Nanocardboard as a nanoscale analog of hollow sandwich plates

Corrugated paper cardboard provides an everyday example of a lightweight, yet rigid, sandwich structure. Here we present nanocardboard, a monolithic plate mechanical metamaterial composed of nanometer-thickness (25–400 nm) face sheets that are connected by micrometer-height tubular webbing. We fabricate nanocardboard plates of up to 1 centimeter-square size, which exhibit an enhanced bending stiffness at ultralow mass of ~1 g m−2. The nanoscale thickness allows the plates to completely recover their shape after sharp bending even when the radius of curvature is comparable to the plate height. Optimally chosen geometry enhances the bending stiffness and spring constant by more than four orders of magnitude in comparison to solid plates with the same mass, far exceeding the enhancement factors previously demonstrated at both the macroscale and nanoscale. Nanocardboard may find applications as a structural component for wings of microflyers or interstellar lightsails, scanning probe cantilevers, and other microscopic and macroscopic systems.

bending stiffness enhancement factor plotted vs. the areal mass density of a beam for three different thicknesses t. b) The same plot as a) though for the optimal spring constant enhancement factor. Note that for both a) and b), these analyzed beams were 2 mm long and 0.5 mm wide. c) Color density plot of the raw spring constant bending stiffness data from the finite element numerical simulations.

Supplementary Tables
Supplementary Table 1: The analytically calculated areal density of the tested nanocardboard plates based on the designed webbing geometry and an alumina density of 3900 kg m -3 . For the experimental samples that we weighed, the actual weight was always lower than this calculated weight by around 10%, therefore suggesting that the density of our atomic-layer-deposited alumina is lower than assumed or some of the alumina was etched during the release.
Areal Density (g m -2 ) Height (μm) 3 10 50 The wafers were then heated on a hotplate at 115 °C for 1 min to harden the resist layer.
In order to transfer the pattern in the photoresist, we performed sequential reactive ion etching processes. The webbing and outline pattern was transferred to the silica/nitride hard mask through CHF3/O2 reactive ion etching (RIE) (Oxford 80 Plus) until all of the silica was removed via inspection with an optical microscope and spectral reflectometer. An RIE of CF4 was used to remove the final oxide and etch slightly into the silicon. The pattern was then transferred into the silicon via deep reactive ion etching (SPTS DRIE) of SF6 and C4F8. The time, number of cycles, and power was calibrated such that the silicon device layer was completely etched through to the buried oxide in a vertical fashion.

Silicon Mold Removal from Handle Wafer
To remove the samples/chips from the wafer, we immersed the wafer upside down in a bath of 49% hydrofluoric acid for >1 hour to etch the oxide hard mask and buried oxide layer. After carefully rinsing with deionized water, a few of the chips self-released from the wafer, though most of the chips remained stuck. To remove the rest of the chips, we slowly inserted a razor blade between the chip and the handle wafer inside of a water bath, lifting the chip from the wafer.
Finally, all of the released chips were air dried.

Atomic Layer Deposition
The structural alumina was deposited with atomic layer deposition on individual chips (Cambridge Nanotech S200 ALD). Chips were taped to a custom glass carrier such that both the top and bottom sides were exposed for deposition. The deposition process was carried out at 250 °C with a pulse of H2O vapor for 0.015 sec, delay of 5 sec, pulse of tetramethylaluminum for 0.015 sec, and delay of 5 sec. Each cycle deposited an incomplete layer of amorphous aluminum oxide of 0.9 to 1.0 angstrom. We measured the final deposited thickness on a separate piece of prime silicon with spectral reflectometry (Filmetrics, F40 model).

Laser Machining of Cantilevers and Mounting
ALD-coated chips were taped to a flat carrier wafer for laser micromachining into individual cantilevers of 2-12 mm length and 0.5 mm width. We used an IPG IX280-DXF green laser at 50% power and 100 kHz rep rate to define the outline of the cantilevers. Between 1 and 250 repetitions of the outline was required completely etch through the chip, depending on the silicon thickness.
After etching, the cantilevers were mounted on glass slides with UV-curing epoxy (Loon Outdoors, High Viscosity). During mounting, we aimed to minimize damage to the cantilever and limit the epoxy creeping along the cantilever beyond the small section of the cantilever that contacted the glass slide.

