nonlinear elasticity of cross-linked networks
TRANSCRIPT
PHYSICAL REVIEW E 87, 042721 (2013)
Nonlinear elasticity of cross-linked networks
Karin John,1,* Denis Caillerie,2 Philippe Peyla,1 Annie Raoult,3 and Chaouqi Misbah1,†1Universite Grenoble 1/CNRS, LIPhy UMR 5588, F-38041 Grenoble, France
2L3S-R, B.P. 53, F-38041 Grenoble Cedex 9, France3Laboratoire MAP5 UMR 8145, Universite Paris Descartes/CNRS, F-75270 Paris Cedex 06, France(Received 18 January 2013; revised manuscript received 12 March 2013; published 26 April 2013)
Cross-linked semiflexible polymer networks are omnipresent in living cells. Typical examples are actinnetworks in the cytoplasm of eukaryotic cells, which play an essential role in cell motility, and the spectrinnetwork, a key element in maintaining the integrity of erythrocytes in the blood circulatory system. We introducea simple mechanical network model at the length scale of the typical mesh size and derive a continuous constitutivelaw relating the stress to deformation. The continuous constitutive law is found to be generically nonlinear evenif the microscopic law at the scale of the mesh size is linear. The nonlinear bulk mechanical properties are ingood agreement with the experimental data for semiflexible polymer networks, i.e., the network stiffens andexhibits a negative normal stress in response to a volume-conserving shear deformation, whereby the normalstress is of the same order as the shear stress. Furthermore, it shows a strain localization behavior in responseto an uniaxial compression. Within the same model we find a hierarchy of constitutive laws depending on thedegree of nonlinearities retained in the final equation. The presented theory provides a basis for the continuumdescription of polymer networks such as actin or spectrin in complex geometries and it can be easily coupled togrowth problems, as they occur, for example, in modeling actin-driven motility.
DOI: 10.1103/PhysRevE.87.042721 PACS number(s): 87.16.Ka, 87.16.A−, 87.10.Pq
I. INTRODUCTION
Cross-linked semiflexible polymer networks play an impor-tant role in the mechanics of living cells. For example, dendriticactin networks are essential during cellular motility [1] andthe so-called spectrin network determines the mechanicalproperties of red blood cell membranes [2]. Experiments showthat most semiflexible polymer networks behave as nonlinearelastic solids [3–5] and exhibit a stress-stiffening behaviorunder a volume-conserving simple shear. At the same time thenetwork tries to contract in the direction normal to the sheardirection, i.e., it develops negative normal stresses.
Previous theoretical studies have successfully obtainednonlinear bulk rheological properties for cross-linked semi-flexible polymer networks [4,6,7] or stiff polymer networkswith flexible cross-linkers [8] by considering the network asa collection of thermally fluctuating filament segments, whichact as entropic springs. Thereby rheological properties haveoften been determined numerically. Other types of studieshave considered, for example, ordered spring networks forthe description of the spectrin network underlying the redcell membrane by focusing on the behavior of the disretenetwork [9]. However, in many situations it is convenientto dispose of a macroscopic constitutive law, but at presentwe are not aware of a presentation of such a law based on amicroscopic network structure.
Our objective is therefore to derive continuous macroscopicconstitutive laws for semiflexible filament networks by usinghomogenization techniques, which capture the nonlinearelasticity of the network and are, at the same time, suitablefor tackling boundary value problems in complex geometries,
*[email protected]†[email protected]
as they arise, for example, in modeling cell shape changesor motion due to site-directed actin polymerization far fromchemical equilibrium. The derivation of a macroscopic law hasseveral advantages. (i) It can be placed in the general contextof existing phenomenological laws. (ii) Unlike phenomeno-logical laws, the coefficients entering the law are expressedin terms of basic microscopic properties (elastic propertiesof filament segments and cross-links, etc.). (iii) It allowsthe fully analytical treatment of limit cases (like linear andweakly nonlinear regimes) and provides some simple results,as exemplified here.
Our approach is based on the definition of a networkstructure at the scale of the mesh size and microscopicmechanical properties based on this structure. Here we restrictourselves to the basic physical ingredients, i.e., we modelfilament segments as linear springs, which can be of thermalorigin (entropic) and/or enthalpic (a change in length mightarise from bond stretching, a fact that may occur understrong solicitations). However, within the same framework,other properties, for example, filament bending modes anda rotational cross-link rigidity, can easily be included. Ingeneral, the network can also have a viscous dissipativeresponse. However, our strategy is to keep the model assimple as possible in this first attempt. Most models of thespectrin network take only the elastic response into account.Furthermore, in cross-linked and bundled actin networks theelastic modulus dominates the mechanical response on thebiologically relevant time scales [6].
The derived constitutive law shows a nonlinear elasticbehavior as found in earlier studies for random networks [3–6,8,10]. Some differences are noteworthy. Since the filamentsegments are modeled as linear springs, the nonlinearity ariseshere from the network topology, and not from the mechanicalproperties of the filaments. Furthermore, the network isisotropic in the linear regime and becomes anisotropic under
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JOHN, CAILLERIE, PEYLA, RAOULT, AND MISBAH PHYSICAL REVIEW E 87, 042721 (2013)
finite deformations. It stiffens and exhibits negative normalstresses when subject to a finite simple shear. This effectagrees with experiments [5]. Under uniaxial compression thenetwork softens for a weak compression and stiffens for astrong compression. Depending on the compression direction,the transition between the two regimes occurs via a localizationof the elastic deformation, a phenomenon already observed fordiscrete spring networks under isotropic compression [9].
II. MODEL
A. Network definition and assumptions
We start out from the hypothesis that most cross-linkedfilament networks are more or less periodic on a mesoscopicscale and act as nonlinear elastic solids. The size of eachelementary cell, i.e., the typical mesh size l (∼100 nm) is smallcompared to the total size L of the structure (∼1–10 μm),which introduces a small parameter η = l/L and allows forthe upscaling of the network to a continuous medium usingthe so-called homogenization method [11].
We consider a planar network consisting of elastic fila-ments, which are stiff on the length scale l, albeit semiflexibleon the macroscale L. The filaments are periodically connectedby nodes. The most simple network can be divided intostructurally identical N1 × N2 elementary cells as shownin Fig. 1. Each elementary cell contains one node andthree filaments, described by the vectors �Bi with i = 1,2,3.Each node is identified by a doublet of integers (n1,n2) ∈{0, . . . ,N1 − 1} × {0, . . . ,N2 − 1}. N1 and N2 are very largeand of the same order. The objective of the homogenizationprocedure is now to approximate the node positions by acontinuous function and to express the filament vectors bymeans of Taylor expansions of the node positions leadingeventually to the expressions given in (2).
