Fiber orientation effects in simple shearing of fibrous soft tissues

Accepted Manuscript Fiber orientation effects in simple shearing of fibrous soft tissues Cornelius O. Horgan, Jeremiah G. Murphy PII: DOI: Reference:

S0021-9290(17)30483-9 https://doi.org/10.1016/j.jbiomech.2017.09.018 BM 8377

To appear in:

Journal of Biomechanics

Accepted Date:

19 September 2017

Please cite this article as: C.O. Horgan, J.G. Murphy, Fiber orientation effects in simple shearing of fibrous soft tissues, Journal of Biomechanics (2017), doi: https://doi.org/10.1016/j.jbiomech.2017.09.018

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Fiber orientation effects in simple shearing of fibrous soft tissues Cornelius O. Horgan(a) School of Engineering and Applied Science, University of Virginia, Charlottesville, VA 22904, USA

Jeremiah G. Murphy Department of Mechanical Engineering, Dublin City University, Glasnevin, Dublin 9, Ireland

ABSTRACT Fiber-reinforcement is a common feature of many soft biological tissues. Continuum mechanics modeling of the mechanical response of such tissues using transverselyisotropic hyperelasticity is now well developed. The fundamental deformation of simple shear within this framework is examined here. It is well known that the normal stress effect characteristic of nonlinear elasticity plays a crucial role in maintaining a homogeneous deformation state in the bulk of the specimen. Here we consider the effect of anisotropy and fiber-orientation on the shear and normal stresses. It is shown that the confining traction that needs to be applied to the top and bottom faces of a block in order to maintain simple shear can be compressive or tensile depending on the degree of anisotropy and on the angle of orientation of the fibers. In the absence of such an applied traction, an unconfined sample tends to bulge outwards or contract inwards perpendicular to the direction of shear so that one has the possibility of both a positive or negative Poynting effect. The results are illustrated using experimental data obtained by other authors for porcine brain white matter. The general results obtained here are relevant to the development of accurate shear test protocols for the determination of constitutive properties of fibrous biological soft tissues.

Keywords: simple shear; fibrous biological soft tissues; fiber-reinforced transversely isotropic materials; fiber orientation; Poynting effect

(a)

Author to whom correspondence should be addressed; electronic mail: [email protected]

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1. Introduction Many biological tissues exhibit fiber-reinforcement, for example, ligaments and tendons (Gardener and Weiss, 2001), myocardial tissue (Humphrey, 2002; Taber, 2004), brain white matter (Feng et al., 2013, 2017), and skeletal muscle (Morrow et al., 2010). Continuum mechanics modeling of the mechanical response of such tissues using hyperelastic transversely-isotropic models is an active area of investigation. Here we treat the fundamental deformation of simple shear within this framework. Some notable applications of simple shear to the biomechanics of soft tissues are, for example, the works of Gardiner and Weiss (2001) on human medial collateral ligaments, Dokos et al (2002) in a study of the shear properties of passive ventricular myocardium, Schmid et al. (2006, 2008) on myocardial material parameter estimation, Guo et al. (2007) on shear testing of porcine skin, Nicolle et al (2012) on shearing of the spleen and Sondergaard et al (2013) on shearing of the cornea. Horgan and Murphy (2011a, b) have considered a model for simple shear of fibrous soft tissues where a single family of parallel fibers has general orientation with respect to the direction of shear. More recently, Murphy (2013) has focused on three distinct modes of simple shear for a cuboid with fibers parallel to the faces of the block. It was shown that it is necessary to include both anisotropic pseudoinvariants in order to obtain physically robust results. Further results on simple shear for strain-energies of the type introduced by Murphy (2013) are given in Destrade et al (2015). Misra et al (2010) have examined simple shear for soft tissues in the context of surgical simulation. Recent contributions to experimental and theoretical analyses of simple shear for both isotropic and anisotropic materials are described in Nunes and Moreira (2013), Moreira and Nunes (2017) and Meng and Terentjev (2016). In this paper we consider the simple shear problem for fiber-reinforced specimens with arbitrary oriented fibers. In particular, the effect of the fiber angle on the lateral normal stress required to maintain simple shear is examined thus determining the character of the Poynting effect. Full generality in the constitutive model for transversely isotropic materials is considered. It is shown that the confining traction that needs to be applied to the top and bottom faces of a block in order to maintain simple shear can be compressive or tensile depending on the degree of anisotropy and on the angle of orientation of the fibers. Inclusion of the second invariant in the isotropic part of the strain-energy used is shown to be of crucial importance in assessing the nature of the confining traction. In the absence of such an applied traction, an unconfined sample tends to bulge outwards or contract inwards perpendicular to the direction of shear so that one has the possibility of both a positive or negative (reverse) Poynting effect. A reverse Poynting effect (i.e., a tensile normal stress) for biopolymer gels in simple shear was observed in experiments by Janmey et al (2007). Similar findings regarding positive or negative Poynting effects were described by Misra et al (2010) in experiments on bovine myocardial tissue and on a silicone dielectric gel used to simulate brain tissue. That paper

