Auxetic polymers

Auxetic polymers

3.1

3

Introduction

Among all types of manmade auxetic materials, auxetic polymers are the first successfully manufactured auxetic materials [1]. Compared to the other types of materials, the polymeric materials have advantages of versatility, easy modification and shape formability etc. [2] and have been studied and developed to fit applications in a large range of fields since last century. Due to their special properties, polymeric materials have become one of the most promising materials for realising auxetic behaviour. In 1987, Lakes [3] successfully manufactured the first auxetic polymeric material by converting a conventional polyurethane (PU) foam through a thermomechanical process. Since then, the development of manmade auxetic materials is widely investigated around the world. This chapter provides an introduction on the basic and novel auxetic conversion methods and mechanical properties of auxetic polymers [1,2,4]. The auxetic polymers can find a large number of applications in industries and societies. In addition, they can be applied with textile materials to create auxetic composites resulting in enhanced properties [5].

3.2

Auxetic polymeric foams

Auxetic foams are the most commonly reported auxetic polymeric materials. Theoretically, auxetic foams can be fabricated through postprocessing of conventional foams or direct synthesising process. However, so far, there are no directly synthesised foams exhibiting auxetic behaviour. In the most cases the auxetic behaviour of foam is realised through changing the conventional honeycomb structure of cells to the auxetic structure. Generally, there are three methods to fabricate auxetic foams, namely thermomechanical method, chemomechanical method and three-dimensional (3D) printing. Also, auxetic foams can be made from various thermoplastic (polyester urethane and polyether urethane), thermosetting (silicon rubber) and metallic (copper) materials [2].

3.2.1 Thermomechanical method The thermomechanical method relates to applying compression force to the conventional cellular foam and then to heating the foam up to the softening temperature,

Auxetic Textiles. DOI: https://doi.org/10.1016/B978-0-08-102211-5.00003-6 © 2019 Elsevier Ltd. All rights reserved.

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thus changing the normal hexagonal cells structure of the foam to the reentrant auxetic hexagonal structure to achieve the auxetic behaviour [3]. The thermomechanical process is the first and most commonly reported method for converting the conventional foam to the auxetic foam. In this method the conventional open-cell PU foam is first put in a mould and triaxially compressed to change the structure of cells inside the foam. Through the triaxial compression, the outward ribs of cells are redirected to protrude to inward forming reentrant structure. Then, the mould is heated up to the temperature between 163
C and 171
C, which is slightly higher than the softening temperature of the PU foam while it remains the compressed state in the mould. The heating process will melt the forced inward ribs and break the molecule connection to help maintain the conversed cells structure. Therefore the foam microstructure is reshaped to the auxetic structure. After that, the mould will be cooled down to the room temperature, and the auxetic foam is created. The cooling effect will store the potential energy in the ribs, which keeps the ribs remaining inward state after the compression force is released. Through these three steps, the auxetic foam is produced. The permanent volumetric compression ratio (VCR) (the ratio of the volume change between the original foam and the compressed foam after the conversion) of the manufactured auxetic foams is between 1.4 and 4 and the foams within this range of VCR exhibit auxetic effect. It was found that the lowest Poisson’s ratio obtained is 20.7 when the VCR is 2. On the other side, the PU foams exhibit auxetic behaviour despite the variation in the cells size (0.3, 0.4 and 2.5 mm) when they share the similar structure, properties and transformation procedure according to Lakes’s research. Although the polymeric foams produced using the above method proposed by Lakes present auxetic behaviour, these auxetic foams have several problems and limitations. For example, the auxetic foams show a long-term instability and tend to recover to their original shape and structure; the foams suffer from severely surface creasing and wrinkles; the thermomechanical method cannot be used to produce large auxetic foam [2]. It was found by Bianchi et al. [6] and Scarpa et al. [7] that after the conversion process the auxetic foams would attempt to reverse to their original shape, and the final VCR of the conversed foams could be decreased by 52% of the initial ratio. The density of the foams also changed up to 30% after 1 week of the conversion process. This recovering behaviour is caused by the surface creasing and wrinkling. According to the characteristics of polymeric materials, the crease and wrinkle occur when the foam is subjected to a long-term stress and the heating process is not sufficient enough. It was also found that the crease and wrinkle would appear at the more deformable region of the auxetic foams. In order to address the problem of the crease and further solve the instability of the conversed foams, several precaution measures could be taken: (1) applying special lubricant (oil-derivative and distilled oil lubricants are not applicable) on the inner surface of the mould; (2) inserting wires or tweezers inside the foam to pull the foam instead of pushing it; (3) redesigning the mould and the compressing method to create an even triaxial compressing; thus no creasing area is formed during the manufacturing [2].

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To find a more effective solution to resolve the instability and crease of auxetic foams and to produce large auxetic foams, the optimised heating temperature and time, the compression ratio and the influence of different materials were examined by Chan and Evans [8]. To do that, two kinds of polyester urethane foam (60 ppi closed-cell and 60 ppi reticulated) and three kinds of polyether urethane foam (10, 30, 60 ppi open-cell) were utilised to fabricate the auxetic foams under various manufacturing conditions. All the foams were cut to the designed size by using a Burgess band saw fitted with a knife blade to guarantee that the foams were cut smoothly, and there were no crease and wrinkles created on the surface. For the heating process, the oven was firstly preheated to 200
C. The prepared foam was put into a square sectional aluminium tube with the help of WD40 lubricant and spatula, which were the keys to eliminate creasing when the foam was compressed, as shown in Fig. 3.1. At this stage, the foam was submitted to the compression in two transverse directions. After that, two aluminium end plates lubricated by WD40 were used to seal the tube and to compress the foam in the longitudinal direction with the help of clamps. In this way, the foam could be compressed in all three directions without crease and wrinkle. The mould with the compressed foam was then placed into the preheated oven for heat-setting process. After that, the mould was taken out of the oven and cooled down at the room temperature for 15 minutes. At last, the foam was removed from the mould and stretched by hand gently in all three directions to prevent adhesion of the foam. It was found that the heating temperature should be 5
C20
C lower than the softening temperature to prevent disorder of the cell ribs, and the heating time was related to the materials and should be long enough to heat warm up the centre of the foam to the required temperature. The work showed that using appropriate heating time and heating temperature is crucial for having auxetic foam without creasing. When compressing a foam block with a larger size such as 300 mm 3 300 mm 3 100 mm, it is unpreventable to create creases and wrinkles on the surface of the foam due to higher compressing force employed and uneven

Figure 3.1 Compression of foam at preparation stage [8].

