Finite Element Analysis of Tunable Composite Tubes Reinforced with Auxetic Structures

A tubular composite structure that is built of two materials, characterized by different Young moduli, is analysed in this paper. The Young’s modulus of one of these materials can be controlled by external conditions e.g., magnetic or electric field, temperature etc. The geometry of the reinforcement is based on typical auxetic re-entrant honeycomb cellular structure. The influence of this external factor on the behaviour of the stretched tube is analysed in this paper. Also, the possibility of creating a tubular composite structure whose cross-section is either shrinking or expanding, while stretching the tube is presented.


Introduction
Auxetics have become more and more popular over the recent years. Not only are they extensively investigated by researchers, but also auxetics or auxetic-like structures are already used commercially. It has been shown that auxetic behaviour i.e., negative Poisson's ratio results from the specific geometry of material structures. Some of the first analyses of re-entrant honeycomb structures were presented by Kolpakov [1] and Gibson [2]. Nowadays, many geometrical patterns are known to be responsible for auxetic behavior [3]. Such behaviour on the molecular level was first presented by Wojciechowski [4,5] and Evans [6], and also by Kang et al. [7] and Grima for single molecules [8], as well as by Goldstein for crystals [9,10]. Macroscopic auxetic structures can be divided into several classes: foams [11] and porous structures [12,13], re-entrant structures [14,15], chiral structures [16], non-chiral structures [17,18] rotating rigid or semi-rigid units [19], and many others.
Auxetic structures can also result from specific material distribution in composite materials, even though all of the constituents are characterized by positive values of the Poisson's ratio, an analysis of such structures was presented by Strek and Jopek [20,21]. Czarnecki et al. [22,23] presented some results concerning composites built by optimal distribution of Young's modulus within elastic and isotropic by means of topological optimization. Auxetic behaviour as a result of hard inclusions in the soft matrix was shown by Pozniak [24] and Kochman [25]. Another class of auxetic composites are multiphase structures, in which at least one constituent exhibit the negative value of the Poisson's ratio. The influence of auxetic fibres reinforcement on the mechanical behaviour of a composite under bending load was studied by Jopek [26]. Bilski and Wojciechowski [27] investigated the possibility of changing the value of effective Poisson's ratio in layered composites with an auxetic layer built of hexamers. The torsional behaviour of composite beams was studied by Strek [28,29].
A special group of materials contains so-called smart materials i.e., materials with controllable properties. In recent years, several articles have been published on using auxetics in smart materials. Ren et al. showed that the auxetic performance can be tuned by the geometry of microstructures, while the strength and stiffness can be tuned by the plasticity of the base material, while maintaining the auxetic performance [30]. Jopek and Strek [21] analysed thermoauxetic composite in which the value