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describes the significance of the Poynting effect in the context of haptic feedback in surgical training. Classical aspects of the Poynting effect for rubber-like isotropic hyperelastic materials are described in Truesdell and Noll (2004). Theoretical analyses based on nonlinear elasticity for such materials have been carried out by several authors in more recent years (see, e.g., Mihai and Goriely (2011, 2013), Horgan and Murphy (2011b), Wu and Kirchner (2010), Wang and Wu (2014 a, b) and Destrade et al (2015) where further references may be found). A negative Poynting effect in the torsion of cylinders was also considered by Mihai and Goriely (2013), Wu and Kirchner (2010), Wang and Wu (2014b) and Horgan and Murphy (2015). 2. Simple shear of incompressible fiber-reinforced hyperelastic materials The simple shear deformation is described by

x1  X1   X 2 , x2  X 2 , x3  X 3 ,

(2.1)

where  X1 , X 2 , X 3  and  x1 , x2 , x3  denote the Cartesian coordinates of a typical particle before and after deformation respectively and   0 is an arbitrary dimensionless constant called the amount of shear. The angle of shear is tan 1  . The deformation gradient tensor F, the left Cauchy-Green strain tensor B = FF T and its inverse are readily calculated and so one finds the three principal invariants of B, defined as

I1  tr B, I 2 

1 2  trB   trB 2  , I3  det B ,  2

(2.2)

to be

I1  I 2  3   2 , I3  1 .

(2.3)

We confine attention here to materials reinforced with one family of parallel fibers alligned at an angle  to the X1 axis in the undeformed state (see Figure 1). Figure 1 here Figure 1. Simple shear of a block with one family of parallel fibers

The unit vector A = (cos  , sin  , 0) in the reference configuration is transformed into

a = (cos  +  sin  , sin  , 0) in the current configuration. We introduce the usual anisotropic invariants I 4  a  a, I 5  F T a  F T a which, for simple shear, have the forms

I 4   c   s   s 2   2 s 2  2 cs  1 , 2

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I5   c   s     c   s   s   3 2 s 2  4 cs  1   2  c   s  , 2

2

2

(2.4)

where we have used the convenient notation c  cos , s  sin  . In this paper, we will only be concerned with the range 0     / 2 so that I 4  1 and so the fibers are always in extension. The constitutive law for the Cauchy stress T for incompressible transversely isotropic hyperelastic materials with W  W I1 , I 2 , I 4 , I 5  is given by (see, e.g., Humphrey, 2002) T   pI  2W1 B  2W2 B 1  2W4 a  a  2W5 a  Ba  Ba  a ,

where Wi  W / I i (i  1,2,4), 

(2.5)

denotes the tensor product with Cartesian

components ai a j and p denotes the arbitrary hydrostatic pressure arising from the incompressibility constraint. It is assumed that the strain-energy and the stress are zero in the undeformed configuration and that, on linearization, the nonlinear theory should be compatible with the theory of incompressible transversely isotropic elasticity for infinitesimal strains (see, e.g., Merodio and Ogden, 2005, Murphy, 2013). In this way, one can deduce that W  W  I1 , I 2 , I 4 , I5  should satisfy the conditions W 0  0, 2W10  2W20  p0 , W40  2W50  0,

2W10  2W20  T , 2W50  L  T , 4W440  16W450  16W550  EL  T  4L ,

(2.6)

where the 0 superscript denotes evaluation at I1  I 2  3, I 4  I5  1, p 0 is the value of the arbitrary pressure field in the undeformed configuration and the positive constants T , L , EL denote three of the four elastic constants of the linear theory, where the subscripts L and T denote longitudinal and transverse directions respectively. The fourth elastic constant ET is related to these by the relation ET  4 EL T /  EL  T  (see Murphy, 2013). There is significant experimental evidence, particularly for skeletal muscles to suggest that L  T , EL  L with an order of magnitude difference recorded in some instances. If the hyperelastic strain-energy depends on

