Table of contents

2.1.Applications of auxetic materials. 8

2.2.Negative Poisson’s ratio. 9

2.3.Types of materials with negative Poisson’s ratio based on their geometry. 11

2.5.Methods of topology optimisation. 15

2.5.1.Homogenisation method. 15

2.5.2.Solid isotropic material with penalisation method (SIMP) 16

2.5.2.1.Mathematical formulation by SIMP method. 16

2.5.3.Evolutionary structural optimisation (ESO) method. 17

2.5.4.Level set method (LSM) 17

2.6.Auxetic polymeric foams. 18

2.7.Additive manufacturing (3D printing) 19

2.8.Mechanical properties of auxetic foams. 20

2.8.1.Toughness and dissipation of energy. 20

2.8.2.Indentation of auxetic foams. 21

2.9.Design of auxetic microstructures by topology optimisation. 22

2.9.1.Two dimensional structures with negative Poisson’s ratio. 23

2.9.2.Three dimensional structures with negative Poisson’s ratio. 25

2.10.Major characteristics of materials with negative Poisson’s ratio during design. 28

2.11.Design of rotating unit auxetic materials. 29

3.2.1.Heat study by heat model 31

3.2.2.Heat study by thermocouples. 32

3.5.Characterisation techniques. 37

3.5.1.Ratio of volumetric compression. 37

3.5.2.Digital volume correlation. 37

# List of figures

Figure 1: A graph representing elastic modulus against elastic strain. 6

Figure 2: Schematic representation of compliant mechanism.. 7

Figure 4:Re-entrant bow tie structure with q as the re-entrant angle. 13

Figure 6: A unit cell with square. 16

Figure 7: SIMP method solutions. 17

Figure 8:The discretisation of design domain into finite elements. 17

Figure 9: Microstructure of conventional and auxetic foam.. 19

Figure 10: Design variable of 2D negative Poisson’s ratio material 24

Figure 11: Design variable negative Poisson’s ratio material with non linear geometry. 25

Figure 12: Design by altering the material properties with respect to q_{1} and q_{2}. 26

Figure 13: Design variable of 3D negative Poisson’s ratio material 27

Figure 14: Effective Young’s modulus with two design variables. 28

Figure 15:Effective Poisson’s ratio with two design variables. 28

Figure 16: Characteristics of the materials with negative Poisson’s ratio. 29

Figure 17: The process of fabrication of auxetic foams. 31

Figure 18: Schematic representation of sample removed from oven. 32

Figure 19: Model for calculation of transfer of heat 33

Figure 20: The 3D printed cell dimensions of auxetic and conventional cells. 34

Figure 21: Solid works design for 3D printed samples for auxetic tensile and auxetic high mass. 36

Figure 22: Schematic representation of drop tower rig. 37

Figure 23:Schematic representation of digital volume correlation. 39

# Abstract

The engineering materials are designed to increase their resistance and stiffness while decreasing the mass of the material. The requirement to minimise or maximise certain quantities is termed as optimisation. The most important approach towards the conceptual stage of structural design is the topology optimisation method where certain parameters can be either maximised or minimised without any constraints. This field of engineering science has undergone stupendous progress during the last decade and has been successfully applied in different engineering problems. The additive manufacturing is another branch of structural design which is also termed as 3D- printing. The thesis comprises a literature review on auxetic materials, topology optimisation and 3D printing methods. The metamaterials are artificially fabricated materials which possess extra-ordinary characteristics as compared to their natural forms. The optimisation of the topology of multi material is generally carried out with level set methods.

The basis of the topology optimisation method is the distribution of the materials in the optimal manner within the design space. The field of materials and the study of cellular materials fall under this category. The group of cells which agglomerate homogeneously are termed as cellular materials which are real materials. The polymeric foam is a type of cellular materials. The topology optimisation of foam is not very suitable as compared to lattice materials which are periodic and regular and can be easily modelled mathematically.

The theoretical optimisation is carried out initially which is later applied physically. The optimised materials are manufactured by 3D printing which is still in an early stage. Thus three steps involved in the complete procedure are theoretical study, design which is followed by manufacturing.

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**Topology optimisation of meta material with negative Poisson’s ratio**

# 1.Background

A variety of materials possessing specific properties are desirable in the arena of engineering applications. Some engineering materials require properties having combination of high thermal diffusivity along with high thermal conductivity, high strength as well as high strain and also materials with low density but high strength.[1] The properties of the materials obtained from nature are limited which are not appropriate for many cases. Thus different materials like foams, alloys, sandwich structures and composites have been designed for manufacturing the products with desired properties.

The materials used in load carrying and structural applications should possess high strength. The flexural strength of a material is compared with the help of a parameter called elastic modulus. The flexural stiffness of structures made of ceramics and alloys are very high but they have low elastic strain. However certain materials like foams and elastomers exhibit high elastic strain but low flexural stiffness. The adaptive structures require both the properties and hence should have the capacity to bear loads as well as alter their size and shape. Therefore the two main properties of adaptive materials are high strain capability and high strength. A graph plotted against maximum elastic strain and elastic modulus demonstrates that the materials can either have high elastic strain or high strength but fails to have both the properties at the same time. Figure 1 However the adaptive materials can be manipulated to exhibit both the properties simultaneously.

