## Abstract

We generalize a theory for modelling the scission and reforming of cross links in isotropic polymeric materials in order to treat anisotropic mechanical behaviour. Our focus is on materials in which elastic fibres are embedded in an elastic matrix. The fibres may have a different natural stress-free configuration than that of the matrix, e.g. the fibres may be initially crimped in the absence of load. The modelling process allows the fibres to dissolve as deformation proceeds and then to immediately reassemble in the current direction of maximum principal stretch. This results in softening, altered mechanical properties and the possibility of permanent set. We illustrate a rich variety of such mechanical behaviours in the context of uniaxial stretch. The phenomena illustrated have important implications for the influence of mechanical factors in the remodelling of fibrous soft matter including biological tissue.

## 1. Introduction

An important class of materials in engineering and medicine consists of polymeric filaments in a soft matrix. Engineering applications include automotive body panels composed of textile composites, i.e. woven fibre laminae in a polymer matrix. Filamentary networks are ubiquitous as reinforcements in biological tissue such as muscle, skin, ligaments and blood vessels. Regarding these as fibre-reinforced elastic materials, various constitutive models in the context of anisotropic hyperelasticity have been explored to describe the mechanical loading response by, for example, Qiu & Pence (1997), Merodio & Pence (2001), Horgan & Saccomandi (2005), Merodio & Ogden (2005) and Guo *et al*. (2007).

These materials, having a macromolecular microstructure, can undergo microstructural changes, such as destruction and reformation of cross links as a result of large deformation, or other causes, such as oxidative scission at high temperatures. Recently, Wineman and collaborators have developed a theoretical framework for modelling the destruction and reformation of cross links in isotropic nonlinearly elastic materials (Wineman & Rajagopal 1990; Shaw *et al*. 2005) and have explored its consequences for mechanical response (Huntley *et al*. 2000; Wineman & Shaw 2007). The destruction of cross links is referred to as scission, which has the effect of degrading the original microstructure network in the material. The reformation of cross links may take place while the material is deformed, in which case the effective new material associated with the new cross links has a different stress-free reference state to that associated with the original network. In this sense, the scission and reformation processes give rise to a multi-network material, and the work referenced above provides a framework for generating constitutive models. Collectively, we shall refer to this work as the *isotropic scission and reformation* theory, or the ISR theory for short.

Fibres in biological tissue have a hierarchical structure. For example, Puxhandl *et al*. (2002) described tendons as an organized array of collagen fibrils connected by macromolecules to an extracellular matrix. This suggests that similar issues should be considered with respect to the destruction and reconstitution of fibrous networks in nonlinearly elastic materials. Indeed, constitutive mathematical models for such processes have been the subject of recent interest (Humphrey 1999; Driessen *et al*. 2008). With this aim in mind, it is our purpose here to generalize the ISR theory to the modelling of fibrous materials. Specifically, we have in mind a material that, at the microstructural level, contains filaments, hence referred to as fibres, embedded in an otherwise isotropic matrix. Although the term *scission* could possibly be used to describe the degradation process in the fibre networks that we consider, we shall reserve its usage so as to indicate a degradation process in the matrix component. The alternative term *dissolution* will be used to refer to the degradation process in the fibrous component. In a similar fashion, the term *reformation* will be taken to refer to the creation of new matrix networks, whereas the alternative term *reassembly* will be used to refer to the creation of replacement fibrous components.

In the treatment presented here, the matrix component will be regarded as microstructurally stable, and it is the fibrous network that undergoes microstructural change. Consequently, the present focus is on dissolution in the absence of scission. In the modelling presented here, we consider those processes for which the total amount of fibrous component is maintained, meaning that, while certain fibrous networks are dissolving, others are reassembling. In general, the direction in which fibre reassembles will be different from the fibre directions that have just undergone dissolution. In particular, we consider reassembly processes that create fibre in the direction in which the underlying material matrix is maximally stretched. Such treatments could describe self-assembling networks in adaptive materials, including biological materials at both the tissue and cell level. The resulting fibrous microstructure that emerges is then dependent on the mechanical deformation history, and in turn contributes to the subsequent mechanical response (*viz*. Driessen *et al*. 2008). Such emergent structural complexity is a characteristic feature of developmental processes in biological systems (e.g. Raeber *et al*. 2008).

Although our purpose is to develop and explore a continuum mechanical model that generalizes the ISR treatment to anisotropic materials, we open, in §2, with a discussion in terms of spring networks that dissolve into component discrete springs and reassemble into new filaments. This allows us to introduce certain basic modelling features in a more concrete setting, and permits later comparison with the continuum framework that is our main focus of attention. We turn to the continuum framework in §3, where we review the original ISR model and then provide certain generalizations so as to treat multiple degradation and reconstitution processes in the context of anisotropic material behaviour. This general framework is then specialized in §4 to the case of fibre networks that dissolve and reassemble within an otherwise inert matrix. The continuum modelling assumptions introduced in §4 mirror those introduced in §2 for the spring network system; however, the mathematical development of §§3 and 4 stands alone in the sense that it requires no reference to §2 development.

The equations presented at the end of §4 provide the basis for a general continuum level treatment of fibre-reinforced materials with dissolving and reassembling filament networks. In §5, we consider in detail the example of uniaxial load, where, among other things, we show how the combined dissolution and reassembly process gives rise to stress-softening, altered mechanical properties and permanent set. Finally, in §6, we relate the continuum treatment of §§3–5 to the spring network treatment of §2. In particular, we show how the continuum treatment captures the same type of dissolution and reassembly phenomena as that put forth in the spring treatment.

## 2. Gedanken experiment: networks assembled from a fixed number of microscale component springs

Certain issues associated with our modelling can be illustrated by considering planar networks of linear springs in an (*X*_{1}, *X*_{2})-plane, especially as regards an eventual continuum interpretation of fibre density and reinforcing moduli. To this end, consider the initially square region 0≤*X*_{1}≤1, 0≤*X*_{2}≤1. The region will be allowed to undergo homogeneous deformations to rectangles such that *X*_{1}→*X*_{1}+*δ*_{1}*X*_{1} and *X*_{2}→*X*_{2}+*δ*_{2}*X*_{2}, where the scalars *δ*_{1} and *δ*_{2} are the functions of time *t*, i.e. *δ*_{1}=*δ*_{1}(*t*) and *δ*_{2}=*δ*_{2}(*t*). We assume that the resistance to deformation is supplied by a network of springs within the square. This network is built from *N* linear springs where *N*≫1. Each such component spring has spring constant *k* and unstretched length *l*_{0}≪1. These microscale component springs are grouped in series into straight multi-spring chains of various lengths and each such filament chain is assumed to be aligned with either the *X*_{1}-direction or the *X*_{2}-direction so as to connect opposite sides of the rectangle. At each instant of time, each of the *N* linear springs participates in exactly one such filament. For any such network, all of the *X*_{1}-aligned filaments taken together give rise to an effective spring in the *X*_{1}-direction. The spring force will be denoted by *F*_{1} and the overall force-free length will be denoted by *L*_{1}. Similarly, all of the *X*_{2}-aligned filaments give rise to an effective spring in the *X*_{2}-direction, with force *F*_{2} and force-free length *L*_{2}.