Composite Structure of the Tube
The reinforcement of the tube is built of auxetic unit cells based on typical re-entrant honeycomb geometry (see Figure 1a). The material of auxetic structure being the reinforcement is denoted as M R (blue colour). The voids within the auxetic structure are completely filled with the matrix material M M (grey colour), so that we obtain a solid bi-material composite layer (L 2 see Figure 1b). The thickness of the reinforcement L 2 layer is denoted as t a , but the reinforcement layer can also be embedded in the M M material so that the matrix creates also inner (L 3 ) and outer (L 1 ) layers, which isolate the reinforcement from the surroundings. Both od these layers are equally thick and their cumulative thickness is denoted as t h . Overall thickness of the tube is defined as t c = t a + t h The curvature of the circular cross-section of the tube is defined by the centre line that divides the thickness of the tube's wall into halves. The radius of this centre line equals R m . Therefore, the inner (R w ) and outer (R z ) radii are defined as follows: R w = R m − (t a + t h )/2, R z = R m + (t a + t h )/2 (see Figure 1b). The geometry of three-dimensional (3D) composite tube is presented in (Figure 1c). It is assumed that the even number N c of unit cells is distributed circumferentially, as well as the even number N z is distributed along the tubes height. Hence, the geometry is symmetrical with respect to the three main planes and only one-eighth part of the tube needs to be analysed. In order to determine the volumetric fraction of each constituent, one can make use of the symmetry of the unit cell and calculate only the quarter of the unit cell whose area is defined as A = ad. The area of the reinforcement can be expressed as follows: and the area of the matrix: So, the share of each constituent in the layer of reinforcement (L 2 ): However, one needs to take into account the thickness of the tube's wall and the fact that the reinforcement could be thinner due to matrix material that surrounds the reinforcement from the inner and outer side. The matrix material that forms the inner and outer layers could isolate the reinforcement that embedded in the tube, but it also diminishes its influence as the share of M R material in the volume of the whole composite decreases. In that case, the wall could be split into three layers: internal layer L 3 built of the matrix material only of volume V 1 , the middle layer L 2 being the mixture of matrix material, and the reinforcement of volume V 2 , and finally the outer layer L 1 , also built only of matrix material only of volume V 3 so the volume of each constituent is defined by the following formulae: where V = V 1 + V 2 + V 3 is the total volume of the tube. Hence volumetric share of each constituents is expressed as: , = , Smart materials whose properties are tunable via external conditions, such as electric or magnetic field, temperature etc. allow for creating more sophisticated structures, which could be also controlled by the same factor. In the case of the considered composite tube, it is built of two materials MR and MM, and the material of the reinforcement is such smart material and its Young's modulus that can be controlled by external factor, while the change of the Young's modulus of the other material is negligibly small. In such a case, it is possible to investigate just the ratio of each constituents Young's modulus instead of the values of each of them.
The reinforcement material MR is characterized by Young Modulus ER and the value of the Poisson's ratio νR, whereas the matrix material is characterized by EM and νM. As it was shown by Milton [58], in the case of composites built of auxetic structure with all of the voids filled, that it is possible to preserve the negative Poisson's ratio of this structure if the structure is stiff and the other material compressible, which means that the ratio of Young's moduli is different from 1. Similar results have been presented by Goldstein [48]. Hence, the value of the effective Poisson's ratio is strongly dependent on the values of Young moduli of constituents. When considering the fact that the key factor in the analysis is the ratio of both materials, Young's moduli, it was assumed that EM = 1 and the value of ER varies in the following range ER = (0.001, 1000), which implies that the ratio ER/EM also changes in the same range. Moreover, it follows that simulation also cover cases in which the Young modulus of the reinforcement