Metagrains

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Investigating Auxeticity and Bistability in

Granular Mechanical Metamaterials

Mees Wortelboer 11241101 July 31, 2020

Report Bachelor Project Physics and Astronomy, size 15 EC, conducted between 30/03/2020 and 31/07/2020

INSTITUTE Van der Waals-Zeeman Institute

Soft Matter Group

Machine Materials Laboratory

SUPERVISOR Corentin Coulais

DAILYSUPERVISOR Daan Haver

EXAMINOR Edan Lerner

UNIVERSITY Universiteit van Amsterdam

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Abstract

Mechanical metamaterials are materials that exhibit exotic properties not found in nature. These properties arise from the geometrical structure of the material, rather than from the properties of its constituents. Virtually every mechanical metamaterial relies on a systematic distribution of unit cells to achieve these exotic properties. In this research, a new class of metamaterials is studied in which the unit cells are distributed randomly: granular mechan-ical metamaterials. Additive manufacturing allows us to design and create single metagrains with exotic properties. Changing the energy absorbing properties of single metagrains could alter the energy dissipating behaviour naturally present in packings of grains. Granular me-chanical metamaterials therefore hold the potential to be a class of materials with unique energy dissipating properties. To achieve temporary energy storage in metagrains during com-pression, auxeticity was investigated. Three designs, based on the unit cells of known auxetic lattices, were compared with the aim of maximizing volume change in compression. To further elaborate the energy storing abilities of the metagrains, slender beam elements were incorpo-rated in two of the auxetic metagrain designs, creating bistable metagrain designs. Bistability allows the metagrains to lock energy into the system even after unloading of the compressive forces. The metagrain designs were numerically simulated and compressed using Finite Ele-ment Analysis. To validate the simulations, the metagrains were 3D-printed and experimen-tally compressed. From both the numerical and experimental analysis, it was observed that the use of internal structures with more degrees of freedom allows for certain design choices that lead to more volume reduction during compression. The maximum observed surface area re-duction was 30.6%. The presence of more degrees of freedom can lead to anisotropic behaviour. Increasing the symmetry of the metagrains decreases the anisotropy of the metagrains. The incorporation of slender beam elements was observed to lead to snap-through deformations and a negative stiffness in the metagrains, but no bistable behaviour. The bistable metagrain designs are found to be very anisotropic, only exhibiting a negative stiffness under specific com-pression directions. The metagrains studied in this research were found to display interesting energy absorbing properties. It can be hypothesized that these will lead to unique interparticle contact, jamming behaviour and energy dissipation in granular packings. This would therefore be a promising field of discovery for future research.

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Samenvatting

Het gebruik van nieuwe technieken zoals 3D-printen, stelt onderzoekers in staat om nieuwe materiaalstructuren te maken met eigenschappen die niet in de natuur voorkomen, zogeheten mechanical metamaterials. De exotische eigenschappen komen voort uit de manier waarop de materialen zijn opgebouwd en niet uit de eigenschappen van de bouwmaterialen zelf. Deze technieken kunnen ook toegepast worden op korrelige materialen. Verzamelingen van korrels hebben interessante eigenschappen op het gebied van schokabsorptie en energie dissipatie. Door de eigenschappen van losse korrels te veranderen, kunnen de interessante eigenschap-pen van verzamelingen van korrels mogelijk veranderd en verbeterd worden. In dit onderzoek zijn een aantal korrels ontworpen met de eigenschap dat ze energie kunnen opslaan. In het eerste deel worden drie korrels onderzocht met de eigenschap dat ze krimpen als ze worden ingedrukt. Hierdoor kunnen deze korrels tijdelijk energie opslaan. De korrels hebben verschil-lende interne structuren. Ze worden met elkaar vergeleken met als doel het maximaliseren van volume verkleining. In het tweede deel van het onderzoek zijn twee van deze korrels verder aangepast om ze de eigenschap te geven dat ze een tweede stabiele toestand bereiken in compressie. Dit betekent dat ze na het indrukken niet meer terug vervormen, maar klein blijven. Hierdoor blijft de energie in de korrels opgeslagen ook nadat de indrukkende krachten zijn losgelaten. Door middel van computersimulaties en experimentele tests zijn deze korrels geanalyseerd. De maximale volume verkleining die is geobserveerd was 30.6%. Door de kor-rels meer symmetrisch te maken, kan worden voorkomen dat ze zich anders gedragen als ze van verschillende richtingen worden ingedrukt. De korrels uit het tweede deel van het onder-zoek vertoonde geen tweede stabiele toestand, maar wel gedrag dat erop lijkt. We zijn erin geslaagd korrels te maken met interessante eigenschapen op het gebied van energieabsorp-tie. Het bestuderen van verzamelingen van dit soort korrels kan uniek gedrag op het gebied van schokabsorptie en energie dissipatie aan het licht brengen en is daarom een interessante richting voor vervolgonderzoek.

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1 Introduction

1.1 Granular Mechanical Metamaterials

Mechanical metamaterials are artificially created materials exhibiting exotic properties not found in conventional materials. These properties arise from the structure of the material rather than from its composition [1]. In virtually every mechanical metamaterial the exotic properties arise from a very systematic distribution of unit cells. One of the motivations for this project is to study metamaterials in which this distribution of unit cells is not systematic but random. To achieve this, we combine metamaterials with granular materials, which are materials where a random distribution of building blocks is inevitably present. The building blocks of a granular material are discrete solid particles: the grains. In the case of a granular mechanical metamaterial, each grain is one unit cell exhibiting the designed property: a meta-grain.

Granular materials are commonly seen in nature and widely used in industry [2]. As such their properties have been studied extensively. Granular materials can undergo a transition from a flowing liquid-like state to a jammed solid-like state. This transition, schematically depicted in Figure 1, is known as the jamming-transition. This transition has been widely studied for conventional granular materials [3], [4], [5], and can be exploited for interesting applications, for example in universal grippers for robotics [6]. This jamming-transition also makes granular materials uniquely efficient in attenuating shockwaves and dissipating energy [7]. When a shockwave hits a granular material, sections of the material will undergo this transition. The many particle collisions in these sections lead to energy losses through interparticle friction

Figure 1: Schematic depiction of the jamming-transition in a granular packing, adapted from [8]. By increasing the packing fractionφ past a critical value φc, the system experiences a transition from an

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Figure 2: Example of an auxetic metamaterial, adapted from [12]. Due to the internal structure this material contracts horizontally when compressed vertically.

and viscoelastic effects [9]. Metagrains can be designed to have energy absorbing properties. Energy absorption by single grains could change the energy dissipating behaviour exhibited by granular packings. This combination of metamaterials and granular materials could therefore lead to unexplored, and potentially very interesting behaviour in terms of energy dissipation. Two interesting properties associated with energy absorption are auxeticity [10] and bistability [11]. In the next section these properties will be further discussed.

