on smaller scales has been eliminated in the process of calibration of the local elastic model. The selection of d is important: it must be small enough for most of the heterogeneity and correlations to be carried over in the continuum model, but must be large enough for a continuum representation to make sense on the respective scale. The resulting continuum is stochastic and has the proper distribution of elastic moduli. Boundary value problems defined over very large domains can be solved using the Stochastic Finite Element method.147 Network failure under tensile stresses has been studied experimentally and numerically (e.g. ref. 148–150). The review of this literature is outside the scope of this review. 5.2 Entangled networks As with cross-linked networks, the discussion of entangled networks can be divided into two parts addressing thermal and athermal systems. The broad class of polymeric melts would fall Soft Matter, 2011, 7, 6768–6785 | 6781


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in the thermal entangled networks category. This topic is vast and its overview may form the subject of a book. Since excellent reviews exist in the polymer literature, polymer melt dynamics is not included in the present discussion. Rather, we prefer to focus on athermal entangled networks, systems made from fibers of diameter larger than approximately one micron, which are relevant for many biological and engineering materials. To date, such systems received much less attention than their thermal counterparts. Entangled networks are fundamentally different from the cross-linked networks discussed above. The central feature governing the mechanical behavior of this system is the formation of temporary contacts between fibers, while in cross-linked networks the cross-links are permanent and the formation of additional contacts (i.e. the effect of the excluded volume of fibers) is usually neglected. The mechanics of athermal entangled networks was studied both experimentally151–154 and by means of modeling and simulation.48,155–158 The focus was in most cases on the response of fiber wads or mats to uniaxial or triaxial compression motivated by industrial compaction processes. The following features are observed experimentally as a wad of fibers is compressed: -Pronounced hysteresis: although the response is elastic, the loading and unloading branches are widely separated. The phenomenon is heuristically attributed to fiber rearrangements and fiber sliding, which are both not happening in cross-linked networks. A stable hysteresis loop develops after a number of loading–unloading cycles. Fig. 11 shows three stable loading– unloading cycles obtained by compressing dry mutton wool at the same strain rate up to increasing levels of strain.151 The loading branches overlap, which indicates that the evolution of the microstructure is similar as the material is cycled up to different levels of strain/stress. -Significant strain rate effects are observed: the slower the test is performed, the larger the strain (the compaction) obtained at the same level of stress. Likewise, relaxation tests have been performed after loading up to a certain strain and, as in many other materials, the degree of stress relaxation is larger when loading is performed at a higher strain rate.151 Fiber reorientation per se is unlikely to produce significant rate effects. These are probably due to viscous processes at contact points.

Fig. 11 Stress–strain curves for uniaxial compression of mutton wool performed at a velocity of 6 mm min1. Adapted from ref. 151.

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Hence, the surface treatment of fibers or simply the level of humidity influences the magnitude of the rate effects observed. -The loading branch of the stress–strain curve is well approximated by a power law. Let us consider a constitutive equation (for the loading branch only) of the form: s ¼ j(rn rnth),

(19)

where r and rth are the current and the stiffness percolation densities. The coefficient j is proportional to the elastic modulus of fibers. The true strain is related to the density as 3 ¼ (1/3) log rth/r in the case of triaxial compression. The effective tangent modulus is expected to scale as E z rn. However, the Young’s modulus needs to be computed from the initial part of the unloading branch and one may attempt to fit to the data a power law with a different exponent, E z rm. Fig. 12 shows a logarithmic plot of the stress versus density obtained for a metallic wool subjected to uniaxial compression.152 The exponent n is seen to increase as the sample mass increases (the mass is reported per unit area projected in the plane perpendicular to the loading direction; this area remains constant during the uniaxial deformation of the sample). Fig. 13 shows the variation of the two exponents, n and m, with the sample mass. Their value increases from 3 to approximately 5. Modeling entangled networks is more difficult (and computationally more expensive) than modeling cross-linked networks due to the need to trace and account for friction at all fiber–fiber contacts. Numerous empirical equations have been proposed based on fitting individual experimental datasets, but these lack generality and transferability to other loading sequences. The classical micromechanics model for network compaction is due to van Wyk.155 This model includes two components of physics: fiber bending and formation of new contacts as the network is compressed. It leads to an expression similar to (19), with n ¼ 3. In 1998, Toll156 confirmed van Wyk’s model for 3D wads and derived an extension for 2D mats in which fibers are preferentially arranged perpendicular to the loading direction before deformation. Hence, less fiber reorientation results due to compaction in this case. The stress variation can be described by

