Delayed buckling of spherical shells due to viscoelastic knockdown of the critical load

Shell structures are lightweight and flexible. Largely owing to their curvature, they offer considerable strength with little material. As a result, shells are abundant in nature (e.g. eggshells and blood vessels) and design (e.g. fuel tanks and soda cans). However, slenderness also brings susceptibility to abrupt and often catastrophic deformations. Clearly, understanding how a thin, curved structure will lose stability – and in particular at what load value this will occur – is crucial.

We restrict our attention to spherical shells in the present work. The first-known quantitative prediction for the critical load $P_{c}$ in a perfect, spherical, elastic shell subjected to uniform pressure was produced by Zoelly in 1915 via linear eigenvalue analysis, and is given as:

$$P_{c}=\frac{2E}{\sqrt{3(1-\nu^{2})}}\eta^{-2}$$

(1)

for a shell with Young’s modulus $E$, Poisson’s ratio $\nu$, and radius ($R$) to thickness ($h$) ratio $\eta\equiv R/h$.

Although this result is still widely accepted today, it severely overpredicts the buckling load observed in experiments. Recognizing this disrepancy, which is due to the extreme sensitivity to imperfections inherent to thin shells, scientists at the space agency NASA and collaborators introduced the knockdown factor ($k_{d}$) in 1930. The quantity is defined as the ratio of the observed critical load $P_{c}^{e}$, to that predicted by the theory, i.e. $k_{d}\equiv P_{c}^{e}/P_{c}$. Based on surveyed experimental results Lee2016 ; Wagner2020 , engineers at NASA settled for the extremely conservative design code of $k_{d}\approx 0.2$ for spherical shell structures NASASP8032 .

Over the decades that followed, extensive work was dedicated to correcting the persistent overprediction of the critical pressure. This involved studies of the post-buckling behavior VonKarman1939 ; Tsien1942 , and the imperfection sensitivity Koiter1945 ; Koiter1969 ; Hutchinson1967 ; Bushnell1967 ; Krenzke1965 ; Koga1969 of thin spherical shells. Yet, marked success arrived only recently, after Lee et al. developed a new fabrication technique for polymeric spherical caps Lee2016a . The authors used this method to create dominant dimple-like defects (larger than those naturally occurring in the shell) with systematic size variations Lee2016 . These experiments, validated with finite element modeling (FEM) and numerical analysis, quantitatively showed for the first time how the imperfection depth lowers the critical load.

Other contributions followed, including studies on spheres with similar dimple defects and sinusoidal equatorial undulations Hutchinson2016 , large-amplitude dimples Jimenez2017 , through-thickness defects Yan2020 (a notable predecessor is Ref. Paulose2013 ), dent defects Gerasimidis2020 , and probing force imperfections Gerasimidis2018 , which collectively clarify the effect of the type of defect on the knockdown factor for spherical shells. More broadly, this long-awaited breakthrough provoked a new surge of progress in spherical shell theory (see e.g. Refs. Hutchinson2016 ; Hutchinson2017 ; Audoly2020 ; HutchinsonEMLoverview ). These developments afford engineers the opportunity to design sturdy structures with more specific – and permissive Wagner2020 – lower bounds on the load carrying capacity. This was the initial goal. In more recent years, though, a community of researchers has adopted, in a sense, the opposite goal: to design structures that buckle and snap on command Holmes2019 ; Reis2015 for functions like colloidal self-assembly Sacanna2010 , encapsulation Shim2012 , inflatable snapping actuation Keplinger2012 ; Gorissen2020 , and artificial muscle actuation Yang2016 .

The new understanding of defect sensitivity in a different light demonstrates the tunability of spherical shell buckling, and thus serves this “buckliphilic” Reis2015 community equally. In particular, the geometry and placement of a dominant defect prescribe, respectively, the buckling strength and the spot where an instability localizes. Recent extensions of this concept couple geometric defects with differentially swelling Lee2019 or magneto-responsive Yan2020b materials to modify the knockdown factor over time. Relatedly, a more general study showed how a homogenous natural curvature – which can be a proxy for nonmechanical stimuli like thermal expansion, changes in pH, or differential growth – acts to raise or lower the knockdown factor in spherical shells Holmes2020 .

