Precision Stress Engineering in Tensegrity‐Inspired Nanoarchitectures Enabled by Size‐Affected Shrinkage

ABSTRACT

Residual stress networks offer a powerful means to enhance mechanical properties, but controlling them at the nanoscale remains challenging. Here, we introduce a method to create prestressed tensegrity‐inspired nanoarchitectures, i.e., nano‐tensegrities, by exploiting a previously uncharacterized size‐affected shrinkage phenomenon. We discover that the shrinkage of acrylate‐based polymers during pyrolysis has a power‐law dependence on size. This size‐effect arises due to increased residual oxygen‐containing groups in larger‐dimension specimens. Leveraging this effect, we use two‐photon lithography to fabricate polymer structures with thicker “bar” and thinner “tendon” members and pyrolyze them to create prestressed glassy carbon nano‐tensegrities. Using combined experiments and numerical modeling, we demonstrate pyrolyzed structures retain their designed state of prestress, which can then be precisely controlled by tuning the bar‐to‐tendon diameter ratio. Prestress is shown to considerably enhance stiffness ‐ up to a two‐and‐a‐half‐fold increase in the structures studied here ‐ but can lead to buckling in excessively stressed slender members. We evaluate the effect of architecture and slenderness on the limits of prestressability and analyze corresponding changes in mechanical performance. This work establishes a method to precisely program 3D residual stress into metamaterials at the nanoscale, enabling a new class of mechanically tunable nanoarchitectures.

This work identifies a size‐dependent shrinkage effect that arises during polymer pyrolysis and uses it to embed residual stress into tensegrity‐inspired nanoarchitectures. By adjusting the dimensions of the structural elements, controlled prestress can be introduced in a simple and repeatable manner, enabling mechanically tunable nanostructures with potential applications in energy storage and MEMS.

1. Main

Stress engineering at both the structural and material level offers a powerful means to tailor mechanical performance. This principle underlies prestressed concrete, where tension in rebar enables stable crack growth in an otherwise brittle material [1, 2]. Residual stresses are commonly used to promote resilience in tempered glass [3, 4] and laser shock peened metals [5], where their compressive surface stress resists cracking. In biology, the musculoskeletal system acts as a tension–compression network, with bones supporting compressive loads created by muscle tension [6, 7]. Similar tensegrity concepts occur in cytoskeletal networks [8], spider silk [9], and avian lung vasculature [10]. Despite their potential benefit, uncontrolled microscale residual stresses can lead to distortion, cracking, and reduced fatigue life in additively manufactured parts [11, 12], composites [13, 14] and metals alike [15]. Many methods exist to quantify residual stress [16, 17], but efficiently controlling its localization remains elusive.

Tensegrities, or tension‐integrity systems, are structures that can be prestressed without altering their shape. They consist of isolated tensile and compressive elements (tendons and bars) and feature at least one inextensional mechanism and one state of self‐stress [18]. Prestress within this self‐stress state resists mechanism motion, allowing stiffness to be tuned by adjusting prestress – similar to tightening the strings on a guitar [19, 20, 21]. Beyond tunable stiffness, tensegrities exhibit exceptional strength‐to‐weight ratios [22, 23], delocalized deformation [24, 25], metamaterial chirality [26, 27], and controllable wave propagation [28, 29]. While tensegrity‐inspired metamaterials have been designed and fabricated at macro‐ and microscales using additive manufacturing [25, 30, 31, 32], introducing prestress to tune their properties has so far required manual intervention.

We present a new class of glassy carbon nano‐tensegrities with controllable prestress enabled by a previously uncharacterized phenomenon of size‐affected shrinkage during pyrolysis. Pyrolyzing polymeric nanostructures at high temperature ($\ge$900) in an inert gas or vacuum produces ultrahigh strength amorphous carbon [33, 34, 35, 36, 37]. We show that axial shrinkage of polymer beams follows a power‐law scaling with diameter, with smaller beams shrinking more. While geometry‐dependent shrinkage has been observed in lithographically fabricated structures [38, 39, 40, 41, 42], it has not been characterized as a structural size effect. Using two‐photon polymerization, we fabricate nano‐tensegrity networks with beams of varying dimensions. During pyrolysis, differential shrinkage introduces tension into thinner tendons and compression into thicker bars while preserving overall geometry. This prestress can enhance stiffness and resilience or, if excessive, cause bar buckling and/or tendon fracture. We establish design principles governing this behavior, including architecture, bar‐to‐tendon ratios, and unit cell size. This precise control over internal stresses establishes the technique as a powerful new tool for applications in energy storage, tissue scaffolds, and C‐MEMS. We note here that our materials are ‘tensegrity‐inspired’ as they are not idealized pin‐jointed networks but they do still follow tensegrity design principles for self‐stable, prestressable network topologies, though material stiffness does contribute to their stability.

