The importance of the second strain invariant in the constitutive modeling of elastomers and soft biomaterials

Mechanics of Materials 51 (2012) 43–52

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The importance of the second strain invariant in the constitutive modeling of elastomers and soft biomaterials Cornelius O. Horgan ⇑, Michael G. Smayda School of Engineering and Applied Science, University of Virginia, Charlottesville, VA 22904, USA

a r t i c l e

i n f o

Article history: Received 30 August 2011 Received in revised form 24 February 2012 Available online 29 March 2012 Keywords: Constitutive models Incompressible isotropic materials Second strain invariant Elastomers and soft biomaterials Fung–Demiray and Vito strain-energy densities

a b s t r a c t The classical phenomenological constitutive modeling of the mechanical behavior of isotropic incompressible rubber-like hyperelastic materials involves strain-energy densities that depend on the first two principal invariants of the strain tensor. For rubber, the most wellknown of these is the Mooney–Rivlin model which has a linear dependence on the two principal invariants and its specialization to the neo-Hookean form which is independent of the second invariant. While each of these models provides a reasonably accurate prediction for the mechanical behavior of rubber-like materials at small stretches, they fail to reflect the strain-stiffening that is observed as the stretch increases. For soft biomaterials, an exponential dependence on the strain-invariants is well known to capture this predominant stiffening effect. The most celebrated of these models is that of Fung and Demiray which depends only on the first strain invariant. In the limit as the strain-stiffening parameter tends to zero, one recovers the neo-Hookean model. A modification of the Fung–Demiray (FD) model that also depends on the second invariant was proposed by Vito. For the Vito model, one recovers the Mooney–Rivlin model as the strain-stiffening parameter tends to zero. It is well known that, in general, inclusion of a dependence on the second invariant models the stress response of rubber-like materials more accurately. More importantly, in the solution of some basic problems involving homogeneous and inhomogeneous deformations, constitutive models that do not include a dependence on the second invariant fail to capture some significant physical effects. These issues for elastomers and soft biomaterials are addressed here and explicitly illustrated by using the FD and Vito models. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction The mechanical modeling of rubber-like materials within the framework of nonlinear elasticity theory is well established. The application of such modeling to soft biomaterials is currently the subject of intense investigation. Numerous constitutive models for incompressible vulcanized rubber have, of course, been developed and widely applied. The kinetic theory of rubber leads to the neoHookean model, for which the strain-energy density per unit-undeformed volume is a linear function of the first ⇑ Corresponding author. Tel.: +1 434 924 7230; fax: +1 434 982 2951. E-mail addresses: [email protected] (C.O. Horgan), smayda@virginia. edu (M.G. Smayda). 0167-6636/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechmat.2012.03.007

strain invariant. A modification of this model to include an additive term linear in the second invariant is the Mooney–Rivlin model (see e.g., Atkin and Fox, 1980; Beatty, 1987; Ogden, 1984). These two classical models are reasonably accurate in predicting the material response of rubber-like materials at small ranges of stretch. However, it is well known that, for larger values of stretch, these classical models fail to predict the strain-stiffening that is observed experimentally. Phenomenological strainstiffening models for rubber have, of course, been developed by many authors, e.g., power law models such as the six parameter model of Ogden (see e.g., Ogden, 1984) and the three parameter model of Knowles (1977). The development of constitutive models that reflect the foregoing phenomena has progressed rapidly in recent years (see

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C.O. Horgan, M.G. Smayda / Mechanics of Materials 51 (2012) 43–52