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only one of the anisotropic pseudo-invariants, it can be seen immediately that (2.6) yield the unrealistic implication L  T (Murphy, 2013). From (2.5) we find that the in-plane Cauchy stresses are given by 2 2 T11   p  2 1   2  W1  2W2  2  c   s  W4  4 1   2   c   s    s  c   s   W5 ,  

T22   p  2W1  2 1   2 W2  2 s 2W4  4  s  c   s   s 2  W5 ,

T12  2 (W1  W2 )  2 s  c   s  W4  2   c   s   s  c   s   2   2    s 2  W5 ,   2

(2.7)

where the derivatives are evaluated at values of the invariants given by (2.3), (2.4). The out-of-plane stress is T33   p  2W1  2W2 . 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 and so p  p 0 , where we recall that p 0 is the value of the hydrostatic pressure in the undeformed state. In the classical formulations of simple shear, plane stress conditions are usually assumed so that T33  0 which determines the pressure to be

p  2W1  2W2 . On substitution of this value of p in (2.7), we obtain T11  2 2W1  2  c   s  W4  4 1   2   c   s    s  c   s   W5 ,   2 2 2 T22  2 W2  2s W4  4  s  c   s   s  W5 , 2

2

(2.8)

T12  2 (W1  W2 )  2s  c   s  W4  2   c   s   s  c   s   2   2    s 2  W5 .   2

Two special cases of the preceding are of particular interest namely when the fibers are parallel or perpendicular to the direction of shear. Longitudinal Shear (   0 ): We find from (2.10) that

T11  2 2W1  2W4  4 1   2 W5 , T22  2 2W2 ,

(2.9)

T12  2 (W1  W2  W5 ), where the derivatives are evaluated at I1  I 2  3   2 , I 4  1, I 5  1   2 . Perpendicular Shear (    / 2 ): Similarly, we get

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T11  2 2 W1  W4  2  2  2 W5  , T22  2 2W2  2W4  4  2  1W5 ,

(2.10)

T12  2 W1  W2  W4   2 2  3W5  ,

where the derivatives are evaluated at I1  I 2  3   2 , I 4  1   2 , I 5  1   2    2 . 2

3.

Stress response for a specific strain-energy

It is instructive to consider a specific form of the strain-energy density that satisfies all the conditions (2.6). We consider

W

T

  I1  3  1    I 2  3   2 

T  L 2

 2 I 4  I5  1 

EL  T  4L 2  I 4  1 , (3.1) 8

where  is a dimensionless constant such that 0    1 . When   1 , this reduces to the modified standard reinforcing model proposed by Murphy (2013). The isotropic part of (3.1) is the usual Mooney-Rivlin strain-energy. The form (3.1) is chosen as it is the simplest form that satisfies all the conditions (2.6) and as we shall see in Section 4, this form and the elastic moduli appearing in (3.1) have been calibrated with data for soft tissue. The importance of including a dependence on the second invariant I 2 will be seen in the sequel. See also Horgan and Smayda (2012) where the role of the second invariant for general deformations is discussed in detail. It can be easily verified that the form (3.1) satisfies all the conditions (2.6) provided that T  2  1  p0 . On using (3.1) in (2.8) we find that the non-dimensional normalized stresses T  T / T are

    1E 4 2 T11   2  2  1  L    c   s   s    c   s     L  L  1  s  c   s   2c   s  , 2  T T  T       1E 4 T22   1     2  2  1  L   s  c   s    L  L  1  s 3  2c   s  , 2  T T  T       1E 4 T12     1  L    c   s  c  2 s    s 2    L  L  1  s 2  c   s  2c   s  . 2  T T  T   (3.2)

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For the special cases of Longitudinal and Perpendicular shear these simplify as follows:

Longitudinal Shear:

   T11     2  2 L   2 , T   T22   1     2 , T12 

(3.3)

L . T

Perpendicular Shear:    3 E T11    2  2 L     L T  2 2 T   5 E T22      L 2 2 T 

 2  , 

 2 2   ,  

(3.4)