Etching of Silicon Mold
The silicon mold was etched with XeF2 vapor (Xactics/SPTS), leaving only the external hollow nanocardboard structure. For the etching of large pieces (> 1 mm on any side), the edges of the chips were slightly fractured with a razor blade as to expose an etch surface of the inner silicon.
They were mounted on a glass slide and covered with aluminum foil as to prevent the nanocardboard device from moving during etching. Full etching required >500 cycles at 60 sec cycle time and 2 T pressure. For nanocardboard cantilevers, the mounted slides were covered with aluminum foil, though etching required fewer cycles as more surface area of the silicon was exposed and less total silicon etching was required. The etching was completed when the nanocardboard became optically translucent and the dark silicon region had clearly disappeared.

Atomic Force Microscopy Characterization
In order to characterize the spring constant of fabricated cantilevers, we made use of an atomic force microscope (AFM) (Asylum MFP-3D) at room temperature and calibrated, commercial AFM probes. In order to determine the spring constant of the AFM probes, we followed the Sader method integrated into the IGOR software (Version 6.37). For each set of data (ie., a displacement curve at a particular length along the nanocardboard cantilever), we determined where the base of the nanocardboard cantilever was through an optical microscope, and then took measurements (by measuring the reaction displacement of the AFM probe as it moved through a z-displacement of 10 μm) every 5 unit cells towards the tip by moving the stage with micrometers. AFM displacement measurements were taken at a speed of 1 or 2 sec per measurement. In order to extract the cleanest measurement, many displacements were required per location before we converged on the expected "hockey stick" shaped curve. When possible, we took measurements at more than one point along the width of the cantilever, and averaged the respective spring constant values. For each of the data points presented in Fig. 2 of the main text, only a single cantilever was tested, but Supplementary Fig. 4 presents the bending testing of many dozen different cantilevers at only the tip to show that the characteristics are consistent and reproducible.
We exported the raw z-stage position and calculated nanocardboard deflection data for each measurement with post-processing.
To calculate the spring constant, we plotted the force displacement graph (raw z-position vs. nanocardboard cantilever force as calculated by the AFM probe calibration) and chose the contact portion of the curve to be fitted with a linear line. Typically, this chosen portion was the first 300 nm of AFM head displacement, as 300 nm was the limit of our simulation capabilities for small deflection. In some cases, the noise in the data was too large to obtain representative data from only the first 300 nm. In these cases, many micrometers of displacement were chosen for the fitting lines as to average out the noise. The slope of the fitting line was the spring constant of the combined nanocardboard cantilever and AFM probe, ktotal, which was then used calculate the individual nanocardboard cantilever spring constant, kcant, as 1/ktotal = 1/kprobe + 1/kcant. The value of kcant was then used to calculate the apparent bending stiffness Dapp of that cantilever at the respective location. The apparent bending stiffness values used in this report were normalized by the width of the cantilevers, which was nominally 0.5 mm, but measured directly with optical microscopy for each sample.

Weight measurements
We weighed a selection of nanocardboard chips on a Perkin Elmer AD4/C655-0001 system.
Samples were loaded onto the scale immediately after XeF2 etching of the silicon mold. The readout resolution was 0.1 microgram, a precision of 0.2 micrograms, and accuracy of +/-6 micrograms. For comparison, our tested samples weighed roughly 50-100 micrograms.

Testing in Liquids
Few samples were introduced to liquid environments after etching of the silicon mold. Samples were inserted into puddles of pure deionized water or acetone for up to 5 min. In many cases, the samples would not sink under their own weight and required manual force in order to fully submerge. The liquid puddle was allowed to naturally evaporate under ambient conditions and then the evaporation of the liquid from the inter-face plate region was observed under optical microscopy.