1.2 Auxeticity

Auxeticity is the property of exhibiting a negative Poisson’s ratio (PR). The PR of a material is defined as follows:

ν= −δ²transverse

δ²axial

, (1)

where νis the Poisson’s ratio, δ²transverse is the strain difference perpendicular to the

direc-tion of compression and _δ²_axial is the strain difference along the direction of compression. A negative PR thus means that compression along one axis leads to contraction along the per-pendicular axis (see Figure 2). This effectively gives the material the property to shrink under external pressure rather than shear.

Auxetic materials have proven to be very suitable for absorbing energy [10]. Through local elastic deformations, energy can be temporarily stored in a system. In this process some en-ergy is dissipated as heat. Upon unloading, the remaining enen-ergy is released as the structure deforms back to its original configuration. Compression limits the space available for a struc-ture to deform. This can quickly halt elastic deformations when the available space is filled. Auxetic deformation has the advantage that it decreases the size of the structure being com-pressed, thus allowing more deformation and energy storage than non-auxetic structures un-der compression. Changing the kinematic behaviour of grains by making them auxetic would

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(a) (b)

Figure 3: (a) Bistable beam: upon constraining and compressing the beam in the way shown in the figure, beams can experience a snap-through deformation from one stable state to the other, known as snapping. Adapted from [11]. (b) Typical force-displacement curves associated with snapping deforma-tions. The red curve shows a bistable snapping, where an amount of energy (Ein− Eout) is locked into

the system. The blue curve shows a metastable snapping. Here, no energy trapping takes place, because the local minimum in force is larger than 0.

therefore in principle allow single grains under compression to temporarily store high amounts of energy.

1.3 Bistability

A bistable structure has two distinct stable configurations. Through deformation, the struc-ture can transition from one stable configuration to the other. This property is often realised in structures by adding slender beam elements [11], [13], [14]. Beams are known to exhibit bistable behaviour when constrained and compressed under specific conditions. An example of a such a beam is shown in Figure 3a, adapted from [11]. When compressed vertically, diagonal beams can deform from one stable state to another by means of a snap-through deformation known as snapping. Figure 3b shows a typical force-displacement curve (Fd-curve) of such a deformation. The descending section of the curve, between the local maximum and the local minimum, indicates the system under consideration exhibits a negative stiffness during this range of compression.

Energy is stored in the system through local elastic deformations. Contrary to monostable structures, bistable structures have the ability to trap part of this energy, such that it is con-tained in the structure even after unloading of the compressive forces. An amount of energy (Ein) is put into the system by deforming parts of it. When a peak in the Fd-curve is reached,

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boundary, is the point where the system snaps to its second stable configuration. This releases a fraction of the input energy (Eout). The remaining fraction of the input energy (Ein− Eout)

is contained in the system. To release this fraction, an amount of energy larger than Eout

has to be put into the system [11]. When a system exhibits a negative stiffness, but the local minimum in the Fd-curve does not reach a negative value, the structure is called metastable (see Figure 3b). Metastable structures can absorb high amounts of energy, and contain it at a relatively low compressive force. However, this energy is not locked into the system. Upon complete unloading, the structure will deform back to its original configuration and the energy will be released.

Changing the energy landscape of single auxetic metagrains by making them metastable or bistable could further enhance their energy storing abilities. Bistable metagrains could be de-signed to lock energy into the system and release it in a controlled manner. This could give rise to interesting, unexplored energy dissipation properties in packings of metagrains.

In this research a selection of energy absorbing metagrains is designed. The first part of this research focusses on altering the kinematics of the metagrains by designing them to be auxetic. This part builds on an earlier study on auxetic metagrains done by Antoine Dop [15], where one design for an auxetic metagrain was proposed. Here, we propose two more designs, with the aim of maximizing volume change during compression. The second part of this research focusses on changing the energy landscape of these auxetic metagrains, by incorporating slen-der beam elements into the designs. The aim for this part is to create metagrains that exhibit bistable behaviour during compression. To study the behaviour of these metagrains, they are simulated and compressed using the Finite Element Analysis (FEA) software Abaqus. To val-idate the simulations, the grains are also 3D-printed and experimentally tested. The meta-grains studied in this research were found to display interesting properties associated with energy absorption. It can be hypothesized that these will lead to unique interparticle contact, jamming behaviour and energy dissipation in granular packings. This would therefore be a promising field of discovery for future research.

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2 Auxeticity - Altering the Kinematics of the Metagrains

2.1 Design

2.1.1 From Lattices to Linkages

Figure 4: Auxetic lattices: (a) re-entrant honeycomb lat-tice, undeformed (left) and compressed (right), adapted from [16]. (b) Kagome lattice, undeformed (left) and com-pressed (right), adapted from [17]. Blue arrows indicate direction of compression, green arrows indicate the re-sulting contraction.

For the design of the auxetic meta-grains, inspiration was drawn from existing auxetic lattices. These lat-tices are shown in Figure 4. In the right hand images, the blue arrows indicate the direction of compression and the green arrows show the result-ing contraction. The lattice shown in Figure 4a is known as a re-entrant honeycomb lattice [16]. The lattice shown in Figure 4b is known as a kagome lattice [17]. Based on the unit cells of these structures, auxetic linkages were designed. These link-ages form the basis for the auxetic behaviour of the metagrains. Fig-ure 5 shows these linkages, highlighted in red, in context of the lattice unit cells.

Figure 5: Auxetic linkages (red) in context of the lattice unit cells: (a) shows a 4-bar linkage based on the unit cell of the re-entrant honeycomb lattice. (b) shows a 6-bar linkage based on the unit cell of the kagome lattice. (c) shows a 6-bar linkage based on the unit cell of the re-entrant honeycomb lattice.

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The linkage shown in Figure 5a is a 4-bar linkage with one degree of freedom. A single de-gree of freedom means that changing one angle determines all the others, provided that the bars are kept undeformed. This linkage is the basis for the earlier proposed auxetic meta-grain design by Antoine Dop. The linkages shown in Figure 5b and 5c are two variations of a 6-bar linkage. These linkages form the basis for the two newly proposed auxetic metagrain design. These structures have three degrees of freedom, giving them more freedom of defor-mation when angles are changed. It is hypothesized that metagrains based on a 6-bar linkage will exhibit more volume change during compression. This extra freedom of movement may also lead to anisotropic behaviour, meaning compression from different sides leads to different deformations.

