Foldable  Auxe,cs  Materials    

Burhan  Saifaddin  Advisor:  Prof.  L.  Mahadevan  

Applied  Math  Lab.    

Folding  in  Nature:  Extreme  Mechanics    

What  Drives  Blooming,  or  Leaves  shape  ?  Liang  and  Mahadevan.  Growth,  geometry,  and    mechanics  of  a  blooming  lily.  ONAS  2011  

Biological  Tissues  :  Gut,  intes,ne,  brain  folds  Savin  et  al.  On  the  growth  and  form  of  the  gut.  PNAS  2011  

Incorporate  foldable  structure  with  electronics  Adv.  Funct.  Mater.  2010,  20,  28–35  

Geological  fold  in  Poland.    

Protein  Folding  from  random  coil    (wikipedia)    

Manufacturing  :  Microrobo,cs  Lab  Whiteside  Lab  

Folding  in  Manufacturing  and  Engineering  

Solar  Cells   Inexpensive  Deployment  of  Solar  Cells  More  than  2/3  of  the  cost  of  PV  systems  is  in    deployment  cost.    

Folding  in  Engineering  :    Foldable  Robo,cs  Foldable  Solar  Cells  Foldable  BaZeries    Paper  Based  Electronics  …..    

Myers,  et  al.  Appl.  Phys.  LeZ.  96,  071902,  2010  

Foldable  3DPV  

=4.1% at normal incidence and efficiency !=6%.19 Testswith a larger number of triangles !up to 1000" in the bound-ing volume did not show significant variation in the optimal3D shape or energy produced. Energy values for 3D struc-tures are lower bounds since we only implemented a singlereflection per ray !to limit computation time" and did notaccount for ground reflections.

We optimized structures with a bounding-volume of areafootprint !base area" 10"10 m2 and height ranging from 2to 10 m. Figure 1 shows the energy generated in a day as afunction of the height of the GA-optimized 3DPV solar cell,compared to that of a flat panel of the same area footprint.The energy generated by the 3D structures scales linearlywith height, thus leading to “volumetric” energy conversion.In addition, the power generated as a function of time duringthe day !inset, Fig. 1" shows a much more even distributionfor 3DPV, due to the availability of cells with different ori-entations within the structure. The increase in power withheight is dominant in the early morning and late afternoon,as expected, although the enhancement is broad in time andremains significant at all times during the day, even at mid-day. This even supply of power throughout the day can be“built-in” to a 3D structure, in contrast to power generatedby a flat panel, which, without dual-axis tracking, decaysrapidly around peak-time.

Interestingly, all the GA structures show similar patternsin their shapes, even for different heights. They contain noholes running across the bounding volume, which is neces-sary to intercept most of the incoming sunlight, and !lessintuitively" they all have triangles coinciding with the 12edges of the bounding box volume, so that they would castthe same shadow on the ground as the open-box. We empha-size that these patterns emerge from randomly generatedstructures, are not artifacts of the simulations, and are a fin-gerprint of emergent behavior resulting from the GAcalculations.20 The primary shape of the GA structure #Fig.2!a"$ is a box with its five visible faces caved in toward themidpoint. A simplified, symmetric version of this was con-structed, as shown in Fig. 2!b"; this idealized structure,which we refer to as the “funnel,” generates only 0.03% lessenergy in the day than the original GA output, and therefore

contains most key ingredients of the complicated GA struc-tures.

We compared the energy generated by simple open-boxshapes and the funnel structures through a figure of merit M,defined as the ratio of the energy produced in a day to thetotal area of active material used, and scaled to one for theflat panel case. As can be seen in Table I, the energy of thefunnel shape outperforms the open-box at all heights, andwhile both structures generate more energy than the flatpanel case, they use excess material for a given energy !i.e.,M#1". For example, for a height of 10 m the open-boxshape generates approximately 2.38" as much energy as theflat panel but requires 9" as much active material !M=0.26". The figure of merit for the open box decreases withheight indicating that such a shape is not ideal for 3DPV interms of efficient materials’ use. On the other hand, the GA-derived funnel shapes maintain a nearly constant figure ofmerit over this height range, with a cross-over to superiormaterials performance compared to the open-box at a heightof %5 m, and 30% higher M at 10 m.