The homogenization procedure starts out by introducingthe contracted coordinates
λη
i = ηni, (1)
B2 B1
B3(n1-1,n2 + 1)
(n1,n2)
(n1,n2 + 1)(n1+1,n2)
FIG. 1. (Color online) Sketch of a small part of the filamentnetwork showing the different types of elementary vectors �Bi of oneelementary cell in different colors along with the node numberingscheme (n1,n2).
with(λ
η
1,λη
2
) ∈ {0,η, . . . ,η(N1 − 1)} × {0,η, . . . ,η(N2 − 1)}and where the superscript η signifies that the new coordinatesare still discrete. The basic idea is, then, that for most ofthe network motions, the positions of the nodes can beapproximated by a continuous function �ψ(λ1,λ2) such that theposition of the node (n1,n2) is �ψ(λη
1,λη
2). For small enoughη the discrete Lagrangian variables (λη
1,λη
2) then becomethe set of continuous Lagrangian coordinates (λ1,λ2) ∈ ω
of the equivalent continuous medium with ω = [0,η(N1 −1] × [0,η(N2 − 1]. In this homogenized network the filamentvectors �Bi are obviously given by a Taylor expansion up toO(η)
�B1 = η∂λ1�ψ, �B2 = η∂λ2
�ψ, and �B3 = �B2 − �B1. (2)
Of course, one could envisage other types of networkstructures, but it will be seen that several features (e.g.,negative normal stresses in response to a shear deformation[5]), obtained within a network model with random filamentorientation [7], are already captured in our simple triangularnetwork model.
B. The elastic energy
Based on the above-defined network structure with thefilament vectors �Bi defined in (2) we now introduce amechanical energy density functional. Neglecting the bendingmodes, the energy density functional at the continuum leveldepends only on ∂λi
�ψ , so that the total energy reads
F [ �ψ] = 1
η2
∫∫ω
f (∂λi�ψ) dλ1dλ2, (3)
with the local energy per unit cell
f = k
2l
3∑i=1
(li − l)2 + βl2
2g. (4)
The first term in Eq. (4) denotes the energy related to extensionor compression of the filaments. Here we treat the filamentsas Hookean springs, whereby li = ‖ �Bi‖ and l denote theactual and the equilibrium distance between two cross-linksconnected by a filament of type i, respectively, and k denotesa spring constant. This constant is, in principle, of entropicorigin. Note that the distance between two cross-links li isshorter than the full contour length of the connecting polymerand that the elasticity originates from shape fluctuations. Whenthe thermal fluctuations of a filament are completely stretchedout by an applied strain, one expects a crossover from entropicto enthalpic extension of the filament, with energy scales muchlarger. We come back to this point in Sec. III B. Because ofthe linear character of the springs, an elementary cell of thenetwork under compression, for example, can collapse, in thatits area may vanish. Due to the excluded volume condition,the collapse of the material cannot occur in reality. To accountfor this condition, we introduce a penalization-like term in thefree energy, (4), that ensures a finite area of the elementary cellg = ‖ �B1 ∧ �B3‖ > 0 and where β denotes a small constant withβ � kl. It is straightforward to enrich further the mechanicalproperties for each filament type and to include other types of
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mechanical ingredients (e.g., a rotational cross-link rigidity,bending of filaments), however, for a proof of concept itis sufficient to limit our analysis to the most simple caseof identical filaments, acting as linear springs supplementedby the excluded volume condition. It turns out that severalobserved features will be captured by the present simple model.
The mechanical equilibrium follows from setting the func-tional derivative δF/δ �ψ = 0 and by applying the appropriateboundary conditions.
Having introduced all physical ingredients of the homog-enized network model we now establish the link between theelastic energy, (3), and the classical Cauchy stress tensor oftenencountered in elasticity [12]. Below we present a short list ofbasic notions of nonlinear elasticity that are necessary in orderto derive the constitutive law.
C. The Cauchy stress tensor and the derivation of theconstitutive law
Given a reference configuration �ψ0 any other configurationof the network �ψ can be expressed using the displacement field�u, which is also a classical notion in linear elasticity. Recallthat �ψ and �ψ0 describe the positions of the material pointsin Eulerian space. Since in linear elasticity, derivatives withrespect to the actual positions of a material point coincide withthe reference position, no care is taken regarding the systemof coordinates. In contrast, in nonlinear elasticity care must betaken, and this is why it is important to specify that each actualmaterial point is linked to a reference point by a function. Wewrite the relation between these points as
�ψ = �ψ0 + �u( �ψ0). (5)
Consequently the derivatives ∂λican be expressed as
∂λi�ψ = (I + ∇0 �u)∂λi
�ψ0 = F∂λi�ψ0, (6)
where I denotes the identity tensor, ∇0 denotes the vectorgradient with respect to the reference configuration �ψ0 (i.e.∇0 = ∂/∂ �ψ0), and F is the deformation gradient. By virtue of(2) and (6) the filament vectors in the deformed configurationare obviously given by
�Bi = F �Bi0, (7)
where �Bi0 denotes the filament vector i in the referenceconfiguration [i.e., they are given by (2), where �ψ is substitutedby �ψ0]. Consequently the length li of the filaments can beexpressed as
li =√
�Bi. �Bi =√
�BTi0C �Bi0, (8)
where C is usually called the right Cauchy-Green strain tensorand is given by
C = FTF = I + ∇0 �u + ∇0 �uT + ∇0 �uT∇0 �u. (9)
We recognize on the right-hand side the classical deformationtensor [13] that measures the change in distance between twomaterial points after the deformation. For small deformationsthe quadratic term in (9) is neglected and the infinitesimal straintensor is given by E = (∇0 �u + ∇0 �uT)/2. This is the classicallinear strain tensor. Using the definitions in this section it is asimple matter to see that the area of the actual elementary cell
g = ‖ �B1 ∧ �B3‖ can be expressed as
g = g0
√detC, (10)
with g0 = ‖ �B10 ∧ �B30‖ being the area of the cell in thereference configuration. Plugging (8) and (10) into the elasticenergy, (4), we straightforwardly obtain that the energydepends only on C for a given reference configuration�ψ0. Hence, we can introduce a local strain energy densityf (C) = f ( �Bi)/g0
f = k
2lg0
3∑i=1
(√ �BTi0C �Bi0 − l
)2 + βl2
2g20
√detC
. (11)
Once the energy is expressed in terms of the appropriatevariables, we can then obtain the constitutive law. For thatpurpose we use the Cauchy stress tensor σ . In linear elasticityit is well known that if f (E) is the strain energy density (withf being a quadratic potential in E), then the Cauchy stresstensor is given by σ = ∂f
∂E in the absence of prestresses. Innonlinear elasticity f depends on C (as shown above forour problem), and the relation between the Cauchy stresstensor and the energy density is not a priori obvious. Forany prestressed configuration the Cauchy stress σ tensor isgiven by the following general relation [12] (see Appendix Afor a short derivation):
σ = 2√detC
F∂f (C)
∂CFT. (12)
Applying (12) to (11) gives, after some simple algebraicmanipulations, the expression
σ = k
lg
3∑i=1
li − l
li�Bi ⊗ �Bi − βl2
2g4
[l23
�B1 ⊗ �B1 + l21
�B3 ⊗ �B3
− �B1. �B3( �B1 ⊗ �B3 + �B3 ⊗ �B1)], (13)
where the strain tensors C and F enter into the highlynonlinear expression, (13), via relations (7), (8), and (10).The derivation of the Cauchy stress tensor, (13), based on themicroscopic mechanical properties of the network constitutesthe first main result of this paper. We write below explicitexpressions in some regimes. Note that the second term in(13) is often negligible, e.g., for small or volume-conservingshear deformations.
