Nanomechanics of individual aerographite tetrapods

ABSTRACT Carbon-based three-dimensional aerographite networks, built from interconnected hollow tubular tetrapods of multilayer graphene, are ultra-lightweight materials recently discovered

and ideal for advanced multifunctional applications. In order to predict the bulk mechanical behaviour of networks it is very important to understand the mechanics of their individual

building blocks. Here we characterize the mechanical response of single aerographite tetrapods via _in situ_ scanning electron and atomic force microscopy measurements. To understand the

acquired results, which show that the overall behaviour of the tetrapod is governed by the buckling of the central joint, a mechanical nonlinear model was developed, introducing the concept

of the buckling hinge. Finite element method simulations elucidate the governing buckling phenomena. The results are then generalized for tetrapods of different size-scales and shapes. These

basic findings will permit better understanding of the mechanical response of the related networks and the design of similar aerogels based on graphene and other two-dimensional materials.

SIMILAR CONTENT BEING VIEWED BY OTHERS MULTI-FUNCTIONAL FLEXIBLE 2D CARBON NANOSTRUCTURED NETWORKS Article Open access 12 October 2020 NANOMECHANICS OF GRAPHENE OXIDE-BACTERIOPHAGE BASED

SELF-ASSEMBLED POROUS COMPOSITES Article Open access 24 September 2020 ANOMALOUS ELASTICITY AND DAMPING IN COVALENTLY CROSS-LINKED GRAPHENE AEROGELS Article Open access 20 January 2022

INTRODUCTION Three-dimensional (3D) cellular materials built from carbon nanostructures are currently under increasing investigation in terms of fabrication and physical properties because

of their significant technological potential for diverse advanced applications, such as biological scaffolds, electrochemical biosensing, supercapacitors, light weight flexible batteries,

and highly efficient oil absorbers1,2,3,4,5,6,7,8,9. Since the introduction of graphene, a large variety of synthesis methods, involving direct growth, wet chemistry and templates, have been

employed for the growth of 3D carbon-based nanomaterials9,10,11,12,13,14,15 and the field is still under development. Nanoscale carbon structures can be built, for example, from carbon

nanotubes (CNTs) or graphene flakes16 exhibiting very high surface to volume ratios from which interesting physical and chemical features originate. But to efficiently access their unique

nanoscopic features, these materials should be preferentially available in a macroscopic 3D form with a sufficient mechanical robustness and stability so that they can be manufactured into

any desired structured shape17. Thus, 3D carbon-based networks comprising both sufficient mechanical strength and very high porosity are desirable, but this is a challenging task in 3D

cellular networks. For instance, it is well known that 3D networks based on CNTs being randomly interconnected and held in place only by van der Waals (vdW) forces are prone to failure when

compressed and several attempts have been made to overcome this drawback18,19,20. In this context, the morphology and interconnections of the nanoscale carbon-based network building units

play a fundamental role. As basic building block of porous 3D networks, the tetrapod geometry is an interesting shape, since when accumulated together their spatially extended arms can

prohibit close packing very efficiently. Recently, a new concept for producing tetrapod-based 3D networks has been introduced by the flame transport synthesis of zinc oxide21,22,23. During

the re-heating at high temperatures the nano- and micro-scale tetrapod arms build interconnections, forming a bridging 3D network which provides necessary mechanical strength and

simultaneously very high porosity (up to ∼98% just by controlling the initial tetrapod template amount in the scaffold)21,24. Furthermore, if these ZnO tetrapods are loaded with other metal

oxide nanostructures, hybrid 3D interconnected networks can be easily realized25, which are suitable for different applications26. In the context of carbon-based networks, the ZnO can also

be exploited as sacrificial template for the growth of ultra-lightweight and highly porous (porosity>99.99%) 3D multilayer aerographite (AG) networks (also known in the literature as

aerographene). In a single-step chemical vapour deposition (CVD) process it is possible to form hollow nano- and micro-tubular multilayer graphene structures, which adopt basic features of

the shape of the ZnO templates. The high temperature, together with the presence of a carbon source and of hydrogen (carbon by precursor and hydrogen by gas) allow the deposition of nm-thick