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collapse of the foam. To solve this problem, Chan and Evans [8] developed a multistage heating and compression process, in which the foam is compressed with a small VCR for each time to prevent creasing, and the overall compression ratio is achieved by conducting four-time compressions. For example, if the conventional foam has a size of 300 mm 3 300 mm 3 100 mm and the targeted compression ratio is 3.4, the first stage will be to compress the foam size to 275 mm 3 275 mm 3 69.5 mm with heat setting, then 250 mm 3 250 mm 3 63 mm with heat setting, followed by 225 mm 3 225 mm 3 56.5 mm with heat setting and finally 200 mm 3 200 mm 3 50 mm with heat setting. After each compression and heating process the mould should be placed in the room temperature for cooling down, and the foam should be removed from the mould following the same procedure described in the manufacturing of small auxetic foam. Fig. 3.2 shows the arrangement of the compression process, and Fig. 3.3 illustrates one stage of the multistage process for compressing the foam. As shown in Fig. 3.3, the whole setup consists of one open mould, two end plates and two clamps. Since the foam is compressed to four different sizes, four corresponding mounds for compression are required. The open mould and the end plates were made from 5 mm thick stainless steel with the inner wall lubricated by WD40. The clamps were used to fix the end plates tightly and provided even compression in the vertical direction. Through

Figure 3.2 Compression process for manufacturing a large auxetic foam block [8].

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Figure 3.3 A stage of compression process [8].

Figure 3.4 One type of arc shape for auxetic foam.

conducting the multistage heating and compression method, the large-size auxetic foam block could be produced with good homogeneity and auxetic effect.

3.2.2 Novel thermomechanical methods Extending from the original thermomechanical conversion process proposed by Lakes, a number of researches have been focusing on optimising the manufacturing process or developing new materials aiming to improve the mechanical properties and the auxetic behaviour of auxetic foams. Bianchi et al. [9] introduced a novel fabricating method for producing auxetic foams with curved and complex shapes such as arc auxetic foam with large size. In this method, a half-mould manufacturing process is adopted, in which the mould is only used on one side as the base and the compressing pressure is applied by vacuum bag on another side. The open-cell PU foam is used in the conversion process. Fig. 3.4 shows the arc circular shape of a special mould for the auxetic conversion. The conventional PU foam pad is firstly put onto the mould, and a thermal sensor is inserted for monitoring the inside temperature of

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the foam. Then, the foam-mould assembly is covered by a thin fluorinated ethylene propylene release film followed with a medium weight polyester nonwoven breather blanket to form a foammouldblanket assembly. Next, the assembly is placed onto a tooling plate for sealing with a dedicated bag and a double-sided sticky tape. After sealing, the assembly is connected with a vacuum pump to reduce the pressure inside the chamber formed by the blanket, foam and mould. The pressure of the chamber is reduced to 0.7 bar and the blanket-film membrane is then drawn to push the foam due to the pressure difference between the inside and outside environments. Thus the foam on the mould is compressed against the mould surface forming a semicircular shape. The whole assembly is then placed into a preheated autoclave for reshaping the internal structure of the foam. During the heating process, the pressure in the oven needs to be kept the same as the pressure before heating. After the foam is evenly heated, the assembly is taken out of the autoclave and the blanketfoammould assembly is removed from the tooling plate and put into the water for cooling with the vacuum state inside the chamber. When the foam reaches surrounding temperature, it is taken out of the bag and stretched randomly. Experiments show that this compressed foam exhibits excellent auxetic behaviour and has high homogeneity of the mechanical properties through the whole foam. Fig. 3.5A displays its stressstrain curves under tensile force along three different directions. It is clearly shown that the foam exhibits the highest stress when stretched along the longitudinal direction (x-direction). On the contrary, the foam exhibits a weak stress behaviour when stretched along the through-the-thickness direction (z-direction). Fig. 3.5B shows the Poisson’s ratio strain curves when the foam is subjected to both the longitudinal and transverse stretches. The auxetic effects are found to be decreased with the increase of the strain in both directions. This phenomenon can be explained by the opening effect of the deformed cell ribs. With the increase of the strain, the ribs will be stretched more, thus decreasing the auxetic performance. The highest auxetic effect is achieved at the strain of 5% with a Poisson’s ratio of 21.26 and 20.96 for stretching along the longitudinal direction and transverse direction, respectively. In addition, it is also found that the

Figure 3.5 (A) Stressstrain curve of the auxetic foam in three directions; (B) Poisson’s ratiostrain curves [9].

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through-the-thickness Poisson’s ratio is 20.15 at the maximum strain of 30%, which proves that the auxetic foam produced possesses auxetic effect in all directions. Meanwhile, with the Poisson’s ratio smaller than 21, the foam is believed to exhibit anisotropic property. Another development for producing auxetic foam using the thermomechanical method was presented by Duncan et al. [10]. In this method, rows of pins are used to control the lateral compression of foam sheets aiming to improve the structural homogeneity and properties. Three types of auxetic foams are produced with different sizes and ways including 150 mm 3 150 mm 3 150 mm cubic foam, 150 mm 3 150 mm 3 30 mm cuboid foams and 150 mm 3 150 mm 3 30 mm pinned cuboid foams for comparison. The first two types of foams are produced using the normal thermomechanical auxetic conversion process, while the third one is controlled by pins during auxetic conversion. The VCR of all foam samples is 2.9. Lubricant is also applied onto the inside surface of the mould for all samples. For the pinned conversion, 16 pins, which are arranged in a square shape with a spacing of 43 mm between each other, are firstly inserted into the foam, as shown in Fig. 3.6A. Then, a lower U-shaped case of a two-part mould is placed under the foam. The U-shaped case has 16 holes with a spacing of 30 mm. The pins penetrated in the foam will pass through the holes of the U-shaped case and further insert into a wooden block beneath to fix the foam and the mould (Fig. 3.6B). Finally, the upper U-shaped case of the mould (with the same holes and arrangements as the lower case) is put onto the foam and fitted by rods. By this way the foam will be compressed in all three orthogonal directions. After the mould is sealed the wooden block is removed from the assembly as presented in Fig. 3.6C. In this development the heating processes are divided into two stages. The cubic mould is put in the oven heating for two 35 minutes periods at 180
C and then an annealing process will be conducted for another 35 minutes at 100
C, which is able to lock the reentrant structure of the cells of the foam. After that, the mould is left in room temperature until it is cooled down to the ambient temperature. Finally, the

Figure 3.6 Manufacturing process of the pinned auxetic foam: (A) foam cuboid with through-thickness pins inserted; (B) foam with pins are inserted in the lower mould; (C) the whole mould assebly [10].

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Figure 3.7 Auxetic foam sample fabricated with through-thickness pins [10].