   1E T12   L   L  3   2   .    T 2  T

4. The shear and normal stresses While our main focus here is on the lateral normal stress required to maintain simple shear, it is of interest to briefly consider the shear stress response. We begin with the special case of Longitudinal Shear for which the nondimensionalized shear stress is given by the third of (3.3). The shear stress is a linear function of  and is simply T12  L , which is the well-known isotropic shear stress response. For Perpendicular Shear, the shear stress is given by the third of (3.4). If EL  3T , this stress is also positive and has a cubic dependence on  for large amount of shear. To investigate the general shear response given by the third of (3.2) estimates of the values for the material constant ratios are needed. We use recent data for porcine brain white matter obtained by Feng et al (2017), i.e., L / T  2 and EL / T  22 . From (3.2) we find that

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15   T12   1   c   s  c  2 s   s 2  s 2  c   s  2c   s   . 2  

(4.1)

A plot of this non-dimensional shear stress versus amount of shear for different angles of orientation is given in Figure 2. It can be seen that, for a given amount of shear  , the smallest shearing stress occurs for   0 (i.e. for Longitudinal shear) while the largest shearing stress occurs at    / 3.

Figure 2 here Figure 2. Shear stress versus amount of shear for various fiber angles for the model (3.1) with data for porcine brain white matter ( L / T  2 and EL / T  22 ).

In general, the shear modulus can be defined as sm  dT12 /d (  0) (see, e.g., Truesdell and Noll, 2004). On using the third of (3.2) we find that for the model (3.1)    sm  L 1  T sin 2 2  ,  2 L 

(4.2)

where the non-dimensional parameter  , assumed positive, is defined by

 1E 4    L  L  1 . 2  T T 

(4.3)

Thus in the range 0     / 2 , the maximum shear modulus occurs at    / 4    with value L 1   T / L   and the minima occur at   0,  / 2 with values  2  L . We turn now to the lateral normal stress. We begin with the special case of Longitudinal Shear for which the non-dimensionalized normal stress is given by the second of (3.3). For 0    1 , this stress is clearly negative and so a compressive normal stress is required to maintain simple shear. In the absence of such a stress the sample would tend to expand laterally and so one has the usual positive Poynting effect. This is the usual result obtained for isotropic materials. Note that the magnitude of the Poynting effect decreases with increasing  and that in the special case when   1 ,

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we have T22  0 and so in this case there is no Poynting effect. For Perpendicular Shear, the relevant normal stress is given by the second of (3.4). For all  in the range 0    1 , if EL / T  5 then this stress is positive. Here, in contrast with the case for longitudinal shear, the magnitude of this stress increases with increasing

 . Thus a tensile normal stress is required to maintain the deformation. In the absence of such a stress, the sample would tend to contract laterally and so one has a negative or reverse Poynting effect (see Destrade et al (2015) for details on using data for skeletal muscles). For general angles of fiber orientation, one would now expect a change in the character of the Poynting effect at some particular angle of orientation. On using (3.2), we write T22 as

        T22    1    2 s 2 1  L    s 4    2cs   s 2  1  L   T     T   

(4.4)

where the positive non-dimensional material parameter  is defined in (4.3). For the case when   1 it may be verified that, for L  T , the normal stress given by (4.4) is positive for all non-zero angles of orientation of the fibers so that one always has a reverse Poynting effect in this case. For the zero angle of orientation, we have T22  0 , as can also be seen directly from (3.4) 2 with   1 . Now suppose that   1 so that   1. Recall from (3.1) that the strain-energy now has a dependence on I 2 . We will now show that the situation is quite different. There are two cases to consider. First suppose that    1    2 s 2  1  L    s 4  0.  T 

(4.5)

Then T22 > 0 for all angles of orientation. Alternatively, suppose that    1    2 s 2  1  L    s 4  0.  T 

(4.6)

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Then we see from (4.4) that T22 changes from tensile to compressive at



   t  2cs   s 2  1  

 L   L  2 4  /  1    2 s 1   s  . T    T  

(4.7)

To obtain explicit results, we use the values of the modulus ratios for porcine brain white matter given earlier, i.e., L / T  2 , EL / T  22 so that   15 / 2 and we take   1/ 2 corresponding to the middle of the allowable range for this parameter. Thus from (4.7) we find that

 t  2cs 15s 2  2  /  1  4s 2  15s 4  .

