3.6. Determining the relaxation of the chosen samples compressed by the rigid plate into the constant deformation value

The sample was compressed by the rigid plate into the constant value atduring 10, 25, 50, and 65% deformation, while measuring the material response to the compression during 3600 s. Resultant courses of the sample force response dependence to time are shown in Figure 12.

The resultant comparison of the course of relaxation of the PU foam sample shows minimum relaxation during low deformation values (10 and 25%); on the other hand, there was an apparent decrease in force over time during 50–65% deformation. The PU foam sample had a decrease in force of 34 N during 65% deformation (initial force value was 118 N and the final force value was 84 N). The drop in force overtime can be converted to the decrease in tension over time, which can then express the values of the relaxation modulus G(t) according to Eq. (14) describing the decrease in tension in the material structure in a time sequence. Values G(t) for each deformation differed in the PU foam sample. This can be explained by the cellular structure with low initial deformation of 10% having greater rigidity and cellular structure with greater initial deformation having less air in its structure. In fact, the structure closes and the permeability decreases, and therefore the air cannot return to the structure. Relaxation module values for the tested samples are shown in Table 2.

Figure 12. Comparison of relaxation experiment of PU foam.


Table 2. Results of relaxation modulus G(t).

• determining the relaxation of the chosen samples compressed by the rigid plate into the

The sample was compressed by the rigid plate into the constant value atduring 10, 25, 50, and 65% deformation, while measuring the material response to the compression during 3600 s. Resultant courses of the sample force response dependence to time are shown in Figure 12.

The resultant comparison of the course of relaxation of the PU foam sample shows minimum relaxation during low deformation values (10 and 25%); on the other hand, there was an apparent decrease in force over time during 50–65% deformation. The PU foam sample had a decrease in force of 34 N during 65% deformation (initial force value was 118 N and the final force value was 84 N). The drop in force overtime can be converted to the decrease in tension over time, which can then express the values of the relaxation modulus G(t) according to Eq. (14) describing the decrease in tension in the material structure in a time sequence. Values G(t) for each deformation differed in the PU foam sample. This can be explained by the cellular structure with low initial deformation of 10% having greater rigidity and cellular structure with greater initial deformation having less air in its structure. In fact, the structure closes and the permeability decreases, and therefore the air cannot return to the structure. Relaxation module values for the tested samples

3.6. Determining the relaxation of the chosen samples compressed by the rigid plate

constant deformation value.

86 Aspects of Polyurethanes

into the constant deformation value

are shown in Table 2.

Figure 12. Comparison of relaxation experiment of PU foam.

• determining the relaxation module for the chosen samples.

### 3.7. Mathematic-physical description of mechanical behavior of PU foam samples

Mechanical properties of the PU foam samples have a non-linear course during compression. This is a viscoelastic behavior with great, almost returnable, transformation, in the order of 92 � 3% [12]. During the force relief on the foam, the hysteresis loop manifests, based on the volume weight, rigidity K, and damping coefficient η<sup>t</sup> with gradual relaxation of the cell structure, that is more significant during the quasi-static compression than dynamic, among others [6, 12]. The inability of rapid recovery after the deformation also materializes. During the compression, the non-linear behavior of PU foam sample is characterized by three areas: first, initial solidification; second, steady course of deformation during the minimum gain in absolute tension; and third, final and significant solidification of the cell structure. Characteristic course of tension in the dependence on the transformation of the tested PU foam sample is shown in Figure 13.