Besides control over when instability occurs – which depends on geometry, loading, and mechanics – mechanical actuators generally rely on reversibility. Repeatable actuation calls for robust, elastic materials, and silicone rubbers like polydimethylsiloxane (PDMS) and vinyl polysiloxane (VPS) have answered this call in mechanics research Reis2015b . In addition to their elastic behavior, these elastomers are readily accessible and allow for fast, easy fabrication Lee2016a .

Recently, Djellouli, et al. combined these ingredients to produce a mechanical swimmer Djellouli2017 . Quasi-static pressure cycles drove the device, a defect-seeded spherical shell made of the elastomer Dragon Skin^TM 30, to propel forward through a viscous fluid by buckling and unbuckling controllably. The authors propose that maintaining dimensionless quantities constant would allow for miniturization, with implications for drug delivery. A natural extension of this work, and the motivation for the present study, is to seek control over the speed of swimming by adjusting the frequency and/or amplitude of pressure cycles. Largely because shell and fluid motion are highly coupled in swimming, and because of possible resonance with postbucking oscillations Mokbel2021 , we expect this phase space to be complex. Thus, we set out to first isolate the shell buckling response, independent of fluid motion, to dynamic loading at pressures in the vicinity of the critical load.

We fabricated imperfect spherical shells like those in Ref. Djellouli2017 , and fixed them in place surrounded by air. A small nozzle allows for internal pressure control, through which we step-load the shells – that is, we abruptly apply, and then maintain, a pressure load. Explicitly, we reduce the pressure inside the shell cavity, creating a negative inside-outside differential pressure. For simplicity, we will refer to this pressure difference in terms of its magnitude. These straightforward experiments produced surprisingly rich results. Even for loads below the experimentally measured elastic critical pressure (which we loosely call “subcritical” herein), we consistently observe buckling. Further, this buckling at subcritical loads occurs abruptly, often after an extended period of very slow deformation perhaps mistakeable for stability. We observe singular thresholds and a delay time which increases monotonically as pressure decreases – this contrasts the findings of a recent numerical study on dynamic step loading of spherical shells which are much thinner than our own Sieber2019 .

As discussed, geometric imperfections can lead to buckling at lower-than-expected loads. In this case, though, the shell defects are already accounted for. Devoid of any plausible geometric explanation for this strange buckling behavior, our results require a closer examination of the materials. Although silicone elastomers are selected precisely because they behave elastically in most settings, they are in fact prone to time-dependent molecular rearrangement, and hence are viscoelastic Lin2009 ; Yuk2017 ; Urbach2020 . Thus, they behave differently depending on how fast they are loaded, and exhibit both stress-relaxation (softening when subjected to a constant strain) and creep (deformation over time under constant stress).

In the present work, we rely on this viscoelasticity to account for our observations, while emphasizing the close connections to elastic spherical shell buckling. The structure is as follows: First, we introduce key aspects of our experiments in Sect. II. Specifically, we report the viscoelastic material properties, describe the geometry of our imperfect spherical shells, and briefly introduce our experimental setup. In Sect. III, we present an overview of our findings. We address the critical pressure conundrum in Sect. IV by defining two pressure thresholds: the elastic critical pressure, and the lower viscoelastic critical pressure, which can be related through the limiting material properties. These thresholds separate three regimes: immediate buckling, delayed buckling, and no buckling (stable). In Sect. V, we introduce an analogy wherein viscoelastic creep deformation lowers the critical load in the same way that a growing dimple-like defect would in an elastic shell. This provides insight about the pre-buckling deformation (Sect.V.1), and reveals how the delay time preceding buckling depends on the imposed pressure, shell geometry and material properties (Sect.V.2). Finally, we offer concluding remarks in Sect. VI.

We have performed dynamic, step-loaded pressure buckling experiments on soft, viscoelastic spherical shells. Here, we briefly summarize the material properties, shell fabrication and geometry, and dynamic loading methods.

For each of our shells, the immediate response to any non-negligible pressure load was qualitatively the same: The shell compresses as soon as the pressure is felt, and deformation quickly localizes at the unclamped pole (in the vicinity of the imperfection), forming a dimple-like depression with a deflection depth $w$ (see Fig. 2 a.) Beyond this early behavior, which occurs in approximately the first 0.05 seconds, three regimes were evident from our experiments.

