## Abstract

There is experimental evidence to suggest that extensible connective tissues are mechanically time-dependent. In view of this, the mechanics of time-dependent lateral stress transfer in skeletal muscle is investigated by employing a viscoelastic shear lag model for the transfer of tensile stress between muscle fibres and the surrounding extracellular matrix (ECM) by means of shear stresses at the interface between the muscle fibre and the ECM. The model allows for both mechanical strains in the muscle as well as the strain owing to muscle contraction. Both the ECM and the muscle fibre are modelled as viscoelastic solids. As a result, time-dependent lateral stress transfer can be studied under a variety of loading and muscle stimulation conditions. The results show that the larger the muscle fibre creep time relative to the ECM relaxation time, the longer it takes for the muscle fibre stress to relax. It also shows that the response of the muscle–ECM composite system also depends on the characteristic time of a strain history relative to the characteristic relaxation time of the ECM. The results from the present model provide significant insight into the role of the parameters that characterize the response of the muscle composite system.

## Introduction

The composite structure of skeletal muscle is composed of muscle fibres and an extracellular matrix (ECM) framework. This framework is associated with different levels of structure: (i) epimysium, which ensheaths the whole muscle, (ii) perimysium, which binds a group of muscle fibres into bundles, and (iii) endomysium, which surrounds the individual muscle fibres. Purslow (2002) and others have provided extensive reviews that discuss the structure, morphology and composition of intramuscular connective tissue (IMCT) that includes epimysium, perimysium and endomysium. The properties of IMCT components and their interaction with muscle fibres determine the overall mechanical properties of the whole muscle. Many observations have shown that, in series-fibred muscle, tension is transmitted laterally from the muscle fibre to the IMCTs, and then to the tendon (Street 1983; Huijing 1999). The IMCT is thus an essential element in the mechanical function of the muscle (Otten 1988; Purslow & Trotter 1994).

Previous studies have experimentally demonstrated that stress could be laterally transmitted through the endomysium, perimysium and the epimysium (Street 1983; Huijing 1998; Maas & Baan 2001). These observations actually depend on the efficient shear linkage between the adjacent muscle fibres (Purslow & Duance 1990; Tidball 1991; Trotter 1993). Since it is known that muscle consists of discontinuous fibres surrounded by a matrix (Trotter *et al.* 1995; Huijing 1999), it is important to properly capture the interaction between these two constituents in describing the mechanism of lateral stress transfer in a muscle fibre–ECM system.

The most widely used model describing load transfer between a discontinuous fibre and matrix is the shear lag model, originally proposed by Cox (1952). This model centres on the transfer of tensile stress between fibres by means of interfacial shear stresses and shear deformation of the matrix. In Cox’s model, the fibre and the matrix are both elastic, and it was found that a minimum fibre length was necessary in order to transmit the force efficiently from the fibre to the matrix. As discussed by Purslow *et al.* (1998), the mechanical behaviour of several classes of extensible soft tissues is time-dependent. Muscle fibres and the ECM, because of their composition and similarity to the constituents within skin and perimyseum, exhibit time-dependent mechanical responses, as shown experimentally by Puxkandl *et al.* (1964). Even though a detailed understanding of the reason for time-dependency in extensible soft tissue, such as skin and perimysium, is still to be investigated, Purslow *et al.* (1998) concluded that time-dependency is primarily because of the molecular re-arrangements within collagen fibres and stress transfer at the interface between the collagen and the ECM. This reasoning motivated the present authors to develop a model that captures the essential features of time-dependent stress transfer between a muscle fibre and the surrounding ECM.

In this paper, a mechanical model is developed to investigate the time-dependent mechanics of lateral stress transfer between activated muscle fibres and the surrounding strained ECM. The results from this model show that time-dependent lateral stress transfer depends on several factors: (i) the amount of stress relaxation and/or creep in the muscle fibre and the ECM, (ii) their characteristic times for stress relaxation and/or creep, and (iii) the time-dependence of the loading and activation processes. The results show that knowledge of only the viscoelastic properties of the muscle fibre or only the matrix is insufficient to gain an understanding of the response of the fibre–matrix system. The properties of both are required to understand their interaction.