The graphic course shown in Figure 13 characterizes nonlinear dependence of tension on the transformation, gained during the experimental measurement of the PU foam sample number 3 (Table 1) compressed to 98 � 2% transformation. The sample number 3 had characteristic area 1 —lasting up to �12 � 3% transformation, which was given by the elastic, almost linear, course with a steep onset of the tension caused by initially fast solidification of the cellular structure, where the inclination of the curve is dependent on the strain rate. Area 2 can be set for the range of 15–50 � 5% transformation, where so-called plateau takes place (even temporary stabilization) meaning the absolute increase in tension depending on the transformation is minimal. In the area 3, approximately from 60 � 7 up to 92 � 3% deformation begins a steep exponential course caused by the final compression of the structure (the structure starts to almost mash). Already, in the year 1969, Rush [13] published analytical model closely describing the behavior of the compressed PU foam. The author based his assumptions on the constitutive equation describing the response of the material after compression, that is characterized by a constant module of flexibility E, the size of deformation transformation ε, and the compression function ΛðεÞ, that can be described by Eq. (11), where E presents module of foam flexibility, ε is deformation, and ΛðεÞ is compression function.

$$
\sigma = \mathbf{E} \cdot \boldsymbol{\varepsilon} \cdot \boldsymbol{\Lambda}(\boldsymbol{\varepsilon}), \tag{11}
$$

Constitutive Eq. (11) was improved upon by Schwaber and Meinecke [14] in 1971 and by Nagy et al. [15] in 1974 by the functional dependence of variable flexibility modulus E on the strain rate ε\_, which is described by Eqs. (12) and (13). These relations can be used to determine immediate rigidity in the structure, because the weight of the structure mðε, ε\_Þ and the flexibility

Figure 13. Nonlinear dependence of stress of PU foam sample (top), and the characteristic courses in three main areas (bottom).

modulus Eðε\_Þ are variable and depend on the immediate state of the deformation or transformation ε.

$$
\sigma = E(\dot{\varepsilon}) \cdot f(\varepsilon) \cdot \varepsilon,\tag{12}
$$

$$
\sigma = \mathfrak{m}(\varepsilon, \dot{\varepsilon}) \cdot f(\varepsilon),
\tag{13}
$$

where fðεÞ describes polynomial function describing the course during compression, mðε, ε\_Þ is the weight of the structure that is variable on deformation ε and the strain rate ε\_.

Characteristic properties of the PU foam sample during compression are influenced mostly by the size of the deformation εðtÞ but also time t that takes to compress the cellular structure. Related to this is a physical effect called creep (thinning the structure); in other words, the structure of the PU foam becomes suppler under constant pressure and the deformation increases <sup>ε</sup>ðt2Þj<sup>σ</sup>¼konst <sup>&</sup>gt; <sup>ε</sup>ðt1Þ. Similarly behaves the structure during relaxation of the material—during constant deformation ε ¼ konst (for example, repeated cyclic compression up to 50% transformation and the material fatigue), the tension gradually decreases. It is possible to describe this physical effect by the value of the relaxation modulus GðtÞ (14) described by the relation between acting tension σðtÞ and a constant GðtÞ deformation ε ¼ konst: In Ref. [1], it is stated that increasing deformation of the PU foam sample decreases the value GðtÞ where a setting of relaxation modulus values materializes during long-term test for values of small and large deformations as stated in Figure 14. This effect is caused by viscoelasticity of the material (this does not have to be true in all viscoelastic materials, and it even can be reversed).

$$G(t) = \frac{\sigma(t)}{\varepsilon\_{k \text{const}}} \tag{14}$$

Mechanical properties of the PU foam are further determined by the temperature T. Authors in [16] already included the influence of temperature T in the analytical model and consecutive constitutive relation is further described by Eq. (15). The relation character (15) was further improved in Ref. [17] by material constants a, b reflecting even the strain rate ε\_ in relation to morphology of the foam, described by Eq. (16). Authors already establish constants in the model, which statistically describe the frequency of air bubbles in the structure. It is necessary to mention that the significant influence of temperature on the foam behavior happens according to the experience of the producers only in foams used to fill the comfort layers of

Figure 14. Relaxation modulus of viscoelastic material depending on time.

modulus Eðε\_Þ are variable and depend on the immediate state of the deformation or transfor-

