Inverted and programmable Poynting effects in metamaterials
Aref Ghorbani, David Dykstra, Corentin Coulais, Daniel Bonn, Erik van der Linden, Mehdi Habibi
IInverted and programmable Poynting effects in metamaterials
Aref Ghorbani, ∗ Erik van der Linden, and Mehdi Habibi † Laboratory of Physics and Physical Chemistry of Foods,Wageningen University, 6708WG Wageningen, The Netherlands
David Dykstra, Corentin Coulais, and Daniel Bonn
Institute of Physics, University of Amsterdam, 1098XH Amsterdam, The Netherlands (Dated: February 23, 2021)The Poynting effect generically manifests itself as the extension of the material in the directionperpendicular to an applied shear deformation (torsion) and is a material parameter hard to design.Here, we engineer a metamaterial that can be programmed to contract or extend under torsion,depending on its architecture. First, we show that our system exhibits a novel type of invertedPoynting effect, where axial compression induces a nonlinear torsion. Then we program the Poyntingmodulus of the structure from initial negative values to zero and positive values via a pre-compressionapplied prior to torsion. Our work opens avenues for programming nonlinear elastic response ofmaterials by rational design.
Introduction .— The Poynting effect is a surprising non-linear elastic effect that makes, in the original experimentof Poynting [1], a hanging piano wire under tension be-come longer when it is twisted (FIG 1a, left). The conse-quence is also that if the distance between the two endsis fixed, the torsion induces a stress normal to the shearplane (normal stresses) that tends to separate the twoends (FIG 1b, left) [2, 3]. Developing normal stresses oraxial deformations under torsion are two equivalent man-ifestations of the Poynting effect. Poynting found thatthe normal stress as a function of shear strain follows aquadratic relation with a positive coefficient [1, 4, 5], nowcalled the Poynting modulus. While conventional mate-rials such as the piano wires of Poynting show a positivePoynting modulus (FIGs 1a and b, left), complex ma-terials such as biopolymer systems [6–10] and designedstructures [11] can exhibit a negative Poynting modulus,causing a shear-induced contraction under a fixed load(FIG 1a, right), or negative normal force at a fixed gap(FIG 1b, right and supplementary video 1 [12]).In this letter, we ask whether one can program thesign of the Poynting modulus using mechanical metama-terials. Designing metamaterials with exceptional me-chanical properties originated from their structure ratherthan their composition has attracted much research indifferent disciplines of science [13, 14]. So far, mechani-cal metamaterials have been studied mostly under com-pression or tension. Although unusual couplings betweentorsion and compression have been uncovered in designedstructures [15–17], the response of metamaterials to di-rect shear [18] and in particular the Poynting effect haveremained largely unexplored. Understanding the com-plex coupling between shear and normal responses inmetamaterials provides insights for harnessing and pro-gramming their Poynting modulus.We design a cylindrical metamaterial that exhibits aninverted Poynting effect under compression, where com-pression induces a torsion, and a programmable Poynt- positive negativePoynting modulus: F i x e d g a p a) F i x e d l o a d b) positive negativePoynting modulus: FIG. 1. Poynting effect. a) Applying torsion under fixedloads causes dilation in a material with a positive Poyntingmodulus (left) and contraction in a material with a negativePoynting modulus (right). b) When the material is confinedunder a fixed gap, dilation and contraction will be manifestedas positive (left) or negative (right) normal forces, respec-tively. ing modulus. The metamaterial is composed of identicalunits (unit-cells) that consist of beams with a suitablydesigned cross-section profile. The sign and magnitudeof the normal and shear responses of the designed struc-ture are tuned by the interplay between buckling insta-bility and self-contact interactions of the beams via apre-compression applied prior to shear deformation. Wepresent a simple spring model that reproduces our exper-imental results and characterizes the essential design pa-rameters to obtain the inverted Poynting and to programthe Poynting modulus. Our findings outline a strategytowards the rational designing of a programmable non-linear elastic response of metamaterials with potentialapplications in engineering biomaterials functioning un-der torsional deformation and designing robot arms ormechanical switches.