For F = I Eq. (13) gives the Cauchy stress σ 0 of thereference configuration �ψ0. Linearizing (13) about σ 0 allowsus to extract the rheological properties of the network, e.g., theelastic moduli, depending on the applied prestresses σ0 andthe reference configuration ψ0.
III. RESULTS AND DISCUSSION
Here we exploit the general law, (13), in order to extractthe bulk rheological properties. Moreover, we see that in theleading nonlinear regime, the constitutive law takes on the formof a known phenomenological constitutive law. Figure 2 showsschematically the orientation of the network in a Cartesiancoordinate frame and the relevant deformations which arestudied.
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We analyze the response of the network under (i) smalldeformations and (ii) finite deformations from the stress-free reference state, where we extract the elastic responsecoefficients for a simple volume-conserving shear and a uniax-ial compression with lateral confinement. Polymer networksunder shear have been studied intensely both experimentallyand theoretically and can serve as a good test for our simplenetwork model. Networks under uniaxial compression havebeen less studied but are of great biological interest, e.g., thecortical actin cytoskeleton, which is crucial for motility, isunder compression. We show that the network behaves as anisotropic material in the linear regime but is anisotropic forfinite deformations. The anisotropy is most pronounced underuniaxial compression: Depending on the compression axis, thenetwork shows a strain localization.
A. Small deformations about the stress-free state
In this section we analyze the general constitutive law, (13),in the weakly nonlinear limit that will allow us to place ouranalysis in the context of known nonlinear phenomenologicalconstitutive laws. Then we show the response of the derivedfully nonlinear law, (13), to various deformations.
Let us choose as the reference configuration a network ofunilateral triangles of vertex length l. The node positions arethen
�ψ0 = L
[(1
2λ1 + λ2
)�ex +
√3
2λ1�ey
], (14)
where �ex and �ey form the basis of a Cartesian coordinate systemas shown in Fig. 2 (center). Consequently, the filament vectorsin this reference configuration �Bi0 [see Eq. (2)] are constantin space. Assuming only small enough deformations aboutthis reference state, the excluded volume contribution in theconstitutive law can be neglected, i.e., β = 0, and the referencestate becomes a stress-free reference state. Up to the leadingorder in G = (C − I)/2, the strain energy of the network, (11),reduces to the following expression:
f =√
3k
4l
[1
2(trG)2 + tr(G2)
]. (15)
Interestingly enough, this equation has exactly the form ofa phenomenological law known as the St. Venant–Kirchhoff
εreference
configuration
(1)
(2)
(3)
(4)
γ
y
xγ ε
shear compression
FIG. 2. (Color online) Network orientation in the Cartesiancoordinate system (center) and applied shear (1, 2) and compressiondeformations (3, 4). Left: Shear deformation in the x (1) and y (2)directions with the shear strain γ . Right: Uniaxial compression in they (3) and x (4) directions with the compressive strain ε < 0.
model and is widely used to describe isotropic compressiblehyperelastic materials [12]. Note that for higher order ex-pansions in G (not shown) the strain energy, (11), be-comes anisotropic and this will be reflected in the elasticresponse functions obtained by analyzing the fully nonlinearproblem, (13).
For readers who are more used to the constitutive laws interms of the stress tensor, we provide below the expression forthe Cauchy stress of the St. Venant–Kirchhoff model obtainedby applying the definition of the stress tensor, (12)–(15):
σ =√
3k
4l√
detB
[(1
2trB − 2
)B + B2
]. (16)
Here we have used the left Cauchy-Green strain tensor B =FFT for practical reasons. In the linear case B ≈ I + 2E, with Ebeing the infinitesimal strain, and the Cauchy stress reduces to
σ =√
3k
4l[(trE)I + 2E]. (17)
From (17) it becomes clear that for small deformationsthe network is described by only one independent variable,i.e., the ratio k/l, and one finds the linear Young’s modulusY = 2k/(
√3l), the linear shear modulus G = √
3k/(4l), andthe two-dimensional Poisson ratio ν = 1/3. These valuescan be compared to the values G = πk/(16l) and ν = 1/2obtained for a two-dimensional Mikado model of randomfilaments of fixed length cross-linked at their intersections inthe regime where filament stretching modes dominate [10].The differences between the homogenization model and theMikado model probably arise from the fact that the filamentorientations and cross-links are completely random in theMikado model, whereas they form a well-ordered structure inour model.
B. Finite simple shear: Stiffening and negative normal stress
Having treated the simple linear or weakly nonlinearisotropic limit, we now consider the fully nonlinear problem,(13). For that purpose we consider a simple volume-conservingshear (i.e., detB = detC = 1, and it follows that the excludedvolume contribution is constant), which has been studiedexperimentally for actin networks with a variety of cross-linkers [4–6]. In general, actin networks show linear behaviorfor small strains but stiffen considerably for larger strainsbefore the network breaks.