A study with a viscoelastic model for the fibre–ECM system is warranted in view of experimental data presented in (Purslow *et al.* 1998). A nonlinear viscoelastic model with complex time-dependent properties would lead to complex equations, warranting advanced methods of solution and possible quantitative agreement with experimental results at the expense of greater computational effort. However, the assumptions contained in the muscle fibre–ECM model presented in this paper, which adopts a linear viscoelastic constitutive model, enable significant insight to be gained into the role of the parameters that characterize the response of the muscle composite system. Thus, the linear viscoelastic treatment presented here is a useful prelude to a study with more modelling complexity, which will follow this initial work.

By modelling the total muscle as a composite element, consisting of the ECM reinforced by muscle fibres that can generate force and bear loads actively, a representative volume element (RVE) of the muscle fibre and the ECM, as shown in figure 1, is considered. This RVE has been used before (Trotter *et al.* 1995; Huijing 1999) to suggest mechanisms of the lateral transmission of force. The contractility of muscle fibres is represented by a contractile activation strain, possibly time-dependent, that accounts for the effect of electrical stimulations. The muscle fibres and the ECM are modelled as linear viscoelastic solids. To simplify the model, the dependency of the properties of the ECM on the collagen fibres is not considered in this paper. The stresses in the muscle are determined for two cases, a step-strain history and a ramp-strain history.

It is noted that the approach adopted in this paper provides a mechanistic basis to determine the ECM–muscle fibre system properties from a knowledge of the constituent properties. This differs from some of the traditional approaches in the biomechanics modelling literature, which adopt combinations of parallel and series spring/dashpot models by fitting experimental data to adjust the constants associated with the springs and dashpots. Here, we provide a synthesis of what exactly constitutes the system response and how each constituent influences the overall system response. Such insight cannot be obtained without adopting a model that uses mechanics as its cornerstone.

## Linear viscoelastic model for muscle fibre and the extracellular matrix

The constitutive equation for a linear viscoelastic material relates the current stress to the preceding strain history. In order to state this constitutive relation, let ‘*t*’ denote the current time and ‘*s*’ denote a generic time, 0 ≤ *s* ≤ *t*. Let ϵ(*s*) and σ(*s*) denote the strain and stress, respectively, at time *s*. It is assumed that the material is initially unstrained and unstressed, i.e. ϵ(*s*)=σ(*s*)=0, *s*<0. The stress at time *t*>0 is expressed in terms of the preceding strain history, ϵ(*s*), 0 ≤ *s* ≤ *t* by Wineman & Rajagopal (2000) as
2.1
where *G*^{R}(*t*) is the stress-relaxation modulus. This material property can, in concept, be measured by subjecting a specimen to a step-strain deformation and measuring the stress-relaxation response. However, it can also be measured in other experiments, such as one in which the strain increases at a constant rate to a specified value and is then held fixed (ramp-strain history). The latter leads to easier characterization.

Equation (2.1) is often written in the form
2.2
in which the term ϵ(0)*G*^{R}(*t*), because of a jump in strain at *t*=0, is accounted for by the lower limit 0^{−}. Equation (2.2) is a Stieltjes convolution that is also denoted using the notation
2.3
Further details of this notation are given in Wineman & Rajagopal (2000).

Corresponding to the stress-relaxation modulus *G*^{R}(*t*), there is a creep compliance, *G*^{C}(*t*), that satisfies the relation
2.4
With this relation, equation (2.2) can be inverted to give the strain at time *t* in terms of the stress history up to time *t*, σ(*s*), 0 ≤ *s* ≤ *t*, as
2.5
The stress-relaxation modulus *G*^{R}(*t*) decays monotonically from its initial value *G*^{R}(0) to its equilibrium value . Of interest is the characteristic time at which substantial stress relaxation has occurred. There are several possible definitions of a characteristic relaxation time. A useful choice is the time corresponding to the centroid of the graph of . In a similar manner, the characteristic creep time is the time at which substantial creep has occurred. One estimate of this time is at the centroid of the graph of .

