Effects of geometrical and processing parameters on mechanical properties of auxetic polyurethane foams

Abstract

This study aimed to investigate the influence of processing parameters on the mechanical properties of auxetic polyurethane foams including Poisson’s ratio and Young’s modulus. 12 different processing scenarios were considered using the method of Plackett–Burman in the design of experiments with three replicates for each one. Eventually, 36 foams were prepared with different densities and initial thicknesses, heating temperatures and times, applied compression ratios, and the rest times between two heating steps. The microstructures of the conventional and auxetic samples were observed by scanning electron microscopy (SEM). All samples were subjected to tensile loading in one direction with two different strain values. The strains of the foams in two other directions were recorded using a digital image correlation method. Also, the required force to create each strain value was recorded. The results showed that depending on the changing parameters, Poisson’s ratio of about 42% and 58% of the samples reduced at the strains of 10% and 20%, respectively. Heating temperature and time, the initial thickness of the foam, and the applied compression ratio were proved to have significant effects on the variations of both Poisson’s ratio and Young’s modulus of the foams. It was concluded that the Poisson’s ratio of foams was reduced at higher heating temperature, time, and applied compression ratio and also a lower foam initial thickness. On the other hand, these changes increased Young’s modulus of the polyurethane foams. The strain energy of the auxetic samples showed higher amounts of energy compared to the other foams.

Article highlights

• The thermo-mechanical procedure was used to manufacture auxetic foams.

• Effect of processing parameters on the foam mechanical properties was investigated.

• Foams Poisson’s ratio and Young’s modulus were measured in two strain values.

1 Introduction

Poisson’s ratio, ν,is one of the most important characteristics of materials. It is defined as the negative transverse strain divided by the axial strain. Most of the regular materials have positive ν values, while ν values of auxetic materials are negative. A negative ν would cause materials to yield special properties such as superior energy absorption, causing broad applications of these materials in sports apparel [1] and fenders [2]. Also, they have high values of strain energy release rate which results in a higher amount of fracture toughness [3]. This property is of great importance in fabricating shock absorbers and fasteners [4]. Also, several numerical studies have shown that auxetic materials can be used for applications that require high energy absorption [5, 6].

Auxetic materials first were reported in 1987 with some experiments on isotropic open-cell foams [7]. Also, the ν for different materials was observed to vary in the range of 0–0.5. Love [8] reported a ν of − 0.14 for single-crystal pyrite. Lakes [7] triaxially compressed an open-cell polyhedron foam that resulted in inward bulking of the cell ribs and formation of the “re-entrant” structure in the foam. The compressed foam was placed in a mold at a temperature of 163–171 °C and then it was cooled down at room temperature. This procedure made a permanent transformation of open-cell foams to re-entrant structures [7].

Later, Evans and Alderson [9] claimed that this property could be found in both natural and artificial materials or could be seen in auxetic geometrical structures. Natural auxetic substances such as arsenic, bismuth, and cadmium were found auxetic by applying stress in their longitudinal direction and evaluating the lateral strain [10, 11]. Negative ν was also observed in cube elemental metals (BCC and FCC crystalline lattices) when they have subjected to tensile loading at [110] direction [12]. In 1992, Haeri et al. [13] reported that the alpha-cristobalite molecular structure is a mechanism responsible for auxetic behavior in materials. They obtained this mechanism by the rotation of SiO₄ tetrahedral units [13]. Moreover, some of the biomaterials such as the load-bearing cancellous bone are known as natural auxetic materials [14]. Keyed brick [15] and polyurethane (PU) foams after specified thermo-mechanical procedures are some examples of artificial auxetic materials [16,17,18] and 2D re-entrant metamaterials are an example of auxetic geometrical structures [19].

Auxetic foams are generally manufactured under a thermo-mechanical process using conventional and common open-cell PU foams [20]. To investigate the effects of processing parameters on the ν value of materials, Bianchi et al. produced an auxetic foam using a thermo-mechanical method [21]. They tested cylindrical PU foam samples with different initial diameters at varying heating and cooling conditions with different Compression ratios (CR) defined using the following equation:

$$ CR = \frac{{\left| {Initial\,\, valume\,\, of \,\,foam - Final\,\, valume \,\,of \,\,foam} \right|}}{Initial \,\,valume\,\, of\,\, foam} $$

(1)

They reported that ν values ranged from − 0.09 to − 0.63. It was concluded that CR had the most significant effect among the processing variables on the ν value. The maximal temperature level and cooling method did not significantly affect the amount of ν [21].