Two types of materials used in adaptive structures are active and passive materials. The materials whose mechanical properties like stiffening and softening can changed due to external energy are termed as active materials. The polymers and shape memory alloys fall into this category. [2]

Figure 1: A graph representing elastic modulus against elastic strain[3]

The materials whose mechanical properties can be altered by compliant mechanism are termed as passive materials. The foams and cellular materials fall into the class of passive materials where the microstructure and the macrostructure are made of compliant joints and links. The compliant members bend alone during deformation thereby allowing a high maximum elastic strain as compared to that of the bulk material. The ratio of the strength to weight of the passive materials is high.[4] Cellular structures and contact aided compliant mechanisms are two passive compliant mechanisms having high strain as well as strength. The compliant mechanism which undergoes self contact during deformation is termed as contact aided compliant mechanism where the concentration of the stress is reduced due to contact. The load carrying applications uses cellular structures which are used in the cases of high elastic strain.[5] Figure 2

Figure 2: Schematic representation of compliant mechanism[6]

# 1.1.Motivation

# 2.Introduction

The materials possessing modified functionalities and extraordinary mechanical properties like negative Poisson’s ratio, negative compressibility and negative elasticity are termed as mechanical metamaterials.[7] Any materials which can be engineered to exhibit unique characteristics are referred to as metamaterials. These materials are also referred to as designer materials as their macro scale properties can be manipulated to design structures with particular physical or mechanical properties. [8] A variety of structures have been designed having complex micro structure by the process of additive manufacturing.[9]

The materials which have a negative Poisson’s ratio are called auxetic materials. These materials are microstructures which become thick in the direction perpendicular to the direction of the applied load when subjected to tensile loading. This generates flexing in the material because of the appearance of artificial hinges. An auxetic structure has repeated patterns of microstructure which is identical. A monolithic body with a particular geometry constitutes each of the microstructure which undergoes a specific motion under a particular loading. Thus the complete body behaves in an auxetic manner as it acts due to compliant mechanism. Figure 3

The mechanical properties like fracture toughness, hardness, sound absorption, indent resistance and shear strength can be enhanced considerably for auxetic materials.[10] Additionally the shape and size of the pores within the molecular sieves can be maintained as well as the double curvature with a honey comb panel is permitted in this type of materials.[11]

The interest in materials with negative Poisson’s ratio was initiated after the synthesis of auxetic re-entrant foam.[12] Other instances of auxetic materials are SiO_{2} polymorph a-cristobalite, complex zeolites and silicates.[13]

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Figure 3: Schematic representation of a. regular behaviour when the structure shrinks due to the applied force, b. auxetic behaviour when the structure enlarges due to the applied force[14]

# 2.1.Applications of auxetic materials

Materials with high value of negative Poisson’s ratio where (*ν *< −0.5), exhibit high values of indent resistance.[15] The foam mattresses possess this property where the support can be enhanced without compromising the comfort. The auxetic foams are used as damping materials during audio recording due to their enhanced acoustic absorptions.[16] The auxetic fabrics prevent the passing of glass and other debris due to their hardness. [17] The auxetic piezoelectric composites are used for generating small scale strains required for the production of electric current.[18] The application of these materials as molecular sieves is significant as the shape of the pores can be altered according to the requirement. [19] These materials have been used for manufacturing bioprostheses, shape memory foams and running shoes.[20]

# 2.2.Negative Poisson’s ratio

The Poisson’s ratio is defined as the ratio of the lateral strain to that of the axial strain.[21] The Poisson’s ratio has interesting properties when it is not within the normal range. This parameter can have a negative value and such materials are termed as auxetic materials.[22] The two parameters –strain and applied stress are responsible for describing the elastic properties of a material. Mathematically the strain and applied stress are represented by the following equations:

Where e = Strain

s = applied stress

DL = extension of length

L_{o} = original length

F = applied force

A = cross sectional area

A material experiences large number of stresses and strains at various cross sectional area. A material subjected to stress will have transverse strain in the perpendicular direction of the force and longitudinal force along the direction of the applied force. For isotropic materials the relation between the stress and strain is termed as elastic moduli. The bulk modulus, Young’s modulus, Poisson’s ratio and shear modulus are all terms of different elastic modulus which are mathematically represented as follows:

Where E = Young’s modulus

n = Poisson’s ratio

K = bulk modulus

G = shear modulus

The elastic properties of anisotropic material cannot be completely described with the above four scalar constants. Thus the strain is expressed in terms of stress which is termed as compliance while the stress is expressed in terms of strain termed as stiffness by using a fourth order tensor.[23] The tensor is presented as 6×6 matrices using Voigt notation which helps in visualisation.[24] The elastic properties are completely defined by a maximum of 21 numbers of coefficients. However there is a decrease in this number as the symmetry of the crystal increases.

The three parameters which describe the Poisson’s ratio of anisotropic materials are a perpendicular lateral vector and two angles describing the axial vector. The complex interdependence of these tensor elements is responsible for the auxetic properties of the material. The elastic properties of the isotropic materials are averaged by four different methods namely a direct averaging method, Reuss,[25] Voigt and Hill[26] method. The values for bulk modulus and shear modulus were analysed by averaging schemes by Voigt and Reuss. The Voigt averages were derived from the stiffness matrix having coefficients C_{ij} as shown below by the following equations:

Where K_{v} = Voigt average of bulk modulus

G_{v} = Voigt average of shear modulus

The Reuss averages were derived from the compliance matrix having coefficients S_{ij} as shown below by the following equations:

Where K_{R} = Reuss average of bulk modulus

G_{R} = Reuss average of shear modulus

The numerical average values of arithmetic mean values of and Reuss and Voigt and the direct method are used in the scheme proposed by Hill. The equations are shown below which aids to find the average values of Poissons ratio and young’s modulus :

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# 2.3.Types of materials with negative Poisson’s ratio based on their geometry

The structures having a negative angle which means that the structure is directed inward are termed as re-entrant structures.[27] Figure 4 The re-alignment, axial deformation as well as deflection of the cell ribs are responsible for the deformations of re-entrant structures.[28]

Figure 4:Re-entrant bow tie structure with q as the re-entrant angle[29]

# 2.3.1.Foams

The designing of re-entrant foams from open cell thermoplastic foams by Lakes was the first example of manufactured auxetic materials.[30] The re-entrant metallic foams are obtained by compressing the foam triaxially followed by heating it just above the softening temperature followed by cooling it to room temperature. Thus the polyhedral unit cells are transformed to their re-entrant unit cells where the cells protrude in an inward direction. Auxetic epoxy foams was fabricated by Quadrini et al where the unit cell structure remains same. [31] However the foams were anisotropic and thus relating the elastic constants poses a difficulty. Other reports revealed the auxetic foams having buckled ribs which initiates the deformation procedure.[32] The Poisson’s ratio was not linearly dependent on the axial strain of the foams due to the presence of a variety of deformation mechanisms during the alignment of the cell ribs at large strains.[33] The Poisson’s ratio as well as Young’s modulus is decreased with an increase in the toughness and shear modulus on increasing the compression ratio. Additionally the energy absorption and strengths of the re-entrant foams are high. [34]