Despite the relatively small increase in energy genera-tion of the GA shapes compared to the open box, these struc-tures shed light on some fascinating aspects of 3DPV andmay give significant practical advantages. The increase inproduced energy of the best-performing GA structures is dueto a decrease in the total power reflected to the environmentand an increase in power generated using light reflected fromother cells. This is easily seen by first disallowing the ab-sorption of reflected rays, which results in a loss of roughlyhalf of the increase in energy production. The remaining dif-ference is eliminated if reflections are completely disabled!case R=0", in which case the open-box and GA structuresgenerate the same energy to within 0.005% agreement.

We also investigated how significant changes in reflec-tance might alter the optimal results. A 3D architecture couldin principle be optimized to capture light using multiple re-flections while preventing shading of the active material,possibly limiting the need for expensive antireflective

FIG. 1. !Color online" Plot of the energy produced in a day by GA-optimized 3DPV structures compared to that of a flat panel in the sameconditions. The inset shows the power generated during the day for the flatpanel compared to the 3DPV at height=10 m.

FIG. 2. !Color online" Schematics of 3DPV structures: !a" GA-optimizedstructure shown with all 64 triangles inside the bounding box; !b" funnel, asimplified version of most GA-optimized structures that retains their supe-rior performance over other shapes.

TABLE I. Energy produced in a day !Ebox and Efunnel" relative to the flatpanel !E0" for the 3D open box and funnel structures, and correspondingfigures of merit !Mbox and Mfunnel" for an area footprint of 10"10 m2.

Height EBox /E0 EFunnel /E0 MBox MFunnel

2 1.29 1.29 0.49 0.364 1.56 1.58 0.37 0.366 1.83 1.87 0.32 0.368 2.11 2.15 0.29 0.35

10 2.38 2.43 0.26 0.34

071902-2 Myers, Bernardi, and Grossman Appl. Phys. Lett. 96, 071902 !2010"

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp

=4.1% at normal incidence and efficiency !=6%.19 Testswith a larger number of triangles !up to 1000" in the bound-ing volume did not show significant variation in the optimal3D shape or energy produced. Energy values for 3D struc-tures are lower bounds since we only implemented a singlereflection per ray !to limit computation time" and did notaccount for ground reflections.

We optimized structures with a bounding-volume of areafootprint !base area" 10"10 m2 and height ranging from 2to 10 m. Figure 1 shows the energy generated in a day as afunction of the height of the GA-optimized 3DPV solar cell,compared to that of a flat panel of the same area footprint.The energy generated by the 3D structures scales linearlywith height, thus leading to “volumetric” energy conversion.In addition, the power generated as a function of time duringthe day !inset, Fig. 1" shows a much more even distributionfor 3DPV, due to the availability of cells with different ori-entations within the structure. The increase in power withheight is dominant in the early morning and late afternoon,as expected, although the enhancement is broad in time andremains significant at all times during the day, even at mid-day. This even supply of power throughout the day can be“built-in” to a 3D structure, in contrast to power generatedby a flat panel, which, without dual-axis tracking, decaysrapidly around peak-time.

Interestingly, all the GA structures show similar patternsin their shapes, even for different heights. They contain noholes running across the bounding volume, which is neces-sary to intercept most of the incoming sunlight, and !lessintuitively" they all have triangles coinciding with the 12edges of the bounding box volume, so that they would castthe same shadow on the ground as the open-box. We empha-size that these patterns emerge from randomly generatedstructures, are not artifacts of the simulations, and are a fin-gerprint of emergent behavior resulting from the GAcalculations.20 The primary shape of the GA structure #Fig.2!a"$ is a box with its five visible faces caved in toward themidpoint. A simplified, symmetric version of this was con-structed, as shown in Fig. 2!b"; this idealized structure,which we refer to as the “funnel,” generates only 0.03% lessenergy in the day than the original GA output, and therefore

contains most key ingredients of the complicated GA struc-tures.