Fig. 12 Stress–density curves for the uniaxial compression of metallic wool. The three sets of data correspond to different sample masses (initial density). The sample mass is reported as mass per unit area projected in the plane perpendicular to the loading direction. This area does not change during compression. Adapted from ref. 152.



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Fig. 13 Variation of the two exponents, m (filled symbols) and n (open symbols) with the sample mass (initial density). The sample mass is reported as mass per unit projected area in the plane perpendicular to the loading direction. This area does not change during compression. Adapted from ref. 152.

an equation similar to (19), with n ¼ 5. This appears to explain the experimentally observed variation of n shown in Fig. 13; the wad evolves from a 3D fiber geometry at the beginning of compaction to a structure similar to a 2D mat at the end of compaction. These models do not attempt to describe statistically the fiber rearrangement and sliding at fiber–fiber contacts and therefore do not predict the effect of friction on the mechanical behavior of the network. The evolution of the fiber structure and the hysteresis of the stress–strain curve were modeled by Carnaby and Pan159,160 who also developed a micromechanical model for shear deformation. A relatively small number of numerical works addressing the mechanical behavior of entangled networks have been published.48,49,158,161 These models are limited in the number and aspect ratio of the fibers that may be considered. Using fibers of aspect ration 38, Durville158 obtained results which generally agree with van Wyk’s theory and observed that the number of contacts increased as r2/3. He also studied the effect of fiber waviness on the stress–density curve, but for the range of parameters considered in his study no definite conclusion can be drawn. The results in ref. 48 confirm these observations. The authors used fibers of aspect ratio as high as 100 and concluded that the stress follows van Wyk’s model predictions (n ¼ 3) and the number of contacts increases linearly with the density. In ref. 49, networks of fibers with aspect ratio up to 100 were subjected to compression, with and without friction between fibers. The focus was on the statistics of sliding events at fiber contacts and the relationship between these and the evolution of stress. It was observed that: (a) once formed, contacts between fibers do not open, (b) significant sliding takes place, in particular in the initial stages of compaction, (c) sliding avalanches are observed; each avalanche produces a drop in the stress–strain curve. Hence, these structures share features with other discrete random systems such as granular materials. Fig. 14 shows the evolution of stress and that of a measure of sliding (the sum of the norm of relative sliding of fibers at all contacts) during compression. The data show a strong correlation between sliding spikes and stress drops, demonstrating the occurrence of avalanches. This journal is ª The Royal Society of Chemistry 2011

Fig. 14 Stress and mean sliding distance versus dilatation strain for a sample of entangled fibers compressed triaxially. The sliding distance refers to the mean of the relative sliding between fibers in contact in each load increment. The stress variation exhibits drops which are correlated with spikes in the magnitude of the relative sliding at fiber–fiber contacts.

6. Closure and outlook The mechanics of fiber networks presents complexities which are not encountered together in other materials. These are due to the random nature of the fiber distribution, their evolution during straining, fluctuations of density and spatial correlations, formation of temporary fiber–fiber contacts, the importance of thermal fluctuations in thermal systems and of friction in athermal systems. Clearly, the current understanding of these systems is incomplete and, although significant progress has been made, more work is needed to reach the stage at which constitutive equations accounting for the network microstructure can be developed. In closure, let us point to some of the outstanding challenges: -In many applications the solution of boundary value problems defined on very large fiber network domains is required. This requires homogenization and adequate multiscale methods. Progress is necessary to determine whether these systems may be homogenized, given the presence of long range correlations and pronounced heterogeneity. -Some fiber networks of biological importance (e.g. the cytoskeleton) contain active elements (myosin motors). Incorporating these in a random network is a challenge. -A better characterization of the microstructure evolution during straining is necessary. In particular, one needs to determine the ‘‘relevant’’ microstructural stochastic quantities which need to be incorporated in a system scale description of network mechanics. This is in the spirit of a broad class of constitutive formulations incorporating ‘‘internal parameters,’’ concept that has proven effective in other fields (e.g. in metal plasticity). -The stability of the network is still an open issue. Many results relevant for stability under small deformations close to percolation, and to stabilization of unstable and marginally stable networks upon large deformations have been obtained. However, the general topic of the stability of materials with Soft Matter, 2011, 7, 6768–6785 | 6783