Figure 13. Nonlinear dependence of stress of PU foam sample (top), and the characteristic courses in three main areas

where fðεÞ describes polynomial function describing the course during compression, mðε, ε\_Þ is

Characteristic properties of the PU foam sample during compression are influenced mostly by the size of the deformation εðtÞ but also time t that takes to compress the cellular structure. Related to this is a physical effect called creep (thinning the structure); in other words, the structure of the PU foam becomes suppler under constant pressure and the deformation increases <sup>ε</sup>ðt2Þj<sup>σ</sup>¼konst <sup>&</sup>gt; <sup>ε</sup>ðt1Þ. Similarly behaves the structure during relaxation of the material—during constant deformation ε ¼ konst (for example, repeated cyclic compression up

the weight of the structure that is variable on deformation ε and the strain rate ε\_.

σ ¼ Eðε\_Þ � fðεÞ � ε, (12)

σ ¼ mðε, ε\_Þ � fðεÞ, (13)

mation ε.

(bottom).

88 Aspects of Polyurethanes

the car seats during high deviations from the room temperature (for example extreme cold �40�C or on the other hand extremely high temperatures above 70�C).

$$
\sigma = h(T, \dot{\varepsilon}) \cdot \mathbf{g}(\rho) \cdot f(\varepsilon),
\tag{15}
$$

$$
\sigma = \sigma\_0(\varepsilon) \cdot h(T) \cdot \left[\frac{\dot{\varepsilon}}{\dot{\varepsilon}\_0}\right]^{a+br} \tag{16}
$$

where hðT, ε\_Þ expresses a step function related to the temperature T and the change in the strain rate ε\_, gðρÞ is experimentally set value related to the structure density, and a, b are material constants ð Þ a, b ≥ 1 .

Viscoelastic behavior of the PU foam can be significantly described using rheological models, i.e. Refs. [8, 14, 16, 17]. Rheology is a science investigating mostly changes in tension σ and transformations in relation to time t and the strain rate ε\_. It stems from the transformation of continuum and therefore does not investigate the structure mechanics (structure morphology and typology, number of air bubbles) compared to the previous relations. Models are created via system of connections of various combinations of Hook elastic elements (springs) and Newton viscose components (damper). They allow for approximate description of non-linear behavior of materials structures including PU foam, by linear components in a various number and combination. It is possible to also study the relaxation and material creep from the obtained relations from the rheological models. The advantage of rheological models is mostly independence on material models in the finite element method programs. In the wide range of publications concerning modeling of PU foam, i.e. Refs. [6, 8], a one-dimensional Kelvin or Maxwell rheological model was used. Authors often mention good congruence of the resultant dependencies of the rheological model in comparison with experimental measurements. Complex cellular structure of the PU foam causes more complicated rheological behavior, where deformation always contains part of elastic, viscose, or sometimes permanent deformation. From the refining values of these one-dimensional rheological models, we can consecutively obtain their n-parameter expansion, shown in Figure 15.

It is possible to create rheological model, which will be getting significantly (in limit) close to the experimentally measured data, by a mutual composition and combination of n-parametric Kelvin and N-parametric Maxwell model, better said by a various number of compounded Hook and Newton elements. According to such model, it will be possible to create a corresponding mathematical expression describing mechanical properties of the PU foam sample, especially the dependence of force/compression or tension/transformation or also dampening and elastic properties (rigidity coefficient K and a dampening coefficient η<sup>t</sup> on the immediate transformation). This can be achieved by a rheological model according to Figure 16. This is a modified n-parametric Tucket model. This model shows practically three characteristic areas. The first part is comprised of a sum of m-number of springs (while m < n), which describes initial solidification characteristic by an elastic deformation. The second part represents an n-parametric Kelvin model, comprised by a sum of m—parallel-connected springs and dampers, which describe the delayed viscoelastic deformation. The third part is then characterized by a sum of m—number of dampers (while m < n), describing final remaining deformation after tension ceases to be applied. The advantage of this model is