We compared the energy generated by simple open-boxshapes and the funnel structures through a figure of merit M,defined as the ratio of the energy produced in a day to thetotal area of active material used, and scaled to one for theflat panel case. As can be seen in Table I, the energy of thefunnel shape outperforms the open-box at all heights, andwhile both structures generate more energy than the flatpanel case, they use excess material for a given energy !i.e.,M#1". For example, for a height of 10 m the open-boxshape generates approximately 2.38" as much energy as theflat panel but requires 9" as much active material !M=0.26". The figure of merit for the open box decreases withheight indicating that such a shape is not ideal for 3DPV interms of efficient materials’ use. On the other hand, the GA-derived funnel shapes maintain a nearly constant figure ofmerit over this height range, with a cross-over to superiormaterials performance compared to the open-box at a heightof %5 m, and 30% higher M at 10 m.

Despite the relatively small increase in energy genera-tion of the GA shapes compared to the open box, these struc-tures shed light on some fascinating aspects of 3DPV andmay give significant practical advantages. The increase inproduced energy of the best-performing GA structures is dueto a decrease in the total power reflected to the environmentand an increase in power generated using light reflected fromother cells. This is easily seen by first disallowing the ab-sorption of reflected rays, which results in a loss of roughlyhalf of the increase in energy production. The remaining dif-ference is eliminated if reflections are completely disabled!case R=0", in which case the open-box and GA structuresgenerate the same energy to within 0.005% agreement.

We also investigated how significant changes in reflec-tance might alter the optimal results. A 3D architecture couldin principle be optimized to capture light using multiple re-flections while preventing shading of the active material,possibly limiting the need for expensive antireflective

FIG. 1. !Color online" Plot of the energy produced in a day by GA-optimized 3DPV structures compared to that of a flat panel in the sameconditions. The inset shows the power generated during the day for the flatpanel compared to the 3DPV at height=10 m.

FIG. 2. !Color online" Schematics of 3DPV structures: !a" GA-optimizedstructure shown with all 64 triangles inside the bounding box; !b" funnel, asimplified version of most GA-optimized structures that retains their supe-rior performance over other shapes.

TABLE I. Energy produced in a day !Ebox and Efunnel" relative to the flatpanel !E0" for the 3D open box and funnel structures, and correspondingfigures of merit !Mbox and Mfunnel" for an area footprint of 10"10 m2.

Height EBox /E0 EFunnel /E0 MBox MFunnel

2 1.29 1.29 0.49 0.364 1.56 1.58 0.37 0.366 1.83 1.87 0.32 0.368 2.11 2.15 0.29 0.35

10 2.38 2.43 0.26 0.34

071902-2 Myers, Bernardi, and Grossman Appl. Phys. Lett. 96, 071902 !2010"

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp

PNAS    March  15,  2005    vol.  102    no.  11  

objects. Thus, programming the object reduces to placing thestickers on top of the sheet. Sticker Programming has three com-ponents: Programmable Matter with Sticker Operating System,Sticker Compiler, and Programmed Sticker device. Algorithmswith the same flavor as the ones described for designing theplacement of the actuators and the control for creating an objectare used to automatically design the base sheet and the stickersthat go with it.

ResultsUsing the two-step planning and fabrication process describedabove, we have designed, fabricated, and tested a thirty-two-tile

sheet, with twenty actuators, five actuator groupings (Groups 1–5),three phases (Phase A–C), and two goal shapes (“boat” and“airplane”). This design is optimized to use the minimum numberof actuators, 14.4% fewer than nonoptimized arrangements.Upon activating Groups 1, 2, and 3 the nine actuators of PhaseA are heated, and the “boat” folds (Fig. 4 A–D). Once unfoldedby hand, Groups 2 and 4 are activated, and Phase B (the trian-gular flaps) of a “paper airplane” folds (Fig. 5 A–C). Finally,Groups 3 and 5 receive current, and Phase C, the second phaseof the “airplane” (backbone and wings), folds (Fig. 5 D–F) (SeealsoMovie S1). One amp is run through the actuators, each with aresistance of 0.25!, resulting in 1–2.25 Wof power depending onthe phase. Each phase takes between 6 and 10 s to fold, requiringbetween 6 and 22.5 Joules of energy for activation. Speed can beincreased, but at the expense of a higher power input.

Discussion and ConclusionWith the concept of self-folding origami, we hope to bring pro-grammable matter into reality in the near future. We havepresented an end-to-end approach to creating self-folding sheetscapable of realizing multiple shapes. In theory, our algorithmscan plan the creation of any three-dimensional shape. In practice,we are limited by actuator size and torque considerations, as wellas control and coordination constraints for multiple actuators.