Li and Zeng [22] developed a low-cost, environmentally friendly, and ultra-fast method to build auxetic foams; a PU foam was compressed in a pressurized tank by spraying the CO₂ gas in the selected pressure and temperature. So, the distance between the foam particles is reduced. After releasing the pressure, this deformation became permanent to form an auxetic structure with a negative ν of about − 0.5. Also, it was concluded that the tensile strain value in the range of 0–0.5 had no significant effect on the ν value [22].

PU foams also, in turn, show different mechanical properties under different conditions. Constantinescu and Apostol [23] analyzed the effect of temperature and strain rate variation on stress response of polyurethane rigid foams under compressive tests and found that at a strain of 60%, these two parameters did not significantly affect the stress response of these foams with a density of less than 93 kg/m³. However, low temperatures and high speeds of testing were found to cause an increase in the absorbed strain energy of polyurethane rigid foams with a density of 200 kg/m³. Deschanel et al. reported tensile failure experiments on polyurethane foams. They showed that the acoustic energy changes as a power of the foam density and/or temperature [24]. Linul et al. investigated the influence of density and loading direction (in-plane or out of plane) on main mechanical properties of polyurethane rigid foams such as Young's modulus and yield strength. They found that these mechanical properties increase significantly with increasing of density [25].

One of the most common thermo-mechanical processes of manufacturing auxetic PU foams is the thickness compression method [26]. This process consists of three main steps including mechanical compression, heating the compressed foam to its softening temperature, and cooling down the foam to maintain its new structure [27]. In this process, the microscopic structure of the foam was shown to be transformed from a semi-regular arrangement with straight and interconnected cell ribs to a denser, more compact, and tortuous cell structure. In this structure, the ribs could rotate around the corners [28,29,30].

As shown in the studies mentioned above, several parameters are involved in manufacturing auxetic foams that could affect the final mechanical properties of the foams. These parameters include applied CR, the heating temperature (T_e) and heating time (T_i) of the process for the compressed foams, the rest time (RT) between two heating steps, and also the foam initial thickness (T₀) and foam initial density (D₀). Previous studies reported inconsistent results on the effects of parameters such as T_e on the ν and Young’s modulus (E) for the processed PU foams [21, 31,32,33]. On the other hand, these studies did not investigate the interactional effects of these parameters on each other that could affect the final mechanical properties of processed PU foams. Also, the effects of D₀, T_0, and the RT on the final properties of the foams have not been investigated. Therefore, this work aimed to investigate the effects of all of the mentioned parameters on the mechanical properties of PU foams (i.e. ν values in two directions and E values of the processed PU foams).

2 Materials and methods

2.1 Obtaining the softening temperature

Open-cell PU foams (Roya mattress corporation, Tehran, Iran) with D₀ of 20 or 40 kg/m³ were used as the conventional foams. These foams were proved to experience four main steps in the heating process; glassy temperature, softening temperature, flow temperature, and breakdown temperature. The softening would be reached in temperature between the glassy and flow temperatures. In the softening state, the foam showed the elastic–plastic behavior and after reaching the flow temperature, the foam entered the plastic state [27].

Three heat tests were performed to obtain the softening temperature of the foam using a smart digital oven (Parsazma Co., Tehran, Iran). The specimens of PU foam of 60 × 150 mm with a thickness of 5 mm were placed in the oven. In the first case, the temperature was increased from 150 °C with an average rate of 5 °C/min and the foam condition was monitored every minute. At 230 °C, some deformation (i.e. shrinking) of the foam was observed. The second sample was placed at 180 °C and heated up to 220 °C with an average rate of 2 °C/min. A lower thermal rate (1 °C/min) also yielded the same outcome as the rate of 2 °C/min did. Finally, it was obtained that the softening temperature was about 220 °C and the thermal rate had no effect on it.

2.2 Design of experiments

Two parameters describing the initial foam characteristics including D₀ and T₀ were examined. Also, four parameters related to the manufacturing process including T_e, T_i, CR applied to the foams and the RT at room temperature between the two steps of oven placement were investigated. Table 1 lists the parameters with their values tested. 12 different experiments were designed using the Plackett–Burman method, and three replicates were considered for each test (Table 1).

