Dynamics of viscoelastic snap-through111© 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
We study the dynamics of snap-through when viscoelastic effects are present. To gain analytical insight we analyse a modified form of the Mises truss, a single-degree-of-freedom structure, which features an ‘inverted’ shape that snaps to a ‘natural’ shape. Motivated by the anomalously slow snap-through shown by spherical elastic caps, we consider a thought experiment in which the truss is first indented to an inverted state and allowed to relax while a specified displacement is maintained; the constraint of an imposed displacement is then removed. Focussing on the dynamics for the limit in which the timescale of viscous relaxation is much larger than the characteristic elastic timescale, we show that two types of snap-through are possible: the truss either immediately snaps back over the elastic timescale or it displays ‘pseudo-bistability’, in which it undergoes a slow creeping motion before rapidly accelerating. In particular, we demonstrate that accurately determining when pseudo-bistability occurs requires the consideration of inertial effects immediately after the indentation force is removed. Our analysis also explains many basic features of pseudo-bistability that have been observed previously in experiments and numerical simulations; for example, we show that pseudo-bistability occurs in a narrow parameter range at the bifurcation between bistability and monostability, so that the dynamics is naturally susceptible to critical slowing down. We then study an analogous thought experiment performed on a continuous arch, showing that the qualitative features of the snap-through dynamics are well captured by the truss model. In addition, we analyse experimental and numerical data of viscoelastic snap-through times reported previously in the literature. Combining these approaches suggests that our conclusions may also extend to more complex viscoelastic structures used in morphing applications.
keywords:Snap-through, Buckling, Viscoelasticity, Bistability, Creep, Snap-through time
1.1 Elastic snap-through
Snap-through buckling is a striking instability in which an elastic object rapidly jumps from one state to another. Such instabilities are familiar from everyday life: umbrellas suddenly flip upwards on a windy day, while the leaves of the Venus flytrap store elastic energy slowly before abruptly snapping shut to catch prey unawares (Forterre et al., 2005). Similarly, snap-through is harnessed to generate fast motions in technological applications ranging from fluidic actuators (Overvelde et al., 2015; Gomez et al., 2017b; Rothemund et al., 2018), micro-scale switches (Krylov et al., 2008; Ramachandran et al., 2016), responsive surfaces (Holmes and Crosby, 2007) and artificial heart valves (Gonçalves et al., 2003). In these applications, snap-through has proved to be particularly useful among other elastic instabilities, such as wrinkling and crumpling, due to its ability to convert energy stored slowly into fast motions in a highly reproducible way.
Despite the ubiquity of snap-through in nature and engineering, its dynamics is not well understood, with classical work focussing on determining the onset of snap-through in simple elastic objects such as plates and shells (Bazant and Cendolin, 1991; Patricìo et al., 1998). Because snap-through generically occurs when a system is initially in an equilibrium state that ceases to exist (a saddle-node/fold bifurcation), standard analytical techniques often cannot be used to study the dynamics. For example, it is generally not possible to perform a linear stability analysis to obtain an eigenvalue (natural frequency) that characterises the growth rate of the instability: beyond the fold point there ceases to be an equilibrium base state from which the system evolves. This is in contrast to the case when snap-through is caused by a bifurcation in which the equilibrium state becomes unstable without ceasing to exist (Pandey et al., 2014; Fargette et al., 2014). The dynamics near a saddle-node bifurcation have been well studied in low dimensional systems, consisting of a few ordinary differential equations (ODEs), in various physical and biological settings (Strogatz and Westervelt, 1989; Trickey and Virgin, 1998; Majumdar et al., 2013) including work on slow-fast systems — see Jones and Khibnik (2012) and chapter of Berglund and Gentz (2006) (and references therein). However, it is much more difficult to extend this to an elastic continuum described by partial differential equations (PDEs). For this reason, previous work has mainly relied on experiments and numerical simulations (e.g. using commercially available finite element packages, or solutions of the governing PDEs using standard numerical methods) to quantitatively model the snap-through dynamics (Diaconu et al., 2009; Santer, 2010; Arrieta et al., 2011; Brinkmeyer et al., 2012; Loukaides et al., 2014). Some progress has also been made using lumped mass-spring models (Carrella et al., 2008), though there remains a general lack of analytical results in the literature, for example closed-form expressions for the time taken to snap-through in terms of the physical system parameters. Analytical insight would be of interest both from the perspective of fundamental science and also for applications of snap-through, as it provides a basis to control the dynamic response and guide more detailed simulations or experiments.
