4.3 Nonhomogeneous Shear Experiment

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To illustrate the differences between the models, the nonhomogeneous shear experiment

investigated in (van den Bogert and de Borst 1994) is discussed. A rubber material is considered

and described by the neo-Hooke, Mooney–Rivlin, Yeoh and Arruda–Boyce model.

The isochoric constants are taken from (Kaliske and Rothert 1997) and were obtained by

fitting tensile test results. The volumetric data are such that νeq = 0.475 at zero deformation.

They satisfy (coefficients Bij and μ in N/mm2, Di in mm2/N, λm is dimensionless)

1. neo-Hooke model

B10 = 0.525,D1 = 0.0952 (4.141)

2. Mooney–Rivlin model

B10 = 0.1486, B01 = 0.4849,D1 = 0.0789 (4.142)

3. Yeoh model

B10 = 0.538, B20 = −0.0685, B30 = 0.0325,

D1 = 0.0929,D2 = 0.0086,D3 = 0.0008 (4.143)

4. Arruda–Boyce model

μ = 0.71, λm = 1.7029,D = 0.1408. (4.144)

When applied to a 1 × 1 ×8 mm3 specimen, we get the force versus stretch curves in

Figure 4.3. According to (Kaliske and Rothert 1997), the experimental results are best fit

by the Yeoh curve exhibiting an S-shape. This typical shape originates from the negative

B20 coefficient. The neo-Hooke model and the Mooney–Rivlin model are not capable of

capturing this effect.

The shear experiment is schematically shown in Figure 4.4. The upper and lower surfaces

are rigid. The lower surface cannot translate or rotate, all degrees of freedom are fixed.

The upper surface can only translate in x-direction and z-direction. A force is applied in

x-direction. A uniform 5 × 5 × 10 20-node brick element mesh was used with reduced

integration.

The displacements in x-direction (Figure 4.5) show similar tendencies as the uniaxial

test data. The Yeoh model predicts more hardening than the neo-Hooke and Mooney–Rivlin

model. However, up to moderate displacements, all models predict similar results closely

fitting the experimental data (overall behavior of the experimental data is symbolized by discrete

symbols). Because of the elongation in x-direction, the specimen shrinks in z-direction

(Figure 4.6). This is reasonably well modeled by the neo-Hooke, Yeoh and Arruda–Boyce

model. The Mooney–Rivlin model, however, shows a completely opposite tendency: the

specimen grows thicker. Notice that the Mooney–Rivlin model is the only model including

the second invariant. It seems that predictions of models that include the second invariant

are not very accurate if model-parameter characterization is based on uniaxial test

results only.

HYPERELASTIC MATERIALS 197

0

1 2

2

4

6

8

10

1.2

neo-Hooke

Mooney–Rivlin

Yeoh

Arruda–Boyce

λ(−)

F(N)

1.4 1.6 1.8

Figure 4.3 Tensile test results

F

x y

z z

Rigid body

20 mm

10 mm

10 mm

Figure 4.4 Nonhomogeneous shear experiment

198 HYPERELASTIC MATERIALS

0

0

0.5

2 4 6 8 10 12 14 16

0.1

0.2

0.3

0.4

0.6

neo-Hooke Mooney–Rivlin

Yeoh

Arruda–Boyce

ux (mm)

F(kN)

Experimental data

Figure 4.5 Horizontal deformation

−0.6 −0.4 0

0

0.5

0.1

0.2

0.2

0.3

0.4

0.4

0.6

0.6

0.8

neo-Hooke

Mooney–Rivlin

Yeoh

Arruda–Boyce

uz(mm)

F(kN)

−0.2

Experimental data

Figure 4.6 Vertical deformation

HYPERELASTIC MATERIALS 199

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