Figure 3.8 Poisson’s ratio of different foams [10].

foam is taken out of the mould and stretched in all three orthogonal directions. For the two cuboidal foam-mould assemblies, the heating and annealing time is reduced to 25 minutes as they are thinner. Fig. 3.7 shows an auxetic foam sample fabricated using through-the-thickness pins. The pinned cuboidal foam clearly exhibits less folding effect on the foam surface as the pins can well control the in-plane compressing process of the foam. Using pins can be useful when fabricating large-size auxetic foams to reduce the surface creasing as described above. Fig. 3.8 shows the comparison of the mean Poisson’s ratio between the conventional foam and the three types of auxetic foams fabricated. Foams 1, 2, 3 and 4 represent the conventional foam, the cubic moulded auxetic foam, the cuboidal moulded auxetic foam and the through-thethickness pinned cuboidal auxetic foam, respectively. It can be seen that the unconverted foam has a positive Poisson’s ratio, and all the converted foams exhibit auxetic behaviour. However, the Poisson’s ratio of all auxetic foams are very close to zero, which means their auxetic effect is not strong. Compared

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Figure 3.9 Cells structure of different foams: (A) cuboidal foam; (B) cubic foam; (C) conventional foam [10].

with the Poisson’s ratio of the foams (with a smaller size) manufactured by Allen et al. [11], the Poisson’s ratio of all samples produced in this method are constrained by relatively high surface friction between the mould and the foam due to a relatively large contact area and low sample thickness. The microscopic images of the cuboidal foam, the cubic foam and the conventional foam are shown in Fig. 3.9. These microscopic images show that the cuboidal foam has distinctive reentrant structure of the cells (Fig. 3.9A), and the cubic foam only has slightly reentrant structure of the cells (Fig. 3.9B). Bianchi et al. [12] have also produced another auxetic foam with large changes in density of the foams. Two types of polymeric foams were used, including a light blue PU-polyethylene(PE) (PU-PE) open-cell foam (with ppi of 2835 and density of 27 kg/m3) and a grey PU-PE open-cell foam (with ppi of 3035 and density of 27 kg/m3). The glass transition temperatures of both the foams are 114
C. The light blue foam is cut to 24 cylindrical samples with a same length of 70 mm but with four different diameters of 58, 48, 37, 30 mm. Each size has six samples. These samples are labelled as batch A, B, C, D and converted to the auxetic foams through the following steps: (1) all the samples are inserted into a special metallic mould which has a 27 mm internal diameter and is sealed by two internal sliding calibrated discs; thus the VCRs of 2, 1.8, 1.4 and 1.1 are achieved for samples A, B, C and D, respectively; (2) the mould is put into an oven which has been preheated at 200
C with a temperature increasing from 20
C to 135
C in 15 minutes; (3) after the centre of the foam is heated up to the required temperature (monitored by a thermocouple sensor), the mould is taken out of the oven and cooled down in the water stream until the temperature is decreased to 20
C; (4) all samples are gently stretched to prevent adhesion of the foam. The grey foam is conversed to auxetic foam samples using another method. The foam is firstly cut to two types of cylindrical shapes, one with a diameter of 30 mm and a length of 180 mm and another with a diameter of 50 mm and a length of 120 mm, which are labelled as batches E and F. Each batch has five specimens. Then, the cut foams are put into a multitube mould with different sizes of each tube. Therefore the samples of batch E are compressed with VCR of 6.4, 6,

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5, 4.5 and 4.1, and the samples of batch F are compressed with VCR of 7.6, 9.2, 11.1, 12.8 and 15.1. After that, the assemblies of moulds and foams are placed in the oven for 45 and 60 minutes at 150
C and 170
C. After the heat-setting process, the foams are immediately removed from the mould and stretched, and then cooled in the room temperature in a released form. It is clear that the cooling process for these foams is different from the auxetic conversion process used for the light blue foam. It is found that the structure, density and surface of novel manufactured grey auxetic foams are different from the blue foams. Since the grey foams are removed from the mould instantly after the heating, the temperature of each foam is not evenly distributed. The out layer of the foam is cooled down faster, and a relatively large temperature difference is created between the core of the foam and the outer layer. Therefore the outer layer of the foam becomes more compact and stiffer, resulting in a higher density in the outer layer of the foam. For the auxetic foams of batch A, B, C and D, they are observed to have a tendency of recovering to their original shape, as shown in Fig. 3.10. However, the specimens of batch E and F do not show this trend. This can be explained by the stiffer outer layer of the novel manufactured auxetic foams, which is able to fix the reentrant shape of the cells inside the foam and to create a stable geometry of the foam. Unlike the auxetic foams fabricated by the normal thermomechanical procedures (the blue auxetic foams), the grey auxetic foams only exhibit auxetic behaviour under compression force. Fig. 3.11A and B shows the Poisson’s ratiocompressive-strain curves of the foams of batch E manufactured at two different final temperatures of 170
C and 150
C, respectively. As observed from Fig. 3.11, the Poisson’s ratio firstly decreases first and then increases with increasing the compressive strain. It can be seen that both the VCR and final

Figure 3.10 Diameter changes versus time for batches A, B, C and D [12].

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Figure 3.11 Poisson’s ratiocompressive-strain curves of batch E samples: (A) foam manufactured under 170
C; (B) foam manufactured under 150
C [12].

heating temperature have obvious effect on the auxetic behaviour of the foams. It is clearly shown that the lowest Poisson’s ratio of 20.34 can be achieved with a heating temperature of 150
C and a compression ratio of 4.5. The effect of the temperature on the auxetic behaviour is also measured. It can be found that the foams produced at 170
C have better auxetic effect when the compressive strain is smaller than 0.35. However, when the compressive strain keeps increasing to above 0.35, the foams produced at 150
C will exhibit better auxetic effect than the foams produced at 170
C.

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3.2.3 Chemomechanical method In addition to the thermomechanical conversion method, the chemomechanical method can also be used for manufacturing auxetic foams. Grima et al. [13] produced a new type of polymeric auxetic foam by using the chemical solvent to replace the heating process. This method is able to reduce the collateral damages caused by the heating process and the energy consumption for the manufacturing process. It is also proven that the auxetic foam manufactured by the chemical procedure can be reconvert to its conventional state. The open-cell 30 ppi PU foam was used for the conversion process. The original foam, the chemomechanical conversed foam, the normal thermomechanical conversed foam and the conventional foam reconverted from the chemomechanical foam are represented by Foam A, Foam B, Foam C and Foam D, respectively. Foam A is tailored in a cylindrical form with a diameter of 40 mm and a length of 84 mm. Then the tailored foam is compressed to a diameter of 26 mm and a length of 55 mm remaining the same configuration. After that, a filter paper is used to wrap the compressed foam to help remain the compressed configuration of the foam. Next, the wrapped and compressed foam is immersed in the acetone for 1 hours and air dried thereafter (remain compressed form). Finally, the auxetic foam is produced without using heating process. In order to compare the auxetic behaviour and mechanical performance between the chemomechanically produced auxetic foam and the normal thermomechanically produced auxetic foam, Foam C is produced from Foam A following the general thermomechanical procedure with the same shape and size of Foam B. For Foam D, some samples of Foam B are put in the acetone in the free state aiming to reconvert Foam B to the nonauxetic foam. When the foam is immersed in the acetone, it is observed that the foams reexpand in all directions. These specimens are then air dried and found to have positive Poisson’s ratio, and they have similar size and shape to the original Foam A. Fig. 3.12 shows the microstructure of all four foams at high magnification. The cells of Foam A have conventional foam model such as honeycomb for twodimensional (2D) and Kelvin cells for three-dimensional (3D) (Fig. 3.12A). Regarding to Foam B and Foam C, their cells have the basic features such as reentrant structure and the rotation rigid structure for having auxetic behaviour (Fig. 3.12B and C). Besides, the cells of Foam B share a very similar geometry of the cells of Foam C, resulting in the similar auxetic performance between the two foams. From the quantified measurements of the PR, Foam B is found to have the Poisson’s ratio values ranging from 20.29 to 20.37 and those of Foam C from 20.32 to 20.36. The similarity of the Poisson’s ratio and the microstructure of these two foams confirm that the chemomechanical method can produce auxetic foams as good as the foams produced by the thermomechanical method. Fig. 3.12D shows that the cells of the reconverted Foam D are straightened again, and the geometries of the cells are similar to those of Foam A, resulting in the nonauxetic performance of Foam D. The chemomechanical method suggests a completely new procedure to produce auxetic foams. Although this method is only at the preliminary stage as only