Table 1 The list of experiments was designed using the Plackett–Burman method

2.3 Thermo-mechanical procedure

All the specimens were cut into rectangular shapes and prepared for the tests [34]. The schematic of the prepared samples is shown in Fig. 1a. A compression device of aluminum was built to compress the foams (Fig. 1b). As shown in Fig. 1a, the x, y, and z axes were defined in the directions of length, width, and thickness of the samples, respectively. Each test sample was first compressed through the z direction as shown in Fig. 1c. The distance between the two plates of the pressure device was measured by a caliper to gauge the amount of compression for all four sides of the specimen to apply a uniform pressure. Then, the whole set was placed in the preheated oven with the desire T_e. After reaching the T_i, the device and foam were removed from the oven, and also the pressure was released from the foam for the RT at room temperature. Previous studies showed that the cooling method did not affect the final mechanical properties of foams [21, 31]. Finally, the entire mechanism including the foam was placed again in the preheated oven at a temperature of 50 °C below that of the first heating step for the same time duration as done for the first heating step [26] (Fig. 2).

Fig. 1

a Schematic of samples with different initial thicknesses. b Schematic of the designed compression device. c The compression device with the compressed foam

Fig. 2

Thermal profile of the process for test No. 4 from the beginning to the end of it

2.4 Mechanical testing and data acquisition

Two fixtures were made to attach the samples to the tensile test machine. All the specimens were lined at both ends equally (i.e. 10 mm) and located precisely in the fixtures (Fig. 3a). They were then subjected to tensile loading utilizing a Zwick-Roell Amsler HCT 25-400 tensile testing machine with a maximum load-cell capacity of 20 kN. Tests were performed with a cross-head speed of 5 mm/s to attain either final strain values of 10% or 20% in the x direction (ε_x).

Fig. 3

a Placing the specimen in the tensile test machine. b Two processed foams with stochastic patterns and different thicknesses to track their dimensional changes during the tensile test. c Using the DIC method to obtain the ε_x, ε_y, and ε_z

The amount of strains in y direction (ε_y) and z direction (ε_z) were measured using the digital image correlation (DIC) method (GOM Correlate software, GOM Metrology Co., Schmitzstraße 2, 38,122 Braunschweig, Germany). In this method, a part of the specimen through its thickness and a part of the surface of the foam was painted with a black marker and a stochastic pattern was created to track foam strains in these two directions (Fig. 3b). The specimens were video recorded at 60 frames per second in both y and z directions during the tests (Fig. 3c). The camera had a 20-megapixel resolution with a focal length of 18–135 mm. The strain results of all three directions obtained from the DIC method were validated by measuring some of the dimensions of the loaded specimen randomly using a digital caliper.

2.5 Statistical analysis methods

The 95% confidence intervals of the results were calculated using Student’s t distribution because each test was repeated three times. Also, the factorial analysis and ANOVA general linear model methods were used to analyze the differences between results and investigate whether the differences were significant. Finally, a P value lower than 0.05 was considered statistically significant.

3 Results

The outputs of the experiments were micrographs of the conventional and processed foams, the final thickness of foams, the amount of ε_y, ε_z, and also the amount of applied tensile load (F) applied on the processed samples to reach the desired ε_x values. Moreover, strain energy was calculated for all specimens.

3.1 SEM examination

Figures 4 and 5 show SEM micrographs of the conventional and processed foam of No. 6 (see Table 1), which had negative ν (see Table 2), in two x–z and x–y planes. It can be seen in Fig. 4 that the quasi-regular convex-shape microstructure of the conventional foam has turned into a complex concave-shape with inwardly-buckled cell walls like re-entrant auxetic structure as Duncan et al. mentioned and, consequently, a negative ν was observed [28]. Figure 5 shows no significant change other than a density increase.

Fig. 4

SEM images of the a conventional and b auxetic foams in the x–z plane (i.e. within the thickness of the foam)

Fig. 5

SEM images of the a conventional and b auxetic foams in the x–y plane

Table 2 The ν values of the processed samples (± 95% Confidence Interval)

3.2 Mechanical properties

The Poisson’s ratios in the y direction (ν_xy) and z direction (ν_xz) were defined using the following equations:

$$ \nu_{xy} = - {\raise0.7ex\hbox{${\varepsilon_{y} }$} \!\mathord{\left/ {\vphantom {{\varepsilon_{y} } {\varepsilon_{x} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\varepsilon_{x} }$}} $$

(2-1)

$$ \nu_{xz} = - {\raise0.7ex\hbox{${\varepsilon_{z} }$} \!\mathord{\left/ {\vphantom {{\varepsilon_{z} } {\varepsilon_{x} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\varepsilon_{x} }$}} $$

(2-2)

where ε_x was either 10% or 20%. The average values of ν_xy and ν_xz are reported in Table 2.