Moreover, some features of snap-through are not understood at a qualitative level, including delay phenomena: snap-through often occurs much more slowly than would be expected for an elastic instability. This slowness is illustrated by children’s ‘jumping popper’ toys, which resemble rubber spherical caps that can be turned inside-out. The inverted configuration remains stable while the cap is held at its edges, but leaving the popper on a surface causes it to snap back to its natural shape and leap upwards. As shown in figure 1, the snap back is not immediate: a time delay is observed during which the popper moves very slowly, apparently close to equilibrium, before rapidly accelerating. The delay can be several tens of seconds in duration — much slower than the estimated elastic timescale, which is on the order of a millisecond (Gomez, 2018).
To explain such discrepancies between estimates of the speed of snap-through and that actually observed, it is commonly assumed that some dissipation mechanism must be present. For example in the Venus flytrap, the estimated elastic timescale is orders of magnitude faster than the observed snap-through time, and air damping is not enough to account for the discrepancy. In this case the proposed mechanism is poroelasticity (Forterre et al., 2005): the snapping leaves are saturated with water and may dissipate energy via internal fluid flow. Similarly, morphing devices that demonstrate delayed snap-through are commonly composed of silicone-based elastomers, which are known to exhibit viscoelastic behaviour (Brinkmeyer et al., 2012). It is also easily demonstrated that holding a popper toy for longer in its inverted state causes a slower snap-back — an observation that is consistent with the importance of viscoelastic effects.
While attributing delayed snap-through to various dissipation mechanisms is natural, we recently demonstrated that anomalously slow dynamics are, in fact, possible in elastic systems with negligible dissipation (Gomez et al., 2017a). In such scenarios, the time delay arises from the remnant or ‘ghost’ of the snap-through bifurcation, reminiscent of the ‘critical slowing down’ observed in other areas of physics such as phase transitions (Chaikin and Lubensky, 1995) and electrostatic ‘pull-in’ instabilities (Gomez et al., 2018): the saddle-node bifurcation continues to attract trajectories that are nearby in parameter space, producing a bottleneck whose duration increases without bound as the distance from the bifurcation decreases. Snap-through then appears to proceed much slower than the elastic timescale. In the process, we were able to propose new analytical formulae for the snap-through time as a function of the material parameters. Nevertheless, a key feature of this slowing down is that the system needs to be very close to the snap-through transition: the amount of delay that is experimentally attainable may in practice be small. Moreover, in viscoelastic systems, it is not clear what role viscoelastic effects play in obtaining anomalously slow snap-through dynamics, as opposed to the purely elastic slowing down. While we have previously considered the influence of external viscous damping (e.g. due to air drag) (Gomez, 2018), viscoelastic behaviour is fundamentally different because it modifies the stability characteristics of structures. Here, therefore, we seek to extend these studies to understand analytically how material viscosity affects the snap-through dynamics.
1.2 Viscoelasticity and pseudo-bistability
Unlike elastic solids, viscoelastic materials generally undergo stress relaxation when subject to a constant strain; this causes the effective stiffness of the structure to evolve in time. If a constant stress is imposed instead, the material may also exhibit a slow creeping motion (Howell et al., 2009). Santer (2010) has demonstrated how these combined effects allow structures to exhibit ‘temporary bistability’ or ‘pseudo-bistability’ during snap-through. The idea of pseudo-bistability is that when a structure is held in a configuration that is near (but just beyond) a snap-through threshold, just as a popper toy may be held inside-out, the change in stiffness associated with stress relaxation may cause the structure to appear bistable (i.e. an elastic structure with the same instantaneous stiffness would be bistable). When the structure is released, the stiffness recovers during a creeping motion, until eventually this bistability is lost and rapid snap-through occurs. Similar to the phenomenon of creep buckling (Hayman, 1978), the total snap-through time is then governed by the viscous timescale of the material and can be very large. This phenomenon may be useful in morphing devices that are required to cycle continuously between two distinct states; for example, dimples proposed for aircraft wings that buckle in response to the air flow to reduce skin friction (Dearing et al., 2010; Terwagne et al., 2014), and ventricular assist devices which use snap-through of a spherical cap under a cyclic pneumatic load to pump blood (Gonçalves et al., 2003). In these applications, pseudo-bistability means that the actuation needed to move the structure between different states can be applied for a shorter duration, which may lead to a significant reduction in the energy consumed (Santer, 2010).