## Formulation

In the RVE shown in figure 1, all of the muscle fibres together are assumed to be one cylinder, with radius *r*, surrounded by the ECM. The distance between the centre line of the fibres and the surface of the ECM is *R*. The total length of the muscle fibres embedded in the ECM is 2*l*. By choosing this RVE, it is assumed that the entire tissue can be thought of as the periodic extension (spatially) of this RVE. Thus, by analysing the mechanics of this RVE, we can infer the behaviour of the entire skeletal muscle.

Shear material properties are introduced for the ECM, and tensile material properties are introduced for the muscle fibre. and denote the shear-stress-relaxation function and shear creep compliance for the ECM. and denote the tensile-stress-relaxation function and tensile creep compliance for the muscle fibre.

In the following development, it is assumed that the muscle is fully activated. Muscle contraction gives rise to a shear stress between the muscle fibre and ECM interface. This shear stress is the physical embodiment of the notion of lateral stress transfer. Let τ_{mi} be the shear stress at the ECM–fibre interface at radius *r*. Let τ_{m} denote the shear stress in the ECM on a surface of radius *z*, *r*≤*z*≤*R*, in the direction of the *x*-axis. Force balance along the *x*-axis at time *t* gives
3.1
It is assumed that radial displacement of the ECM can be neglected when compared with the displacement along the *x*-axis. Let *u*_{m} denote the displacement of a point of the ECM at radius *z*, in the direction of the *x*-axis. The shear strain in the ECM is
3.2
Using equations (2.2) and (2.5), the shear response of the ECM is described by
3.3
and the inverse relation by
3.4
Substitute equations (3.1) and (3.2) into equation (3.4) gives
3.5
Next, integrate equation (3.5) at time *t* with respect to *z* from the interface radius *r* to the outer radius *R* of the ECM. The resulting equation is
3.6
where *u*_{m}(*R*,*x*,*t*) is the displacement along the *x*-axis at time *t* in the ECM at radius *R* and *u*_{m}(*r*,*x*,*t*) is the displacement along the *x*-axis at time *t* at the ECM–muscle fibre interface at radius *r*.

It will be convenient to invert equation (3.6) to get the following expression for τ_{mi} at time *t*:
3.7
It is commonly assumed that the normal stress in the ECM on surfaces normal to the *x*-axis can be neglected since the axial extensional modulus of the fibre is much larger than that of the ECM if the muscle is fully activated. Assume that there is no shear stress in the muscle fibre and that there is perfect bonding between the muscle fibre and the ECM. Let σ_{f} denote the tensile stress in the muscle fibre and denote the shear stress at the muscle fibre–ECM interface in the direction of the *x*-axis. Then, axial force balance for the muscle fibre implies
3.8
Traction continuity at the muscle fibre–ECM interface implies
3.9
Combining equations (3.7)–(3.9) gives
3.10
Differentiating equation (3.10) with respect to *x* provides
3.11
Let the notation ∂*u*_{m}(*R*,*x*,*s*)/∂*x*=ϵ_{m}(*R*,*x*,*s*) and ∂*u*_{m}(*r*,*x*,*s*)/∂*x*=ϵ_{m}(*r*,*x*,*s*), be introduced. ϵ_{m}(*R*,*x*,*s*) and ϵ_{m}(*r*,*x*,*s*) are the elongational strains in the ECM along the *x*-axis at radius *R* and at the ECM–muscle fibre interface at radius *r*, respectively. Since the displacement is continuous at the ECM–muscle fibre interface and there is no shear in the muscle fibre, ϵ_{m}(*r*,*x*,*s*)=ϵ_{f}(*x*,*s*), the axial strain in the muscle fibre. Let it be assumed that the ECM is uniformly stretched at *R* so that ϵ_{m}(*R*,*x*,*s*)=ϵ_{mR}(*s*). Then, equation (3.11) can be restated as
3.12
The notation introduced in equation (2.3) is used here for convenience in the following development.