# 2.3.2.Honeycombs

An array of similar re-entrant hexagonal cells in a three dimensional space with negative Poisson’s ratio and high anisotropy are termed as re-entrant hexagonal honeycombs. They exhibit high value of transverse shear modulus and transverse Young’s modulus. [35] The Poisson’s ratio increases and the stiffness decreases with a decrease in the thickness of the vertical rib. However the Young’s modulus reduces due to the decrease in the thickness of the diagonal rib with an increase of the in plane Poisson’s ratio. The ratio of the cell rib length and the re-entrant angle is responsible for the degree of auxeticity. Introduction of a narrow rib and an enhancement of the base angle of the re-entrant honeycombs increased the negative Poisson’s ratio. Other re-entrant modified honeycomb geometries are auxetic arrowhead structure, Lozenge grid, square grid and star shaped structures. Figure 5

# 2.4.Topology optimisation

The parameters like location, number and shape of holes as well as the distribution of materials over a certain range can be optimised with a tool called topology optimisation. [36] It is an effective tool used in computer where the problem of design is placed as a standard distribution of material which satisfies the functional requirement with the help of a standard layout of material. The strain energy or compliance of a structure for a given volume fraction dictates the distribution of material in a body. The objective function in the optimisation of topology is analysed with the aid of finite element analysis. The elastic light weight structures used for carrying weight has been intensively subjected to the process of topology optimisation in the previous years.[37]

Figure 5: Re-entrant honeycomb structures. A. arrowhead, B. Loznge and square grid, C. star shaped structure[38]

The topology estimation has recently been carried out in the field of systematical design of metamaterials. The metamaterials are characterised by multi physics coupling and non linear behaviour. The inverse homogenisation theory was employed by Sigmund for fabricating microstructures having constitutive parameters.[39] The truss frame microstructures having negative Poisson’s ratio was obtained in 2D and 3D with the help of inverse homogenisation method. The elastic properties were calculated and a dual optimisation algorithm was applied to fabricate microstructure with desirable elastic properties based on strain energy method by Zhang et al.[40] Another report stated the design of 3D microstructures with elastic properties by modified SIMP method.[41]

# 2.5.Methods of topology optimisation

# 2.5.1.Homogenisation method

The relation between the properties at the macro and micro level is analysed by the homogenisation theory. This theory was used for the determination of topology by combining it with finite element method. Materials are added or removed to create microstructures within a design domain in the process of topology design. The micro-structures can possess voids with different shapes. Figure 6 The parameters which define the voids are width, height and the angle of rotation. The micro structure of each unit cell within the domain lies between a pure void and a pure solid. The finite element analysis is responsible for modification of the design variables. The homogenisation theory updates the analysed microstructures from the former iteration and the process continues until the satisfaction of the convergence criteria.

Figure 6: A unit cell with square[42]

# 2.5.2.Solid isotropic material with penalisation method (SIMP)

The solid isotropic material with penalisation method is an adaptation of the homogenisation method where variant removes the topologies which cannot be manufactured. The density of the design variables lying between 0 and 1 are removed by this method. The density of the element is increased to an exponential factor in the objective function which filters out the variants. The penalisation factor in the original homogenisation method is set up as 1 and as it increases the intermediate variant densities gets removed. Figure 7

Figure 7: SIMP method solutions[43]

# 2.5.2.1.Mathematical formulation by SIMP method

The problem of topology optimisation is a 0-1 integer optimisation problem. The design domain is broken into finite elements initially. Figure 8

Figure 8:The discretisation of design domain into finite elements[44]

A design variable x_{e} is assigned in each finite element e. The material present in that area is indicated by a cell which painted in black. The material is present in the cell only when the density is equal to one. The cell is void if there is no material in the cell. The integer values can be replaced by continuous values which can be powered in a penalty value p>3. Thus the design variables are approximately close to the ideally discrete values of 0 and 1. The overall flexibility of the structure is minimised which is the objective of the function by optimisation and is described by the following equation:

Where K = global stiffness matrix

K_{o}= stiffness matrix of the element that is full of material

# 2.5.3.Evolutionary structural optimisation (ESO) method

The theory of evolutionary structural optimisation method is based on the fact that the inefficient materials will be removed with the evolution of the optimal structure. A structure when subjected to excessive strain or stress fails while a material fails if it is under low stress or strain. The level of stress should be equal throughout the structure and thus the materials with low stress will be removed gradually. Other design constraint parameters like frequency, buckling load and stiffness are imposed and finally removed.

# 2.5.4.Level set method (LSM)

The boundary is represented by a scalar function of high dimension in the level set topology method. The structural boundary is set at the zero level set of a function with higher dimensional level set. The topology like smooth boundary and proper interface can be optimised by this method.

# 2.6.Auxetic polymeric foams

The polyurethane foams are used in coatings, medical devices and interiors of automobiles. The smart polymeric and metallic foams were manufactured from low density open cell polymeric foams by altering the microstructure of the conventional foams. Figure 9

Figure 9: Microstructure of conventional and auxetic foam[45]

There are many reports on the improved mechanical properties like fracture toughness, indentation resistance, sound absorption and viscoelastic loss.[46] The mechanism of rotation of rigid units was applied for introducing the auxetic behaviour in cellular materials of foam. The ribs of the cells exhibited the auxetic nature which was thick at the joints than at the centre.[47] The gradient properties and homogeneity of the auxetic foams were reported which revealed that the concavity of the cells are responsible for imparting auxetic nature. Additionally they reported that the dissipation of energy by the auxetic foams was more as compared to that of the conventional foam.[48] The complex microstructure aids in the heterogeneous strain distribution of foam polymers.[49] It was reported that the microstructure of the auxetic polyurethane foams remains unaltered in presence of solvent or high temperature in a contained state as expansion is not allowed under this condition.[50] Computer simulation studies showed that foams with hard joints exhibit more strong auxetic character as compared to that of soft and normal joints.[51]