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Figure 3.12 Microstructures of all samples: (A) Foam A; (B) Foam B; (C) Foam C; (D) Foam D [13].

acetone and PU are used for conversion and the manufacturing process is not fully optimised, it is a highly promising method for manufacturing auxetic foams. The preliminary work by Grima et al. [13] has proven the possibility of producing auxetic foams with the same auxetic performance and probably better mechanical performances by using purely chemical and mechanical processes.

3.2.4 Three-dimensional printing 3D printing is an additive manufacturing method in which the materials are deposited layer by layer based on the structure designed by the computer-aided design (CAD) software to create 3D products. To date, 3D printing technology is widely applied in art, engineering, medical and manufacturing fields etc., and it can be used to produce both polymeric and metallic products. Therefore it is possible to utilise 3D printing to create polymeric objects with auxetic geometries and to achieve auxetic behaviour of products. Through implementing 3D printing in the foam manufacturing, the random distribution and orientation of cells can be prevented as the whole structure is designed and realised by the computer. At present, numerous papers have reported the geometrical models of auxetic structures such as reentrant model through computer simulation. Based on this digital reentrant model, Critchley et al. [14] proposed a 3D reentrant model and therefore produced pliable auxetic foam by using 3D printing method. The cellular structure of the auxetic foam is designed using CAD software. Fig. 3.13A illustrates a 2D geometry of the computer-designed single cell of the foam. This 2D cell is

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Figure 3.13 (A) 2D model of auxetic cell; (B) 3D model of auxetic cell; (C) designed auxetic foam structure [14]. 2D, Two-dimensional; 3D, three-dimensional.

then transformed to a 3D cell which is merged with an identical 3D cell to form a single symmetrical 3D auxetic cell of the foam (Fig. 3.13B). Thereafter, the symmetrical 3D cells are arranged and layered to form a whole foam structure. The first layer is formed with the arrangement of 5 3 5 followed by the second layer with the arrangement of 4 3 5. This double layer structure is then repeated for five times in the height direction to create the whole auxetic foam structure and two end plates

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with a dimension of 18 3 18 3 1 mm are inserted to the two sides of the model. Fig. 3.13C shows the entire cells arrangement and the auxetic structure of the foam with a dimension of 18 3 18 3 44 mm. According to this design, two batches of three samples are manufactured using a Connex 350 Objet printer. One of the batches is made from polymeric composite material TangoBlack, and another batch is made from a different polymeric composite TangoBlack85. Both TangoBlack and TangoBlack85 are made from combinations of various organic chemicals. During the manufacturing process, a support material is deposited throughout the samples, which requires removal after the fabricating process. Therefore after the printing, all samples will firstly be put in a potassium hydroxide (KOH) solution and shaken for 24 hours. Then the samples will be rinsed in water and dried. This cleaning process will be repeated several times until the support material is completely cleaned from the foam. In order to examine the accuracy and quality of the printing, both macroscopic images and scanning electron microscope (SEM) images of the foam are taken. Fig. 3.14A shows the original computer-designed 3D structure of the foam, while Fig. 3.14BD shows the macroscopic, computerised tomography (CT) and SEM images of the foam, respectively. From the macroscopic image of the foam (Fig. 3.14B), it can be seen that the physical model of the foam has the same cellular structure as the CAD designed model. However, some damages such as broken ribs are found throughout the whole sample from the CT image (Fig. 3.14C)

Figure 3.14 Images of 3D printed auxetic foam structure; (A) computer-designed structure; (B) macroscopic image; (C) CT image; (D) SEM image [14]. 3D, Three-dimensional; CT, computerised tomography.

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and the SEM image (Fig. 3.14D). These damages are found to be caused during the cleaning stage in which the foam is placed in the KOH solution multiple times. Consequently, the water permeates into the polymeric material and causes its swelling. The swelling effect will increase the stress between the cells and finally result in breaking and cracking of ribs. Besides, the printing accuracy is also an important parameter as it has direct effect on the overall performance of the 3D printed product. In this study, it is found that all samples have an accuracy error of 6 0.1 to 0.3 mm, indicating the unavoidable structure distortion and inaccurate production. Fig. 3.15 presents the Poisson’s ratio values obtained at different tensile strains for all the 3D printed foam samples. It can be seen that all the foams exhibit auxetic behaviour at a strain of 0.05, and the lowest Poisson’s ratio is found to be 21.18. However, the negative Poisson’s ratio effect decreases with increasing the strain. Some of the samples are even found to have positive Poisson’s ratio as the base material (0.3), when stretched at high strain. Meanwhile, the predicted Poisson’s ratio of these samples obtained from the mathematical model suggest a linear relationship between the Poisson’s ratio and the strain, which is almost in agreement with the experimental measurements, though there are errors of the predictions caused by the broken ribs and error of the structure, etc.

3.2.5 The mechanical properties of auxetic foams The auxetic conversion not only changes the Poisson’s ratio of the polymeric foam materials, also can largely enhance their mechanical properties such as shear

Figure 3.15 Poisson’s ratio versus strain data for three TangoBlack foams (KW&), three TangoBlack85 foams (◇% 3 ) and the mathematical model (▼) [14].