(4.8)

For these values of the material parameters, the inequality (4.6) reads

1  4s 2  15s 4  0

(4.9)

which can be shown to hold for non-zero angles of orientation  in the range 0    m

(m  23 o ).

(4.10)

Thus, in the range (4.10), there is now a transition from a tensile to a compressive normal stress at a transition amount of shear  t given by (4.8). In Figure 3, in addition to the case   0 (Longitudinal shear) for which the normal stress given in (3.3)2 is always compressive, plots which show such transitions are given for realistic amounts of shear. Outside the range (4.10), the normal stress is always tensile (not shown in Figure 3).

Figure 3 here Figure 3. Normalized normal stress versus amount of shear for various fiber angles for the model (3.1) on using data for porcine brain white matter ( L / T  2 and EL / T  22 so that

  15 / 2 ) and

  1/ 2. 10

5. Conclusions We have examined the effects of fiber orientation on the shear and lateral normal stress response in simple shear of fibrous soft tissues. The normal stress effect plays a crucial role in maintaining a homogeneous deformation state in the bulk of a specimen under simple shear. The conventional wisdom from rubber elasticity is that this stress is generally compressive. However, several recent studies have shown that a negative or reverse Poynting effect (i.e., a tensile normal stress) can occur in some soft solids. Here we have focused on fibrous soft tissues with a general fiber orientation and used a fully general theory for fiber-reinforced incompressible transversely-isotropic hyperelastic materials. We have examined the effect of anisotropy and fiber-orientation on the normal stress response. The results are illustrated for a specific strain-energy density that is consistent with the linearized theory of elasticity for transversely isotropic materials. Specific elastic moduli for porcine brain white matter recently obtained by Feng et al (2017) are used to quantify the results. It was shown that the confining traction that needs to be applied to the top and bottom faces of a specimen in order to maintain simple shear can be compressive or tensile depending on the degree of anisotropy and on the angle of orientation of the fibers. The results are relevant to the development of accurate shear test protocols for the determination of constitutive properties of fibrous biological soft tissues. Conflict of interest statement The authors have no conflict of interest to declare Acknowledgments We are grateful to the reviewers for constructive helpful comments on an earlier version of the manuscript.

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References

Destrade, M., Horgan, C. O., Murphy, J. G., 2015. Dominant negative Poynting effect in simple shearing of soft tissues. J. Eng. Math. 95, 87-98. Dokos, S., Smaill, B. H., Young, A. A., LeGrice, I. J., 2002. Shear properties of passive ventricular myocardium. Am. J. Physiol. Heart Circ. Physiol. 283, H2650-H2659. Feng, Y., Okamoto, R.J., Namani, R., Genin, G.M., Bayly, P.V., 2013. Measurements of mechanical anisotropy in brain tissue and implications for transversely isotropic material models of white matter. J. Mech. Behav. Biomed. Mater. 23, 117-132. Feng, Y., Lee, C-H., Sun, L., Ji, S., Zhao, X., 2017. Characterizing white matter tissue in large strain via asymmetric indentation and inverse finite element modeling. J. Mech. Behav. Biomed. Mater. 6 5 , 4 9 0 - 5 0 1 . Gardiner, J. C., Weiss, J. A., 2001. Simple shear testing of parallel-fibered planar soft tissues. J. Biomech. Eng. 123, 170-175. Guo, D-L., Chen, B-S., Liou, N-S., 2007. Investigating full-field deformation of planar soft tissue under simple-shear tests. J. Biomech. 40, 1165-1170. Horgan, C. O., Murphy, J. G., 2011a. Simple shearing of soft biological tissues. Proc. Roy. Soc. Lond. A 467, 760-777. Horgan, C. O., Murphy, J. G., 2011b. On the normal stresses in simple shearing of fiberreinforced nonlinearly elastic materials. J. Elasticity 104, 343-355. Horgan, C. O., Murphy, J. G., 2015. Reverse Poynting effects in the torsion of soft biomaterials. J. Elasticity 118, 127-140. Horgan, C. O., Smayda, M., 2012. The importance of the second strain invariant in the constitutive modeling of elastomers and soft biomaterials. Mech. of Materials 51, 43-52. Humphrey, J.D., 2002. Cardiovascular Solid Mechanics. Springer, New York. Janmey, P. M., McCormick, M. E., Rammensee, S., Leight, J. L., Georges, P. C., MacKintosh, F. C., 2007. Negative normal stress in semiflexible biopolymer gels. Nature Materials 6, 48-51. Meng, F., Terentjev, E. M., 2016. Nonlinear elasticity of semiflexible filament networks. Soft Matter 12, 6749-6756.