We now allow the network to modify itself as a function of the deformation. Specifically, we consider the dissolution and reassembly of filaments, such that:

Dissolution takes place over some interval of elongational deformation. This results in a progressive loss of filaments and hence a progressive loss of stiffness in the direction of original filament alignment. Stopping the elongational deformation halts the dissolution. Reversing the loading after such a halt does not reverse the dissolution, i.e. there is no healing associated with load reversal. There is, of course, the reassembly, but this reassembly makes no reference to loading direction and so is not viewed as a healing process due to the load reversal.

Reassembly takes place so as to immediately reincorporate all released springs into new network filaments. These new filaments are assumed to form in the direction that is maximally stretched. This direction of maximal stretch may, but need not, coincide with the original fibre direction. At the instant of reassembly, the new filaments are force free. However, additional stretching causes them to exert a restoring force so that these new filaments enhance the stiffness in their alignment direction.

For simplicity, assume that there is initially no deformation (i.e. *δ*_{1}(0)=*δ*_{2}(0)=0). Assume also that the original network involves all filaments aligned with the *X*_{1}-direction with all filaments consisting of unstretched springs at *t*=0, so that initially *L*_{1}=1. Thus, each filament in the initial network will consist of 1/*l*_{0} springs in series so that each such filament has spring constant *l*_{0}*k*. Accounting for all springs indicates that there are *Nl*_{0} parallel filaments in the network. Consider now a deformation such that *δ*_{1}=*c*_{1}*t* and *δ*_{2}=*c*_{2}*t*, where *c*_{1}>0 and *c*_{2}>0 are constants. We now turn to consider the dissolution and reassembly of the filament chains as the deformation proceeds.

Regarding dissolution, we assume that each filament dissolves into its component springs if these component springs are stretched too severely. Upon dissolution, each component spring that had been in the filament then returns to its unstretched length. Specifically, for each filament, we suppose that the dissolution takes place with some probability distribution over the interval of filament end displacements *u*_{s}<*δ*_{1}<*u*_{f}, where *u*_{f}>*u*_{s}>0 are material constants. We may therefore introduce an associated dissolution probability *ϕ*(*δ*_{1}). This dissolution probability is positive for *u*_{s}<*δ*_{1}<*u*_{f} and zero otherwise, such that(2.1)Since the filaments comprising the original network are taken to be identical to each other, the same dissolution probability *ϕ*(*δ*_{1}) and threshold values *u*_{s} and *u*_{f} are taken to govern each of the original filaments.

Consider this dissolution process over the time interval from *t* to *t*+d*t*, for some *t*∈(*t*_{s}, *t*_{f}). The number of filaments that dissolve over this time interval is *Nl*_{0}*ϕ*(*δ*_{1})d*δ*_{1}, with *δ*_{1}=*c*_{1}*t* and d*δ*_{1}=*c*_{1} d*t*. Each dissolved filament releases all 1/*l*_{0} of its component springs into solution, so that, in total, *Nϕ*(*δ*_{1})d*δ*_{1} component springs are released during the time interval d*t*.

Regarding reassembly, we assume that the component springs released into solution immediately reform into new filaments that are initially force free and aligned with the direction of maximum stretch. Thus, the newly assembled filaments will align with the *X*_{1}-direction if *c*_{1}>*c*_{2}, and will align with the *X*_{2}-direction if *c*_{2}>*c*_{1}. For the present discussion, we assume the latter to be the case for now so that the filaments reassemble in the *X*_{2}-direction. At time *t*∈(*t*_{s}, *t*_{f}), the length of such a filament is (1+*δ*_{2}) with *δ*_{2}=*c*_{2}*t*, and so is assembled from (1+*δ*_{2})/*l*_{0} springs. Thus, the *Nϕ*(*δ*_{1})d*δ*_{1} springs that are released during the time interval d*t* gives rise to *Nl*_{0}(1+*δ*_{2})^{−1}*ϕ*(*δ*_{1})d*δ*_{1} filaments in the *X*_{2}-direction, each with stiffness (1+*δ*_{2})^{−1}*l*_{0}*k*. The aggregate stiffness of all the filaments that are assembled over the time interval d*t* is therefore . Unlike the original filaments in the *X*_{1}-direction, which all had a common force-free length, the various filaments assembled in the *X*_{2}-direction will have a force-free length that is dependent on the time of their formation. Those filaments that formed when *δ*_{2} had the specific value will have force-free length . Thus, at a later time, when *δ*_{2} has a different value, these filaments will not be at their force-free length and so will develop a restoring force according to their overall endpoint displacement . The restoring force developed by this portion of the new filament network is therefore(2.2)where it is to be emphasized that and refer to the past deformation at the time that this portion of the network was created, whereas *δ*_{2} corresponds to the current deformation.

The above considerations place us in a position to summarize the force that is needed to support the deformation associated with displacements *X*_{1}→*X*_{1}+*δ*_{1}*X*_{1} and *X*_{2}→*X*_{2}+*δ*_{2}*X*_{2}, with *δ*_{1}=*c*_{1}*t*, *δ*_{2}=*c*_{2}*t* and *c*_{2}>*c*_{1}>0. There are three time intervals of interest: *t*<*t*_{s}, during which the original spring network is still completely intact; *t*_{s}≤*t*≤*t*_{f}, during which the original network dissolves and new network forms; and *t*>*t*_{f}, during which the original network is gone and the secondary spring network is complete. We continue to use and to indicate the deformations associated with a past time at which some portion of the dissolution and reassembly occurred. Unadorned *δ*_{1} and *δ*_{2} will refer to the current values of deformation.