graphene flakes on ZnO (ref. 27). Then ZnO is chemically reduced to metallic Zn, whereas Zn evaporates and is removed by the carrier gas (Ar). This process had been introduced in 2012

together with model of a belt-like growth27. Meanwhile, the working principle of the CVD synthesis was confirmed by other authors28,29,30. By modifying the CVD parameters several variants

can be synthesized, which differ in walls morphology (closed or open) and/or inner graphitic fillings27. Some further AG variants do not have closed graphene-based shells, but just consist

of narrow carbon filaments on the former tetrapod surfaces and thus possess an extreme high level (>99.99%) of porosity27. This kind of hierarchical networks containing carbon filaments

is even more attractive in terms of porosity and was used to realize flexible and semiconducting composites which could be exploited as next generation materials for electronic, photonic and

sensors applications17,31. Apart from being highly porous and extremely lightweight, AG exhibits very interesting specific mechanical properties such as remarkable specific tensile strength

(σ/_ρ_) and Young’s moduli (_E_/_ρ_)27, being in principle ideal candidates for impact protection and shock absorption. Some variants show also self-stiffening in cyclic mechanical

loading27. To use AG for different applications, and thus predict the overall mechanical properties of its 3D networks, a detailed understanding of the mechanical behaviour of its individual

building blocks is necessary. So far, even nanoscale hollow carbon tetrapods synthesized using ZnO tetrapods templates have been studied with respect to their different properties but their

constitutive mechanical response has not been discussed yet. Here we investigate and model the mechanical behaviour of single tetrapods, building a base for the future prediction of the

mechanical behaviour and properties of AG networks with different densities and known tetrapod characteristics. Single arms of AG tetrapods were bent _in situ_ (with other arms fixed) inside

a scanning electron microscope (SEM) in a controlled manner by a soft atomic force microscopy (AFM) cantilever. Buckling of the central joint was seen to be the governing mechanism of the

behaviour of the single tetrapod. We then propose an analytical nonlinear nanomechanical model for describing the formation of a ``buckling hinge'' at the tetrapod central joint or

within the arms. Complementary, finite element method (FEM) simulations were carried out, assuming identical conditions of the tetrapod arms corresponding to the _in situ_ atomic force and

scanning electron microscopy experiments. The results from _in situ_ experiments, analytical calculations and FEM simulations are in good agreement. The observed internal tetrapod

deformation mechanisms and the constitutive behaviour are then generalized for different size-scales and shapes, that is tube cross-section aspect ratios. Thus, the results can help in the

design of aerogel tetrapodal networks of different materials and structures. RESULTS MORPHOLOGY OF AEROGRAPHITE TETRAPODS First, the morphology of the AG tetrapods was studied. In Fig. 1a

the conversion principle of tetrapodal ZnO into hollow and tubular tetrapodal AG (t-AG) during the CVD process is schematically illustrated. Furthermore, the panel displays representative

SEM images of a typical ZnO tetrapod (Fig. 1b) and of an AG tetrapod (Fig. 1c), respectively, before and after the CVD conversion within the network. The ZnO tetrapod template used in this

study, and thus the resulting graphene counterpart, has four arms which are interconnected together with a mutual dihedral angle of ∼106° via a central joint, resulting in a 3D spatial

shape27. Thus, their geometry can be defined, in a good approximation, from the vertexes and the centroid of a regular tetrahedron. In the variant used here, the ZnO tetrapods exhibit

uniform hexagonal cylindrical arms narrowing towards their tips (see SEM image in Fig. 1a). The arms typically have a diameter of ∼1, 5 μm at their tip and joint, respectively. The arm

length is in the range of 15–30 μm. The morphology of the t-AG arms is strongly influenced by the growth parameters during the CVD process and, if required, t-AG variants with hollow tubular

arms, a closed shell and low aspect ratios can be grown27. The arms of the AG tetrapods used for the _in situ_ bending experiments exhibit a hollow tubular morphology with diameters being

equally in the range of 1–5 μm and wall overall thicknesses of the graphene flakes of ∼15 nm, see energy filtered transmission electron microscopy (TEM) picture in the Supplementary Fig. 1.