e.g., the review articles by Boyce and Arruda (2000) and Horgan and Saccomandi (2006)). However, for soft biomaterials, and in particular for soft tissues, exponential models have been shown to be particularly useful. Viable alternatives are strain-energies with a logarithmic dependence on the strain invariants that lead to a locking stretch, for example the Gent model (1996) originally developed for rubber but also appropriate for soft tissues (see e.g., Horgan and Saccomandi, 2002, 2003, 2006). Soft tissues are known to undergo strain-stiffening at much smaller values of stretch than is the case for rubber (see e.g., Humphrey, 2002; Holzapfel, 2005 for illuminating discussions). It is well known that, in general, inclusion of a dependence on the second invariant models the stress response of rubber-like materials and soft biomaterials more accurately. More importantly, in the solution of some basic problems involving homogeneous and inhomogeneous deformations, constitutive models that do not include a dependence on the second invariant fail to capture some significant physical effects (see e.g., Wineman, 2005 and references cited therein). In this paper, as illustrative models to demonstrate the significance of inclusion of the second invariant, we consider two of the simplest strain-stiffening models that have exponential dependence on the strain invariants, namely the celebrated FD model due to Fung (1967) and Demiray (1972) and the generalization proposed by Vito (1973). The first of these models, which depends only on the first strain-invariant, involves just two constitutive parameters, namely the shear modulus at infinitesimal deformations and a dimensionless parameter measuring the extent of strain-stiffening. The second is a three parameter model that has an additional parameter reflecting the degree of dependence on the second strain invariant. Our objective in this paper is to compare the stress response of both models in some basic homogeneous and inhomogeneous deformations in order to illustrate the effects of inclusion of the second invariant in the constitutive modeling of soft biomaterials. The outline for this paper is as follows. In Section 2, we describe in detail the two specific constitutive models mentioned above. In Section 3, we examine the stress response for the basic homogeneous deformations of simple extension, equi-biaxial extension and simple shear and compare results with emphasis on examining the effects of including the dependence on the second invariant. It will be seen that the FD and Vito models predict a qualitatively similar stress response in simple extension and equi-biaxial extension. However, in simple extension, for a fixed value of the strain-stiffening parameter the former model predicts a stiffer stress response especially noticeable at small values of the strain-stiffening chain parameter. Conversely, for equi-biaxial extension, the character of the relative response is reversed. In simple shear, the shear stress response for both models is identical. However, the normal stress perpendicular to the direction of shear is zero for the FD model while this is not the case for the Vito model. This component of stress plays an important role in experimental protocols for assessing the stress response of soft tissues in shear. In particular, this stress component plays a key role in the occurrence of a Poynting-type effect in simple

shear. In Section 4, we discuss some problems involving inhomogeneous deformations where inclusion of the second invariant also has a major influence on the stress response. As an illustrative example, results for the torsion of a solid circular cylinder are described. 2. The Fung and Vito constitutive models Here we describe the two constitutive models of primary concern and examine the predicted response for the basic homogeneous deformations of simple extension, equi-biaxial extension and simple shear. First we recall the kinematic quantities (see e.g., Beatty, 1987; Ogden, 1984 or Taber, 2004)

F ij ¼

@xi ; @X j

B ¼ F FT ;

I3 ¼ det B;

I1 ¼ tr B;

I2 ¼ I3 trðB1 Þ;

J ¼ det F;

ð1Þ

where Fij, B, J, I1, I2, and I3 are the deformation gradient, left Cauchy–Green tensor, the Jacobian, and the first, second, and third strain-invariants respectively. The physical interpretation of the third invariant is immediate since I3 = J2 and so I3 provides a measure for the change in volume. The first invariant provides a measure for the average over all possible orientations of three times the squared stretches of an infinitesimal line element while the second invariant measures the corresponding quantity for an area element (see e.g., Kearsley, 1989 for a discussion). The strain-energy density function per unit undeformed volume for an incompressible isotropic elastic material is given as W = W(I1, I2) where det F = 1 so that I3 = 1. The neo-Hookean strain-energy

W NH ðI1 Þ ¼

l 2

ðI1  3Þ;

ð2Þ

is the simplest model for non-linear elastic behavior of isotropic incompressible materials. Useful at small stretches, the neo-Hookean model involves a single constitutive constant (the shear modulus l for infinitesimal deformations) and has a linear dependence on the first strain-invariant. A classic modification of (2) is the two constant Mooney– Rivlin model

W MR ðI1 ; I2 Þ ¼

l 2

½aðI1  3Þ þ ð1  aÞðI2  3Þ;

ð3Þ

where the non-dimensional parameter a is such that 0 < a 6 1. When a = 1 in (3), one recovers (2). The Fung–Demiray (FD) strain-energy

W FD ðI1 Þ ¼

l 2b

ebðI1 3Þ  1



ð4Þ

was proposed by Demiray (1972) following the seminal development by Fung (1967), where the non-dimensional positive constant b provides a measure of the strainstiffening characteristic of soft biomaterials. In the limit as b ? 0 in (4), one recovers the neo-Hookean model (2). The two constant FD model involves the shear modulus l and the dimensionless parameter b. Note that WFD depends only on the first strain-invariant and so belongs to the class of generalized neo-Hookean materials.