Using finite element simulations, Santer (2010) has demonstrated pseudo-bistability in a single-degree-of-freedom truss-like structure, as well as spherical caps similar to the jumping popper toy of figure 1. The phenomenon has been observed experimentally in spherical caps (Madhukar et al., 2014) and truncated conical shells (Urbach and Efrati, 2017), and generically appears to occur only in a narrow parameter range near the transition to bistability, i.e. the threshold at which snap-through no longer occurs. Brinkmeyer et al. (2012) performed a systematic study of the snap-through dynamics of viscoelastic spherical caps, using a combination of finite element simulations and experiments. Continuing this work, Brinkmeyer et al. (2013) studied the pseudo-bistable effect in viscoelastic arches. In these studies the phenomenon is found to have a number of common features, including (i) to obtain any time delay the structure needs to be held for a minimum period of time in an inverted state before release; and (ii) the resulting snap-through time depends sensitively on the parameters of the system and appears to diverge at the bistability transition. However, these basic features are not well understood quantitatively despite having important implications for applications of pseudo-bistability. The sensitivity of the snap-through time, for instance, means the system needs to be precisely tuned to obtain a desired response time. For this reason direct comparison between experiments and finite element simulations has revealed large quantitative errors (Brinkmeyer et al., 2012, 2013).
In addition, the numerical simulations referred to above are all based on two key assumptions regarding the viscoelastic response: (i) the material behaves as though elastic with an effective stiffness that evolves in time, and (ii) the response during recovery is the reverse of the response during relaxation, i.e. once the structure is released, the stiffness smoothly recovers to its initial, fully unrelaxed, value. These assumptions mean that modelling the dynamics is relatively simple compared to more general viscoelastic models, and the resulting equations are more easily implemented in commercially-available finite element packages. Furthermore, the different dynamical regimes can often be inferred by considering the elastic response in which the stiffness is fully unrelaxed and fully relaxed, as the instantaneous stiffness must be bounded between these two extremes (Santer, 2010). However, the validity of these assumptions, and whether they can be justified from first principles, remains unclear.
An alternative approach is to start from the constitutive law of a viscoelastic solid, and derive the equations of motion that couple the stress to the deformation of the structure. While this approach is significantly more complicated, it eliminates the need to make any additional assumptions regarding the behaviour of the stiffness. This method has previously been used to obtain analytical expressions for the snap-through loads of simple viscoelastic structures (Nachbar and Huang, 1967), as well as the conditions under which creep buckling occurs (Hayman, 1978). More recently, Urbach and Efrati (2018) developed a general theoretical framework for modelling viscoelastic snap-through based on a metric description of the constitutive equations. While this approach yields insight into the phenomenon of pseudo-bistability, the dynamics are modelled quasi-statically by neglecting the system inertia, so that it is unclear precisely when pseudo-bistable behaviour is obtained. Elsewhere, due to the inherent complexity of viscoelastic effects, it is unknown what role inertia plays in the dynamics and why the snap-through time appears to diverge near the snap-through transition. Are we simply observing another instance of critical slowing down, similar to the purely elastic dynamics studied by Gomez et al. (2017a)?