At the moment, the self-folding devices we have created arelimited to a finite set of shapes that can be created with one-way actuators. This limitation is primarily due to the physicalrealization of the sheet (one-way actuators, fixed wiring scheme,etc.), which is specified by the algorithm given the desired shapes.This limitation could be alleviated in one of two ways: (i) bidirec-tional actuation at each joint, each individually addressable by thecontroller or (ii) calculation of a large number of mutually com-patible basis shapes from which goal shapes could be composed.The former would involve only the first two steps of the algorithm(the third and fourth are required to place multiple goal shapes ina predetermined manner on a single sheet). The latter is a naturalextension of our current methods but with larger sheets with moreactuation. The result would be a single sheet that can be foldedinto any shape up to a prescribed resolution.

One important consideration for the practicality of thismethod is the scalability of the self-folding sheet, both in termsof resolution of the individual elements (tiles) and the number ofelements that can be effectively combined in a single sheet. First,the resolution of the tiles is limited by the scalability of eachelement. The current materials and methods use to create thesubstrate can be reasonably shrunk to create tiles on the orderof a few millimeters in largest dimension. Below this threshold,lithography and micromolding techniques (for example, moldingelastomers using capillary action (25)) could be used to reducefeature sizes further. Scaling of actuation has two considerations:

Fig. 3. SMA actuator unfolded after annealing (top). The actuator is“stapled” to the sheet in this position, with each of the three legs fittedthrough a hole in the sheet. Heat is applied at each red arrow to fold thethree distal joints (middle). This causes the “heat-activated staple” to clampthe bottom side of the sheet. When the entire actuator is Joule heated, thecentral joint bends, folding the sheet 180° (bottom).

Fig. 4. Simulation (left) and experiments (right, with time shown in lower right—mm:ss.s) of a self-folding “boat.”(A). All actuators receiving current(B). Immediately before magnetic closures engage (C). Finished boat on side (D).

12444 ! www.pnas.org/cgi/doi/10.1073/pnas.0914069107 Hawkes et al.

Programmable  Magne,c  Self-­‐assembly  PNAS  ∣  July  13,  2010  ∣  vol.  107  ∣  no.  28  ∣  

How  Auxe,cs  (NPR)  Materials  work  

so the overall Poisson’s ratio is almost alwayspositive.

The elastic behaviour of the membranesstudied by Bowick et al.1 is said to be ‘uni-versal’ because the authors require only asparse set of assumptions to predict the Poisson’s ratio. They start with a simple net-work of nodes, resembling a fishing net withfixed connections, which they model using a Monte Carlo simulation. Bowick et al.show that a negative Poisson’s ratio is a uni-versal property of such systems, whether themembrane is dominated by rigid bonds thatresist bending or by ‘self-avoiding’ inter-atomic forces that prevent portions of thestructure overlapping.

This unusual form of elasticity may alsoarise in biological processes because themembranes considered by Bowick et al. aresimilar to the protein skeletons8 of biological

membranes. The overall elasticity of a cellmembrane results from both the proteinskeleton and the high lipid content, but therelative contributions are not yet known.Even so, Bowick and colleagues’ results areprovocative. If our usual expectations abouthow things deform do not apply to biologicalmembranes then we may need to reconsiderthe influence of membrane mechanics7 onthe shape of cells, the formation of vesicles,and the deformation of cells during lifeprocesses. For example, red blood cells areroutinely deformed when they pass throughfine blood capillaries. As they deform, themembrane skeleton can unfold, whichmight help to transport large molecules orexpose reactive chemical groups. Similarconcepts may be used in the design of indus-trial membranes for responsive filtration or catalysis. Materials with specific Poisson’sratios are always useful — cork, for example,has a ratio of almost zero, making it ideal forsealing wine bottles. For membranes with anegative Poisson’s ratio, the applications aresure to keep expanding. !

Roderic Lakes is in the Department of EngineeringPhysics, Engineering Mechanics Program at theUniversity of Wisconsin-Madison, 1500 EngineeringDrive, Madison, Wisconsin 53706-1687, USA.e-mail: lakes@engr.wisc.edu

1. Bowick, M., Cacciuto, A., Thorleifsson, G. & Travesset, A. Phys. Rev. Lett. 87, 148103 (2001).

2. http://silver.neep.wisc.edu/~lakes/Poisson.html3. Kolpakov, A. G. Prikl. Mat. Mekh. 59, 969–977 (1985).4. Gibson, L. J., Ashby, M. F., Schajer, G. S. & Robertson, C. I.