Using the network model we have calculated the shearstress σS and the stress normal to the shearing direction σN
for homogeneous shear deformations, as shown schematicallyin Fig. 2 (cases 1 and 2). The corresponding deformationgradients take on the form
∇0 �u =[
0 γ
0 0
]and ∇0 �u =
[0 0
γ 0
], (18)
where γ is constant. In addition, we have calculated the(differential) shear modulus G = dσS/dγ .
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0 0.2 0.4 0.6 0.8 1γ
0
0.2
0.4
0.6
0.8
1σ s
| σ N
[uni
ts o
f k/l]
σS(1)σS(2)σS(SVK)σN(1)σN(2)σN(SVK)
(a)
10-2 10-1 100
σS [units of k/l]
10-3
10-2
10-1
σ N [u
nits
of k
/l]
σN (1)σN (2)σN (SVK)
-1 -0.5 0 0.5 1σS
-0.8
-0.4
0
-σN
~σS2
~σS
(b)
FIG. 3. (Color online) Rheology of the network under a simpleshear for two shearing directions (cases 1 and 2 in Fig. 2) and the St.Venant–Kirchhoff model (SVK) as indicated in the legend. (a) Theshear and normal stress depending on the applied shear strain γ and(b) the normal stress σN depending on the applied shear stress σS .The inset in (b) shows −σN dependent on σS using a linear scale forbetter comparison with the experimental Fig. 4(b) in Ref. [5]. Thefully nonlinear analytical expressions for cases 1 and 2 are given inAppendix B 1.
Within the St. Venant–Kirchhoff model, (16), the shear andnormal stresses are
σS =√
3k
4l
(γ + 3
2γ 3
), (19)
σN = 3√
3k
8lγ 2. (20)
The relevant analytical expressions for the fully nonlinearproblem, (13), are given in Appendix B 1.
Figure 3(a) shows the dependence of the shear stressesand the normal stresses on the shear strain γ for the twoshearing directions and in the isotropic limit of the St. Venant–Kirchhoff model. The normal stresses are lower than the shearstresses, but for high strains the shear and normal stressesare of the same order of magnitude. The normal stressesare positive. Note that we have calculated the Cauchystress, which describes the external forces acting on thematerial, and that therefore a positive σN corresponds to thefact that the material tends to contract in the direction normal tothe shear direction as observed experimentally in [5] (see Fig. 2there). Consequently, a positive normal stress in our definitioncorresponds to a negative one in the definition in Ref. [5]. Moreprecisely, the authors of [5] measured the normal stress which
0.01 0.1 1σS [units of k/l]
1
2
4
G/G
0
G1G2GSVK
FIG. 4. (Color online) Rheology of the network under a simpleshear. Shown is the nonlinear shear modulus G normalized by thelinear shear modulus G0 dependent on the applied shear stress σS fortwo shearing directions, G1 [see Eq. (B6)] and G2 [see Eq. (B9)],for a shear corresponding to cases 1 and 2 in Fig. 2, respectively orobtained from the St. Venant–Kirchhoff model (GSVK) as indicatedin the legend.
the material was exerting on the rheometer; i.e., they measured−σN < 0. Therefore the expression “negative normal stress” isequivalent to a positive normal Cauchy stress and describes thefact that the material tries to contract in the normal direction.
For small strains γ the shear stress σS increases linearly withγ and the relation σs = √
3k/(4l)γ holds. For higher strainsthe shear stress becomes anisotropic and depends nonlinearlyon γ . The normal stresses σN scale with γ 2 up to a strain γ ≈0.5. However, even for small strains the behavior is anisotropicand we find σN = 3
√3k/(32l)γ 2 for a shear deformation
corresponding to Fig. 2 (case 1) and σN = 9√
3k/(32l)γ 2
for a shear corresponding to Fig. 2 (case 2). The quadraticdependence for small strains is to be expected from thesymmetry σN (−γ ) = σN (−γ ). The above-described behaviorcan also be seen in Fig. 3(b). It shows the dependence of thenormal stresses σN on the shear stress σS . For small stressesthe dependence is quadratic in σS as found experimentally(for comparison, see Fig. 4(b) in [5] for a fibrin network)and predicted from more involved models [5,7]. For extremestrains we find that σN → √
3k/(l) tends to a constant valueand σs → √
3kγ /l increases linearly with γ . Within theisotropic limit of the St. Venant–Kirchhoff model we findeasily from (19) and (20) that σN ∼ σ
2/3S for extreme strains.
For intermediate strains the scaling of σN is nontrivial and forstrains in the range of γ = 1 one finds an approximately linearscaling (see Fig. 3(b) and Fig. 4(b) in Ref. [5]).
Figure 4 shows the dependence of the nonlinear shearmodulus G = dσS/dγ on the shear stress σs for the two shear-ing directions and the St. Venant–Kirchhoff model. The term“nonlinear modulus” here refers to the fact that the materialproperties are determined at nonzero residual stress σ 0, asopposed to the linear modulus, which is calculated at σ 0 = 0.
As is already evident in Fig. 3(a) the network showslinear isotropic behavior for small stresses (strains) withthe linear modulus G0 = √
3k/(4l) but starts to deviate fromthe linear isotropic behavior for σS > 0.02k/l (correspondingto γ ≈ 0.05). Shearing in the x direction [Fig. 2 (case 1)]leads first to a small softening and then stiffening of the
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JOHN, CAILLERIE, PEYLA, RAOULT, AND MISBAH PHYSICAL REVIEW E 87, 042721 (2013)
network, whereas shearing in the y direction [Fig. 2 (case 2)]immediately leads to a stiffening behavior. From (19) it followsthat within the St. Venant–Kirchhoff model the shear modulusis GSV K =
√3k
4l(1 + 9
2γ 2). Within this limit one therefore findsa linear behavior for small strains (stresses) and a stiffeningbehavior for larger strains (stresses).