During the first time interval *t*<*t*_{s}=*u*_{s}/*c*_{1} it is the case that(2.3)During the second time interval *u*_{s}/*c*_{1}=*t*_{s}≤*t*≤*t*_{f}=*u*_{f}/*c*_{1} the applied forces must be(2.4)Here, we note that in (2.4)_{2} is a function of , and, in this particular example, is given by for the purposes of evaluating the integral.

During the third time interval *t*>*t*_{f}=*u*_{f}/*c*_{1}, the applied forces must be(2.5)and, as in (2.4)_{2}, one must take to be a function of in (2.5)_{2}.

Alternatively, one may consider the case in which *c*_{1}>*c*_{2} so that the springs reassemble in the *X*_{1}-direction. In this case, the *X*_{2}-direction remains inert, giving *F*_{2}=0, while one finds that(2.6)Figure 1 shows the mechanical response given by (2.6) for the dissolution probability(2.7)The graphs in figure 1 involve *u*_{s}=0.3 and various values for *u*_{f}. The response is non-monotone during the dissolution and reconstitution of the fibres.

For either the case *c*_{2}>*c*_{1} with loading response (2.3)–(2.5) or the case *c*_{1}>*c*_{2} with loading response (2.6), one may now inquire into the associated unloading response. We here consider the case *c*_{1}>*c*_{2} for a loading that is interrupted when, say, and which is then followed by a decreasing *δ*_{1}. Such unloading does not induce microstructural change, so it follows that if , then unloading proceeds down the original loading curve. If, however, , then unloading is governed by the modified spring network. In particular, if , then the unloading response is given by with effective spring constant and new natural length *L*_{1} given by(2.8)and(2.9)These formulae also govern unloading for , provided that in the above expressions is replaced by *u*_{f}. Such unloading curves are shown as dashed curves in figure 1. Reloading after any such unloading retraces the unloading curves until these curves rejoin the loading curves, whereupon further loading response proceeds as if the unloading had never occurred.

The analysis leading to (2.3)–(2.6) is easily generalized for the case in which the *N* identical component microscale springs with force-free length *l*_{0} obey a nonlinear relationship between the force *f*_{spring} and the spring end displacement *ξ*, say *f*_{spring}=*f*(*ξ*). For example, (2.6) generalizes to(2.10)In particular, (2.10) retrieves (2.6) if *f*(*ξ*)=*kξ*.

A generalization that goes beyond that considered above is one in which the original square has some intrinsic ‘background’ elasticity independent of the actively changing spring network. This would then provide resistance to the deformation, even if there are no filaments currently aligned with certain directions of the deformation (*viz*. (2.3)_{2} and (2.5)_{1}). We do not, however, pursue such a modification in this section, although the analogue of these and other generalizations will arise in the continuum framework to which we now turn our attention.

## 3. Continuum framework for dissolution and reassembly

To model the type of processes described in §2 in the context of a finite deformation continuum framework, we begin by reviewing the ISR treatment of scission and re-cross-linking. Let ** X** be the position vector of a material particle in a reference configuration and let the deformation be described by a placement mapping

**=**

*x**Χ*(

**,**

*X**t*). It will be convenient to take this reference configuration to be the actual configuration at

*t*=0, whence

*Χ*(

**, 0)=**

*X***. The deformation gradient is**

*X***=∂**

*F***/∂**

*x***. We recall the right and left Cauchy–Green deformation tensors =**

*X*

*F*^{T}

**and =**

*F*

*FF*^{T}, with principal scalar invariants

*I*

_{1},

*I*

_{2},

*I*

_{3}and principal stretches

*λ*

_{3}≥

*λ*

_{2}≥

*λ*

_{1}. Thus, (

*i*=1, 2, 3) are the eigenvalues of and with corresponding unit eigenvectors and

*n*_{i}, respectively.

Prior to the microstructural change, the materials are considered to be incompressible, isotropic and hyperelastic, so that det ** F**=1, and there is a stored energy density function

*W*

^{(1)}(

*I*

_{1},

*I*

_{2}), such that the Cauchy stress tensor is given by(3.1)Microstructural change involves the degradation of the original material network, described by a function

*b*

^{(1)}(

*t*), and the creation of new material network, described by a function

*a*(

*t*). In particular,

*b*

^{(1)}(

*t*) gives the fraction of original network remaining at time

*t*, and

*a*(

*t*) gives the amount of new network that is created at time

*t*. This new network could also be degraded, giving rise to another degradation function that denotes the fraction of new network remaining at time

*t*, which was created at time . The ISR framework can treat a variety of physical situations, including various possibilities for the dependence of

*b*

^{(1)},

*a*and

*b*

^{(2)}upon temperature and deformation by virtue of different types of kinetic mechanisms. Any new network is assumed to be stress free at the time of formation, and is described in terms of its own reference configuration, which is taken to be the current configuration at the time of its creation. For new networks created at time , this reference configuration is described by position vectors . Let(3.2)The notation follows that of Wineman (2001), whereas the notation is introduced here for convenience. In view of the appearance of in the respective denominator and numerator of and , one might consider reversing this notation convention; however, then we are at odds with the use of in Wineman (2001) and subsequent papers, and so we take (3.2) in what follows. It is then also convenient to take , , , etc. In a similar fashion, we define , etc. Note that is the deformation gradient at time , and so quantities, such as and , etc., are the standard deformational quantities at the past time . Thus, we could dispense with the notation by carefully noting past times in the expressions. However, we find that the use of the notation is a good reminder of the ‘historical’ aspect of these deformational quantities. This is also the sense in which the notation was used in the discussion of spring networks in §2. By contrast, the notation denotes the deformational quantities referenced, not with respect to

**, but rather with respect to , the position vector at time . Thus, quantities depend upon both the current time**

*X**t*and the past time , in contrast to quantities that only depend on the past time .

