## Abstract

Shearing is induced in soft tissues in numerous physiological settings. The limited experimental data available suggest that a severe *strain-stiffening* effect occurs in the shear stress when soft biological tissues are subjected to simple shear in certain directions. This occurs at relatively small amounts of shear (when compared with the simple shear of rubbers). This effect is modelled within the framework of nonlinear elasticity by consideration of a class of incompressible *anisotropic* materials. Owing to the large stresses generated for relatively small amounts of shear, particular care must be exercised in order to maintain a homogeneous deformation state in the bulk of the specimen. The results obtained are relevant to the development of accurate shear test protocols for the determination of constitutive properties of soft tissues. It is also demonstrated that there is a fundamental ambiguity in determining the *normal* stresses in simple shear when soft tissues are modelled as incompressible hyperelastic materials owing to the arbitrary nature of the hydrostatic pressure term. Two physically well-motivated approaches to determining the pressure are presented here, and the resulting hydrostatic stresses are compared and contrasted. The possible generation of cavitational damage owing to critical hydrostatic stress levels is briefly discussed.

## 1. Introduction

In the mechanics of rubber-like solids modelled by nonlinear elasticity theory, the classical problem of *simple shear* has played an important role as a basic canonical problem involving a *homogeneous* deformation (where the stresses are constant) that is rich enough to illustrate several key features of the nonlinear theory, most notably the presence of normal stress effects that are absent in the linear theory. Indeed, no less an authority than Truesdell (1966, p. 115) has stated that ‘simple shear is the most illuminating homogeneous static deformation’. From a practical point of view, the study of simple shear is motivated by its importance as an approximation to the deformation that arises in the relative quasi-static motion of parallel planes of elastomeric materials, which is an important mode of deformation encountered in practical applications. The hope is that the essential qualitative features of the actual stress field generated by the relative motion of parallel planes will be captured by the simple shear stress distribution in the bulk of the material.

As was remarked by Taber (2004, pp. 81–82), even though shearing is induced in soft tissues in numerous physiological settings, the study of shear deformation had received relatively little attention in the biomechanics literature compared with extension or compression, perhaps owing to the fact that testing in shear is more difficult to implement. The paper of Sacks (1999) was cited by Taber (2004) as an indication that the testing of shear properties of soft tissues was undergoing rapid development in recent years. While it is outside the scope of the present paper to discuss in detail the developments that have taken place in this area, we note that the review article by Sacks & Sun (2003) on biaxial testing concludes with a summary of the work of Dokos *et al.* (2000) on a *shear* test device that was subsequently used by Dokos *et al*. (2002) to measure shear properties of passive ventricular myocardium. Six possible modes of *simple shear* for a block of myocardium were examined by Dokos *et al.* (2002). A structurally based constitutive model of orthotropic hyperelasticity was developed by Holzapfel & Ogden (2009*a*, 2010) that reflects the basic features of these shear test results. See also Holzapfel & Ogden (2009*b*) for a discussion of planar biaxial testing of anisotropic nonlinearly elastic materials. Other examples of applications of *simple shear* to the biomechanics of soft tissues are the works of Schmid *et al.* (2006, 2008) on myocardial material parameter estimation and that of Gardiner & Weiss (2001) on human medial collateral ligaments.

Material anisotropy is an essential component of any mathematical model of soft tissue that has good predictive capability. Although transverse isotropy is the simplest type of anisotropy that can be considered, it has been widely used to model fibrous soft tissue. Some examples include the recent study of Destrade *et al.* (2008) on surface instabilities in the shearing of soft tissues, the work of Humphrey (2002) and Taber (2004) where papillary muscles are modelled, the research of Taber (2004) on myocardium and that of Ning *et al.* (2006) on brainstem.