Measurement and Numerical Modeling of Mechanical Properties of Polyurethane Foams http://dx.doi.org/10.5772/intechopen.69700 91

Figure 15. N-parametric rheological models: Maxwell model (left) and Kelvin model (right).

the car seats during high deviations from the room temperature (for example extreme cold

<sup>σ</sup> <sup>¼</sup> <sup>σ</sup>0ðεÞ � <sup>h</sup>ðTÞ � <sup>ε</sup>\_

where hðT, ε\_Þ expresses a step function related to the temperature T and the change in the strain rate ε\_, gðρÞ is experimentally set value related to the structure density, and a, b are

Viscoelastic behavior of the PU foam can be significantly described using rheological models, i.e. Refs. [8, 14, 16, 17]. Rheology is a science investigating mostly changes in tension σ and transformations in relation to time t and the strain rate ε\_. It stems from the transformation of continuum and therefore does not investigate the structure mechanics (structure morphology and typology, number of air bubbles) compared to the previous relations. Models are created via system of connections of various combinations of Hook elastic elements (springs) and Newton viscose components (damper). They allow for approximate description of non-linear behavior of materials structures including PU foam, by linear components in a various number and combination. It is possible to also study the relaxation and material creep from the obtained relations from the rheological models. The advantage of rheological models is mostly independence on material models in the finite element method programs. In the wide range of publications concerning modeling of PU foam, i.e. Refs. [6, 8], a one-dimensional Kelvin or Maxwell rheological model was used. Authors often mention good congruence of the resultant dependencies of the rheological model in comparison with experimental measurements. Complex cellular structure of the PU foam causes more complicated rheological behavior, where deformation always contains part of elastic, viscose, or sometimes permanent deformation. From the refining values of these one-dimensional rheological models, we can consecutively

It is possible to create rheological model, which will be getting significantly (in limit) close to the experimentally measured data, by a mutual composition and combination of n-parametric Kelvin and N-parametric Maxwell model, better said by a various number of compounded Hook and Newton elements. According to such model, it will be possible to create a corresponding mathematical expression describing mechanical properties of the PU foam sample, especially the dependence of force/compression or tension/transformation or also dampening and elastic properties (rigidity coefficient K and a dampening coefficient η<sup>t</sup> on the immediate transformation). This can be achieved by a rheological model according to Figure 16. This is a modified n-parametric Tucket model. This model shows practically three characteristic areas. The first part is comprised of a sum of m-number of springs (while m < n), which describes initial solidification characteristic by an elastic deformation. The second part represents an n-parametric Kelvin model, comprised by a sum of m—parallel-connected springs and dampers, which describe the delayed viscoelastic deformation. The third part is then characterized by a sum of m—number of dampers (while m < n), describing final remaining deformation after tension ceases to be applied. The advantage of this model is

ε\_ 0 <sup>a</sup>þb<sup>τ</sup>

σ ¼ hðT, ε\_Þ � gðρÞ � fðεÞ, (15)

(16)

�40�C or on the other hand extremely high temperatures above 70�C).

obtain their n-parameter expansion, shown in Figure 15.

material constants ð Þ a, b ≥ 1 .

90 Aspects of Polyurethanes

Figure 16. N-parametric Tucket model allowing description of nonlinear behavior of PU foam.

mainly the fact that the faster the deformation is supposed to happen, the greater the inhibition effect of the viscoelastic element; therefore, greater force must be applied to achieve desired deformation. The model therefore describes the behavior of viscoelastic material that increases resistance to compression of the applied force by an inner viscose medium. After the force ceases to be applied (final value of compression δj <sup>τ</sup>¼max), the deformation stays maintained in a limit moment εðtÞ � δj <sup>t</sup>¼0, and after some time, recovery follows.