The new network considered in the ISR theory is also regarded as isotropic at the time of its formation and is also taken to deform in an isochoric fashion. In view of all of these assumptions, it follows that and . Upon further deformation, the stored energy density of the new network formed at time is . Following the development based on superposition of the various network stresses, the ISR model leads to the following expression for the Cauchy stress tensor,(3.3)While the development presented above is motivated by the consideration of networks that form in an isotropic fashion, there is no essential difficulty in generalizing the framework to the consideration of anisotropic, incompressible materials. In this regard, objectivity then demands that the original network stored energy density *W*^{(1)} is a function of , and that the newly formed network stored energy density *W*^{(2)} is a function of . An immediate generalization of (3.3) is(3.4)with ** F**=

**(**

*F**t*), =(

*t*), and . The associated stored energy density is(3.5)Certain care must be exercised with respect to this last equation. This is because any further degradation of the primary network causes the energy that had been stored in the now degraded network to be lost. This is reflected in the observation that the ISR framework naturally gives rise to phenomena, such as permanent set, that are associated with energy dissipation. Hence, the

*W*given by (3.5) is the energy storage density that is retrievable only if no additional microstructural change takes place. Detailed discussion of the thermodynamic consequences of this observation is beyond the scope of the present work, as are also the thermodynamical issues associated with the creation of secondary networks. However, with respect to the purely mechanical consequences of the present development, we do wish to point out that, since , one may rewrite (3.5) as(3.6)By writing

*W*=

*W*(,

*t*) in (3.6), we express the dependence of

*W*upon the current deformation gradient

**since =**

*F*

*F*^{T}

**and upon time**

*F**t*by virtue of the changing microstructure. This changing microstructure is, in turn, dependent on the overall history of the deformation gradient, as given by for all . Hence, on writing

*W*=

*W*(,

*t*), the functional dependence of

*W*upon the history of the deformation gradient is subsumed into the dependence upon

*t*. For that matter, since

**is given by with , the question arises as to the purpose of even acknowledging in writing**

*F**W*=

*W*(,

*t*) in (3.6). Certainly, given a deformation gradient history with , the expression (3.6) is formally a function of and

*t*, and this expression gives the current stored energy density. A more important point, however, is based upon the observation that the relations(3.7)give(3.8)in (3.4). This, in turn, indicates that (3.4) follows from (3.6) by means of the standard connection(3.9)This familiar formula from hyperelasticity is applicable in the present context in the sense that the differentiation with respect to in (3.9) is at fixed

*t*, and so corresponds both to ‘fixing the history of the deformation gradient’ that resulted in the current microstructure and then to ‘freezing the microstructure’ in its current form. On the other hand, it is to be emphasized that such an energetic connection, while often useful, is not essential in the sense that the constitutive treatment could proceed on the basis of the stress relations alone (i.e. (3.4)). Indeed, the previously cited papers that develop the ISR constitutive theory for matrix scission and reformation do so solely on the basis of a stress superposition, such as (3.3), without any connection to energy storage, although such a connection to energy storage by means of (3.9) similarly follows for the original ISR framework as well.

Returning now to the issue of generalizing the ISR constitutive theory so as to treat anisotropic materials, we observe that the constitutive description represented by (3.4) posits that the same degradation and recreation functions *b*^{(1)}, *a* and *b*^{(2)} apply to all aspects of the mechanical response. Hence, for example, with regard to a fibre–matrix composite, the same degradation and recreation kinetics would govern both the fibre and matrix components for the model governed by (3.4). If, on the other hand, one wishes to model the fibre and matrix components which degrade and recreate in a manner that is different from each other, then further generalization is required. One such generalization involves a case in which the equivalent of *W*^{(1)} and *W*^{(2)} in (3.4) are each the sum of a collection of component stored energy densities denoted, say, through the use of the subscript *j* with *j*=1, …, *N*_{comp}. For each of the *N*_{comp} stored energy components, there would then be an associated set of degradation and recreation functions: ; *a*_{j}; and . In this way, (3.4) further generalizes to(3.10)The above expression also follows from (3.9) provided that *W* in (3.6) is similarly generalized to(3.11)This framework is well suited to the consideration of fibre–matrix materials that undergo microstructural change. For such a material, *N*_{comp}=2 in (3.10) and (3.11), whereupon it makes sense to simply take the subscript *j* as either f or m rather than either 1 or 2.

## 4. Fibre dissolution and reassembly

Further development of the model in this paper is based on a number of specific assumptions, many of which are analogous to those presented in the spring network discussion of §2. The present section, however, proceeds solely on the basis of the continuum treatment of §3 and so does not make use of the development in §2. Instead, the connection between the separately derived approaches will be explored further in §6.

The first specific assumption that we now place upon the general treatment of §3 is a hypothesis that microstructural evolution is restricted to the fibre phase. Hence, we formally take and *a*_{m}≡0, so that attention is restricted to , *a*_{f}≡*a* and . Under these specializations, (3.10) becomes(4.1)In addition, using , it is found that (3.11) becomes(4.2)Furthermore, the possibility that the secondary network could also experience dissolution is captured in (4.1) and (4.2) by virtue of the function *b*^{(2)} in the integral expressions for the secondary network contribution. To the extent that this could, in turn, lead to the formation of a tertiary network is not captured in (4.1) and (4.2); however, such tertiary network formation (and dissolution) could be accounted for by adding an additional integral term to both (4.1) and (4.2). Obviously, higher order networks beyond the tertiary could also be considered by evermore book-keeping integrals.

The material description prior to any microstructural change is governed by the energy densities and in (4.1) and (4.2). These can be taken to depend on the scalar invariants associated with the matrix and fibre deformation as (*viz*. Holzapfel 2000)(4.3)where *μ*, *γ* and *λ*_{nat} are positive material constants; ** M** is a unit vector giving the direction of reinforcing in the reference configuration; and

*I*

_{4}=

**.**

*M***=**

*CM***.**

*FM***and**

*FM**I*

_{5}=

**.**

*M*

*C*^{2}

**=**

*M***.**

*CM***. Note that is the fibre stretch as measured from the reference configuration. It shall, therefore, be convenient to use the additional notation . The constants**

*CM**μ*and

*γ*in (4.3) have units of stress and represent material moduli for the matrix and fibre constituents, respectively, and

*λ*

_{nat}is dimensionless and represents the stretch from the reference configuration that puts a fibre at its natural length (a length at which it exerts no force). With respect to the natural length, the reciprocal 1/

*λ*

_{nat}can be viewed as a fibre prestretch, with

*λ*

_{nat}<1 corresponding to a ‘prelengthening’ and

*λ*

_{nat}>1 corresponding to a ‘preshortening’. For example, if

*λ*

_{nat}<1, then the fibre must be contracted from its reference configuration length to attain its natural length. This means that the fibre's reference length is longer than its natural length. In particular, elongating a natural length fibre by the factor 1/

*λ*

_{nat}>1 then puts the fibre at its reference configuration length. Similar considerations apply to the preshortened case

*λ*

_{nat}>1. The simplest case to consider is of course

*λ*

_{nat}=1, whereupon the original fibres are at their natural length in the reference configuration.