Simple shear of transversely isotropic biomaterials will be studied here. One would intuitively expect that because of the fibres, there will be some directions in which severe resistance to shearing will be encountered. This is manifested in the simple shear experimental data of Dokos *et al.* (2002). A simple constitutive model within the framework of nonlinear elasticity will be used here to reflect this feature. It will be shown that there is a narrow band of angles of orientation of the fibres for which unlimited shear is possible (shearing in the approximate direction of the fibres) while, for *all*other angles, there is essentially the *same* limit as to the amount of shear allowed. We anticipate that this model will have a wide applicability in biological systems where shearing is a dominant mode of deformation. The constitutive model used reflects the stretch-induced stiffening of collagen fibres with increased shear loading. The corresponding strain–energy densities are of logarithmic form in the anisotropic invariant and are a viable alternative to the exponential forms usually used in biomechanical modelling of soft tissues.

Even for rubbers, accurate experimental simple shear data are difficult to obtain and some protocols have been developed to deal with these difficulties. No such protocols exist for biomaterials, as far as the authors are aware. The need for such guidance is demonstrated by our results, which indicate that large stresses are generated in sheared blocks of soft biological tissue for relatively small amounts of shear when compared with the shearing of rubber blocks and consequently large tractions must be applied on the inclined faces of the block to maintain homogeneity of deformation. These tractions are never applied in practice and thus the question arises as to the conditions under which the assumption of simple shear is appropriate when one face of a block of biological tissue is moved relative to the parallel face. Some guidance will be provided here based on shear test methods for *rubbers* and it is hoped that the results obtained will be relevant to shear test protocols for the determination of constitutive properties of soft tissues.

In §§6 and 7 of this paper, attention is drawn to some ambiguities in the formulation of simple shear for an incompressible hyperelastic material arising from the determination of the arbitrary hydrostatic pressure term in the normal stresses. A discussion of the various options for an *isotropic* material is given in our recent paper (Horgan & Murphy 2010). Here, we consider analogous issues for *transversely isotropic* materials that have been used to model the mechanical behaviour of soft biological tissues. The basic issue is the same in both settings: there is an arbitrary pressure term in the constitutive relations that must be determined from the boundary conditions. Within the context of nonlinear elasticity for *isotropic* materials, one of the earliest treatments of the deformation of simple shear (see equation (2.1) for the definition) was given by Rivlin (1948), who favoured the assumption of plane stress to the determination of the pressure. An alternative approach has been advocated by Gent *et al.* (2007), who favour the assumption of zero normal traction on the inclined faces of the sheared block. Both of these assumptions will be considered here for anisotropic materials. It is important to emphasize that, somewhat surprisingly, the normal stress distribution in simple shear for both isotropic and anisotropic materials depends on the method of determination of the hydrostatic pressure. Thus, it will be shown here that the level of hydrostatic stress depends crucially on the method used for the determination of the pressure, with the two physically well-motivated approaches adopted here sometimes yielding qualitatively different hydrostatic stresses. The accurate determination of hydrostatic stress is important because if large hydrostatic stresses are developed within the block there is the potential for cavitational damage to be generated.

## 2. Simple shear of fibre-reinforced elastic materials

The deformation known as simple shear has the mathematical representation2.1where (*X*_{1},*X*_{2},*X*_{3}) and (*x*_{1},*x*_{2},*x*_{3}) denote the Cartesian coordinates of a typical particle before and after deformation, respectively, and *κ* > 0 is an arbitrary dimensionless constant called the amount of shear. The *angle of shear* is . The usual interpretation of simple shear is *two-dimensional* where a rectangular specimen, whose dimensions are all of the same order, is deformed into a parallelogram. On assuming that the block is composed of an incompressible isotropic homogeneous nonlinear hyperelastic material, Rivlin (1948) obtained the corresponding stress distribution necessary to sustain this deformation. We follow this approach here to develop results for transversely isotropic fibre-reinforced materials. For the simple shear deformation (2.1), the deformation gradient tensor ** F**, the left Cauchy–Green strain tensor

**=**

*B*

*FF*^{T}and its inverse are given by2.2The three principal invariants of

**are defined as2.3so that for simple shear we have2.4We confine attention here to materials reinforced with one family of parallel fibres aligned at an angle**

*B**θ*to the

*X*

_{1}axis in the undeformed state (figure 1).