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random structure is not fully understood theoretically and the application to large deformations network mechanics is expected to be challenging. It is our hope that this review will facilitate future progress in these and related directions.


References 1 K. Buet-Gautier and P. Boisse, Exp. Mech., 2001, 41, 260. 2 M. Alava, P. Nukkala, S. Zapperi, Y. A. Bahei-El-Din, A. M. Rajendran and M. A. Zikry, Int. J. Solids Struct., 2004, 41, 2307. 3 P. Boisse, B. Zouari and A. Gasser, Compos. Sci. Technol., 2005, 65, 429. 4 S. V. Lomov, D. S. Ivanov, I. Verpoest, M. Zako, T. Kurashiki, H. Nakai and S. Hirosawa, Compos. Sci. Technol., 2007, 67, 1870. 5 J. Cao, R. Akkerman, P. Boisse, J. Chen, H. Cheng, E. F. de Graaf, J. L. Gorczyca, P. Harrison, G. Hivet, J. Launay, W. Lee, L. Liu, S. V. Lomov, S. Long, E. de Luycker, F. Morestin, J. Padvoiskis, X. Q. Peng, J. Sherwood, T. Stoilova, X. M. Tao, I. Verpoest, A. Willems, J. Wiggers, T. X. Yu and B. Zhu, Composites, Part A, 2008, 39, 1037. 6 M. Ostoja-Starzewski, Appl. Mech. Rev., 2002, 55, 35. 7 H. Hatami-Marbini and R. C. Picu, Eur. J. Mech. Solid., 2009, 28, 305. 8 M. de Berg, C. Otfried, M. van Kreveld and M. Overmars, Computational Geometry: Algorithms and Applications, SpringerVerlag, Berlin, 2008. 9 B. Wagner, R. Tharmann, I. Haase, M. Fisher and A. R. Bausch, Proc. Natl. Acad. Sci. U. S. A., 2006, 103, 13974. 10 C. P. Broedersz, C. Storm and F. C. MacKintosh, Phys. Rev. Lett., 2008, 101, 118103. 11 M. Wyart, H. Liang, A. Kabla and L. Mahadevan, Phys. Rev. Lett., 2008, 101, 215501. 12 S. K. Patel, S. Malone, C. Cohen, J. R. Gillmor and R. H. Colby, Macromolecules, 1992, 25, 5241. 13 This expression is to be used in 3D. The corresponding expression for 2D is P2(q) ¼ 2 cos2 q 1. 14 O. Kallmes and H. Corte, Tappi J., 1960, 43, 737. 