2.1.2 From Linkages to Metagrains

The step-by-step design process of the auxetic metagrains is schematically depicted in Figure 6. To create metagrains from the chosen linkages (Figures 6a, 6b & 6c), circles were drawn around these structures, defining the bodies of the grains. The body of each metagrain is then a circular disk with a linkage-shaped hole in it (Figures 6d, 6e & 6f). In Figure 6f the shaded triangles were also filled with body material. This does not change the nature of the internal 6-bar linkage. To allow auxetic compression, triangular cuts were made in the bodies of the metagrains (Figure 6g, 6h & 6i). These triangular cuts are referred to as the mouths of the grains.

Figure 6: Step-by-step depiction of the design process for the auxetic metagrains. (a), (b) & (c) show the linkages taken as internal structure for the grains. (d), (e) & (f) show the circular bodies of the grains, drawn around the linkages. The shaded area in (f) was also filled with body material. This does not change the nature of the internal 6-bar linkage. (g), (h) & (i) show the finalized metagrain designs with triangular cuts to allow auxetic compression.

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(a) (b) (c)

Figure 7: Auxetic metagrain designs as designed in Abaqus: (a) Grain 1 ’Pacman’, adapted from [15], (b) Grain 2 ’Star Fighter’, (c) Grain 3 ’Shuriken’. The green sections define the bodies of the grains. The black lines covering the metagrains define elements used by Abaqus to perform the calculations.

These design steps lead to the final auxetic metagrain designs shown in Figure 7: Grain 1 ’Pacman’ (Figure 10a) adapted from the research of Antoine Dop, Grain 2 ’Star Fighter’ (Figure 10b) and Grain 3 ’Shuriken’ (Figure 10c). To simplify simulations, the grains were simulated in 2D. The grains were designed with a radius of 2.5 mm. [Due to a mistake in the design process Grain 3 was designed with a radius of 0.25 mm. However, this does not change the results of the simulations if a scaling of 1/10 is employed where necessary.] The hinges, the thinnest parts of the grains where most of the deformations will occur, were designed with a thickness of 0.3 mm. The mouths of the grains were designed to be 1.38 mm wide at the edge of the grains. This width was chosen in consideration of experimental constraints that arise when treating packings of these metagrains. When the mouths are too wide, the mouths of different grains will interlock and prevent auxetic compression.

2.1.3 Metagrain Material

The material of the metagrains was simulated by a Neo-Hookean model for hyperelastic ma-terials [18]. Two parameters used by Abaqus to describe this class of mama-terials are the shear modulus G and the bulk modulus K of the material. The shear modulus is a measure of how much a material deforms under shearing forces. The bulk modulus is a measure of how resis-tant a material is to volume change under compression [19]. The material was given a shear modulus of 2 P a and a bulk modulus of 20, 000 P a. This regime of a bulk modulus several orders of magnitude higher than the shear modulus puts the material in the incompressible limit, meaning that total material volume has to be conserved in deformation. This is a good approximation for rubbers. Incompressible materials have a PR very close to 0.5 [20]. From the shear and the bulk modulus a third modulus can be calculated: the Young’s modulus E. The Young’s modulus is a measure for how much a material elongates under tensional stresses

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[19]. It can be calculated using:

E = 9KG

G + 3K. (2)

This gives the material in the numerical models a Young’s modulus of 5.9998 P a.

2.2 Numerical Analysis

2.2.1 Compression

The auxetic metagrains are simulated using FEA software Abaqus. FEA is a simulation method used to predict the reaction of a system when it is subjected to real-world effects such as forces, vibrations, displacements, heat, etc. Structures can be designed, given specific mate-rial properties and be subjected to boundary conditions and external forces. A so-called mesh is used to divide the system into small elements. Abaqus solves equilibrium equations for each of these elements to calculate their behaviour as a result of the prescribed boundary conditions and forces. Adding up the results of the equilibrium equations for all of these elements, gives the behaviour of the entire system under consideration. Abaqus thus allows us to numerically compress these grains and see how they would react in real-world situations.

To mesh the auxetic metagrains, the mesh seeds were distributed evenly over the edges of the grains and set to be 0.15mm apart. A plane stress model was used in the 2D simulations. This means the stress is assumed to be distributed only in-plane. This assumption is best suited for thin models such as our metagrains. The precise mesh element type used, is described by the following code: CPS4R.

(a) (b)

Figure 8: Grain 1 showing the important nodesets: (a) Pressure Nodes, used to compress the grains. (b) Measure Nodes, used to track the area of the grains during compression.

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Numerical compression of the grains was done by employing a prescribed displacement on a set of points. Both at the top and bottom of the grains, a number of points were defined as Pressure Nodes (PN’s), with the middle PN’s directly above and below the centre point of the grain. Figure 8a shows the PN’s as defined on Grain 1. They are evenly spaced at 0.1mm. This set of points was taken as an approximation of a pressing surface. This was done to simplify the simulations. By default, five points were taken on the top and bottom. In the case of Grain 1, this led to excessive distortion and excessively high and localized stresses around these points during compression. These extreme values make Abaqus unable to solve the equilibrium equations and cause the simulation to be aborted. To prevent this, seven points were taken at the top and bottom as PN’s for Grain 1. By moving the top PN’s down and the bottom PN’s up, the metagrains are compressed. To prevent unwanted rotation, the middle PN’s were restricted from sideways movement. Abaqus was instructed to do this in a minimum of 21 evenly spaced incremental steps. When excessive distortions or stresses occur during an increment, Abaqus reduces the size of the increment and recalculates, in order to avoid aborted simulations. This ultimately results in more increments for these simulations. The auxetic metagrains were compressed by 0.3 times their diameter, corresponding to an axial strain of −0.3.

2.2.2 Measuring Volume Change and other Figures of Merit

To test the hypothesis that Grains 2 and 3 will exhibit more volume change during compression than Grain 1, the volume of the metagrains was calculated during each increment of compres-sion. Since the metagrains were simulated in 2D, this reduces to calculating the surface area. To gain more insight on the behaviour of these metagrains, three other figures of merit were defined: an effective PR, the force required for compression and the effective stiffness of the metagrains were calculated during each increment of compression. These four figures of merit were plotted against the axial strain. Furthermore, it was observed from the simulations that the behaviour of the grains exhibits strong changes at the increment where the mouths com-pletely close or the internal structure is fully folded onto itself. The strain at these increments was defined as the critical strain and identified for each of the auxetic metagrains.

To calculate the area of the metagrains during compression, a set of points was defined as Measure Nodes (MN’s) (see Figure 8b). This set contains 16 points, located on the edge of the metagrains. Abaqus tracks the displacement of these points. This data was extracted and used to calculate the position of each point during each increment of compression. An ellipse was fitted through these points at each increment of compression to approximate the area of the metagrains. This area was normalized by the initial, uncompressed area.