System and procedure .— We design a hollow cylindri-cal shell composed of an array of unit-cells, which pro-vides a network of non-uniform beams (FIG 2a) capa-ble of side-buckling and self-contacting under compres-sion [19]. The cylindrical shell has an outer radius of R max = 12 . mm , and an inner radius of R min = 7 . mm . a r X i v : . [ c ond - m a t . s o f t ] F e b We 3D print the structure with an elastomer and conductthe experiments using an Anton Paar MCR302 rheometer(further experimental details in [12]). The compressionstrain is defined as δ = | h − h | /h , where h = 40 . mm is the initial height of the cylindrical shell and h is itsheight after the pre-compression, under the compressionforce, F , with ± .
2% experimental error. We first con-duct two series of compression experiments with differentboundary conditions: in the first series the bottom sideof the shell is clamped and the top side is free to ro-tate (‘torsion-free’), while in the second series rotationis not allowed at both sides (‘clamped’). Then we usethe clamped boundary condition and perform two seriesof torsion experiments under fixed loads and fixed gaps.Torsional angle, φ , develops by applying shear force, F s ,and induces the axial deformation of δ n under a fixedload or the normal force of F n under a fixed gap. Inisotropic elastic materials, the shear stress, σ s , is propor-tional to the shear strain, γ , σ s = G s γ , and the normalstress induced by shear follows a quadratic relation as afunction of shear strain, σ n = G n γ [1, 5]. Here, we char-acterize the normal and shear responses of the structurewith a Poynting modulus, G n , and a shear modulus, G s ,respectively. Normal and shear force responses of a cylin-drical shell under torsion are given by F n = G n J ( φ/h ) ,and F s = τ /R = G s Jφ/Rh , respectively, where τ is thetorque around the axis of the shell, R = ( R max + R min ) / J = π ( R max − R min ) is the sec-ond moment of area of the shell. Inverted Poynting effect and three regimes of structuralrearrangements .— Our designed cylindrical metamaterialis capable of inducing torsion under compression (FIGs2b and c). We dub this phenomenon as the invertedPoynting effect since the manifestation of the Poyntingeffect in Poynting’s original experiment was the inverseof the present case: the applied torsion caused elonga-tion along the torsional axis. As shown in FIG 2b inthe torsion-free compression, we observe the rotationalbuckling with an affine torsional deformation across theheight of the sample. However, in the clamped compres-sion experiment, the torsional deformation accumulatedat the middle of the structure (FIG 2c). In both com-pression experiments, three distinct regimes of configu-rational changes are observable in the structure: (i) pre-buckling (FIG 2a), (ii) buckling of the beams (FIGs 2band c, left), and (iii) self-contact (FIGs 2b and c, right).FIG 2d, shows the compression stress, σ , rescaled byYoung’s modulus of the elastomer, Y = 944 ± kP a [12],as a function of compression strain, δ , for the torsion-free (dashed curve) and clamped (solid curve) experi-ments, respectively. The three regimes have different ef-fective stiffnesses due to their structural configurations.The effective stiffness of the system in each regime, Y eff ,is determined by the slope of the curve times Young’smodulus of the elastomer. In the pre-buckling regime( δ < δ b , where δ b = 0 .
012 is the compression strain for h a) ( ⅱ ) Buckling ( ⅲ ) Self-contact( ⅱ ) ( ⅲ )( ⅱ ) ( ⅲ )( ⅱ ) ( ⅲ )( ⅰ ) ( ⅰ ) d) e) c) FF b) FF Contact points
FIG. 2. Experimental setup and compression tests. a) 3Dprinted structure is clamped between two plates. Unit-cellsand nonuniform beams are magnified. b,c) The compressedcylinder at the buckling (left) and self-contact (right) regimesin torsion-free (b) and clamped (c) compression experiments.Deformed unit-cells are magnified. d) Rescaled nominal nor-mal stress, σ/Y , as a function of compression strain, δ , fortorsion-free (dashed curve) and clamped (solid curve) exper-iments. e) Torsional angle, φ , as a function of compressionwith square root (blue) and linear (red) fits, in the bucklingand self-contact regimes, respectively. Insets are the predic-tions of the model and have the same axes and units as themain plots, except for the vertical axis in the inset of (e),which represents the shear strain in the 2D model, γ . γ canbe converted to the equivalent torsion in our cylindrical shellvia φ = ( h /R ) γ . Vertical dashed blue and red lines showthe onset of buckling and self-contact regimes, respectively.Three regimes are marked with (i), (ii) and (iii). the onset of the buckling regime) the vertical beams arestable and resist buckling (FIG 2a), resulting in a rela-tively high stiffness, with Y eff = 0 . Y , in both com-pression tests. In the buckling regime ( δ b ≤ δ < δ c ,where δ c = 0 .