The occurrence of normal stresses, which tend to contractthe network, as well as the stiffening of the network is atypical response of semiflexible filament networks such asactin. Previous theoretical studies have reported that normalstresses arise either from a nonlinear force-extension relationof the filaments [5] or from the buckling of filaments undercompression [7]. The stiffening of the network is also attributedto a nonlinear force-extension relation of the filaments [4,6].While in real networks these two contributions undoubtedlyplay a role and the nonlinear force-extension relation leadsnotably to the scaling G ∼ σ
3/2S [6], we have shown here that
normal stresses and stiffening are already present in a simpletriangular network structure composed of linear springs, wherefilaments are straight. The fact that our simple picture capturesthe complex nonlinear behavior points to the robustness of thematerial behavior and does not depend on the various detailsof the models; our model simply uses in this regime the pureharmonic approximation between nodes [since under shear, asstated above, the β term in (13) is irrelevant].
Note, however, that it is straightforward to include anonlinear force-extension relation or filament bending onthe macroscale into the network model to adjust the modelbehavior to the rheological data of interest. In fact, assuming adiverging force-extension curve for the filaments of the form∼(li − lc)−2, where lc denotes the fully extended filament, it isstraightforward to show (see Appendix B 1 for details) that thenonlinear shear modulus scales in our model as G ∼ σ
3/2S as
previously shown in [6] and that σN ∼ σS as li → lc as in [5].
C. Uniaxial compression: Strain localization
In a biogical context the network response under compres-sion is of special interest. For example, polymerizing dendriticactin networks exert forces on the plasma membrane and aretherefore assumed to be under compression. We have usedEq. (13) to calculate the nonlinear response of the modelnetwork under uniaxial compression with lateral confinement[see Fig. 2 (cases 3 and 4) for the directions of compression].The corresponding deformation gradients are
∇0 �u =[
ε 0
0 0
]and ∇0 �u =
[0 0
0 ε
](21)
for a compression in the x and in the y direction, respectively.Thereby the strain ε is constant and negative. Figure 5shows the dependence of the compressive Cauchy stress σC
on the applied compressive strain ε for the two directionsof compression. All analytical expressions can be found inAppendix B 2. Here we show the expression of the compressivestress in the y directions (case 3 in Fig. 2) since it showsnontrivial behavior,
σc = σyy =√
3k
l
l1 − l
l1(1 + ε) − 2β
3l2(1 + ε)2, (22)
with l1 = l3 = l2
√1 + 3(1 + ε)2.
0 0.2 0.4 0.6 0.8 1-εA -εB-ε0
0.2
0.4
0.6
0.8
1
-σC [u
nits
of k
/l]
σC (3)σC (4)
BA-σCP
FIG. 5. (Color online) Uniaxial compression of the network inthe x and y directions with a confinement in the direction normalto the compression axis (see Fig. 2, cases 3 and 4). Shown is thecompressive stress σC dependent on the macroscopic strain ε for thetwo directions of compression as indicated in the legend [see alsoEqs. (22) and (B16)]. The horizontal dotted line connecting pointsA and B for a compression in the y direction (case 3) indicates thephase transition, i.e., the compressive stress σCP, where the strainlocalization occurs as described in the text. The remaining parameteris β/(kl) = 10−3.
The response of the network is highly anisotropic. For acompression in the x direction [Fig. 2 (case 4) and Eq. (B16)]the stress increases monotonously with the strain. The behavioris dramatically different for a compression in the y direction[Fig. 2 (case 3) and Eq. (22)]. For weak and strong compressionthe stress increases with the strain. However, for intermediatestrains the dependence is nonmonotonic. The appearance of aregion with a negative slope [see solid (black) line in Fig. 5],i.e., a “negative” elastic modulus, indicates a spontaneousinstability. This leads to the appearance of bands of localizedstrain, transverse to the direction of compression. In theregion of strain localization a highly compressed network is inequilibrium with a less compressed network. The stress-straincurve of the network for low and intermediate strains up to−ε = 0.9 is determined only by the first term in Eq. (22),i.e., the shortening and reorientation of filaments 1 and 3. Thestiffening response for very high strains −ε > 0.9 is governedby the second term in Eq. (22), i.e., the steric constraint, whichprevents the complete collapse of the network.
The coexistence of the two phases can be calculated byminimizing the integral, (3), with f given in (4) for a fixedmacroscopic compressive strain. Thereby we assume thatwithin the two phases the deformation gradient is constantand of the same form as in (21) on the right-hand side. Thenecessary conditions which hold at the phase boundary canbe found in [14]. The above-outlined procedure is identicalto the lever rule or Maxwell construction typically used forcalculating the liquid-gas phase transition. Figure 6 shows asan illustration the local strain energy f [Eq. (4)] dependenton the area g of the elementary cell. Since in our problem thelateral length of the elementary cell is fixed, the compressiveCauchy stress is simply given by σc = ∂f
∂g. This is the two-
dimensional analog of the relation p = − ∂f
∂Vof the liquid-gas
phase transition, where p is the pressure and V is thevolume. At coexistence σc = σCP is the same for the two
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NONLINEAR ELASTICITY OF CROSS-LINKED NETWORKS PHYSICAL REVIEW E 87, 042721 (2013)
0 0.4 0.8g [units of l2]
0
0.1
0.2
0.3
0.4f [
units
of k
l]
0 0.2 0.4 0.6 0.80
0.2
0.4f(g)Maxwell construction
gA gB
FIG. 6. (Color online) Energy f [Eq. (4)] of the network depen-dent on the area of the elemenentary cell g for a compression in the y
direction (case 3 in Fig. 2) and the Maxwell construction as indicatedin the legend. The areas gA =
√3
2 (1 + εA)l2 and gB =√
32 (1 + εB )l2
correspond to the areas of the elementary cells in the two-phaseregion. The remaining parameter is β/(kl) = 10−3.
phases as represented by the common tangent constructionin Fig. 6.
In the two-phase coexistance region the stress is inde-pendent of the macroscopic strain as shown in Fig. 5. Formacroscopic strains −εA < −ε < −εB the stress response ofthe network is given by the plateau stress σCP connectingpoints A and B (see dotted line in Fig. 5). Increasingthe macroscopic strain in this region results in an increasein the size of the collapsed network phase at a constantcompressive stress σCP until the whole network is in thecollapsed state. Further compression leads to a steep increasein the stress. As an illustration, Fig. 7 shows schematicallythe appearance of a strain localization band transverse to thedirection of compression. A similar behavior of deformationlocalization has been described before for triangular networksunder isotropic compression [9] and has been observedexperimentally for elastomeric cellular solids [15]. However,we are not aware of any macroscopic constitutive law derivedfrom microscopic considerations [i.e., Eq. (22) following fromthe general law, (13)] that produces the localization. It wouldbe an interesting task for future experimental investigations to
y
x
σCP
FIG. 7. (Color online) Schematic of the bands of localized straintransverse to the direction of compression with the compressive stressσCP.