If the imposed pressure is high, the initial fast rate of pole deformation is maintained, and the shell quickly buckles – that is, the pole inverts, driving global collapse (Fig. 2, i). In postbuckling, which we do not study in detail here, the pole region is completely inverted and oscillations occur before stability is reached again when $w\approx 2R$. We refer to this regime, wherein the shell behaves elastically throughout deformation, as the immediate buckling regime. The lowest pressure at which we observe this buckling behavior defines the experimental elastic critical load $P_{c}^{e}$. Due to the seeded defects and the non-negligible thickness of our shells, the experimentally-determined value $P_{c}^{e}$ differs from the theoretical critical pressure $P_{c}$ (Eq. (1)) for thin, perfect elastic shells. For details, see Sect.  VII.3.

At slightly lower pressures, the deformation rate slows considerably following the fast response to loading, at a transition time ($t\approx 0.05$ s). The dimple slowly deepens (i.e. $w$ increases; See Fig. 2, ii), before an abrupt acceleration after some time $t_{c}\in[0.09$ s, $17.09$ s$]$ signifies buckling. We define this intermediate regime as the delayed buckling regime. At still-lower pressures, slow pole motion eventually stops, and the shell settles into indefinite stability for as long as the load is maintained. We call this third regime the stable regime (Fig. 2, iii).

The value of the pressure which separates the delayed buckling and stable regimes, and hence marks the boundary of whether collapse will occur, is clearly of interest. This lower pressure threshold was more or less constant for all of our shells when normalized by the elastic load. In other words, the reduced critical pressure is independent of geometry. With this nudge toward the materials, we proceed to rationalize these findings.

Since we know our materials are viscoelastic, we can presume that the slow pole deformation under constant pressure is an exhibition of creep. This would situate our observations in the terrain of creep buckling. Creep buckling was introduced in the literature in 1951 Rosenthal1951 . The bulk of the work in this field was developed in the thirty or so years that followed, and was aimed at understanding creep collapse that occured on timescales of hours or even days in metallic, mono-resin materials, and reinforced concrete. However, general theories emerged, which are illuminating when applied to our elastomer shells. In particular, Hayman Hayman1978 ; Hayman1981 and others Hoff1973 ; Minahen1993 proposed that a viscoelastic structure that buckles due to creep may be treated as an equivalent elastic structure with a lower critical load. The main result is that the lower threshold – which we will henceforth refer to as the viscoelastic critical pressure $P_{c}^{v}$ – is directly related to the long-term modulus. For spherical shells, the lower critical pressure $P_{c}^{v}$ may be found by simply replacing Young’s elastic modulus $E$ with $E_{\infty}$ in calculating the elastic critical load (Eq. (1)). Denoting normalization by the experimentally measured elastic critical pressure $P_{c}^{e}$ with an overbar, i.e. $\bar{P}\equiv P/P_{c}^{e}$ and $\bar{P}_{c}^{v}\equiv P_{c}^{v}/P_{c}^{e}$, that is:

$$\bar{P}_{c}^{v}=\bar{E}.$$

(5)

In Fig. 3, we show that (5) agrees with our experimental data very well, solidifying the notion that viscoelasticity is indeed the cause for the subcritical buckling we observe. The dashed line marks the theoretical lower limit where creep bucking may be observed, $\bar{P}_{c}^{v}=\bar{E}$, which for our materials (see Sect.II.1 & Sect. VII.1) is $0.91$.

Explicitly, Eq. (5) ignores the actual mechanism that leads to instability, creep deformation, in favor of a straightforward way to determine the minimum buckling pressure for creep-limited materials, which is perhaps the most crucial information for any design goal. Given the elastic (viscoelastic) critical pressure and the experimentally-determined instantaneous and long-term moduli, the ratio in Eq. (5) readily predicts the viscoelastic (elastic) critical pressure.

This view is effective and completely general with respect to geometry and material properties. However, it provides no information about deformation or the time delay that preceeds buckling. Besides that these features are of fundamental interest, understanding this delay mechanism will offer a route to including controllable delays in elastomer device design. We address these open questions in the following section.

