www.sciencemag.org/content/360/6389/632/suppl/DC1

Supplementary Materials for

Handedness in shearing auxetics creates rigid and compliant structures

Jeffrey Ian Lipton*, Robert MacCurdy, Zachary Manchester, Lillian Chin, Daniel Cellucci, Daniela Rus*

*Corresponding author. Email: jlipton@mit.edu (J.I.L.); rus@csail.mit.edu (D.R.)

Published 11 May 2018, Science 360, 632 (2018)

DOI: 10.1126/science.aar4586

This PDF file includes:

Materials and Methods Figs. S1 to S10 Tables S1 and S2 References Captions for Movies S1 to S3

Other Supplementary Materials for this manuscript include the following: (available at www.sciencemag.org/content/360/6389/632/suppl/DC1)

Movies S1 to S3

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Materials and Methods

To create the composite cylinders (Fig. 2), we cut four 10-inch strips and four 20-inch strips out of 1/16th in. thick spring steel with an Omax waterjet. Each strip was 4 mm wide, with through-holes to fit M2 screws and lock nuts to connect two strips in place. Composite structures were created by sliding one complete cylinder inside another and either using a ziptie to connect the two cylinders or putting both within the same aluminum cap.

To create the linear actuator (Fig. 3), a 223 pattern was laser cut using a rotary engraving attachment on a Universal 120W laser cutter on 1 in. diameter, 1/16th in. thick Teflon rods. These rods were then bolted into 3d printed caps made from a Stratasys Fortus printer and driven with multi-turn HS-785 HB servos using an off-the-shelf servo controller.

Mechanical testing of the composite structures was completed via an Instron 3344 machine as seen in Movie S3. The structure was placed within two water-jet-fabricated aluminum caps by sliding the individual struts through the cap and then bolting them in place. Each aluminum cap was then attached to 3d printed adapters that allowed the Instron to clamp the piece securely. One adapter was fully static while the other incorporated two press-fit bearings to allow rotational movement. Each sample was compressed by 50 mm, with load (N) and compression (mm) being measured. For each category of sample (L, R, LR, LLL, LRL), three samples were measured, with each sample being measured three times.

To calculate the effective stress and effective strain, we used the initial diameter (101 mm) and initial length (254 mm) of the spring steel composite, treating these as the diameter and length of the material we were simulating. To get a sense of effective stiffness, we selected the most linear section of the stress-strain curve to calculate the tangent modulus through least-squares regression. A summary of the effective stiffness of all materials is presented in Table S2.

During mechanical testing, we noticed that upon compression, several different modes of behavior were apparent depending on how much torsional preload was applied to the structure – namely, buckling, bulging and twisting compression (Fig.S9 and movie S3). Composites that had high rotational stiffness (LR, LRL) exhibited a combination of bulging and buckling modes, while composites with low rotational stiffness (L, R, LLL) could experience the twisting mode, see Fig. S10. In this paper, data was collected with minimal preload and did not account for which modes were present for the average.

Supplementary Text

Conditions for Shearing on Auxetics

Tiling the plane with a pattern causes the points within this pattern to repeat with a translational offset. At each repeated point, we place two vectors and , pointing towards another repeated point, as seen in Figure S1. The lengths of the vectors are proportional to the distances between the points, and defines the translations of the tiling. These define the matrix , where the determinate of is the signed area of the unit cell. It cannot

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change sign since that would require det 0, which only occurs when is parallel to , implying that the tiling is one dimensional. The unit cells will transform according to a

single internal parameter of the auxetic tiling (27). As the unit cell transforms, the vectors

will have associated derivatives such that ⋅ where defines the instantaneous transformation of the tiling, and is the system matrix. can be used to map the vectors of the unit cells for a given internal angle to another such that

⋅ since lim→∏ ∗ ⋅ . To insure that the

periodicity of the tiling is not broken, all auxetic tiling patterns must undergo transformations of the unit cell such that transformations of are affine. Two types of affine transformations can occur: continuous and discrete.

The continuous transformations are defined by the auxetic trajectories that the unit cells undergo as varies (27). Continuous transformations include shearing and scaling. As the tiling undergoes continuous transformation, the affine transforms of scaling and shearing must be driven by the same internal parameter . is of the form

0 , where is a continuous function, shears in one direction, and scales

in both simultaneously. Since is an upper triangular matrix, is upper triangular and is upper triangular. Since and must be the same sign to have the unit cell expand

or contract, the system is auxetic whiledet 0. When G 0 the system is expanding, when 0 the system is contracting. Therefore defines a trajectory in the 1,1 Lie group and defines the derivative of the trajectory in the tangent space. is an auxetic trajectory whiledet 0 and is a shear auxetic while is upper triangular.