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C.O. Horgan, M.G. Smayda / Mechanics of Materials 51 (2012) 43–52

  1 bðk2 þ2k1 3Þ e T 11 ¼ l k2  k

The three constant Vito model

W V ðI1 ; I2 Þ ¼

l 2b

 eb½aðI1 3Þþð1aÞðI2 3Þ  1 ;

ð5Þ

was proposed by Vito (1973) as a generalization of (4) to include dependence on the second invariant. The dimensionless parameters a and b have the same interpretation as before. When a = 1 in (5), we recover (4). In the limit as b tends to zero in (5), one recovers the Mooney–Rivlin model (3). The model (5) has received comparatively little attention in the literature on biomaterials modeling compared with the classic model (4). The response of an incompressible isotropic elastic material can be determined by applying the standard constitutive law (see e.g., Atkin and Fox, 1980; Ogden, 1984; Beatty, 1987; Taber, 2004)

T ¼ p1 þ 2

@W @W 1 B2 B ; @I1 @I2

ð6Þ

where p is a hydrostatic pressure term associated with the constraint I3 = 1 and T denotes the Cauchy stress. For the Vito model (5), we have

@W V la b½aðI1 3Þþð1aÞðI2 3Þ ¼ ; e @I1 2 @W V lð1  aÞ b½aðI1 3Þþð1aÞðI2 3Þ e ¼ ; 2 @I2

ð7Þ

with corresponding results for the FD model following on setting a = 1. Thus from (6) we get

T FD ¼ p1 þ lebðI1 3Þ B; T V ¼ p1 þ ½aB  ð1  aÞB1 leb½aðI1 3Þþð1aÞðI2 3Þ ;

ð8Þ

respectively. It is sometimes convenient to use a nominal stress based on the reference configuration of the body. This stress (first Piola–Kirchhoff stress) in the case of incompressible materials is given by

S ¼ F 1 T:

ð9Þ

3. Response in basic homogeneous deformations 3.1. Simple extension In simple extension, the deformation of a rectangular block whose long sides are along the X1-direction is given by

x1 ¼ kX 1 ;

1

x2 ¼ k2 X 2 ;

1

x3 ¼ k2 X 3 ;

k > 1;

ð10Þ

and one has

2 I1 ¼ k2 þ ; k

I2 ¼

1 k2

þ 2k;

I3 ¼ 1:

ð11Þ

It is well known (see e.g., Atkin and Fox, 1980) that the axial stress T11 in simple extension of an isotropic incompressible hyperelastic material is given by

   1 @W 1 @W ; T 11 ¼ 2 k2  þ k @I1 k @I2

ð12Þ

where the derivatives of W are evaluated at the values of the strain invariants given in (11). Thus, for the FD model, we have

ð13Þ

while for the Vito model, we have

     1 1 þ ð1  aÞ k  2 T 11 ¼ l a k2  k k  eb½aðk

2

þ2k1 3Þþð1aÞðk2 þ2k3Þ

:

ð14Þ

When a = 1, the stresses T11 given in (14) reduce to (13). In the limit as b ? 0 in (13) and (14) one recovers the wellknown stresses for the neo-Hookean and Mooney–Rivlin models respectively. In Fig. 1, we plot the non-dimensional Cauchy stress T11/l in simple extension for the two models under consideration. In this figure, the values chosen for the stiffening parameter are b = 1.0, 1.7 and 5.0. For the thoracic artery of a 21-year old male, it was shown in Horgan and Saccomandi (2003) that b  1 when the FD model is fitted with experimental data while for a 70-year old male one obtains b  5.5, the larger value reflecting the higher degree of strain-stiffening of the older artery. Values of this parameter for the FD model of similar order are given in Ho and Kleiven (2007) for human cerebral veins (b  1.7) and cerebral arteries (b  4.4). In a recent paper by Destrade et al. (2009), tensile tests on a pig thoracic aorta were described where a value of b  1.3 was found to fit well with the FD model (see Fig. 1 of Destrade et al., 2009). A comparable value of b  1.5 for hydrated pig muscle was obtained by Martins et al. (2006). Thus the range of values chosen in Fig. 1 for b reflect a typical range for soft tissues and the solid curves show the response for the FD model. Except for some data on the aorta of a dog given in Vito (1973) where it is proposed that b = 0.325, a = 0.932, we are unaware of comparable detailed experimental data that has been used to fit the Vito model. Furthermore, to compare the responses of both models, it makes sense to use the same values of b as were used above for the FD model. Values of the additional parameter in the Vito model are chosen as a = 0.25, 0.5, and 0.75 respectively. The degree of dependence on the second invariant is measured by this parameter, where the smallest of these values of a reflects the largest influence of this invariant. We have not attempted to choose these values to match any experimental data on particular biomaterials so the parameter values chosen above are merely representative. Clearly a = 0.5 corresponds to equal dependence on each invariant in (5) and the other values of a represent symmetric departures from this equi-dependence. From Fig. 1, it is clear that for a given value of the strainstiffening parameter b, the FD model predicts a stiffer stress response than the Vito model especially noticeable at the smaller values of the strain-stiffening parameter. Consider, for example, the far right dashed red curve corresponding to b = 1 and a = 0.25 where this value of a reflects the largest dependence on the second invariant for this value of b. At a stretch of 1.8, we see from Fig. 1 that T11/l = 5 for the Vito model whereas T11/l = 10 for the FD model. Thus there is approximately a 100% difference between the predictions of the models at this value of stretch. For larger values of b, such a comparison at a given stretch would yield an even greater difference. Also notice from Fig. 1