According to the following rheological model (Figure 16), we can construct a corresponding mathematical expression of the compressed PU foam sample by the Eqs. (17)–(24) describing the dependence of the force on compression, for example, rigidity, dampening, or creep suppleness.

$$\sum\_{i=1}^{m} \left( K\_{PL}(\tau\_1) \cdot \overline{\delta}(\tau\_1) + \frac{d}{d\tau\_1} \left( \eta\_{IPL}(\tau\_1) \overline{\delta}(\tau\_1) \right) \right)\_t = F\_z(\tau\_1) \text{ pro } m < n, \tau\_1 < t,\tag{17}$$

$$\sum\_{i=1}^{m} \left( K\_{\text{Pl}}(\tau\_2) \cdot \overline{\delta}(\tau\_2) + \frac{d}{d\tau\_2} \left( \eta\_{\text{IP}1}(\tau\_2) \overline{\delta}(\tau\_2) \right) \right)\_t = F\_o(\tau\_2) \text{ pro } m < n, \tau\_2 < t \tag{18}$$

$$\overline{\delta}(\tau\_1) = \delta(\tau\_1) - \frac{F\_z(\tau\_1)}{K\_{oPlI}} - \frac{1}{\eta\_{tPlI}} \int\_0^{\tau\_1} F\_z(\tau\_1) dt\_\prime \tag{19}$$

$$\overline{\delta}(\tau\_2) = \delta(\tau\_2) - \frac{F\_o(\tau\_2)}{K\_{oPL}} - \frac{1}{\eta\_{tPL}} \int\_0^{\tau\_2} F\_o(\tau\_2) dt\_\prime \tag{20}$$

where Fzðτ1Þ describes stress force in time τ<sup>1</sup> < t, Foðτ2Þ describes relieving force in time τ<sup>2</sup> < t, δðτ1Þ describes the duration of the material compression, that is different in time during hysteresis (significantly longer, shorter, or negligible), among others due to resistance of the material compared with the relief time δðτ2Þ, KPUðτ1, <sup>2</sup>Þ is the immediate value of rigidity of the PU foam sample during compression, and stress relief ηtPUðτ1, <sup>2</sup>Þ is the immediate value of PU foam sample dampening during compression and stress relief.

The functional dependence of the total rigidity and total dampening of the structure can be consequently described.

$$K\_{\rm PL}(t) = \sum\_{i=1}^{n} \left( \frac{\overline{\delta}^{(\tau\_2)} \cdot F\_z + \frac{\overline{\delta}^{(\tau\_1)} \cdot F\_o}{\tau\_1}}{\left( \frac{\overline{\delta}(\tau\_1)}{\tau\_1} + \frac{\overline{\delta}(\tau\_2)}{\tau\_2} \right) \cdot \overline{\delta}(t)} \right)\_t \tag{21}$$

$$\eta\_{tPl1}(t) = \sum\_{i=1}^{n} \left( \frac{\frac{\overline{\delta}(\tau\_1)}{\tau\_1}}{\left( \frac{\overline{\delta}(\tau\_1)}{\tau\_1} + \frac{\overline{\delta}(\tau\_2)}{\tau\_2} \right) \cdot \overline{\delta}(t)} \Big|\_{0}^{t} (F\_z(\tau\_1) - F\_o(\tau\_2)) dt \right)\_t \tag{22}$$

where KPUðtÞ describes total rigidity of the PU foam sample, and ηtPUðtÞ describes total dampening of the sample.

Also through the work difference, it is possible to obtain the relation for the dissipated energy ϑðt, δ, TÞ, which describes energy that can be absorbed by the material.

$$\mathcal{S}(t,\delta,T) = \sum\_{i=1}^{n} (W\_z - W\_o)\_t = \sum\_{i=1}^{n} \oint (F\_z - F\_o)\_t d\mathbf{l}\_\prime \tag{23}$$

where Wz, Wod describe work that the material does during compression and unloading.