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resistance, indentation resistance, fracture toughness, compression performance, shear modulus, stiffness, acoustic dampening, dynamic performance and viscoelastic loss [1,2,5,1520]. Many of these property changes are caused by the reconstruction of the cellular structure and redistribution of the material density during the conversion process. More specifically, these improvements are due to the variations of Young’s modulus (E), shear modulus (G), bulk modulus (K) and Poisson’s ratio (v) which are the four elastic constraints of the material and define the stiffness, rigidity, compressibility and volumetric change under strain, respectively. The four elastic constraints have interaction effect to each other, which has a great effect on the overall mechanical properties of the material. This influence is illustrated by Eqs. (1.11), (1.14), (1.15) and the following equation:
 1 3K 2 2G v5 2 3K 1 G

(3.1)

These equations can briefly illustrate the mechanical properties enhancements of the auxetic materials theoretically. However, the changes of mechanical properties are complicated, which are hard to be simply described by these equations. This section provides a deliberately understanding of the mechanical property variations with respect to the deformation behaviour and all major mechanical properties. During tensile or compressive loadings, the deformation process of the auxetic foam is different from conventional foam as the cellular structure has been changed. Chan and Evans [21] first discussed the deformation behaviour of the auxetic foams through microscopic examination of the microstructure. Due to the complex reentrant geometry of the cells, during stretching or compression, the internal structures of auxetic foams tend to hinge and flexure rather than stretching. Fig. 3.16AC shows the SEM pictures of the reticulated auxetic PU foam at unloaded, tension loaded and compression loaded states, respectively. The figures indicate that the deformation mechanism of the auxetic foam is similar to that of conventional foam. However, as auxetic foam cells have the reentrant shape, the cells and the foam become wider in tension and narrower in compression. It should be noted that since the cells of auxetic foams deform through hinging and flexure, the strain distribution is different when the foam is under tensile or compressive loadings. To measure the strain distribution of the auxetic foams, Pierron et al. [22,23] used X-ray CT and digital volume correlation for the images collection. The volumic strain maps of the conventional and auxetic foams are obtained under tension loading by several load steps. Fig. 3.17A and B shows the strain maps of conventional foam and auxetic foam specimens at the loading step 1, respectively. These strain map images indicate that the standard foam has reasonably homogeneous strain distributions with Poisson’s ratio, but the auxetic foam specimen exhibits an opposite heterogeneous strain distributions. This uneven distribution of the strain is probably due to the complex and random cellular structure. Since the cells are not in the idealised state and arrangement, the auxetic effect is

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Figure 3.16 SEM images of the reentrant cells of the auxetic foam: (A) unloaded state; (B) under tension loading; (C) under compressive loading [21].

not uniform throughout the whole foam, which is demonstrated by zones of positive and negative transverse strains on the strain maps. In addition to the strain distribution, the strain maps confirm that the auxetic conversion process is not uniform and some parts of the foam is not conversed or the cells are conversed to the direction which cannot bring auxetic effect to the foam. Clearly, two basic mechanical properties (tensile and compressive) of the foams are affected by the auxetic conversion. To describe other mechanical properties, Scarpa et al. and Pastorino et al. [7,24,25] carried out a number of tensile tests including quasistatic tensile tests, dynamic tensile tests and tensile fatigue tests for the auxetic PU foam. The stressstrain curves of the auxetic and conventional foams under the quasistatic tensile tests are shown in Fig. 3.18. For the auxetic foams, all samples have different initial sizes (same diameter with different lengths), but the same compressed size; therefore the all auxetic foams have different compression ratios. Sample 1L1, 2L1, 3L1, 4L1 and 5L1 have the compression ratios of 93.6, 92.8, 91.7, 90.4 and 88.4, respectively. It is observed that all auxetic conversed foams have better tensile properties than the base foams. The majority of the conventional foams have a maximum stress of 23 kPa, while the auxetic specimens show higher maximal stress. For example, the auxetic sample 1L1 shows a

Figure 3.17 Strain maps of foams at tension loading step 1: (A) conventional foam; (B) auxetic foam [23].

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Figure 3.18 Quasistatic tensile stressstrain curves of auxetic foams and conventional foam [7].

stress of 2.5 MPa. Also, the VCRs of the auxetic foams lead to the enhancement of mechanical properties in most situations. Scarpa et al. [7] also pointed that the tensile tests of the auxetic foams do not exhibit the classical elastic-plateau region in the stressstrain curve but having an almost continuous linear slope pattern. This reflects the influence of the auxetic conversion to the overall stressstrain pattern of the foam. The dynamic tensile property of the auxetic foam and its base foam is measured through conducting the tensile test with a constant strain rate. The constant strain rate value is given by the following equation: ε5

v l0

(3.2)

where v is the imposed velocity on the sample, l0 is the initial length of the specimen. Experiments (Fig. 3.19) show that the dynamic tensile property is similar to the quasistatic property, this is, auxetic foam has better mechanical stiffness and presents the stressstrain curve without plateau area. It is also found by Bezazi and Scarpa [25] that the tensile fatigue resistance of the auxetic foam is better than conventional foam. Fig. 3.20 shows a comparison of the maximum loading degradation F/F0 between the auxetic foam and conventional foam at two different loading levels (r), where F and F0 are the maximum load at each testing cycle and the maximum load at the first cycle, respectively, and r is the ratio between the maximum displacement at a specific level and the maximum displacement at failure of the foam. It is clearly shown that the auxetic foam provides better tensile fatigue resistance compared to the conventional foam. Bezazi and Scarpa [25] also found that, at a low loading level, the difference between the

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Figure 3.19 Constant strain-rate tensile stressstrain curves: (A) auxetic foam; (B) conventional foam [7].

auxetic and conventional foams in fatigue resistance is small, but that difference becomes big at high loading level. The compressive behaviour of the auxetic foam is also found to be enhanced compared with conventional foam by Cadamagnani et al. [26], Bezazi and Scarpa [27], Chan and Evans [28], Frioui et al. [29] and Pastorino et al. [24]. The compressive stressstrain curves of one type of auxetic and conventional foams are shown in Fig. 3.21. These auxetic foams are manufactured from the conventional foams with the same initial dimension (length of 180 mm and diameter of 30 mm), but they are converted to the auxetic foams with different lengths (70, 75, 90, 100 and 110 mm) by keeping the diameter constant (19 mm). The foams with these different lengths are nominated as 2C8, 3C8, 4C8 and 5C8. It can be seen that the auxetic specimen has higher compressive stress versus strain, and the variation trend is

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Figure 3.20 Stiffness degradation versus number of cycles for both auxetic and conventional foams at different tension loading levels [25].

Figure 3.21 Compressive stressstrain curves of auxetic foams and conventional foam [26].

similar to that of the tensile curve. The auxetic foams do not exhibit the obvious plateau region neither in the compressive curve. Besides, the compressive fatigue resistance between the auxetic foams and the conventional foam shows an opposite trend with the tensile fatigue resistance. In this case the conventional foam loses less stiffness with the increase of using cycles in all loading levels. One example is shown in Fig. 3.22 when the compressive loading level r is 0.95. Extending from the discussion about the basic mechanical properties, the indentation resistance (also refers to hardness) of the auxetic foams is found to be largely increased due to the negative Poisson’s ratio [3,3035]. When the collision happens between an object and the auxetic foam in one direction, the auxetic foam will contract laterally as shown in Fig. 1.2, which indicates the material

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Figure 3.22 Stiffness degradation versus number of cycles for different foams under compressive loading [27].