0.1 1-σC [units of k/l]
0.1
1
10
K/K
0
K3K4
FIG. 8. (Color online) Uniaxial compression of the networkwith lateral confinement. Shown is the compression modulus K
(normalized by the linear modulus K0) dependent on the appliedcompressive stress σC for two directions of compression, K3 [seeEq. (B15)] and K4 [see Eq. (B17)], corresponding to cases 3and 4 in Fig. 2, respectively. The discontinuity of K3 at thephase transition is marked by circles. The remaining parameter isβ/(kl) = 10−3.
see whether polymer networks under a load exhibit this type oflocalization.
Figure 8 shows the dependence of the differential compres-sion modulus on the compressive stress for the two directionsof compression [see Eqs. (B15) and (B17)]. Here we have usedthe definition for K as K = dσc
dε= dσc
dε(1 + ε), where ε denotes
a small strain of the precompressed network with prestrain ε
(see Appendix B 2).For both directions of compression the network exhibits
a linear regime (i.e., the compression modulus is constant)with the modulus K0 = 3
√3k/(4l), followed by a softening
behavior for a weak compression. The softening of themodel network is due to the fact that the compression leadsto a reorientation of the filaments away from the loadingdirection and, therefore, a decreased capacity to withstandthe compression. For a compression in the y direction theelastic modulus (K3 in Fig. 8) shows as discontinuity atthe stress σCP where strain localization occurs. The highlycompressed network stiffens due to the excluded volumecontribution.
An experiment on the uniaxial compression of an actinnetwork nucleated from a functionalized cantilever of anatomic force microscope [16] shows a linear behavior forsmall compressive stresses, followed by a stiffening and a(reversible) softening behavior. However, recent experimentshave shown that actin networks nucleated from functionalizedsurfaces are typically surrounded by a brush of unbranchedfilaments, which are visible with total internal reflectionfluorescence microscopy [17,18] and as “fishbone tails” withconventional microscopy [19]. These long and unbranchedfilaments of variable length could account for the stiffening be-havior observed in [16]. Therefore, compression experimentsunder conditions more controlled than those in Ref. [16] areneeded to determine the mechanical response of semiflexiblefilament networks.
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JOHN, CAILLERIE, PEYLA, RAOULT, AND MISBAH PHYSICAL REVIEW E 87, 042721 (2013)
B1 B3
B2
θ1 θ2
FIG. 9. (Color online) Elementary cell of a network with filamentvectors �Bi and angles θ1 and θ2 associated with the rotational rigidityof the cross-link.
D. The effect of cross-link rigidity
In a cortical actin network cross-linked by an Arp2/3 proteincomplex, most filaments are oriented away from the directionof compression. Therefore one would expect a response similarto the compression of the model network in the y direction.However, the Arp2/3 cross-link is not completely flexible butexhibits a rotational rigidity [20]. Here we briefly demonstratehow to include a rotational cross-link rigidity in the model andshow its effect on a network under uniaxial compression withlateral confinement.
The rotational cross-link rigidity can by included in thelocal strain energy, (4), in the following manner:
f = k
2l
3∑i=1
(li − l)2 + βl2
2g+ δ
2
[(θ1 − π
3
)2
+(
θ2 − π
3
)2
+(
θ1 + θ2 − 2
3π
)2], (23)
where δ denotes the rotational rigidity related to the angles θ1,θ2, and θ1 + θ2 as shown in Fig. 9. The angles θ1 and θ2 arerelated to the filament vectors �Bi by
θ1 = arccos
( �B1. �B2
l1l2
)and θ2 = arccos
( �B2. �B3
l2l3
).
(24)
The linear constitutive law in the vicinity of the referencestate, (14), is isotropic and is given by
σ =√
3
4
[(k
l− 3δ
l2
)(trE)I + 2
(k
l+ 3δ
l2
)E
]. (25)
Recall, that for the linear law we neglect the second term in (23)related to the kinematic constraint, which is only relevant forhighly compressed networks. Introducing the dimensionlessparameter δ = δ/(kl) one obtains the Young’s modulus,
Y = 2k(1 + 3δ)√3l(1 + δ)
, (26)
and the Poisson ratio,
ν = 1 − 3δ
3(1 + δ). (27)
Note that for δ > 1/3 the Poisson ratio is negative. For auniaxial compression of the network with a lateral confinementin the direction normal to the compression axis, the linearcompression modulus is given by K0 = 3
√3k(1 + δ)/(4l).
0 0.2 0.4 0.6 0.8 1-ε
0
0.4
0.8
1.2
-σC [u
nits
of k
/l]
0.00.010.1
(a)
δ/(kl)
0.1 1-σC [units of k/l]
0.1
1
10
K/K
0
(b)
FIG. 10. (Color online) Uniaxial compression of the network inthe y direction with a lateral confinement in the x direction (see Fig. 2,case 3) for various rotational rigidities as indicated in the legend. (a)Compressive stress σC dependent on the strain ε. The thin dottedhorizontal line shows the dependence of the stress on the strain at thephase transition. (b) Nonlinear compression modulus K (normalizedby the linear modulus K0) dependent on the compressive stress σC .The discontinuity of the modulus at the phase transition is marked bycircles. The remaining parameter is β/(kl) = 10−3.
Figure 10(a) shows the dependence of the compressiveCauchy stress σC on the applied strain ε for a compressionin the y direction (case 3 in Fig. 2) for various values ofδ as indicated in the legend. For small δ the dependenceis nonmonotonic, suggesting the appearance of bands oflocalized strain, transverse to the direction of compression.
The elastic modulus, shown in Fig. 10(b), has a disconti-nuity at the compressive stress, where the strain localizationoccurs. For smaller stresses the network shows a softeningbehavior, and for larger stresses a stiffening behavior. Thesoftening is due to the reorientation of the load-bearingfilaments of types 1 and 3 away from the compressionaxis and thus their load-bearing capacity is decreased. Thestiffening is due to the strong volume-excluding forces for ahighly compressed network. Increasing the cross-link rigiditybeyond a critical value δc/(kl) ≈ 0.042, the strain localizationbehavior vanishes and the dependence of the stress on thestrain becomes monotonic [see Fig. 10(a)].