Using this model, it is possible to describe creep suppleness ΘðtÞ in Eq. (24).

$$\Theta(t) = \frac{1}{E\_0} (1 - e^{\frac{t}{\tau}}) + \sum\_{i=1}^{m} \Theta(m) \cdot \left( 1 - e^{\left(-\frac{t}{\tau}\right)^m} \right) \text{ for } m < n, \tau\_2 < t,\tag{24}$$

where E<sup>0</sup> is the initial stiffness module.

Xm i¼1

92 Aspects of Polyurethanes

Xm i¼1

consequently described.

dampening of the sample.

KPUðτ1Þ � <sup>δ</sup>ðτ1Þ þ <sup>d</sup>

KPUðτ2Þ � <sup>δ</sup>ðτ2Þ þ <sup>d</sup>

dτ<sup>1</sup> <sup>η</sup>tPUðτ1Þδðτ1<sup>Þ</sup> � � � �

dτ<sup>2</sup> <sup>η</sup>tPUðτ2Þδðτ2<sup>Þ</sup> � � � �

<sup>δ</sup>ðτ1Þ ¼ <sup>δ</sup>ðτ1Þ � Fzðτ1<sup>Þ</sup>

<sup>δ</sup>ðτ2Þ ¼ <sup>δ</sup>ðτ2Þ � Foðτ2<sup>Þ</sup>

foam sample dampening during compression and stress relief.

KPUðtÞ ¼ <sup>X</sup><sup>n</sup>

0

BBB@

<sup>η</sup>tPUðtÞ ¼ <sup>X</sup><sup>n</sup>

i¼1

i¼1

δðτ1Þ τ1 þ δðτ2Þ τ2

ϑðt, δ, TÞ, which describes energy that can be absorbed by the material.

i¼1

<sup>ϑ</sup>ðt, <sup>δ</sup>, TÞ ¼ <sup>X</sup><sup>n</sup>

0

BBB@

δðτ1Þ τ1

� �

KoPU

KoPU

where Fzðτ1Þ describes stress force in time τ<sup>1</sup> < t, Foðτ2Þ describes relieving force in time τ<sup>2</sup> < t, δðτ1Þ describes the duration of the material compression, that is different in time during hysteresis (significantly longer, shorter, or negligible), among others due to resistance of the material compared with the relief time δðτ2Þ, KPUðτ1, <sup>2</sup>Þ is the immediate value of rigidity of the PU foam sample during compression, and stress relief ηtPUðτ1, <sup>2</sup>Þ is the immediate value of PU

The functional dependence of the total rigidity and total dampening of the structure can be

δðτ2Þ

δðτ1Þ τ1 þ δðτ2Þ τ2

<sup>τ</sup><sup>2</sup> � Fz <sup>þ</sup> <sup>δ</sup>ðτ1<sup>Þ</sup>

� �

� δðtÞ

where KPUðtÞ describes total rigidity of the PU foam sample, and ηtPUðtÞ describes total

Also through the work difference, it is possible to obtain the relation for the dissipated energy

ð Þ Wz � Wo <sup>t</sup> <sup>¼</sup> <sup>X</sup><sup>n</sup>

where Wz, Wod describe work that the material does during compression and unloading.

ðt

0

i¼1

∮ ð Þ Fz � Fo <sup>t</sup>

<sup>τ</sup><sup>1</sup> � Fo

� δðtÞ

1

CCCA

ð Þ Fzðτ1Þ � Foðτ2Þ dt

t

t

t

τð1

0

τð2

0

� <sup>1</sup> ηtPU

� <sup>1</sup> ηtPU

¼ Fzðτ1Þ pro m < n, τ<sup>1</sup> < t, (17)

¼ Foðτ2Þ pro m < n, τ<sup>2</sup> < t (18)

Fzðτ1Þdt, (19)

Foðτ2Þdt, (20)

, (21)

, (22)

1

CCCA

t

dl, (23)

Results of the n-parametric Tucket model expressed as the dependence of the tension on the transformation are shown in Figure 17 where courses are in a very good agreement with the experiment. Correlation coefficient comparing between surveyed courses has a value of �0.978. Certain difference is probably given by a fact that the rheological model does not include the structure morphology, i.e., the cell walls bend and from a certain phase of compression create friction between one another. According to Eqs. (21) and (22), it is possible to determine the course of rigidity and dampening of the PU foam sample (Figure 18).