Microscopy
Optical microscopy was performed with Zeiss Smartzoom5 2D/3D Optical Microscope and Zeiss

Supplementary Note 1: Face sheet wrinkling and perforation pattern
The wrinkling or buckling of face sheets in sandwich structures, a well-known occurrence in macroscopic sandwich plates, 1 can limit the predictability of the mechanical properties and the overall stiffness. For some designs of the nanocardboard, in which the webbing was a hexagonallyperiodic array of cylinders ( Supplementary Fig. 2a), we observed the wrinkling of the face sheet for thicknesses of 100 nm or less ( Supplementary Fig. 2b). The wrinkles formed over entire cantilevers because the webbing cylinders were disconnected (not forming a continuous array like in honeycomb cores), and furthermore were widely spaced. Supplementary Figure 2c provides a diagram showing how a straight line can be drawn through the face sheet without intersecting any webbing cylinders. The face sheets can therefore buckle or wrinkle along these straight lines.
Accordingly, the wrinkling was experimentally observed along the 0°, 60° and 120° directions of the hexagonal lattice ( Supplementary Fig. 2d). During mechanical characterization with an atomic force microscope probe, we observed irreproducible stress-strain curves which were highly nonlinear. The wrinkles moved and reoriented during each test, resulting in inconsistent mechanical responses.
In order to satisfy the "no-straight-line rule" and prevent wrinkling, we replaced the hexagonal cylinder motif with a simple basketweave motif for all subsequent experimental samples and numerical simulations. Interestingly, in our previous corrugated plates, 2 we used a similar nostraight-line rule to increase the stiffness of corrugated single-layer plates. However, in nanocardboard, the no-straight-line rule does not necessarily increase the bending stiffness.
Instead, its main purpose is to prevent the formation of wrinkles along straight lines that pass between the webbing cylinders and therefore produce plates with consistent and predictable mechanical properties. Supplementary Figure 2e shows how a straight line is not able to be drawn through a basketweave face sheet without being interrupted by the webbing rectangles, thus the design eliminated wrinkles.
There are other possible webbing patterns that one could formulate for the nanocardboard structure. The general requirements for the pattern include satisfying the no-straight-line rule, ensuring that the face sheet is continuous (or the perforations must not connect with one another), and the perforations must be as small as possible (generally ensuring high stiffness of the face sheet). Recently, researchers have reported on the mechanical properties of thins sheets with high aspect-ratio periodic perforations, specifically investigating conditions with negative Poisson's ratio. The square basketweave pattern and the hexagonal kagome pattern are the two most studied patterns for their nearly-isotropic controllable elastic modulus and Poisson's ratio. [3][4][5][6] Supplementary Table 2

Supplementary Note 2: Supplement bending stiffness data and discussion
While we successfully fabricated nanocardboard cantilevers and larger pieces with an alumina thickness of 25 nm, the bending stiffness for this thickness was not fully characterized due to lower repeatability and increased measurement noise; however, a few data points are provided in Supplementary Fig. 4 for reference. Samples with thicknesses of roughly 50, 100 and 400 nm and heights of roughly 3, 10 and 50 μm were fully characterized with AFM bending measurements.
In addition to the bending stiffness (Dxx) calculated from the bending tests performed along the length of cantilevers, we also measured the apparent bending stiffness of many samples solely at the tips of cantilevers. While this data was not used to calculate the reported Dxx and G, it does follow similar trends to the "along the length" data shown in Fig  Lastly, nanocardboard cantilevers with low ℎ/ , or low , had relatively fewer unsaturated data points available for accurate fitting. As for (Fig. 2d in the main text), the simulation fittingdetermined data points match the theoretical trends within 40% relative error except for: the nanocardboard cantilevers with 400 nm thickness, again where the fitting was error-prone due to the relatively few data points in the low , shear-dominated region. The experimental fittingdetermined data points only match (relative error < 32%) with our theoretical model for cantilevers with large and high (ℎ = 50 , ℎ/ ≥ 500). We attribute this discrepancy to insufficiently stiff clamping of the cantilevers at the base since the discrepancy was observed for the stiffest and shortest cantilevers and inadequate clamping is known to reduce the measured spring constant for short beams. For the numerical simulation data presented in Figs. 2b-d of the main text, the cantilever was displaced with a given load of 1 × 10 -12 N at different lengths, as described in the previous paragraph. The displacement of the cantilever was measured to extract the spring constant, and ultimately calculate the apparent bending stiffness for each particular condition. As was described in Supplementary Note 2, the curve of apparent bending stiffness vs. length was plotted and fitted.
From the fitted function, we extracted the simulated true bending stiffness (N m) and shear modulus (N m -2 ). For many geometric conditions, we also verified the fitted bending stiffness by applying a pure bending moment in place of an out-of-plane force load, and measuring the resulting curvature of the deformed cantilever. Comparable results were obtained between the force-and moment-induced bending stiffnesses.
Supplementary Figure 7d presents the effect of the pattern angle, i.e., the angle between the repeated unit and nanocardboard length, on the deviation of simulated bending stiffness. The pattern angle was varied from 0° to 90°, and three thicknesses were investigated (100, 300, and 400 nm). Symmetric variations are obtained from all three cases. In particular, the minimum bending stiffness occurs when the pattern angle is 30° and 60°, with a relatively stiffening at 45°.
The deviation of up to 15% was within the range of our experiment error, and therefore we did not attempt to validate this angle-dependent trend with fabricated samples. Further investigation, both experimental and theoretical, will better elucidate the anisotropic nature of the bending stiffness for the basketweave and other webbing patterns.