As we have seen, the efficient, modulus-based approach in Sect. IV connects the limiting critical pressure of a viscoelastic shell to that of the equivalent elastic shell. It leaves questions, however, about the time it takes a subcritically-loaded shell to buckle, and the underlying pre-buckling deformation. Traditional analytical approaches to capture the critical time and/or deflection for creep buckling involve incorporating calculated quantities for stress and strain into the constitutive model (in our case Eq. 3). Instability may be identified by solving the eigenvalue problem of the governing differential equations, or by the quasi-static “critical strain approach” Gerard1958 wherein the critical strain must be known or assumed a priori, and the corresponding time is directly solved for Miyazaki2015 . These methods require precise representations of the stresses and strains throughout deformation, and have met moderate success in capturing experimental behavior for simple structures like columns Hoff1954 ; Hoff1956 , trusses and arches Huang1967 , plates and even cylinders Gerard1958 . (See Ref. Miyazaki2015 for a review of the relatively recent work on creep buckling of shell structures, or Ref. Hoff1973 for an earlier review on creep buckling of plates and shells.)

A clear problem with these approaches is that it is generally assumed that a shell undergoing creep will lose stability at the same strain as its elastic counterpart Gerard1956 ; Hoff1975 . However, it has been noted that this assumption often leads to underprediction of the critical displacement and time Obrecht1977 . Indeed, although the immediate and delayed buckling regimes appear qualitatively very similar in terms of deformation in our experiments (see Fig.  2, i. & ii.), we observe that shells which creep for longer sustain more deformation before buckling.

Complex geometries like imperfect spherical shells and their nonlinear deformations introduce significant analytical difficulties of their own. Creep buckling in spherical caps and complete spherical shells has primarily been studied with numerical analyses Huang1965 ; Shi1970 ; Jones1976 ; Miyazaki1977 ; Xirouchakis1980 , which do not produce closed-form solutions for when instability occurs. Few experiments exist for comparison to these results, and attempts to replicate limited experimental creep buckling behavior for spherical shells have largely been unsuccessful Shi1970 ; Leckie1976 .

A more enlightening approach relies on the observation that the pre-buckling deformation approximately amplifies the initial defect (see Fig. 4a). This suggests that we may be able to draw an analogy between creep deformation and a growing imperfection in an elastic shell. This concept was proposed by Hayman in 1981, but the author conjectured that while his “locus of critical points” approach offered intuition, it lacked predictive power for all but simple, statically determinate structures Hayman1981 . To our knowledge, the approach has not been implemented besides in the original work, when it was validated against a small number of experiments on concrete three-pin arches.

One key challenge was that the effect of the defect must be quantifiable. Indeed, as discussed in Sect. I, the imperfection sensitivity of shell structures has only recently become experimentally tractable. In what follows, we rely on the work of Lee and collaborators Lee2016 , and center our analysis upon quantities that are readily determined through experiments, to derive a practical and predictive model for creep buckling in our shells.

Knowing that an imperfection “knocks down” the critical load in an elastic structure, in this view quasi-static creep deformation has the same effect. In other words, creep deformation behaves like an evolving imperfection by progressively knocking down the critical load. It follows that creep collapse occurs when the critical load associated with the “imperfect” creep-deformed structure falls to the value of the applied load. Conversely, the imperfection size associated with a given applied load should correspond to the critical deformation at which creep buckling occurs. It is important to note that this was buried but implied in the approach of Sect. IV; the two arguments are complementary.

To make this analogy quantitative, we turn to the recent literature on geometric imperfections in elastic spherical shells. The true defects in our shells are characterized by a local reduction in thickness as in Ref. Yan2020 (see Sect. VII.3). The pole deflection generated during creep, however, is qualitatively more similar to a dimple-like imperfection where the thickness does not change, but the curvature of the shell midline does. Because the midline curvature of our shells is unaffected by the thickness reduction except at the discontinuity at either edge of the defect profile, we consider our shells initially “perfect” in the dimple sense. For the present analysis, we rely on the findings of Lee et al. Lee2016 , which are in agreement with Refs. Hutchinson1967 ; Hutchinson2016 ; Jimenez2017 .