# 2.7.Additive manufacturing (3D printing)

The unconventional process for manufacturing auxetic foams is termed as additive manufacturing or 3D printing. It generates physical models from digital CAD data by adding layers of thin cross sectional material in order to produce a similar design. The thinner the layers used for creating the model, more approximate is the model to the original design. The thickness of the film for the systems is around 0.1 mm normally. The main asset of additive manufacturing technique is that there is no limitation to design. Thus the technique has been successfully applied in duplicating the auxetic foam polymers and also has been used for creating new polymeric foams. [52] The mechanical properties, accuracy of the final model, cost are dictated by factors like thickness of the layers, bonding of the layers and process of the creation of layers.[53] The additive manufacturing technologies are encapsulated in table 1

Serial no | Technology of additive manufacturing | Materials |

1 | Fused deposition modelling | Polycarbonate, acrylonitrile, butadiene, styrene, polyphenylsulfone |

2 | Direct metal laser sintering | Metal alloys |

3 | Electron beam melting | Metal alloys |

4 | Selective laser melting | Polymers, metals, ceramics |

5 | Selective heat sintering | Thermoplastics |

6 | Selective laser sintering | Plastics, polymers, metals, composites, ceramics |

7 | Laminated object manufacturing | Paper, composites, polymers, metals |

8 | Stereolithography | Photopolymers |

9 | Digital light processing | Photopolymers |

10 | 3DP | Elastomers, composites, ceramics, composites |

11 | Prometal | Ceramics, metal |

12 | Polyjet | Photoploymers |

13 | LENS | Metals |

Table 1: The different additive technologies[54]

# 2.8.Mechanical properties of auxetic foams

The cellular structure of the material is altered as the density of the material gets enhanced due to the applied volumetric compression ratio which improves the mechanical properties of the auxetic foams.[55]

# 2.8.1.Toughness and dissipation of energy

A most significant mechanical property of auxetic polymeric foams is toughness. The porous polymers are responsible for absorbing the amount of energy applied per unit volume. The mathematical expression for the stress strain curve is given by the following equation:[56]

Where e = strain

e_{f} = strain upon failure

s = stress

The Poisson’s ratio indicates the toughness of the auxetic polymeric foams. As the value of becomes -1 the toughness of the material increases. The critical tensile stress is given by the following equation:[57]

Where T = surface tension

r = radius of the circular crack

The other two parameters affecting the toughness and Young’s modulus of the material are structural aspects and non linear properties[58]. The initial stiffness of the re-entrant foam is less but the density of energy is high as the deformation is high.

A fatigue teat was conducted by Bezazi et al[59] for studying the toughness of the material under quasi-static cyclic loading which revealed that the auxetic polymeric foams exhibited low loss of rigidity, high degradation of stiffness as well as high absorption of energy as compared to normal foams after a large number of cycles. The energy dissipated per unit volume for N number of cycles is given by the following equations:

Where e_{max} = maximum strain

e_{min} = minimum strain

E_{d} = energy dissipated per unit volume

Another study[60] revealed that the energy dissipation of the auxetic foams was reduced as the negative Poisson’s ratio reached zero during the cyclic and quasi static loading under tension. During a four phase fabrication process of polymeric foam the energy dissipation during tension and compression is highly affected. The dissipation of energy is less in the first phase as compared to the second phase. A lower mean value of energy dissipation was observed for the specimen with returned phase specimen both in tension loading and compression.

# 2.8.2.Indentation of auxetic foams

The indentation of a material with a value of Poisson’s ratio near to -1 is a very difficult task. However the material can be easily compressed at this stage as the value of bulk modulus is lower than that of the shear modulus. This behaviour is obtained due to the following mathematical relation:

However a material where the Poisson’s ratio is nearing 0.5 cannot be compressed since the shear modulus is less than that of the bulk modulus. Lakes described the indentation rigidity by the following equation:[61]

Where a = radius of the circular localised pressure

P = localised pressure

w = indentation depth

The circular pressure distribution for small impacts is given by the following equation:

Where F = indentation force

u = maximum displacement

The circular pressure distribution for large impacts is given by the following equation:

Where H = thickness of the mat

# 2.9.Design of auxetic microstructures by topology optimisation

The distribution of optimum material within a fixed design domain with reference to a set of constraints and a specific objective function is termed as topology optimisation. The goal is predefined with a given set of boundary conditions which creates the resulting structure. The finite element method is used for the iterative design process based on various optimisation techniques.

# 2.9.1.Two dimensional structures with negative Poisson’s ratio

A two dimensional material with negative Poisson’s ratio having a generalised configuration of a unit cell having three design variables is shown in figure 10.

Figure 10: Design variable of 2D negative Poisson’s ratio material[62]

Where q_{1} = design variable representing V angle formed by two adjacent stuffer members

q_{2} = design variable representing V angle formed by two adjacent tensor members

h = distance between the two vertex C and D

The geometry of the unit cell is defined by the three design variables which finally gives the basic configuration of the negative Poisson’s ratio material. The variation of the material along its length, shape of the cross section, material properties of the tensor members and the stuffer members dictates the material properties of the design. The materials of the stuffers and tensors can be varied while designing which helps to design a function oriented design for different applications.

The materials designed by assuming linear constitutive material response with effects of non linear geometry are shown in figure 11

Figure 11: Design variable negative Poisson’s ratio material with non linear geometry[63]

The figure shows the effective material properties as well as effective deformation shape of two designs having effective Young’s modulus and effective Poisson’s ratio. The deformed shape is indicated by solid lies while the solid lines represent the undeformed shape. It can be observed that the pattern of deformation is different in both the cases when subjected under the same loading condition. The load is applied on top of the sample. Both the materials have different values of two parameters –Young’s modulus and Poisson’s ratio thus indicating that the materials can be fabricated according to the desired requirements.

Another report reveals the alteration of the effective material properties like Poisson’s ratio and Young’s modulus within the design domain with respect to the design variables q_{1} and q_{2}. Figure 12

Figure 12: Design by altering the material properties with respect to q_{1} and q_{2}[64]

The alteration of the two design variables can change the effective Young’s modulus from 60.3 MPa (at θ_{1} = 160° and θ_{2} = 170° ) to 6.1 GPa (at θ_{1} = 10° and θ_{2} = 170° ). Additionally any desired value within the range can be obtained by manipulating the design variables. The minimum value of the effective Poisson’s ratio can be -59.5 (when θ_{1} = 10° and θ_{2} = 20°). Other higher values can be obtained manipulation of the two design variables.

# 2.9.2.Three dimensional structures with negative Poisson’s ratio

A 3D design of material with negative Poisson’s ratio is shown in figure 13 which has five design variables. The variation of the material along its length, shape of the cross section, material properties of the tensor members and the stuffer members dictates the material properties of the design.

Where q_{1x}, q_{1y} = angular design variables related to stuffer members

q_{2x}, q_{2y} = angular design variables related to tensor members

h = distance between the two vertex E and F

Figure 13: Design variable of 3D negative Poisson’s ratio material^{64}

The variation of the effective Young’s modulus with respect to the design variables q_{1} = q_{1x}= q_{1y} and q_{2} = q_{2x}= q_{2y} are shown in figure 14 The parameters for material properties of the tensor members and stuffer members are set with a particular design. The effective Young’s modulus alters from 23.2 MPa (at θ_{1} = 150° and θ_{2} = 160° ) to 6.1 GPa (at θ_{1} = 50° and θ_{2} = 60° ). Additionally it can have any value within the range.

Similarly the variation of effective Poisson’s ratio with respect to the design variables q_{1} = q_{1x}= q_{1y} and q_{2} = q_{2x}= q_{2y} are shown in figure 15 The minimum value of the effective Poisson’s ratio can be -77 (when θ_{1} = 40° and θ_{2} = 50°). Other higher values can be obtained manipulation of the two design variables.

Figure 14: Effective Young’s modulus with two design variables

Figure 15:Effective Poisson’s ratio with two design variables^{64}

# 2.10.Major characteristics of materials with negative Poisson’s ratio during design

The three major characteristics of materials with negative Poisson’s ratio are concentration of the material, bulging effect and improvement of the impact force. Figure 16

Figure 16: Characteristics of the materials with negative Poisson’s ratio^{64}

The figure exhibits the effect on the material concentration after the application of pressure where the solid lines indicate the deformed material after simulation while the dotted line indicates the original configuration of the material. The effect of the negative Poisson’s ratio helps in the concentration of the material into the local area where the force is applied thereby converting the area into strong and stiff material. Thus the external load can be resisted by accumulation of more material and thus harder material can be designed.

The bulging effect aids in creating a reactively adaptive deflector. During the attack by a blast wave the original flat structure gets deformed into the shape of a bulging arc which aids in the deflection of the blast force and thereby decreases the blast load on the structure. The property of blast wave deflection can be enhanced by stuffing the structure with negative Poisson’s ratio with materials which can absorb energy. The materials like foam, polymer and jelly are used as stuffing materials which undergo permanent deformations on absorbing the energy.

# 2.11.Design of rotating unit auxetic materials

The connection of units of equilateral triangle and square by hinges at the vertices form the rotating unit auxetics. [65] The total motion of the rotating units initiated by pulling of the material in a certain direction forces the material to increase in the transverse direction. This mechanism makes the material to have a negative value of Poisson’s ratio. The planar sheet can be converted into rotating unit auxetics by various methods. The sheets maybe perforated randomly,[66] symmetrically[67] or with fractal patterns with long incisions.[68] The most significant advantage of these structures for stretchable and flexible materials is that they can be isotropically tuned at the target where expansion takes place. The joining of squares and rectangles of different sizes generate anisotropic rotating unit auxetic structures. [69] The auxetic metamaterials can also be designed by kirigami[70] and origami as in Ron Resch’s pattern.[71] The small modes of deformation which are responsible for the elastic instabilities are utilised for imparting the transformation of shape in these materials. It is a difficult task for obtaining stable modification of shape as the system remains in a pre-stressed state.

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# 3.Methodology

**3.1.Manufacture of auxetic foams**

The foam samples having dimension of (height = , width = , length = ) were cut from a polyurethane sheets. The foam was allowed to undergo tri axial compression of heating and cooling. The aluminium moulds with length ……. and cross sectional area …….were used for bi-axial compression of the foam samples. The sample was subjected to tri-axial compression by compressing it along the third axis with the help of two aluminium compression blocks which are placed at either ends of the mould. Two plates made of steel are used to held the blocks in place. The length of the sample affects the volumetric compression ratio due to the applied tri axial compression.Figure 17

Figure 17: The process of fabrication of auxetic foams[72]

The aluminium moulds were cooled to room temperature after being removed from the oven. The auxetic microstructure gets fixed and on cooling which is removed from the mould and stretched by hand longitudinally. Figure 18

Figure 18: Schematic representation of sample removed from oven[73]

# 3.2.1.Heat study by heat model

A two dimensional heat transfer model helps to study the variation of temperature during the designing of the auxetic structures. The model was generated with the help of COMSOL Multiphysics Version 4.4 software. The three separate regions of the model had their own specific properties which was coupled with a source for heating. The model had a hot region at 200^{o}C and cold region where convection took place between the cold mould and hot oven. Figure 19

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Figure 19: Model for calculation of transfer of heat[74]

The heat equation used by the model for calculation of temperature with respect to time is given by the following equations:

Where = rate of change of temperature with time

= temperature as a function of space and time

** ** a = thermal diffusivity

k = thermal conductivity

c_{p} = specific heat capacity

The influence of the heating time on the volumetric compression ratio of the polymeric foam was predicted from the model.

# 3.2.2.Heat study by thermocouples

A thermocouple was inserted directly into the triaxially compressed foam sample in the middle and the samples were placed in the centre of a……………… oven at 200^{o}C for two hours. The rate of temperature change was recorded by a DAQ card. The saples were allowed to cool to room temperature.

# 3.3. 3D printing

The cellular structures of the polymeric foam were designed with the aid of coputer software Solidworks. The cells were modelled as 2D structures which were combined to form symmetrical 3D cells. The initial base measurement of the cell size was ……… which after trial and error method was used for obtaining the desired size. Figure 20

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Figure 20: The 3D printed cell dimensions of auxetic and conventional cells[75]

The modelled 3D cells are perpendicularly fused with other similar unit cell to form3D cells having certain symmetry. The symmetrical 3D cells are arranged into different layers for generating samples ready for impact testing and tensile testing. A flat interface is provided by introducing endplates of thickness …… at each end of the unit cell. These endplates bind the polymer foam samples for impact and tensile testing. A printer uses the the foam structures by converting them to a single part and the samples are designed from the polymeric composites. Figure 21

Figure 21: Solid works design for 3D printed samples for auxetic tensile and auxetic high mass[76]

The support material gets deposited on the sample during the process of printing which is a major drawback of this procedure. The support material was removed chemically by placing the materials in aqueous KOH solution at room temperature. The samples are stirred with the help of electronic rollers followed by rinsing with tap water. The samples are finally allowed to dry.

# 3.4.Compression test

The Instron test machine was used for conducting the uniaxial compression tests and analysing the compressive properties of auxetic foam polymer. The 3D printed samples were tested by taking the compression along the z axis.

Impact testing

The auxetic polymeric foam was tested by a free fall drop tower where the sample was impacted only for a single time. Figure 22 A steel cylinder datum bar equipped with a semi sphere end was used to deliver the compressive load which is dropped from a particular height on the sample placed on Roma Plastilina Clay. The clay was covered by an aluminium box. Kinetic energy is transferred to the clay body due to the interaction of the datum and the sample along with the clay. Craters are formed in the clay body due to the transfer of the kinetic energy. The realtion between the volume of the crater and impact energy is given by the following equation:

Where E _{unit }= absorption of energy per unit volume of clay

E _{impact }= impact energy

V _{crater}= volume of a crater

Figure 22: Schematic representation of drop tower rig[77]

The energy absorbed by the sample is calculated by the direct impact of the sample with the clay body. Accorging to assumptions there is no loss in energy due to frictionwith the environment. The energy absorbed can be calculated since the volume of the crater and potential energy of the datum are known. The condition under which the complete energy is absorbed by the clay or the sample is described by the following equation:

Thus the total energy of absorption by the sample is given as:

# 3.5.Characterisation techniques

# 3.5.1.Ratio of volumetric compression

The actual volume compressed ratio as well as the imposed volume compressed ratio are given by the following equations:

Where VCR _{imposed }= imposed volume compressed ratio

VCR _{actual }= actual volume compressed ratio

V _{mould}= internal volume of the mould

** **V _{actual}= volume of the sample after conversion

** **

# 3.5.2.Digital volume correlation

The mapping and analysing of full field deformations of the 3D volumes are done by digital volume correlation method through pattern correlation algorithm. Local correlation and global correlation are two different approaches for analysing continuum level strain measurement. The reference volume and the deformed volume are broken into sub volumes followed by independent correlation between them through a fast Fourier transform in the local correlation process. Figure 23

** **

** **

Figure 23:Schematic representation of digital volume correlation[78]

# 4. Conclusion

# 5. References

** **

[1]. Ashby, M. F. (2005) Materials Selection in Mechanical Design, 3 ed., Elsevier Butterworth-Heinemann, Boston, MA.

[2]. Kornbluh, R. H. Prahlad, and R. Pelrine (2004) “Rubber to Rigid, Clamped to Unclamped: Towards Composite Materials with Wide Range of Controllable Stiffness and Damping,” in Smart Structures and Materials 2004: Industrial and Commercial Applications of Smart Structures Tech nologies, *Proceedings of SPIE* Vol. 5388, Bellingham, WA, pp. 372–386

[3]. Henry, C. and G. McKnight (2006) “Cellular Variable Stiffness Materials for Ultra-Large Reversible Deformations in Reconfigurable Structures,” in Smart Structures and Materials 2006: Active Materials: Behavior and Mechanics, *Proceedings of SPIE* 6170, pp. 1–12.

[4]. Gibson, L. J. and M. F. Ashby (1997) Cellular Solids – Structure and Properties, 2 ed., Cambridge University Press, Cambridge, UK.

[5]. Olympio, K. R. and F. Gandhi (2009) “Flexible Skins for Morphing Aircraft using Cellular Honeycomb Cores,” *Journal of Intelligent Material Sys- tems and Structures*, 0, pp. 1–17.

[6]. Mehta, V., M. Frecker, and G. A. Lesieutre (2009) “Stress Relief in Contact-aided Compliant Cellular Mechanisms,” ASME Journal of Mechan- ical Design, 31(9), pp. 1–11.

[7]. Grima, J. N. and Caruana-Gauci, R. (2012) Mechanical metamaterials: materials that push back, *Nat. Mater*., 11(7), 565–566.

[8].Zadpoor, A. A. (2016). Mechanical meta-materials, *Mater. Horiz*., 3, 371–381.

[9].Yang, L., Harrysson, O., Cormier, D., West, H.,Gong, H. and Stucker, B. (2015) Additive manufacturing of metal cellular structures: design and fabrication, *JOM*, 67(3), 608– 615.

[10] .Scarpa, F., Ciffo, L. and Yates, J. (2004) Dynamic properties of high structural integrity auxetic open cell foam. *Smart Materials and Structures*, 13(1): p. 49.

[11] . Rasburn, J., et al. (2001). Auxetic structures for variable permeability systems. *AIChE Journal, * 47(11): p. 2623-2626.

[12]. Lakes, R. (1987) Foam structures with a negative Poisson’s ratio*. **Science*, 235.

[13]. Yeganeh-Haeri, A., D. Weidner, and J. Parise. (1992) Elasticity of a-Cristobalite : A Silicon Dioxide with a Negative Poisson ‘ s Ratio. *Science*, 257(5070): p. 650-652.

[14]. Kaminakis, N. T., Drosopoulos, G. A., Stavroulakis, G. E. Design and verification of auxetic microstructures using topology optimization and homogenization. *Arch Appl Mech* DOI 10.1007/s00419-014-0970-7

[15]. Evans, K.E. and A. Alderson. (2000) Auxetic materials: functional materials and structures from lateral thinking. *Advanced Materials*, p. 617-628.