1.1. Size‐Affected Shrinkage During Pyrolysis

We investigated the size‐dependent shrinkage during pyrolysis of an acrylate‐based photoresist (IP‐Dip, Nanoscribe GmbH). Arrays of cylindrical pillars were written using two‐photon polymerization (TPP) with initial diameters ($D_i$) ranging from 1 to 15$\mu \mathrm{m}$. Parameter studies were performed to investigate the effect of length‐to‐diameter aspect ratios, prebaking, polymer degree of conversion (DC), and heating rates, further details of which can be found in the Supplementary Information.

Initial “as‐written” and final “post‐pyrolysis” pillar lengths, $L_i$ and $L_f$ respectively, were measured using a scanning electron microscope (SEM). The axial shrinkage resulting from the pyrolysis process, calculated as ${\epsilon }_{ax}=\frac{(L_i-L_f)}{L_i}$, was found to have a power law dependence on the initial beam diameter as,

$${\epsilon }_{ax}=\beta D_i^{\alpha }$$ (1)

where $\alpha$ is a size‐affected shrinkage coefficient and $\beta$ is a constant representing the maximum effective shrinkage.

The results from representative experiments are shown in Figure 1B. The shrinkage coefficient, $\alpha =-0.040\pm 0.005$, was consistent across all batches tested and did not show any dependence on ramp rate or prebaking time. The effective shrinkage constant showed some batch‐to‐batch variation and had a moderate dependence on the prebaking step, where samples without prebaking had a $\beta =84.8\pm 0.9$ and samples with a one hour prebaking at 200 had a $\beta =88.5\pm 1.0$. The increased shrinkage observed in pre‐baked specimens is attributed to additional molecular rearrangement and partial conversion occurring prior to the main degradation window, leading to increased volatile release and densification during subsequent heating. Although prebaking was performed only on pillars with diameters less than 7 $\mu \mathrm{m}$, similar trends of enhanced shrinkage with pre‐baking are expected to persist across length scales, consistent with prior studies [43, 44].

Figure 1.

Size‐affected shrinkage and nano‐tensegrity design. (A) Pillar arrays before and after pyrolysis. (B) Semilog plot of axial shrinkage versus initial pillar diameter for varying pyrolysis conditions. (C) Illustration of differential axial‐shrinkage (${\epsilon }_{ax}$) in pillars. (D) Illustration of size‐affected shrinkage in beams in a four‐bar tensegrity. (E–J) SEM images and CAD models of individual unit cells: (E,H) icosahedron, (F,I) twisted truncated octahedron, and (G,J) four‐bar. (K,L) SEM images displaying tensegrity towers obtained by tessellating (K) twisted truncated octahedron, and (L) four‐bar unit cells. Scale bars are 50$\mu \mathrm{m}$ in A, 5$\mu \mathrm{m}$ in E–G, and 2$\mu \mathrm{m}$ in K and L.

These results are markedly different from similar literature on the shrinkage during carbonization of SU‐8, which showed minimal dependence on feature size [38, 39]. The power law dependence of shrinkage on feature size in this work was clearly observed across all experiments, and no parameter had as strong or consistent an effect on the axial shrinkage as the diameter. Interestingly, the size‐affected shrinkage phenomenon observed here is not restricted to TPP architectures. As demonstrated in Figure S1, this effect was observed at the macroscale in 3D printed materials made using DLP, which exhibit the same power‐law dependence of shrinkage on feature diameter, details of which are included in the Supporting Information.