15 M. Rigdahl and H. Hollmark, in Paper Structure and Properties, ed. J. A. Bristow and P. Koleth, Marcel Dekker, 1986, ch. 12, p. 241. 16 S. Goudsmit, Rev. Mod. Phys., 1945, 17, 321. 17 R. E. Miles, Proc. Natl. Acad. Sci. U. S. A., 1964, 52, 901. 18 R. E. Miles, Proc. Natl. Acad. Sci. U. S. A., 1964, 52, 1157. 19 P. I. Richards, Proc. Natl. Acad. Sci. U. S. A., 1964, 52, 1160. 20 C. T. J. Dodson and W. W. Sampson, J. Pulp Pap. Sci., 1996, 22, 165. 21 W. W. Sampson, J. Mater. Sci., 2001, 36, 5131. 22 D. Stoyan, W. S. Kendall and J. Mecke, Stochastic Geometry and its Applications. Probability and Mathematical Statistics, Wiley, Berlin, 1987. 23 S. J. Eichhorn and W. W. Sampson, J. R. Soc., Interface, 2005, 2, 309. 24 C. T. J. Dodson, J. Roy. Stat. Soc., 1971, 33, 88. 25 M. Deng and C. T. J. Dodson, Paper: an Engineering Stochastic Structure, Atlanta Tappi Press, Atlanta, 1994. 26 R. C. Picu and H. Hatami-Marbini, Comput. Mech., 2010, 46, 635. 27 B. H. Kaye, A Random Walk through Fractal Dimensions, VCH Publishers, New York, 1989. 28 S. Feng and P. Sen, Phys. Rev. Lett., 1984, 52, 216. 29 M. B. Isichenko, Rev. Mod. Phys., 1992, 64, 961. 30 D. Stauffer and A. Aharony, Introduction to Percolation Theory, Taylor & Francis, London, 1994. 31 J. Wilhelm and E. Frey, Phys. Rev. Lett., 2003, 91, 108103. 32 G. E. Pike and C. H. Seager, Phys. Rev. B: Solid State, 1974, 10, 1421. 33 D. A. Head, A. J. Levine and F. C. MacKintosh, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2003, 68, 061907. 34 M. Latva-Kokko, J. Makinen and J. Timonen, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2001, 63, 046113. 35 J. C. Maxwell, Philos. Mag., 1864, 27, 27. 36 C. R. Calladine, Int. J. Solids Struct., 1978, 14, 161. 37 M. Kellomaki, J. Astrom and J. Timonen, Phys. Rev. Lett., 1996, 77, 2730.