Heating the compressed samples to temperatures near to the softening temperature of the foams (i.e. 220 °C) reduced their thicknesses. The percentage of change in foam thickness (T_c) was defined using the following equation:

$$ T_{c} = \frac{{T_{f} - T_{0} }}{{T_{0} }} \times 100 $$

(3)

where T_f is the final thickness of the foams. The T_c values are reported in Table 3. Also, changing the thickness of the samples changed their densities. The percentage of change in foam density (D_c) was defined using the following equations assuming that the variations in foam mass and surface area during the manufacturing process were negligible:

$$ D_{c} = \frac{{D_{f} - D_{0} }}{{D_{0} }}*100 $$

(4-1)

$$ D_{f} = D_{0} *{\raise0.7ex\hbox{${T_{0} }$} \!\mathord{\left/ {\vphantom {{T_{0} } {T_{f} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${T_{f} }$}} $$

(4-2)

Table 3 The T_c, D_c, and E value of processed samples (± 95% Confidence Interval)

where D_f is the final density of the foams. The D_c values are also reported in Table 3.

The load–displacement graphs of tensile tests were registered using a Zwick-Roell load-cell and a magnetic-based displacement sensor and are shown for one of the processed samples (i.e. sample number 9 with ε_x = 20%) in Fig. 6. The graphs were used to measure the amount of F to reach the deformation related to the desired ε_x. The next output of the study was E (i.e. Young’s modulus of the foams). This property was calculated using the following equation [35]:

Fig. 6

The load–displacement graph of sample number 9

foams (i.e. the conventional foams before the thermo-mechanical procedure) with different D₀s and T₀s in ε_x of 10% and 20% and were reported in Table 4.

$$ E = \frac{Tensile\,\, Stress}{{\varepsilon_{x} }} = \frac{F}{{T_{f} *w*\varepsilon_{x} }} $$

(5)

Table 4 The mechanical properties of conventional foams

where w is the width of the sample (i.e. 60 mm). The E values for both ε_x values of 10% and 20% are reported in Table 3. Also, ν_xy, ν_xz, and E values were calculated for the conventional.

According to the results reported in Tables 2, 3, and 4, it is seen that two processed samples (i.e. samples 4 and 6) had a negative ν_xz in both ε_x values of 10% and 20%. Also, all of the ν_xy values remained positive. ν_xy of 42% of processed samples reduced for both ε_x values. On the other hand, 42% of the processed samples at the ε_x of 10% and 58% of samples at the ε_x of 20% had fewer ν_xz values than the related conventional ones. Also, the E value of 92% of processed samples increased in the tensile direction in both 10% and 20% ε_x values. The P value for the effect of each geometrical or processing parameter was calculated using the ANOVA method and reported in Table 5. Table 6 summarizes the influential parameters on the properties of the foams that will be discussed in the following section.

Table 5 P values for the effect of each geometrical or processing parameter on the outputs

Table 6 A summary of the influential parameters

3.3 Strain energy and ν _xz calculation

As seen in Fig. 7, the slope of the stress–strain curves was linear and the foams were in the elastic region within the strain range of 20% consistent with the observations of ref. [21].

Fig. 7

stress–strain curve of foams in the tensile test at a 10% and b 20% strain

For 3D materials, the stress–strain equation is obtained from Hooke's law [36]:

$$ \left[ {\begin{array}{*{20}c} {\sigma_{1} } \\ {\sigma_{2} } \\ {\sigma_{3} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {c_{11} } & {c_{{12{ }}} } \\ {c_{12} } & {c_{22} } \\ \end{array} \begin{array}{*{20}c} {{ }c_{13} } \\ {{ }c_{23} } \\ \end{array} } \\ {c_{13} c_{{23{ }}} { }c_{33} } \\ \end{array} } \right] \left[ {\begin{array}{*{20}c} {\varepsilon_{1} } \\ {\varepsilon_{2} } \\ {\varepsilon_{3} } \\ \end{array} } \right] $$

(6)

Strain energy density is obtained from the following equation [36]:

$$ u = \frac{1}{2}c_{ij} \varepsilon_{i} \varepsilon_{j} { }\left( {i,j = 1,2} \right) $$

(7)

The strain energy can be also calculated from the area under the stress–strain curve. Since all the specimens were only compressed in the z-direction in the fabrication process, the mechanical properties of the other two directions of the samples were assumed to remain uniform in the energy calculations. By comparing the energies obtained from each sample in both ways (Fig. 8), it is seen that Eq. 7 was able to calculate the strain energy with high accuracy (i.e. within 7% of the measured values). Therefore, Eq. 7 could be used to calculate the strain energy density of auxetic foams in the elastic region.

