3. Measurement of properties of selected samples during static compression

Figure 2. (a) Inner structure (low-porous envelop). (b) Inner structure of PU foam with characteristic shape of cell (left)

Figure 3. The volume of air and material in the structure of polyurethane foams depending on the specific weight of pure

and detail of inner structure (right).

76 Aspects of Polyurethanes

polymer.

Mechanical properties of selected samples of PU foams show a stress dependent on the strain rate, which is accompanied by a change of stiffness. The stiffness of the sample can be experimentally determined as a slope of the tangent of the force depending on compression or deformation. The most significant change of K is in the initial phase (area no. 1, Figure 13). The material damping η<sup>t</sup> is a mechanical variable that is difficult to measure. It can be approximated for example by energy dissipation in a hysteresis curve. But it should be understood that the obtained values will vary depending on the strain rate and the geometry of the loading body. Generally, two foams with the same density may not have the same stiffness. Comparative parameter may be contact pressures and the transmission characteristic that the value of the resonant frequency determines. There are numerous of methodologies describing the mechanical properties of PU foam, but they are not standardized. Therefore, own measurement methodology was designed and implemented for a measurement of properties under static and dynamic compression.

### 3.1. Determination of mechanical properties of samples of PU foams under static compression

For obtaining mechanical properties of selected materials under static or quasi-static compression, the measurements were made on samples compressed with rigid steel plate with dimensions of 200 � 200 � 50 mm. The specimen having areal dimension 100 � 100 mm was placed on the rigid support. Samples change mechanical properties with the thickness that is reflected in the changing of the stiffness and damping. These properties have measured on the PU foam with thickness 60, 40 and 20 mm (sample No. 3). For the tests (Figure 4) an universal testing machine Labortech 2.050 was used. A tension load cell with loading capacity 1 kN was placed on a moving part of the measuring device. For the measurement, up to 50% deformation in accordance with the standard DIN 54 305 was applied. This standard is used for quasi-static compression tests of bulky fibrous structures, PU foam, and similar materials. The strain rate was set 60 mm min�<sup>1</sup> . After deformation, achieving the unloading phase up to 0% deformation

Figure 4. Determination of mechanical properties of samples of polyurethane foams during static compression: (a) scheme, (b) realization of the measurement.

follows. This represents one cycle that is four times repeated. A loading signal has a triangular wave shape. During the test, the force depending on compression is recorded. The results of PU foam samples with different thickness are shown in Figure 5. The results have confirmed that with decreasing thickness, the force required to compression of the sample to the desired deformation increases. It is also seen that between loading and unloading cycle is a hysteresis, and between first and second cycles, a significant loss of force occurs (relaxation of the material). The stiffness of samples was measured at the 5th cycle (Figure 6). The stiffness of samples with thickness 60 and 40 mm in the range between 30 and 50 % exhibits a difference about 20 % (2000 N/m), while highest force at 50% deformation differs only about 30 N. A sample with a thickness of 20 mm exhibited an increase in strength of 80 N in comparison with a sample thickness of Measurement and Numerical Modeling of Mechanical Properties of Polyurethane Foams http://dx.doi.org/10.5772/intechopen.69700 79

Figure 5. Dependence of force on the deformation of PU foam samples with dimensions 100 � 100 � 60, 40 and 20 mm during cyclic compression.

60 mm, and increase in strength of 50 N in comparison with a sample having a thickness of 40 mm. However, the stiffness of the sample having a thickness of 20 mm compared with the samples having a thickness of 40 and 60 mm increased about approximately 11,000 N/m. There can be applied the inequality α<sup>60</sup> < α<sup>40</sup> < α20, that describes tangent slope of initial stiffness of the cellular structure of a given thickness. It can be concluded that the targeted reduction in the thickness of the polyurethane foam (the idea of new seat and back car seats design) is not useful for the quality of the seating. It is not suitable especially for safety reasons. For example, currently manufactured headrests, where the structure of comfortable filler is made of PU foam having the thickness less than 20 mm exhibit at high strain rate significant hardening of the structure [3].

### 3.2. Determination of mechanical properties of PU samples during dynamic compression

follows. This represents one cycle that is four times repeated. A loading signal has a triangular wave shape. During the test, the force depending on compression is recorded. The results of PU foam samples with different thickness are shown in Figure 5. The results have confirmed that with decreasing thickness, the force required to compression of the sample to the desired deformation increases. It is also seen that between loading and unloading cycle is a hysteresis, and between first and second cycles, a significant loss of force occurs (relaxation of the material). The stiffness of samples was measured at the 5th cycle (Figure 6). The stiffness of samples with thickness 60 and 40 mm in the range between 30 and 50 % exhibits a difference about 20 % (2000 N/m), while highest force at 50% deformation differs only about 30 N. A sample with a thickness of 20 mm exhibited an increase in strength of 80 N in comparison with a sample thickness of

Figure 4. Determination of mechanical properties of samples of polyurethane foams during static compression: (a) scheme,

(b) realization of the measurement.