For no other reason other than to simplify the presentation, it is convenient, in this paper, to consider the materials for which *Φ*_{m} is also independent of *I*_{2} and for which *Φ*_{f} is also independent of *I*_{5}. Thus, we henceforth replace (4.3) with(4.4)This allows us to present simpler formulae in this paper, and also motivates the shorthand notation(4.5)for differentiation with respect to the first slot. The more general case associated with (4.3) involves formulations that are more complicated algebraically, but are otherwise not difficult to obtain. A standard expression for a *Φ*_{m} consistent with (4.4)_{1} is the well-known neo-Hookean form(4.6)Useful models for *Φ*_{f} of the form (4.4)_{2} include(4.7)and(4.8)as considered in Demirkoparan & Pence (2007). In-depth discussion of these models for the case *λ*_{nat}=1 can be found in Qiu & Pence (1997) and Guo *et al*. (2007), respectively. Both of the models (4.7) and (4.8) allow the fibres to support compressive stress. A modification of, for example, (4.8) that only permits the fibres to support tensile stress is simply(4.9)where *H*(.) is the unit step function. We shall use the matrix energy form (4.6) in conjunction with both of the fibre energy forms (4.8) and (4.9) in the examples presented later in the paper.

It shall be assumed that a fibre contraction does not lead to microstructural change, but that sufficiently large fibre elongation generates fibre dissolution. Thus, we assume that the dissolution of fibres in the original network takes place over the fibre stretch interval *λ*_{s}≤*i*_{4}≤*λ*_{f}, where *λ*_{f}>*λ*_{s}>1 are material constants. This will be described by the function *β*^{(1)}(*i*_{4}), which is monotonically decreasing from one to zero on *λ*_{s}≤*i*_{4}≤*λ*_{f}, in accordance with the fraction of original fibres that remain in the material.

With respect to the time parametrization in (4.1) and (4.2), consider the fibre dissolution and reassembly process at some past time . The fibre stretch at is . The description, therefore, involves the connection(4.10)Upon dissolution, we consider a model for which all degraded fibre immediately reassembles into new fibre. This new fibre will have a new natural length, and will probably also lie in a different direction. To account for the immediate reassembly, we restrict attention to dissolution that is confined to the original fibre network. In other words, we suppose that the fibres in the secondary network do not dissolve, which in turn gives that .

In a similar manner to the dissolution function in (4.10), we also introduce *α*(*i*_{4}), such that(4.11)in (4.1) and (4.2). It is important to remark that the use of (4.10) and (4.11) in (4.1) and (4.2) complicates the differentiation process with respect to the connection (3.9). Namely, for the purposes of this differentiation, the energy density *W* is regarded as a function of and *t*, and so the differentiation with respect to in (3.9) is at fixed *t*. Here, *t* parametrizes the changing microstructure, so, as discussed in §3, the differentiation with respect to in (3.9) is viewed as a differentiation at fixed microstructure. Operationally, the use of (4.10) and (4.11) in (4.2) may lead to a final expression for *W* that masks the microstructural evolution contribution to the analytical form for *W*. An operationally correct implementation of (3.9) in this case is obtained by holding both *α* and *β* fixed in the differentiation process.

The fibre reassembly is described by means of *α*, and this reassembly must account for all of the dissolved fibre material. In the original ISR development, a similar conservation issue with respect to network scission and reformation is considered, as, for example, indicated in the discussion leading to eqns (2.4) and (2.5) in Wineman (2001). Adapting the same type of argument, the appropriate conservation requires that(4.12)where ′ denotes differentiation with respect to the argument.

A reasonable and useful form for *β*^{(1)}(*i*_{4}) is the reversed sigmoidal function(4.13)with *λ*_{p}≡(3*λ*_{s}−*λ*_{f})/2<*λ*_{s}. It then follows that (4.12) gives (*viz*. both (2.7) of §2 and also eqn (4.5) of Wineman 2001)(4.14)The mechanical properties of the secondary network are determined by the associated energy density *W*^{(2)} in (4.1) and (4.2). This secondary network, since it forms with respect to a deformed matrix, will have fibres of different natural lengths than those of the primary network. The reassembled fibres will also exhibit a different alignment. After accounting for these differences, we shall here assume that the secondary network is assembled from constituents that have the same mechanical properties as the primary network. In other words, the mechanical properties of the primary network with respect to the original placements ** X** will be assumed to be identical to the mechanical properties of each portion of the secondary network with respect to the placements that held sway when that portion of the secondary network was created.

Such a description requires the knowledge of the direction in which the fibres reassemble. We shall assume that when the fibre reassembly takes place, it does so in the direction at which the matrix, at that instant, is maximally stretched. This assumption is adopted here in the absence of a more specific reassembly law as determined, for example, from the study of a particular biological process. The current framework is, however, sufficiently general so as to allow for the consideration of other reassembly rules when such rules become known or become sufficiently accepted by the biological community.

For example, in the case of a biological cell, there are three families of intercellular fibres comprising the cytoskeleton: the actin filaments; the microtubules; and the intermediate filaments, which control the cell shape changes by active polymerization (Kumar *et al*. 2006). One might use a model of the present type to describe the microstructural evolution of one such filament family in which case the remaining two filament families, along with the cytoplasm and other organelles, would comprise the matrix. Then, it is further conceivable that the sensing aspect in the above description could be accomplished either by the cytoskeleton filament family that is undergoing the change or else by some other entity, possibly including one of the remaining two cytoskeleton filament families that are taken as fixed in the present modelling.

The maximal stretch in the matrix at time is the principal stretch , whose square is the maximal eigenvalue of both and . With respect to the placements , the direction of secondary fibre alignment is then given by the unit eigenvector of . When mapped back to the original reference configuration, these secondary fibres align in the direction given by the unit eigenvector of .

We here consider a process such that the fibre reassembly results in fibres that are prestretched by amount with respect to the current value of stretch in the formation direction at the time of the fibre formation. Thus, with respect to the original reference configuration with placements ** X**, a stretch of amount would put the newly created fibre at its natural length.

It is appropriate to clarify these concepts in a bit more detail by taking a point of view where one may identify any stretch in terms of a ratio of appropriate lengths. In this case, we imagine a portion of original network fibre having length *l*_{ref} in the reference configuration. If this portion has a force-free length *l*_{nat}, then *λ*_{nat}=*l*_{nat}/*l*_{ref}. Similarly, if a portion of a secondary network fibre is, at its instant of formation, of length , and is such that its force-free length is , then . Furthermore, since the reassembled fibres are aligned with the direction of maximum stretch, and since this maximum stretch is at time , it follows that , where is notation for the length that the secondary fibre portion would have if it were to be mapped back to the original reference configuration with placements ** X**. Hence,(4.15)which is in keeping with the above statement that the combined stretch takes the secondary fibre portion from its length in the original reference configuration to its natural length.