The unit vector in the reference configuration is transformed into in the current configuration. We introduce the usual anisotropic invariants (e.g. Spencer 1972; Holzapfel 2000)2.5which, for simple shear, have the forms2.6Here, we have introduced the convenient notation2.7and the symbol *λ* for the stretch in the fibre direction. There is no obvious physical interpretation for the symbol *α* in simple shear. In this paper, we will only be concerned with the range 0 ≤ *θ* ≤ *π*/2 so that *I*_{4} ≥ 1 and so the fibres are always in extension.

The constitutive law for the Cauchy stress ** T** for incompressible transversely isotropic hyperelastic materials with

*W*=

*W*(

*I*

_{1},

*I*

_{2},

*I*

_{4},

*I*

_{5}) is given by, for example, Ogden (2001), Horgan & Saccomandi (2005) and Holzapfel & Ogden (2009

*a*) as2.8where

*W*

_{i}=∂

*W*/∂

*I*

_{i}(

*i*=1,2,4,5) and ⊗ denotes the tensor product with Cartesian components

*a*

_{i}

*a*

_{j}. It is sufficient for our purposes here to consider the special case of equation (2.8) for strain-energies of the form

*W*(

*I*

_{1},

*I*

_{4}). The general stress–strain relation (2.8) then reduces to2.9

For simple shear, the in-plane Cauchy stresses are therefore given by2.10The out-of-plane stress is2.11where the derivatives in equations (2.10), (2.11) are evaluated at values of the invariants given by equations (2.4) and (2.6)_{1}. In equations (2.10), (2.11), the quantity *p* is the arbitrary hydrostatic pressure arising owing to the incompressibility constraint. Since the deformation (2.1) is *homogeneous*, the equilibrium equations in the absence of body forces are satisfied if and only if *p* is a constant.

As is well known, in general, surface tractions have to be applied to the inclined faces of the deformed specimen in order to maintain the deformation (2.1). On resolving these tractions into components tangential and normal to the surfaces on which they act in their deformed state (denoted by *S* and *N*, respectively) one finds that, irrespective of the constitutive law,2.12For the class of materials considered here for which the stresses are given by equation (2.10), it can be verified that2.13On using equation (2.13) in equation (2.12), we can write the tangential and normal components of the tractions on the leading and trailing edges as2.14and so2.15

## 3. Shear stress response for soft tissues

We will consider the class of strain-energy densities3.1where *F*(1)=0 and *F*′(1)=0 so that *W*=0 in the undeformed state and the stress is purely hydrostatic there. We also assume that *F*′(*I*_{4}) > 0 for *I*_{4} > 1, which ensures that the axial stress is tensile for simple extension along the fibre direction. For the model (3.1), the shear stress has the form3.2

The general class of models (3.1) has been widely used in continuum mechanics modelling of soft tissues. Perhaps the most well-known specific form of model (3.1) is the *standard reinforcing*model3.3where *E* is a non-negative material modulus that measures the *degree of anisotropy*. The model (3.3) with quadratic nonlinearity in *I*_{4} was proposed initially by Polignone & Horgan (1993*a*,*b*) in the context of cavitation problems for rubber-like materials (e.g. Horgan & Polignone 1995 for a review). It was shown in Polignone & Horgan (1993*a*) that this quadratic nonlinearity represents the *simplest polynomial form* for *F*(*I*_{4}) in the model (3.1) that satisfies the conditions on *F* listed after equation (3.1). The form (3.3) has subsequently been adopted by many authors (e.g. Qiu & Pence 1997; Merodio & Pence 2001; Merodio & Ogden 2005; Merodio *et al.* 2006 and references cited therein). The quadratic nonlinearity in model (3.3) is used to reflect the presence of oriented collagen fibres in an elastin matrix. The fibres are stiffer than the matrix but are extensible.