is constant and the Young's modulus of the matrix is influenced by external factors. Either way, the ratio of Young's moduli can be tuned. Although the effect of an external field on the Young's modulus of magneto-and electro-elastic materials is usually not that great, it is possible to obtain the change of the Young's modulus by two or three orders of magnitude with the change of temperature, hence, such wide range of analysis seems justified. Smart materials whose properties are tunable via external conditions, such as electric or magnetic field, temperature etc. allow for creating more sophisticated structures, which could be also controlled by the same factor. In the case of the considered composite tube, it is built of two materials M R and M M , and the material of the reinforcement is such smart material and its Young's modulus that can be controlled by external factor, while the change of the Young's modulus of the other material is negligibly small. In such a case, it is possible to investigate just the ratio of each constituents Young's modulus instead of the values of each of them.
The reinforcement material M R is characterized by Young Modulus E R and the value of the Poisson's ratio ν R , whereas the matrix material is characterized by E M and ν M . As it was shown by Milton [58], in the case of composites built of auxetic structure with all of the voids filled, that it is possible to preserve the negative Poisson's ratio of this structure if the structure is stiff and the other material compressible, which means that the ratio of Young's moduli is different from 1. Similar results have been presented by Goldstein [48]. Hence, the value of the effective Poisson's ratio is strongly dependent on the values of Young moduli of constituents. When considering the fact that the key factor in the analysis is the ratio of both materials, Young's moduli, it was assumed that E M = 1 and the value of E R varies in the following range E R = (0.001, 1000), which implies that the ratio E R /E M also changes in the same range. Moreover, it follows that simulation also cover cases in which the Young modulus of the reinforcement is constant and the Young's modulus of the matrix is influenced by external factors. Either way, the ratio of Young's moduli can be tuned. Although the effect of an external field on the Young's modulus of magneto-and electro-elastic materials is usually not that great, it is possible to obtain the change of the Young's modulus by two or three orders of magnitude with the change of temperature, hence, such wide range of analysis seems justified.
In case of the experimental investigation, magnetorheological elastomers could be used. These materials are built of a polymeric matrix with magnetic micro-or nanoparticles embedded. The structure that is considered could by obtained if the particles were distributed so that they create a re-entrant auxetic pattern. Another possibility is to use two polymers that can be bonded together, assuming that the Young modulus of one of the materials changes significantly with the change of temperature, while the other is almost constant in the same range of temperature e.g., two types of silicone rubber.
For simulation, it was assumed the basic values of ν M and ν R were assumed as 0.3, although the simulation for several different values of ν R was performed in order to investigate its impact as well. Symmetry boundary conditions are used on Γ x , Γ y , and Γ z1 boundaries and prescribed displacement along the z-axis is assumed on the Γ z2 see Figure 1d, the prescribed displacement ∆h = 0.1h.
The change of E R results in the change of effective Poisson's ratio of the composite, which in the case of flat geometry would by described as ν = − ε t ε l , where ε t -transverse strain, ε l -longitudinal strain along the stretching/compression direction. However, when considering the tube, the effective Poisson's ratio of stretched (compressed) tube is calculated on the basis of the change of the length of its circumference, which could be expressed simply by its diameter or radius. Hence, the effective Poisson's ratio of the axially stretched tube is given by the following [39]: where ε c -average circumferential strains and ε c = ∆R R , and ε z = ∆h h . Given the fact that the composite material properties are not homogeneous in the cross-sectional area, one should expect that the deformed cross-section could diverge from its initial circular shape. Hence, both longitudinal and circumferential strains are averaged.