The key finding of their work, for our purposes, is that the knockdown factor for a given shell $k_{d}=P_{c}^{e}/P_{c}$ is a function of $\bar{\delta}=\delta/h$ which initially decreases for increasing $\bar{\delta}$, then reaches a plateau. The authors present an empirically-determined function describing the lower bounding envelope over the range of $\lambda$ they study, which takes the form $k_{d}=a+b/(c+\bar{\delta}$). We find that this functional form describes their individual curves sufficiently well.

Primarily because $\lambda$ for our thick shells is below the range studied in Ref. Lee2016 , we cannot directly extract the results relevant to our work. Instead, we expect that these general trends will hold. We assume that we can simply replace the imperfection depth $\delta$ with the pole deflection $w$. Because we only consider the additional knockdown due to creep deformation, and not that due to the inital defect, we take the reference pressure as the experimental critical pressure $P_{c}^{e}$ that corresponds to the initially imperfect shell. Then we define a general viscoelastic knockdown function

$$k_{d}^{v}(w)=\bar{P}_{c}(\bar{w})=a+\frac{b}{c+\bar{w}},$$

(6)

where $\bar{w}=w/h$, and $a$, $b$, and $c$ are yet unknown.

Since $w$ increases according to the creep strain rate, an alternative form of Eq. (6) specifies the time dependence. Approximating the magnitude of the circumferential strain to first order as $\varepsilon\approx w/R$, we can say $\bar{w}\approx\eta\varepsilon(t)$. From Eq. (3) this means $\bar{w}\approx\eta\sigma J(t)$, which is approximately $\eta\sigma/E(t)$ since the creep compliance function $J(t)$ and the inverse of the relaxation modulus $1/E(t)$ are nearly indistinguishable for our material. At early times the shell behaves elastically, so we assume Hooke’s Law applies, i.e. $\sigma\approx E_{0}\epsilon(t_{t})$ at a transition time $t_{t}$ between the elastic and creep stages of deformation. Further, we assume $\epsilon(t_{t})\approx\epsilon_{c}^{e}$, where $\epsilon_{c}^{e}\approx w_{c}^{e}/R$ is the critical strain corresponding to $\bar{P}=1$, when the shell buckles immediately following elastic deformation. Then by Eq. (2), the normalized pole deflection increases with time following the relation

$$\bar{w}(t)\approx\frac{E_{0}\>\bar{w}_{c}^{e}}{E_{\infty}+E_{1}e^{-t/\tau_{\sigma}}}.$$

(7)

Note that due to our simplified representations of stresses and strains in Eq. (7), all relations that follow are approximations, despite that we present them as equalities for simplicity. Substituting Eq. (7) in Eq. (6), gives a time-dependent version of the generalized viscoelastic knockdown function:

$$k_{d}^{v}(t)=\bar{P}_{c}(t)=a+\frac{b}{c+\frac{E_{0}\>\bar{w}_{c}^{e}}{E_{\infty}+E_{1}e^{-t/\tau_{\sigma}}}}.$$

(8)

It remains to determine the three unknown quantities, which should explain how sensitive the critical pressure is to deformation (Eq. (6)) and how quickly the critical pressure decreases (Eq. (8)) for each shell. To do so, we constrain the functions, enforcing what we know about the limiting behavior. The critical deflection required for buckling at the elastic limit where $\bar{P}=1$ is an experimentally-determined value, $w_{c}^{e}$, which differs for each shell based on geometry. Since for buckling to occur, $P_{c}(w)=P$, from Eq. (6) this condition is stated as:

$$1=a+\frac{b}{c+\bar{w}_{c}^{e}}.$$

(9)

We also know from Eq. (5) that buckling will not occur below $k_{d}=\bar{P}_{c}^{v}=\bar{E}$. Taking the limit of the right hand side of Eq. (8) as $t$ approaches infinity gives a second constraint:

$$\bar{E}=a+\frac{b}{c+\frac{\bar{w}_{c}^{e}}{\bar{E}}}.$$

(10)

Solving Eqs. (9) & (10) simultaneously gives

$$b=\frac{(a-1)(a-\bar{E})\bar{w}_{c}^{e}}{\bar{E}}$$

(11)

and

$$c=\frac{-a\bar{w}_{c}^{e}}{\bar{E}},$$

(12)

which we insert into Eq. (6) to arrive at:

$$\bar{P}_{c}(w)=a+\frac{(1-a)(\bar{E}-a)\bar{w}_{c}^{e}}{\bar{E}\bar{w}-a\bar{w}_{c}^{e}}$$

(13)

which describes the critical pressure for a given degree of pole deflection. If $\bar{w}$ is large enough that $\bar{P}_{c}$ is lowered to the imposed dimensionless pressure $\bar{P}$, in theory buckling will occur.