The AG variant which was used here, possesses tubular and smooth walls (see t-AG SEM images in Fig. 1c–e). Other crumpled variants may arise from thickness-dependent growth processes which

could be influenced by several parameters, such as surface energies, defects and internal mechanical stresses during the conversion process in the CVD chamber. TETRAPOD BENDING EXPERIMENTS

During the first type of _in situ_ experiment a tetrapod arm was bent inside a SEM with a soft AFM cantilever while the other three arms were attached to a substrate, following a

well-established procedure32,33,34. It was repeatedly observed via instant video recording (Supplementary Movie 2), that the free standing AG tetrapod arms tend to preferentially rotate

around the central joints and to experience there localized elastic instability, while the arm itself behaves very rigidly. Thus the tetrapod joint appears to be the most compliant location

of the arm, as demonstrated later. In general, when a tube starts to buckle its stiffness is significantly lowered35. To examine and quantify this stiffness reducing effect of the

buckling-hinge, the free-standing AG tetrapod arm depicted in Fig. 2a,b was deflected to an angle _α_=0.6 rad with the help of the aforementioned AFM cantilever tip from the right towards

left side of the image, parallel to the surface. The angle increment Δα at the buckling joint is a function of the resulting moment and of the joint rotational stiffness _D_, thus where F is

the external applied force on one of the arms and R its lever arm with respect to the computing point (inset in Fig. 2c). Being in the elastic regime, we assume _D_ as constant until the

applied moment _M_=|R||ΔF|sin_θ_ is lower than the buckling threshold _M_bh which is a function of the joint cross-section geometry and of the material elastic properties. The resulting

non-linear moment–rotation curve experimentally measured is shown in Fig. 2c. As expected, it reveals progressively decreasing rotational arm stiffness for higher deflection angles. We

propose a nonlinear equation for describing the formation of a buckling hinge in the tetrapod central joint or along the arm length analogously to the moment-rotation curve observed during

the formation of a plastic hinge in elastic-plastic bent beams. In both cases (buckling and yielding) the involved sections at large load possess very low rotational stiffness: a large local

deformation arises with small increment of load and the local curvature _χ_ goes to infinite (the radius of curvature _ρ_=1/_χ_→0). Considering a homogeneous linear elastic perfectly

plastic isotropic material, an initial linear regime occurs. Then, if the section is sufficiently thick and the material ductile, localized plastic deformation starts with the formation of

the so-called plastic hinge36. On the contrary, if the section is very thin, as in the case of tubular sections, the local elastic buckling may forego yielding or fracture. When _M>M_bh

the following non-linear buckling-hinge equation enters into play (see ‘Methods’ section for its derivation): where _α_bh is the joint (hinge) rotation at the buckling onset and is the model

parameter which describes the evolution of the hinge, with _M_u being the maximum asymptotic bending moment that the hinge is able to carry. Note that the equation (1) is the generalization

of the one describing the evolution of the bending moment after a formation of a plastic-hinge: for example when _γ_=1/2 it represents the plastic behaviour of a filled rectangular

cross-section. The analogy holds just in the monotonic loading regime: in fact the buckling hinge can be completely reversible. Eventually, different values of _γ_ could be estimated for

different cross-section and different causes of the joint rotation. We then simulated the _in situ_ experiment presented in Fig. 2a in which the geometry of the tetrapod was highly regular

and clearly visible from the SEM, being its bending not covered by the AFM cantilever. The length of each arm was derived from the _in situ_ SEM videos (the detailed procedure is described

in the Supplementary Note 1) and found to be ∼27 μm (Fig. 2b). This value was set as the distance from the base of the tetrapod arm (thus not the central joint) and the top face of the

circular tapered cone defining the arm end (Fig. 2b). The diameters of the cone at the tetrapod central joint and at the arm end are respectively _d_1=5 μm and _d_2=3 μm and each tetrapod

arm is capped at the end with a hemispherical shell of diameter _d_2 (Fig. 2b). We assumed a wall thickness _t_=15.3 nm, namely corresponding to 45 graphene layers, as suggested from an

energy filtered TEM image taken from a representative tetrapod arm (see Supplementary Fig. 1). Figure 3 shows the normalized moment–rotation curves of the analysed tetrapod, comparing the

experimental results with the curves defined by the non-linear buckling-hinge model and FEM simulation. The analytical curve is obtained from the best-fit of the experimental data