The total extensional strain in the muscle fibre, ϵ_{f}, is the superposition of mechanical strain, ϵ_{fm}, and the activation strain generated by fibre contraction, ϵ_{a}, ϵ_{a} < 0,
3.13
Recalling equation (2.5), the extensional mechanical strain ϵ_{fm} can be expressed in terms of the tensile stress in the muscle fibre, σ_{f}, and the extensional creep compliance by
3.14
Let ϵ_{f} from equation (3.13) be substituted into equation (3.12), and then ϵ_{fm} from equation (3.14) be substituted into the result. This leads to
3.15
It can be shown (Wineman & Rajagopal 2000) that
3.16
where
3.17
This is a composite property that represents the complex interaction between shear-stress relaxation in the ECM and extensional creep in the muscle fibre. This significant result shows that the composite property of the whole muscle fibre–ECM system is influenced by the relaxation properties of the ECM and the creep properties of the muscle fibre in a very specific manner. The composite property, A, in equation (3.17) has interesting clinical implications. For instance, certain medications can be designed that selectively affect the chemical structure of the fibre or the ECM, which in turn affects the time-dependent mechanical properties of these constituents. If these effects are quantified, then the present model describes how the composite time-dependent properties are affected.

Equation (3.15) can be rewritten using equations (3.16) and (3.17) as 3.18 The terms in equation (3.18) that involve the Stieltjes convolution can be expressed in terms of integrals by using equations (2.1)–(2.3). The result is 3.19 In deriving equation (3.19), use was made of the result 3.20 which is obtained using an integration by parts.

Equation (3.19) is a partial differential Volterra integral equation for σ_{f}(*x*,*t*). Its solution shows how the spatial variation of tensile stress along the fibre varies with time, and how this depends on the viscoelastic properties of the fibre, the ECM and the activation strains. The formulation is completed by assuming, as is usually done in a shear lag analysis, that σ_{f} is symmetric about *x*=0 and σ_{f}=0 at the end of the muscle fibre.

## Stresses at *t*=0 and at long-time equilibrium

The matrix strain ϵ_{mR} and the activation strain ϵ_{a} are assumed to reach fixed values in some finite time interval. Accordingly, it is also assumed that creep and stress-relaxation processes reach their long-time values (steady states) and so does the muscle fibre stress, σ_{f}. The notation is used here to denote such a time when the muscle fibre–ECM system is in its long-time equilibrium state.

Analytical solutions to the boundary value problem for σ_{f} can be obtained at *t*=0 and in the limit . These solutions are discussed in this section so as to obtain insight into how the amount of stress relaxation and creep affect the stress in the muscle fibre and the lateral transmission of force at the muscle fibre–ECM interface. The influence of the time-dependent processes of stress relaxation and creep in the evolution of muscle fibre tensile stress and shear stress at the muscle fibre–ECM interface is treated in the next section.

Let *t*=0 in equations (3.17), (3.19) and (2.4), with and .

Then,
4.1
and
4.2
It is assumed that the time-dependent physical quantities introduced in §2 are bounded as . Upon taking the Laplace transforms of equations (3.17) and (3.19), and then applying the asymptotic theorems for the Laplace transform (see Wineman & Rajagopal 2000), one obtains
4.3
and
4.4
Equations (4.2) and (4.4) have similar mathematical forms and hence similar solutions. The solutions to equations (4.2) and (4.4) are
4.5
and
4.6
where
4.7
ζ=*l*/*r* and *C*_{1}, *C*_{2}, and are integration constants. When the boundary conditions on σ_{f} are imposed, equations (4.5) and (4.6) reduce to
4.8
and
4.9
The shear distributions at the muscle fibre–ECM interface at *t*=0 and as are found by applying equations (3.8) and (3.9) at *t*=0, as and making use of equations (4.8) and (4.9),
4.10
and
4.11
In order to discuss the variation along the muscle fibre of the tensile stress in the fibre and the shear stress at the muscle fibre–ECM interface, it is convenient to define the following functions:
4.12
and
4.13
Figures 2 and 3 show, respectively, plots of and versus *x*/*l* for different values of λζ. Values for the geometric parameters ζ and were determined with *l*=6.5 mm, *r*=0.744 mm and *R*=1.05*r* (Gao 2007). These values were based on the measurement of the extensor digitorum longus (EDL) muscle. Values for the material parameters were MPa (Lieber *et al*. 2003) and MPa (Gao 2007). Then, λ=0.67, ζ=8.74 and λζ=5.8. It is not clear from the conditions under which they were determined whether these values for and are to apply at *t*=0 or at . However, they do provide a baseline value for λζ. The results in figures 2 and 3 were obtained using values of λζ about this baseline.