Proc. R. Soc. Lond. A 382, 25–42 (1982).5. Lakes, R. S. Science 235, 1038–1040 (1987).6. Lakes, R. S. J. Mat. Sci. 26, 2287–2292 (1991).7. Lipowsky, R. Nature 349, 475–481 (1991).8. Schmidt, C. F. et al. Science 259, 952–955 (1993).9. Wen, X. et al. Nature 355, 426–428 (1992).10.Kantor, Y., Kardar, M. & Nelson, D. R. Phys. Rev. Lett. 57,

791–794 (1986).11.Aronovitz, J. A. & Lubensky, T. C. Phys. Rev. Lett. 60, 2634–2637

(1988).12.Baughman, R. H., Shacklette, J. M., Zakhidov, A. A. &

Stafström, S. Nature 392, 362–365 (1998).

news and views

504 NATURE | VOL 414 | 29 NOVEMBER 2001 | www.nature.com

Daedalus

The art of copyingThe libraries of the ancient world survivedonly by continual copying, and the slowcrumbling of paper threatens ours as well.So Daedalus wants to be able to copy aclosed book at speed. Ideally, the librarianshould be able simply to load a block ofbooks onto the copier, which would thentransfer their contents to compact disk.

A paper page is about 0.1 mm thick, atypical infrared wavelength. It is composedof fibres of much finer diameter, so itshould be pretty transparent to light with awavelength of 0.1 mm or less. At such awavelength, each page should be opticallydistinguishable from the pages on eitherside. A pulsed infrared laser beamlaunched perpendicularly into the side, topor bottom of the book, or into all threesimultaneously, should illuminate onepage only, and would follow it even if thepage was a little bent.

A sufficiently narrow laser beam cantrace a single line of type, or pick out singleletters from successive lines. Each letter isan opaque indentation of the paper; it willabsorb the light pulse a little, and thissharp attenuation will return a reflection,or ‘echo’, characteristic of the shape of theletter. A beam-splitter could then separatethe stream of returning echoes into aphotocell for computer analysis.

Unfortunately, a page is printed onboth sides. At any printed site, the papermay be indented by two opaque letters,both of which will contribute to the echo.But this is where beams from all three sidescome in. Each could be swung through anangle to illuminate any of several files ofletters. Cunning software with linguisticknowledge will be needed to resolve themultiple ambiguities. But a closed page litby pulsed lasers should be readable.

DREADCO physicists and librariansare now at work. Their prototype stepsthrough a block of books at high speed, a page at a time. The final device willprobably send its output to a compactdisk, archival microfilm or miniprint onalkaline paper. Photographs and diagramsare a problem, but they could be identifiedby the software, and the book opened andphotographed the hard way.

The new DREADCO book-copier willtransform the task faced by the world’slibrarians. Instead of having to open everybook at every page, they will merely haveto load books in bulk into the copier. Itsoutput will be far smaller, neater and moreeasily stored. The mighty achievements ofthe nineteenth and twentieth centurieswill be faithfully recorded for futurehistorians. David Jones

Figure 2 An artificial membrane with a negativePoisson’s ratio. The local roughness of a ‘flat’fixed-connectivity membrane is due to thermalfluctuations. Stretching the membrane flattensthe roughness, causing lateral expansion in theplane of the membrane. In their simulation,Bowick et al.1 find that all fixed-connectivitymembranes, possibly including some biologicalones, share this property. (After M. Bowick, withpermission.)

Figure 1 Positive and negative Poisson’s ratios. Stretching these two-dimensional hexagonalstructures horizontally reveals the physical origin of Poisson’s ratio. a, The cells of regularhoneycomb or hexagonal crystals elongate and narrow when stretched, causing lateral contractionand so a positive Poisson’s ratio. b, In artificial honeycomb with inverted cells, the structuralelements unfold, causing lateral expansion and a negative Poisson’s ratio.