Results
Numerical analysis was performed with the use of Comsol Multiphysics FEM software. Linear elastic materials are assumed for both of the materials. The mesh was built of about 20,000 tetrahedral elements. Geometrical parameters of the tube: R m = 1, N c = 6, N z = 10, t a = 0.1, t h = {0, 0.2}. The aim of the analysis is to present the influence of the change of reinforcement elastic modulus on the behaviour of the tube.

Case 1
In the first case, the results were obtained for the case, in which t h = 0, which means that the tube's wall is of the same thickness as the reinforcement layer so no additional inner and outer layer occur. The volumetric share of the phase of the reinforcement S V,R = 0.43. The change of the inner radius R w of the stretched tube with respect to the Young's moduli ratio is presented in the Figure 2. Several values of the Poisson's ratio of the reinforcement material are presented in order to investigate the influence of this parameter, as well while the value of the Poisson's ratio of the matrix is kept constant ν M = 0.3. The dashed line in the plot represents the inner radius of the tube before deformation, just for the reference. If the ratio E R /E M = 1 and ν R = ν M , then the tube is built of isotropic homogenous material and it behaves as such.
One can observe that as the E R /E M diverge from 1 the radius of the tube also starts changing. However, the influence of the ratio E R /E M is far more significant if the ratio is greater than 1, which means that the material of the reinforcement is stiffer. This effect results from the fact that the reinforcement material creates auxetic structures, and the stiffer the reinforcement, the stronger it influences the behaviour of the whole tube. It is clearly noticeable that it is possible to obtain a tube that would either shrink or expand depending on the external factor that would change the reinforcement Young's modulus. Moreover, such change is possible even for a relatively small change of this Young's modulus, If E R is greater than E M by about one order of magnitude, then we can obtain a tube that would change its radius from 0.93 up to 0.97, which means that the cross-section of the tube would change by almost by 10%. Furthermore, a very interesting result is obtained in the case in which the ratio is changed to its reciprocal so that the matrix is stiffer than the auxetic reinforcement. The tube behaves similarly to the one with stiffer reinforcement, although the effect of changing the Young's moduli ratio is smaller.
can obtain a tube that would change its radius from 0.93 up to 0.97, which means that the cross-section of the tube would change by almost by 10%. Furthermore, a very interesting result is obtained in the case in which the ratio is changed to its reciprocal so that the matrix is stiffer than the auxetic reinforcement. The tube behaves similarly to the one with stiffer reinforcement, although the effect of changing the Young's moduli ratio is smaller. The value of the Poisson's ratio of the reinforcing material was also studied, and, naturally, it also influences the behaviour of the whole composite. The impact of changing the value of νR decreases as the ratio ER/EM diverge from 1. For a completely isotropic material of the tube, the diameter of the stretched tube is minimal, however, this relation is dependent on the Poisson's ratio. In the case of νR = 0.49, the minimum value of the radius is obtained if the value of the ratio ER/EM < 1. For all the values of νR < 0.3 the minimum radius of the deformed tube is obtained if ER/EM > 1. It is clearly noticeable that the value of νR is also significant, and it enhances the auxetic behaviour of the reinforcement structure as νR tends to −1. In consequence, lower values of νR diminish the effect of the changing ER/EM ratio. For clarity reasons it was assumed that the external factor influences only the ER/EM ratio. So, in each case of νR, the ER/EM ratio was the only control variable. It is important, however, to remember that the Poisson's ratio can be also dependant on such phenomena as a magnetic field, electric field, or temperature. Hence, the radius of the tube could change in a wider range if both Young's modulus ratio and the values of the Poisson's ratio were controlled simultaneously by this external factor.
The shape of the deformed cross section is presented for two cases. In Figure 3, the cross-sectional shape of the tube is presented for the following set of parameters: ER/EM = 0.001, νR = νM = 0.3, th = 0. The cross-sectional shape for the same parameters but the ratio of Young's moduli changed ER/EM = 1000 is presented in the Figure 4. The selected cross-section lays in the plane of tube symmetry. The stiffness of the auxetic reinforcement is far smaller than the stiffness of the matrix material. The stresses are concentrated in the area of the matrix in the case of ER/EM = 0.001, while for ER/EM = 1000, the highest values of von Mises stresses occur at the corners of the auxetic structure. Additional red arrows show the directions of displacements so that it is clearly noticeable that the first tube shrinks while the latter expands. The value of the Poisson's ratio of the reinforcing material was also studied, and, naturally, it also influences the behaviour of the whole composite. The impact of changing the value of ν R decreases as the ratio E R /E M diverge from 1. For a completely isotropic material of the tube, the diameter of the stretched tube is minimal, however, this relation is dependent on the Poisson's ratio. In the case of ν R = 0.49, the minimum value of the radius is obtained if the value of the ratio E R /E M < 1. For all the values of ν R < 0.3 the minimum radius of the deformed tube is obtained if E R /E M > 1. It is clearly noticeable that the value of ν R is also significant, and it enhances the auxetic behaviour of the reinforcement structure as ν R tends to −1. In consequence, lower values of ν R diminish the effect of the changing E R /E M ratio. For clarity reasons it was assumed that the external factor influences only the E R /E M ratio. So, in each case of ν R , the E R /E M ratio was the only control variable. It is important, however, to remember that the Poisson's ratio can be also dependant on such phenomena as a magnetic field, electric field, or temperature. Hence, the radius of the tube could change in a wider range if both Young's modulus ratio and the values of the Poisson's ratio were controlled simultaneously by this external factor.
The shape of the deformed cross section is presented for two cases. In Figure 3, the cross-sectional shape of the tube is presented for the following set of parameters: E R /E M = 0.001, ν R = ν M = 0.3, t h = 0. The cross-sectional shape for the same parameters but the ratio of Young's moduli changed E R /E M = 1000 is presented in the Figure 4. The selected cross-section lays in the plane of tube symmetry. The stiffness of the auxetic reinforcement is far smaller than the stiffness of the matrix material. The stresses are concentrated in the area of the matrix in the case of E R /E M = 0.001, while for E R /E M = 1000, the highest values of von Mises stresses occur at the corners of the auxetic structure. Additional red arrows show the directions of displacements so that it is clearly noticeable that the first tube shrinks while the latter expands.