According to figure 2 and equations (4.8), (4.9) and (4.12), the tensile stress in the muscle fibre is a maximum at *x*=0 and then decreases monotonically to zero at *x*/*l*=±1. According to figure 3 and equations (4.10), (4.11) and (4.13), the shear stress transferred from the muscle fibre to the ECM increases monotonically from zero at *x*=0 to a maximum at *x*/*l*=±1. The ratio of the maximum shear stress to the maximum tensile stress is . Values of λζ used in figures 2 and 3 are λζ=15, λζ=10 and λζ=5.8. For these values, the ratio is approximately λ/2≈0.34. Thus, the maximum shear stress at the muscle fibre–ECM interface is smaller than the maximum tensile stress in the muscle fibre. Let *x*_{ T}(*t*)/*l* denote the dimensionless stress transfer length, that is, the length of the portion of the fibre at time *t* over which the tensile stress builds up to 99 per cent of its maximum value at that instant of time. figure 2 shows that this portion is at the end of the fibre and that *x*_{ T}(*t*)/*l* increases as λζ decreases. figure 3 shows that this is also the portion over which the shear stress is transferred from the muscle fibre to the ECM.

It is seen from equation (4.7) that the stress transfer lengths at *t*=0 and as depend on the ratio of the ECM shear stiffness to the muscle fibre tensile stiffness. Note that
4.14
4.15
According to equation (4.14), the stress transfer length as will differ from that at *t*=0 when the ECM–fibre stiffness ratio changes. The stress transfer length will also change from *t*=0 to if the shear relaxation ratio in the ECM, differs from the tensile relaxation ratio in the muscle fibre, .

## Viscoelastic response: time-dependent properties

The essential features of the viscoelastic properties of the ECM or muscle fibre can be accounted for if each is modelled as a three-parameter or standard linear solid. These features are the initial values, the long-time equilibrium values and the characteristic times. Thus, the shear-stress relaxation function of the ECM is
5.1
The tensile creep compliance of the muscle fibre is
5.2
The long-time equilibrium values are and . The initial values are and . τ_{mR} is a characteristic shear relaxation time and τ_{fC} is a characteristic tensile creep time. Further, is a measure of the amount of shear-stress relaxation in the matrix. is a measure of the amount of creep in the fibre.

Using equation (2.4), it can be shown that the tensile-stress-relaxation property has the form
5.3
in which the long-time equilibrium value , the initial value and characteristic tensile-stress-relaxation time τ_{fR} is
5.4
More complicated expressions could be introduced for use in obtaining the numerical results presented in the subsequent sections. However, this would obscure the influence of the parameters in the above expressions.

Using equations (5.1) and (5.2), the interaction property *A*(*t*) defined in equation (3.17) is found to be given by
5.5
in which
5.6
and
5.7
where .

The interaction property *A*(*t*) depends on the shear stiffness of the ECM relative to the tensile stiffness of the fibre, as indicated by its variation from to . Its time course depends on the ECM stress-relaxation time, τ_{mR}; the fibre stress-relaxation time, τ_{fR}; the amount of stress relaxation in the fibre, ; and the amount of stress relaxation in the ECM, . figure 4 shows a plot of *A*(*t*) versus *t*/τ_{mR} for several values of τ_{fR}/τ_{mR}. Note that *A*(*t*) is not monotonic in time. Depending on whether τ_{fR}/τ_{mR} ≥1 or τ_{fR}/τ_{mR}≤1, *A*(*t*) initially decreases or increases with time and then approaches its long-time value. The time course of *A*(*t*) plays an important role in determining the stresses produced in a muscle. An important property of a muscle–ECM system is that its mechanical response depends on the characteristic times of the activation strain, ϵ_{a}(*t*), and applied strain, ϵ_{R}(*t*), relative to the characteristic response time of the muscle system. As alluded to earlier, the effect of medications in selectively altering the viscoelastic properties of the muscle fibre and/or the ECM can have significant effects on the entire muscle behaviour, as seen through the time history of *A*(*t*).