13 is the compression strain for the onset ofthe self-contact regime), the stress remains almost con-stant and the structure softens ( Y eff = 0 . Y ), due tothe buckling instability in the beams (FIGs 2b and c,left). Finally, in the third regime ( δ ≥ δ c ), the struc-ture becomes stiff again, with Y eff = 0 . Y , due to theself-contact between the beams (FIGs 2b and c, right).In the torsion-free experiments, compression inducesshear deformation, with distinct behaviors at bucklingand self-contact regimes (FIG 2e and supplementaryvideo [12]). The coupling between the compression andtorsion, in the buckling regime, is nonlinear and the bestfit to the experimental torsional angle as a function ofcompression gives | φ | = 4 . √ δ − δ b (blue curve in FIG2e). The emergence of this square root relation is due tothe buckling of the internal beams [20]. However, in theself-contact regime, compression and torsion are linearlyproportional (red line in FIG 2e). Whereas linear cou-pling between compression and torsion has been achievedbefore for chiral structures [15, 17, 21], a coupling be-tween compression and torsion in achiral structures hasnot been observed so far. Thus, the inverted Poyntingeffect translates an axial compression to a nonlinear orlinear torsion depending on the amount of compression.In the following, we investigate the Poynting response ofthe clamped structure under different loading conditions. Poynting modulus under fixed loads/fixed gaps .— Toquantify both manifestations of the Poynting effect forour metamaterial, we apply torsional deformation underfixed loads (FIG 1a) or fixed gaps (FIG 1b) and followits normal responses. The first loading scenario is equiv-alent to Poynting’s original experiment [1]. Initially, theclamped structure is loaded under a force F , resultingin an axial compression strain, δ . Then the torsion isapplied on the top boundary while F remains constant,causing an axial strain of δ n . The axial strain and ap-plied shear forces are shown as a function of torsion for arange of loads in FIGs 3a and b, respectively. To quantifythe second manifestation of the Poynting effect (torsionunder fixed gap), we apply torsion on the pre-compressedshell while the height of the structure is fixed at h . Here, F is the force needed for the pre-compression and F n isthe torsion-induced normal force; F + F n is the total axialforce response of the pre-compressed shell under torsion.FIGs 3c and d show the normal forces and applied shearforces as a function of torsion under fixed gaps, respec-tively (see supplementary video 3 [12]). For both series,the normal responses behave quadratically (FIGs 3a, c)while the shear responses behave linearly (FIGs 3b, d)as a function of torsion in the limit of small torsionaldeformation, φ < . rad . Initially, the non-compressedcylinder shows a contraction/negative normal force un-der torsion. While not a common material response, thenegative Poynting modulus has been observed and inves-tigated in biopolymer networks [6–8]. The origin of thenegative Poynting modulus in biopolymers is rooted inthe expulsion of water from their porous networks underdeformation allowing them to shrink [10]. Similarly, inour metamaterial, the presence of voids allows to circum-vent the volume conservation and to stretch the beamsunder torsion, which leads to negative normal responses.By applying a pre-compression on our metamaterial thecurvatures of the normal response curves in both fixedload and fixed gap experiments (FIGs 3a and c) changetheir sign and magnitude similarly. This indicates thatthe Poynting response of the structure is independentof whether the gap or the load was fixed, however, itssign and magnitude can be tuned by the level of pre-compression. Programmable Poynting and shear moduli .— To de- a) b)c) d)e) f)
Fixed loadFixed gap Fixed loadFixed gap