‘moves’ to the impact point and increases the local density. Consequently, the auxetic foam exhibits higher indentation resistance. Lakes [3] theoretically studied the effects of indentation on an auxetic foam. It is found that for isotropic materials, the indentation is proportional to (1 2 v2)/E when the indentation object has localised pressure distribution. Consequently, for a material with a Poisson’s ratio approaching the thermodynamic limit v 5 21, indentation is extremely difficult to happen. Meanwhile, this material is very compressible as the shear modulus is higher than the bulk modulus. This phenomenon is explained by the relationship among the shear modulus, bulk modulus and Poisson’s ratio as described below: K5

2Gð1 1 vÞ 1 2 2v

(3.3)

Conversely, according to Eq. (3.3), a material with the Poisson’s ratio close to 0.5 such as rubber is incompressible as the bulk modulus is larger than the shear modulus. Smith et al. [31] also showed that the indentation resistance of auxetic foam is dependent on the bulk density and modulus. When the indentation happens, the density of auxetic foam and the strain field under the indenter will become larger due to the improved shear modulus. Lakes [36] studied the effect of indentation on an auxetic wrestling mat, and the indentation was assumed to be indentation rigidity: P E 5 ð1 2 v2 Þ w 2a

(3.4)

where a is the radius of a circular localised pressure, P is the localised pressure and w is the indentation depth.

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The indentation processes are divided into small and large impacts. For small impacts, the elastic half space yielding is taken as the circular pressure, and the following equation is derived:   F Gan 5 u narrow 1 2 v

(3.5)

where F is the indentation force, u and an are the maximum displacement and the radius for narrow indentation, respectively. On the other hand, for the wide indentation (the impacts greater than the mat thickness), the compression and the indentation force are assumed to be uniformly distributed and the following equation is obtained:   Ga2w F 2hð1 1 vÞ 5 u wide 1 2 2v

(3.6)

where h is the mat thickness measured in mm, and aw is the radius for wide indentation. From Eqs. (3.4)(3.6), Lakes concluded that the continuum theory of elasticity may not be enough to describe the auxetic behaviour of the auxetic foam and suggested to apply Cosserat theory of elasticity which considers both the maximum stresses and a natural length scale. Therefore the microstructural size of the material can be taken into account in the prediction of failure. Later, Evans and Alderson [37] confirmed Lakes’s assumption by assuming the indentation of auxetic foams as an effect of uniform pressure distribution and proposed that the indentation could be proportional to [(1 2 v2)/E] 2 1. Based on the classic elasticity theory, it is further noticed that the indentation resistance increases towards infinity for a negative Poisson’s ratio. The fracture toughness of the auxetic foams is improved with the negative Poisson’s ratio effect. A discussion on the fracture toughness for both the regular tetrakaidecahedron and reentrant tetrakaidecahedron foams is briefly introduced here by citing the work of Choi and Lakes [38]. The fracture toughness of the regu lar tetrakaidecahedron, KIC , is given as a function of relative density:  KIC ρ pffiffiffiffiffi 5 0:19 ρs σf πl

(3.7)

where ρ denotes the density of the foam, ρs is the density of the solid from which the foam is made；σf is the fracture strength of the cell rib and l is its length. Following the same procedure which was used for conventional foams, the fracture toughness of reentrant foams is given as follows: NPR KIC

pffiffiffiffiffi 5 σf πl

0:1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi 1 1 sin π2 2 ϕ ρ 1 1 cos 2ϕ

ρs

  where ϕ π=4 , ϕ , π=2 is the rib angle (see Fig. 3.23).

(3.8)

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Figure 3.23 (A) Structural and continuum views of a crack in a conventional foam; (B) collapsed tetrakaidecahedron model for a reentrant foam cell; (C) cross-section view of the reentrant cell used for analysis [38].

From Eqs. (3.7) and (3.8), the following equation can be obtained: NPR KIC  KIC

5 0:53

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi 1 1 sin π=2 2 ϕ 1 1 cos 2ϕ

(3.9)

Experimental results for the fracture toughness of both the conventional and reentrant foams are displayed in Fig. 3.24, and compared with the analytical model, that is Eq. (3.9), which gives the ratio of fracture toughness of the reentrant foam to that of the conventional one and proves better fracture toughness of the auxetic one. For the energy absorption, it has been proven in many experiments that auxetic foams have overall superior energy absorption capacity. The auxetic foams have also demonstrated their better absorption of ultrasonic, acoustic and damping compared to conventional foams. Scarpa et al. [39] reported that the auxetic foam case possesses excellent dynamic crushing properties, while the resilience of the conventional foam at high constant strain rate loading is relatively low. Other experimental results also showed the remarkable quasistatic energy absorption behaviour [2]. It is known that the toughness is a critical mechanical property for the energy absorption performance of polymeric porous materials as it decides the maximum energy absorption ability of foam per unit. In 1987, Lakes [3] found that the toughness of the auxetic foam would be influenced by the Poisson’s ratio. The relationship between the toughness and the Poisson’s ratio can be explained by the following equation: σ5

πET 2r ð1 2 v2 Þ

(3.10)

where σ is the stress, T is the surface tension and r is the circular crack radius. From the above equation, it can be seen that, when the Poisson’s ratio approaches 21, the stress can be infinitely high, which suggests the materials can be extremely tough. Choi and Lakes [40] also reported the relationship between the toughness and the VCR when auxteic foams were subjected to both tensile and compressive

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Figure 3.24 Experimental fracture toughness: open symbol, conventional foam; solid symbols, reentrant foam. The fracture toughness of reentrant foams is normalised with that of conventional foams, so the ordinate is dimensionless. Analytical result [solid circle, Eq. (3.9)], is shown for comparison [38].

strains and found that the toughness of the foams manufactured with an increased VCR was increased to a large extent. The energy absorption performance of auxetic foam was also studied by Bezazi and Scarpa [25] through the quasistatic cyclic loading fatigue test. It was found that the auxetic foam exhibits excellent energy absorption over a large number of loading cycles. The energy dissipated by per unit volume of the auxetic foam is defined as follows: Ed 5

ð εmax εmin

σdx

(3.11)

where Ed is the energy dissipated per unit volume, εmax is the maximum strain and εmin is the minimum strain. The auxetic and conventional foams are subjected to cyclic loading under different loading levels to measure the energy absorption, and the correspondent hysteresis loops are shown in Fig. 3.25. The graphical data indicate that the hysteresis cycle areas for all samples increase with increasing the loading level and the auxetic foams dissipate around three times more energy than the conventional foam, indicating greater energy absorption performance of the auxetic foams. The auxetic foams also have unique acoustic absorption properties compared with their base material [39,41,42]. The acoustic absorption data exhibited in Fig. 3.26 show that the auxetic conversion process significantly changes the

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Figure 3.25 Load versus displacement at different loading levels: (A) auxetic; (B) conventional foams [25].

acoustic properties of the foam. In the lower frequency range from 0 to 1000 Hz, the auxetic foam has higher acoustic absorption coefficient, which indicates better acoustic absorption performance. For example, at 500 Hz the auxetic foam has an absorption coefficient of 0.4 while the conventional foam only possesses 0.3. However, at higher frequencies, the situation becomes more complicated. For the conventional foam, the absorption coefficient will fluctuate with the increase of the

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Figure 3.26 Acoustic absorption coefficient of auxetic foam and conventional foam [41].

frequency after 1000 Hz. But for the auxetic foam, its performance is relatively stable and the absorption coefficient remains around 0.85. It can be seen that the conventional foam has better sound absorption (absorption coefficient around 1) when the frequency ranges between 10002300 Hz and is higher than 3800 Hz. For the bending property, when the conventional foams are subjected to an outof-plane bending force, the conventional foams will be slightly stretched in one direction and the foam of the opposite direction will be naturally contracted [1]. Therefore the conventional foams will exhibit anticlastic curvature as the edges of the foam will curl. This curl is caused by the unbalanced force distribution as shown in Fig. 1.3A. On the contrary, for the auxetic foams, the bending force will lead to a synclastic curvature forming a dome shape (Fig. 1.3B) without the help of any other machine or force [43]. The synclastic curvature of the auxetic foams can give an edge of being used for making protective device such as knee pads. Overall, the auxetic conversion can endow a number of unique and superior properties to the polymeric foams, which allows the auxetic foams to replace the conventional materials used in different applications as the next generation materials.