Using appropriate relations, we can determine absolute value of deformation energy EðμÞ or deformation work W(μ) that is necessary to apply to compress such structures. Fibrous structures are not conservative, therefore during the compression depending on the character of the deformation from the initial to final state, according to Ref. [18], it can be considered that

Figure 17. Comparison of nonlinear dependence of stress on strain of PU foam, experiment (line), n-Tucket n-parametric model (dotted line), and correlation coefficient 0.978.

Figure 18. N-parametric Tucket model: course of stiffness and damping depending on compression.

elementary work growth dW used to compress structure is directly proportional to the elementary gain of the deformation energy dE according to the Eq. (25) where elementary energy gain is a total function differential EðμÞ ¼ EμðεiÞ, which can be described by Einstein summary convention according to Eq. (26). Deformation into individual directions ε<sup>i</sup> is possible to further describe using the filling μ according to Eq. (25). It is then possible to gain derivation change (transformation) of the filling in dependence to the deformation ε<sup>i</sup> as per Eq. (26).

$$d\mathcal{W}(\mu) = \mathbb{C}dE(\mu) \text{ pro } \mathbb{C} \ge 1,\tag{25}$$

$$dE = \frac{\partial E}{\partial \varepsilon\_i} d\varepsilon\_i \text{ pro i } = 1, \dots, 3,\tag{26}$$

$$\mu = \frac{V\_V}{V\_{\mathcal{C}}} = \frac{V\_V}{(1+\varepsilon\_i)(1+\varepsilon\_j)(1+\varepsilon\_k)} = \frac{\mu\_0}{(1+\varepsilon\_i)(1+\varepsilon\_j)(1+\varepsilon\_k)}\tag{27}$$

$$\frac{\partial \mu}{\partial \varepsilon\_i} = \frac{\partial}{\partial \varepsilon\_i} \left[ \frac{\mu\_0}{(1 + \varepsilon\_i)(1 + \varepsilon\_j)(1 + \varepsilon\_k)} \right] = \frac{-\mu\_0}{(1 + \varepsilon\_i)^2(1 + \varepsilon\_j)(1 + \varepsilon\_k)},\tag{28}$$

where <sup>W</sup>ðμ<sup>Þ</sup> is a deformation work (W(μ) = <sup>Ð</sup> σdε), C is a constant of proportionality, εi,j, <sup>k</sup> describes deformation into the main direction of extension, and μ<sup>0</sup> is the initial filling, i.e. μ<sup>0</sup> ¼ VV.

The structure creates resistance during deformation against the compression described by the distribution of Cauchy (real) stress tensor σii related to the areas in the deformed continuum,

Measurement and Numerical Modeling of Mechanical Properties of Polyurethane Foams http://dx.doi.org/10.5772/intechopen.69700 95

Figure 19. Elementary continuum of compressed structure.

elementary work growth dW used to compress structure is directly proportional to the elementary gain of the deformation energy dE according to the Eq. (25) where elementary energy gain is a total function differential EðμÞ ¼ EμðεiÞ, which can be described by Einstein summary convention according to Eq. (26). Deformation into individual directions ε<sup>i</sup> is possible to further describe using the filling μ according to Eq. (25). It is then possible to gain derivation change (transforma-

dWðμÞ ¼ CdEðμÞ pro C ≥ 1, (25)