(corresponding to the ones reported in Fig. 2), while the FEM simulations are calibrated assuming as fixed degrees of freedom, the buckling point (_M_bh, αbh) and the ultimate hinge moment

_M_u. From the buckling-hinge model we estimated _γ_=0.44, and _D_=0.85 pN m rad−1 in the elastic regime (_α_<_α_bh), while from FEM simulation we determined as best-fit of the AFM

experiment a Young’s modulus of _E_=270 GPa (refs 37, 38), which was not known _a priori_. Notice that the _γ_ for a thin circular elastic–plastic section undergoing yielding would be

_γ_≈0.27. FEM images of the tetrapod deformation at three different stages are depicted as inset in Fig. 3 displaying stress distribution within the tetrapods (see also Supplementary Movie

3). These FEM pictures confirm that, prior to buckling, the response is governed by a transverse deformation of the adjacent arms nearby the joint and that in the end it merges in the

central joint buckling (see FEM third stage image of Fig. 3b) and that bending deformation of the loaded arm has a negligible contribution. This can also be theoretically claimed

approximating the arm as a bent cantilever of length _l_ under a concentrated force at the free end: indeed, assuming by absurd that the arm tip displacement is due to the elastic bending of

the arm, the materials Young’s modulus can be derived as _E_=_lD_/(3_J_), where _J_ is the average cross section moment of inertia of the tapered arm. The corresponding calculated value

would be for our case _E≈_20 MPa, which is very low referring to nominal properties of multi-layer graphene39. Consequently, for our thin-walled tube tetrapods, the pure elastic arm bending

is negligible when compared to the most compliant buckling-hinge section, either is represented by the central joint or by an intermediate arm section. In a second type of _in situ_

experiment the AG arm was isolated from the tetrapodal structure and thus from the central joint and was instead placed in between two gold tips (see Supplementary Fig. 2). In this way,

without the AFM cantilever tip masking the buckling location, a better visual evaluation of the hinge formation was realized by constructing a deformation situation in which it was more

probable to observe buckling at the most compliant position along the arm length. Figure 4 shows a series of SEM micrographs from the buckling of a single tube of AG bent in between the two

gold tips, from the undeformed state (Fig. 4a) to a state in which the tube has started to buckle (position indicated by the circle in Fig. 4b), to a heavily buckled state in which the

stiffness of the tube is dramatically decreased due to buckling (Fig. 4c). This confirms that buckling can occur even on the arm provided that the joint rotational stiffness is sufficiently

high (here the joint is not present and the left-hand side extremity can be assumed fully clamped, thus analogous to a rigid joint). Interestingly, the tube recovered elastically to its

original shape without any visible damage after flexure folding (Fig. 4d and Supplementary Movie 5). This property has already been reported by Falvo _et al_.40 for multiwalled CNTs, but in

contrast to these CNTs the diameter of the examined AG tubes is about three orders of magnitude bigger. However, these results indicate similarities between the elastic buckling of AG tubes

and multiwalled CNTs. The buckling-hinge model is likewise applicable to this case. We assumed that the buckled section takes an elliptic shape, this can be computed by imposing two

conditions: (1) the perimeter of the tube section must keep constant under ovalization and (2) the cross-section moment of inertia is related step by step to the current value of the joint

rotational stiffness _D_(_α_) (see ‘Methods’ section for its analytical derivation). Figure 4e shows the curve obtained from FEM simulation and the fit obtained with the buckling-hinge model

(_E_=270 GPa, _d_1=0.5 μm, _d_2=0.75 μm, _t_=15.3 nm and _l_=2.6 μm). The buckling-hinge model validity is also confirmed by the good agreement between the simulation observed cross-section

shape at the buckling hinge and its analytically derived counterpart (Fig. 4e). SCALING LAWS These results can be generalized to tetrapods of different size-scale and shape, namely aspect

ratio _t_/_d_. Indeed, it is acknowledged that for thin-walled tubes, such as ours, the critical compressive local strain _ɛ_bh, corresponding to the buckling condition under bending, is

given by the following relation41,42: where, in our case, _κ_≈1.178 assuming a Poisson’s ratio _ν_=0.2 for the graphite tube walls and _η_ is an adimensional factor theoretically equal to 1.

