FEM model of pneumatic spring assembly

2.1. Testing of silicone rubber membrane

For investigation of properties of the silicone rubber membrane the bulge test was used (Fig. 2). Regarding the scheme in Fig. 3 the assumption of spherical shape of bulge was used. The deformation of the specimen in dependence on internal pressure is described by height $h$ and radius $R$ of bulge window.

In accordance with [2], the stress $\sigma$ in a spherical membrane can be expressed by the formula:

where $r$ is the radius of the sphere, $p$ is the applied internal pressure and $t$ is the thickness of the membrane.

The thickness $t$ is given by equation [3]:

where $t_0$ is initial thickness of the membrane.

From the geometry of deformed membrane (Fig. 3) follows the relation Eq. (3) and Eq. (4) for radius $r$ and angle $\theta$:

In this case engineering strain $\lambda$ has to be derived by this way [3]:

where: $l_0$ is the initial length and $l$ instantaneous length of meridian curve of deformed membrane, $r_0$, ${\theta }_0$ is the initial value of radius $r$ and angle $\theta$ respectively.

Fig. 2Experimental setup for testing of silicone membrane

Fig. 3Setting up bulge test

Fig. 4Experimental setup measurement of the a) foam and b) the tape

a)

b)

2.2. Uniaxial compression test of foam

The strain energy function of finite hyperelasticity of compressible media is used to describe the elastic properties of open cell soft foams. To determine the constitutive equation the experimental data from a uniaxial compression test is used. Tested specimen was taken from a car seat cushioning and has dimensions 100×100×80 mm. The tests was conducted by means of Instron loading machine. Fig. 4(a) shows the experiment setup. The test was carried out under displacement control at a constant loading rate of 1 mm/s up to a given deformation 60 mm.

2.3. Uniaxial tensile test of the tape

The cover layer of pneumatic spring is made by several layers of adhesive tape along the silicone tube in order to limit its deformation. The specimen of the tape was 0.16 mm thick, 48 mm wide and 220 mm long. Uniaxial tensile test was conducted by means of Tira test 2810 machine showed in Fig. 4(b).

4.1. Mathematical model of silicone rubber

In according with [4], rubber can be defined by a stored energy function as hyperelastic material. Conventional hyperelastic material models such as Ogden model works very well for silicone rubber. Ogden designed the energy function as separable function of principal stretch in generalized form:

where ${\mu }_n$ and ${\alpha }_n$ are material constants

$K$ is the initial bulk modulus, $J$ is the volumetric ratio, defined as $J={\lambda }_1^2{\lambda }_2^2{\lambda }_3^2$. Under consideration of incompressible material ($J=$1) and Ogden model selection for silicone rubber we can use a 3-term formulation with engineering stress- engineering strain curve for biaxial mode with engineering strain ${\lambda }_1={\lambda }_2=\lambda$, ${\lambda }_3={\lambda }^{-2}$ [5], thus:

According to [5], in the material fitting data option in MSC.Marc the biaxial deformation mode is selected. The software automatically calculates results as follows: ${\mu }_1=$1.21738×10⁸ Pa, ${\mu }_2=$7.7123×10⁶ Pa, ${\mu }_3=$117.45 Pa, ${\alpha }_1=$0.0110147, ${\alpha }_2=$0.0123274, ${\alpha }_3=$15.898, $K=$3.5946×10⁹.

4.2. Mathematical model of a foam

In accordance with [4], the foam is considered as compressible materials and thus suitable material model us represented by function:

where ${\mu }_n$, ${\alpha }_n$ and ${\beta }_n$ are material constants and second term represents volumetric change, $J$ is the Jacobian measuring dilatancy defined as the determinant of deformation gradient $f$ ($f=\partial x/\partial X$, where $x$ and $X$ refer to the deformed and original coordinates of the body) with ${\lambda }_1=1$, ${\lambda }_2=1$, ${\lambda }_3=L/L_0$ [5]. For this deformation state we have: ${\lambda }_1{\lambda }_2{\lambda }_3=L/L_0=V/V_0$.

In the material fitting data option, uniaxial deformation mode was selected with the value of fictive poison’s ratio 0.1, material model is Foam with 3 terms ($n=$3). The results obtained are as follows: ${\mu }_1=$4688.15Pa, ${\alpha }_1=$9.72738, ${\beta }_1=$–1.21592, ${\mu }_2=$360734 Pa, ${\alpha }_2=$–0.01998, ${\beta }_2=$0.002497, ${\mu }_3=$–0.02778 Pa, ${\alpha }_3=$–6.18115, ${\beta }_3=$0.772644.

4.3. Mathematical model of the tape layer.

In case of the tape the Mooney-Rivlin material model can used in form [5]:

Experimental verification using uniaxial extension [5]: ${\lambda }_1=\lambda$, ${\lambda }_2={\lambda }_3={\lambda }^{-2}$. Hence:

The results obtained from material fitting data option are as follows: $C_{10}=$–2.55107×10⁸, $C_{01}=$ 2.9003×10⁸, $C_{11}=$1.18362×10⁸, $C_{20}=$–8.66098×10⁸.

Result of repeat plot of engineering stress-engineering strain curve and behavior of materials is shown in Fig. 8 and Fig. 9 respectively.

Fig. 8Repeat plot of engineering stress-engineering strain curve of materials

a) Rubber silicone

b) Foam

c) Tape

Fig. 9Repeat plot of engineering stress- engineering strain curve of materials

a) Rubber silicone

b) Foam

c) Tape