4.7 Anisotropic Hyperelasticity

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Recently developed materials are frequently anisotropic, such as fabrics embedded in a

matrix material (Reese et al. 2001). For these applications, the theory of the previous

section has to be extended. Anisotropic materials are characterized by the fact that

_(F ·Q) = _(F) (4.289)

does not apply for arbitrary rotation tensors Q ∈ SO(3) (the group of all rotations without

reflection in three-dimensional space). The group G for which Equation (4.289) applies, if

any, is called the material symmetry group and characterizes the material:

G = _Q|_(F ·Q) = _(F)_ ⊂ SO(3). (4.290)

For instance, transverse isotropic materials are characterized by a preferred unit direction

A about which the material is isotropic, that is,

G = _Q(α,A)|0 < α < 2π_ (4.291)

where Q(α,A) denotes a rotation about A covering an angle α. Anisotropic materials are

frequently characterized by so-called structural tensors M, which are invariant under the

material symmetry group:

Q·M = M, ∀Q ∈ G. (4.292)

For transversely isotropic materials, there is one structural tensor defined by

M := A ⊗ A, _A_ = 1. (4.293)

Indeed,

Q·M = Q· (A ⊗ A) = (Q· A) ⊗ A

= A ⊗ A = M. (4.294)

The behavior of anisotropic materials is not invariant under the proper orthogonal group

of transformations SO(3). However, if the structural tensors are also transformed, that is,

the preferred material directions undergo the same rotation, one obtains

_(F,M) = _(F ·Q,QT ·M ·Q) (4.295)

and taking objectivity into account,

_(C,M) = _(QT · C ·Q,QT ·M ·Q). (4.296)

Consequently, _ is an isotropic function of C and M. Here, M stands for all structural

matrices appropriate for the material. An application of the above concept to anisotropic

viscoplasticity can be found in (Schr¨oder et al. 2002). In what follows, we will concentrate

on transversely isotropic materials and deal with only one structural tensor. For further

details, the reader is referred to (Schr¨oder and Neff 2001).

HYPERELASTIC MATERIALS 217

4.7.1 Transversely isotropic materials

To proceed, we will express _ as a function of a polynomial basis. For isotropic scalar

functions of two symmetric tensors, such a basis consists of the following terms (Spencer

1971):

J1 := trC (4.297)

J2 := trC2 (4.298)

J3 := trC3 (4.299)

J4 := tr(C ·M) (4.300)

J5 := tr(C2 ·M) (4.301)

J6 := tr(C ·M2) (4.302)

J7 := tr(C2 ·M2). (4.303)

Since M2 = M, we have J6 = J4 and J7 = J5. One recognizes J1, J2 and J3 as invariants

of C, although more frequently I1, I2 and I3 are used, defined by

I1 = J1 = _F_2 (4.304)

I2 = tr(CofC) = 12

(J 2

1

− J2) = _CofF_2 (4.305)

I3 = detC = 16

(2J3 + J 3

1

− 3J1J2) = (detF)2. (4.306)

In previous sections, it was shown that I1, I2 and I3 are polyconvex functions. The invariant

J4 can be expressed as

J4 = tr(FT · F · (A ⊗ A))

= AT · FT · F · A

= _F · A_2. (4.307)

This is a convex function of F due to the norm properties and the convex monotonic

increasing behavior of x2 for x ≥ 0. Accordingly, J4 is convex.

It can be proved that J5 is not polyconvex (Schr¨oder and Neff 2001). However, it is

clear that

K1 := _CofF · A_2 (4.308)

is polyconvex. K1 can also be written as

K1 = AT · (CofF)T · (CofF) · A

= AT · Cof(FT · F) · A

= tr [(CofC) · (A ⊗ A)]

= tr [(CofC) ·M] . (4.309)

218 HYPERELASTIC MATERIALS

The tensor C satisfies its characteristic equation, accordingly,

C3 − I1C2 + I2C − I3 = 0

_

C2 ·M − I1C ·M + I2M − I3C

−1 ·M = 0

_

tr(C2 ·M) − I1tr(C ·M) + I2 = tr(CofC ·M)

_

J5 − I1J4 + I2 = K1 (4.310)

which shows that K1 can be written as a function of J5. Notice the nice analogy between

J4 and K1, and I1 and I2 respectively (Equations (4.304), (4.305), (4.307) and (4.308)).