FIG. 3. Poynting and shear moduli. a,b) Fixed-load exper-iments: axial strain, δ + δ n , (a) and shear force, F s , (b) asa function of torsional angle, φ , while the structure is loadedunder a fixed force ( F , color scale). For the sake of clarity,only data for five experiments are shown. c,d) Fixed-gap ex-periment: compression plus normal forces, F + F n , (c) andshear force, F s , (d) as a function of φ at different levels ofpre-compression strain ( δ , color scale). e,f) Poynting (e) andshear (f) moduli rescaled by Young’s modulus of the bulk, Y ,as a function of pre-compression strain, δ , for both loadingscenarios calculated by fitting (solid curves) data points in a-d. Insets are the modeling results in the fixed gap boundarycondition and have the same axes and units as the main plots. termine how to program the nonlinear moduli of themetamaterial, we quantify shear and Poynting modulias a function of pre-compression. For torsion under fixedgap experiments, the coefficients of the fits to the nor-mal force, F n , and shear force data, F s , in FIGs 3cand d, rescaled by J/h and J/Rh , give the Poynting( G n ) and shear moduli ( G s ), respectively. For torsionunder fixed load experiments, we define the coefficientof the quadratic fits in FIG 3a rescaled by J/A s Y eff h as the Poynting modulus, where A s is the cross-sectionarea. FIGs 3e and f show the Poynting and shear modulirescaled by Young’s modulus of the elastomer as a func-tion of pre-compression strain for both loading scenar-ios. The Poynting modulus in the pre-buckling regime isnegative. For the intermediate pre-compressions (buck-ling regime), the Poynting modulus becomes zero. How-ever, by approaching the self-contact regime, it rapidlyincreases and reaches a maximum value at δ ≈ δ c ,where the transition from the buckling to the self-contactregime occurs. The obtained moduli from both loadingscenarios coincide as expected, apart from a deviationoccurring at this transient deformation. By further in-creasing the strain G n decays sharply and approacheszero. Both shear moduli calculated from the two loadingscenarios coincide as well (FIG 3f). The shear modulus, G s , first decreases to zero after the transition from thepre-buckling to the buckling regime. Subsequently, G s in-creases sharply at the onset of the self-contact regime andkeeps increasing at higher pre-compressions but with alower rate. Whereas incompressible isotropic solids suchas rubber have G n /G s = 5 / Oscillatory Poynting modulus in large deformation. —In FIGs 3a-d, we observe deviations from the quadraticresponse with a nonmonotonic behavior at large tor-sions. To understand this behavior at large deforma-tions, we follow the structural changes and mechanicalresponses under large torsional deformation at a fixedpre-compression in the self-contact regime ( δ c ). FIG 4ashows sequential images of the structural changes in ourexperiment. In FIG 4b, we observe a periodic oscilla-tion in both normal (solid line) and shear (dashed line)forces as a function of torsion amplitude. In the courseof torsion, one layer slides over another layer by snap-ping its beams from tilted to vertical and again tiltedconfigurations, causing a local maximum in the normalforce. Since the rearrangement occurs layer by layer, thenumber of the peaks of the normal response is set by thenumber of layers ( l = 4). For large deformation exper-iments we determine the Poynting and shear moduli as G n = ( h / J )( ∂ F n /∂φ ), and G s = ( Rh/J )( ∂F s /∂φ ).FIG 4d shows both rescaled G n (solid line, left axis) and G s (dashed line, right axis) oscillate between positive andnegative values. Negative values of G s accompanied bynegative slopes in the curves of shear force indicatingthe occurrence of the snap instability (see supplementaryvideo 4 [12]). Thus, snap instability and self-contact un-der large torsional deformation result in oscillatory non-linear moduli that are not allowed in conventional ma-terials and rare in metamaterials. In the following, weexplain a Hookean spring model that we use to repro-duces the experimental results theoretically. Modeling. — We model the cylindrical metamaterialas a 2D square network of elastic beams that eitherstretch or contract, while at each connecting node bend-ing can occur. Bending can initiate the self-contact atthe bending angle θ c = π − α , where 2 α is the an-gle the beam profile, as shown in supplementary mate-rials [12], that increases the elastic energy of the sys-tem. By considering Hookean elasticity for contract-ing and bending elements, we can write the energy of Experiment Model d) a) δ hz φ + − e) b) c) ×10 -2 ×10 -1 ×10 -2 ×10 -1 max FIG. 4. Periodic Poynting moduli in large strain deforma-tion under the pre-compression of δ = 0 .