Similarly, inserting the expressions for $b$ and $c$ in Eq. (8) gives

$$\bar{P}_{c}(t)=a+(a-1)(a-\bar{E})\bigg{(}\frac{E_{0}\bar{E}}{E_{\infty}+E_{1}e^{-\frac{t}{\tau_{\sigma}}}}-a\bigg{)}^{-1}$$

(14)

which specifies the time-dependence, according to the SLS model, of the knockdown to the critical pressure that occurs as the pole deflection progresses. Again, if $t$ is such that $\bar{P}_{c}=\bar{P}$, we expect collapse to occur.

Note that from Eq. (14), it is clear that any explicit dependence on geometry is contained in $a$. When $a$ is left as a fitting parameter for the curves defined by Eq. (16) (see Fig. 6a), we find that it is a decreasing function of the geometric defect parameter $\lambda$, which is in general agreement²²2We fit a function of the form $k_{d}=a+b/(c+\bar{\delta})$ to the numerical data over $\lambda\in[1,5]$ in Fig. 6b of Ref. Lee2016 , and found $a\approx 0.5-0.14\lambda$. with Ref. Lee2016 . In particular, we determine $a\approx 1.08-0.44\lambda$ (Fig. 4b) for our shells. We refer to this linear function henceforth as $f(\lambda)$, and substitute it accordingly for $a$ to emphasize the deduced connection between our fitting parameter and the defect geometry.

The deflection (Eq. (13)) and time (Eq. (14)) forms of the viscoelastic knockdown relations lead to expectations for how the critical displacement and critical time should depend on the material properties, geometry and the imposed pressure. We assess both in the following subsections.

In summary, we subjected thick, spherical, defect-seeded viscoelastic shells to step pressure loading. We observed three regimes: When the pressure load was at or above the experimentally determined elastic critical load, we observed predictible elastic behavior, i.e. prompt buckling. At intermediate loads – below the elastic critical pressure – the shell buckled, albeit after a time delay during which deformation slowly progressed. At still-lower pressures, the shell deformed but collapse never occured. Our aim in this work was to rationalize our findings in a way that maintains close ties to elastic shell buckling, and is readily useable for experiment or design goals. To this end, we demonstrated that the load thresholds, critical deflection, and critical time may all be captured by a framework that treats creep deformation like an evolving defect in an elastic shell.

In particular, the ratio of the long-term modulus to the short-term (elastic) one is the same as as the ratio of the two critical pressures. This result is rooted in elastic shell theory, but practically, the material properties alone can explain the two pressure thresholds. This finding was suggested in various theoretical works on creep buckling Hayman1978 ; Hayman1981 ; Hoff1973 and is independent of geometry, and hence is completely general.

We used this fact and existing work Lee2016 on defects in spherical shells to discern an expression for how deformation due to creep acts to “knock down” the critical pressure. In this view, the shell loses stability when creep deformation, which localizes at pole and amplifies the initial imperfection, progresses enough to reduce the critical pressure to the value of the imposed one. This allowed us to capture the dependence of the critical deflection on the imposed pressure (normalized by the experimentally-determined elastic critical pressure), the modulus ratio, and the defect geometry (Eq. (15)). This offers an explanation rooted in elastic shell behavior for a decidedly viscoelastic phenomenon: in the delayed buckling regime, higher deflection is required for instability as the pressure decreases.