## Influence of viscoelasticity

The viscoelastic response of the muscle fibre–ECM structure is studied by considering different histories of the mechanical and activation strains. The stress response is obtained by solving equation (3.19). Analytical solutions to this equation have been found at *t*=0 and . Although there are analytical methods for solving equation (3.19), they lead to expressions that are not easily amenable to evaluation. Hence, in this paper, equation (3.19) will be solved numerically.

Let the following dimensionless variables be introduced into equation (3.19):
6.1
Equation (3.19) becomes
6.2
where λ_{0} is defined in equation (4.7). In the remainder of this section, the superposed bar is deleted for notational convenience and the notation was introduced for notational convenience.

In carrying out numerical simulations, the geometric parameters ζ and were determined with *l*=6.5 mm, *r*=0.744 mm and *R*=1.05*r* (Gao 2007). The material parameters were *G*_{1}=0.6 (Purslow *et al.* 1998) and *K*_{1}=−0.4 (Wolff *et al.* 2006); thus, MPa, MPa and . Then, ζ=8.74 and λ_{0}=0.64.

### Step mechanical and activation strain histories

Let ϵ_{a}(*t*) and ϵ_{mR}(*t*) both be step-strain histories, and ϵ_{mR}(*t*)=ϵ^{0}_{R}, *t*≥0. Equation (6.2) reduces to
6.3

Let *x*_{n}=(*n*−1)Δ*x*, *n*=1,…,*N*+1, where *N*Δ*x*=1 and *t*_{p}=*p*Δ*t*, *p*=0,1,… denote the set of spatial points and times, respectively, at which the solution is to be found. Let σ_{np}=σ_{f}(*x*_{n},*t*_{p}). The spatial derivatives at interior points are approximated by a central difference scheme, and the time integral is approximated as a discrete sum,
6.4
This can be rearranged so as to express the nodal values of stresses at current times σ_{np} in terms of nodal values at previous times, σ_{jk}, *k*=1,2,…,*p*−1,
6.5
The matrix form of the above equation can then be written as
6.6
The boundary condition that σ_{f} is symmetric about *x*=0 implies that dσ_{f}/d*x*=0 at *x*=0. This was implemented by letting σ_{1p}=σ_{2p}. The boundary condition that σ_{f}=0 at the end of the muscle fibre was implemented by letting σ_{(N+1)p}=0.

Let *K*_{1}=(−1/Δ*x*^{2})−(λ_{0}ζ)^{2}, *K*_{2}=(−2/Δ*x*^{2})−(λ_{0}ζ)^{2} and α=1/Δ*x*^{2}. Equation (6.6) can be rewritten as
6.7
Calculations were carried out with ϵ_{mR}=0.2 and ϵ_{a}=−0.2. The accuracy of the results depends on the values of Δ*x* and Δ*t*. It was found that when Δ*x* or Δ*t* were reduced to 0.0005, the maximum change in the value of the stress was 0.05 per cent. Therefore, excellent accuracy in the numerical solution and reasonable CPU time was balanced with Δ*x*=0.001 and Δ*t*=0.001. The tensile stress distribution in the muscle fibre at several times is shown in figure 5. The stress transfer length *x*_{ T}(*t*)/*l* decreases as time increases. figure 6 shows that τ=τ_{fC}/τ_{mR} significantly affects the relaxation of the tensile stress in the muscle fibre at *x*=0. This relaxation takes longer for larger values of τ=τ_{fC}/τ_{mR}.

### Step activation strain and ramp mechanical strain histories

Let ϵ_{a} be a step-strain history and let ϵ_{mR}(*t*) be a ramp-strain history that reaches the constant strain ϵ^{0}_{mR} at time *t*=*T**, with strain rate α=ϵ_{mR}/*T**. These histories are shown in figure 7. For *t*<*T**, equation (6.2) becomes
6.8
and at time *t*>*T**, equation (6.2) becomes
6.9

Since equations (6.8) and (6.9) are in terms of dimensionless variables, *T** now represents the dimensionless rise time *T**/τ_{mR}.