[16]. Scarpa, F., Ciffo, L. and Yates, J. (2004) Dynamic properties of high structural integrity auxetic open cell foam. *Smart Materials and Structures*, **1**3(1): p. 49.

[17]. Miller, W., et al., (2009) The manufacture and characterisation of a novel, low modulus, negative Poisson’s ratio composite*. **Composites Science and Technology*, 69(5): p. 651-655.

[18]. Smith, W. (1991) Optimizing electromechanical coupling in piezocomposites using polymers with negative Poisson’s ratio. in Ultrasonics Symposium, 1991. Proceedings*., IEEE 1991*.

[19]. Alderson, A., et al., (2005) Modelling of the mechanical and mass transport properties of auxetic molecular sieves: an idealised organic ( polymeric honeycomb) host-guest system*. **Molecular Simulation*, 897-905.

[20]. Scarpa, F. (2008) Auxetic materials for bioprostheses, *IEEE Signal Process Mag*., 25(5), 125–126.

[21]. Mcclintock, F.A. and A.S. Argon, (1966) *Mechanical Behaviour of Materials.*

[22]. Evans, K.E., et al., (1991) Molecular network design*. **Nature*, 353(6340), 124-124.

[23]. Nye, J.F., (1985) *Physical Properties of Cristals: Their Representation by Tensors and Matrices*: Oxford University Press.

[24]. Voigt, W., (1928) Lehrbuch der Kristallphysik*. *Leipzig Teubner, 1928, 962.

[25]. Reuss, A., (1929) Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. ZAMM – *Journal of Applied Mathematics and Mechanic*s / Zeitschrift für Angewandte Mathematik und Mechanik, 9(1): p. 49-58.

[26]. Hill, R., (1952) The elastic behaviour of a crystalline aggregate*. **Proceedings of the Physical Society. Section A*, 65(5): p. 349.

[27]. Reentrant, Merriam-Webster.comn.d.

[28]. Smith, C. W., Grima, J. and Evans, K. (2000) A novel mechanism for generating auxetic behaviour in reticulated foams: missing rib foam model, *Acta Mater*., 48(17), 4349–4356.

[29]. Evans, K. E. and Alderson, A. (2000) Auxetic materials: functional materials and structures from lateral thinking!, *Adv. Mater*., 12(9), 617–628.

[30]. Lakes, R. (1987) Foam structures with a negative Poisson’s ratio, Science, 235(4792), 1038–1040.

[31]. Quadrini, F., Bellisario, D., Ciampoli, L., Costanza, G. and Santo, L. (2015) Auxetic epoxy foams produced by solid state foaming, *J. Cell. Plast*., 0021955X15579456.

[32]. Chan, N. and Evans, K. (1999) The mechanical properties of conventional and auxetic foams. Part I: compression and tension, *J. Cell. Plast*., 35(2), 130–165.

[33]. Choi, J. and Lakes, R. (1992) Non-linear properties of metallic cellular materials with a negative Poisson’s ratio, *J. Mater. Sci*., 27(19), 5375–5381.

[34]. Choi, J. and Lakes, R. (1995) Analysis of elastic modulus of conventional foams and of re-entrant foam materials with a negative Poisson’s ratio, *Int. J. Mech. Sci*., 37(1), 51–59.

[35]. Smith, F. C., Scarpa, F. L. and Burriesci, G. (2002) Simultaneous optimization of the electromagnetic and mechanical properties of honeycomb materials, SPIE’s 9th Annual International Symposium on Smart Structures and Materials, International Society for Optics and Photonics, 582–591.

[36]. Bendsoe, M. P. and Sigmund, O. (2003) Topology Optimization Theory, Methods and Applications, Springer, Germany.

[37]. Sigmund O and Maute K. (2013) Topology optimization approaches. *Struct Multidiscip Optim* , 48, 1031–55.

[38]. Grima, J. N., Gatt, R., Alderson, A. and Evans, (2005) On the potential of connected stars as auxetic systems, *Mol. Simul*., 31(13), 925–935.

[39]. Sigmund O. (1994) Materials with prescribed constitutive parameters: An inverse homogenization problem. *Internat J Solids Structure, s* 31, 2313–29.

[40]. Zhang W, Dai G, Wang F, Sun S, Bassir H. (2007) Using strain energy-based prediction of effective elastic properties in topology optimization of material microstructures. *Acta Mech Sin , *23, 77–89.

[41]. Andreassen E, Lazarov BS, Sigmund O. (2014) Design of manufacturable 3D extremal elastic microstructure. *Mech Mater* , 69, 1–10.

[42]. Nishiwaki, S., Min. M. I. and Kikuchi, N. (1998) Topology optimization of compliant mechanisms using the homogenization method. *International Journal for Numerical Methods in Engineerin*g, 42(3), 535- 539.

[43]. Nishiwaki, S., Min. M. I. and Kikuchi, N. (1998) Topology optimization of compliant mechanisms using the homogenization method. *International Journal for Numerical Methods in Engineering*, 42(3), 535- 539.

[44]. Kaminakis, N. T., Drosopoulos, G. A., Stavroulakis, G. E. Design and verification of auxetic microstructures using topology optimization and homogenization. *Arch Appl Mech* DOI 10.1007/s00419-014-0970-7

[45]. Bhullar** , S. K. **(2015) Three decades of auxetic polymers: a review. *e-Polymers* 15(4), 205–215.

[46]. Li D, Dong L, Lakes R.S. (2013) The properties of copper foams with negative Poisson’s ratio via resonant ultrasound spectroscopy, *Phys Stat Sol B,* 250(10), 1983–7.

[47]. Chiang F. (2009) Major Accomplishments in Composite Materials and Sandwich Structures. In: Daniel IM, Gdoutos EE, Rajapakse YDS, editors. Heidelberg, Germany: Springer; 779–98 pp.

[48]. Lisiecki J, Klysz S, Blazejewicz T, Gmurczyk G, Reymer P. (2014) Tomographic examination of auxetic polyurethane foam structures. *Phys Stat Sol* *B* , 251(2), 314–20.