In modelling the reassembly process, it is necessary to provide a specification for the value . Here, we assume that the value is the same as that considered previously in connection with *Φ*_{f} in (4.3), i.e. we take . This is consistent with the original network being created at or before *t*=0 by the same type of kinetic process as that governing reassembly. As before, *λ*_{nat}=1 admits to the easiest interpretation, in that the newly formed fibre is then at its natural length at the time of its creation. A preshortened fibre (i.e. *λ*_{nat}>1) might occur, for example, if the fibre creation mechanism creates initially limp fibres. Less obvious is the need for considering fibres that are created in a state of pre-elongation (i.e. *λ*_{nat}<1), although biological fibre creation offers certain paradigms. Here, one might consider either some form of pretensioning mechanism via a novel fibre substrate scaffold or else some form of post-tensioning mechanism via protein motors that tighten ‘turnbuckle elements’ on the newly created fibre.

Under the above set of assumptions, it follows that the energy density in (4.1) and (4.2) is of the form(4.16)It will be useful to express the quantities appearing in (4.16) with respect to the original reference configuration. To this end, we note (e.g. Holzapfel 2000) that the polar decomposition gives the eigenvector connection , so that . This, in turn, yields , which, in conjunction with , provides the useful expression(4.17)The stored energy density *W*, as given by (3.11), is now(4.18)prior to any dissolution of the original fibre network. The energy density is(4.19)while the primary fibre network is dissolving and the secondary network is assembling. Finally, the energy density is(4.20)after reassembly is complete. Here, it is to be noted that both and in the integration kernels require a parametrization in terms of .

In a similar fashion, the Cauchy stress (4.1) is given by(4.21)prior to the microstructural change. The stress is given by(4.22)during the fibre dissolution and reassembly. Finally, the stress is given by(4.23)after reassembly is complete.

Equations (4.18)–(4.23) in conjunction with (4.12) comprise the basic constitutive model for the fibre dissolution and reassembly process. In this section, we have chosen to provide a purely continuum mechanical exposition in developing this constitutive framework. An obvious question that arises is whether or not the continuum model embodied in (4.12) and (4.18)–(4.23) is consistent with the filament dissolution and reassembly processes, as described with respect to the spring network model given in §2. Here, it is useful to point out that the reassembled springs in the §2 spring network model involve what can be regarded as a loss in stiffness due to a secondary filament assembly process that occurs at a stretch that is greater than that associated with the originally unstretched primary filaments. This, for example, gave rise to the term in the denominator of the integral expressions in (2.6), where one of these two factors is due to the fewer number of filaments, and the other is due to the greater number of component microscale springs in each filament. The nonlinear spring expression (2.10) shows this loss in stiffness as well, although in that case, the filament number effect appears outside of the *f*, whereas the component spring effect appears in the argument of *f*.

In view of these observations, it is not obvious that the continuum model obtained in this section similarly replicates these loss-in-stiffness effects. We examine this issue in §6 in which we establish the appropriate connections for the case of uniaxial load. Uniaxial load in the spring network model gave rise to (2.6) for the linear treatment and to (2.10) for the nonlinear treatment. Accordingly, we now turn to consider uniaxial loading behaviour in the context of the continuum model (4.18)–(4.23) with (4.12). Thus, in §5, we eventually obtain (5.4)–(5.6) as the continuum analogue of (2.10). We then show on this basis in §6 that the two approaches are equivalent. In this process of establishing this equivalence in §6, we provide an alternative justification for the relation (4.12).

## 5. Example: uniaxial extension in the fibre direction

As a basic illustration of the continuum model, we consider uniaxial loading in the original fibre direction. This gives the simplification that the direction of maximum principal stretch is the same as the original fibre direction. We take ** M**=

*e*_{3}, and consider(5.1)Thus, and

*i*

_{4}=

*λ*

_{3}. Consider a hypothetical process in which

*λ*

_{3}in (5.1)

_{1}is monotonically increasing with

*t*. The material exhibits classical hyperelastic response as long as

*λ*

_{3}<

*λ*

_{s}. The dissolution and reassembly processes of interest then take place as

*λ*

_{3}traverses the interval

*λ*

_{s}≤

*λ*

_{3}≤

*λ*

_{f}. During the dissolution and reassembly, it follows that and that and are also given by the form in (5.1)

_{1}, provided that

*λ*

_{3}is replaced by and , respectively. It then further follows that and that .Under these specializations, the stresses prior to the dissolution follow from (4.21) as(5.2)Also observe from (4.22) and (4.23) that, for this example, the more complicated expressions for

**, during the dissolution/reassembly and after reassembly is complete, will differ from (5.2) only in the**

*T*

*e*_{3}⊗

*e*_{3}component of

**.**

*T*For ** T** given by (5.2), the condition (5.1)

_{2}requires that

*e*_{1}.

*Te*_{1}=

*e*_{2}.

*Te*_{2}=0, so that (5.2) gives(5.3)Moreover, this value of

*p*also holds during the dissolution/reassembly and after reassembly is complete by virtue of the fact that these processes, for this example, only affect the

*e*_{3}⊗

*e*_{3}component of

**. Hence, in all the cases, the pressure is given by (5.3) and the stress is given by (5.1)**

*T*_{2}, with

*T*given as follows.

Prior to the dissolution/reassembly, i.e. when *λ*_{3}<*λ*_{s}, it follows that(5.4)During the dissolution/reassembly, i.e. when *λ*_{s}≤*λ*_{3}≤*λ*_{f}, it follows that(5.5)After reassembly is complete, i.e. when *λ*_{3}>*λ*_{f}, it follows that(5.6)We now provide a variety of examples of this stress–stretch behaviour for the case *λ*_{nat}=1 using (4.6) for the stored energy density of the matrix component, and using (4.8) for the stored energy density of the fibre component. The dissolution and reassembly functions will be given by (4.13) and (4.14). Under these circumstances, the integral expressions in (5.5) and (5.6) can be explicitly evaluated, although the final cumbersome expressions are not displayed here. Figure 2 shows the basic relationship between the stress *T* and the stretch *λ*_{3} for a case in which *λ*_{s}=1.3, *λ*_{f}=1.5 and *γ*=*μ*. The main curve (black) shows a softening response on the interval *λ*_{s}<*λ*_{3}<*λ*_{f}, which in this case leads to a non-monotone graph. In the absence of dissolution and reassembly, the mechanical response in figure 2 would continue along the dark blue curve. Dissolution in the absence of reassembly is given by the magenta curve on the interval *λ*_{s}<*λ*_{3}<*λ*_{f} which, as one would expect, is below the response curve when reassembly takes place. For dissolution in the absence of reassembly, the response becomes that of the matrix by itself for *λ*_{3}>*λ*_{f}. This ‘matrix-only’ response is shown by the red curve in figure 2.