Figure 5a shows the simulations results in terms of critical buckling stress _σ_bh for tetrapods of different scales _ζ_=_d_/_d_0=_l_/_l_0 both for constant and variable aspect ratios

_t_/_d_. These are compared to the analytical predictions of equation (2), according to _σ_bh=_Eɛ_bh, from where a very good agreement is observed, considering _η_≈0.787 as derived from the

best fit of numerical simulations. Buckling stresses of the order of the gigapascal emerge. The small difference of this factor from the theoretical unit value, which corresponds to the case

of simple tubular section41,42, can be imputed to the higher complexity of the buckling deformation mechanism at the central joint, which involves also multiple layers constituting the tube

wall. Figure 5b shows the dimensionless moment versus rotation curves for all the analysed cases compared with the analytical prediction obtained inserting _M_bh=2_σ_bh_J_/_d_ into equation

1, with _γ_=0.44. The collapse of all the curves into a single master curve confirms the validity of this last scaling-shape law. It could also be used to include statistical variation in

the tetrapods geometry for the modelling of realistic networks. COMPRESSIVE AND TENSILE BEHAVIOUR OF TETRAPODS With FEM simulations we performed further mechanical characterizations on the

same tetrapod geometry under pure compression or tension, with fixed or sliding boundary conditions. We subjected the central joint of the tetrapod to an imposed displacement orthogonal to

the substrate. The results are reported in Fig. 6. Figure 6a depicts the compression behaviour: the buckling-hinge local instability leads to a global snap-through instability. For sliding

boundary conditions the buckling-hinge appears at the central joint while for fixed boundary conditions early buckling-hinges appear on each arm near the clamps (deformation level ), being

there also a bending moment. At larger joint displacement the central buckling-hinge occurs (level ) while the arm hinges disappear. After the snap-through the three base arms are under

tension and a further increase in the force is observed (level ). Regarding the tensile behaviour depicted in Fig. 6b, the fixed boundary conditions are able to prevent buckling and the

tetrapod behaviour is governed by the elastic bending of the arms attached to the substrate, thus resulting in much higher overall stiffness and bearing capacity with respect to the sliding

boundary conditions. In the latter, the formation of the buckling-hinge at the central joint is observed, representing an example of buckling in tension43; at very large displacements the

tetrapod starts to stiffen, being governed by the arms axial rather than bending stiffness. The slope of the force–displacement curve is nearly the same in both tension and compression as

expected, and depicted in the Supplementary Fig. 3. The four in-silico tests—related Supplementary Videos 6–9 are provided—, which could be considered as limiting cases of real scenarios

where mixed boundary conditions are expected (compliant clamps), are all in agreement with the buckling-hinge model prediction, as demonstrated in the Supplementary Fig. 4, confirming the

generality of the proposed approach. DISCUSSION Complex shaped hollow nano- and micro-structures, for instance the here considered tetrapods, enable the tunable fabrication of advanced 3D

highly porous materials with unique mechanical specific properties. The non-linear constitutive law of these modular networks is mainly dictated by the mechanical behaviour of the individual

network building blocks, which themselves strongly depend upon their morphology. In particular, the mechanics of single hollow AG tetrapods with hollow arms, that in the present work have

been successfully synthesized from sacrificial ZnO tetrapods in a single-step CVD conversion process, is governed by the buckling-hinge formation at the central joint or along its arms,

rather than by the elastic deformation of the arms, as dictated by its thin walls. This mechanism, which clearly emerges from experiments and simulations, is reversible and allows high

overall deformation without damage under extreme and cyclic loads, as confirmed and visible by experiments. The developed analytical model, which describes the mechanical behaviour of the

tetrapod buckling-hinges with three parameters (the arm rotation at buckling onset _α_bh, the hinge elastic rotational stiffness _D_ and the buckling-hinge parameter _γ_), represents the

essential basis for understanding the mechanical behaviour of AG networks as a whole. The nonlinear softening of tetrapods suggests that the relative network under compression may experience

an analogous behaviour before its stiffening due to material densification. Indeed, this is in agreement with the experimentally observed behaviour for networks, which shows a change in the