Unstretched Stretched

b

a

© 2001 Macmillan Magazines Ltd

ν  =  –et/el  =  –(ΔD/D)/  (ΔL/L)      Two  dimensional  structures:    a)  Honycomb  cells    b)  Inverted  honycomb  cells  

Miura-­‐ori  fold  

silver.neep.wisc.edu/~lakes/Poisson.html  

Proper,es:    Absorb  Energy,  resist  fracture.  Some  applica,ons:  body  armor,  shock  absorber,  packing  material,  knew  and  elbow  pads,  sponge  mops.      

Metallic  Foam  

Poisson  Ra,o  is  a  very  important  mechanical  proper,es    

•  ν  =  [3(B/G  –  2)]/[6(B/G  +  2)]    •  ν  =  [1⁄2(Vt/Vl)2  –  1]/[(Vt/Vl)2  –  1]  •  isotropic  range  of  –1  ≤  ν  ≤  1⁄2  for  0  ≤  B/G  <  ∞  at  small  strains  •  Nonlinear  regime  ν  <  -­‐1.  

Poisson’s  ra*o  and  modern  materials.  Nature  Materials.  G.  N.  Greaves,    A.  L.  Greer,  R.  S.  Lakes  and  T.  Rouxel  24  Oct  2011    

826 NATURE MATERIALS | VOL 10 | NOVEMBER 2011 | www.nature.com/naturematerials

Considering polycrystalline solids, these are macroscopically isotropic and have Poisson’s ratios of !C(0.2)!<!!Si(0.22)!<!!Ge(0.278) for tetravalent elements, and for the hcp polymorphs of !Zn(0.25)! <! !Cd(0.3) or !Ti(0.32)! <! !Zr(0.36)! <! !Hf(0.37), just to mention a few. Lead (!Pb!=!0.44) and thallium (!Tl!=!0.45) with high Z have remarkably high Poisson’s ratios. In the case of crystalline structures, the valence electron density plays a key role, and ab-initio calculation is becoming an increasingly popular approach follow-ing the early work of Cohen27. Subsequently, the cases of diamond semiconductors (!Si!=!0.22 and "Ge =!0.28) were reported28, as were the fullerites29, molybdenum40 and most recently dental amalgams41. #ere are, however, isolated cases where the interpretation is not straightforward. Most data are measured under ambient condi-tions, whereas ! is known to generally increase with temperature (see Fig.!4b). High melting points thus favour low Poisson’s ratio (everything else remaining unchanged) and vice versa. #is is pos-sibly why tungsten, being more refractory than molybdenum, has a smaller Poisson’s ratio and why beryllium, combining low Z with a

comparatively high melting point (1,560!K), exhibits a remarkably small Poisson’s ratio of 0.032!and is also very brittle.

Paradoxically, complex materials such as polycomponent glasses follow simple monotonic trends. In this case the atomic packing density Cg is de$ned as the ratio between the minimum theoreti-cal volume occupied by the ions and the corresponding e%ective volume of the glass (Fig.! 3). For instance, Cg is about 0.52! for a standard window glass and 0.45!for silica glass. An estimation of the packing density in metallic glasses can be obtained using the atomic radii of the corresponding constituent metals. Poisson’s ratio, which covers a wide range for inorganic glasses, increases almost monot-onically with Cg (Fig.!3). Eventually, within limited compositional ranges, linear trends can be observed42. Much earlier, Makishima had proposed that !!=!& ' 1/7.2Cg for silicate glasses43 but without knowledge of accurate atomic glass network parameters, such as interatomic distance and coordination number.

Nevertheless, a direct comparison of Poisson’s ratio for specimens of identical composition and initial structure but submitted

bBulk modulus B

Stable

Anti-rubberyDilationalAuxetic

Rubbery Sti!