Supplementary Note 4: Sharp bending of nanocardboard cantilevers
The large-deformation bending of the nanocardboard cantilevers was carried out for select samples in order to determine the minimum radius of curvature that could be attained without apparent fracture or plastic deformation. As has been shown with other bulk mechanical metamaterials, the use of elements with nanoscale thickness can allow for much larger failure-free deformations than would be expected from similar macroscale materials. 7,8 Similarly, our nanocardboard plates were able to recover to their original position after being bent to have a radius of curvature < 100 μm, as show in Fig. 1h-k  The finite element models of sharply bent cantilever used the same two steps as the experiments: 1) bending to introduce buckling imperfection, and 2) compression. To fully investigate the large deformation effects, the geometric nonlinearity was taken into account. In order to ensure convergence of the model, displacements were placed at the free end of the cantilever. In particular, a displacement in the transverse direction of half of the cantilever width was added in the bending step, and a displacement in the longitudinal direction of the cantilever length was placed for the compression step. Supplementary Figures 8b, 8d, and 8g  Comparing with the perforated structure, the large deformation-induced absolute strains of the continuous structure are approximately 5~10 times higher, which results in permanent damages to the face sheet material. In contrast, the FE model studies suggest that the maximum local strain for the nanocardboard structure with the perforated face sheets was close to 1%, hence free from material strain failure as was the case in the experimental bending.

Supplementary Note 5: Theoretical modeling and geometric optimization
Theoretical scaling laws for the deformation characteristics of nanocardboard We use the theory of an elastic double-stranded rod 9 to explore the effect of dimensions and elastic properties of the webbing on the macroscopic deformation of nanocardboard through a model biplate.

Sandwich Structure
In our experiments the nanocardboard consists of two parallel alumina plates that were connected to each other through a webbing of hollow rectangular columns. This geometry is akin to sandwich panel plates, except our plates were made of a brittle material and their dimensions in the nanometer to micrometer range. Sandwich panels have been extensively studied in the literature, both analytically 10 and through finite element simulations. 11,12 Finite element simulations of our nanocardboard are described in Supplementary Note 3; here we will focus on our analytical model.
For the purposes of deriving scaling laws we will treat our two-dimensional nanocardboard (or biplates) as a one-dimensional bi-rod since they are loaded like a cantilever. Also, we are not aware of a theory of bi-plates, while there exists a well-established continuum theory for bi-rods (including large deformations) 9 which can account for complex geometries of the web. In contrast, analytical models for sandwich plates treat the webbing as a homogeneous elastic continuum even though in practical applications the web could be granular, like a foam. 13,14 Our idealization of the bi-plate consists of two thin outer face sheets connected by a web as in Supplementary Fig. 9. The web itself consists of two parallel sheets of thickness t (denoted as tcant in the following section) with separation h (denoted as hcant in the following section) since the web in our nanocardboard is hollow. We need to estimate the transverse shear stiffness and flexural stiffness of the structure and examine the effect of parameters t, h and s (the spacing between webs) on it. This exercise has been performed for corrugated plates with a Z-core and for foam filled sandwich plates that are of central importance from the standpoint of structural engineering.
Various approaches based upon the arguments from the strength of materials have been used to obtain these estimates. 10,11,13,14 In such structures, the core and the outer face sheets consist of the same thickness and the bending stiffness of both the members scale as ∼ t 3 . However, in our biplate t << d<< h ≈ s, as such the bending stiffness of the core ∼ th 2 and that of outer plates ∼ t 3 .
The fact that s >> d >> t and a granular web (with zero core stiffness at some places and finite at others) imparts a distinctive staircase structure to the resultant displacement profiles as shown in Fig. 2a. A homogenized modulus for the core cannot capture this. Moreover, for plane strain loading we can describe the resultant behavior using the theory of a double-stranded elastic rod. 9 The principal advantage of this theory is that we can account for the geometry of the hollow web including its granularity exactly.