6784 | Soft Matter, 2011, 7, 6768–6785

38 M. Latva-Kokko and J. Timonen, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2001, 64, 066117. 39 D. A. Head, F. C. MacKintosh and A. J. Levine, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2003, 68, 025101(R). 40 S. Alexander, Phys. Rep., 1998, 296, 65. 41 R. Connelly and W. Whiteley, SIAM J. Discrete Math., 1966, 9, 453. 42 D. A. Head, J. Phys. A: Math. Gen., 2004, 37, 10771. 43 C. Moukarzel and P. M. Duxbury, Phys. Rev. Lett., 1995, 75, 4055. 44 S. P. Obukhov, Phys. Rev. Lett., 1995, 74, 4472. 45 H. Corte, Statistical Geometry of Random Fibre Networks, Structure, Solid Mechanics and Engineering Design, Proceedings of the Southampton 1969 Civil Engineering Materials Conference, Part I, ed. M. Te’eni, Wiley-Interscience, New York, 1969, p. 341. 46 Y. B. Yi, L. Berhan and A. M. Sastry, J. Appl. Phys., 2004, 96, 1318. 47 R. D. Cook, M. E. Malkus and M. E. Plesha, Concepts and Applications of Finite Element Analysis, John Wiley & Sons, New York, 1989. 48 D. Rodney, M. Fivel and R. Dendievel, Phys. Rev. Lett., 2005, 95, 108004. 49 G. Subramanian and R. C. Picu, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2011, in press. 50 K. Kremer and G. S. Grest, J. Chem. Phys., 1990, 92, 5057. 51 X.-J. Luo, S. Triantafillos, V. H. Barocas and M. S. Shephard, Eng. Comput., 2009, 25, 87. 52 K. L. Billiar and M. S. Sacks, J. Biomech. Eng., 2000, 122, 327. 53 P. L. Chandran and V. H. Barocas, J. Biomech. Eng., 2006, 128, 259. 54 M. Rubinstein and R. Colby, Polymer Physics, Oxford Univ. Press, Oxford, 2003. 55 The definition of lp is based on computing the correlation function of tangent unit vectors along the filament, r(s). This can be expressed as hr(s) r(0)i z exp(s/lp), with s being the curvilinear coordinate along the filament. Note that for short chains for which the condition Rmax [ lp does not hold, the decay of this correlation function is not exponential and lp cannot be defined in this way. 56 C. Bustamante, J. F. Marko, E. D. Siggia and S. Smith, Science, 1994, 265, 1599. 57 J. F. Marko and E. D. Siggia, Macromolecules, 1995, 28, 8759. 58 T. Odijk, Macromolecules, 1995, 28, 7016. 59 M. Fixman and J. Kovac, J. Chem. Phys., 1973, 58, 1564. 60 K. Kroy and E. Frey, Phys. Rev. Lett., 1996, 77, 306. 61 D. Morse, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 1998, 58, 1237R. 62 C. Storm, J. J. Pastore, F. C. MacKintosh, T. C. Lubensky and P. A. Janmey, Nature, 2005, 435, 191. 63 J. Wilhelm and E. Frey, Phys. Rev. Lett., 1996, 77, 2581. 64 A. Ott, M. Magnasco, A. Simon and A. Libchaber, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 1993, 48, 1642R. 65 J. Hendricks, T. Kawakatsu, K. Kawasaki and W. Zimmermann, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 1995, 51, 2658. 66 T. Van Dillen, P. R. Onck and E. Van der Giessen, J. Mech. Phys. Solids, 2008, 56, 2240. 67 A beam of length lp subjected to bending and storing kBT strain energy has a deflection on the order of lp. 68 P. R. Onck, T. Koeman, T. Van Dillen and E. Van der Giessen, Phys. Rev. Lett., 2005, 95, 178102. 69 A. Kabla and L. Mahadevan, J. R. Soc., Interface, 2007, 4, 99. 70 F. C. MacKintosh, J. Kas and P. A. Janmey, Phys. Rev. Lett., 1995, 75, 4425. 71 L. R. G. Treloar, The Physics of Rubber Elasticity, Clarendon, Oxford, 1975. 72 H. M. James and E. Guth, J. Chem. Phys., 1943, 11, 455. 73 W. Kuhn and F. Groan, Kolloid-Z., 1942, 101, 248. 74 M. C. Wang and E. J. Guth, J. Chem. Phys., 1952, 20, 1144. 75 P. J. Flory and J. Rehner, J. Chem. Phys., 1943, 11, 512. 76 L. R. G. Treloar, Trans. Faraday Soc., 1946, 42, 83. 77 L. R. G. Treloar, Trans. Faraday Soc., 1954, 50, 881. 78 L. R. G. Treloar and G. Riding, Proc. R. Soc. London, Ser. A, 1979, 369, 261. 79 P. D. Wu and E. van der Giessen, Mech. Res. Commun., 1992, 19, 427. 80 P. D. Wu and E. van der Giessen, J. Mech. Phys. Solids, 1993, 41, 427.