Fig. 8

Bar chart of the Strain energy density of the specimens obtained from Eq. 7 and the area under the stress–strain curve for strains of a 10% and b 20%

The elastic stiffness constants of $c_{ij} { }\left( {i,j = 1,2,3} \right){ }$ can be obtained by applying small deformations and calculating the resultant stress vector or differentiating energy per unit volume, $u$, with respect to strain (see Eq. 7). Then, ν_xz can be derived by the following expressions (the proof of this equation is given in “Appendix A”):

$$ {\upnu }_{xz} = \frac{{c_{13} \left( {c_{12} - c_{11} } \right)}}{{{\text{c}}_{13}^{2} - c_{33} c_{11} }} $$

(8)

After obtaining the elastic stiffness constants, the Poisson's ratio values of − 0.223 and − 0.326 for samples 4 and 6 were obtained, respectively for the strain of 10%. These values are consistent with the ones that were experimentally measured (see Table 2).

4 Discussion

4.1 Effects of geometrical and processing parameters on ν _xz changes

According to Table 5, the T_e, the T₀, the T_i, and the CR applied to the foam had significant effects on the variations of ν_xz for both ε_x values of 10% and 20%. It contradicts some previous observations that claimed only the CR value had a significant effect on ν; i.e. no significant effect of the T_e level on ν was observed [21, 31, 37, 38]. However, some investigations showed a substantial effect of T_e and T_i on this property [32, 33]. It is believed that these discrepancies in the observations of different studies are due to the interactional effects of some parameters on each other that were not incorporated in previous studies. The effects of D₀ and RT on ν_xz were found insignificant.

The influence of each parameter on ν_xz changes for different ε_x values is shown in Fig. 9. It can be seen that in both ε_x values, increasing the values of the T_e, T_i, and CR reduced the ν_xz. It is believed that when these parameters increased, the foam formed to a permanent new re-entrant structure with a reduced ν value. The effect of T₀ was the opposite so that when it increased, the amount of ν_xz increased. It is believed that is because of the interactional effects of T₀ and processing parameters; as the T₀ increased, higher values of T_e, T_i, and CR should be applied to the samples to decrease their ν_xz values. Also, it should be noted that the ε_x value did not influence the ν_xz variations significantly and the results were almost the same for both ε_x values (P value = 0.847).

Fig. 9

Effect of different parameters on the ν_xz changes for ε_x of a 10% and b 20%

4.2 Effects of geometrical and processing parameters on ν _xy changes

As reported in Table 5, the rest time, RT, between two heating stages had the most significant effect on the variations of ν_xy for both values of ε_x. Also, T₀ and applied CR for ε_x of 10% and D₀, applied CR, and T_i for ε_x of 20% significantly affected the ν_xy changes. Nevertheless, the T_e had no significant effect on ν_xy variations in any of the ε_x values. Therefore, the effects of the varying parameters on ν_xy changes were almost different from their effects on ν_xz variations. It is believed that this difference was observed because the compression was only applied in z direction and the structure of the foam in the y direction did not significantly change during the thermo-mechanical procedure.

As seen in Fig. 10, T_i, T₀, and the applied CR had a direct relationship with the ν_xy changes for both values of ε_x. But this correlation for the RT and foam D₀ was inverse. It could be noted that the changes that occurred in ν_xy and ν_xz could be somewhat independent of each other by changing the initial and processing parameters. As reported in Table. 5 and Fig. 10, the most effective parameters on ν_xy variations were the RT and D₀ that did not affect ν_xz. Also, the most influential factors on ν_xz changes had less effects on ν_xy variations. It can be seen in Fig. 10 that the effect of the other parameters was almost independent of the ε_x value (P value = 0.747).