Spongy

Unstable bydomain

formation ! = 1/2 ! = 0.3

! = 0

! = –1

–4G/3 < B < 0 ! = –2

! = 2 ! = 1

Shear modulus G

Unstable by volume changebut stable if constrained

a

0.6

0.4

0.2

0.0

–0.2

–0.4

–0.6

–0.8

–1.0

–1.2

Metallic glassesLead

GelsConcrete

BoneCork

sheetsCarbon nanotube Laminates

Laser-cooled crystals

Gases

Liquids

Critical fluids

Dense

r pac

king

Re-entrant polymer foams

Bi, As!-cristobalite

Unscreened metalsInc

rease

d con

necti

vity

Colloidal crystals

SteelOxide glasses

0.001 0.01 0.1 1B/G

10 100

RubberDental composites

HoneycombZeolitesCartilage

c

00

50 100 150 200G (GPa)

250

400

350

300

250

450

200

150

100

50

B (G

Pa)

Ductile

Unreachable

Brittle

Ideal isotropic solidB/G = 5/3

fcc metalsbcc metalshcp metals

Fe–Ni–

Pd–

Cu–

RE- Zr–

Mg–

fccBMG

bcc

hcpB/G = 2.4

Figure 1 | Poisson’s ratio: physical significance, materials characteristics and the Milton map. a, Numerical window of Poisson’s ratio !, from –1!to ", plotted as a function of the ratio of the bulk and shear moduli B/G for a wide range of isotropic classes of materials (see text for details). b, Milton map11 of bulk modulus versus shear modulus, showing the regimes of !, and the di#erences in material characteristics. The stability boundaries are pertinent to phase transformations11,12. c, Jiang and Dai plot32 of bulk modulus versus shear modulus for metallic glasses and polycrystalline metals. Straight lines refer to thresholds for brittle–ductile transition in metals. The ratio B/G !=!5/3 (!!=!0.25) relates to the lower limit of Poisson’s ratio for most metals. RE, rare earth. Ni- indicates nickel-based, and so on. Figure adapted with permission from: a, ref.!11, © 1992 Elsevier; ref.!12, © 1993 Wiley; c, ref.!32, © 2010 Taylor & Francis.

REVIEW ARTICLE NATURE MATERIALS DOI: 10.1038/NMAT3134

nmat_3134_NOV11.indd 826 12/10/2011 16:02

Foam  Structures  with  a  Nega*ve  Poissson's  Ra*o.  Lakes.  science.    1987.    

bulk  modulus  B    -­‐  change  in  size    shear  modulus  G  –  change  in  shape    

Project    

•  Make  Mouri-­‐Ori  Fold.    •  Vary  angle  to  vary  mechanical  proper,es.  •  How  does  the  rela,on  look  like.    •  Measure  Poisson  ra,os.    

   Miura-­‐ori  paZern  is  a  Rhomboidal  (2  angles  ,  2  lengths).    

•  They  developed  computa,onal  tool  to  simulate  the  stretching,  bending  and  folding  of  thin  sheets  of  material  to  predict  its  mechanical  proper,es.    

Pleated  and  Creased  Structures.  Levi  Dudte,  Zhiyan  Wei,  L.  Mahadevan.  APS  2012  Mee,ng.    2:30  PM–5:30  PM,  Tuesday,    February  28,  2012  Room:  153C  

Fabrica,on  and  Measurements    

Versalaser  VLS2.30  (Universal  Laser  Systems)  -­‐1.2  -­‐1.1  -­‐1  

-­‐0.9  -­‐0.8  -­‐0.7  -­‐0.6  -­‐0.5  -­‐0.4  -­‐0.3  -­‐0.2  -­‐0.1  

0  0.1  0.2  0.3  0.4  0.5  

1   2   3   4   5  

Poisson  ra1o  v  

Angles  in  radians  

Series2  

Possion  ra,o  measurements    Strain  Vs.  strain  plots  for  SU-­‐8  microstructures  to  measure  NPR  (no  varying  of  angles)  ~100  μm  

Adv.  Mater.  1999,  11,  No.  14  

Acknowledgment    

•  Applied  Math  Lab  – Advisor  and  PI:  L.  Mahadevan  –     Levi  Dudte  –     Zhiyan  Wei            

•  Microrobo,cs  Lab  – PI:  Robert  Woods  – Lab  Manger:  Michael  Smith.    

Influence  of  membrane  mechanics  on  Cells  shape    

Problems  faced      

•  Project  start  Monday,  Feb  6  2012:  17  days  only  and  talked  to  students  much  later.  

•  No  sotware  access  (Corel  Draw,  AutoCAD)  •  Hard  access  to  Laser  Lab    •  Access  to  lab  to  measure  passion  ra,o.    

Tensile  Tes,ng  structures  to  measure  the  Poisson's  Ra,o