The effect of the relationship between the fibre modulus *γ* and the matrix modulus *μ* is shown in figure 3, where again *λ*_{s}=1.3 and *λ*_{f}=1.5. As *γ*/*μ* becomes larger, the non-monotonicity becomes more pronounced. Conversely, *γ*/*μ* sufficiently close to zero gives a monotonic response curve, although this is not evident from the graphs featured in figure 3.

For fixed *γ*/*μ*, one finds that the distinction between a monotone and a non-monotone response is dependent upon the values of *λ*_{s} and *λ*_{f}. Figure 4 displays several response curves for *γ*=*μ* and *λ*_{s}=1.3 corresponding to different values of *λ*_{f}. In particular, for *λ*_{f} near *λ*_{s} the response is non-monotone. On the other hand, for sufficiently large *λ*_{f} the stress response remains monotone during dissolution and reassembly. In this latter case, the dissolution and reassembly process is spread over a larger stretch interval, giving a sufficiently gradual effect so as not to overwhelm the ‘background response’ provided by the matrix.

The fibre model (4.8) used in the above examples, as well as the fibre model (4.7), place no limit on the ultimate fibre extensibility. The effect of a limiting fibre extensibility in conventional hyperelasticity has been investigated by Horgan & Saccomandi (2005), and such models could be incorporated into the present framework involving dissolution and reassembly. Here, we point out that the dissolution model embodied in (4.10) presumes that all original fibres will have dissolved if *λ*_{3} exceeds *λ*_{f}, thus rendering the limiting fibre extensibility concept moot as regards the present model. On the other hand, one could certainly generalize the model presented here so that some remnant of the original fibre network does not dissolve, in which case the issue of a limiting filament extensibility becomes significant. We do not develop this aspect of the model here.

We now turn to consider the unloading response. Here, we recall both the unloading curves from the spring network theory of §2 as well as from the ISR theory as shown, for example, in Huntley *et al*. (2000). In both the cases, there is residual deformation upon the removal of load if a new network has formed. This is also evident in the present example, as shown by the various unloading curves in figure 5. As in the spring network of §2 and as also in the original ISR treatment, unloading involves no microstructural change, in which case the material is formally hyperelastic.

Coiling, wrinkling or buckling of the fibres may prevent them from contributing much in the way of a compressive stress. Such considerations are dependent upon any inherent bending stiffness in the fibres and the extent to which the matrix is capable of providing lateral support to the filaments. If such stiffness or support effects are either negligible or absent, then the fibres store little if any energy in contraction. This gives *Φ*_{f}=0 when *i*_{4}<*λ*_{nat} and, as mentioned previously in connection with (4.9), can be treated in the present framework through the use of the step function *H*. Consider, therefore, the modified energy density (4.9) and let us continue to consider the case with *λ*_{nat}=1, *λ*_{s}=1.3 and *λ*_{f}=1.5. There is no distinction between the mechanical response of (4.8) and (4.9) for processes in which all the fibres experience stretch at or beyond each fibre's natural length. This includes the loading curves depicted in figures 2–4. A distinction arises in the unloading behaviour so that, for example, the unloading curves in figure 5 do not apply for (4.9). The correct type of unloading curves for (4.9) is shown in figure 6.

It is worth noting that, in contrast to figure 5, the unloading response in figure 6 does not show a permanent set at zero stress. This is because, upon unloading, all secondary fibres in this example are shorter than their natural length as soon as *λ*_{3}<*λ*_{s}. Thus, there is no contribution of the secondary network to the unloading curves on the interval 0≤*λ*_{3}≤*λ*_{s}. In particular, if the maximum stretch exceeds *λ*_{f} prior to unloading, so that none of the primary network remains, then the unloading behaviour will involve only a matrix contribution for unloading as soon as *λ*_{3}<*λ*_{s}. Hence, the rightmost unloading curve in figure 6 merges with the matrix-only response curve for 0≤*λ*_{3}≤*λ*_{s}.

In all of the examples discussed so far in this section, we have restricted attention to the case *λ*_{nat}=1. We conclude this section by considering the case *λ*_{nat}>1 which, as remarked previously, means that the fibre is initially contracted and so only attains its natural length if it is extended by the factor *λ*_{nat}. For fibre models such as (4.7) and (4.8), which support a compressive stress when contracted, the present modelling framework then gives rise to secondary network fibres that, at least initially, contribute a compressive stress. Figure 7 shows the loading response for the case using (4.6) for the stored energy density of the matrix component, and using (4.8) for the stored energy density of the fibre component with *λ*_{nat}=1.2, *λ*_{s}=1.3 and *λ*_{f}=1.5. Thus, the only difference between the models associated with figures 2 and 7 is the value of *λ*_{nat}. For *λ*_{nat}=1.2, the primary fibres act in compression for 1<*λ*_{3}<*λ*_{nat}=1.2, and so the response curve is below the matrix-only curve in this regime. Dissolution and reassembly begins at *λ*_{3}=*λ*_{s}=1.3 and the secondary network fibres initially act in compression. Thus, the response curve is now below the magenta curve, which again corresponds to a hypothetical process in which the primary network dissolves but does not reassemble. All of the reassembled fibres are at less than their natural length for 1.3=*λ*_{s}<*λ*_{3}<*λ*_{s}*λ*_{nat}=1.56. Conversely, all the reassembled fibres are elongated beyond their natural length for *λ*_{3}>*λ*_{f}*λ*_{nat}=1.8. In the intermediate loading range 1.56=*λ*_{s}*λ*_{nat}<*λ*_{3}<*λ*_{f}*λ*_{nat}=1.8, the most recently formed fibres are still contracted, but the earlier formed fibres are elongated. We note from figure 7 that the compressive stress contribution of the contracted fibres in this example depresses the response curve below that of the matrix-only response for an interval of stretches *λ*_{3}.