A number of more general choices for *F*(*I*_{4}) were considered by Horgan & Saccomandi (2005) that reflect the *stretch-induced stiffening of collagen* *fibres* as they are loaded. One of the simplest of these models is given by3.4where the *additional* parameter *J* is a positive dimensionless parameter that measures the rapidly increasing stiffness of the fibres with increasing stretch. In order for the logarithm function in the model (3.4) to be well defined, it is required that the deformation satisfy the constraint3.5or equivalently3.6where the fibre stretch *λ* is defined in equation (2.6). As is discussed by Horgan & Saccomandi (2005), the model (3.4) was motivated by the Gent model (Gent 1996) of rubber elasticity that reflects *limiting chain extensibility* of the molecular chains; see e.g. Horgan & Saccomandi (2002, 2006) for a review of such models and their applications to strain-stiffening rubber-like and biological tissues. Application of the Gent model to the mechanics of arterial walls is discussed by Horgan & Saccomandi (2003), Holzapfel (2005) and Ogden & Saccomandi (2007). The Gent model has been shown to be a viable alternative to the classical Fung exponential model that has had widespread application in the biomechanics of soft tissues. In the limit as in model (3.4), we recover the *standard reinforcing*model (3.3). The new model (3.4) has the *additional* feature of measuring the increased stiffness of the collagen fibres with deformation. The constraint (3.6) provides a value for the maximum fibre stretch (or *locking stretch*) as A generalization of model (3.4) in which the quadratic dependence on *I*_{4} is replaced by an arbitrary power-law was also proposed by Horgan & Saccomandi (2005) and was shown there to reflect the gradual *engagement* of crimped collagen fibres on loading. An application of this generalized model to soft-cuticle biomechanics is given in Lin *et al.* (2009). Another model proposed by Horgan & Saccomandi (2005) similar to model (3.4) was used by Sadovsky *et al.* (2007) to study the plant mechanics of angiosperm roots. The *isotropic* part of the model (3.4) can also be generalized to reflect increased stiffening of the elastin matrix (e.g. Ogden & Saccomandi 2007). For exposition purposes, however, we shall confine attention to the model (3.4), for which3.7where *λ* is defined in equation (2.6). Thus, regardless of the choice of the hydrostatic pressure, we see from equation (3.7) and equation (2.10) that the in-plane stresses have a singularity as , reflecting the ultimate stiffness of the fibres. On using equation (2.6), we write the constraint (3.6) as3.8Thus, for a given material parameter *J* and a given fibre angle *θ*, the constraint (3.8) restricts the amount of shear that the specimen can undergo. Specifically, *κ* must satisfy the constraint3.9
where 0 < *θ* ≤ *π*/2. The limiting shear *κ*_{m} will be called the *locking shear*.

If *θ*=0, then *I*_{4}=1, the constraint (3.8) is automatically satisfied and it is easily verified that, in this case, the results coincide with those for an isotropic neo-Hookean material. The other limiting case of *θ*=*π*/2 yields3.10and therefore, if one has accurate experimental data on shearing the material perpendicular to the direction of the fibres, the crucial parameter *J* can be easily determined. However, there are practical difficulties in obtaining such accurate data and these are discussed later.

## 4. Effects of fibre orientation

Based on the experimental work of Lawton & King (1951) on oscillations of human thoracic aorta segments, Horgan & Saccomandi (2003) proposed values of the limiting chain parameter in the isotropic Gent model. In the absence of experimental data corresponding to the new model (3.4), here, we simply use the *same* values for the parameter *J* as were proposed by Horgan & Saccomandi (2003) for the Gent model. We anticipate that this will simply provide a crude estimate for realistic values of this parameter for soft tissues. Thus, for a 21 year old male, we adopt the value *J*=2.3 while for the stiffer aorta of a 70 year old male, we take the value *J*=0.4, on rounding off the values proposed by Horgan & Saccomandi (2003) to one decimal place. Plots of the locking shear versus the angle of orientation of the fibres for these two values of *J* are given in figure 2.

The striking feature of both curves in figure 2 is their ‘L’-shaped character, which is especially notable for the smaller value of *J*. These plots reveal that the simple model (3.4) has a complex response in shearing. Specifically, this model allows virtually unrestricted shear when shearing almost parallel to the direction of the fibres and when shearing in *any* other direction there is a limit to the possible amounts of shear, with this limit being essentially the same for all directions of shearing. We note that, from equation (3.9) and figure 2, this locking shear decreases with decreasing values of *J*. This is consistent with the smaller value of *J* reflecting a stiffer tissue.

The shear stress for the model (3.4) is4.1where here, and henceforth, the superposed bar denotes stresses normalized with respect to the shear modulus *μ* and we have introduced the dimensionless parameter *γ*=*E*/*μ*. It follows from equation (4.1) that the shear stress becomes unbounded as the amount of shear approaches the locking shear value. This behaviour provides an idealization of the rapid stiffening produced by collagen fibres in soft tissues under shear load.

The effect of the fibre angle on the shear stress response will now be investigated. We choose *J*=2.3 and set *γ*=20, a value used by Destrade *et al.* (2008) reflecting experimental data of Ning *et al.* (2006) for brainstems of four week old pigs. A plot of the shear stress versus the amount of shear for different angles of orientation is given in figure 3.

The striking feature of the plots in figure 3 is that there are essentially two distinct modes of shear response: the response for fibres with angles of orientation close to 0^{°} and the response for all other angles. This mirrors the previously observed dependence of the locking shear on the angle of orientation. We note from equation (4.1) that a 0^{°} orientation is equivalent to the neo-Hookean *isotropic* response mode for the shear stress with a linear dependence on the amount of shear. Finally, we note that, *for small amounts of shear* *κ* < 0.2, the 90^{°} orientation stress response is most like the linear isotropic response corresponding to a fibre angle of 0^{°}. The corresponding plots for the smaller value of *J*=0.4 with *γ*=20 are given in figure 4. Essentially, the same qualitative features are again present.

In general, the shear modulus can be defined as4.2It follows from equation (4.1) that for the model of interest here4.3Thus, in the range 0 ≤ *θ* ≤ *π*/2, the maximum shear modulus occurs at *θ*=*π*/4 with value *μ*(1+*γ*/2) and the minima occur at *θ*=0,*π*/2 with values *μ*.

## 5. The shear stress on the inclined faces

Despite Truesdell’s assertion quoted in §1, there are few experimental data on the simple shear of rubber in the literature and even less data available for soft tissue (the most widely cited data for biological materials are those of Dokos *et al.* (2002) for passive myocardial tissue). This reflects the difficulty in performing simple shear experiments in comparison to the standard material characterization tests of simple and biaxial tension, and pure shear. An immediate practical issue that arises in designing and performing experiments based on the deformation field (2.1) is maintaining the constant dimension of the block in the plane of shear. On assuming that this issue can be resolved (a particularly elegant solution to this problem for rubber can be found, for example, in Brown (2006)), there are other difficulties that need to be overcome and these are particularly acute for soft tissue.

Although both rubbers and soft biological tissue exhibit a severe strain-stiffening effect, they are distinguished by the fact that this occurs at much lower strain values for soft tissue than for rubber-like materials (e.g. Holzapfel 2005). In the context of simple shear experiments, this can be easily seen by comparing the plots of shear stress in Andreev & Burlakova (2007) with those in Dokos *et al.* (2002). In one of the few simple shear experiments on elastomers reported in the literature, Andreev & Burlakova (2007) sheared a layer of a rubber-like polymer (plastisol) of thickness 5 mm between platens of size 7.5×4 cm, which yields a ratio of the dimensions in the plane of shearing of 1:15. This aspect ratio was used presumably to ensure homogeneity of deformation for the range of shear considered. Amounts of shear up to 1.2 were studied and it is easily seen from their fig. 2 that the shear stresses remain bounded with a maximum of only 7 kPa. By contrast, fig. 6 of Dokos *et al.* (2002) shows that some of the shear stresses reported for passive ventricular myocardium are increasing very rapidly at amounts of shear of the order 0.5. Thus, close to the values of the locking shears for soft tissue, the sizes of the shear and normal components of the traction on the inclined faces for soft tissue, necessary to maintain the homogeneous deformation (2.1), are likely to be an order of magnitude*greater* than those for rubber for the same amount of shear.

In practice, however, tractions are never applied to the inclined faces and consequently homogeneity of the deformation is lost. This loss of homogeneity of deformation has long been recognized as a problem when rubbers are sheared (e.g. Sommer & Yeoh 2001) and was also discussed by Schmid *et al.* (2008) in the context of passive myocardial material parameter estimation. To circumvent this difficulty, rectangular specimens of rubber are usually cut so that the *X*_{2} dimension is *smaller* than the other two, a ratio of at least 1:4 being recommended. We note that the experiments of Dokos *et al.* (2002) were performed on *cuboid* samples. There is no guidance in the literature as to the *range of shear stress* for which homogeneity is maintained for blocks proportioned in this way, although it seems implicit in the data and experimental protocol of Andreev & Burlakova (2007) that a range bounded above by the value of the shear modulus *s*_{m} defined in equation (4.2) is appropriate. This will be assumed here.

It was observed in §4 that there are essentially only two modes of shear stress response in the simple shear of fibre-reinforced materials modelled by equation (3.4), one of which is the neo-Hookean isotropic shear stress response corresponding to *θ*=0^{°}. Since it follows from equation (2.6)_{1} and equation (3.7) that *W*_{4}(*θ*=0^{o})=0, the shear traction on the inclined faces for fibre-reinforced materials for this angle of orientation is the same as the corresponding neo-Hookean *isotropic* shear traction given by5.1Thus, it would be expected that for specimens of aspect ratio of at least 1:4, homogeneity will be maintained for shear stresses up to the order of the shear modulus which, from equation (4.3), has the value *μ*. The shear stress response is qualitatively the same for all other fibre-orientations. Dokos *et al.* (2002) only sheared specimens for which *θ*=0, 90^{°} and consequently the shearing of specimens for which *θ*=90^{°} will only be considered further in this section. For this angle, it follows from equation (2.15)_{1} and model (3.4) that again the shear traction necessary to maintain the simple shear is given by the isotropic relation (5.1). Thus, again, it would be expected that for specimens of aspect ratio of at least 1:4, homogeneity will be maintained for shear stresses up to the order of the shear modulus *s*_{m}, which from equation (4.3) again has the value *μ*. The important difference between the two shearing modes is that the *amount of shear* *κ* for which homogeneity is preserved for *θ*=90^{°} is *significantly smaller* than the allowable range of shear for *θ*=0^{°}. This is most easily seen from the plots of shear stress given in §4. Recall that the plotted shear stress is normalized with respect to *μ* so that the maximum allowable shear stress value is 1. It is expected that homogeneity is preserved up to this stress value for properly proportioned specimens. It is seen from figures 3 and 4 that the allowable amount of shear when *θ*=90^{°} is in the approximate range (0,0.4) whereas for *θ*=0^{°} the corresponding range is (0,1).

## 6. The normal stress response

In the classical formulations of simple shear, physical intuition has motivated the choice of boundary conditions to determine the arbitrary pressure term in the normal stresses. Consider then a specimen of soft tissue whose initial configuration is defined by 0 ≤ *X*_{1} ≤ *A*,0 ≤ *X*_{2} ≤ *B*,0 ≤ *X*_{3} ≤ *C*. Intuitively, one would expect that *only* a shear traction would need to be applied on the face *X*_{2}=*B*, while keeping the face *X*_{2}=0 fixed. Therefore, it will be assumed here that the corresponding shear stress has been specified as a function of *κ*. It remains to determine the arbitrary pressure *p*. As described in detail in Horgan & Murphy (2010) for the case of an isotropic solid, this can be done in several different ways, with the ‘best’ being those that appeal both on the basis of physical intuition and mathematical simplicity. We now show that there is also the same fundamental ambiguity in determining the pressure term in the normal stresses for anisotropic materials and that different seemingly well-motivated assumptions yield qualitatively different normal stress distributions. The actual stress field generated when soft tissues are sheared is therefore open to interpretation. The usual plane stress assumption is one of the approaches adopted here to determine the pressure. The second approach assumes that the normal traction on the inclined faces of the block is identically zero.