Equivalence between a bi-rod and a bi-plate
Our primary concern in this section is to show that we can apply a one-dimensional (beam) theory to study the deformation of plates away from the free edges if the dimension along the transverse direction W (denoted as Wcant in later sections) is much larger than the thickness of the plate t. We follow ref 15 . The axial stress σzz in the bulk of a beam or a plate is directly proportional to the distance from the neutral axis or neutral plane, so that In the case of a beam, W ∼ t and φ = 1, while for a plate W >> t and φ = 1/(1−ν 2 ) away from the edges 15 where ν is the Poisson ratio of the material. We show that in the case of a cantilever plate we recover the solutions for an Euler beam with E replaced by E/(1 − ν 2 ). Consider a cantilever plate in the bottom of Supplementary Fig. 8 which is bent by the force pxe1 -pze3 and moment mye2 at the edge z = L. Let the displacement of the neutral plane be w(y, z). Furthermore, we assume that the plate is inextensible. For small deformations |w| << 1, the elastic energy can be obtained using the Kirchoff's theory of plates.
The work done by the external forces and moments at z = L is given by If the transverse dimension of the plate W >> t, then away from edges w(z, y) = w(z). Due to anticlastic curvature, this assumption breaks down close to the edges. We substitute this into The potential energy is the difference of the elastic energy and external work done, hence Setting δPE = 0, we obtain and = , = , at = .
Another set of boundary conditions is w = wz = 0 at z = 0. Thus, we recover the governing equation for the bending of an Euler-Bernoulli beam with E replaced by E/(1−ν 2 ) which is valid at constant loading sufficiently far from the free edges. Since the bi-plate shown in Supplementary Fig. 8 is an assembly of three such beams with W >> t, and the load is constant in the transverse direction in our experiments, the equations of a bi-rod should accurately describe the mechanics of a biplate away from the free edges.

Assumptions of the bi-rod model
The geometry of the problem is presented in Supplementary Fig. 8. The bi-rod consists of:  Outer Strands: The outer strands denoted by ± are elastic rods capable of undergoing bending, extension and shear deformation. Any quantity, say a, pertaining to ± strands is denoted by a ± . For instance, displacement of the upper strand in the direction of e 1 is u .
The two strands have identical elastic properties.
 Inner Web: The outer strands are connected to each other via an elastic web which is capable of transferring both forces and moments. The web is capable of undergoing extension, shear and bending.

Kinematics
We consider a general planar deformation in e1, e2 plane and specialize the equations of Moakher and Maddocks. 9 For the ± strand the kinematic variables u ± and u ± denote the displacements in e1 and e3 directions, respectively and θ ± denotes the orientation of the cross-section.
r + and r − are position vectors of a point located at z.

Governing Equations
We assume n + and n − are the internal force vectors in the outer strands and m + and m − are the internal moments in the outer strands, then − = 0 (9a) Here f and c are the force and moment exerted by + strand on − strand. These are related to the deformations of the web via elastic constitutive laws.