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C.O. Horgan, M.G. Smayda / Mechanics of Materials 51 (2012) 43–52

Fig. 1. Cauchy stress response in simple extension: comparison between models. The red, green and blue curves (colors, ordered from right to left, refer to online version) correspond to the values b = 1.0, 1.7 and 5.0 respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 2. Each curve shown provides a parameter pair (b, a) in the Vito model for which the stress response in simple extension coincides with that of the FD model for b = 1.0, 1.7 and 5.0.

that for b = 1.0 in the FD model and b = 1.7, a = 0.25 in the Vito model the stress responses are virtually identical. Thus, given an FD model with prescribed value of b, one can find parameters in the Vito model so that the stress responses are virtually coincident. This is further illustrated in Fig. 2 where for each value of b chosen for the FD model in Fig. 1, denoted by bFD in Fig. 2, each curve plotted in the (b, a) plane provides a parameter pair for the Vito model for which the stress responses coincide. As we shall see at the end of this Section and in Section 4, models that include a dependence on the second invariant in general predict

more realistic physical behavior so that use of the Vito model to fit experimental data is, in general, preferable to using the FD model. We remark that one of the motivations given in Vito (1973) for the model (5) was that this model provides a better fit than the FD model to experimental data in simple extension for a dog aorta. On using (9), the corresponding plots for the nominal stress can be generated but we shall not include these here. The results for the nominal stress are given in Horgan and Smayda (2012) and applied there to the analysis of the trousers test for tearing of soft biomaterials.

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C.O. Horgan, M.G. Smayda / Mechanics of Materials 51 (2012) 43–52

3.3. Simple shear

3.2. Equi-biaxial extension

Simple shear is an isochoric deformation defined by

Here we have

x1 ¼ kX 1 ;

x2 ¼ kX 2 ;

x3 ¼ k2 X 3 ;

ð15Þ

so that

I1 ¼ 2k2 þ k4 ;

I2 ¼ k4 þ 2k2 :

ð16Þ

In this case, we find that T11 = T22 = T, where

  1 2 4 T ¼ l k2  4 ebð2k þk 3Þ ; k

 eb½að2k

þk

4

4

3Þþð1aÞðk þ2k

2

3Þ

;

x2 ¼ X 2 ;

x3 ¼ X 3 ;

ð19Þ

where k > 0 is the amount of shear in the X1  X2 plane. The angle of shear is arctan (k). For the case of simple shear, the well-known stress field (see e.g., Atkin and Fox, 1980; Horgan and Murphy, 2010) for an incompressible isotropic elastic material is given by

2

ð17Þ

3 2 2kðW 1 þ W 2 Þ 0 2k W 1 6 7 T ¼ 4 2kðW 1 þ W 2 Þ 2k2 W 2 0 5; 0

     1 1 T ¼ l a k2  4 þ ð1  aÞ k4  2 k k 2

x1 ¼ X 1 þ kX 2 ;

ð18Þ

for each of the two models respectively. These results are plotted in Fig. 3 for the same values of the parameters that were used in Fig. 1. On comparing Figs. 1 and 3, we see that each of the models capture the intuitive result that the extensibility of soft tissue is smaller under equi-biaxial stretching than in simple extension (see e.g., Boyce and Arruda, 2000; Gent, 2005 for a discussion of this effect in rubber). Also from Fig. 3, in contrast to the situation found for simple extension in Fig. 1, it is clear that for a given value of the strain-stiffening parameter b, the Vito model predicts a stiffer stress response in equi-biaxial extension. We observe also in Fig. 3 a similar feature to that found in Fig. 1 namely that for appropriate choices of the parameters, in this case for b = 1.7 in the FD model and b = 1.0, a = 0.50 in the Vito model, the response for both models are virtually identical. A detailed review of biaxial testing of soft biomaterials is given in Sacks (2000) and Sacks and Sun (2003).

0

ð20Þ

0

where the subscripts on W denote differentiation with respect to the corresponding invariants and these invariants are evaluated at I1 = I2 = 3 + k2. For the FD and Vito models we thus obtain

2 2

T ¼ lebk

2

k 6 4k

0

3 k 0 7 0 0 5;

ð21Þ

0 0

and

2 2

T ¼ lebk

3 2 k a k 0 6 7 2 4k k ð1  aÞ 0 5; 0

0

ð22Þ

0

respectively. We see from (21) and (22) that the shear stress for both models coincides and is bk2

T 12 ¼ lke

:

ð23Þ

This result can also be deduced from (20) and the use of (7) to get

Fig. 3. Cauchy stress response in equi-biaxial extension: comparison between models. The red, green and blue curves (colors, ordered from right to left, refer to online version) correspond to the values b = 1.0, 1.7 and 5.0 respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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C.O. Horgan, M.G. Smayda / Mechanics of Materials 51 (2012) 43–52

W V1 þ W V2 ¼

l 2

eb½aðI1 3Þþð1aÞðI2 3Þ ¼

l 2

ebðI1 3Þ ¼ W FD 1 ;

ð24Þ

where we have used I1 = I2 for the deformation at hand. In the limit as b ? 0 in (23), i.e. for the neo-Hookean and Mooney–Rivlin models, we obtain T12 = lk which exhibits a linear dependence on the amount of shear. This is also the result one has in linear elasticity. However, unlike linear elasticity, normal stresses are necessary to maintain a state of simple shear in nonlinear elasticity as can be seen directly from (20). In fact, these normal stresses reflect many of the interesting and significant features of nonlinearity. From (21) and (22), we see that a normal stress T11 occurs for each of the models of concern which in the limit as b ? 0, i.e. for the neo-Hookean and Mooney–Rivlin models is T11 = lk2 and T11 = lak2 respectively. More interesting is the normal stress component T22 which represents the confining traction at the top and bottom faces of the specimen under shear that are necessary to maintain the simple shearing deformation (19). From (20), it is clear that such a component of traction is predicted to be necessary only for models that have a dependence on the second invariant. If no such traction is applied, a specimen under shear will in general tend to exhibit a lateral expansion or contraction perpendicular to the direction of shear. In rubber elasticity this well-known phenomenon is called a Poynting-type effect and has been observed in experiments. An analogous effect for soft materials has been reported in Janmey et al. (2007). For further discussion, see Horgan and Murphy (2010, 2011a,b). Thus materials that exhibit such an in-plane lateral expansion or contraction under simple shear cannot be realistically modeled by isotropic strain-energies of the form W = W(I1). In particular, the FD model does not reflect a Poynting-type effect in simple shear. In contrast, for the Vito model we find from (22) that a compressive traction 2

2

T 22 ¼ lk ð1  aÞebk

ð25Þ

must be applied to the top and bottom faces of the specimen in order to maintain simple shear. In the limit as b ? 0 in (25), one recovers the well-known result that T22 = lk2(1  a) for the Mooney–Rivlin model (3). Of course, the deformation (19) of simple shear represents an idealization of the complex inhomogeneous deformation that occurs when an elastomer bonded between two parallel planes is deformed by the relative displacement of the confining planes (see e.g., Horgan and Murphy, 2010, 2011a,b, 2012a and references cited therein). Indeed, as remarked in the concluding sentence of Horgan and Murphy (2010), ‘‘simple shear is not so simple’’. In particular, the homogeneous deformation (19) is likely to be valid only near the top and bottom boundary planes. The onset of instabilities such as out-of-plane buckling in the case of thin samples might also need to be taken into account. Such considerations are beyond the scope of the present paper. We refer to Sommer and Yeoh (2001) and Brown (2006) for a discussion of typical experimental protocols for shear testing of elastomers. Even though shearing is induced in soft tissues in numerous physiological settings, until quite recently the study of shear deformation had received relatively little attention in the biomaterials literature com-

pared to extension or compression perhaps due to the fact that testing in shear is more difficult to implement. The review article by Sacks and Sun (2003) on biaxial testing concludes with a summary of 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. 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, that of Gardiner and Weiss (2001) on human medial collateral ligaments and that of Guo et al. (2007) on porcine skin. The latter reference in particular points out the difficulty in maintaining the assumed homogeneous state (19) in shearing of biomaterials. See Horgan and Murphy (2011a,b) for a recent study of simple shearing of soft tissues where material anisotropy is also taken into account. In these references, a commonly used continuum mechanics based model for such tissues is used that models the matrix elastin material as isotropic which is reinforced by collagen fibers giving rise to overall anisotropy. For such models, inclusion of the second invariant in the isotropic matrix phase was shown by Horgan and Murphy (2011b) to play a key role in determination of the character of the transverse normal stress component T22. It was shown that this stress can be compressive or tensile depending on the degree of anisotropy and the angle of orientation of the collagen fibers. In the latter case, in the absence of such a tensile confining traction, the sample would tend to contract inwards perpendicular to the direction of shear thus exhibiting a reverse Poynting-type effect. Such behavior was observed experimentally by Janmey et al. (2007) in simple shear of semi-flexible biopolymer gels. We have seen in the preceding that the second invariant plays a particularly important role in the deformation of simple shear. Some other homogeneous deformations where consideration of the second invariant is relevant are described in Wineman (2005). These include unequal biaxial extension in plane stress and simple shear superimposed on triaxial extension. In the next section we describe some results involving non-homogeneous deformations that arise in problems of biomechanical relevance where consideration of the second invariant is also particularly significant.

4. Non-homogeneous deformations Non-homogeneous deformations are those for which the stresses vary with position and the solution of such problems in general requires consideration of nonlinear differential equations. For a variety of problems involving such deformations, the pioneering work of Rivlin furnished solutions for all incompressible isotropic materials (see Barenblatt and Joseph, 1997 for the collected works of Rivlin). Here we briefly describe a representative problem of this type that is pertinent to the mechanical behavior of soft biological tissues and explicitly illustrate the results for the FD and Vito models. We will focus attention primarily on the classic problem of pure torsion of a solid circular cylinder composed of an incompressible isotropic hyperelastic material by

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C.O. Horgan, M.G. Smayda / Mechanics of Materials 51 (2012) 43–52

an applied twisting moment at its ends. This problem has been extensively studied in the literature on rubber elasticity both from a theoretical and experimental viewpoint (see e.g., Atkin and Fox (1980) for a concise summary of some of this work). One of the most interesting features of the solution of this problem is that a compressive resultant axial force proportional to the squared amount of twist must be applied to the ends of the cylinder in order to maintain pure torsional deformation. In the absence of such a force, the cylinder would elongate on twisting. In rubber elasticity, this is the celebrated Poynting effect that has been observed experimentally for isotropic materials. It is an inherently nonlinear effect. The pure torsion problem is concerned with a long solid circular cylinder of radius A subjected to a twist at its ends. On using cylindrical coordinates (R, H, Z) in the undeformed configuration and (r, h, z) in the current configuration, one has

h ¼ H þ sZ;

r ¼ R;

z ¼ Z;

ð26Þ

where s denotes the twist per unit length. Classical results of Rivlin (see Barenblatt and Joseph, 1997) yield the non-zero Cauchy stresses as (see e.g., Kanner and Horgan, 2008a)

Z

T rr ¼ 2s2

sW 1 ðsÞds; R

T hh ¼ 2s

A

Z

A

ð27Þ

sW 1 ðsÞds  2s2 R2 W 2 ;

ð28Þ

R

T zh ¼ 2sRðW 1 þ W 2 Þ;

ð29Þ

where the subscript on W denotes differentiation with respect to the corresponding invariant and these derivatives are evaluated at 2 2

I1 ¼ 3 þ s R ;

2 2

I2 ¼ 3 þ s R :

ð30Þ

The resultant applied moment and axial force necessary to maintain the deformation are

M¼

Z 2p Z 0

A

Z

T zh R2 dRdh ¼ 4ps

0

A

R3 ðW 1 þ W 2 ÞdR;

ð31Þ

0

T FD hh

ð34Þ

and it is easily verified that for the Vito model one has

la 

2 2 2 2 ebs R  ebs A ; 2 2 i la h 2 2 ¼ 1 þ 2bs2 R2 ebs R  ebs A : 2b

T Vrr ¼ T Vhh

2b

ð35Þ

ð36Þ

as can be seen directly from (27) and (28) whereas this is not the case when the more general model including a dependence on the second invariant is considered. Thus the axial stress for the FD model is identical to the first of (34) while for the Vito model, one finds that

T Vzz ¼

la  2b

ebs

2 R2

 ebs

2 A2



2 R2

 lð1  aÞs2 R2 ebs

:

ð37Þ

The shear stress is identical for both models and is

T zh ¼ lsRebs

2 R2

ð38Þ

:

The argument given in connection with (24) for simple shear can be used here in (29) to verify this equivalence. From (31), or directly from (38) we get

M¼

i lp h 2 2 bs2 A2 bs A  1 e þ1 2 3 b s

ð39Þ

for both models while (32) yields

and

N¼

2 2 2 2 ebs R  ebs A ; 2b 2 2 i l h 2 2 ¼ 1 þ 2bs2 R2 ebs R  ebs A ; 2b

T rr ¼ T zz

sW 1 ðsÞds þ 2s2 R2 W 1 ;

R

T zz ¼ 2s2

l

T FD rr ¼

As pointed out in Kanner and Horgan (2008a), for generalized neo-Hookean materials one has

A

Z

2

neo-Hookean materials. As was discussed in Horgan and Saccomandi (1999) and in Wineman (2005), experimental data for a particular hyperelastic material can be used to check if (33) holds and thus determine whether a dependence on the second invariant is necessary to model that material. For example, it is pointed out in the aforementioned references that (33) is not satisfied in experiments on cylinders composed of sulphur vulcanized natural rubber and peroxide vulcanized natural rubber. For a given soft biomaterial, violation of (33) in torsional testing would imply in particular that a FD model is not appropriate. The local stresses (27)–(29) and the global quantities (31) and (32) for the FD model are given in Kanner and Horgan (2008a). It was found that

Z 2p Z 0

A

0

2

T zz RdRdh ¼ 2ps

Z

A

NFD ¼

3

R ðW 1 þ 2W 2 ÞdR; 0

ð32Þ respectively. The normal stresses (27) do not depend on the second invariant while the axial stress (28) and its resultant (32) clearly do. For generalized neo-Hookean materials W = W(I1), it was shown by Horgan and Saccomandi (1999) (and as can be seen immediately from (31) and (32) that

N ¼ sM=2:

ð33Þ

This result does not depend on the strain-energy density and is a global universal relation valid for all generalized

i 2 2 lp h 1  bs2 A2 ebs A  1 2 2 2b s

ð40Þ

for the FD model and

NV ¼

i 2 2 lpð2  aÞ h 1  bs2 A2 ebs A  1 : 2 2 2b s

ð41Þ

for the Vito model (see also Eqs. (4.31) and (4.32) in Kanner and Horgan (2008a) for the FD results). From (39) and (41), we see that

NV ¼ 

sð2  aÞ 2

M:

ð42Þ

In contrast with (33), this is not a universal relation as it depends on the constitutive parameter a.

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From (40) and (41) it can be easily verified that these resultant axial forces are negative i.e. compressive and that

NV ¼ ð2  aÞNFD :

ð43Þ

This relationship could also have been deduced from (32) on using (24). (We remark that a relationship identical to (43) also occurs between the axial forces for the Mooney–Rivlin and neo-Hookean models). The very simple relationship (43) allows one to immediately assess the effect of including a dependence on the second strain invariant. Since 0 < a 6 1, we find from (43) that

NV P NFD ;

ð44Þ

so that for given material parameters b and l and total angle of twist sA, the magnitude of the resultant axial force necessary to maintain pure torsion for the Vito model is larger than that for the FD model. In Fig. 4, we plot the non-dimensional resultant applied moment M/lpA3 versus the total angle of twist for the same values of b as used in Section 3. The compressive non-dimensional resultant axial forces N/lpA2 necessary to maintain pure torsion are plotted in Fig. 5 for the same values of the parameters b and a as used in Section 3. As remarked earlier, in the absence of such a force, the cylinder would elongate on twisting. Thus the usual Poynting effect occurs for both models. This is in contrast with the situation for simple shear described in Section 3 where a Poynting-type effect did not occur for the FD model. A generalization of the foregoing problem is that of torsion superimposed on axial extension. Results analogous to (27) and (32) for this problem are given in Kanner and Horgan (2008a) and specialized for the FD and Gent models and a generalization of the Gent model that includes a dependence on the second invariant. The influence of the axial stretch is quite complex in these cases. Comparable results for the Vito model may be readily obtained but

we shall not pursue the details here. The problems of torsion and of torsion superimposed on extension have application, for example, to the response of papillary muscles of the heart. Papillary muscles are approximately cylindrical extensions of the ventricle and play an essential role in the operation of both the mitral and tricuspid valves in the heart. Coupled with the obvious contraction and expansion of ventricular walls, there is an associated torsional effect, where the valvular plane is twisted relative to a plane through the base of the ventricle. There is considerable interest in this twisting motion of the ventricular walls as an indicator of cardiac disease (see e.g. Tibayan et al., 2002). In vivo observations by Gorman et al. (1996) of papillary muscles confirm that the papillary muscle is deformed in the same way. The use of nonlinear elasticity in modeling the passive response of papillary muscles in torsion is described, for example, in the books of Humphrey (2002) and Taber (2004), in the work of Humphrey et al. (1992) and in the recent paper by Horgan and Murphy (2012b). Many other problems involving inhomogeneous deformations have been investigated where the second invariant has been shown to play a significant role. For example, a membrane inflation problem with implications for material identification in biomaterials was considered by Wineman (2005). The shearing (axial or circular) of a hollow circular cylinder composed of strain-stiffening rubber-like materials is investigated in Kanner and Horgan (2008b). The effect of the second invariant in a related circular shear problem has been examined by Wineman (2005). Shearing problems are relevant to cardiovascular mechanics (see e.g., Humphrey, 2002; Taber, 2004). For example, long cylindrical specimens are readily excised from papillary muscles and have been important sources for biomechanical data. It is noted in Taber (2004) that transverse shear of the heart wall may play a role in proper functioning of the left ventricle. The bending of a rectangu-

Fig. 4. Non-dimensional resultant applied moment versus the total angle of twist for both the FD and Vito models (ordered from right to left in order of increasing b).

C.O. Horgan, M.G. Smayda / Mechanics of Materials 51 (2012) 43–52

51

Fig. 5. Non-dimensional resultant axial forces necessary to maintain pure torsion: comparison between models. The red, green and blue curves (colors, ordered from right to left, refer to online version) correspond to the values b = 1.0, 1.7 and 5.0 respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

lar strip composed of strain-stiffening materials including soft tissues has been investigated in Kanner and Horgan (2008c) and some results pertaining to inclusion of the second invariant are described there. It was pointed out in Taber (2004) that such bending problems arise, for example, in cardiac looping of an embryonic heart. 5. Concluding remarks In this paper, we have drawn attention to the importance of including the second invariant in constitutive modeling of elastomers and soft biomaterials within the framework of the theory of nonlinear elasticity for incompressible isotropic materials. Even within this simple framework, we have seen that the second invariant plays a significant role in both homogeneous and non-homogeneous deformation problems. While some soft biomaterials, for example, brain and liver tissue and biogels have been shown to be isotropic (see e.g., Pervin et al., 2011; Janmey et al., 2007; Dobrynin and Carrillo, 2011) it is also well known that many other soft tissues such as arterial walls, skin and tendons demonstrate anisotropic behavior reflecting their fibrous nature. As pointed out at the end of Section 3, a commonly used continuum mechanics based model for such tissues is to model the matrix elastin material as isotropic which is reinforced by collagen fibers giving rise to overall anisotropy thus requiring consideration of additional invariants reflecting anisotropy (see e.g. Horgan and Murphy, 2011a,b and references cited therein). In such models, inclusion of the second invariant in the isotropic matrix phase retains the significance demonstrated here. For example, in the context of simple shear of fibrous biomaterials, it was shown in Horgan and Murphy (2011b) that, for a class of transversely isotropic models with isotropic matrix phase modeled by the Mooney–Rivlin form (3), the transverse normal stress can

be compressive or tensile depending on the degree of anisotropy and the angle of orientation of the collagen fibers. In the latter case, in the absence of such a tensile confining traction, the sample would tend to contract inwards perpendicular to the direction of shear thus exhibiting a reverse Poynting-type effect. Such behavior was observed experimentally by Janmey et al. (2007) in simple shear of semi-flexible biopolymer gels. In contrast, if the isotropic matrix phase is modeled by the neo-Hookean form (2), it is shown in Horgan and Murphy (2011b) that no such change in sign is predicted and the transverse normal stress is always compressive. Acknowledgements COH is pleased to acknowledge helpful discussions with Dr. J.G. Murphy of Dublin City University, Ireland regarding some of the issues addressed here. The authors are grateful to a reviewer for constructive comments on an earlier version of the manuscript. References Atkin, R.J., Fox, N., 1980. An Introduction to the Theory of Elasticity. Longman, New York (Reprinted by Dover (2000)). Barenblatt, G.I., Joseph, D.D. (Eds.), 1997. Collected Papers of RS Rivlin, vols. 1–2. Springer, New York. Beatty, M.F., 1987. Topics in finite elasticity: hyperelasticity of rubber, elastomers, and biological tissues-with examples. Appl. Mech. Rev. 40, 1699–1733 (Reprinted with minor modifications as ‘‘Introduction to nonlinear elasticity’’. In: Carroll, M.M., Hayes, M.A., 1996 (Eds.), Nonlinear Effects in Fluids and Solids, Plenum Press, New York, pp. 16–112). Boyce, M.C., Arruda, E.M., 2000. Constitutive models of rubber elasticity: a review. Rubber Chem. Technol. 73, 504–523. Brown, R., 2006. Physical Testing of Rubber, fourth ed. Springer, New York. Demiray, H., 1972. A note on the elasticity of soft biological tissues. J. Biomech. 5, 309–311.

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