The physical significance of J4 and K1 is also noteworthy: Equation (1.30) reveals that J4

is a measure of the change of length of a unit vector along the structural axis, whereas

Equation (1.65) shows that K1 can be interpreted as the area change of a unit area perpendicular

to the structural axis. On the basis of these physical observations, sometimes the

change in length of vectors perpendicular to the material axis, and the change in area of

area elements whose normal is perpendicular to the material axis are considered. They are

defined by

K2 := tr(C · D) = I1 − J4 = _F_2 − _F · A_2 (4.311)

K3 := tr[(CofC) · D] = I1J4 − J5 = _CofF_2 − _(CofF) · A_2 (4.312)

where

D := G_ −M. (4.313)

It can be proved that K2 and K3 are polyconvex (Schr¨oder and Neff 2001). Indeed,

K2 = tr[FT · F · (G_ −M)]

= _F · (G_ −M)_2 (4.314)

since

(G_ −M) · (G_ −M) = G_ −M (4.315)

and

(G_ −M)T = (G_ −M). (4.316)

Similarly, one finds

K3 = _CofF · (G_ −M)_2. (4.317)

HYPERELASTIC MATERIALS 219

Analogous to the proof leading to Equation (4.64), one can prove that

J4

(detF)2/3 (4.318)

and

J

3/2

4

(det F)2 (4.319)

are polyconvex as well as the same terms with J4 replaced by K1, K2 or K3. In particular,

J 4 := J4

(det F)2/3 (4.320)

K

3/2

1 :=       K1

(detF)4/3

3/2

(4.321)

K2 := K2

(det F)2/3 (4.322)

and

K

3/2

3 :=       K3

(detF)4/3

3/2

(4.323)

are polyconvex and consequently, also J

n

4, K

3n/2

1 , K

n

2 and K

3n/2

3 (Theorem 4.1.8) (notice

that J4, K1, K2, K3 ≥ 0). For C = G, one obtains

J4 = J 4 = 1 (4.324)

K1 = K1 = 1 (4.325)

K2 = K2 = 2 (4.326)

K3 = K3 = 2 (4.327)

but J 4 ≥ 1 is not guaranteed, hence we cannot argue that terms of the form (J 4 − 1)k are

polyconvex. On the other hand, terms such as

e(J 4−1), e(K

3/2

1

−1), e(K2−2), e(K

3/2

3

−2

√

2) (4.328)

are polyconvex, since ex is a convex, monotonic increasing function in R.

4.7.2 Fiber-reinforced material

In this section, an anisotropic hyperelastic model for fiber-reinforced materials will be

discussed. It is a model that was developed for arteries by Holzapfel (Holzapfel et al.

2000) but which seems promising for other applications as well. It consists of an isotropic

neo-Hooke part superimposed by strengthening terms in the fiber direction (the volumetric

220 HYPERELASTIC MATERIALS

−1 −0.5 0

0

0.5

0.5

1

1

1.5

2

exp(< x >2) − 1

x

Figure 4.11 Generic form of the anisotropic term

term is not a part of the original Holzapfel model):

_ = B10(I 1 − 3) + 1

D1

(J − 1)2 +

N

_

i=1

k1i

2k2i

_ek2i<J 4i−1>2 − 1 (4.329)

where

< x > = x for x > 0

=0 for x ≤ 0.

(4.330)

and

k2i > 0. (4.331)

Notice that the anisotropic term applies only if the fibers are extended. Under compression

the fibers do not contribute any strength. Under tension, however, the strengthening is

exponential. There are as many terms as there are fiber directions, each with its own

constants k1i and k2i . Notice that ea<x>2 − 1 (a > 0) is a C1 monotonically increasing

convex function (Figure 4.11). Accordingly, the anisotropic terms in Equation (4.329) are

polyconvex.