Case 2
In the second analysis, the same auxetic structure is embedded in the matrix material so that th = 0.2 and the volumetric share of the reinforcement material , = 0.15. The change of the inner radius of the tube Rw with respect to ER/EM ratio for selected values of νR is presented in the Figure 5.

Case 2
In the second analysis, the same auxetic structure is embedded in the matrix material so that th = 0.2 and the volumetric share of the reinforcement material , = 0.15. The change of the inner radius of the tube Rw with respect to ER/EM ratio for selected values of νR is presented in the Figure 5.

Case 2
In the second analysis, the same auxetic structure is embedded in the matrix material so that t h = 0.2 and the volumetric share of the reinforcement material S V,R = 0.15. The change of the inner radius of the tube R w with respect to E R /E M ratio for selected values of ν R is presented in the Figure 5. For such a composite, it is still possible to influence its behaviour by changing the E R /E M ratio, however, significant change is obtained only if E R /E M > 1. Since the share of the reinforcement phase is much smaller than in the first case, the effect of its auxetic structure is suppressed if it is made of softer material i.e., the ratio E R /E M < 1. Hence, the effect of increasing the inner radius R w if the E R /E M tends to 0.001 is not observed in this case. Furthermore, the influence of the ν R diminishes with the decrease of E R /E M , and it becomes negligibly small for E R /E M < 0.1. Likewise, the first case of the composite tube, this one is also capable of changing the inner radius by ca. 10% if the E R /E M change by about one order of magnitude (e.g., from 12 to 120). It is worth noting that this behaviour is only slightly influenced by the value of ν R , although the value of the inner radius depends on it. For such a composite, it is still possible to influence its behaviour by changing the ER/EM ratio, however, significant change is obtained only if ER/EM > 1. Since the share of the reinforcement phase is much smaller than in the first case, the effect of its auxetic structure is suppressed if it is made of softer material i.e., the ratio ER/EM < 1. Hence, the effect of increasing the inner radius Rw if the ER/EM tends to 0.001 is not observed in this case. Furthermore, the influence of the νR diminishes with the decrease of ER/EM, and it becomes negligibly small for ER/EM < 0.1. Likewise, the first case of the composite tube, this one is also capable of changing the inner radius by ca. 10% if the ER/EM change by about one order of magnitude (e.g., from 12 to 120). It is worth noting that this behaviour is only slightly influenced by the value of νR, although the value of the inner radius depends on it. The shape of the deformed tube's cross-section is presented in the Figure 6 in the case of ER/EM = 0.001, whereas Figure 7 shows the cross-sectional shape for ER/EM = 1000 is presented in Figure 7. The highest values of stresses concentrate in the area of the matrix along the interface between phases in the case of ER/EM = 0.001. If the Young's modulus of the reinforcement is higher than the one of the matrix i.e ER/EM = 1000, then the highest values of the stresses occur at the corners of the auxetic reinforcement, as it occurred for the geometry with th = 0.
The values of effective Poisson's ratio of the composite were also calculated (see Figure 8). Since the value of the effective Poisson's ratio is strongly related to the change of the tube's radius, the plot in the Figure 8 resembles the plot in the Figure 5. The shape of the deformed tube's cross-section is presented in the Figure 6 in the case of E R /E M = 0.001, whereas Figure 7 shows the cross-sectional shape for E R /E M = 1000 is presented in Figure 7.
The highest values of stresses concentrate in the area of the matrix along the interface between phases in the case of E R /E M = 0.001. If the Young's modulus of the reinforcement is higher than the one of the matrix i.e E R /E M = 1000, then the highest values of the stresses occur at the corners of the auxetic reinforcement, as it occurred for the geometry with t h = 0.
The values of effective Poisson's ratio of the composite were also calculated (see Figure 8). Since the value of the effective Poisson's ratio is strongly related to the change of the tube's radius, the plot in the Figure 8 resembles the plot in the Figure 5.

Conclusions
In this paper, a composite tube that is reinforced with auxetic structure is studied by means of Finite Element Analysis. The reinforcement of the tube is assumed to be made of the material whose Young's modulus can be controlled by external factor e.g., magnetic field, temperature, or other physical phenomena, while the other Young's modulus is constant. Hence, the ratio of both Young's moduli changes as well. The change of this ratio allows for controlling the deformation of the composite tube. The shape of reinforcement is designed as auxetic re-entrant structure. Two cases of the tube are considered. In the first case, the tube's wall is only as thick as the reinforcement layer, and the matrix materials fill the voids of the auxetic structure. The results indicate that the diameter of the stretched tube can change about 10% as a result of the external field applied. In the second analysis, the reinforcement is embedded in the matrix material so the influence of its auxetic geometry is diminished. Yet, the results obtained indicate that it is also possible to control the diameter of such a tube in the range of about 10%, although the changes are limited to the case in which the value of Young's modulus of the reinforcement is greater than the Young's modulus of the matrix. Further research will be conducted in order to investigate the behaviour of such structure under different types of loading.
The reinforcement can be completely embedded in the matrix i.e., it is possible to isolate it from the surroundings if necessary.

Conclusions
In this paper, a composite tube that is reinforced with auxetic structure is studied by means of Finite Element Analysis. The reinforcement of the tube is assumed to be made of the material whose Young's modulus can be controlled by external factor e.g., magnetic field, temperature, or other physical phenomena, while the other Young's modulus is constant. Hence, the ratio of both Young's moduli changes as well. The change of this ratio allows for controlling the deformation of the composite tube. The shape of reinforcement is designed as auxetic re-entrant structure. Two cases of the tube are considered. In the first case, the tube's wall is only as thick as the reinforcement layer, and the matrix materials fill the voids of the auxetic structure. The results indicate that the diameter of the stretched tube can change about 10% as a result of the external field applied. In the second analysis, the reinforcement is embedded in the matrix material so the influence of its auxetic geometry is diminished. Yet, the results obtained indicate that it is also possible to control the diameter of such a tube in the range of about 10%, although the changes are limited to the case in which the value of Young's modulus of the reinforcement is greater than the Young's modulus of the matrix. Further research will be conducted in order to investigate the behaviour of such structure under different types of loading.
The reinforcement can be completely embedded in the matrix i.e., it is possible to isolate it from the surroundings if necessary.