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This area was also used to calculate the effective PR of the metagrains during compression. The ellipses used to approximate the surface area of the metagrains were, more often than not, tilted by some angle. This makes it very difficult to determine what δ²transverse and δ²axial

are. To overcome this, an approximation has been used in determining these values. This approximation is the reason we defined an effective PR. This effective PR was calculated in four steps. First the axial strain was calculated using:

δ²axial=δP N− D0

D0

, (3)

where δP N is the distance between the middle top and bottom PN’s at a certain increment

of compression and D₀ is the initial diameter of the metagrains. Then, the area strain was calculated using:

δ²area=δA− A0

A0

, (4)

where δA is the area at a certain increment of compression and A0 is the initial area of the

metagrains. From this, the transverse strain was approximated using:

δ²transverse=δ²area−δ²axial. (5)

The effective PR of the metagrains during each increment of compression was then calculated using:

P R_{e f f} _{= −}δ²transverse

δ²axial

. (6)

The force required to move the PN’s in the defined way is tracked by Abaqus. This data was exported and plotted against the axial strain to generate Fd-curves for the auxetic metagrains. The force was normalized by dividing it by the initial radius, the thickness and the Young’s modulus of the material. This normalization was done to obtain the Fd-curves in dimensionless units and to make them better comparable to experimental data. Abaqus sets the thickness of 2D structures to 1 by default. The slope of the Fd-curves was taken as the effective stiffness of the metagrains. We defined an effective stiffness because this is technically a material property and here we assign it to structures. The effective stiffness was normalized by the Young’s modulus of the material.

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(a) (b) (c)

Figure 9: Compression directions of the auxetic metagrains. (a) Grain 1 was rotated 4 times in equal steps of 22.5° (b) Grain 2 was rotated 3 times in equal steps of 10°. (c) Grain 3 was rotated 4 times in equal steps of 22.5°. Further rotation was not necessary due to the symmetry of the metagrains.

2.2.3 Measuring Isotropy

In a granular packing, each grain would be oriented differently and compressed from different sides. The metagrains are not isotropic in design. This might lead to anisotropic behaviour. Grain 1 has an internal linkage with one degree of freedom. This means that Grain 1 should behave perfectly isotropically, were the body of the grain undeformable except for the hinges. Since the body of the grain is simulated to be a deformable material, some deformation will take place in the body, outside of the hinges. These out-of-hinge deformations will require more energy than in-hinge deformations, since the hinges were designed to be the thinnest part of the grains. Isotropic behaviour is therefore energetically favourable to anisotropic behaviour in the case of Grain 1. Grains 2 and 3 have an internal linkage with three degrees of freedom. This means that these grains would behave anisotropically even in the theoretical limit where only in-hinge deformations occur. It is therefore hypothesized that Grain 1 will exhibit more isotropic behaviour than Grains 2 and 3.

To measure the anisotropic behaviour of the grains, the metagrains were numerically com-pressed from different sides. This was done by rotating the grains around their centre point, while redefining the PN’s directly above and below this centre point for each simulation. The symmetry of the grains was exploited to limit the amount of orientations tested. In Figure 9 the grains are shown with the black dashed lines indicating the directions along which they were compressed. When the placement of the mouths of the grains hindered the placement of the PN’s above or below the centre point, two PN’s were defined at the points where the mouth meets the edge of the grain. One of these PN’s was restricted from sideways movement.

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In a large packing of grains, all metagrain orientations will be equally represented. The be-haviour of each individual metagrain can therefore be approximated by a mean bebe-haviour of the different orientations. Due to the symmetry of the metagrains, some orientations will occur more often than others. This is shown in Figure 10. In Grains 1 and 3, the 0° and 90° orienta-tions occur once. The 22.5°, 45° and 67.5° orientaorienta-tions for Grain 1 and Grain 3 have a mirrored counterpart, which is essentially the same orientation. As such these orientations occur twice. In calculating a mean, this has to be taken into account. Grain 2 has three axes of symmetry. The pattern of red dashed lines repeats itself in the other sections of this grain. The black dotted lines and the red dashed line with a yellow circle occur half as much as the red dashed lines with red and blue circles. This effectively means that the 0° and the 30° orientations have to be counted once and the 10° and 20° orientations for Grain 2 have to be counted twice when calculating a mean for the figures of merit.

(a) (b) (c)

Figure 10: Repetition of orientations due to symmetry that has to be accounted for in calculating a mean. (a) Grain 1: 0° (vertical dotted line) and 90° (horizontal dotted line) orientations have to be counted once. Orientations in between have to be counted twice. After 90°: coloured circles show mirrored pairs. (b) Grain 2: Black lines show symmetry axes. Pattern of red dashed lines is repeated. 0° (vertical dotted line) and 30° (red dashed line yellow circle) orientations have to be counted once. Other orientations have to be counted twice. After 30°: coloured circles show mirrored pairs. (c) Grain 3, same as Grain 1: 0° (vertical dotted line) and 90° (horizontal dotted line) orientations have to be counted once. Orientations in between have to be counted twice. After 90°: coloured circles show mirrored pairs.

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2.2.4 Numerical Results

The results of the area change and the effective PR measurements are shown in Figure 13. Some important numbers are shown in table 2.2.4. The critical strain was clearly visible in the simulations. When either the mouths were fully closed or the internal linkages were fully folded onto themselves, self-contact caused dramatic changes in the behaviour of the meta-grains. It can be observed in both the strain curves and the PR-strain curves. In the area-strain curves this is the point where the curves level out, indicating the grains have stopped shrinking. In the PR-strain curves this is the point where the curves level out above 0, in-dicating the grains are no longer auxetic. Grain 2 shows the lowest mean area minimum, corresponding to the largest mean area change. Both Grains 2 and 3 show more area change than Grain 1. The rate of area change is about the same for all three grains, but Grains 2 and 3 change area over a larger range of strains. This can be seen explicitly by looking at the critical strain, which is higher for Grains 2 and 3. This confirms our hypothesis that the use of an internal 6-bar linkage leads to more volume change during compression compared to an internal 4-bar linkage. Through the use of a 6-bar linkage, more freedom of deformation is created. By exploiting this freedom through the strategic placement of multiple mouths, the metagrains are allowed to compress auxetically over a larger range strains, leading to more volume reduction.

Grain 1 ’Pacman’ Grain 2 ’Star Fighter’ Grain 3 ’Shuriken’

mean critical strain 0.125 0.2 0.21

range critical strain 0.09 - 0.15 0175 - 0.22 0.19 - 0.26

minimum mean area 0.819 0.694 0.702

(divided by initial area)

(a) (b)

Figure 11: Two orientations of Grain 3 at the point of critical strain. (a) 0° orientation, (b) 45° ori-entation. The difference in behaviour is due to the freedom of deformation the internal 6-bar linkage provides.

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The PR plots confirm that the grains are auxetic up until a little before the critical strain. The one exception is the 45° orientation of Grain 3. This orientation starts out with an effective PR of 0.57 and this value never drops below 0. Figure 11 shows the 0° orientation (11a) and the 45° orientation (11b) of Grain 3 at the point of critical strain. It clearly shows the anisotropy of Grain 3 due to the extra degrees of freedom of the internal 6-bar linkage. The use of an inter-nal 4-bar linkage, with a single degree of freedom, in Grain 1 led to almost uniform behaviour in terms of area change and effective PR. Although Grain 2 also contains an internal 6-bar linkage, it displayed almost the same amount of isotropy as Grain 1. This difference between the isotropy of Grain 2 and Grain 3 can be explained by the symmetry of Grain 2. Grain 2 has three axes of symmetry while Grain 3 only has two. This higher degree of symmetry in Grain 2 means that the different orientations are closer together in rotation and thus behave more similarly in compression. This partially confirms our second hypothesis, concerning the effects of the internal linkages on the isotropy of the grains. Although more degrees of freedom led to more anisotropy, the symmetry of the linkage also has a significant effect where more symmetry leads to less anisotropy.

The results for the force and stiffness measurements are shown in Figure 14. The critical strain, at which self-contact in the metagrains leads to dramatic changes in behaviour, can again be identified in both the Fd-curves and the stiffness-strain curves. In the Fd-curves this is the point where the slope of the mean curve increases, indicating a strong increase in the force required for further compression. In the stiffness curves this is the point where the curves roughly levels out after a sudden increase. All three Fd-curves and therefore also the stiffness curves show the same trend. The metagrains start out with a relatively low stiffness, which decreases a little at first when the grain is auxetically deforming. After the point of critical strain is reached, the metagrains show a dramatic increase in stiffness. In all three grains the mean stiffness increases over 10-fold from the lowest measured mean value.

Figure 12: Grain 1, 0° orientation at -0.3 axial strain

Remarkable to see is the extreme behaviour of the Fd-curve and stiffness Fd-curve of the 0° orientation of Grain 1. Figure 12 shows this orientation at ²axial = −0.3.

After the point of critical strain, the grain starts to be-have as a uniform piece of material and the PN’s are pressed straight into the body of the grain. In the other orientations of Grain 1 and in Grains 2 and 3, the com-ponents have the freedom to slide over each other, re-ducing the stiffness of the grains.

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(a) (b)

(c) (d)

(e) (f)

Figure 13: Results of the area change and effective PR measurement on the auxetic metagrains: (a) and (b) show the results for Grain 1, (c) and (d) show the results for Grain 2 and (e) and (f) show the results for Grain 3. The black dashed line shows the mean. The shaded green area gives a measure for the anisotropy of the metagrains. A bigger green area means more anisotropy.

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(a) (b)

(c) (d)

(e) (f)

Figure 14: Results of the force and effective stiffness measurement on the auxetic metagrains: (a) and (b) show the results for Grain 1, (c) and (d) show the results for Grain 2 and (e) and (f) show the results for Grain 3. The forces F are normalized by the initial radius R0 and the Young’s modulus E. The stiffness k is normalized by the Young’s modulus. The black dashed line shows the mean. The shaded green area gives a measure for the anisotropy of the metagrains. A bigger green area means more anisotropy.

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2.3 Experimental Analysis

In an experimental setting, a lot of real-world factors come into play that were disregarded in the simulations. Examples are out-of-plane deformations, viscoelastic effects, frictional forces and manufacturing imperfections. These factors were disregarded to simplify the simulations and because they are expected to only have a very small effect on the results of our analyses. To validate this approach, the grains were 3D-printed and experimentally tested. Fd-curves obtained from these tests were compared to the corresponding numerically obtained Fd-curves.

2.3.1 Printing and Compression

The metagrains were manufactured with a diameter of 2cm and a thickness of 0.4cm, using the Formlabs 3D-printer Form 3. Figure 15 shows the printed metagrains. The material used for printing was the Formlabs Elastic 50A resin. The exact value of the Young’s modulus for a print made of this material depends on the structure. However it can be estimated using:

E = exp((Shore − A Durometer) ∗ 0.0235 − 0.6403), (7) where E is in MP a and Shore − A Durometer is a rating on the Shore Durometer scale. This is a method for testing the hardness of a material [21]. The Elastic 50A resin has a Shore-A Durometer rating of 50, leading to an approximate Young’s modulus of 1.7 MP a. The measured force is normalized in the same way as in the numerical analysis, using this estimation of the Young’s modulus and the dimensions of the metagrains. This will not give an exact comparison between experiments and simulations due to this estimated Young’s modulus, but it will suffice for validating the simulations. The grains were compressed, oriented as shown in Figure 15, to an axial strain of −0.25 using the Instron 5943 single column testing system with a 500N load cell with an accuracy of 0.05N.

(a) (b) (c)

Figure 15: Snapshots of the printed auxetic metagrains: (a) Grain 1 ’Pacman’, (b) Grain 2 ’Star Fighter’, (c) Grain 3 ’Shuriken’. The metagrains were printed with a diameter of 2cm and a thickness of 0.4cm.

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2.3.2 Experimental Results

Figure 16: Grain 3 com-pressed to the point of crit-ical strain. (a) shows the 3D-printed and experimentally compressed metagrain. (b) shows the simulated meta-grain.

The results of the experimental tests are in good qualitative agreement with the simulations. In deformation, the experi-ments showed a close match to the simulations up to the point of critical strain. This is exemplified in Figure 16, where Grain 3 is shown printed (16a) and simulated (16b) at the point of critical strain. When the strain exceeded the critical value, the experiments started to slightly deviate from the sim-ulations in terms of deformation. A possible explanation for this, is the difference between numerical compression and ex-perimental compression. In the simulations, a pressing sur-face was approximated by a set of pressing points. After the point of critical strain, the PN’s are pressed into the body of the grains. In the experiments the grains were compressed with a pressing surface. After the point of critical strain, the contact area between the grains and the pressing surface starts to get bigger due to the non-auxetic compression of the grains. This difference only becomes apparent after the point of critical strain and could explain the difference in deforma-tion.

The Fd-curves obtained from the experimental tests are shown in Figure 17 together with the Fd-curves of the corresponding simulations. Figure 17a shows the results for Grain 1, 17b for Grain 2 and 17c for Grain 3. The red curves show the experimental Fd-curves and the blue

(a) (b) (c)

Figure 17: Experimentally obtained Fd-curves (red) compared to the numerically obtained Fd-curves (blue) of Grain 1 (a), Grain 2 (b) and Grain 3 (c).

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curves show the numerical Fd-curves. The experimental curves show the same trend as the numerical curves, with a strong increase in force after the points of critical strain. For Grains 1 and 3 this point seems to be at almost the exact same strain. In Grain 2 there is a slight dif-ference most likely due to the difdif-ference in compression described in the previous paragraph.

These results validate our numerical simulations at least up to the point of critical strain. The extreme behaviour seen in the simulations after this point is less apparent in the experiments, but the same trend was observed. Our hypothesis concerning the maximization of volume change by use of an internal 6-bar linkage has been confirmed both numerically and experi-mentally. The volume changing ability of the metagrains is heavily dependent on the closing of the mouths, as is illustrated by the dramatic changes in behaviour at the point of critical strain. The volume changing ability of Grain 1 could be enhanced by widening the mouth, but then the experimental constraint of limited mouth width would no longer be met. The use of an internal 6-bar linkage allows the implementation of multiple mouths, giving the metagrains more volume changing ability while ensuring experimental constraints are met.

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3 Bistability - Shaping the Energy Landscape of the Metagrains

3.1 Incorporating Bistable Beams

Figure 18: Diagonal beam showing vari-ables. θ is the beam angle, t is the thick-ness and L is the length of the beam.

The approach for designing bistable metagrains is to incorporate slender beam elements in the auxetic metagrain designs. These beams have to be ori-ented in such a way that compression of the grains leads to a loading condition as shown in Figure 18 by the arrow. To this end, one beam was placed in the mouth of Grain 1 to create Grain 4 ’Ms Pacman’ (Figure 19a), and two beams were placed inside the internal 6-bar linkage of Grain 3 to create Grain 5 ’Butterfly’ (Figure 19b). To ensure the beams have room to snap, the bodies of the grains were altered slightly.

An earlier study was done on a metamaterial that relied on these diagonal beams to achieve bistability [11]. In this study, a lattice was designed consisting of unit cells that contained two of these beams. Tests were done on these unit cells to determine the influence of the beam angleθ and the thickness to length ratio t/L (see Figure 18) on the snap-through behaviour of the unit cells. The results of this study are shown in Figure 20. The colour scale indi-cates the amount of energy trapped in the system. The black dashed line indiindi-cates an experi-mentally observed transition between metastable behaviour above the line, where no

energy-(a) (b)

Figure 19: Bistable metagrains: (a) Grain 4 ’Ms Pacman’ with_{θ = 60° and t/L = 0.14 , (b) Grain 5} ’Butterfly’ with_{θ = 70° and t/L = 0.10}

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Figure 20: Mechanical response of a constrained, tilted elastic beam: Effect of θ and t/L on the energy absorbed by the elastic beam (Ein).

The black dashed line indicates an experimen-tally observerd transition between metastable haviour (no energy-trapping) and bistable be-haviour. Adapted from [11]

trapping occurred and bistable behaviour be-low the line.

In this research we will investigate the influ-ence of the beam angleθand the ratio t/L on the Fd-curve characteristics of our bistable metagrains. Based on the results of [11], a set of test values for_θ and t/L was chosen, with the aim to see how these results of isolated sets of beams translate to the environment of our bistable metagrains. For Grain 4, the test values forθare 50°, 60° and 70° and the test values for t/L are 0.08, 0.10, 0.12, 0.14 and 0.16. For Grain 5, the test values forθare 60° and 70° and the test values for t/L are 0.06, 0.08, 0.10, 0.12 and 0.14. A metagrain with a certain combination ofθand t/L is referred to as a configuration of this general metagrain design. Grain 5 configurations with θ_{= 50°} were omitted because there was not enough room to incorporate two beams under this an-gle in the current Grain 5 design.

3.2 Numerical Analysis

The bistable metagrains were simulated in a 2D plane stress model with a radius of 2.5cm. To reduce the chance of simulation failure, the mesh was altered to give smaller elements in the hinges and beams, where most of the deformation and stresses will occur. The mesh seeds in the hinges and beams were placed 0.5mm apart. The mesh seeds in the rest of the body were placed 7.5mm apart. The precise mesh element type used, is described by the code: CPS4R. The bistable metagrains were compressed in the same way as the auxetic metagrains: a set of PN’s was defined above and below the centre point of the grains and a prescribed displacement was employed on these points. Grain 4 was compressed to -0.25 axial strain. Grain 5 was compressed to -0.225 axial strain. The snapping of the beams was observed to happen before these values were reached and further compression past these values resulted in aborted sim-ulations.

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(a) (b)

Figure 21: Compression directions of the bistable metagrains. (a) Grain 4 was rotated 3 times in equal steps of 22.5° and 3 times in equal steps of −22.5°. (b) Grain 5 was rotated 4 times in equal steps of 22.5°. Further rotation was not necessary due to the symmetry of the metagrains.

To study the behaviour of the bistable metagrains, Fd-curves were created for all Grain 4 and Grain 5 configurations during compression. Three figures of merit were defined to characterize these Fd-curves: the peak force, the size of the force drop and the difference in strain between the local maximum and the local minimum.

To gain insight on the anisotropy of the bistable metagrains, one configuration of each grain was chosen to compress from different sides. Only one configuration was chosen for each grain due to limited time and because it is expected that the other configurations will show approx-imately the same level of anisotropy. For Grain 4 the θ= 60° t/L = 0.12 configuration was chosen. For Grain 5 the θ_{= 70° t/L = 0.10 configuration was chosen. The symmetry of the} metagrains was exploited to limit amount of orientations. The compression directions for the bistable metagrains are shown in Figure 21.

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3.2.1 Numerical Results

Figure 22: Two configurations of Grain 4 show-ing different kinds of deformations. (a) shows a deformation resembling the Euler Buckling in the _{θ = 70° t/L = 0.08 configuration. This} buckling can be identified by the presence of three bending points in the beam. (b) shows a snapping deformation in theθ = 70° t/L = 0.14 configuration. A snapping beam can be iden-tified by the presence of two bending points in the beam.

The Fd-curves of the bistable metagrains are shown in Figure 25. Curves that terminate before reaching the end of compression indicate a simu-lation that was aborted due to excessive distortion or stresses. None of the configurations exhibited bistable behaviour. However, metastability was observed in most of the configurations of Grain 4 and Grain 5. This is most likely due to the stiff-ness of the bodies of the metagrains. The negative stiffness of the beams is counteracted by the pos-itive stiffness of the metagrain bodies, resulting in metastable behaviour rather than bistable be-haviour. The Grain 4 configurations that finished the simulation all showed metastable behaviour and a beam exhibiting a snapping deformation. An example of such a deformation in a Grain 4 configuration is shown in Figure 22b. It can be identified by the presence of two bending points in the beam. The Grain 4 configurations whose sim-ulations were aborted, all exhibited a deformation resembling the so-called Euler buckling. This de-formation is shown in Figure 22a in context of a Grain 4 configuration. It can be identified by the presence of three bending points in the beam. In

the bistable metagrains, a buckling beam led to excessive stresses and distortions, causing the simulations to be aborted. It occurs when the beams are loaded along the axis of the beam rather than under an angle. This type of deformation is typically associated with a plateau in the Fd-curve but not with a force drop. When the beams were thick enough, they deformed the top of the grain, resulting in the force being applied more vertically and ultimately in a snapping deformation.

This distinction between different types of beam deformations was also visible in Grain 5, al-though a buckling deformation in one of the beams in the Grain 5 configurations, still resulted in metastable behaviour. The one exception was the_θ_{= 60° t/L = 0.10 configuration. This}

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con-(a) (b)

Figure 23: Two configurations of Grain 5 showing different kinds of deformations. (a) shows a snapping deformation in both beams of the θ = 70° t/L = 0.10 configuration. A snapping deformation can be identified by the presence of two bending points in the beam. (b) shows a snapping deformation in the left beam and a buckling deformation in the right beam of the_{θ = 60° t/L = 0.12 configuration. A} buckling deformation can be identified by the presence of three bending points in the beam.

figuration did not show metastable behaviour. The Grain 5 configurations where both beams showed a snapping deformation are shown by the red and blue curve in Figure 25b and the red, blue and yellow curve in Figure 25d. In the other configurations, at least one beam showed a buckling deformation. An example of a Grain 5 configuration with two snapping beams is shown in Figure 23a. An example of a Grain 5 configuration with only one snapping beam is shown in Figure 23b. The fact that a force drop is still observed for these configurations is most likely due to the snapping deformation by the second beam (left beam in Figure 23b). It appears that the Grain 5 design only leads to a snapping deformation in both beams when the beams are thin enough. Thicker beams require more force to snap. A buckling deformation is apparently energetically more favourable. The freedom of deformation stemming from the use of the internal 6-bar linkage, makes Grain 5 unstable and allows the grain to deform asymmet-rically, leading to buckling.

The results of the figure of merit measurements are shown in table 3.2. The green elements show the maximum value of the force drop and the peak force and the minimum value of the strain difference. In general, a higher_θ and a higher t/L both yielded a higher peak force. A higherθyielded a bigger force drop for the same t/L. A higher t/L did not always yield a bigger force drop. There seems to be an optimal region in the middle of the tested values of t/L. In Grain 4 the largest force drop was obtained by theθ= 70° t/L = 0.14 configuration. In Grain 5 the largest force drop was obtained by theθ_{= 70° t/L = 0.10 configuration.}

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(a) (b)

Figure 24: Results for the anisotropy measurements of Grain 4 (a) and Grain 5 (b). Theθ = 60° t/L = 0.12 configuration of Grain 4 was tested. The_{θ = 70° t/L = 0.10 configuration of Grain 5 was tested. Curves} that show a notable force drop are shown by the thick lines.

Figure 24 shows the results of the anisotropy measurements of Grains 4 and 5. For Grain 4, three orientations showed metastable behaviour between 0 and −0.25 strain. Only the 0°, −22.5° and −45° orientations exhibited a snapping deformation in the beam. The placement of the beam hinders the closing of the mouth, which causes out-of-hinge deformations. Because the mouths do not properly close, the beams do not properly snap within the prescribed range of strains. Only negatively angled Grain 4 configurations showed a snapping beam. In positively angled orientations, the out-of-hinge deformations mostly occurred in the top half of the meta-grain, where the body of the grains was altered to leave room for the snapping of the beam. Because there is less material here, the out-of-hinge deformations were more extreme. In the 22.5° orientation, this caused the beam to snap later, as can be seen in the Fd-curve which starts to descend but does not yet reach a local minimum. In the other negative orientations and in the 67.5° orientation, this prevented the beam from snapping at all.

For Grain 5, only the 0° orientation showed metastable behaviour. The other orientations all showed buckling deformations in the beams. Snapping deformations require a higher amount of force as can be seen by the high peak force in the Fd-curve of the 0° orientation. The extra degrees of freedom in Grain 5 allow the beams to buckle even when no out-of-hinge deforma-tions occur. As buckling deformadeforma-tions seem to be energetically more favourable than snapping, and they do not require any out-of-hinge deformations, buckling will occur more frequently than snapping. Only when the beam is loaded in a specific way, i.e. as in the 0° orientation, will snapping of both beams occur.

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Table 3.2: Results of the beam parameter variations in the Grain 4 and Grain 5 configurations. The green elements show the maximum value for the force drop and the peak force, and the minimum value for the strain difference. The configurations that did not show metastable behaviour have been left empty.

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(a) (b)

(c) (d)

(e)

Figure 25: Numerically obtained Fd-curves for Grain 4 and Grain 5. Beam angleθ is varied between the graphs. Thickness to length ratio t/L is varied between different curves in the same graph. (a), (c) and (e) show the results for Grain 4. (b) and (d) show the results for Grain 5. When a curve exhibits a force drop, metastability was achieved in this configuration. When a curve terminates before reaching −0.25 axial strain, the simulation of this configuration was aborted.

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(a) (b)

Figure 26: Snapshots of the printed bistable metagrains: (a) Grain 4 ’Ms Pacman’, (b) Grain 5 ’Butterfly’. The metagrains were printed with a diameter of 2cm and a thickness of 0.4cm.

3.3 Experimental Analysis

To see if our numerical results are robust enough to translate to an experimental setting, one configuration of each bistable metagrain was manufactured and experimentally compressed. The bistable metagrains were 3D-printed and compressed using the same methods, machines and material as the auxetic metagrains. Both grains were printed with a diameter of 2cm and a thickness of 0.4mm. Grain 4 was printed with θ_{= 70° and t/L = 0.14 (Figure 26a). Grain} 5 was printed with _θ_{= 70° and t/L = 0.17 (Figure 26b). This configuration of Grain 5 was} chosen before the numerical simulations were completed. To improve the printability of the metagrains, thin components were avoided. This is the reason a high t/L was chosen for Grain 5, even though simulations later showed thick beams in Grain 5 lead to buckling rather than snapping deformations in the beams.

(a) (b)

Figure 27: Fd-curves obtained from experiments. (a) shows the experimental results for Grain 4 and the numerically obtained Fd-curve from the corresponding simulation. (b) shows only the experimental results for Grain 5, since this particular configuration was not numerically tested.