Dendritic actin networks, which provide the protrusiveforces during cell motility, are cross-linked by the so-called Arp2/3 complex. Its rotational cross-link rididity is≈10−19 J rad−2 [20]. Assuming that kl ∼ L2
pkBT /l2 [6], withthe persistence length Lp = 10 μm and the mesh size l =200 nm, one finds for the dimensionless parameter δ/(kl) =δl2/(kBT L2
p) ≈ 0.01, which is still smaller than the critical
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NONLINEAR ELASTICITY OF CROSS-LINKED NETWORKS PHYSICAL REVIEW E 87, 042721 (2013)
rigidity δc/(kl) ≈ 0.042 to suppress the strain localization butwhich has about the right order of magnitude to play a role inthe mechanical behavior of the network.
IV. CONCLUSIONS
We have presented a simple two-dimensional filamentnetwork model using a homogenization approach which allowsus to derive a nonlinear constitutive law depending on themechanical network properties on the scale of the typicalmesh size. Despite the simplicity of the model, the resultingmacroscopic bulk rheological properties are nonlinear andare in qualitative agreement with the experimental data onpolymer networks, such as actin. The description of three-dimensional networks or two-dimensional surfaces (e.g., redcell membranes with an underlying spectrin network) usingthe homogenized network model can be performed in the samespirit as in the two-dimensional case and does not present anynew challenge. However, it will be interesting to see the effectof the third spatial dimension on the scaling of the macroscopicelastic constants. Likewise, the addition of more complexmechanical properties on the micro scale is straightforward.Here in this paper we have used the homogenization approachto describe a highly ordered microscopic structure, which isisotropic for small deformations but anisotropic for largerdeformations. Other types of networks, e.g., rectangularnetworks or networks where the properties of the elementaryfilaments vary, are expected to be anisotropic, even in thelinear regime. An interesting future task is to investigate howthe homogenization approach can be employed for the coarse-grained description of disordered structures. This seemsespecially relevant for biological applications, e.g., the growthof dendritic actin networks under stress in complex geometries.
The homogenization model is not restricted to the descrip-tion of passive networks, but it can also be applied to growingor so-called active networks. The problem formulation in termsof a strain energy in a Lagrangian frame permits us to derivethe elastic part of the chemical potential for the networkgrowth at the interfaces. This contribution is, for example,essential to study the effect of mechanical stresses on thepolymerization/depolymerization dynamics of dentritic actinnetworks, as it occurs during cell motility. The addition ofstresses produced in the bulk by molecular motors will allowthe description of the complex morphodynamics encounteredin living cells [21,22] and active gels [23] beyond the linearelastic regime.
ACKNOWLEDGMENTS
C.M. acknowledges financial support from CNES, ESA,and ANR (project MOSICOB).
APPENDIX A: EXPRESSION OF THE CAUCHY STRESSTENSOR AS A FUNCTION OF THE ELASTIC ENERGY
In this Appendix we provide a derivation of relation (12).Let us consider an element of an elastic solid with volume dV
(actual volume). The elementary energy δfV is nothing but thenegative work performed by the elastic force per unit volume,denoted hereafter F . Let the displacement of the material
element be denoted δu; we have δfV = −F . δu. Equally, thenegative work performed by the traction forces T acting on thesurface of the solid is δfA = −T . δu. The total elastic energyis given by the elastic energy stored in the volume V less thework performed by the external forces on the surface A:
δF =∫
V
δfV dV −∫
A
δfAdA
= −∫
V
F . δu dV +∫
A
T . δu dA
= −∫
V
Fiδui dV +∫
A
Tiδui dA. (A1)
According to Cauchy the elastic force density is given byF = div(σ ) (or Fi = ∂xj
σij ), where σ is the Cauchy stresstensor. Upon integration by parts, one obtains
δF =∫
V
σij ∂xjδui dV +
∫A
(Ti − σijnj )δui dA, (A2)
where the nj denote the components of the outside normalvector �n on the surface element dA. The first term representsthe elastic energy stored in the bulk. The second term cancelsdue to the identity T = σ �n, which is nothing but the boundarycondition at the surface. The above considerations are classicaland can be found in textbooks [13].
The next step is to transform the system of coordinates fromthe actual ones �x into the coordinates of the reference state �X.Note that here we have used the notation �x and �X for the actual(deformed) state and the reference state, respectively, whichcorrespond to the coordinates �ψ and �ψ0 used in the networkmodel. Let us first introduce some identities:
d �x = ∂ �x∂Xi
dXi = Fd �X and d �X = ∂ �X∂xi
dxi = F−1d �x.
(A3)
Consequently,
∂δui
∂xj
= ∂δui
∂Xl
∂Xl
∂xj
= δFilF−1lj . (A4)
Let dV0 be the volume in the reference configuration; it isrelated to dV by dV = det(F)dV0. We thus have, from (A2),
δF =∫
V0
δf dV0 =∫
V0
σij F−1lj det(F)δFildV0
≡∫
V0
[det(F)σF−T] : δFdV0. (A5)
It follows that
σ = 1
det(F)
∂f
∂FFT. (A6)
From the definition C = FTF we deduce det(F) = √det(C).
Using f (F) = f (FTF) = f (C) and the fact that C is symmet-ric we find
∂f
∂F= 2F
∂f
∂C. (A7)
It follows the desired relation, (12):
σ = 2√det(C)
F∂f
∂CFT. (A8)
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JOHN, CAILLERIE, PEYLA, RAOULT, AND MISBAH PHYSICAL REVIEW E 87, 042721 (2013)
APPENDIX B: NONLINEAR BULK RHEOLOGYOF THE NETWORK
For all following calculations we assume the reference state
�ψ0 = L
[(1
2λ1 + λ2
)�ex +
√3
2λ1�ey
], (B1)
where �ex and �ey form the basis of a Cartesian coordinate system(see Fig. 2). The reference state corresponds to a network ofequilateral triangles of vertex length l. For β/(kl) � 1 thisstate is stress-free. The filament vectors are constant and aregiven by Eq. (2) by substituting �ψ0 for �ψ :
�B10 = l2 [(�ex + √
3�ey], �B20 = l�ex, �B30 = l2 [(�ex − √
3�ey].
