4.2 Isotropic Hyperelastic Materials

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In this section, frequently used stored-energy potentials for hyperelastic materials are

treated. These include the Arruda–Boyce, the Mooney–Rivlin, the neo-Hooke, the polynomial,

the reduced polynomial, the Yeoh and the Ogden model. The preferred form involves

a split into an isochoric part and a volumetric part (ABAQUS 1997), (Kaliske and Rothert

1997), (Stor˚akers 1986). For a treatise on volumetric strain-energy functions, see (Doll

and Schweizerhof 2000). This implies that we will use the reduced invariants I 1 and I 2

and the reduced principal stretches λ1, λ2, and λ3. Unfortunately, these forms do not fit

the generic form of Equation (4.103). However, in some cases we can prove explicitly

that John Ball’s conditions are satisfied. Notice that the use of the reduced quantities in

polynomial-type functions automatically implies that _ grows beyond bounds as J → 0

(Equation (4.19)).

The split of the stored-energy function in an isochoric part and a volumetric part finds

its origin in the near isochoric behavior of most rubber materials. The isochoric coefficients

are usually determined by simple tests such as the uniaxial, equibiaxial or planar

HYPERELASTIC MATERIALS 191

test (ABAQUS 1997). The compressibility coefficients are derived by volumetric compression

tests. Depending on the model, a linear or a nonlinear least-squares procedure is

used to find the coefficients (Hartmann 2001a), (Hartmann 2001b). In general, different

types of tests are needed for a good description of the material. Even then, extrapolation

to stretches significantly exceeding the range of the test data can lead to wildly erroneous

behavior. If only one test type is performed (nearly always a uniaxial test), the neo-Hooke

and the Arruda–Boyce models seem to perform well because of the physical foundations

of these models. More complex phenomenological models such as the Ogden model and

the polynomial model with many terms require the availability of different test type data

to perform well.

Another issue is the stability of the models. Several criteria exist, such as the Baker–Ericksen

inequality and the incremental stability. The Baker–Ericksen inequality states that if a

Cauchy principal stress σi exceeds another Cauchy principal stress σj , the corresponding

stretch λi should exceed λj as well. The incremental stability requires that the incremental

power ˙S : ˙E be positive. For details, the reader is referred to (Hartmann 2003) and (Reese

1994). For most models, stability requirements imply limits on the coefficients.

Taking a stored-energy functional of the form _(C, θ,X), we are interested in the

stress (Equation (4.2))

SKL = 2

∂_

∂CKL

(4.113)

and the tangent stiffness (Equation (3.6)):

_KLMN = 2

∂SKL

∂CMN

= 4

∂2_

∂CKL∂CMN

. (4.114)

For isotropic materials, _ will be of the form _(I 1, I 2, J, θ,X) or of the Ogden form

_(λ1, λ2, λ3, J, θ,X), where the dependence on θ and X is hidden in the coefficients of

the models.

4.2.1 Polynomial form

The general polynomial stored-energy function takes the form

_ =

N

_

i+j=1

Bij (I 1 − 3)i(I 2 − 3)j +

N

_

i=1

1

Di

(J el − 1)2i (4.115)

where

J el = J

J th (4.116)

with J el the elastic Jacobian of the deformation, J the total Jacobian and J th the thermal

Jacobian,

J th = (1 + αT )3 (4.117)

192 HYPERELASTIC MATERIALS

cf Equation (1.449). Notice that the polynomial form is not polyconvex unless j = 0 (since

I 2 is not polyconvex) and Bij,Di ≥ 0. Special forms are the Mooney–Rivlin strain-energy

potential

_ = B10(I 1 − 3) + B01(I 2 − 3) + 1

D1

(J el − 1)2 (4.118)

the neo-Hooke strain potential

_ = B10(I 1 − 3) + 1

D1

(J el − 1)2 (4.119)

the Yeoh form

_ = B10(I 1 − 3) + B20(I 1 − 3)2 + B30(I 1 − 3)3

+ 1

D1

(J el − 1)2 + 1

D2

(J el − 1)4 + 1

D3

(J el − 1)6 (4.120)

and the reduced polynomial form

_ =

N

_

i=1

Bi0(I 1 − 3)i +

N

_

i=1

1

Di

(J el − 1)2i . (4.121)