sign of the stress–strain curve second derivative (non-linear softening followed by stiffening)27. We believe that our findings on the dominant deformation mechanisms of individual AG

tetrapods can lead to a more profound understanding of the mechanical behaviour of the 3D interconnected t-AG. Moreover, due to the proved generality of the buckling-hinge model, not

restricted to the specific geometry, loading and boundary conditions, size-scale, and shape of the tetrapod, our work is expected to be useful in the design and optimization of aerogels and

foams44 in different fields, from materials science to scaffold medical engineering. METHODS PRODUCTION OF AEROGRAPHITE The carbon-based AG tetrapods with hollow microtubular arms were

synthesized by direct conversion of sacrificial zinc oxide nano- and micro-tetrapods27 at 760 °C by a CVD process described in a previous work15. In the CVD process, networks of μm-sized

tetrapodal ZnO (92% porosity)21 were converted into AG networks (_V_∼1 cm3) equipped with nm-thick, closed multilayer graphene shells. The morphology of the AG network was studied using an

Ultra Plus Zeiss SEM (7 kV). Several carbon tetrapods were carefully selected from a single AG 3D network sample for _in situ_ bending investigations in the SEM. The bending experiments were

done using _in situ_ methods in a Hitachi SEM S-4800. For manipulation of the free-standing tetrapods a customized piezo-driven SmarAct 13D nanomanipulation system45,46,47 was used. To

measure their mechanical properties, soft AFM cantilevers BL-RC-150VB from Olympus (spring constant _k=_2.9–50 pN nm−1) were utilized. Two different types of bending experiments were

performed which are described below. SINGLE TETRAPOD BENDING EXPERIMENTS In the first _in situ_ experiment, tetrapods were dropped on a Si/SiO2 wafer after scratching them from a bulk AG

sample. For mechanical measurements, only the tetrapods which were strongly adhering by vdW forces with the wafer substrate were chosen, that is, those attached with three arms to the

substrate and thus did not change position during the bending experiments. When the proper geometrical alignment was achieved, the tip of the cantilever was moved towards the free arm of the

tetrapod and the bending of both tetrapod and cantilever was observed. The whole motion was captured in a video (see Supplementary Movies 1 and 2), which was analysed frame-by-frame, to

quantify the deflections of both, tetrapod and cantilever, respectively (calculation of the torque–deflection-curves from the _in situ_ video data is described in detail in the Supplementary

Note 1). ISOLATED TETRAPOD ARM BENDING EXPERIMENTS In the second _in situ_ experiment, tetrapods were glued, using a micromanipulator, with a conductive epoxy (CW2400) to an

electrochemically etched gold tip, thus forming a small network of AG at the apex of the tip. Afterwards a part of a single arm of a tetrapod was picked up from the network with another gold

tip, which actually was possible due to the strong vdW interactions between the gold tip and the AG tube. Then the single free tetrapod arm was brought into contact with a third Au tip and

the buckling experiment was performed. In these measurements, the tetrapod was buckled against two approaching gold tips (Fig. 4a–d) and the mechanical deformation was observed and evaluated

by analysing the videos recorded during the experiment (see Supplementary Movie 4 for the _in situ_ experiment and Supplementary Movie 5 for its simulation). BUCKLING-HINGE MODEL The model

was build making the analogy with the plastic hinge formation in bent beams. In the most general expression the behaviour of a section forming a local buckling-hinge can be expressed

substituting plastic characteristic thresholds with the buckling counterparts: where _α_bh is the joint (hinge) rotation at the buckling onset, _M_u is the ultimate asymptotic moment that

the hinge is able to carry, _δ_>0 and since _M_ (_α_=_α_bh) ≡ _M_bh for continuity. Thus we can compute the evolution of the joint stiffness in the nonlinear regime as derivative of the

_M_–_α_ relationship. It follows: Note that and that for guaranteeing continuity of the curve slope _D_(_α_) at the buckling onset (_α_=_α_bh) it must hold _δγ_=1. We come then to the final

formulation of the nonlinear buckling-hinge law of equation (1). For modelling the evolution of the shape of the arm cross-section which is assumed to be elliptical with major and minor

semi-axes _a_ and _b_, respectively, the two conditions of section perimeter conservation and the relation of the buckled cross section inertia to the current value of the joint rotational

stiffness _D_ are expressed by the following equations, that must hold for each arm bending angle: where _d_ is the diameter of the tube before buckling in the section where the hinge forms.