13 in the self-contactregime. a) Sequence of images showing the internal deforma-tions of the cylindrical structure during one cycle of a largeamplitude torsion experiment. b,c) Normal response, F + F n ,(solid) and shear force, F s , (dashed) as a function of torsionunder large torsional deformations, for experiment (b) andmodel (c). d,e) Rescaled Poynting, G n /Y , (solid) and shear, G s /Y , (dashed) moduli as a function of torsion for experi-ment (d) and model(e). (c) and (e) are predictions of themodel only for positive shear deformations ( γ ≥
0) with afixed gap boundary. The results of negative shear are mirrorimages of these curves, with four peaks in one deformationcycle. a vertical array of l beams as E = ka (cid:80) li =1 e i + k b (cid:80) li =1 θ i + k c (cid:80) li =1 ( θ i − θ c ) H [ θ i − θ c ] . Here, a is theinitial length, k is the stretching coefficient, and k b is thebending coefficient of the beams. e i is the local strain and θ i is the deformation angle of the beam i . We include theself-contact mechanism by introducing the Heaviside stepfunction, H [ x ]. This term represents the self-contact in-teraction of the initially vertical beams with the horizon-tal ones, with the coefficient of k c , and is nonzero when θ i ≥ θ c . Considering n unit-cells in each layer with a pe-riodic boundary condition in the vertical direction, thetotal energy of the structure is given by E tot = nE (fur-ther details in supplementary materials [12]). We min-imize the energy of the system for a range of positiveshear deformation under fixed gap boundary conditionsat different compression strains. The results predicted bythe model successfully reproduce all the qualitative fea-tures of the experiments, as shown in the insets of FIGs2d, 2e, 3e, 3f, 4c, and 4e, with minor quantitative differ-ences. Therefore, the simple linear spring model identi-fies buckling and self-contact as the minimum ingredientsto achieve programmable/inverted Poynting effects andconfirms the structural origin of the nonlinear responses.In conclusion, we have engineered a cylindrical meta-material with programmable Poynting and shear re-sponses. We showed that our designed structure is ca-pable of exhibiting the inverted Poynting effect by trans-lating an axial compression to a nonlinear torsion, incontrast to conventional elastic materials. We also suc-ceeded in programming the Poynting modulus by varyingthe level of pre-compression/loading prior to torsion. Weswitched the sign of the Poynting modulus and tuned itsvalue over a wide range, including even eliminating it.Furthermore, we successfully modeled and studied thesystem using an energy minimization method. The modelidentifies buckling of the ligaments and self-contact asthe essential design elements to achieve programmableand inverted Poynting in a metamaterial. Our analyti-cal approach opens avenues for bottom-up programmingof the shear and normal mechanical responses of meta-materials based on self-contact as a mechanical feedbackmechanism. Considering the fundamental importance ofshear and normal moduli in mechanics, the ability to pro-gram them inspires applications in diverse fields from softrobotics to engineering biological tissues and implantsfunctioning under torsion and compression. In addition,our system is capable of translating a unidirectional mo-tion into torsion and of switching the mechanical forces.As these are relevant features of machines, we anticipateapplications in designing machine materials, robot arms,and mechanical switches.M. H. acknowledges funding from the Netherlands Or-ganization for Scientific Research NWO, through NWO-VIDI grant No. 680-47-548/983. C.C. acknowledgesfunding from the European Research Council, throughthe Starting Grant No. 852587. ∗ Email: [email protected] † Email: [email protected][1] J. H. Poynting, On Pressure Perpendicular to the ShearPlanes in Finite Pure Shears, and on the Lengthening ofLoaded Wires When Twisted, Proceedings of the RoyalSociety A: Mathematical, Physical and Engineering Sci-ences , 546 (1909).[2] C. Truesdell, The Mechanical Foundations of Elasticityand Fluid Dynamics, Journal of Rational Mechanics andAnalysis , 125 (1952).[3] E. W. Billington, The Poynting effect, Acta Mechanica , 19 (1986).[4] R. S. 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A. Bessa, P. Glowacki, and M. Houlder, Bayesian Ma-chine Learning in Metamaterial Design: Fragile BecomesSupercompressible, Advanced Materials , 1904845(2019). SUPPLEMENTARY MATERIAL
Metamaterial design and experimental procedure .— Weuse the following polar function to create shape of thepores with a 4-fold symmetry contour s ( u ) = c [(1 − ( a + b ))+ a cos(4 u )+ b cos(8 u )], where, s ( u ) is the radius at thepolar angle u , x = s ( u ) cos( u ), and y = s ( u ) sin( u ). Inthis equation, a and b are the shape tuning parameters,and c sets size of the pore. Considering a = b = 0, we cancreate a circle with a radius of c . Overvelde et al. showedthat a 2D network with the flowing parameters for thepore shape of the unit-cell, a = − .