If one were to try to reproduce simple shearing in a laboratory, one would not expect to have to apply forces in the out-of-plane direction. It follows from equations (2.1) and (2.9) that *T*_{13}=*T*_{23}=0 in agreement with this observation. If therefore, one were to insist on plane stress conditions, then it must be assumed that *T*_{33}=0 which then, on using equation (2.11), determines the pressure to be *p*=2*W*_{1}. This is the most commonly used assumption made in the literature when discussing simple shear. On substitution of this value of *p* in equation (2.10), we obtain6.1The special case of an *isotropic* material can be obtained by formally setting *W*_{4}=0 in equation (6.1). It follows from equation (2.15)_{2} that the normal component of the traction on the inclined faces is given by6.2The shear component of the traction is independent of the pressure term and is given by equation (2.15)_{1}.

An alternative approach was first suggested by Rivlin (1948, p. 394) for the isotropic case, namely that the normal component of the traction on the inclined faces should be identically zero. Motivated by an analysis of rubber shear springs, this alternative has been advocated more emphatically as a physically realistic assumption by Gent *et al.* (2007). On making this assumption for the anisotropic case of concern here, we set *N*=0 and then obtain the pressure determined from equation (2.15)_{2} as6.3On substitution of this value of *p* into equations (2.10) and (2.11), we obtain the stresses as6.4and6.5respectively. On formally setting *W*_{4}=0 in the above equations, we recover the results of Horgan & Murphy (2010) for the isotropic case. Similar to observations made there, our intuition suggests that equation (6.4) is a closer approximation to the actual stress field generated when one plane of a rectangular specimen of soft tissue is moved relative to a parallel plane than that given in equation (6.1), which is generated under plane stress conditions. Additionally, one anticipates that equation (6.4) will be very close to the real stress field in sheared soft tissue near the leading and trailing edges of the specimen. Again, we observe that in the zero normal traction formulation the shear component of the traction acting on the inclined face is given by equation (2.15)_{1}.

We observe that in both sets of in-plane stresses (6.1) and (6.4) there is a non-zero normal stress *T*_{22}, the sign of which depends on the constitutive model employed. This gives rise to the ‘normal stress effect’ characteristic of nonlinear elasticity of rubber and also well-known in the mechanics of non-Newtonian fluids.

## 7. Hydrostatic stress response for the model (3.4)

As we have seen in §6, there is a basic ambiguity involved in obtaining the forms of the in-plane normal stresses. On assuming a model of the form (3.4), the plane stress formulation (6.1) yields7.1while for the zero normal traction formulation, we find from equations (6.4) and (6.5) that7.2where we recall that *γ*=*E*/*μ*. On formally letting *γ* → 0 in equations (7.1) and (7.2) we recover results for the isotropic case obtained in Horgan & Murphy (2010) while as we obtain results for the standard reinforcing model (3.3).

Instead of comparing the individual normal stresses, it is more instructive to compare their sum, i.e. the hydrostatic stress, which has also an important physical interpretation. The classical experiments of Gent & Lindley (1958) have identified large hydrostatic stress as a significant contributor to the damage and failure of *rubbers*. The failure of isotropic rubber blocks owing to large hydrostatic stress in simple shear was recently discussed in Gent *et al.* (2007), where it was suggested that a possible mechanism of failure was *cavitation*. In cavitation, pre-existing (micro) voids grow and coalesce to form micro-level cracks and fissures once a critical level of hydrostatic pressure is reached. Increasing levels of high hydrostatic pressure then induce these micro-level imperfections to coalesce to macro-level voids and thence failure of the specimen. Since soft biological tissues are also usually modelled as being incompressible, it seems possible that the mechanisms of damage identified for rubbers could also be valid for soft tissues. High levels of hydrostatic stress have not previously been identified as a possible cause of damage of soft tissue. See, however, the recent paper by Zimberlin *et al.* (2007) where a technique of cavitation rheology testing (CRH) was proposed for hydrogels that serve as model materials for tissue scaffolds and soft biological tissues. In this work, cavitation is induced at the tip of a syringe needle and measurement of the critical hydrostatic tensile stress at which instability occurs is used to infer the material moduli. If hydrostatic stress *is* considered of critical importance in the understanding of the damage and degradation of soft tissue, our results show that there is a fundamental ambiguity in quantifying its value in simple shear. The hydrostatic stress is given by7.3so that for the plane stress formulation of the problem, we use equation (7.1) to get the normalized hydrostatic stress7.4On using equation (7.2), the zero normal traction formulation yields7.5As before the two cases of *θ*=0^{°}, 90^{°} will be taken as representative. For the plane stress formulation, the hydrostatic stress for these two orientations are given, respectively, by7.6while for the zero normal traction formulation, the hydrostatic stresses are given by7.7As before, we will consider the case where *J*=2.3 and *γ*=20 and in figure 5 we plot these hydrostatic stresses.

The first point to note from equations (7.6) and (7.7) is that, except for very small values of *κ*, both the plane stress and zero normal traction formulations predict approximately the same hydrostatic stress response when *θ*=90^{°}, and consequently only one curve is plotted. Since the *θ*=90^{°} response is typical of non-zero orientation angles, we thus conclude that the hydrostatic stress response is virtually independent of the determination of the pressure for such angles, although the stresses become unbounded at a finite amount of shear. By contrast, for angles close to 0^{°}, there is a marked difference in the hydrostatic stress response for the two formulations as can be seen from figure 5.

The experimental findings of Gent & Lindley (1958) agree well with a theoretical prediction that the critical load for cavitation in *isotropic neo-Hookean* elastic rubber spheres under dead loading is 5/2 (recall that all stresses have been normalized with respect to the infinitesimal shear modulus). See the review article by Horgan & Polignone (1995) for details. This result has been used in implementation of the CRH technique for hydrogels proposed by Zimberlin *et al.* (2007). Although the effects of *anisotropy* on the critical cavitation load have also been investigated *analytically* by several authors for fibre-reinforced models of the type considered here (e.g. Polignone & Horgan 1993*a*,*b*; Horgan & Polignone 1995; Horgan & Saccomandi 2005; Merodio & Saccomandi 2006), there are no experimental data known to the authors that provide critical loads for cavitation in soft tissue. In the absence of experimental evidence of cavitation in such tissues, it is hypothesized here, based on the plots in figure 5, that *if* cavitational damage were to occur in *simple shear*, only small amounts of shear would be possible before damage would occur for non-zero angles of orientation and in particular for *θ*=90^{°}. The picture is much less clear-cut if one shears a block with angles close to *θ*=0^{°}. Since, for a fixed amount of shear, the hydrostatic stress for the zero normal traction formulation is *larger* than that for the plane stress formulation, it appears that cavitational damage is more likely to occur if the former formulation is adopted. Since this formulation presumably has greater validity in a boundary layer close to the leading and trailing edges, our analysis thus predicts that cavitation should occur in these layers *first* before appearing in the bulk of the specimen. The presence of boundary-layer effects in simple shear was emphasized by Gent *et al.* (2007) within the context of the simple shearing of rubbers.

## Acknowledgements

The work of C.O.H. was supported by the US National Science Foundation under grant CMMI 0754704. This research was completed while this author held a Science Foundation Ireland E. T. S. Walton Fellowship at Dublin City University. We are grateful to the reviewers for their constructive remarks on an earlier version of the manuscript.

- Received June 7, 2010.
- Accepted August 3, 2010.

- This journal is © 2010 The Royal Society