Boundary Value Problem
We substitute the elastic constitutive relations for the outer strands and the web in the governing equations to get:  For the + rod: and Boundary conditions are: Adding and subtracting the governing equations given above and setting Boundary conditions can be similarly obtained.
The governing Supplementary Equations 10 and boundary conditions Supplementary Equations 11 are solved using MATLAB to get the transverse displacement u1 (z = L) = δ (say) at z = L which is then used to calculate the effective bending stiffness using = . (20)

Results
In order to respect the granular nature of the web, we assume that the elasticity of the web is a function of the arclength parameter.
Here f(z) is a modulating function with period s. For convenience, we choose We then solve the system of ordinary differential equations of the previous section to find the deformation of the bi-rod in response to loading of F = 5 µN as in a cantilever of W = 1 m. The resultant displacement profiles exhibit a distinctive staircase character as shown in Fig. 2a.
The staircase-like displacement profile of the end-loaded cantilever is a result of the granularity of the web which a sandwich plate model with a homogeneous web cannot capture. From our experiments and finite element calculations we know that the deformation of our nanocardboard could be shear-dominated or bending-dominated depending on the geometry of the specimens. For a Timoshenko beam, the displacement at the end z = L contains contributions both from shear and bending, so that In the above, EI is the bending stiffness of the Timoshenko beam and GA is its shear stiffness. For such a system, we say that the effective bending stiffness is Kb which is calculated as follows: As the length of the plate keeps increasing the effective bending stiffness increases and eventually saturates. L90 is the length of the plate at which Kb = 0.9EI.
In order to calculate L90 for a bi-plate, we need the effective bending stiffness EIeff and effective shear stiffness GAeff which replace EI and GA, respectively in the above equation. We can plot u(z = L) vs. length of rod L and fit a cubic polynomial through the points and extract the effective bending stiffness EIeff and effect shear stiffness GAeff from the fit.
From the fit, we find that a2 ≈ a0 ≈ 0. Note that a3 and a1 are proportional to P which makes EIeff and GAeff independent of P. In case of a homogeneous web, we compared our result for L90 using the above methodology with the numerical result obtained by integrating the ordinary differential equations with various lengths L of the bi-rod, computing their Kb using Supplementary Equation 24, and then getting L90 from a plot of Kb vs. L. We found that L90 computed using both methods agreed very well. Thus, we apply the method based on fitting with a cubic polynomial to compute L90 for a bi-rod with a granular web.
Variation with the thickness t, height h, and period s We obtained the results in Supplementary Fig. 10  With the above information taken into account, we found the following scaling relationships: ~ℎ , ~ℎ , ~ . These scaling relationships are graphed and represented in Fig. 2 of the main text. Note also that ∝ .

Optimal Design of the Cantilever Problem Statement
For cantilevers and plate structures, designing for the minimum deflection or maximum resistance to deflection is often desirable in a wide variety of situations, including transportation, construction materials, and biological skeletons or shells. Therefore, the optimization in this section seeks to maximize the spring constant of the cantilever for any chosen areal density AD . Since the shear component of deflection tends to dominate for cantilevers on the order of a few millimeters, the optimization is not as simple as maximizing the bending stiffness . In the following, we analytically design the stiffest and lightest nanocardboard cantilever considering both bending and shear deformations. The optimization focuses on only the basketweave patterns that can be reliably fabricated and do not exhibit spontaneous wrinkling, but the derivation also provides insight into the maximum achievable enhancement factor of the structure and how the optimal designs scale for different geometric or density constraints.
Two design variables of the presented cantilever are particularly investigated, i.e., the cantilever height ℎ , and the rectangle length of the basketweave pattern. Increasing the bending stiffness requires increasing the cantilever height, which also increases the mass of the cantilever.
Increasing the shear stiffness, however, requires decreasing the rectangle length of the basketweave pattern, which also increases the mass of the cantilever. Therefore, for a fixed mass or areal density, there is a tradeoff between increasing the height to reduce the bending displacement and decreasing the rectangle length to reduce the shear displacement.
Mathematically, the optimization can be described as where is an arbitrary value of the areal density of the cantilever.
The Lagrange multiplier method is used to optimize the cantilever. The Lagrange expression is defined as where represents the Lagrange multiplier. Therefore, the optimization in Supplementary   Equation 30 is rewritten as To derive the optimal ℎ and , we need to use the expressions for the spring constant and areal density of the cantilever, described in the following sections.

