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

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. 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. 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.


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 SiO 4 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 thermomechanical 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: 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 CO 2 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/m 3 . 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/m 3  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 (T 0 ) and foam initial density (D 0 ). 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 D 0 , 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).

Obtaining the softening temperature
Open-cell PU foams (Roya mattress corporation, Tehran, Iran) with D 0 of 20 or 40 kg/m 3 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.

Design of experiments
Two parameters describing the initial foam characteristics including D 0 and T 0 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 Table 1 The list of experiments was designed using the Plackett-Burman method 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).

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).

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 ). 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.

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.

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.  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.

Mechanical properties
The Poisson's ratios in the y direction (ν xy ) and z direction (ν xz ) were defined using the following equations: 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) 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 reduced their thicknesses. The percentage of change in foam thickness (T c ) was defined using the following equation: 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:  Table 2 The ν values of the 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]: foams (i.e. the conventional foams before the thermomechanical procedure) with different D 0 s and T 0 s in ε x of 10% and 20% and were reported in Table 4.
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 Table 3 The T c , D c, and E value of processed samples (± 95% Confidence Interval)  . 6 The load-displacement graph of sample number 9 Table 4 The mechanical properties of conventional foams   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.

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]. For 3D materials, the stress-strain equation is obtained from Hooke's law [36]: Strain energy density is obtained from the following equation [36]: 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.
The elastic stiffness constants of c ij (i, j = 1, 2, 3) 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 These values are consistent with the ones that were experimentally measured (see Table 2).

Effects of geometrical and processing parameters on ν xz changes
According to Table 5, the T e , the T 0 , 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 D 0 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 reentrant structure with a reduced ν value. The effect of T 0 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 T 0 and processing parameters; as the T 0 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).

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, T 0 and applied CR for ε x of 10% and D 0 , 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 , T 0 , 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 D 0 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 D 0 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).

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 , T 0 , 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 T 0 on T c was the opposite; when T 0 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 T 0 and other effective parameters that should be investigated in the future. Also, it is shown in Fig. 11 that the effects of T e , T i , T 0 , 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 T 0 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 D 0 on the D c value was observed.

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 D 0 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  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 T 0 and the D 0 of samples. It is believed that with increasing the D 0 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 .

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.

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.

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 T 0 , 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, T 0 , 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 , T 0 , 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. Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
Availability of data and material All authors make sure that all data and materials as well as software application or custom code support their published claims and comply with field standards.

Data availability
The datasets generated during and/or analysed during the current study are not publicly available due to the large amount of data but are available from the corresponding author on reasonable request.
Code availability Not applicable.

Declarations
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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: where: The mechanical properties (Young's modulus and Poisson's ratios) are the same in both the x and y directions.
Therefore, Hoke's low becomes: where: Then: Also, the stiffness matrix is symmetric as follows: Therefore: Thus, Eq. 8 is obtained:  Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons. org/licenses/by/4.0/.