Only the neo-Hooke, the Yeoh and the reduced polynomial form are polyconvex because

of the absence of I 2. Since I 2 is difficult to determine experimentally and its inclusion in

the stored-energy function does not necessarily improve its predictive quality (Kaliske and

Rothert 1997), it is advisable to start off with a dependence only on I 1. This especially

applies if only uniaxial data are available. Among the models that depend only on I 1, the

neo-Hooke type assumes a special position. Indeed, using Gaussian statistical thermodynamics,

its constant can be linked to the molecular chain density of the material (Treloar

1975).

Since _ is linear in the coefficients Bij , a linear least-squares procedure suffices to determine

them. The Baker–Ericksen inequality is assured if all Bij ≥ 0 (sufficient condition)

SKL and _KLMN take the form

SKL = 2



N

_

i+j=1

Bij

_

i(I 1 − 3)i−1(I 2 − 3)j ∂I 1

∂Ckl

+j (I 1 − 3)i(I 2 − 3)j−1 ∂I 2

∂Ckl

_

+

N

_

i=1

2i

Di

(J el − 1)2i−1 ∂Jel

∂CKL

_

(4.122)

HYPERELASTIC MATERIALS 193

_KLMN = 4



N

_

i+j=1

Bij (I 1 − 3)i−2(I 2 − 3)j−2 _

i(i − 1)(I 2 − 3)2 ∂I 1

∂CKL

∂I 1

∂CMN

+ ij (I 1 − 3)(I 2 − 3)

∂I 1

∂CKL

∂I 2

∂CMN

+ i(I 1 − 3)(I 2 − 3)2 ∂2I 1

∂CKL∂CMN

+ ij (I 1 − 3)(I 2 − 3)

∂I 2

∂CKL

∂I 1

∂CMN

+ j (j − 1)(I 1 − 3)2 ∂I 2

∂CKL

∂I 2

∂CMN

+ j (I 1 − 3)2(I 2 − 3)

∂2I 2

∂CKL∂CMN

_

+

N

_

i=1

2i(Jel − 1)2i−2

Di

           

(2i − 1)

∂Jel

∂CKL

∂Jel

∂CMN

+ (J el − 1)

∂2J el

∂CKL∂CMN

_

.

(4.123)

The derivatives of the invariants with respect to C are treated in Section 4.4.

4.2.2 Arruda–Boyce form

This potential function satisfies

_ = μ

            12

(I 1 − 3) + 1

20λ2

m

(I

2

1

− 9) + 11

1050λ4

m

(I

3

1

− 27)

+ 19

7000λ6

m

(I

4

1

− 81) + 519

673 750λ8

m

(I

5

1

− 243)

 

+ 1

D

_(J el)2 − 1

2

− ln J el_

, μ,D ≥ 0. (4.124)

Notice that all terms are polyconvex and that

lim

J el→0

_ = +∞ (4.125)

lim

J el→∞

_ = +∞ (4.126)

lim

I1→∞

J el<M

_ = +∞. (4.127)

where M is some positive real number. Accordingly, the physical requirements in Section

4.1.1 are fulfilled. The Arruda–Boyce model is based on an 8-chain representation of

the macromolecular network of rubber and is extensively described in (Arruda and Boyce

1993). For this model, the Baker–Ericksen inequality is satisfied. The determination of

194 HYPERELASTIC MATERIALS

the coefficients requires a nonlinear least-squares procedure. The second Piola–Kirchhoff

stress and tangent stiffness satisfy

SKL = 2

_

μ

_1

2

+ 1

10λ2

m

I 1 + 33

1050λ4

m

I

2

1

+ 76

7000λ6

m

I

3

1

+ 2595

673 750λ8

m

I

4

1

_ ∂I 1

∂CKL

+ 1

D

_

J el − 1

J el

_ ∂Jel

∂CKL

 (4.128)

_KLMN = 4

_

μ

_ 1

10λ2

m

+ 66

1050λ4

m

I 1 + 228

7000λ6

m

I

2

1

+ 10 380

673 750λ8

m

I

3

1

_ ∂I 1

∂CKL

∂I 1

∂CMN

+μ

_1

2

+ 1

10λ2

m

I 1 + 33

1050λ4

m

I

2

1

+ 76

7000λ6

m

I

3

1

+ 2595

673 750λ8

m

I

4

1

_ ∂2I 1

∂CKL∂CMN

+ 1

D

_1 + 1

(J el)2

_ ∂Jel

∂CKL

∂Jel

∂CMN

+ 1

D

_

J el − 1

J el

_ ∂2J el

∂CKL∂CMN

 

.