The system of the two previous equations can be solved numerically state by state providing the evolution of the cross sectional shape after buckling (Fig. 4e). The result at buckling onset

is analogous to the one that can be derived by different method presented elsewhere35. FINITE ELEMENT SIMULATIONS FEM models of the tetrapods (results reported in Figs 2, 4 and 6) were

built associating the arms extremities and the central joint of the tetrapods to the vertexes and centroid of a regular tetrahedron, respectively. The tube walls were modelled with thin

shell elements with selective-reduced integration48, while the spurious modes effects were properly controlled. For the bending experimental set-up three arms were fully clamped at the end

accounting for the adhesion to the substrate, while both fixed and sliding boundary conditions were considered for both compression and tensile tests. The constraint is applied to a set of

nodes rather than a single point, to avoid undesired stress localization and to properly account for moments at the clamped restrains. The arm deflection simulating the AFM load is obtained

imposing the displacement at the end of the arm, in order also to maximize and quantify the contribution of the arm bending stiffness with respect to the one of the rotational stiffness of

the tetrapod central joint. In compression and tension tests the displacement is imposed at the intersection node of the arms axes (tetrahedron centroid). The total applied force, and the

bending moment at the joint are computed from resultant at the restrained node with the substrate. Self-contact is implemented to avoid walls interpenetration at the buckling/folding sites

due to large displacements and properly evaluate the post-buckling contribution. The model for the single arm buckling (second type of _in situ_ experiment) follows the same procedure, with

the arm modelled as a clamped cantilever at one of the ends (see Supplementary Movie 5) and subjected to a transversal imposed displacement at the tip simulating the action of the gold

manipulator. The critical buckling point (_M_bh, _α_bh) is determined looking at the evolution of the tetrapod deformation energy _U_, in particular it corresponds to the drop in the local

derivative of the _U_–_α_ curve42. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding authors upon request. ADDITIONAL INFORMATION HOW TO

CITE THIS ARTICLE: Meija, R. _et al_. Nanomechanics of individual aerographite tetrapods. _Nat. Commun._ 8, 14982 doi: 10.1038/ncomms14982 (2017). PUBLISHER’S NOTE: Springer Nature remains

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_Nonlinear Finite Elements For Continua And Structures_ John Wiley & Sons (2013). Download references ACKNOWLEDGEMENTS R.A., O.L. and K.S. would like to thank the German Research

Foundation (DFG) for the financial support under schemes AD 183/17-1 and SFB 986-TP-B1, respectively, and the Graphene FET Flagship. R.M. and D.E. would like to thank for financial support

from Latvian Council of Science, no. 549/2012. N.M.P. is supported by the European Research Council (ERC PoC 2015 SILKENE no. 693670) and by the European Commission H2020 under the Graphene

Flagship (WP14 ‘Polymer Composites’, no. 696656) and under the FET Proactive (‘Neurofibres’ no. 732344). S.S. acknowledges support from SILKENE. AUTHOR INFORMATION Author notes * Raimonds

Meija and Stefano Signetti: These authors contributed equally to this work AUTHORS AND AFFILIATIONS * Institute of Chemical Physics, University of Latvia, Raina bulvāris 19, Rīga, LV-1586,

Latvia Raimonds Meija & Donats Erts * Department of Civil, Laboratory of Bio-Inspired & Graphene Nanomechanics, Environmental and Mechanical Engineering, University of Trento, via

Mesiano 77, Trento, I-38123, Italy Stefano Signetti & Nicola M. Pugno * Functional Nanomaterials, Institute for Materials Science, Kiel University, Kaiserstraße 2, Kiel, D-24143, Germany

Arnim Schuchardt, Kerstin Meurisch, Daria Smazna, Oleg Lupan, Yogendra Kumar Mishra & Rainer Adelung * Institute for Polymers and Composites, Hamburg University of Technology,

Denickestraße 15, Hamburg, D-21073, Germany Matthias Mecklenburg, Karl Schulte & Bodo Fiedler * School of Engineering and Materials Science, Queen Mary University of London, Mile End