78 Aspects of Polyurethanes

Mechanical properties during dynamic compression relate to the ability of the material to dampen incoming vibrations with a given frequency and amplitude. It is caused by the reorganization of the structure, in this instance cellular, in which entering mechanical energy is being transformed to heat in a short time interval. Great amount of the dissipated mechanical energy ϑðtÞ that was

Figure 6. Dependence of stiffness on the deformation of PU foam samples with dimensions 100 � 100 � 60, 40 and 20 mm.

described by Eq. (23) is proportional to the area of hysteresis curve that describes the relation between tension and the relative deformation during one cycle of harmonic stress. Generally, with viscoelastic structures, it is true that in the harmonic excitation, the structure stress σðtÞ and deformation εðtÞ change in time, while εðtÞ has certain phase delay to the applied stress σðtÞ, which is defined by Eqs. (4) and (5). Phase shift φðtÞ between the tension and relative deformation lies during the harmonic excitation in the interval φðtÞ ∈ð Þ 0, π=2 .

$$\sigma(t) = \sigma \cdot \cos\left(\omega \cdot t + \phi\right) = \sigma \cdot \cos\phi \cdot \cos\left(\omega \cdot t\right) + \sigma \cdot \sin\phi \cdot \cos\left(\omega \cdot t + \pi/2\right) \tag{4}$$

$$
\varepsilon(t) = \varepsilon \cdot \cos(\omega \cdot t),
\tag{5}
$$

Eq. (4) describing time dependency of the tension during harmonic compression can be further described in Eq. (6) expressing components of the dynamic module of the material structure.

$$\sigma(t) = \stackrel{\circ}{E\_P} \cdot \varepsilon \cdot \cos\left(\omega \cdot t\right) + \stackrel{\circ}{E\_P} \cdot \varepsilon \cdot \cos\left(\omega \cdot t + \pi/2\right),\tag{6}$$

where E 0 <sup>P</sup> is a real component of the dynamic flexibility module describing durability properties of the material, and E 00 <sup>P</sup> is imaginary component of the dynamic flexibility module describing dissipation of energy (loss module). Both modules are described by Eqs. (7) and (8), from which it is possible to obtain complex dynamic module E D <sup>P</sup> according to Eq. (9).

Measurement and Numerical Modeling of Mechanical Properties of Polyurethane Foams http://dx.doi.org/10.5772/intechopen.69700 81

$$E\_P' = \frac{\sigma\_0}{\varepsilon\_0} \cdot \cos\phi,\tag{7}$$

$$E\_P'' = \frac{\sigma\_0}{\varepsilon\_0} \cdot \sin \phi\_{\prime} \tag{8}$$

$$E\_P' = \frac{\sigma\_0}{\varepsilon\_0} \cdot \cos \phi,\tag{9}$$

where E D <sup>P</sup> is a complex dynamic module and i represents imaginary component.

To obtain mechanical properties during dynamic compression of the selected PU foams, measurements with 100 � 100 � 40 mm samples were conducted. All observed properties from these measurements can be summarized in these points:


### 3.3. Determining the mechanical properties of selected samples with dynamic compression against a rigid plate without the initial deformation

described by Eq. (23) is proportional to the area of hysteresis curve that describes the relation between tension and the relative deformation during one cycle of harmonic stress. Generally, with viscoelastic structures, it is true that in the harmonic excitation, the structure stress σðtÞ and deformation εðtÞ change in time, while εðtÞ has certain phase delay to the applied stress σðtÞ, which is defined by Eqs. (4) and (5). Phase shift φðtÞ between the tension and relative deformation

Figure 6. Dependence of stiffness on the deformation of PU foam samples with dimensions 100 � 100 � 60, 40 and

Eq. (4) describing time dependency of the tension during harmonic compression can be further described in Eq. (6) expressing components of the dynamic module of the material structure.

ing dissipation of energy (loss module). Both modules are described by Eqs. (7) and (8), from

00

<sup>P</sup> is a real component of the dynamic flexibility module describing durability proper-

<sup>P</sup> is imaginary component of the dynamic flexibility module describ-

D

<sup>P</sup> � ε � cos ðω � tÞ þ E

σðtÞ ¼ σ � cos ðω � t þ φÞ ¼ σ � cos φ � cos ðω � tÞ þ σ � sin φ � cos ðω � t þ π=2Þ (4)

εðtÞ ¼ ε � cos ðω � tÞ, (5)

<sup>P</sup> � ε � cos ðω � t þ π=2Þ, (6)

<sup>P</sup> according to Eq. (9).

lies during the harmonic excitation in the interval φðtÞ ∈ð Þ 0, π=2 .

σðtÞ ¼ E 0

00

which it is possible to obtain complex dynamic module E

where E 0

20 mm.