(4.129)

4.2.3 The Ogden form

The Ogden form resembles the stored-energy function in Theorem 4.1.12; however, the

principal stretches are replaced by their reduced form:

_ =

N

_

i=1

2μi

α2

i

(λ

αi

1

+ λ

αi

2

+ λ

αi

3

− 3) +

N

_

i=1

1

Di

(J el − 1)2i (4.130)

where

λi := λi

J 1/3

= λ

2/3

i

λ

1/3

j λ

1/3

k

j, k _= i. (4.131)

For αi = 2, one obtains I 1, for αi = −2, the invariant I 2 emerges. Since I 2 is not polyconvex,

Equation (4.130) is not necessarily polyconvex. SKL and _KLMN satisfy

SKL = 2

_ N

_

i=1

2μi

αi

_ 3

_

k=1

λ

αi−1

k

∂λk

∂CKL

_

+

N

_

i=1

2i

Di

(J el − 1)2i−1 ∂Jel

∂CKL

_

(4.132)

_KLMN = 4

_ N

_

i=1

2μi

αi

_

(αi − 1)

_ 3

_

k=1

λ

αi−2

k

∂λk

∂CKL

∂λk

∂CMN

_

+

3

_

k=1

λ

αi−1

k

∂2λk

∂CKL∂CMN

_

+

N

_

i=1

            2i(2i − 1)

Di

(J el − 1)2i−2 ∂Jel

∂CKL

∂Jel

∂CMN

+ 2i

Di

(J el − 1)2i−1 ∂2J el

∂CKL∂CMN

_

.

(4.133)

HYPERELASTIC MATERIALS 195

4.2.4 Elastomeric foam behavior

Whereas the potentials in the previous sections are frequently used for materials that are

nearly incompressible, such as rubber, elastomeric foams are very compressible. The general

form satisfies

_ =

N

_

i=1

2μi

α2

i

_ˆλαi

1

+ ˆλαi

2

+ ˆλαi

3

− 3 + 1

βi

_(J el)

−αiβi − 1_ (4.134)

where

ˆλ

i := λi

(J el)1/3 . (4.135)

This form comes close to the Ogden form defined in Theorem 4.1.12 if αi,μi > 0. Indeed,

for αi > 0, the first three terms correspond to I1(Cαi /2). For αi < 0, however, they correspond

to

I2(C

−αi /2)

I3(C

−αi /2)

, (4.136)

which is not compatible with the Ogden form. Since the second derivative of the volumetric

term satisfies

_

__

vol

=

N

_

i=1

(αiβi)(αiβi + 1)

βi

2μi

α2

i

(J el)

−αiβi−2 (4.137)

convexity is guaranteed if

μiαi(αiβi + 1) ≥ 0 (4.138)

(sufficient but not necessary condition). The derivatives yield

SKL = 2

N

_

i=1

2μi

αi

_ 3

_

k=1

ˆλ

(αi−1)

k

∂ˆλk

∂CKL

− (J el)

−αiβi−1 ∂Jel

∂CKL

_

(4.139)

_KLMN = 8

N

_

i=1

μi

αi

_

(αi − 1)

3

_

k=1

ˆλ

(αi−2)

k

∂ˆλk

∂CKL

∂ˆλk

∂CMN

+

3

_

k=1

ˆλ

(αi−1)

k

∂2ˆλk

∂CKL∂CMN

+ (αiβi + 1)(J el)

−αiβi−2 ∂Jel

∂CKL

∂Jel

∂CMN

− (J el)

−αiβi−1 ∂2J el

∂CKL∂CMN

_

. (4.140)

196 HYPERELASTIC MATERIALS

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