Spring Constant
Following Timoshenko beam theory, the deflection of the cantilever consists of two components, i.e., bending-induced and shear-induced deflections, which are given as and the total cantilever deflection is The spring constant of the cantilever is, therefore, given by: where E is the Young's Modulus of the solid material and the shear modulus is where , , and refer to the shear modulus constant of the plate metamaterial (obtained empirically by fitting experimental and finite element results), webbing rectangle length and webbing rectangle width of the basketweave, respectively. To simplify the optimization, we will sometimes assume below that the rectangle width is negligible ( = 0) while the gap is half of the rectangle length ( = ). These are reasonable approximations of the actual nanocardboard webbing design implemented in experiments and numerical simulations, which used finite but small rectangle width. We note that these idealized parameters still satisfy the no- 38) The areal density AD of the cantilever is given as where is the density of the cantilever material. α and β, the unit areal density constants for the face sheets and core of the cantilever, respectively, are determined by the basketweave design pattern, as shown in Supplementary Fig. 1 (shaded in blue), where = 2 = . (42) Optimization of the Enhancement Factor for the Bending Stiffness, We define the enhancement factor of the bending stiffness as for any pair of a nanocardboard cantilever and a solid rectangular beam that have the same length, width, mass, and, therefore, the same areal density, AD = AD . EF is similar to the bending shape factor Φ used to described the enhanced bending stiffness incurred with reforming a solid cylindrical beam into a different shape with the same cross-sectional area. 16 where for simplicity we neglected the Poisson ratio correction factor of up to for wide beams.
According to the numerical calibration, we find that the bending stiffness of the nanocardboard cantilever is 30% of the ideal sandwich beam, which can be written as where the bending stiffness of the ideal sandwich beam is given by = ℎ .
Therefore, the bending stiffness of the nanocardboard cantilever is Clearly, if the aim is to maximize the bending stiffness alone, both the plate height and the rectangle length should be made as large as possible. If, however, the rectangle length is fixed due to some practical considerations, the enhancement factor can be maximized with respect to the cantilever height, yielding which predicts an enhancement factor of approximately 33 333 for the rectangle length of 50 μm and thickness of 50 nm. An example plot of EF vs. AD is given in Supplementary Fig. 11a for a cantilever length L of 2 mm. Note that the maximum EF occurs at the peak of each thickness curve.
For the actual basketweave rectangle width and gap used in experiments, the absolute maximum enhancement factor is somewhat smaller as illustrated by Fig. 3b in the main text.
In summary, the bending stiffness can in principle be increased indefinitely by increasing both the height and rectangle length. However, the resulting cantilever will become extremely soft with respect to shear displacements. Therefore, the optimization of the bending stiffness alone makes sense only in cases where the cantilever is very long, and therefore bending dominated, and the scale of the basketweave pattern is fixed (note that the period of the basketweave pattern is equal to twice the rectangle length).

Enhancement Factor for the Spring Constant
Similar to Supplementary Equation 43, we define the enhancement factor of spring constant as where represents the spring constant of the nanocardboard cantilever given in Supplementary   Equation 38, refers to the spring constant of a solid beam that has the same optimal areal density as the cantilever, AD = AD . Given that AD = ℎ , the height of the solid beam with equivalent mass is To compare the spring constant between the cantilever and solid beam, we assume the structures have equivalent material density . Since the equivalent solid beam will be much longer than it is thick, the shear deflection can be neglected and the deflection of the solid beam subjected to tip load F is then where = , = and = . Therefore, the spring constant of the equivalent solid beam is In order to obtain the optimal enhancement factor EF , , the spring constant of the cantilever is maximized with respect to the cantilever height ℎ and rectangle length . In Therefore, the ratio of the face sheets areal density to the core areal density for the optimal cantilever is = 2 .
The optimal areal density ratio in Supplementary Equation 63 is identical to the optimal weight ratio of the face weight to the core weight for the optimal web-core sandwich structures. 17 This is because the rectangle width of the cantilever is assumed zero, i.e., = 0, which results in the face areal density equivalent to the areal density of the web-core sandwich structures. However, the bending stiffness of the cantilever is still 30% of the ideal sandwich beam (as discussed in Supplementary Equation 47), given the cuts of the basketweave pattern on the face sheets.
The maximum enhancement factor of the spring constant is