Fig. 10

Effect of different parameters on the ν_xy changes in the ε_x of a 10% and b 20%

4.3 Effects of geometrical and processing parameters on T _c and D _c

According to Table 5, the influential parameters on the T_c and D_c of the samples were similar to those that affected the value of ν_xz. T_e, T_i, T₀, and applied CR had the most significant effects on both T_c and D_c. As it is shown in Fig. 11, increasing T_e, T_i, and also the applied CR caused T_c to become more negative and, therefore, T_f value decreased. It is believed that applying compression in temperature near the softening point of the foams (i.e. 220 °C) would cause plastic deformation in them. So when the T_e increased to a temperature near the softening temperature (i.e. 200 °C or 150 °C) and also CR and T_i increased, the T_c value (i.e. the plastic deformation of foams) would grow too. But the effect of T₀ on T_c was the opposite; when T₀ increased, the T_c value became closer to zero and the permanent deformation ratio in the thickness of the foam decreased. It is hypothesized that this is due to the interactional effects of T₀ and other effective parameters that should be investigated in the future.

Fig. 11

Effect of different parameters on a T_c and b D_c

Also, it is shown in Fig. 11 that the effects of T_e, T_i, T₀, and applied CR on D_c opposed their effects on T_c. This was predictable due to Eqs. 4-1 and 4-2 where it was shown that with a decline in the amount of T_f (i.e. T_c being more negative), the D_f and D_c would increase. It could be also observed that when the T_e and T_i were elevated or the T₀ value declined, the D_c value of the samples increased. Also, consistent with the observations of ref. [31], D_c and D_f values became larger when the applied CR increased. It should be noted that no significant effect of D₀ on the D_c value was observed.

4.4 Effects of geometrical and processing parameters on E changes

The statistical analysis reported in Table 5 showed that for both ε_x values of 10% and 20%, the most influential parameters on the variations of E value were T_e, T_i, T_0, and D₀ of the foams. Also, the applied CR in the ε_x value of 10% affected the changes in the E value in the thermo-mechanical procedure significantly. But the RT did not affect E variations significantly in any strain value. Figure 12 shows that the changes in the E value of the samples are directly related to T_e and T_i and also the applied CR on the foam. This is consistent with the findings of Refs. [31, 33, 37] where an increase in CR was shown to increase the E value. It is hypothesized that with the growth of these parameters (i.e. reduction of T_f), the processed foam would have a more compact and denser structure (see Sect. 4.3) and the amount of porosity in the structure would reduce, rendering an increase in the E variations and so the E value.

Fig. 12

Effect of different parameters on the E changes in the ε_x of a 10% and b 20%

Figure 12 shows a direct relationship between T_e and changes of E. But previous investigations have reported different conclusions on this correlation [31, 32]. It is believed that if the T_e increases to a value more than the softening temperature, the foam will enter the flow temperature region and will have plastic behavior and its E value will reduce [27]. On the other hand, the E variations were found to be inversely related to the T₀ and the D₀ of samples. It is believed that with increasing the D₀ of samples and reduction of their porosity, their behavior would become closer to elastic conditions and, hence, the changes in their E value would decrease.

Finally, it is seen in Fig. 12 that the ε_x value did not significantly change the effect of the parameters on E (P value = 0.839). It is believed that this is due to the elastic behavior of conventional and processed PU foams in ε_x values of lower than 20% [39]. This behavior of the foam material is also illustrated in Fig. 6 that shows an almost linear relationship between the applied load and the displacement of the foam. According to Tables 5 and 6, it could be observed that the parameters that affect the E and ν_xz variations are the same. This implies these two properties could be dependent on each other. Also, Figs. 9 and 12 show that the effect of parameters on the E was the opposite of their effect on the ν_xz.

4.5 Comparison of strain energies and Poisson's ratios in theoretical and experimental calculations

It was observed that the strain energy value in foams with a negative Poisson's ratio (i.e. samples No. 4 and 6) was higher than that for the other specimens. However, the foams with a reduced Poisson's ratio showed increased strain energy compared to the conventional specimens. The difference of the calculated Poisson's ratio with the measured value was approximately 8% for samples No. 4 and 6. It shows that Poisson's ratio of auxetic foams can be obtained using a stress–strain curve of the tensile test.

4.6 Limitations of the study and future studies

This investigation aimed to observe the effects of six different factors on several output properties such as ν in two directions and the E value of the products. A large number of experiments were performed and each sample required a long time for the thermo-mechanical procedure. Also, performing the tensile test on each sample in two ε_x values and data acquisition with the DIC method in two directions increased the required time to perform each test. Therefore, this study used a Plackett–Burman design of experiments method to reduce the number of the required tests that were suitable to detect the most effective parameters on the results. Also, this study identified the factors that had almost insignificant effects on the desired properties to eliminate them from future studies. Another experiment design would be needed to investigate the exact effect and interactions of the identified significant parameters to optimize the properties.