View Article Online 81 M. C. Boyce and E. M. Arruda, Rubber Chem. Technol., 2000, 73, 504. 82 A. J. Staverman, Adv. Polym. Sci., 1982, 44, 73. 83 B. Erman and J. E. Mark, Structures and Properties of Rubber-like Networks, Oxford Univ. Press, Oxford, 1997. 84 M. C. Boyle, D. M. Parks and A. S. Argon, Mech. Mater., 1988, 7, 15. 85 E. M. Arruda and M. C. Boyce, Int. J. Plast., 1993, 9, 697. 86 E. M. Arruda and M. C. Boyce, J. Mech. Phys. Solids, 1993, 41, 389. 87 J. Gao and J. H. Weiner, Macromolecules, 1989, 22, 979. 88 J. Gao and J. H. Weiner, J. Chem. Phys., 1989, 90, 6749. 89 J. Gao and J. H. Weiner, Macromolecules, 1991, 24, 1519. 90 J. Gao and J. H. Weiner, Science, 1994, 266, 748. 91 R. C. Picu and M. C. Pavel, Macromolecules, 2003, 36, 9205. 92 R. C. Picu, Macromolecules, 2001, 34, 5023. 93 C. T. J. Dodson and P. Herdman, Mechanical Properties of Paper, in Handbook of Paper Science, ed. H. F. Rance, Amsterdam, Elsevier, 1980, vol. 2, p. 71. 94 T. Komori and M. Itoh, Text. Res. J., 1991, 61, 588. 95 T. Komori and M. Itoh, Text. Res. J., 1991, 61, 420. 96 D. H. Lee and G. A. Carnaby, Text. Res. J., 1992, 62, 185. 97 X.-F. Wu and Y. A. Dzenis, J. Appl. Phys., 2005, 98, 093501. 98 O. J. Kallmes, I. H. Stockel and G. A. Bernier, Pulp Pap. Mag. Can., 1963, 64, T449. 99 D. A. Head, A. J. Levine and F. C. MacKintosh, Phys. Rev. Lett., 2003, 91, 108102. 100 H. Hatami-Marbini and R. C. Picu, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2008, 77, 062103. 101 Note that the deformation of regular network structures, i.e. those in which a repeat unit (or ‘‘unit cell’’) may be identified, is affine on scales equal and larger than that of the unit cell. 102 J. Liu, G. H. Koenderink, K. E. Kasza, F. C. MacKintosh and D. A. Weitz, Phys. Rev. Lett., 2007, 98, 198304. 103 F. Leonforte, A. Tanguy, J. P. Wittmer and J. L. Barrat, Phys. Rev. B: Condens. Matter Mater. Phys., 2004, 70, 014203. 104 C. Heussinger and E. Frey, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2007, 75, 011917. 105 C. Heussinger and E. Frey, Phys. Rev. Lett., 2006, 96, 017802. 106 B. A. DiDonna and T. C. Lubensky, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2005, 72, 066619. 107 M. L. Gardel, J. H. Shin, F. C. MacKintosh, L. Mahadevan, P. Matsudaira and D. A. Weitz, Science, 2004, 304, 1301. 108 R. W. Perkins and R. E. Mark, J. Appl. Polym. Sci., 1983, 37, 865. 109 V. Raisanen, M. Alava, K. Niskanen and R. Nieminen, J. Mater. Res., 1997, 12, 2725. 110 The exponent m1 corresponding to situations in which r is not close to rth is not necessarily identical to the exponent h listed in Table 1 for densities r / rth. 111 H. Hollmark, H. Andersson and R. W. Perkins, Tappi J., 1978, 61, 69. 112 E. M. Huisman, T. van Dillen, P. R. Onck and E. Van der Giessen, Phys. Rev. Lett., 2007, 99, 208103. 113 J. Ferry, Viscoelastic Properties of Polymers, Wiley, Indianapolis, 1980. 114 D. J. Bergman and E. Duering, Phys. Rev. B: Condens. Matter Mater. Phys., 1986, 34, 8199. 115 R. L. Satcher and C. F. Dewey, Biophys. J., 1996, 71, 109. 116 M. F. Thorpe and I. Jasiuk, Proc. R. Soc. London, Ser. A, 1992, 438, 531. 117 C. Huet, J. Mech. Phys. Solids, 1990, 38, 813. 118 R. Hill, J. Mech. Phys. Solids, 1963, 11, 357. 119 Two second order tensors A and B are in the relation A < B if vAv < vBv for any vector v. 120 H. Hatami-Marbini and R. C. Picu, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2009, 80, 046703. 121 M. Perez, Tappi J., 1970, 53, 2237.