The discrete transformations of the unit cell that are discrete affine transformations of represent discrete symmetries of the pattern, such as the cyclic groups and reflections. The rotation operators of a cyclic group are ,cos , sin ,

sin , cos ,, where , . We can define the reflection operator

about a line of angle as cos 2 sin 2sin 2 cos 2 . The continuous and discrete

transformations must form a group for the symmetries to be compatible with tiling the plane and being part of an auxetic trajectory, so , and must be compatible with shearing.

The family of continuous shear transformations, S, in 2D is represented as the

operators 10 1

in matrix form, where ∈ . S forms a group since ⋅

. There are other operators that perform shearing: , 10 1

,

and 10 1

. These operators along with form a group we call

SABE, whose multiplication table is Table S1. In SABE, and form a subgroup, and form a subgroup, and and form a subgroup.

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For the operators of to form a group with the shear operators, , ⋅ must equal either , , , , , or . This only occurs if , is 0 or ; so, shear transforms are only compatible with or cyclic groups. In fact , and , , . Therefore any group formed with and has symmetry, and those with without has only symmetry.

For a flip operator to form a group with S, ⋅ must equal

eitherF , , , , or . This only occurs at 0 or with the operators ↑

1 00 1

and →1 00 1

that represent mirrors about the X and Y-axis. In fact ↑

and → . and either ↑ or → can form the group, and both F operators form the dihedral group .

Therefore, for a given basis, we have 5 sets of transformation operators that

combine shearing with discrete transforms: , , 2,1 , , ↑ , , → , and , , , ↑, → . These sets each have a group structures and are subgroups of the SABE group.

The group with elements we call the group S. The group with operators , , we call SA. The groups , ↑ and , → both represent with a shear, therefore they are the same group we call SB/E. We therefore have 4 shear compatible symmetry groups, S with no discrete symmetries, SA with symmetry, SB/E with symmetry and SABE with symmetry.  

The SABE and SB/E groups raise the question: how are reflection symmetries compatible with net shearing? While a tiling of rhombi, i.e. parallelograms with symmetry, can each individually shear, the net tiling has aligned lines of reflection. By placing the repeated points at the intersection of lines of reflection, there is clearly no net shear, simply scaling. In fact, any reflections or glide reflection symmetries in the tiling prevent the system from having a net shear since every right-handed shear is paired with a left-handed shear. Therefore, or symmetry cannot be present in the tiling throughout the auxetic trajectory and result in a shear auxetic. However, and symmetries can exist for a single of the auxetic trajectory.

If one can apply a reflection operator on any arbitrary point along an auxetic trajectory and reach another point on the same auxetic trajectory, than either that point ( ) must map onto itself or another point ( ) along the trajectory. If it maps onto itself then the unit cell is symmetric under reflections at that point. If it maps onto another point thendet det . Because the reflection operators are their own inverse, must map to under reflection, creating a bijection. Therefore one-half of the range of must map onto the other half, and there must be one point that maps to itself or each point on the trajectory maps to itself. If half the trajectory maps onto the other, then the

function det is symmetric around . The point therefore must be where 0.

This means that a shearing auxetic cell cannot shear and expand through a point where it develops a reflection symmetry. SB/E, and SABE therefore represents auxetic trajectories that develop momentary lines of reflection.

We can use this to determine the wallpaper groups that are compatible with shearing. S, with no discrete symmetries, is part of the o pattern in orbifold notation. SA,

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with only symmetries represents the 2222 pattern. SB/E, with a single reflection direction, represent o patterns that become ** or *x. SABE has a 2222 pattern which at a single is *2222, 2*22, or 22*. S and SA are inherently handed since they are not symmetric under reflection. SB/E, and SABE are unhanded since they are symmetric under reflection. Examples of each pattern are shown in Fig. S2.

Auxetic Trajectory conditions

Analyzing the auxetic trajectory can provide insights into the symmetries of the system and the system’s maximum extent. When det 0, three things can happen. First, there can be a momentary zero crossing, 0 and return to their previous sign as varies, and the system remains auxetic and expanding/contracting. The second option is

0 and both switch sign (positive to negative or negative to positive). In this case the system switches between shearing and expanding to shearing and contracting. Lastly, the third option is 0 and and take on different signs from each other, and the system stops being auxetic at that point. This is connected to det , the area of the unit cell, by its derivative.

detdet ∗ det ∗ det ∗

For a trajectory in 1,1 , 0 only occurs if 0. So if

det 0 then =0. To determine if the unit cell will remain auxetic and continue

expanding or contract at a point we must look at the curvature (second derivative) of det at that point:

detdet ∗

det ∗

If

0 , 0 at that point, then the curvature of det changes at that

point, and the system remains auxetic and continues expanding/contracting as before.

Should

0 at that point, then the system must switch between expanding and

contracting.

These conditions are not specific to handed shearing auxetics. The double arrowhead model (28), the herringbone model (29), and the inverted honeycomb (30) all have a point along their auxetic trajectory where they are at maximal auxetic extension and cease to be auxetic. The oblique box structure (29) and those of the achiral expanding polyhedra (31) have points where they reach maximal auxetic extension and then switch between expansions and contraction. . Producing Shearing and Handedness

Given the symmetries of shearing auxetics, it is possible to turn conventional auxetic patterns into shearing auxetics. In Fig. 1A, we see how conventional auxetics and

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both handed and unhanded shearing auxetics transition into each other by breaking or adding symmetries. In Fig. S3, we see how conventional auxetic patterns have handed shearing auxetic counterparts. We modify the double arrowhead models (28), chirals (32), kagome lattice (7), and rotating squares (8). The moving elements, highlighted in orange, either are arranged into a 2222 symmetry, or have their internal symmetries broken to produce 2222 or o symmetric tilings.

To make a handed shearing auxetic pattern from an unhanded shearing auxetic pattern, the reflection symmetry of must be broken. This is done in one of two ways. One way is to change the symmetries of the constituent elements to eliminate emergent lines of reflection. In Fig S3A, we see that replacing p4 symmetries of one of two different rectangles with d4 symmetries turns an SABE pattern into an SA pattern. In Fig S2, SB is turned into S by eliminating the reflection symmetry of one of its triangles. The other way is to restrict the range of to exclude , or to be asymmetric around . In Fig. S4, we see that this is how handedness emerges in chiral auxetics (32). Right and left-handed chiral auxetics are part of the same auxetic trajectory. Tetrachirals (33) are symmetric around a point with *332 symmetry. When fabricated, the links of the chiral auxetics (32) are designed to prevent them from reaching the *332 point, and therefore are handed. It should be noted that simply breaking the diagonal symmetries but keeping the vertical symmetries produces anti-tetrachiral (33) auxetics.

Handedness on a cylinder is generated by either the alignment of the tiling on the surface, or the nature of the pattern being tiled. A handed planar shearing auxetic tiling will produce a handed cylinder since the auxetic trajectory is asymmetric. An unhanded tiling such as SABE, SB, or SE, can generate a handed auxetic by aligning at an angle to the circumferential directions at . In Fig. S5A, we see a tiling of a cylinder using a planar SABE tiling. The cylinder is tiled with the same number of rectangular unit cells of height H and width W along the circumferential and axial directions. As varies, H and W change to maintain the tiling. The unit cell of the unhanded tilings is symmetric in around

along the direction. In fact, the direction is a constant of the shearing tiling since is always in line with . If at , is at an angle to the circumferential direction, then it will spiral around the cylinder with a right or left handed direction. If one were to reflect the pattern along the unhanded shearing auxetic’s lines of reflection at that point, would switch chirality around the cylinder. The pattern cannot shear to become the other chirality since no other along the auxetic trajectory yields the same area as , so if could reach the reflected angle, the pattern would not be the same. Therefore, handed cylinders can be made from planar unhanded shearing auxetic patterns.

We can define a cylinder from a planar shearing auxetic pattern with two vectors. In Fig. S5A, we identify a series of nodes such that each node is an integer number of and

away from an origin, so each node is at . Two vectors and that

each point from the origin to another node so that and

define the node points A and D. For the case in Figure S5, 0.

The point B is defined by . The line is parallel to and equal length, and is parallel and equal in length to . Because O,A,D and B are defined by the

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nodes along and as changes, the lines will keep these relations. So we can identify and with each other to make the cylinder. The bottom of the cylinder is then along and the top is along . Therefore the circumference of the cylinder is

| | | | 2 | || |cos the N of the symmetry group is therefore

min , so . The area of the cylinder is so the height is .

min , , so

.

It should be noted that a shearing auxetic cylinder cannot be approximated by a prism or made using rigid links. The sides of the prism are flat and have no curvature in any direction. The edges have infinite curvature along the circumferential direction and zero curvature along the axial direction. An auxetic pattern could have a point on an edge provided it moves only along the edge. Shearing auxetics would require a point on the edge to move across the edge, putting a discontinuity into a rigid link. Therefore for a point on an edge to move circumferentially, the edge must twist. When the face with a shearing auxetic pattern expands, it will deform from a rectangle into a parallelogram, moving the points on the edge of the face circumferentially, or shifting the edge vertically. If the edge moves circumferentially, they twist to form a continuous helix. However if the parallelogram is between two helixes, then the faces are no longer flat, and you no longer have a prism. If the edges move vertically and the faces shear in the same direction, the pattern can only loop around the structure if it is constructed from separate helices made from discrete bends, since each successive edge of the faces must compound the vertical movement of the edge. Therefore, you cannot make a shearing auxetic prism with net handedness. If a net handedness is not desired, you can mirror the faces at the edge, to produce a structure that has no net handedness, but is made from shearing auxetic patterns on the faces. This requires an even number of faces since each right-handed shear, must be counter balanced by a left-handed shear.

The method of manufacture of a cylinder can bias it along the auxetic trajectory. In Fig. S6. We see two HSA structures at their stable configurations at the opposite ends of the auxetic trajectory. The inner cylinder was bent to a specific radius and biases inwards. The outer cylinder was made from flat unbent strips. The pattern cut into a cylinder using laser cutting can also bias the cell to a point along the auxetic trajectory. Previous work on deformable non-auxetic lattices on cylinders generated bistable structures similar to the t4 bacteriophage by using carbon fiber strips formed to a cylindrical mold (34,35).

Extending to spheres

Imagine a series of identical parallelograms that wrap around a sphere, pictured in Fig S7A. The sides form two circles connected with links. As they shear, the circles would rotate relative to each other. The great circle in the middle of the parallelograms defines an equator on the sphere. Poles are formed by tracing along great circles from the corners of the parallelograms: the point where they all intersect is a pole. The number of parallelograms along the great circle (N) would form a rotational symmetry at the poles. Therefore, you can have arbitrary rotational symmetry at the poles. Based on the conditions from shear, shearing is only compatible with or symmetry. Therefore, shearing auxetics on a sphere can only exist for patterns with 22N or NN symmetry.

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Previous auxetic sphere patterns were made from cast silicone in a pattern that was 432 symmetric with a where the symmetries were *432 (12). Similar to the chiral auxetics, limiting the range of to exclude would produce a non-shearing handed auxetic from the structure. In this design the twisting is local and each rotating section is surrounded by counter rotating structures. Antipodal structures on the sphere rotate relative to each other, making a local twist, but there is no net rotation of the structure.

In Figure S7D and movie S1, we see a 22N symmetry shearing auxetic tiling that is unhanded. The system starts to the left, twists until it becomes a *22N then contracts and becomes rightward. The repeated unit of the pattern is in Fig S7A. Using the spherical trigonometry we derive the angle A of the unit as a function of r, the normalized radius

.

2 cos1

√21 cot

2cos

2 1

Since is a 1-1 function, we can use A to define r. We can then use this to describe the change in radius as a function of the change in angle, with being the point where .

We can use this unit linkage to define a NN symmetric sphere, seen in Fig. S7C. of those linkages of different lengths F and E are separated by a fixed angle at one pole, and a variable angle at the other pole. The Δ of a link can be calculated using spherical trigonometry such that

sin sin2

1 sin2

2

Δ 4 ∗ tan tan14

1sin 1

2 2

sin 12 2

3

Because all links repeat in the direction after , Δ Δ . If

then the sphere is at maximum extent at Δ 0. If we define 0 at then,

Δ | . Since for most , the system is NN symmetric. If , then the

system becomes 22N symmetric. As seen in Fig. S7E, it is a handed shearing auxetic.

Unlike a cylinder, all forms of handed spherical auxetics can be used to make a locking structure. Handed structures with local or global twists can be composited to make LRL or RLR structures. On a cylinder, handed chiral patterns would not generate a twist on the ends of the structure, so attaching the ends together would not cause the system to lock. Unhanded hemi-spheres can connect to handed cylinders to produce handed capsules as seen in Fig S8.

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Fig. S1. Illustration of auxetic lattice

Translational cells for isotropic and shearing auxetics. A unit cell (highlighted blue) is the minimal structure that repeats in the tiling of the plane. The cells transform as a function of the internal parameter . Translational cells are marked with black lines and dots at the intersections. The vectors and at each point denote the translations of the cell. Isotropic auxetics (A) have derivatives of the translation in line with the translation.

Shear auxetics (B) have in line with but is not in line with .

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Fig. S2 Illustration of shearing auxetic groups.

The 4 symmetry groups that generate shearing auxetics. The translational cells are marked in orange and yellow, the symmetry unit cells are marked in blue, and lines of reflection are green. The S and SA groups are handed auxetics, shearing and expanding, and then become non-auxetic at maximum extent. The SB/E, and SABE groups develop reflections at maximum extent and are unhanded. S, SB/E, are symmetric and SA and SABE are symmetric

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Fig. S3 Auxetic and handed shearing auxetic counterparts

Auxetic structures become Shear Auxetic through symmetry breaking. Breaking symmetries can make shearing auxetics from links (A), chiral (B), triangles (C), and quadrilaterals (D) auxetics.

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Fig. S4 Chiral 2D auxetics

Right and left chiral auxetics. The handedness of the chiral auxetic emerges from limiting the auxetic trajectory to exclude .

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Fig. S5 Patterning shearing auxetics to produce handedness

Mapping shearing auxetics to make handed cylinders. (A) A section of the plane tiled with a 2222-SABE shear auxetic pattern (black) with internal parameter . It is mapped onto a cylinder as a parallelogram (red) or as a rectangular section (blue highlight). The alignment of the axial direction (green) and circumferential direction (orange) relative to the direction of at (cyan) is determined by the angle . (B) Along the compacted direction, you must have a frieze pattern, whose unit cell is highlighted in yellow. In this instance, it is 224 pattern.

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Fig. S6 Biasing to points on auxetic trajectory

Cylinder Biasing. Cylinders can be biased by the manufacturing method. The inner cylinder was bent to a radius and the outer cylinder was made from straight strips.

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Fig. S7 Spherical shearing auxetics

(A) Net shearing on a sphere produced a global twist. (B) The linkage can be tiled to produce an unhanded shearing auxetic. (C) Two linkages of different lengths can produce a handed shearing spherical auxetic. (D) Unhanded 22N shearing auxetics twist to develop reflections, and then continue twisting to contract. (E) Handed shearing auxetics have a maximal extent with no lines of reflection.

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Fig. S8 Handed shearing auxetic capsule

(A) A collapsed 224 capsule made from a 224 cylinder and an unhanded sphere. (B) The extended capsule.

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Fig. S9 Cylinder buckling modes.

(A) Uncompressed left-handed shear auxetic with the three possible modes of compression behavior shown – (B) buckling, (C) bulging, and (D) twisting compression. Which mode is exhibited is significantly affected by the torsional preload present in the system.

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Fig. S10 Comparison of composite cylinder behaviors.

Composites of two or more individual cylinders exhibit a combination of compression modes. In (A), a left-handed and right-handed composite, the internal cylinder buckles while the external cylinder bulges. In (B), the triple left-handed composite, all three cylinders buckle similarly while in (C), the left-right-left handed components act antagonistically, causing distinct buckling modes through the material.

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Table S1. The Cayley table for the S A B and E shearing operators.

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Sample Effective Stiffness (MPA) Ultimate Compressive Strength (N)

Left 0.19 ± 0.01 8.6 ± 0.97

Right 0.20 ± 0.01 9.4 ± 1.3

Left-Right 0.43 ± 0.01 24.6 ± 2.8

Left-Left-Left 1.10 ± 0.05 42 ± 4.3

Left-Right-Left 1.16 ± 0.34 62 ± 8.0

Table S2. The stiffness and compressive strength of various composites made from HSA

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Movie S1

Various handed and unhanded cylinders and composite cylinders and spheres

Movie S2

The linear and four degree-of-freedom actuators

Movie S3

Compression test of various cylinders

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