7. Results of simulations

behavior of the modified Kelvin model that allows to express Cauchy stress σ<sup>i</sup> in the

where E is modulus of elasticity, εiðtÞ and ε\_iðtÞexpress the strain and strain rate in a single

• Another suitable material can be material model 37—Viscoelastic Ogden Rubber for Solid Elements—which allows to describe not only viscoelastic but also hyperelastic material properties (suitable for the study of rubber, polymers, fibers, foams, etc.). This is based on the description of the functional dependence of the strain energy density Eðλ1, λ2, λ3Þ defined by Eq. (42), expressing the energy that is required for the structure deformation.

3

μp αp

where i ¼ 1, …, 3, Eðλ1, λ2, λ3Þis the strain energy density, μ<sup>p</sup> and α<sup>p</sup> are material constants,

<sup>2</sup> <sup>¼</sup> <sup>G</sup>, where <sup>G</sup> is the shear modulus defined by Eq. (43), and <sup>λ</sup><sup>∝</sup><sup>σ</sup>

� <sup>X</sup> 3

i¼1

∂Eðλ1, λ2, λ3Þ ∂λ <sup>∝</sup> <sup>σ</sup> i

λ <sup>∝</sup><sup>σ</sup> <sup>i</sup> � 3 !

<sup>2</sup> � ð Þ <sup>1</sup> <sup>þ</sup> <sup>v</sup> , (43)

] Initial module E [MPa] Poisson ratio γ [�] Damping coefficient

(42)

(44)

<sup>i</sup> vectors are

i¼1

<sup>G</sup> <sup>¼</sup> <sup>F</sup>

<sup>σ</sup><sup>i</sup> <sup>¼</sup> pk <sup>þ</sup> <sup>λ</sup> <sup>∝</sup><sup>σ</sup>

The input material parameters of the simulation model are given in Table 4.

Rigid plates Linear elastic 7850 210,000 0.3 – PU foam mat. 45 50.16 2.6 – 0.2

Using strain energy Eðλ1, λ2, λ3Þ, the Cauchy stress in principal directions σ<sup>i</sup> can be expressed

<sup>i</sup> �

σ<sup>i</sup> ¼ E � εiðtÞ þ η<sup>t</sup> � ε\_iðtÞ, (41)

loading axis in accordance with Eq. (41).

direction, and η<sup>t</sup> is a damping of the material.

Practically, it is analogy to Eqs. (25) and (30).

where E is elastic modulus and ν is Poisson's ratio.

and let <sup>X</sup><sup>n</sup>

102 Aspects of Polyurethanes

in Eq. (44).

p¼1

μ<sup>p</sup> � α<sup>p</sup>

elongation in principal directions.

where pk is compression stress.

Part Material model Density [kg m�<sup>3</sup>

Table 4. Material properties of FEM model dynamically loaded sample.

<sup>E</sup>ðλ1, <sup>λ</sup>2, <sup>λ</sup>3Þ ¼ <sup>X</sup>

The input signal and a material response are shown in Figure 23. The results of simulations for frequency 5 Hz are shown in Figure 24. In comparison with the real experiment, simulated values have a high correlation coefficient (0.961) up to 37.5% deformation (compression of 15 mm). The correlation coefficient between the model and the real measurement when compressed to 50% deformation exhibits a correlation of 0.932. The results of dynamically compressed samples against the rigid plate without the initial deformation with parameters according to Table 4 are in good agreement with experiments.

Figure 23. FEM model: excitation signal (left), and the response of the material on compression (right).

Figure 24. Comparison of the courses of experiment and FEM models.

The distribution of strain in the X and Y directions (plane perpendicular to the axis of compression) at a 37.5% deformation of the sample showed (Figure 25) that a sample of the PU foam in these directions is significantly deformed, because in these areas, there is highest stress. The maximum value of displacement vectors is 6404 mm, which is approximately 15% deformation of the sample. This leads to the fact that the structure is substantially pushed out from the sample. In contrast, non-polyurethane sample no. 11 is not practically in the plane perpendicular to the axis of compression deformed (Figures 26–28), because the maximum value of the displacement vectors is 0.034 mm.

Strain distribution also influences the value of the maximum principal stress during compression, as shown in Figures 26–28.

Results of the main stress shown in Figure 28 indicate that material of the PU foam sample is pushed out already at 12.5% deformation. This phenomenon confirms that the PU foam with increasing strain rate increases the stiffness and the foam is pushed out. These results were further studied and tested in Refs. [3, 12]. The simulation of contact pressures has shown that the strain of the foam in directions perpendicular to the direction of compression leads to uneven stress distribution, which is reflected by the uneven distribution of contact pressures; however, these results cannot be obtained experimentally under dynamic loading. That is why


Figure 25. FEM model of PU foam sample, distribution of displacement vectors at 37.5% deformation.

Measurement and Numerical Modeling of Mechanical Properties of Polyurethane Foams http://dx.doi.org/10.5772/intechopen.69700 105

Figure 26. Deformation 37.5%: FEM model of PU foam sample.

The distribution of strain in the X and Y directions (plane perpendicular to the axis of compression) at a 37.5% deformation of the sample showed (Figure 25) that a sample of the PU foam in these directions is significantly deformed, because in these areas, there is highest stress. The maximum value of displacement vectors is 6404 mm, which is approximately 15% deformation of the sample. This leads to the fact that the structure is substantially pushed out from the sample. In contrast, non-polyurethane sample no. 11 is not practically in the plane perpendicular to the axis of compression deformed (Figures 26–28), because the maximum

Strain distribution also influences the value of the maximum principal stress during compres-

Results of the main stress shown in Figure 28 indicate that material of the PU foam sample is pushed out already at 12.5% deformation. This phenomenon confirms that the PU foam with increasing strain rate increases the stiffness and the foam is pushed out. These results were further studied and tested in Refs. [3, 12]. The simulation of contact pressures has shown that the strain of the foam in directions perpendicular to the direction of compression leads to uneven stress distribution, which is reflected by the uneven distribution of contact pressures; however, these results cannot be obtained experimentally under dynamic loading. That is why

Figure 25. FEM model of PU foam sample, distribution of displacement vectors at 37.5% deformation.

value of the displacement vectors is 0.034 mm.

sion, as shown in Figures 26–28.

104 Aspects of Polyurethanes

Figure 27. Deformation 25%: FEM model of PU foam sample.

Figure 28. Deformation 12.5%: FEM model of PU foam sample.

the modeling is a suitable tool for obtaining of information that is not possible to achieve by the experiments (Figure 29).

The results of loading at frequency 5 Hz and 37.5% deformation are summarized in Table 5.

Figure 29. FEM model: comparison of contact pressures distribution at 37.5% deformation.


Table 5. Results of stress in dynamically compressed sample.