5 Conclusions

The main goal of this study was to detect influential parameters and compare their effects on different mechanical properties of PU foams after the thermo-mechanical process of manufacturing auxetic materials. The tensile test and DIC of the samples were used to obtain stress–strain curve, ν and E values for all specimens. It was concluded that T₀, T_e, T_i, and applied CR had the most significant effects on ν_xz, E, and D_c. But the effect of these parameters on E was the opposite of their effect on ν_xz and D_c. On the other hand, RT, T₀, and CR had the most significant effects on ν_xy. Generally, the most effective parameters on almost all of these outputs (i.e. ν_xz, ν_xy, E, T_c, and D_c) were T_e, T_i, T₀, and applied CR on them during the heating procedure. Strain energy values were calculated in both experimental measurements and theoretical relationships, and the two auxetic specimens showed a remarkably higher stored energy compared to the non-auxetic ones. These investigations on the effective parameters would help to optimize the final mechanical properties (i.e. cost function) of the PU foams such as ν_xz, E, or energy dissipation depending on the desired applications.

Appendix A

By convention, the 9 elastic constants in orthotropic constitutive equations are comprised of 3 Young's moduli, E_x, E_y, E_z, 3 Poisson's ratios, ν_yz, ν_zx, ν_xy, and 3 shear moduli, G_yz, G_zx, G_xy that are assumed to be zero. Therefore, the stiffness matrix takes the form:

$$ \left[ {\begin{array}{*{20}c} {\sigma_{xx} } \\ {\sigma_{yy} } \\ {\sigma_{zz} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {\begin{array}{*{20}c} {\frac{{1 - \nu_{yz} \nu_{zy} }}{{E_{y} E_{z} \Delta }}} \\ {\frac{{\nu_{xy} + \nu_{xy} \nu_{zy} }}{{E_{z} E_{x} \Delta }}} \\ {\frac{{\nu_{xz} + \nu_{xy} \nu_{yz} }}{{E_{x} E_{y} \Delta }}} \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\frac{{\nu_{yx} + \nu_{zx} \nu_{yz} }}{{E_{y} E_{z} \Delta }}} \\ {\frac{{1 - \nu_{zx} \nu_{xz} }}{{E_{z} E_{x} \Delta }}} \\ {\frac{{\nu_{yz} + \nu_{xz} \nu_{yx} }}{{E_{x} E_{y} \Delta }}} \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\frac{{\nu_{zx} + \nu_{yx} \nu_{zy} }}{{E_{y} E_{z} \Delta }}} \\ {\frac{{\nu_{zy} + \nu_{zx} \nu_{xy} }}{{E_{z} E_{x} \Delta }}} \\ {\frac{{1 - \nu_{xy} \nu_{yx} }}{{E_{x} E_{y} \Delta }}} \\ \end{array} } \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\varepsilon_{xx} } \\ {\varepsilon_{yy} } \\ {\varepsilon_{zz} } \\ \end{array} } \right] $$

(A1)

where:

$$ \Delta = \frac{{1 - \nu_{xy} \nu_{yx} - \nu_{yz} \nu_{zy} - \nu_{zx} \nu_{xz} - 2\nu_{xz} \nu_{yz} \nu_{zx} }}{{E_{x} E_{y} E_{z} }} $$

(A2)

The mechanical properties (Young's modulus and Poisson's ratios) are the same in both the x and y directions.

$$ \nu_{xy} = \nu_{yx} ,\nu_{zx} = \nu_{zy} , \nu_{xz} = \nu_{yz} , E_{x} = E_{y} $$

(A3)