3.3

Other types of auxetic polymers

3.3.1 Microporous polymers In 1989, Caddock and Evans [44,45] firstly investigated the auxetic microporous polymeric material. An expanded form of polytetrafluoroethylene (PTFE) is found to exhibit a highly anisotropic negative Poisson’s ratio as low as 212. This high auxetic behaviour can be explained by its complex nodulefibril microstructure. It is observed that all the nodules in the PTFE polymer are interconnected by many

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85

fibrils; thus the deformation behaviour of the structure is nodule translation through hinging of fibrils, which can reproduce the similar deformation behaviour of the reentrant structure. Through the investigation on the microstructure of the expanded PTFE, the concept of producing other polymeric materials having the similar microstructure to obtain auxetic performance has been proposed. Various processes including compaction, sintering and extrusion have been developed for the manufacturing of the microporous polymers such as ultrahigh molecular weight PE, polypropylene and nylon [4648]. The compaction stage gives structural integrity to the extrusion, which is a crucial stage to bring proper mechanical properties to the polymer during manufacturing. It has been observed that without compaction process, the polymeric materials can have auxetic effect but the mechanical properties and density are very low. The most distinctive features of the microporous auxetic polymers are the superb indentation resistance and the absorption of ultrasound. Up to four times improvement of the indentation resistance of the auxetic microporous polymers has been found when compared with conventional polymers [46]. In addition, the fibrillar form of the internal structure of the microporous polymers are proven to be extremely good at absorbing vibration signal such as ultrasound signal which becomes undetectable after passing through the microporous polymers. Therefore the auxetic microporous polymeric materials can be applied for various applications. Although the auxetic microporous polymers have several superb advantages compared with their counterparts, most of these polymers are made into cylindrical form which is difficult to be used in the application-based work. Thus auxetic polymeric fibres such as polypropylene, polyester and nylon having the microporous structure are also produced [49]. These microporous polymeric fibres can find a range of applications in different applications. For example, they can be used to produce fibre reinforced composites and technical textile applications, which will be specifically discussed in Chapter 4, Auxetic fibres and yarns, Chapter 9, Auxetic fibre reinforced composites, and Chapter 10, Applications of auxetic textiles.

3.3.2 Auxetic natural polymers Apart from those manmade auxetic polymeric materials, several types of natural materials have also been found. The first auxetic natural polymeric material found is the cat skin, which was discovered by Veronda and Westmann [50]. Then cow teat skin was found to exhibit negative Poisson’s ratio by Lees et al. in 1991 [51]. To verify auxetic behaviour, the cow teat skin was cut in different sizes with different aspect ratio (the ratio of length to width), and these cut skin samples were subjected to uniaxial and biaxial tensile tests. It was found that the cow teat skin would have auxetic effect when the aspect ratios were 1.4 and 2.46, as shown in Fig. 3.27, and the best auxetic performance was obtained when the aspect ratio was 1.4 at a strain of 0.1. In addition to the animal skins, other types of auxetic natural material have also been found. For example, Peura et al. [52] has recently reported that crystalline cellulose Iβ exists negative Poisson’s ratio in the xz plane when the loading is along

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Figure 3.27 Poisson’s ratio of skin strips cut from cow teat [51].

the cellulose chain direction. Cellulose II has also been reported to have in-plane auxetic performance when loaded in the cellulose chain direction [53]. It is believed that the negative Poisson’s ratio of these natural materials comes from their molecular structures.

3.3.3 Auxetic nanopolymers In order to further improve the mechanical performance such as strength and stiffness of auxetic polymeric materials, operating their microstructure at molecular or nanoscale is necessary [1,2]. So far, although various molecular models to achieve auxetic effect have been proposed, no practical auxetic polymers by operating the microstructure at molecular or nanolevel have been made. Therefore only the modelling studies of auxetic polymers are presented here. The first molecular model was the reentrant honeycomb structure derived from the macroscopic reentrant structure [5456]. As shown in Fig. 3.28, the reentrant structure of the molecular model can realise negative Poisson’s ratio, but it is too heavily cross-linked to achieve in practice. In addition, the auxetic effect of the reentrant structure can only be realised in one single molecular layer as the molecule may be randomly arranged for the bulk material. Fig. 3.29 shows another molecular model structure for realising auxetic effect proposed by Wei [57]. It is based on the flexyne/reflexyne networks to form a selfassembled copolymer with the hydrogen-bonded polymer network. The molecule structure of the copolymer is formed with a double arrowhead shape ‘hard’ block and a spring shape ‘soft’ segment. This molecular structure can be self-assembled into 2D molecular network. The potential polymer having this molecular network is predicted to have auxetic effect and to be easier to fabricate real polymer. Grima and Evans [58] developed a series of self-expanding molecular networks based on the polyphenylacetylene arrows. These networks have similar shape as the graphite network which is an infinite planar network as shown in Fig. 3.30. The

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Figure 3.28 The reentrant structure at molecular level [56].

Figure 3.29 Double arrowhead molecular structure [57].

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Figure 3.30 Structure of the self-expanding model [58].

Figure 3.31 Double calix structure [59].

simulation models of these networks suggest very high in-plane auxetic effects (around 20.97) due to the cooperative rotation of the interconnected triangles. Grima et al. [59] also proposed another auxetic molecular structure based on the calix arene molecular building blocks. The proposed molecular network is shown in Fig. 3.31, from which it can be seen that this structure cannot be easily realised. To date, the most simple and promising approach to successfully fabricate molecular-level polymer is the concept of liquid crystalline polymer (LCP) (Fig. 3.32) proposed by He et al. [60,61]. The auxetic effect of LCP is dependent on the reorientation of the side-connected rods in the main molecular chain. The LCP compromises chains of rigid rod molecules interconnected by flexible spacer groups along the whole molecule chain. There are two types of rigid rods. Some of

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Figure 3.32 LCP polymer molecular chains: (A) at unloaded state; (B) at loaded state. LCP, liquid crystalline polymer.

them are terminally attached at the ends of the rods, while others have lateral attachment. When there is no stress applied, all the rigid rods are oriented along the main chain direction, whereas in the stressed situation, the extension of the molecular chains will force the laterally attached rods to rotate to the direction vertically and to push neighbouring chain apart, thus causing an expansion of the polymer. The computer modelling of this structure has been studied, and the auxetic behaviour has been predicted. Meanwhile, the practical synthesis of the LCP polymer has been widely studied.