We attribute the mechanism of this size‐affected shrinkage to the incomplete removal of oxygen‐containing species in larger diameter species. This phenomenon of incomplete oxygen removal has been described in prior studies [43], albeit without any reported size‐affected change in shrinkage. In this work, energy‐dispersive X‐ray spectroscopy (EDS) performed on pyrolyzed DLP specimens showed a clear increase in the residual oxygen, 4.0% residual ${\mathrm{O}}_2$ in a 0.6 mm diameter bar versus 8.8% in a 1 mm diameter bar. Raman spectroscopy performed on these specimens showed no appreciable shift in peak intensity, indicating no change in the molecular structure or graphitization of the amorphous carbon. Additional SEM analysis of broken beams in the post‐test specimens showed no porosity, eliminating density difference as a potential source of size‐affected shrinkage. These results are presented in S2 and examined in more detail in the Supporting Information.

We observed no correlation between the shrinkage and the pillar aspect ratio. While there has been a demonstrated dependence of shrinkage on aspect ratio in literature [39, 45, 46], the beams studied here fell outside any aspect ratio‐dependent shrinkage regimes. There was also no dependence of TPP pillar shrinkage on ramp rates below 20/min, but there was a notable distortion and bubbling of written structures at ramp rates $\ge$35/min, likely due to the inability of generated gases to escape at free surfaces [47]. Finally, while we observed a shrinkage of the as‐written polymer with changing DC – as has been well documented in literature [48] – there was no correlation between ${\epsilon }_{ax}$ and DC, though we acknowledge that features written with a lower DC would be smaller relative to the “as‐designed” dimensions due to initial shrinkage of the polymer.

One notable finding was a size‐affected change in the shrinkage rate. While the exact shrinkage rates of individual members are difficult to quantify experimentally, it was observed at low ramp rates that slender features shrank and viscoplastically stretched, causing them to unpredictably buckle as larger features more slowly shrank (Figure S3). This is examined in more detail in the Supporting Information. To accommodate this size‐dependent shrinkage rate effect, a fast ramping rate of 20/min was used for all nano‐tensegrities in this study.

1.2. Controlled Prestress in Nano‐Tensegrities

Three different tensegrity unit cells were created using TPP to investigate the utility of prestress for augmenting mechanical performance: an icosahedron [49], a twisted truncated octahedron [24], and a stacked four‐bar simplex [50] (Figure 1E–J). The unit cells were designed with smaller diameter tendons ($D_t$) connected to larger diameter beams ($D_b$), to promote tensile and compressive loading in each respectively. The $D_b/D_t$ ratios were kept between 1 and 2.25 with the as‐printed bar diameter fixed at 4.5$\mu \mathrm{m}$. Additional fabrication details can be found in the Supporting Information.

The four‐bar, twisted truncated octahedron and icosahedron unit cells were written with heights of 60$\mu \mathrm{m}$, 45$\mu \mathrm{m}$, and 30$\mu \mathrm{m}$, respectively, while an additional icosahedron unit cells with more slender bars was written with a height of 45$\mu \mathrm{m}$. The resulting shrinkage of the unit cells was found to be $\sim$81%, consistent with the pillar shrinkage, producing nano‐tensegrities that had a height of $\sim 12\mu \mathrm{m}$, $\sim 9\mu \mathrm{m}$ and $\sim 6\mu \mathrm{m}$, respectively. Note that the twisted truncated octahedron and icosahedron constitute class‐1 tensegrities while the four‐bar is a class‐2 tensegrity, where a class‐k tensegrity has a maximum of “k” bars intersecting at any given node.

Due to the size‐dependent axial shrinkage, prestress can be predictably controlled by changing the diameter ratio between the constituent bars and tendons (Figure 1C,D). The difference in axial strain between the bars and tendons is calculated to be:

$$\mathrm{\Delta }\epsilon ={\epsilon }_t-{\epsilon }_b=\beta D_t^m\left(1-{\left(\frac{D_b}{D_t}\right)}^m\right)$$ (2)

where ${\epsilon }_t$ and ${\epsilon }_b$ are the unconstrained tendon and bar shrinkage, respectively.

If the tendons and bars remain connected, this differential strain will produce stress as the structure comes into a new stable equilibrium. The actual stress values will depend on the tensegrity architecture, particularly the nodal connectivity and angles between bars and tendons. We develop an approximation here for a single bar and tendon connected in‐line. The equilibrium strain (${\epsilon }_c$) in this one‐dimensional analysis is found by taking a volume weighted average of strains to be:

$${\epsilon }_c=\frac{D_t^2{\epsilon }_t+D_b^2{\epsilon }_b}{D_t^2+D_b^2}={\epsilon }_b+\mathrm{\Delta }\epsilon \left(\frac{D_t^2}{D_t^2+D_b^2}\right)={\epsilon }_t-\mathrm{\Delta }\epsilon \left(\frac{D_b^2}{D_t^2+D_b^2}\right)$$ (3)

The final stress in the bars and tendons then depends on the difference between the free and confined strain as ${\sigma }_t$ = $E({\epsilon }_t$ ‐ ${\epsilon }_c$) and ${\sigma }_b$ = $E({\epsilon }_b$ ‐ ${\epsilon }_c$), where $E$ is the Young's modulus of the material. Taking these with Equation (3), we obtain

$$\begin{array}{c}\hfill {\sigma }_t=E\mathrm{\Delta }\epsilon \left(\frac{D_b^2}{D_t^2+D_b^2}\right)\end{array}$$ (4)

$$\begin{array}{c}\hfill {\sigma }_b=-E\mathrm{\Delta }\epsilon \left(\frac{D_t^2}{D_t^2+D_b^2}\right)\end{array}$$ (5)

While these are oversimplified models of the bar and tendon stresses, they provide a useful baseline for estimating the effect of differential shrinkage on prestress. Architecture influences mechanical response, but the magnitude of prestress is primarily determined by differential shrinkage between members. As the bar‐to‐tendon diameter ratio approaches $D_b/D_t$ = 1, there will be no difference in shrinkage, i.e., $\mathrm{\Delta }\epsilon$ = 0, and therefore no prestress. As $D_b$ $\gg$ $D_t$, the tendon stress will monotonically increase but the bar stress will decrease as the tendons become too thin to effectively apply load. Note that there is a practical limit here where the tendons will yield or break during pyrolysis due to high stresses, but this point is difficult to estimate due to the dynamically evolving yield behavior during pyrolysis.

We note that achieving perfectly uniform shrinkage in high–aspect ratio, free‐standing structures during pyrolysis remains challenging. Shrinkage‐induced stresses can become spatially non‐uniform due to slight asymmetries in feature diameter, local material conversion, or surface roughness introduced during fabrication, which may be amplified during thermal treatment and result in macroscopic deviations such as tilting or bending. This effect is most pronounced in tall, slender structures, like in the case of the towers shown in Figure 1K,L. The applied thermal protocol was originally optimized for single unit cells, and extending it to taller, tessellated architectures likely contributes to asymmetric stress evolution. Importantly, these distortions do not alter size‐affected shrinkage but rather highlight practical fabrication limits that must be mitigated through writing uniformity, reduced aspect ratio, or geometry‐specific thermal protocols.

Finite element (FE) models for each unit cell were developed using Abaqus (Dassault Systèmes) to more accurately predict the effective stress in the members and the corresponding structural response to applied loads, details of which are provided in the Supporting Information. Since direct experimental measurement of residual stress in structures at this scale is not feasible, the FE models also serve as the primary means of estimating internal stress states resulting from size‐affected shrinkage. A thermal expansion field with a strain $\mathrm{\Delta }\epsilon$ was applied to the tendons to mimic the effect of the differential shrinkage. The resulting minimum bar principal stress and maximum tendon principal stress (${\sigma }_p$) along with the average in each member are shown in Figure 2 as a function of the bar‐tendon diameter ratio.

Figure 2.

Shrinkage‐induced residual stresses in nano‐tensegrities. (A–C) Plots show average prestress vs bar‐tendon diameter ratio with insets of FE‐derived stress contours displaying minimum principal stress in bars and maximum principal stress in tendons (${\sigma }_p$) in the (A) twisted truncated octahedron, (B) four‐bar, and (C) icosahedron (9$\mu \mathrm{m}$ and 6$\mu \mathrm{m}$ unit cells). (D) Plot of compressive stress in bars vs bar‐tendon diameter ratio ($D_b/D_t$) in the 9$\mu \mathrm{m}$ icosahedron unit cell; Insets show SEM images of the unit cell demonstrating buckling in bars as prestress increases (Scale bars are 2$\mu \mathrm{m}$).

The residual stresses in the FE model show a reasonable agreement with the 1D analytical model both quantitatively and qualitatively. The tendons in all architectures show a monotonic increase in stress with increasing $D_b/D_t$ ratios, while the bar stress begins to plateau in all architectures around a ratio of $D_b/D_t$ $\approx$ 2. In the icosahedron and twisted truncated octahedron, all the tendons are identical and experience the same stress, while the changing tendon lengths and orientations in the four‐bar result in three distinct tendon stress values.

Notably, the average stress in the bars and tendons is found to be the same irrespective of unit cell size (Figure 2C). From this, we can infer that although the prestress magnitude is independent of unit cell size, larger unit cells will be more prone to buckling. An estimate of the critical buckling stress (${\sigma }_{cr}$) can be found using a classical Euler beam buckling model as:

$${\sigma }_{cr}=\frac{{\pi }^{2}EI}{A{(kL)}^2}=\frac{{\pi }^{2}E}{k^2{\lambda }^2}$$ (6)

where $E$ is a Young's modulus, $I$ is a second moment of area, $A$ is a cross‐sectional area, $L$ is the bar length, and $\lambda =\sqrt{A{L}^2/I}$ is the bar slenderness.

By approximating the tensegrity members as pin‐jointed beams with $k=1$, we can obtain predicted buckling limits, values of which are shown in Figure 2. The prestress generated in most of the unit cells in this study are insufficient to cause buckling, and we correspondingly do not observe any significant bending or buckling in most unit cells here. However, the bars in the 9$\mu \mathrm{m}$ icosahedron are predicted to experience buckling due to prestress at $D_b/D_t$ $\ge$1.8. This value is shown to coincide closely with an experimentally observed buckling point in Figure 2D. This observation not only confirms that there is prestress, but it validates our ability to consistently and predictably engineer stress states into amorphous carbon nano‐tensegrities.

1.3. Mechanical Response and Stiffness Amplification of Nano‐Tensegrities

To quantify changes in mechanical response due to prestress, compression testing was performed on the fabricated nano‐tensegrities using an in‐situ nanoindenter (ASA, Alemnis AG) in an SEM. Testing was done in a displacement controlled mode at a strain rate of ${10}^{-3}$ (see Supplementary Information for additional details). Representative data and video stills are shown in Figure 3, complete recordings are provided in Movies S1–S3. All specimens exhibited a linear‐elastic behavior prior to the onset of failure, and the stiffness ($S$) of each unit cell was taken to be the slope in the elastic region. Unlike ideal tensegrity structures, where failure occurs exclusively through bar buckling or tendon yielding, tensegrities failed primarily via fracture at the nodes. This can be attributed to the more complex stress states that arise from the differential shrinkage during pyrolysis, with higher stresses localizing at nodes (Figure 2). The post‐peak load behavior also differed as a function of $D_b/D_t$. Specimens with a lower $D_b/D_t$ ratio showed a more catastrophic, brittle failure, with multiple bars and tendons breaking simultaneously, while specimens with a higher $D_b/D_t$ ratio showed a more gradual failure progression with clear failure of individual members (Figure 3).

Figure 3.

Nano‐Tensegrity mechanical testing. Representative load‐displacement compression data and accompanying video frames for (A) four‐bar, (B) icosahedron, and (C) twisted truncated octahedron at $D_b/D_t$ = 1 (left) and $D_b/D_t$ = 1.8 (right) (Scale bars are 2.5$\mu \mathrm{m}$).

The stiffness of all the nano‐tensegrities as a function of $D_b/D_t$ are shown in Figure 4, and the peak loads are shown in Figure S5. While prestress can lead to an increased tensegrity stiffness [18, 51], prestress in this study comes at the cost of thinner tendons and thereby results in a drop in density. These competing effects of reduced stiffness from reduced density and increased stiffness due to prestress make it impossible to analytically determine the change in stiffness of a prestressed nano‐tensegrity. Instead, the above developed FE models were employed to isolate and quantify the effect of prestress on the stiffness. In this, the compressive stiffness of prestressed and non‐prestressed specimens was determined numerically, the results of which are shown alongside experimental results in Figure 4A–C. The FE‐predicted stiffness of the prestressed nano‐tensegrities showed strong agreement with experiments for all architectures in this study, providing a quantitative metric that prestress is augmenting mechanical properties.

Figure 4.

Prestress‐induced stiffening in nano‐tensegrities. (A–C) Experimental and FE‐derived stiffness of nano‐tensegrities vs bar‐tendon diameter ratio ($D_b/D_t$) for (A) four‐bar, (B) twisted truncated octahedron, and (C) icosahedron. (D–F) Stiffness amplification factors ($\mathrm{\Omega }$) vs bar‐slenderness ($\lambda$) at fixed bar‐tendon diameter ratios of (D) $D_b/D_t$ = 1.125, (E) $D_b/D_t$ = 1.5, and (F) $D_b/D_t$ = 2.0.

The FE models importantly allow for the calculation of a “stiffness amplification factor,” $\mathrm{\Omega }$, defined as the ratio between the prestressed and non‐prestressed stiffness. The models predictably show that $\mathrm{\Omega }$ = 0 when $D_b/D_t$ = 1, and there is a progressive increase in $\mathrm{\Omega }$ as $D_b/D_t$ increases. FE models were run for bar‐tendon diameter ratios up to $D_b/D_t$ = 2.0, which is approximately the maximum bar‐tendon diameter ratio that was examined experimentally. The maximum predicted stiffness amplification for the experimentally realized four‐bar and twisted truncated octahedron with a $D_b/D_t$ = 2 are $\mathrm{\Omega }$ = 1.78 and 1.14 respectively, and the icosahedron with a $D_b/D_t$ = 1.8 is $\mathrm{\Omega }$ = 1.10. These results thereby reveal two key insights: 1) the ability of prestress to increase stiffness is architecture dependent, and 2) the slenderness of the members has a considerable impact on the stiffness amplification.

To systematically decouple the effect of slenderness and architecture on stiffening, a parametric FE study was performed over a wide range of slenderness values, $\lambda$. The bar slenderness was varied from $\lambda =20-80$ by changing its length while keeping diameter constant to ensure that the applied prestress (Equation (1)) remained consistent. This slenderness range encompassed all of the experimentally realized values (${\lambda }_{exp}$) and was selected to avoid buckling at the upper bound and to prevent member overlap at the lower bound. $D_b/D_t$ ratios of 1.125, 1.5, and 2.0 were investigated to decouple the effect of applied prestress and slenderness.

The computed $\mathrm{\Omega }$ values are shown in Figure 4D–F. These results reveal a consistent trend across all nano‐tensegrities examined here – stiffness amplification due to prestress was significantly greater with increasingly slender structures, i.e., at lower relative densities. There was little to no stiffness amplification for non‐slender structures, and a linear increase in $\mathrm{\Omega }$ with increasing bar slenderness. Architecture also had a pronounced effect on the degree of prestressed stiffening, with the four‐bar unit cell showing the strongest stiffening response, followed by the icosahedron, and finally the twisted truncated octahedron showing minimal change in the stiffness.

Many analytical frameworks [18, 51, 52, 53, 54, 55] have been formulated to understand and quantify the effect of prestress on the mechanical behavior of tensegrities. These studies have shown that prestress primarily stiffened tensegrities by suppressing bending, shear, and torsional modes in individual members and eliminating infinitesimal inextensional mechanisms in the structure. Such analyses assume ideal cable‐strut members with pin‐jointed connections, where nodes do not transfer bending moments between members and contribute no intrinsic stiffness [24, 28].

To assess how these idealized predictions translate to the nano‐tensegrities in this study, we conducted a prestressed structure matrix analysis for all the architectures in accordance with methods developed by Pellegrino [18, 56] and Guest [51, 57] (see Supporting Information for additional details). The analysis confirmed that the shrinkage‐induced prestress state does stiffen infinitesimal modes in all three tensegrities, although to varying degrees. The icosahedron and four‐bar were found to have one and three infinitesimal modes, respectively. We experimentally observed one of these mechanisms – a twisting mode about the central axis – in both nano‐tensegrities (See Movies S1 and S2, Supporting Information). The twisted truncated octahedron, in contrast, is highly under‐constrained and has eighteen infinitesimal modes, making them difficult to visualize qualitatively. Using this framework, we show that increasing prestress stiffens the infinitesimal modes, but it grossly underpredicts the stiffness as compared to experiments. This is unsurprising given that the nano‐tensegrities have rigid‐jointed nodes, making the pin‐jointed assumption inappropriate. More robust models using rigid‐jointed beams in FE analysis showed a much closer agreement with experimental results but still underpredicted the experimental stiffness (see Supporting Information).

It is necessary to fully model the 3D architecture in an FE model to capture the effect of the solid nodes. The nodes shorten the effective length of the beams and increase stiffness independent of any applied prestress, a phenomenon that has been well explored in lattices [58]. The increased nodal stiffness additionally makes theoretically‐predicted zero‐stiffness modes inaccessible by hindering beam‐rotation, leaving fewer pathways for prestress to augment stiffness. Thus, prestress is most effective in low‐density architectures where compliant modes remain accessible, but its impact diminishes in dense structures dominated by geometric rigidity.

We can fully understand how prestress affects the stiffness of the nano‐tensegrities by considering the mechanism analysis alongside FE results. The twisting mode in the four‐bar and icosahedron structures was observed to be progressively suppressed as prestress increased, directly contributing to the observed stiffness amplification. In contrast, the self‐stress in the twisted truncated octahedron had little effect on constraining the multiple potential mechanisms, leading to minimal stiffening. This difference was further influenced by each architecture's load paths, member redundancy, and dominant deformation modes [24]. The four‐bar, which is a class‐2 tensegrity, features well‐oriented members and limited redundancy, promoting effective load‐transfer between bars and higher global‐stiffening [59, 60]. The twisted truncated octahedron contained multiple redundant members and deformation pathways that relaxed the applied prestress under compression, minimizing its effect on stiffness. Overall, the results in this study are consistent with prior analytical predictions, but were captured with greater fidelity through FE, which captured geometric non‐nonlinearities and member interactions that idealized models often neglect.

2. Summary

We demonstrate a unique method to introduce prestress into nano‐tensegrity architectures by exploiting size‐dependent shrinkage during pyrolysis. This approach enables precise control of internal stress by changing feature dimensions, allows prestress to become a design parameter for tuning mechanical response. Our experiments and simulations confirm that prestress significantly amplifies stiffness – an effect that is amplified in low‐density, mechanism‐rich architectures – while also influencing failure modes by promoting gradual, delocalized damage rather than catastrophic collapse.

This work represents a new paradigm for stress engineering in architected materials. By controllably embedding prestress at the nanoscale, we can create materials with programmable stiffness, enhanced energy absorption, and improved resilience under extreme loading. Such properties are highly desirable for lightweight structural components, impact‐resistant coatings, and adaptive metamaterials. Beyond structural applications, prestress tuning could enable acoustic and elastic wave control, paving the way for reconfigurable phononic crystals and acoustic lenses with unprecedented precision. In MEMS/NEMS, where performance often hinges on stiffness and damping, prestressed glassy carbon architectures could deliver high‐Q resonators, precision sensors, and energy harvesters with tunable dynamic response [61]. Integrating prestress control with multi‐material printing, active materials, or stimuli‐responsive polymers could lead to adaptive, self‐tuning systems capable of responding to environmental cues. This work thus establishes a foundation for a new class of nanoarchitected materials where geometry, material, and internal stress are co‐designed to achieve properties beyond the limits of conventional materials.

3. Methods

3.1. Fabrication

All specimens were written in this work using a commercial two‐photon polymerization (TPP) direct laser writing (DLW) tool (Photonic Professional, Nanoscribe, GmbH). Samples were printed using a proprietary acrylate‐based photoresist (IP‐Dip, Nanoscribe, GmbH) onto diced silicon chips. Substrates were etched using an oxygen plasma for 10 minutes to activate the surface and subsequently functionalized with 3‐(Trimethoxysilyl) propyl methacrylate to promote adhesion and prevent specimen peel‐off during pyrolysis.

Pillars were made with 1–15$\mu \mathrm{m}$ diameters and aspect ratios ranging from 3 to 15. Nano‐Tensegrities were designed and sliced using a custom script written in Python with a fixed bar diameter of 4.5$\mu \mathrm{m}$ and a bar‐tendon diameter ratio ranging from 1‐2.25. Voxel lines were written at a laser power of 20 mW with scan speeds of 10 mm/s, a hatching distance of 0.15$\mu \mathrm{m}$ and a layer spacing of 0.6$\mu \mathrm{m}$, corresponding to a $36\%$ degree of conversion of the photopolymer. Samples were developed in a PGMEA bath followed by 3 ultrapure IPA baths. Owing to their fragility, the nano‐tensegrities were dried using supercritical ${\mathrm{CO}}_2$ (Tousimis Autosamdri 931).

After development, samples were pyrolyzed in an ultra high vacuum (UHV) ($\le 1.6\mathrm{x}$ ${10}^{-6}$ mbar) tube furnace (STF16/180, Carbolite Gero) to a maximum temperature of 900. All samples were held at the maximum temperature for one hour. Some of the pillars and all the nano‐tensegrities had an additional one hour prebaking step at 200. For the pillars, we investigated the effect of heating rates ranging from 1 to 35 /min. Samples were imaged used Scanning Electron Microscopy (Thermo‐Fisher Scientific Apreo) at various stages of fabrication. Pillars were imaged before and after pyrolysis to measure dimensions for shrinkage calculations.

3.2. Mechanical Testing

Uniaxial compression tests on the nano‐tensegrities were performed using a flat‐tip displacement‐controlled nanoindenter (ASA, Alemnis AG). Testing was done at a rate of 50 nm/s and specimens were taken to failure. All tests were conducted in‐situ in an SEM. To avoid thermal drift, the setup was installed in the SEM 12 hours before testing. During data processing, any remnant drift was subtracted assuming a linear‐relationship with time. The raw data was filtered using a fast‐Fourier transform (FFT)–based low‐pass filter. Each signal was transformed to the frequency domain, and Fourier components above a cutoff frequency of approximately 5 Hz were removed to suppress high‐frequency noise arising from the tool noise. The filtered signal was then reconstructed by inverse FFT. A subsequent fifth‐order Savitzky–Golay filter was applied to further reduce residual fluctuations while preserving the overall shape of the loading curve.

3.3. Numerical Modeling

FE modeling was done using the Abaqus/standard software package. Bars and tendons were imported as individual parts and later connected using a tie constraint at contact surfaces. All members were modeled using identical elastic properties, as glassy carbon produced from the same precursor and thermal protocol is not expected to exhibit intrinsic size‐dependent material heterogeneity, consistent with prior studies [62]. The material was modeled as elastic homogeneous solid with a Young's modulus of 25 GPa [63, 64], a Poisson's ratio of 0.17 [65]. Expansion was applied to the tendon members using a temperature field and a thermal expansion coefficient as defined above. Note that no transient thermal modeling was performed, the temperature field was only used to introduce expansion. The seed size for the tetrahedral mesh was determined by conducting mesh convergence studies. Displacement‐controlled uniaxial compression tests were modeled in three consecutive steps where the base nodes were constrained, a temperature field ($\mathrm{\Delta }\mathrm{T}$) was introduced in the tendons, and the top nodes were compressed to half of the structure's height.

3.4. Raman Spectroscopy

Raman spectroscopy was performed using a Renishaw InVia Raman microscope equipped with a 532 nm laser and a 1200 lines/mm grating. Spectra were acquired at a laser intensity of 5% and an exposure time of 10 s, averaged over three acquisitions per measurement to improve signal‐to‐noise. For comparison across samples, all Raman spectra were baseline‐corrected and normalized by their maximum intensity.

3.5. Energy Dispersive X‐Ray Spectroscopy

Energy‐dispersive X‐ray spectroscopy (EDS) was conducted using an Oxford Instruments EDS system integrated into a Thermo Fisher Scientific Apreo‐S scanning electron microscope. Measurements were performed at an accelerating voltage of 5 kV using a silicon drift detector. Distinct acquisition parameters were used for as‐printed polymer and pyrolyzed carbon samples to account for differences in conductivity and count rate. For each material, spectra were collected under identical conditions and normalized to total counts to enable relative comparison of elemental composition.

Author Contributions

The following author contributions have been classified following the CRediT taxonomy. Conceptualization, A.R.M., C.W., L.R.M.; Data curation, A.R.M., R.V., A.C., L.R.M.; Formal analysis, A.R.M.; Funding acquisition, L.R.M.; Investigation, A.R.M., C.W., R.V., Z.S.P., A.C., M.L.; Methodology, A.R.M., C.W., A.C., L.R.M.; Project administration, L.R.M.; Supervision, L.R.M.; Visualization, A.R.M.; Writing, A.R.M, L.R.M.

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

Supporting File 1: smll72947‐sup‐0001‐SuppMat.pdf.

Supporting File 2: smll72947‐sup‐0002‐MovieS1.avi.

Supporting File 3: smll72947‐sup‐0003‐MovieS2.avi.

Supporting File 4: smll72947‐sup‐0002‐MovieS3.avi.

Data Availability Statement

All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data and code will be deposited in a public repository upon publication.