Therefore, Hoke’s low becomes:

$$ \left[ {\begin{array}{*{20}c} {\sigma_{xx} } \\ {\sigma_{yy} } \\ {\sigma_{zz} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {\begin{array}{*{20}c} {\frac{{1 - \nu_{xz} \nu_{zx} }}{{E_{x} E_{z} \Delta }}} \\ {\frac{{\nu_{xy} + \nu_{xy} \nu_{zx} }}{{E_{z} E_{x} \Delta }}} \\ {\frac{{\nu_{xz} + \nu_{xy} \nu_{xz} }}{{E_{x} E_{x} \Delta }}} \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\frac{{\nu_{yx} + \nu_{zx} \nu_{xz} }}{{E_{x} E_{z} \Delta }}} \\ {\frac{{1 - \nu_{zx} \nu_{xz} }}{{E_{z} E_{x} \Delta }}} \\ {\frac{{\nu_{yz} + \nu_{xz} \nu_{yx} }}{{E_{x} E_{x} \Delta }}} \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\frac{{\nu_{zx} + \nu_{yx} \nu_{zx} }}{{E_{x} E_{z} \Delta }}} \\ {\frac{{\nu_{zx} + \nu_{zx} \nu_{xy} }}{{E_{z} E_{x} \Delta }}} \\ {\frac{{1 - \nu_{xy} \nu_{yx} }}{{E_{x} E_{x} \Delta }}} \\ \end{array} } \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\varepsilon_{xx} } \\ {\varepsilon_{yy} } \\ {\varepsilon_{zz} } \\ \end{array} } \right] $$

(A4)

where:

$$ \frac{{\nu_{yz} }}{{E_{y} }} = \frac{{\nu_{zy} }}{{E_{z} }}, \frac{{\nu_{zx} }}{{E_{z} }} = \frac{{\nu_{xz} }}{{E_{x} }} , \frac{{\nu_{xy} }}{Ex} = \frac{{\nu_{yx} }}{{E_{y} }} $$

(A5)

Then:

$$ \left[ {\begin{array}{*{20}c} {\sigma_{xx} } \\ {\sigma_{yy} } \\ {\sigma_{zz} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {\begin{array}{*{20}c} {\frac{1}{{E_{x} E_{z} \Delta }} - \frac{{\nu_{xz}^{2} }}{{E_{x}^{2} \Delta }}} \\ {\frac{{\nu_{yx} }}{{E_{z} E_{x} \Delta }} + \frac{{\nu_{xy}^{2} }}{{E_{z}^{2} \Delta }}} \\ {\frac{{\nu_{xz} \left( {1 + \nu_{yx} } \right)}}{{E_{x}^{2} \Delta }}} \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\frac{{\nu_{yx} }}{{E_{x} E_{z} \Delta }} + \frac{{\nu_{xz}^{2} }}{{E_{x}^{2} \Delta }}} \\ {\frac{{1 - \nu_{zx} \nu_{xz} }}{{E_{z} E_{x} \Delta }}} \\ {\frac{{\nu_{xz} \left( {1 + \nu_{yx} } \right)}}{{E_{x}^{2} \Delta }}} \\ \end{array} } \hfill & {\begin{array}{*{20}c} {\frac{{\nu_{xz} \left( {1 + \nu_{yx} } \right)}}{{E_{x}^{2} \Delta }}} \\ {\frac{{\nu_{zx} + \nu_{zx} \nu_{xy} }}{{E_{z} E_{x} \Delta }}} \\ {\frac{{1 - \nu_{yx}^{2} }}{{ E_{x}^{2} \Delta }}} \\ \end{array} } \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\varepsilon_{xx} } \\ {\varepsilon_{yy} } \\ {\varepsilon_{zz} } \\ \end{array} } \right] $$

(A6)

Also, the stiffness matrix is symmetric as follows:

$$\left[ {\begin{array}{*{20}{c}}{{\sigma _1}}\\{{\sigma _2}}\\{{\sigma _3}}\end{array}} \right] = \left[ {\begin{array}{*{20}{l}}{{c_{11}}}&{{c_{12{\rm{~}}}}}&{{c_{13}}}\\{{c_{12}}}&{{c_{11}}}&{{c_{13}}}\\{{c_{13}}}&{{c_{13{\rm{~}}}}}&{{c_{33}}}\end{array}} \right]~\left[ {\begin{array}{*{20}{c}}{{\varepsilon _1}}\\{{\varepsilon _2}}\\{{\varepsilon _3}}\end{array}} \right]$$

(A7)

Therefore:

$$ c_{11} = c_{22} $$

(A8)

$$ c_{12} = c_{21} $$

(A9)

$$ c_{13} = c_{23} = c_{31} = c_{32} $$

(A10)

Thus, Eq. 8 is obtained:

$$ {\upnu }_{xz} = \frac{{c_{13} \left( {c_{12} - c_{11} } \right)}}{{{\text{c}}_{13}^{2} - c_{33} c_{11} }} $$

(8)