<sup>¼</sup> <sup>μ</sup><sup>0</sup>

<sup>¼</sup> �μ<sup>0</sup> ð1 þ εiÞ 2

dε<sup>i</sup> pro i ¼ 1, …, 3, (26)

ð1 þ εjÞð1 þ εkÞ

σdε), C is a constant of proportionality, εi,j, <sup>k</sup> describes

<sup>ð</sup><sup>1</sup> <sup>þ</sup> <sup>ε</sup>iÞð<sup>1</sup> <sup>þ</sup> <sup>ε</sup>jÞð<sup>1</sup> <sup>þ</sup> <sup>ε</sup>k<sup>Þ</sup> (27)

, (28)

tion) of the filling in dependence to the deformation ε<sup>i</sup> as per Eq. (26).

<sup>¼</sup> VV

<sup>μ</sup> <sup>¼</sup> VV VC

where <sup>W</sup>ðμ<sup>Þ</sup> is a deformation work (W(μ) = <sup>Ð</sup>

∂μ ∂ε<sup>i</sup> ¼ ∂ ∂ε<sup>i</sup>

94 Aspects of Polyurethanes

dE <sup>¼</sup> <sup>∂</sup><sup>E</sup> ∂ε<sup>i</sup>

Figure 18. N-parametric Tucket model: course of stiffness and damping depending on compression.

ð1 þ εiÞð1 þ εjÞð1 þ εkÞ

deformation into the main direction of extension, and μ<sup>0</sup> is the initial filling, i.e. μ<sup>0</sup> ¼ VV.

The structure creates resistance during deformation against the compression described by the distribution of Cauchy (real) stress tensor σii related to the areas in the deformed continuum,

μ0 ð1 þ εiÞð1 þ εjÞð1 þ εkÞ � � which is well described in Ref. [19]. This is described in Figure 19, which shows base configuration of elementary continuum cube, where ε<sup>i</sup> ¼ 0j <sup>t</sup>¼t<sup>0</sup> transforms during increases in deformation into the deformed shape ε<sup>i</sup> 6¼ 0j t>t<sup>0</sup> . Using Cauchy stress tensor σii, it is possible to describe Eq. (25) by the sum of contributions to the main directions of deformation, which is given by Eq. (29). Total stress σHMH described by the von Mises hypothesis (Eq. (30)) based on normal parts of Cauchy stress tensor σii simply describes the compression pressure pk.

$$\sum\_{i=1}^{3} \sigma\_{\text{ii}} d\varepsilon\_i = \mathbb{C} \sum\_{i=1}^{3} \frac{\partial E}{\partial \varepsilon\_i} d\varepsilon\_i \tag{29}$$

$$
\sigma\_{\rm HMH} = \sqrt{\frac{1}{2} \left[ \left( \sigma\_{11} - \sigma\_{22} \right)^2 + \left( \sigma\_{22} - \sigma\_{33} \right)^2 + \left( \sigma\_{11} - \sigma\_{33} \right)^2 \right]} \tag{30}
$$

where σHMH is a total (reduced) tension according to the hypothesis HMH (Huber, von Mises, Hencky), σ11, σ22, σ<sup>33</sup> are the Cauchy stresses in the main directions of the basic coordinate system Xi.

### 3.8. Summary of the property analysis of selected PU foam samples

The property analysis of the chosen PU foam construction samples determined that the tested samples have a low-permeable shell caused by the filled structure in form, and inner structure is permeable. Depending on the specific weight ρ, which can move from 47 to 51 kg m�<sup>3</sup> ,the packing density Ψ (1) increases while influencing the average cell size of the material structure (Table 1). Cellular structure of the PU foam sample influences strain-stress curve during compression. Typical course is characterized by the initial rigidity depending on the strain rate, given stable area so-called plateau and the final steep exponential increase in force.

Mechanical properties are significantly influenced by the character of the cellular structure, and they can be summarized as follows:

