An auxetic structure expands in all directions when it is stretched in one direction; it shrinks in all directions when it is compressed in one direction. Tom’s models are auxetic only in very special circumstances, but this might be a result of his choice of materials. Still to be investigated: auxetic properties of highly prestressed tensegrity structures built of components that have very little give to them.
Context: Tom Flemons Archive
An auxetic structure expands in all directions when it is stretched in one direction; it shrinks in all directions when it is compressed in one direction. This is illustrated by Henry Segerman’s 2018 video of hinged 3D auxetic mechanisms, and by this video of a tensegrity icosahedron (aka expanded octahedron tensegrity).
Tensegrity spheres exhibit auxetic properties in some orientations and not in others
(April 28, 2016) Spherical tensegities built as diamond patterns have a certain origami quality to them – the tensional diamonds are dihedral and thus can be ‘unfolded’ in ways reminiscent of the valley and mountain folds of origami patterns. Pulling on the struts causes them to expand outward and pushing on them cause them to compress inward. But this only happens when circumferential struts are simultaneously manipulated.
(April 28, 2016) Spherical tensegrities generally act like inflated prisms – facets composed of struts rotate forces out along the struts which follow chordal paths to the sphere’s ‘surface’. This has the effect of sending forces to the circumference which expands even as the poles contract towards each other. An exact analogy is an inflated exercise ball. Sit on it and it becomes a flattened or oblate spheroid – the exact opposite of auxetic behavior. In the diamond form tensegrity cubo-octahedron 12 struts form 4 triangular circumferences. if all 3 struts of any triangle are compressed and pulled simultaneously then it too exhibits auxetic attributes. But if like the expanded octahedron tensegrity, it is compressed or pulled apart between two opposite facets then it again becomes an oblate spheroid and does not behave like an auxetic structure.
(April 28, 2016) Similarly the 30 strut icosadodecahedron is built from 6 pentagonal rings all intersecting each other. Each ring of 5 struts approximately describes an equator. However this structure is usually built using a circuit pattern whereby the diamond tension relationship is replaced by a single tension line. Because there are no origami-like folds to expand or contract it does not possess any auxetic characteristics. If it is built using the diamond approach, then simultaneous compression or expansion of any five circumferential struts will exhibit auxetic properties. By extension the act of inflating an exercise ball is equivalent to expanding simultaneously all circumferential sets of struts of any sized tensegrity sphere and thus auxetic behavior is observed. If the ball has some elastic quality then inflating and deflating it will resemble a pump that perhaps could be compared to pumps in the body. But the instant it is asymmetrically compressed (e.g. sitting on it) it no longer behaves auxetically.
(June 7, 2016) Auxetic structures require that the surface of the structure have wrinkles or folds that can expand or contract. Only tensegrities built using the diamond pattern as delineated by Pugh have the possibility of being auxetic and they will only behave auxetically if they are compressed or expanded in a specific way. For the most part they act like balloons which are clearly not auxetic.
Tensegrity prisms and tensegrity masts are not auxetic; this explains why the 6-strut tensegrity ball is not auxetic in certain orientations
(March 23, 2016) Tensegrity prisms and by extension tensegrity masts made of stacked prisms are not auxetic. If you pull on a mast from either end it gets narrower in its girth, not fatter.
(April 28, 2016) The expanded octahedron tensegrity exhibits auxetic properties if two parallel struts are pulled apart or pushed together. However if it is resting on one of its triangular facets and a force is applied to the opposite triangular facet on the top it does not act auxetically. In this orientation the expanded octahedron tensegrity can be understood to be a two layer threefold tensegrity prism – and all tensegrity prisms or masts derived from them are not auxetic. If you push down on them their girth widens. If you pull up and lengthen them they also get narrower. It is not the case as Steve mentioned that tensegrity masts auxetically increase their diameter as they are stretched – quite the opposite occurs. Please see attached images [the pairs of images below] that illustrate this.
Illustration of a non-auxetic response: Tom pushes down on a triangular face and the structure becomes wider. In contrast, pushing opposite struts towards each other produces an auxetic response: the structure becomes smaller in all dimensions.
More image pairs showing non-auxetic response as Tom pushes or pulls on his models.
To sum up: only some tensegrities under certain conditions exhibit auxetic behavior
(April 28, 2016) So to sum up – only some tensegrities under certain conditions exhibit auxetic behavior. All prisms, and masts are not auxetic and spheres are only conditionally auxetic provided that they are built as diamond models and that their circumferences are compressed or expanded equally simultaneously.
(May 8, 2016) To reitierate: tensegrities cannot be said to be auxetic structures except under very narrow circumstances. A tensegrity sphere (i.e. a multistrut regular geometrical structure that approaches sphericity) in the form of a skinned geodesic dome will oscillate (expand and contract) like a balloon and exhibit auxetic behavior because of the variable air pressure differential between the inside and outside. A tensegrity sphere without a skin does not behave auxetically without simultaneous and equal pull on all struts towards the centre. Any other configuration of forces e.g. gravity or a push or impact on one place will demonstrate non-auxetic behaviour. Tensegrity prisms and masts (which are linkages of prisms) are in no way auxetic. If you stretch or compress it the girth narrows or expands respectively.
Open questions
Youtube video: Tom Flemons in conversation with Steve Levin ©2005 Steve Levin They compare the auxetic response of a compliant model (at 5:00) to a stiff model (at 5:20).
Interesting research questions remain. Eleven years after the above video was filmed, Tom was still unsure:
(March 24, 2016) All biologic material is auxetic? Really? If I pull on my ear lobe I’m pretty sure it gets thinner as it gets longer….
(Nov 22, 2016) I can show that tensegrities built with some compliance in them are not necessarily auxetic but he [Steve Levin] points out that this is an artifact of my model technique i.e. their elasticity and not tensegrities in general. I still dispute this but admit that I have to look closer at taut models. The problem though is that living tissue is compliant and so should look and act a lot more like my models than taut ones. He would respond that stress/strain ratios in tensegrities and living tissue follow a ‘J’ curve (easy to change and then harder) while elastic materials follow an ‘S’ curve (hard to change at first, then easy, then hard again – i.e. blowing up a balloon). This is impossible to prove without a lot of research which hasn’t been done. I’m not sure I agree with this characterization of elastic cord anyway. It’s easy to stretch and then gets harder – a classic ‘J’ curve. We can never nail down this difference because living tissue is opaque to observation at this scale.