Because deformation occurs on a timescale sufficiently well-described by our chosen viscoelastic material model (SLS), a time-dependent form of the viscoelastic knockdown function immediately follows. From this we devise an expression for how the pre-buckling delay time depends on the same quantities: the modulus ratio, the dimensionless pressure, and the defect geometry (Eq. (16)). The buckling time increases monotonically but non-linearly as the pressure decreases, which is generally captured by our model. While the modulus ratio $\bar{E}$ was fixed in our experiments, we expect that our model is valid for any material with relatively low relaxation strength (i.e. the relaxation modulus and creep compliance functions do not differ significantly). Further, we note that the success of our model does not depend on the approximation $J^{-1}(t)\approx E(t)$ that we have employed. Rather, if one were to obtain the creep compliance function in experiments, using this directly in place of the relaxation modulus would likely improve the accuracy of predictions.

While viscoelastic shells behave elastically during buckling, viscous effects re-enter in later stages of postbuckling, as discussed in Ref. Coupier2019 . Unbuckling was studied in detail in Ref. Djellouli2017 , as the non-reciprocal nature of the buckling-unbuckling cycle is the source of motility. We did not examine unbuckling in the present work. However, we could reasonably expect that another viscoelastic delay phenomenon, termed “pseudo-bistability” Brinkmeyer2012 (first introduced in Ref. Santer2010 as “temporary bistability”), could be seen in our shells.

Pseudo-bistability refers to the delayed “snap-back” instability that occurs in the unloaded state, following a loading-unloading sequence that induces both stress-relaxation and creep Gomez2018 . Early works modeled viscoelasticity via an evolving stiffness, acheiving qualitative agreement with experiments Santer2010 ; Brinkmeyer2012 . A recently-proposed metric framework introduces viscoelasticity as a temporally evolving (fictitious) reference length instead, and among its merits is the ability to predict delayed snap-back instability Urbach2020 ; Urbach2018 . Delayed snap-back instability occurs in the unloaded state, and in our setting would require maintaining the pressure load for a sufficiently long time while the shell is fully buckled before unloading. As such, creep instability and pseudo-bistable snap-back can in theory be induced in the same thick structure, with one or the other surpressed at will.

Our findings highlight the importance of the load rate, and the sensitivity of the shell to pressure variations in the vicinity of the elastic critical load (when approached from below). These are important considerations for structural designs using silicone rubber where either stability or elastic behavior is desired. For instance, knowledge of the critical load thresholds is clearly important when designing an efficient spherical swimmer, or indeed any viscoelastic structure that should undergo oscillatory instability. If the goal is fast motility, it would likely be desirable to minimize the time delay before buckling by avoiding the delayed buckling (intermediate pressure) regime altogether.

In other settings, viscoelastic behavior can enhance the functionality of reversibly actuatable structures. This has been demonstrated recently in designs that rely on pseudo-bistable snap-back, e.g. 3D-printed viscoelastic metastructures whose time-dependent properties are tunable based on temperature Che2019 , with even more flexibility afforded by using multiple viscoelastic materials Che2018 . Another study examines the interplay between viscous dissipation and geometric hysteresis, as a function of the strain rate, for the design of optimal energy dissipating metamaterials Dykstra2019 . Introducing tunable delays via creep buckling has not yet been explored, but the potential is vast: A switch or capsule could be loaded subcritically, so that the shell has time to move to a desired location before buckling occurs. Over time as deformation reduces the critical load and thus the energy barrier Hutchinson2018 , a much smaller probing force or other perturbation could trigger buckling. This could be useful for pneumatic gripping. A mechanical signal of fixed input frequency (and varying amplitude) could produce a varied output frequency, which has implications for mechanical computing. Because our analysis relies only on quantities that are straightforward to determine in experiments, our findings are especially amenable to accessing such tunability. As we have shown, these possibilities are achievable with the nearly-elastic materials that are already common and cherished in mechanics research.

In 1956, Gerard wrote about creep buckling that “there are almost more theories than reliable test points which can be used to check the theories” Gerard1956 . While a limited number of experimental contributions have come about since, to our knowledge our experiments on full spheres are novel to the existing creep buckling literature. Further, we have demonstrated that some concepts central to general creep buckling theories, which previously were mostly tested on metallic, mono-resin materials, and reinforced concrete Miyazaki2015 , are indeed applicable to soft, rubbery materials. We expect that the concepts we have studied can be extended to other geometries, other loading methods, and similar polymers.