122 M. Perez and O. J. Kallmes, Tappi J., 1965, 48, 601. 123 D. H. Page and R. S. Seth, Tappi J., 1980, 63, 99. 124 C. Heussinger, B. Schaefer and E. Frey, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2007, 76, 031906. 125 D. A. Head, A. J. Levine and F. C. MacKintosh, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2005, 72, 061914. 126 Y. C. Lin, N. Y. Yao, C. P. Broedersz, H. Herrmann, F. C. MacKintosh and D. A. Weitz, Phys. Rev. Lett., 2010, 104, 058101. 127 P. A. Janmey, E. Amis and J. Ferry, J. Rheol., 1983, 27, 135. 128 P. A. Janmey, U. Euteneuer, P. Traub and M. Schliwa, J. Cell Biol., 1991, 113, 155. 129 M. D. Bale and J. D. Ferry, Thromb. Res., 1988, 52, 565. 130 Y. Tseng, K. M. An, O. Esue and D. Wirtz, J. Biol. Chem., 2004, 279, 1819. 131 J. F. Leterrier, J. Kas, J. Hartwig, R. Vegners and P. A. Janmey, J. Biol. Chem., 1996, 271, 15687. 132 L. L. Ma, J. Xu, P. A. Coulombe and D. J. Wirtz, J. Biol. Chem., 1999, 274, 19145. 133 J. Xu, Y. Tseng and D. Wirtz, J. Biol. Chem., 2000, 275, 35886. 134 M. L. Gardel, J. H. Shin, F. C. MacKintosh, L. Mahadevan, P. Matsudaira and D. A. Weitz, Science, 2004, 304, 1301. 135 B. A. DiDonna and A. J. Levine, Phys. Rev. Lett., 2006, 97, 068104. 136 R. Tharmann, M. M. A. E. Claessens and A. R. Bausch, Phys. Rev. Lett., 2007, 98, 088103. 137 C. Heussinger and E. Frey, Phys. Rev. Lett., 2006, 97, 105501. 138 H. Hatami-Marbini and R. C. Picu, Acta Mech., 2009, 205, 77. 139 D. H. Wachsstock, W. H. Schwartz and T. D. Pollard, Biophys. J., 1994, 66, 801. 140 J. A. Astrom, P. B. Sunil-Kumar, I. Vattulainen and M. Karttunuen, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2008, 77, 051913. 141 H. L. Cox, Br. J. Appl. Phys., 1952, 3, 72. 142 J. A. Astrom, J. P. Makinen, J. M. Alava and J. Timonen, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 2000, 61, 5550. 143 J. A. Astrom, S. Saarinen, K. Niskanen and J. Kurkijarvi, J. Appl. Phys., 1994, 75, 2383. 144 C. Heussinger and E. Frey, Eur. Phys. J. E: Soft Matter Biol. Phys., 2007, 24, 47. 145 T. Stylianopoulos and V. H. Barocas, Comput. Meth. Appl. Mech. Eng., 2007, 196, 2981. 146 M. A. Soare and R. C. Picu, Int. J. Numer. Meth. Eng., 2008, 74, 668. 147 R. G. Ghanem and P. D. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, New York, 1991. 148 P. Isaksson and R. Hagglund, Int. J. Fract., 2009, 156, 1. 149 P. Isaksson and R. Hagglund, Int. J. Solids Struct., 2009, 46, 2320. 150 P. Isaksson, Eng. Fract. Mech., 2010, 77, 1240. 151 D. Poquillon, B. Viguier and E. Andrieu, J. Mater. Sci., 2005, 40, 5963. 152 J. P. Masse, L. Salvo, D. Rodney, Y. Brechet and O. Bouaziz, Scr. Mater., 2006, 54, 1379. 153 D. Poquillon, B. Viguier and E. Andrieu, Compression des Fibres Enchevetreescalibrees en Nylon et Aluminium. Experiences et Modelisations, 18 Congres Francais de Mecanique, Grenoble, 2007. 154 S. Bergonnier, F. Hild, J. B. Rieunier and S. Roux, J. Mater. Sci., 2005, 40, 5949. 155 C. M. Van Wyk, J. Text. Inst., 1946, 37, T285. 156 S. Toll, Polym. Eng. Sci., 1998, 38, 1337. 157 M. Baudequin, G. Ryschenkow and S. Roux, Eur. Phys. J.B, 1999, 12, 157. 158 D. Durville, J. Mater. Sci., 2005, 40, 5941. 159 N. Pan and G. A. Carnaby, Text. Res. J., 1989, 59, 285. 160 G. A. Carnaby and N. Pan, Text. Res. J., 1989, 59, 275. 161 B. Pourdeyhimi, B. Maze and H. V. Tafreshi, J. Eng. Fibers Fabr., 2006, 1, 47. 162 M. Sahimi, Phys. Rep., 1998, 306, 213.

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