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3.3.1 Experimental Results

Figure 28: Bistable metagrains in compres-sion. (a) shows 3D-printed and experimentally compressed Grain 4. (b) shows the simulated Grain 4. (c) shows 3D-printed and experimen-tally compressed Grain 5.

The Fd-curves obtained from the experimental tests of the bistable metagrains are shown in Figure 27. Figure 27a also shows the Fd-curve of the corresponding simulation. Figure 27b only shows the experimental data because the printed Grain 5 configuration was not numeri-cally tested. The experimental test of Grain 4 show good qualitative agreement with the sim-ulations. The Fd-curves show the same trend with a peak force at an axial strain of ap-proximately −0.1 and a minimum at an axial strain between approximately −0.17 and −0.19. There is a scaling difference of about a fac-tor 2. This is most likely due to a com-bination of the approximated Young’s modulus in the experimental curve and the 2D approx-imation in the simulations. In deformation, there was also a good agreement between ex-periments and simulations. This is shown in Figure 28, where the printed metagrain (28a) and the simulated metagrain (28b) are shown at approximately the same strain. The de-formation of the top of the mouth, seen in the simulations and needed to achieve a snap-ping deformation, was also seen in the experi-ments.

Grain 5 did not exhibit metastable behaviour. This can be seen in Figure 27b as the Fd-curve does not show a force drop. The deformation of this grain, shown in Figure 28, showed the same characteristics as the simulated configurations that did not show two snapping beams. The force needed to produce snapping deformations in both beams was too high, making the metagrain unstable and causing it to deform to the side, which in turn caused the left beam to buckle rather than snap. This is in agreement with the numerical finding that Grain 5 configurations with thick beams show this kind of behaviour.

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4 Conclusion

4.1 Summary

The fields of granular materials and mechanical metamaterials are combined to create meta-grains with interesting properties. Contrary to virtually every mechanical metamaterial, gran-ular mechanical metamaterials consist of a random distribution of building blocks instead of a systematic distribution. This class of metamaterials has not been studied so far and might prove to be very interesting in terms of energy absorption and dissipation. The energy dissi-pating behaviour naturally present in granular materials, could be changed by creating meta-grains capable of energy absorption and energy trapping. The combination of these two mate-rial fields might lead to a new class of matemate-rials, extremely adapted to shock absorption and energy dissipation.

In the first part of this research, auxetic metagrains were investigated. By designing the meta-grains to be auxetic, their kinematic behaviour is altered, causing the metameta-grains to shrink under compression and thus allowing them to temporarily store energy. To this end, an earlier proposed design for an auxetic metagrain, relying on an internal 4-bar linkage for auxetic-ity, was compared to two newly proposed designs, relying on an internal 6-bar linkage for auxeticity. The new metagrains were designed with the aim of maximizing volume reduction during compression. The metagrains were numerically simulated using Finite Element Analy-sis software. To validate the simulations, the metagrains were 3D-printed and experimentally compressed. The use of an internal 6-bar linkage allowed for certain design choices in the new metagrains. This caused these metagrains to compress auxetically over a longer range of strains, ultimately resulting in more volume reduction.

In the second part of this research, the energy landscape of the metagrains was changed. Slen-der beam elements were incorporated in two of the auxetic metagrain designs, with the aim of creating bistable metagrains. Through bistability, the metagrains would be able to trap the absorbed energy even after unloading of the compressive forces. Two beam parameters were varied to investigate their influence on the behaviour of the metagrains. The metagrains did not exhibit bistable behaviour, but a negative stiffness was achieved in both proposed bistable metagrain designs. The negative stiffness of the slender beam elements is counteracted by the positive stiffness of the body of the metagrains, resulting in metastable behaviour rather than bistable behaviour, i.e. a local force minimum above zero rather than below zero. These metagrains exhibited very anisotropic behaviour and metastable behaviour was only observed under specific compression directions.

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4.2 Outlook

Figure 29: Buckliball in different stages of compression. This mech-anism would be very suitable to im-plement in the field of granular me-chanical metamaterials

This research has laid out groundwork for experimen-tal testing of metagrains. Further optimization of the metagrain designs can be done to enhance the exotic properties of the metagrains. For the auxetic meta-grains, a next step in design could be the expan-sion to three dimenexpan-sional metagrains. Promising de-signs by the name of Buckliballs have already been proposed and printed [22]. Figure 29 shows such a Buckliball. The mechanism used here is very suit-able to employ in the field of granular mechanical meta-materials. The proposed bistable metagrain designs can be further optimized. The body of the meta-grains can be altered to create a more optimal en-vironment for the snapping deformations of the slen-der beam elements. By reducing the stiffness of the body of the metagrains, and leaving more room for deformation, the negative stiffness behaviour of the slender beam elements could become more promi-nent, potentially leading to bistability in the meta-grains. Repositioning of the slender beam elements could reduce the anisotropy of the bistable metagrain de-signs.

Another interesting direction for future research is the study of packings of metagrains. The current research has resulted in auxetic metagrain designs with comparable levels of isotropy but significantly different volume changing abilities. The effect of volume change by single grains on the properties of granular packings in the jamming-transition regime, can be studied and might produce interesting results with respect to the critical packing density and particle interactions. A snapshot of a preliminary test with a packing of auxetic metagrains, done by Antoine Dop, is shown in Figure 30a. Preliminary tests with packings of the bistable metagrain designs can be done. Due to the anisotropy of these designs, a random distribution will likely tamp out the negative stiffness behaviour of the metagrains. However, a careful placement in for example one dimensional packings could lead to interesting results. An example of such a one dimensional packing is shown in Figure 30b.

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(a) (b)

Figure 30: Packing experiments with metagrains. (a) shows a packing experiment done by Antoine Dop with a packing of auxetic metagrains. (b) shows a preliminary experiment with a 1D packing of different configurations of the Grain 4 bistable metagrain design.

5 Acknowledgements

I would like to thank my supervisor Corentin Coulais for proposing this project and for his guidance and helpful feedback during our weekly meetings throughout the project. I would also like to thank my daily supervisor Daan Haver for his input in the design process, his help in conducting both the numerical and experimental research and the many brainstorm sessions we had in the course of this project. I would like to thank Martin Brandenbourger and David Dykstra for explaining how to operate the machinery used in the experiments and more generally I would like to thank all members of the Machine Materials Laboratory for being so welcoming and making this such a good experience.

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Metagrains - Investigating Auxeticity and Bistability in Granular Mechanical Metamaterials

Metagrains

Investigating Auxeticity and Bistability in

Granular Mechanical Metamaterials

Abstract

Samenvatting

Contents

1

Introduction

2

Auxeticity - Altering the Kinematics of the Metagrains

3

Bistability - Shaping the Energy Landscape of the Metagrains

4

Conclusion

5

Acknowledgements

References