Calculations were carried out with the same values of ϵ^{0}_{mR} and as in the previous case. figure 8 shows results for several values of the rise time *T**. As the rise time decreases relative to the ECM relaxation time, the mechanical strain history approaches a step-strain history and the stress response is seen to approach the response to a step-strain history.

## Summary and concluding remarks

A mechanical model is presented for the study of the influence of time-dependent material properties on the transfer of tensile stress between muscle fibres by means of interfacial shear stresses in the ECM. The model allows for both mechanical strains in the muscle as well as the activation strain owing to muscle contraction. As a result, stress transfer can be studied under a variety of loading and muscle stimulation conditions.

Both the muscle fibre and the ECM are treated as linear viscoelastic solids in this initial investigation, the muscle responding in tension and the ECM in shear. As stated earlier, there is strong experimental evidence to suggest that time-dependent mechanical response of soft tissues emanates from the time-dependent mechanical behaviour of the constituents (in this case the muscle fibres and the ECM and the interface between the two; Purslow *et al.* 1998). The significant aspects of the time-dependent properties are the amount of stress relaxation in the fibre and ECM, and the characteristic stress-relaxation times of the fibre and ECM.

The equation for the tensile stress in the muscle depends on the contractile activation strain in the muscle and a composite property that represents the interaction between shear-stress relaxation in the ECM and tensile creep in the muscle. Once tensile stress in the muscle has been found, the shear stress at the muscle–ECM interface can be readily calculated.

The study is carried out in two steps. Analytical solutions are obtained at *t*=0, representing the response to step changes in deformation, and , representing the equilibrium state at large times. Figures 2 and 3 show how the amount of stress relaxation or creep alters the variation of stress along the fibre, the magnitude of the stress and the stress transfer length. In particular, the results depend on the shear stiffness of the ECM relative to the tensile stiffness in the muscle fibre at *t*=0 and at . These results can also be thought of as depending on the amount of tensile-stress relaxation in the fibre relative to the amount of shear stress relaxation in the ECM.

The second part of this study addresses the time dependence of the material properties. Owing to the complexity of the governing equations, a numerical method of solution is introduced. Two examples are presented: (i) step changes in the mechanical and contractile activation strains and (ii) a ramp mechanical strain and a step contractile activation strain. The results show that the relaxation of the maximum tensile stress in the fibre is strongly dependent on the creep time in the fibre relative to the relaxation time in the ECM. Figure 6 shows that the larger the fibre creep time relative to the ECM relaxation time, the longer it takes for the fibre stress to relax. This result is a consequence of the interaction between two different viscoelastic materials forming the muscle composite system. It points out that knowledge of the viscoelastic properties of one of the constituents is not sufficient to gain an understanding of the response of the muscle composite system. This motivates the need for separate experiments on the muscle fibres and the ECM, along the lines of those attempted by Mosler *et al.* (1985) and Purslow *et al.* (1998). Figure 8 shows that the response of the muscle composite system also depends on the characteristic time of a strain history relative to the characteristic relaxation time of the ECM. As the rise time *T** gets smaller relative to the relaxation time, the stress is seen to approach the response to a step mechanical strain history.

The modelling of the muscle fibre and ECM as linear viscoelastic materials led to the formulation of equation (3.19), a linear equation from which two sets of analytical results were obtained for the tensile stress in the fibre and the shear stress in the ECM at *t*=0 and as . These explicitly show how material properties affect the lateral transmission of force. Use of a nonlinear viscoelastic model would lead to replacing equation (3.19) by a system of nonlinear equations and it is unlikely that explicit analytical results would be obtained, making it difficult to discern the influence of a particular material property or its interactions with other material properties. The results in this study, therefore, using linear viscoelasticity, have set the stage for an advanced study using realistic nonlinear time-dependent material models.

## Footnotes

- Received February 3, 2009.
- Accepted April 17, 2009.

- © 2009 The Royal Society