Reference [1] Alderson A, Alderson K. Auxetic materials. Proc Inst Mech Eng, G: J Aerospace Eng 2007;221(4):56575. [2] Critchley R, Corni I, Wharton JA, Walsh FC, Wood RJ, Stokes KR. A review of the manufacture, mechanical properties and potential applications of auxetic foams. Phys Status Solidi (B) 2013;250(10):196382. [3] Lakes R. Foam structures with a negative Poisson’s ratio. Science 1987;235:103841. [4] Saxena KK, Das R, Calius EP. Three decades of auxetics research 2 materials with negative Poisson’s ratio: a review. Adv Eng Mater 2016;18(11):184770. [5] Liu Y, Hu H. A review on auxetic structures and polymeric materials. Sci Res Essays 2010;5(10):105263. [6] Bianchi M, Scarpa F, Smith C. Shape memory behaviour in auxetic foams: mechanical properties. Acta Mater 2010;58(3):85865. [7] Scarpa F, Pastorino P, Garelli A, Patsias S, Ruzzene M. Auxetic compliant flexible PU foams: static and dynamic properties. Phys Status Solidi (B) 2005;242(3):68194. [8] Chan N, Evans K. Fabrication methods for auxetic foams. J Mater Sci 1997;32 (22):594553.

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[30] Alderson A. A triumph of lateral thought. Chem Ind 1999;10:38491. [31] Smith C, Lehman F, Wootton R, Evans K. Strain dependent densification during indentation in auxetic foams. Cell Polym 1999;18(2):79101. [32] Mohsenizadeh S, Alipour R, Rad MS, Nejad AF, Ahmad Z. Crashworthiness assessment of auxetic foam-filled tube under quasi-static axial loading. Mater Des 2015;88:25868. [33] Scarpa F, Yates J, Ciffo L, Patsias S. Dynamic crushing of auxetic open-cell polyurethane foam. Proc Inst Mech Eng, C: J Mech Eng Sci 2002;216(12):11536. [34] Zhang P, Wang Z, Zhao L. Dynamic crushing behavior of open-cell aluminum foam with negative Poisson’s ratio. Appl Phys A 2017;123(5):321. [35] Duncan O, Foster L, Senior T, Alderson A, Allen T. Quasi-static characterisation and impact testing of auxetic foam for sports safety applications. Smart Mater Struct 2016;25(5):054014. [36] Lakes R. Design considerations for materials with negative Poisson’s ratios. J Mech Des 1993;115(4):696700. [37] Evans KE, Alderson A. Auxetic materials: functional materials and structures from lateral thinking!. Adv Mater 2000;12(9):61728. [38] Choi J, Lakes R. Fracture toughness of re-entrant foam materials with a negative Poisson’s ratio: experiment and analysis. Int J Fract 1996;80(1):7383. [39] Scarpa F, Bullough W, Lumley P. Trends in acoustic properties of iron particle seeded auxetic polyurethane foam. Proc Inst Mech Eng, C: J Mech Eng Sci 2004;218 (2):2414. [40] Choi J, Lakes R. Non-linear properties of polymer cellular materials with a negative Poisson’s ratio. J Mater Sci 1992;27(17):467884. [41] Chekkal I, Remillat C, Scarpa F. Acoustic properties of auxetic foams. High Perform Struct Mater VI. 2012;124:119. [42] Scarpa F, Smith F. Passive and MR fluid-coated auxetic PU foam  mechanical, acoustic, and electromagnetic properties. J Intell Mater Syst Struct 2004;15(12):9739. [43] Evans K. Tailoring the negative Poisson’s ratio. Chem Ind 1990;20:654. [44] Evans K, Caddock B. Microporous materials with negative Poisson’s ratios. II. Mechanisms and interpretation. J Phys D: Appl Phys 1989;22(12):1883. [45] Caddock B, Evans K. Microporous materials with negative Poisson’s ratios. I. Microstructure and mechanical properties. J Phys D: Appl Phys 1989;22(12):1877. [46] Alderson K, Kettle A, Neale P, Pickles A, Evans K. The effect of the processing parameters on the fabrication of auxetic polyethylene. J Mater Sci 1995;30(16):406975. [47] Pickles A, Webber R, Alderson K, Neale P, Evans K. The effect of the processing parameters on the fabrication of auxetic polyethylene. J Mater Sci 1995;30(16):405968. [48] Neale P, Pickles A, Alderson K, Evans K. The effect of the processing parameters on the fabrication of auxetic polyethylene. J Mater Sci 1995;30(16):408794. [49] Alderson K, Evans K. The fabrication of microporous polyethylene having a negative Poisson’s ratio. Polymer 1992;33(20):44358. [50] Veronda D, Westmann R. Mechanical characterization of skin—finite deformations. J Biomech 1970;3(1):111 IN9123-122124. [51] Lees C, Vincent JF, Hillerton JE. Poisson’s ratio in skin. Bio-med Mater Eng 1991;1 (1):1923. [52] Peura M, Grotkopp I, Lemke H, Vikkula A, Laine J, Mu¨ller M, et al. Negative Poisson ratio of crystalline cellulose in kraft cooked Norway spruce. Biomacromolecules 2006;7(5):15218. [53] Nakamura KI, Wada M, Kuga S, Okano T. Poisson’s ratio of cellulose Iβ and cellulose II. J Polym Sci, B: Polym Phys 2004;42(7):120611.

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[54] Evans KE, Alderson A, Christian FR. Auxetic two-dimensional polymer networks. An example of tailoring geometry for specific mechanical properties. J Chem Soc Faraday Trans 1995;91(16):267180. [55] El-Sayed FA, Jones R, Burgess I. A theoretical approach to the deformation of honeycomb based composite materials. Composites 1979;10(4):20914. [56] Evans KE, Nkansah M, Hutchinson I, Rogers S. Molecular network design. Nature 1991;353(6340):124. [57] Wei G. Design of auxetic polymer self-assemblies. Phys Status Solidi (B) 2005;242 (3):7428. [58] Grima JN, Alderson A, Evans K. Auxetic behaviour from rotating rigid units. Phys Status Solidi (B) 2005;242(3):56175. [59] Grima J, Williams J, Gatt R, Evans K. Modelling of auxetic networked polymers built from calix [4] arene building blocks. Mol Simul 2005;31(13):90713. [60] He C, Liu P, Griffin AC. Toward negative Poisson ratio polymers through molecular design. Macromolecules 1998;31(9):31457. [61] He C, Liu P, McMullan PJ, Griffin AC. Toward molecular auxetics: main chain liquid crystalline polymers consisting of laterally attached para-quaterphenyls. Phys Status Solidi (B) 2005;242(3):57684.