80 Aspects of Polyurethanes

ties of the material, and E

The experiment took place in the hydrodynamic laboratory (HDL). The measuring device was comprised of a hydraulic cylinder with an attached contraption to insert the sample. The contraption consists of two vertical supporting tubes put on the circular plate that were connected by a crosspiece from the top. In the middle of the crosspiece, an immovable tube pole was attached, with a 0.5 kN sensor placed on it. The sample was put between the upper and lower rigid plates. The arrangement of the conducted experiment is described in Figure 7. Input excitation harmonic signal (stationary periodical) was defined by Eq. (10). This signal is suitable for more than a study and experimental comparison of the material samples of different structures, because it is a base signal for the comparison and optimization of competent car seats [6].

$$y\_{(z)} = A\_{(z)} \cdot \sin\left(at\right),\tag{10}$$

where <sup>y</sup>ðz<sup>Þ</sup> is a defined cylindrical lift, <sup>A</sup>ðz<sup>Þ</sup> is the input amplitude, and <sup>ω</sup> <sup>¼</sup> <sup>2</sup>π<sup>f</sup> is an angular velocity.

There are two possible approaches to define the input excitation of the hydraulic cylinder:


Measurements of the selected samples were conducted according to the method number 2 measurement with a constant initial frequency value. In total, seven measurements with a gradually rising frequency f were conducted, starting from 0.5, 1, 2, 3, 4, 5, and 8 Hz with constant

Figure 7. Determination of mechanical properties of samples of polyurethane foams during dynamic compression: (a) scheme, (b) realization of the measurement.

value of the amplitude Aðz<sup>Þ</sup> ¼ 20 mm (i.e. up to 50% deformation) for the evaluation of five consecutive cycles. The difference of the harmonic course of a given input frequency is described in Figure 8, describing how the rising frequency value also raises the value of a phase shift (oscillation) of the hydraulic cylinder. Measurements were repeated three times. The resultant courses of the tested PU foam sample for individual frequency values during the fifth cycle are shown in Figure 9. The resultant courses of the dependence between pressure forces applied on the PU foam samples that were compressed against a rigid plate without the initial deformation for selected frequency values, which were 0.5, 2, and 4 Hz, are presented in Figure 11. The order of experiment where initial deformation is applied is shown in Figure 10.

The results of harmonic compression without the initial deformation in the fifth cycle of the PU foam sample (Figure 9) show that the change in frequency changes hysteresis force dependence on the pressure and relief. Looking at the PU foam sample, it is apparent that increasing frequency value 0.5, 2, and 4 Hz also increases the value of the force necessary to compress the material, but on the other hand starts dropping for frequencies 5 and 8 Hz. Maximum force value necessary to compress the PU foam sample was 191 N during 4 Hz frequency, while frequency 5 Hz slightly decreases the force value to 182 N and frequency 8 Hz requires only 165 N. This shows that the cell structure of the PU foam sample changes mechanical properties with the speed of deformation and change in frequency. It is apparent from the results that the PU foam sample changes its properties based on the speed of deformation, while the main influence can be seen in the presence of air in the foam structure. The air is not capable of getting back into the structure during unloading after reaching a certain strain rate; therefore, its influence is not as significant and the force value also decreases during compression.

Measurement and Numerical Modeling of Mechanical Properties of Polyurethane Foams http://dx.doi.org/10.5772/intechopen.69700 83

Figure 8. Input harmonic signals for measuring of dynamically compressed samples without initial deformation.

value of the amplitude Aðz<sup>Þ</sup> ¼ 20 mm (i.e. up to 50% deformation) for the evaluation of five consecutive cycles. The difference of the harmonic course of a given input frequency is described in Figure 8, describing how the rising frequency value also raises the value of a phase shift (oscillation) of the hydraulic cylinder. Measurements were repeated three times. The resultant courses of the tested PU foam sample for individual frequency values during the fifth cycle are shown in Figure 9. The resultant courses of the dependence between pressure forces applied on the PU foam samples that were compressed against a rigid plate without the initial deformation for selected frequency values, which were 0.5, 2, and 4 Hz, are presented in Figure 11. The order of experiment where initial deformation is applied is

Figure 7. Determination of mechanical properties of samples of polyurethane foams during dynamic compression: (a)

The results of harmonic compression without the initial deformation in the fifth cycle of the PU foam sample (Figure 9) show that the change in frequency changes hysteresis force dependence on the pressure and relief. Looking at the PU foam sample, it is apparent that increasing frequency value 0.5, 2, and 4 Hz also increases the value of the force necessary to compress the material, but on the other hand starts dropping for frequencies 5 and 8 Hz. Maximum force value necessary to compress the PU foam sample was 191 N during 4 Hz frequency, while frequency 5 Hz slightly decreases the force value to 182 N and frequency 8 Hz requires only 165 N. This shows that the cell structure of the PU foam sample changes mechanical properties with the speed of deformation and change in frequency. It is apparent from the results that the PU foam sample changes its properties based on the speed of deformation, while the main influence can be seen in the presence of air in the foam structure. The air is not capable of getting back into the structure during unloading after reaching a certain strain rate; therefore, its influence is not as

significant and the force value also decreases during compression.

shown in Figure 10.

scheme, (b) realization of the measurement.

82 Aspects of Polyurethanes

Figure 9. Dependence of force on deformation of dynamically compressed samples without initial deformation.

Figure 10. Determination of mechanical properties of samples of polyurethane foams during dynamic compression with initial deformation: (a) scheme, (b) realization of the measurement.

Figure 11. Dependence of force on deformation of dynamically compressed samples with initial deformation.