[49]. Pierron F, McDonald SA, Hollis D, Fu J, Withers PJ, Alderson A. (2013) Comparison of the mechanical behaviour of standard and auxetic foams by x-ray computed tomography and digital volume correlation. *Strain* , 49(6), 467–82.

[50]. Gatt R, Attard D, Manicaro E, Chetcuti E, Grima JN. (2011) On the effect of heat and solvent exposure on the microstructure properties of auxetic foams: a preliminary study. *Phys Stat Sol B ,*248(1), 39–44.

[51]. Pozniak AA, Smardzewski J, Wojciechowski KW. (2013) Computer simulations of auxetic foams in two dimensions. *Smart Mater Struct*. 22(8), 1–11.

[52]. Yang, L. (2011) Structural Design, Optimization and Application of 3D Re-entrant Auxetic Structures. PhD Thesis. North Carolina State University.

[53]. Gibson, I., Rosen, D. W., Stucker, B. (2010) Additive Manufacturing Technologies: Rapid Prototyping to Direct Digital Manufacturing. Springer New York: Springer.

[54]. Gibson, I., Rosen, D. W., Stucker, B. (2010) Additive Manufacturing Technologies: Rapid Prototyping to Direct Digital Manufacturing. Springer New York: Springer.

[55]. Scarpa, F., Tomlin, P. J. (2000) On the Transverse Shear Modulus of Negative Poisson’s Ratio Honeycomb Structures. *Fatigue & Fracture of Engineering Materials*. 23(8), 717-20.

[56]. Gibson, L. J., Ashby, M. F*. *(1988) Cellular Solids: Structure and Properties. London: Cambridge University Press; 516.

[57]. Sneddon, I. N. (1951) Fourier Transforms. New York: McGraw-Hill, Inc. 542 p.

[58]. Lakes, R. (1991) Deformation Mechanisms in Negative Poisson’s Ratio Materials: Structural Aspects. *Journal of Material Science*. 26(9), 2287-92.

[59]. Bezazi, A. and Scarpa, F. (2007) Mechanical Behaviour of Conventional and Negative Poisson’s Ratio Thermoplastic Polyurethane Foams Under Compressive Cyclic Loading, *International Journal of Fatigue.* 29(5), 922-30.

[60]. Bianchi, M., Scarpa, F. L. and Smith, C. W. (2008) Stiffness and Energy Dissipation in Polyurethane Auxetic Foams, *Journal of Materials Science*. 43(17), 5851-60.

[61]. Lakes, R. S. (1993) Design Considerations for Materials with Negative Poisson’s Ratios, *Journal of Mechanical Design*. 115(4), 696-700.

[62]. Larsen, U. D., Signund, O., and Bouwsta, S., (1997) “Design and fabrication of compliant micromechanisms and structures with negative Poisson’s ratio”, *Journal of Micro electromechanical Systems*, 6, 99-106.

[63]. Ma, Z.-D., Wang, H., Kikuchi, N., Pierre, C. and Raju, B. (2003) “Function-Oriented Material Design for Next Generation Ground Vehicles”, ASME International Mechanical Engineering Congress & Exposition, November 15-1, 2003, Washington, DC.

[64]. Ma, Z. D., Bian, H., Sun, C.,Hulbert, G. M., Bishnoi, K. and Abadi, F. R. (2009) Functionally-Graded NPR (Negative Poisson’s Ratio) Material for a Blast-Protective Deflector Proceedings of the 2009 Ground Vehicle Systems Engineering and Technology Symposium.

[65]. Grima. J. N. and Evans, K. E. (2000) Auxetic behavior from rotating squares, *J. Mat. Sci. Lett*. 19, 1563-1565.

[66]. Grima, J. N., Mizzi, L., Azzopardi, K. M. and Gatt, R. (2015). Auxetic perforated mechanical metamaterials with randomly oriented cuts. *Adv. Mater*. 28, 385-389.

[67]. Shan, S., Kang, S. H., Zhao, Z., Fang, L. And Bertoldi, K. (2015). Design of planar isotropic negative Poisson’s ratio structures. *Ext. Mech. Lett*. 4, 96-102.

[68]. Cho, Y., Shin, J. H., Costa, A., Kim, T. A., Kunin, V., Li, J., Yeon, S., Lee, Yang, S., Han, H. N., Choi, I. S. and Srolovitz, D. J. (2014) Engineering the shape and structure of materials by fractal cut, *Proc. Natl. Acad. Sci.* USA 111, 17390-17395.

[69]. Grima, J. N., Manicaro, E. and Attard, D. (2011). Auxetic behaviour from connected di_erentsized squares and rectangles. *Proc. R. Soc. A* ,467, 439-458.

[70]. Blees, M. K., Barnard, A. W., Rose, P. A., Roberts, S. P., McGill, K. L., Huang, P. Y., Ruyack, A. R., Kevek, J. W., Kobrin, B., Muller, D. A. and McEuen, P. L. (2015) Graphene kirigami, *Nature *524, 204-207.

[71]. Dudte, L. H., Vouga, E., Tachi, T. and Mahadevan, L. (2016) Programming curvature using origami tessellations, *Nat. Mat*. 15, 583-588.

[72]. Lakes, R. (1986). Foam Structures with a Negative Poisson’s Ratio, *Science, * 4792, 1038-40.

[73]. Gaspar, N., Smith, C. W., Evans, K. E. (2009) Auxetic Behaviour And Anisotropic Heterogeneity. *Acta Materialia*., 57(3) , 875-80.

[74]. John, F. (1991) Partial Differential Equations. 4th Edition ed: Springer.

[75]. Lakes, R.(1991) Deformation Mechanisms in Negative Poisson’s Ratio Materials: Structural Aspects, *Journal of Material Science*. 26(9), 2287-92.

[76]. Critchley, R. (2015). The preparation and characterisation of auxetic foams for the application of trauma attenuating backings. Thesis

[77]. US-Army. (2010)Testing of Body Armor Materials for Use by the U.S. Army–Phase II: Letter Report. Washington DC, New York: US Army, Defence Do; Report No.: ISBN: 0-309-15222-4.

[78]. Starley, D. (1999). Determining the technological origins of iron and steel, *Journal of Archaeological Science , * 26(8), 1127-33.