A useful model is that with *λ*_{nat}>1 for a fibre energy density *Φ*_{f} that only contributes a tensile load, such as the previously considered (4.9). This could serve as a model for a fibre reassembly process that initially creates limp fibres that are only recruited to support load once *λ*_{3} exceeds , where is again the past value of *λ*_{3} at the time of the fibre reassembly. Such a situation might occur, for example, if the reassembly creates either wavy or coiled filaments that must first be extended so as to either straighten or uncoil before they are in a position to support any load. Figures 8 and 9 show this loading response for the cases in which *λ*_{s}*λ*_{nat} is, respectively, less than and greater than *λ*_{f}. In the first case, the response curve remains above the matrix-only curve once dissolution and reassembly takes place. In the second case, the response curve coincides with the matrix-only curve for a certain portion of loading, specifically for *λ*_{f}≤*λ*_{3}≤*λ*_{s}*λ*_{nat}. The transition case in which *λ*_{f}=*λ*_{s}*λ*_{nat} gives a loading curve that touches the matrix-only curve at a single point. For comparison's sake, figures 8 and 9 also show the loading response for the fibre model (4.8) in which the contracted fibres contribute a compressive stress, in which case the associated curves are able to dip below the matrix-only response.

Unloading for a case with *λ*_{nat}>1 for a fibre energy density *Φ*_{f} that only contributes a tensile load is depicted in figure 10. The earlier discussion in regard to figure 6 applies here, except that the contribution of the fibres to the stress *T* becomes zero at *λ*_{3}=*λ*_{nat}>1. The remaining unloading response for 0≤*λ*_{3}<*λ*_{nat} is solely due to the matrix contribution.

## 6. Relating the spring network treatment to the continuum treatment

In this section, we return to the issue raised at the end of §4 regarding the extent to which the continuum description (4.18)–(4.23) with (4.12) is faithful to the notion of spring dissolution and reassembly, as presented in the somewhat more concrete setting of §2. The main result of this section is to verify this equivalence in the context of uniaxial load.

Consider a unit cube in the reference configuration. In the spring network treatment, it is regarded as containing *N* nonlinear springs of length *l*_{0}, each with a force law *f*_{spring}=*f*(ξ), and all of which originally form force-free filaments in the *X*_{1}-direction, as described in §2. The cube is subject to a displacement with associated deformation gradient(6.1)Hence, the force *F*_{1} in the *X*_{1}-direction is described by (2.10). The Cauchy stress in the load direction is given by this *F*_{1} divided by the current area . The Cauchy stress in the load direction is therefore(6.2)Each component spring experiences an end displacement *ξ*=*l*_{0}*δ*_{1}, so that the total energy stored in the spring network from this deformation before any dissolution and reassembly is simply(6.3)Our aim is to compare this spring network treatment with the continuum treatment in §5 for the special case in which the matrix exhibits no mechanical effect. Formally, this corresponds to taking *μ*=0 in the various formulae in §5. Hence, the formulae (5.4)–(5.6) describe the mechanical response provided that the stretch(6.4)Similarly, the displacement thresholds *u*_{s} and *u*_{f} correspond to stretch thresholds *λ*_{s} and *λ*_{f} via *λ*_{s}=1+*u*_{s} and *λ*_{f}=1+*u*_{f}. The relationship (6.2) between the force and the Cauchy stress also applies in the continuum treatment, which, in turn, gives *F*=*T*/*λ*_{3}. We also take *λ*_{nat}=1 so that the continuum treatment mirrors the notion in the spring network treatment that the filaments are force-free at the instant of their formation. For ease of notation in this section, the fibre energy density term that appears in the formulae of §5 will, in this section, be replaced by the simpler notation . Under these replacements and specializations, the associated force *F*_{1} on the *X*_{1} face of the cube that supports this deformation in the continuum treatment is as follows.

For 0≤*δ*_{1}<*u*_{s}, which corresponds to 1≤*λ*_{3}≤*λ*_{s}, it follows from (5.4) that(6.5)For *u*_{s}≤*δ*_{1}≤*u*_{f}, which corresponds to *λ*_{s}≤*λ*_{3}≤*λ*_{f}, it follows from (5.5) that(6.6)For *δ*_{1}>*u*_{f}, which corresponds to *λ*_{3}>*λ*_{f}, it follows from (5.6) that(6.7)Since this deformation is homogeneous and the domain is a unit cube in the reference configuration, the total energy stored in this continuum treatment before any dissolution and reassembly is simply(6.8)We now compare this continuum treatment described by the above equations (6.5)–(6.8) with the spring network treatment as described by (2.10) and (6.3). Comparison of the energy expression (6.3) with (6.8) gives the relationship between the component spring force function *f* in the spring network treatment and the continuum fibre energy density *Φ*_{f} as(6.9)For the purposes of using (6.9) in (2.10), we note from (6.4) that(6.10)We shall also introduce(6.11)Together, (6.10) and (6.11) when used in (2.10) indicate that the spring network treatment gives the expressions for the force *F*_{1} as follows.

For 0≤*δ*_{1}<*u*_{s}, which corresponds to 1≤*λ*_{3}<*λ*_{s}, it follows that(6.12)For *u*_{s}≤*δ*_{1}≤*u*_{f}, which corresponds to *λ*_{s}≤*λ*_{3}≤*λ*_{f}, it follows that(6.13)For *δ*_{1}>*u*_{f}, which corresponds to *λ*_{3}>*λ*_{f}, it follows that(6.14)In particular, we verify that (6.5) matches (6.12). In addition, since in (6.6) and in (6.13) are dummy integration variables, it follows that these equations are equivalent to each other provided that(6.15)and(6.16)Furthermore, the relations (6.15) and (6.16) also give the equivalence of (6.7) with (6.14). Eliminating *q*(*λ*_{3}) between (6.15) and (6.16) now retrieves (4.12), which verifies the equivalency of the two approaches for this class of deformations. In particular, the loss-in-stiffness effect due to the conservation of microscale springs in the spring network treatment, which was remarked upon at length at the end of §4, is indeed accounted for in the continuum treatment, which is the main object of this paper. More generally, this effect is captured by the connections (3.8) and (4.12), which are central to the present framework.

## Acknowledgments

This material is based upon the work that is partially supported by Carnegie Mellon University in Qatar, under the sponsorship of the Qatar Foundation, and by the US National Science Foundation under grant no. 0510600 to Michigan State University.

## Footnotes

- Received September 3, 2008.
- Accepted October 31, 2008.

- © 2008 The Royal Society