Road, London, E1 4NS, UK Nicola M. Pugno * Ket-Lab, Italian Space Agency, via del Politecnico snc, Roma, I-00133, Italy Nicola M. Pugno Authors * Raimonds Meija View author publications You

can also search for this author inPubMed Google Scholar * Stefano Signetti View author publications You can also search for this author inPubMed Google Scholar * Arnim Schuchardt View author

publications You can also search for this author inPubMed Google Scholar * Kerstin Meurisch View author publications You can also search for this author inPubMed Google Scholar * Daria

Smazna View author publications You can also search for this author inPubMed Google Scholar * Matthias Mecklenburg View author publications You can also search for this author inPubMed 

Google Scholar * Karl Schulte View author publications You can also search for this author inPubMed Google Scholar * Donats Erts View author publications You can also search for this author

inPubMed Google Scholar * Oleg Lupan View author publications You can also search for this author inPubMed Google Scholar * Bodo Fiedler View author publications You can also search for this

author inPubMed Google Scholar * Yogendra Kumar Mishra View author publications You can also search for this author inPubMed Google Scholar * Rainer Adelung View author publications You can

also search for this author inPubMed Google Scholar * Nicola M. Pugno View author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS A.S., Y.K.M., R.A.

and N.M.P. designed the study. A.S., D.S., M.M., K.S., O.L., B.F., Y.K.M. and R.A. fabricated AG samples. A.S., R.M. and D.E. performed the _in situ_ mechanical AFM/SEM experiments. A.S.,

K.M., Y.K.M., R.A., S.S. and N.M.P. analysed the _in situ_ mechanical data. S.S. performed the numerical FEM simulations and N.M.P. developed the theoretical model. Y.K.M., K.M., S.S., R.M.,

R.A. and N.M.P. finalized the study and wrote the paper. All the authors have reviewed the manuscript. CORRESPONDING AUTHORS Correspondence to Yogendra Kumar Mishra, Rainer Adelung or

Nicola M. Pugno. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no competing financial interests. SUPPLEMENTARY INFORMATION SUPPLEMENTARY INFORMATION Supplementary Figures,

Supplementary Notes and Supplementary References (PDF 719 kb) SUPPLEMENTARY MOVIE 1 First type of in situ experiment showing the formation of a buckling-hinge at the central joint of a

tetrapod with three arms fixed at a substrate and one being bent by an AFM cantilever. This video was used as basis for the curves shown in figures 2 and 3 in the main text. (AVI 20369 kb)

SUPPLEMENTARY MOVIE 2 Another example for the first type of in situ experiment executed on a second tetrapod obtained from the same sample. (AVI 33956 kb) SUPPLEMENTARY MOVIE 3 FEM

simulation corresponding to the experiment shown in Supplementary Video 1 (AVI 459 kb) SUPPLEMENTARY MOVIE 4 Second type of in situ experiment showing the formation of a buckling-hinge in a

single AG tube clamped between two gold tips. (MOV 255 kb) SUPPLEMENTARY MOVIE 5 FEM simulation corresponding to the experiment shown in Supplementary Video 4 (AVI 8189 kb) SUPPLEMENTARY

MOVIE 6 FEM simulation of tetrapod test in compression with fixed boundary conditions. (AVI 484 kb) SUPPLEMENTARY MOVIE 7 FEM simulation of tetrapod test in compression with sliding boundary

conditions. (AVI 574 kb) SUPPLEMENTARY MOVIE 8 FEM simulation of tetrapod test in tension with fixed boundary conditions. (AVI 223 kb) SUPPLEMENTARY MOVIE 9 FEM simulation of tetrapod test

in tension with sliding boundary conditions. (AVI 426 kb) RIGHTS AND PERMISSIONS This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other

third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative

Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Reprints and permissions ABOUT THIS ARTICLE CITE THIS ARTICLE Meija, R., Signetti, S., Schuchardt, A. _et al._ Nanomechanics of individual aerographite tetrapods. _Nat Commun_ 8, 14982

(2017). https://doi.org/10.1038/ncomms14982 Download citation * Received: 13 September 2016 * Accepted: 19 February 2017 * Published: 12 April 2017 * DOI: https://doi.org/10.1038/ncomms14982

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