Differentiation of Equation (4.329) leads to SKL and _KLMN,

SKL = B10

∂I 1

∂CKL

+ 1

D1

(1 − I

−1/2

3 )

∂I3

∂CKL

+

N

_

i=1

k1i (J 4i − 1) _ek2i (J 4i−1)2 ∂J 4i

∂CKL

(4.332)

HYPERELASTIC MATERIALS 221

_KLMN = ∂2_

∂CKLCMN

= B10

∂2I 1

∂CKL∂CMN

+ 1

2D1

I

−3/2

3

∂I3

∂CKL

∂I3

∂CMN

+ 1

D1

(1 − I

−1/2

3 )

∂2I3

∂CKLCMN

+

N

_

i=1

k1i _ek2i (J 4i−1)2 ·

·

_ ∂J 4i

∂CKL

∂J 4i

∂CMN

_1 + 2k2i(J 4i − 1)2_ + (J 4i − 1)

∂2J 4i

∂CKLCMN

_

(4.333)

and the derivatives of J4i and J 4i yield

J4i = MIJ

i CIJ (4.334)

∂J4i

∂CKL

= MKL

i (4.335)

∂2J4i

∂CKL∂CMN

= 0 (4.336)

J 4i = I

−1/3

3 J4i (4.337)

∂J 4i

∂CKL

= −13

I

−4/3

3 J4i

∂I3

∂CKL

+ I

−1/3

3

∂J4i

∂CKL

(4.338)

∂2J 4i

∂CKL∂CMN

= 49

I

−7/3

3 J4i

∂I3

∂CKL

∂I3

∂CMN

− 13

I

−4/3

3

_ ∂J4i

∂CMN

∂I3

∂CKL

+ ∂J4i

∂CKL

∂I3

∂CMN

_

− 13

I

−4/3

3 J4i

∂2I3

∂CKL∂CMN

+ I

−1/3

3

∂2J4i

∂CKL∂CMN

. (4.339)

To investigate the effect of the anisotropic terms on the initial stiffness, the limit C →G

is taken

J4i |

C=G

= 1 (4.340)

∂J4i

∂CKL

____

C=G

= MKL

i (4.341)

∂2J4i

∂CKL∂CMN

____

C=G

= 0 (4.342)

J 4i_

_C=G

= 1 (4.343)

∂J 4i

∂CKL

_____

C=G

= MKL

i

− 13

GKL (4.344)

222 HYPERELASTIC MATERIALS

F

h

8 h h

α

Figure 4.12 Geometry of the cantilever beam

∂2J 4i

∂CKL∂________CMN

_____

C=G

= 19

GKLGMN + 16

(GKMGLN + GKNGLM)

− 13

(MMN

i GKL +MKL

i GMN). (4.345)

Substitution in Equation (4.333) yields for the anisotropic terms

_KLMN_

__C=G,anisotropic

=

N

_

i=1

k1i(MKL

i

− 13

GKL)(MMN

i

− 13

GMN). (4.346)

For a fiber aligned with the 1-direction (M11 = 1, all other MKL = 0) one obtains

_KLMN_

_

_C=G,anisotropic

=



4/9 −2/9 −2/9

−2/9 1/9 1/9

−2/9 1/9 1/9

_0_3×3

_0_3×3 _0_3×3



k1i . (4.347)

Accordingly, the initial stiffness in fiber direction is increased by 49

k1i . The parameter k1i

has the unit of stress, k2i is dimensionless and governs the strength increase at increasing

deformation.

Consider the cantilever beam in Figure 4.12 subject to a force F evenly distributed

at its end. The force keeps its magnitude and direction during deformation. The material

consists of an isotropic neo-Hooke substrate strengthened by fibers. It is assumed to satisfy

Equation (4.329) with constants:

B10h2/F = 0.192505 (4.348)

D1F/h2 = 0.26 (4.349)

N = 1 (4.350)

k11h2/F = 0.23632 (4.351)

k21 = 0.8393. (4.352)

There is only one layer of fibers making an angle α with the axis of the beam. The

relative axial displacement ua/h and transversal displacement 100ut/h at the end of the

HYPERELASTIC MATERIALS 223

0

0

2

4

6

8

10

10

12

14

16

20 30 40 50 60 70 80

α(

◦

)

Axial displacement

Transversal displacement

90

ua/h, 100ut/h

Figure 4.13 Longitudinal and transversal displacement of the end of the beam

beam are shown in Figure 4.13. If the fibers are parallel to the axis of the beam, the

transversal displacement is zero and the axial displacement exhibits a minimum because of

the strengthening effect of the fibers. As the angle with the axis increases, the strengthening

effect decreases steadily. The transversal displacement exhibits a maximum at about 27◦

because of the asymmetry induced by the fibers.

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