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Merodio, J., Ogden, R. W., 2005. Mechanical response of fiber-reinforced incompressible nonlinear elastic solids. Int. J. Nonlinear Mech. 40, 213-227. Mihai, L. A., Goriely, A., 2011. Positive or negative Poynting effect ? The role of adscititious inequalities in hyperelastic materials. Proc. Roy. Soc. London A, 467, 3633-3646. Mihai, L. A., Goriely, A., 2013. Numerical simulation of shear and the Poynting effects by the finite element method: an application of the generalized empirical inequalities in nonlinear elasticity. Int. J. Nonlinear Mech. 49, 1-14. Misra, S., Ramesh, K. T., Okamura, A. M., 2010. Modeling of nonlinear elastic tissues for surgical simulation. Comput. Methods Biomech. Biomed. Engin. 13, 811-818. Moreira, C. S., Nunes, L.C.S., 2017. Simple shearing of rubberlike materials filled with parallel fibers at large strain. J. Composite Mater. 51, 1643-1651. Morrow, D. A., Haut Donahue, T.L., Odegard, G.M., Kaufman, K.R., 2010. Transversely isotropic tensile material properties of skeletal muscle tissue. J. Mech. Behav. Biomed. Mater. 3, 124-129. Murphy, J. G., 2013. Transversely isotropic biological soft tissue must be modeled using both anisotropic invariants. Eur. J. Mech. A/Solids 42, 90-96. Nicolle, S., Noguer, L., Palierne, J.-F., 2012. Shear mechanical properties of the spleen: Experiment and analytical modeling. J. Mech. Behav. Biomed. Mater. 9, 130-136. Nunes, L. C. S., Moreira, D. C., 2013. Simple shear under large deformation: experimental and theoretical analyses. Eur. J. Mech. A/Solids 42, 315-322. Schmid, H., Nash, M. P., Young, A. A., Hunter, P.J., 2006. Myocardial material parameter estimation-a comparative study for simple shear. J. Biomech. Eng. 128, 742750. Schmid, H., O’Callaghan, P., Nash, M. P., Lin, W., LeGrice, I. J., Smaill, B. H., Young, A. A., Hunter, P. J., 2008. Myocardial material parameter estimation-a non-homogeneous finite element study from simple shear tests. Biomech. Model. Mechanobiol. 7, 161-173. Sondergaard, A. P., Ivarsen, A., Hjortdal, J., 2013. Corneal resistance to shear force after UVA-riboflavin cross-linking. Investigative Opthalmology and Visual Science 54, 5059-5069. Taber, L. A., 2004. Nonlinear Theory of Elasticity: Applications in Biomechanics. World Scientific, Singapore. Truesdell, C., Noll, W., 2004. The Non-linear Field Theories of Mechanics, Handbuch der Physik (S. Flugge, ed.,), Vol. III/3, 3rd edition, Springer, Berlin.

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Wang, D., Wu, M.S., 2014a. Generalized shear of a soft rectangular block. J. Mech. Phys. Solids 70, 297-313. Wang, D., Wu, M.S., 2014b. Poynting and axial force-twist effects in nonlinear elastic mono- and bi-layered cylinders: torsion, axial and combined loadings. Int. J. Solids Struct. 51, 1003-1019. Wu, M. S., Kirchner, H. O. K., 2010. Nonlinear elasticity modeling of biogels. J. Mech. Phys. Solids 58, 300-310.

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Figure Legends

Figure 1. Simple shear of a block with one family of parallel fibers Figure 2. Shear stress versus amount of shear for various fiber angles for the model (3.1) with data for porcine brain white matter ( L / T  2 and EL / T  22 ). Figure 3. Normalized normal stress versus amount of shear for various fiber angles for the model (3.1) on using data for porcine brain white matter ( L / T  2 and

EL / T  22 so that   15 / 2 ) and   1/ 2.

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X2

x2

 X1

x1

25 20 0

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π/6

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π/4 π/3

5

π/2

0 0

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amount of shear

0.8

1

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15o 0.4 0.2

10o normal stress

0 0

0.2

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-0.2

5o

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0o

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amount of shear