(B2)
For any given deformation the filament vectors can beexpressed as �Bi = F �Bi0 = (I + ∇0 �u) �Bi0 and the lengths canbe expressed as li =
√ �BTi0C �Bi0. In the following we calculate
the Cauchy stress σ using Eq. (13) for a variety of deformationgradients ∇0 �u. Most of the time we are only interested in oneor two components of the stress tensor. To avoid any ambiguitywe show here σ in component notation,
σ =[
σxx σxy
σyx σyy
], (B3)
where σxy = σyx .
1. Finite simple shear
For a pure shear corresponding to case 1 in Fig. 2 thedeformation gradient is ∇0 �u = γ �ex ⊗ �ey , where γ denotes theapplied shear strain. Since the shear is volume conserving,the area of the unit cell g = √
3l2/2 does not depend on γ ,and we need only to consider the first sum in Eq. (13). Theshear stress is given by
σS = σxy = k
2l
[(1 +
√3γ )
l1 − l
l1− (1 −
√3γ )
l3 − l
l3
]
(B4)
and the normal stress is
σN = σyy =√
3k
2l
[l1 − l
l1+ l3 − l
l3
], (B5)
with l1 = l2
√(1 + √
3γ )2 + 3 and l3 = l2
√(1 − √
3γ )2 + 3.The nonlinear shear modulus G1 = dσS/dγ is
G1 =√
3k
2l
[2 − 3l3
4l31
− 3l3
l33
]. (B6)
The calculation for a pure shear corresponding to case2 in Fig. 2 can be performed in an analogous manner. Thedeformation gradient is ∇0 �u = γ �ey ⊗ �ex , which results in theshear stress
σS = σyx = 2k√3l
[(γ +
√3)
l1 − l
4l1+ (γ −
√3)
l3 − l
4l3
+ γl2 − l
l2
](B7)
and the normal stress
σN = σxx = 2k√3l
[l1 − l
4l1+ l3 − l
4l3+ l2 − l
l2
], (B8)
with l1 = l2
√1 + (γ + √
3)2, l2 = l√
1 + γ 2, and l3 =l2
√1 + (γ − √
3)2. The nonlinear shear modulus G2 =dσS/dγ is given by
G2 = 2k√3l
[3
2− l3
16l31
− l3
16l33
− l3
l32
]. (B9)
In the following we briefly demonstrate the effect of thenonlinear force-extension curve due to the transition fromentropic to enthalpic elasticity on the scaling of the shearmodulus. It has been proposed that for full filament extensionthe force fi scales as fi ∼ (lc − li)−2 [24], where lc denotesthe length of the fully extended filament and fluctuations havebeen stretched out completely.
For the shearing direction shown in Fig. 2 (case 1) thefilaments of type 1 are the most extended filaments. In thevicinity of the critical shear strain where filament 1 reaches itsfull extension the response of the network is therefore governedonly by the mechanical properties of this filament. For a sheardeformation gradient of the form ∇0 �u = γ �ex ⊗ �ey one easilyfinds that in the case l1 → lc,
σS ∼ 1 + √3γ
l1(lc − l1)2∼ 1
(lc − l1)2. (B10)
Expression (B10) can be obtained by substituting the constitu-tive law of filament 1 (l1 − l)
�B1l1
by (lc − l1)−2 �B1l1
in Eq. (13) andretaining only the term related to the extension of filament 1.Consequently, the nonlinear shear modulus dσS/dγ scales as
G1 ∼ 1
l1(lc − l1)2+ (3l1 − lc)l2
4l31(lc − l1)3
(1 +√
3γ ) ∼ 1
(lc − l1)3,
(B11)
which gives the scaling G1 ∼ σ3/2S as found in [6]. For the
normal stress one finds from Eq. (13) the scaling
σN ∼ 1
(l1 − lc)2l1∼ 1
(l1 − lc)2. (B12)
Comparing (B10) and (B12) in the case l1 → lc one findsthat the linear dependence σN ∼ σS holds as found in [5].The same type of argument is valid for a shear of the formγ �ey ⊗ �ex (case 2 in Fig. 2).
2. Uniaxial compression under lateral confinement
Here we calculate the network response to a uniaxialcompression with a confinement in the direction perpendicularto the compression axis. For a compression in the y directioncorresponding to case 3 in Fig. 2 the deformation gradientis ∇0 �u = ε�ey ⊗ �ey , where ε < 0 is constant and denotes theuniaxial strain. The compressive Cauchy stress σC can becalculated from (13),
σC = σyy =√
3k
l
l1 − l
l1(1 + ε) − 2β
3l2(1 + ε)2, (B13)
with l1 = l3 = l2
√1 + 3(1 + ε)2.
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NONLINEAR ELASTICITY OF CROSS-LINKED NETWORKS PHYSICAL REVIEW E 87, 042721 (2013)
The nonlinear compression modulus is obtained by ap-plying infinitely small perturbations about the precompressedstate. To that end we decompose ε = ε + δε, where ε denotesthe strain of the precompressed state (with respect to the stressfree state) and δε denotes a small perturbation of the strain. Toobtain the compression modulus of the precompressed systemwe then apply the differential strain ε = δε/(1 + ε) to theprecompressed network. Since
σC = σC(ε) = σC(ε + δε) = σC[ε + ε(1 + ε)], (B14)
the compression modulus of the precompressed state in thelimit of ε → 0 is given by
K3 = dσC
dε= dσC
dε(1 + ε)
=√
3k
l(1 + ε)
[l1 − l
l1+ 3l3(1 + ε)2
4l31
]
+ 4β
3l2(1 + ε)2. (B15)
The compressive stress and the modulus for a compressionin the x direction corresponding to case 4 in Fig. 2 canbe calculated in the same way by applying the deformationgradiend ∇0 �u = ε�ex ⊗ �ex . The compressive stress is
σC = σxx = 2k√3l
(1 + ε)
(l1 − l
2l1+ l2 − l
l2
)− 2β
3l2(1 + ε)2,
(B16)
with l1 = l3 = l2
√(1 + ε)2 + 3 and l2 = l|1 + ε|. The com-
pression modulus is
K4 = dσc
ε(1 + ε)
= 2k√3l
(1 + ε)
[l1 − l
2l1+ l2 − l
l2+ (1 + ε)2
(l3
8l31
+ l3
l32
)]
+ 4β
3l2(1 + ε)2. (B17)
The compressive stresses and compression moduli are shownin Fig. 5.
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