21 and b = 0 . l layersleaves us with a cylindrical network of nonuniform beamscomposed of 8 × l unit-cells. We set the pore size with c = 4 . mm , which gives the minimum beam thickness of0 . mm at the outer surface of the shell. We create extrahalf layers on top and bottom and clamp the structurewith two O-rings with the same maximum and minimumradii as the shell and height of 3 mm (FIG 1 SM. c). Thesefeatures enable us to clamp two sides of the cylindricalmetamaterial during the measurements.We 3D print the designed cylindrical metamaterial us-ing a Formlab Form2 3D printer and elastic resin v1,with a resolution of 0 . mm . We conduct the compres-sion and torsion experiments on our cylindrical metama-terial via an Anton Paar MCR302 rheometer with accura-cies of 1 µm (longitudinal displacements), 0 . µrad (an-gular displacements), 0 . N (normal forces), and 1 nN m (torques). Using a custom made plate-plate geometry,we clamp the upper and bottom sides of the shell (FIGs2a-c). The deformation angle, φ , is positive when thetorsion is clockwise. We 3D print a cube of 1 cm , and bycompression test we determine the Young’s modulus ofthe bulk as Y = 944 kP a . Details of the modeling .— We model the cylindricalmetamaterial as a 2D square network of Hookean elasticbeams with the length of a that could either stretch orcontract. The beams can bend at the connecting nods(FIG 2 SM.). Each beam has a pair of arc-shaped elasticarms, which are placed at the distance r from each end ofthe beam and symmetrically spread by ± α (FIG 2 SM.a). We study this model system in a 2-step deformationprocedure: first, compression along the vertical axis, z (FIG 2 SM. c), and then shear along the horizontal axis( x ). The connecting arms to the beams are designed to a) b) c) z FIG. 5. SM. a) The pore contour. b) CAD model of thevoid, created by extruding the pore contour in radial directiontoward z axis. An isometric view of the CAD model of thecylindrical metamaterial. mimic the nonuniformity of the beams’ profile, and theycan deliver self-contact under deformation. Self-contactslead to elastic contractions of the arms that produce thereaction forces.The total elastic energy due to deformation of eachbeam is composed of stretching energy, E s = ka e , andbending energy, E b = k b θ . To calculate the contributionof self-contact in the elastic energy we assume that thearc-shaped arms have the same Hookean coefficient asthe straight parts of the beam, k , and when subjected toself-contact, their curvature remains constant but theirlength, s , changes through change of the arc angle, 2 α ;thus, δs = 2 rδα . Since the deflection of the beam isdivided between to contacting arcs, we can write δα = δθ/
2, where δθ = θ − θ c is the deflection of the beam afterself-contact at θ c = π − α . Due to such deformation, theenergy of each arm changes by E c = kr δθ = k c δθ ,which gives the elastic coefficient of the self-contact as k c = kr . So the total energy for l layers of the verticalbeams is: E = k b l (cid:88) i =1 θ i + 12 ka l (cid:88) i =1 e i + kr l (cid:88) i =1 δθ i H [ δθ i ] . (1)Since bending and self-contact happen symmetrically atboth sides of the beam, their energy terms do not havethe pre-factor 1 /
2. Heaviside step function, H , representsthe self-contact interactions; H [ x ] is 0 for x < x ≥ n unit-cells in each layer is r a α θ θ Self-contact a) b) δ h c) xz FIG. 6. SM. a) Schematics of a non-uniform beam withassociated parameters in our model. b) Schematic of the net-work of the beams in 4 layers. c) Under the pre-compression,beams are tilted by θ i and when θ i ≥ π − α self-contactoccurs. E tot = nE .We use n = 8, l = 4, α = 0 . rad , and r = 4 . mm ,mimicking our experimental structure. We estimate thestretching coefficient from the force response in the com-pression experiment before buckling as k = 331 N/m . Toestimate the bending stiffness, k b , we consider a beamwith a rectangular cross-section and use k b = M/θ = Y I/r c θ ≈ (2 Y I/l b ), where I is the moment of inertia, l b is the length of the beam, and r c is its bending curva-ture. This estimation predicts an order of magnitude of10 − N m for the bending stiffness of the system. We fi-nally calibrate the bending stiffness at k b = 2 × − N m to obtain the buckling at the same compression strain asin the experiment ( δ b = 0 . δ c ≈ (1 − cos( π/ − α )) + δ b ≈ .
13, in a good agreementwith the experimental observations. Moreover, we ana-lytically predict the square root coupling of compression-torsion observed in the inverted Poynting experiment as γ = (cid:112) δ − δ b ) (FIG 2e, inset). Since shear deformationrepresents itself as a torsion on the cylindrical shell, byconverting shear strain to torsion using φ = ( h /R ) γ , wepredict a coefficient of 5 . ..