1.3 Summary and structure of this paper
In this paper, we aim to provide analytical understanding of the dynamics of viscoelastic snap-through, and in particular the features of pseudo-bistability. We consider a thought experiment in which a structure is indented to a specified displacement, and allowed to undergo stress relaxation before the indenter is abruptly removed. While we are motivated by continuous viscoelastic structures such as shells and arches, we first study a Mises truss for simplicity. This is a single-degree-of-freedom structure that exhibits bistability and snap-through, and enables us to make significant analytical progress. Focussing on the limit in which the timescale of viscous relaxation is much larger than the characteristic elastic timescale, we obtain three key results. (1) Inertial effects immediately after the indentation force is removed play an important role in determining when snap-through and pseudo-bistability occur. (2) While the intuitive picture of pseudo-bistability as being caused by a temporary change is stiffness is correct, the assumption of reversibility made in previous numerical studies (i.e. that the stiffness smoothly reverses back to its fully unrelaxed value when the indenter is removed) leads to significantly different predictions of when snap-through occurs, compared to our first principles analysis. (3) Pseudo-bistability is a type of creeping motion governed by the viscous timescale, and so does not rely on critical slowing down to obtain slow dynamics, unlike purely elastic snap-through (Gomez et al., 2017a). Nevertheless, this creeping motion may be very slow indeed as the system may, in addition, be subject to critical slowing down in the pseudo-bistable regime. We then study a pre-buckled viscoelastic arch as an example of a more realistic structure that is used in morphing applications. Using direct numerical solutions, we show that the predictions of the truss model are qualitatively accurate for the arch system. This suggests that the analytical insight gained from the truss model may apply more broadly to the complex viscoelastic structures used in applications of snap-through.
The remainder of this paper is organised as follows. We begin in §2 by deriving the equations governing the motion of the Mises truss. We then discuss the equilibrium states and the stress relaxation during indentation. In §3, we analyse the snap-through dynamics when the indenter is released, focussing on the limit in which the timescale of viscous relaxation is much longer than the characteristic elastic timescale. Using direct numerical solutions, we identify the different dynamical regimes and explain these asymptotically using the method of multiple scales. In §4, we perform simulations of a viscoelastic arch, showing that its snap-through behaviour is well captured by the truss model. In §5, we compare our predictions to experimental and numerical data of pseudo-bistable snap-through times reported in the literature. Finally, in §6, we summarise our findings and conclude.
2 A simple model system: Mises truss
As a first step towards understanding the dynamics of viscoelastic snap-through, we follow Brinkmeyer et al. (2013) and consider a Mises truss (also referred to as a ‘von Mises truss’). In its simplest (elastic) form, this features two central springs, assumed to be linearly elastic, that are pin-jointed at their ends and inclined at a non-zero angle to the horizontal in their natural state. To give the system inertia, we place a point mass where the springs meet, and restrict the mass to move only in the vertical direction; see figure 2a.
The truss in its current form is bistable: as well as the undeformed or ‘natural’ state, the truss may be in equilibrium in a reflected state, where the length of each spring is unchanged from its rest length. However, by connecting an additional spring of sufficient stiffness to the point mass (figure 2b) (Krylov et al., 2008; Panovko and Gubanova, 1987), the inverted state ceases to be a stable equilibrium: in an experiment in which the truss is held fixed in an inverted position using an indenter, the truss will immediately snap back to its natural state when the indenter is released, independently of how long it is held. This snap-through is reminiscent of a spherical cap; in fact, we may consider the truss as a lumped model for a generic, continuous elastic structure that features an ‘inverted’ state that snaps back to a ‘natural’ state. The central springs represent the membrane (stretching) stiffness, since these springs can be viewed as corresponding to the midsurface of the structure. The vertical spring models the bending stiffness (Krylov et al., 2008): this spring penalises rotating the truss about its pin-jointed ends, mimicking bending the structure about its edges as it is loaded to an inverted position.
We now suppose that the vertical spring is viscoelastic. To choose an appropriate viscoelastic model, we note that a typical snap-through experiment includes both displacement-control and force-control: during indentation we impose a given displacement, but releasing the structure corresponds to imposing zero indentation force. It is therefore insufficient to describe the viscoelastic response using a Kelvin-Voigt or Maxwell model, since these fail to accurately capture both stress relaxation (under displacement-control) and creep (under force-control) behaviour. Instead, we shall use the constitutive law of a standard linear solid (SLS), which is the simplest model that describes both of these effects (Lakes, 1998). Physically, the SLS model is equivalent to placing a linear spring in parallel with a Maxwell element that features a second spring and a dashpot in series; see figure 2c. While we could also incorporate viscoelasticity of the central springs in our formulation, this would introduce additional viscous timescales and hence make it much more difficult to make analytical progress; we will show that our simpler model successfully captures snap-through and pseudo-bistability without additional complexity.
When the truss is indented and held for a specified time, stress relaxation causes the effective stiffness of the SLS element to decrease, so that the behaviour upon release is no longer obvious: the truss may immediately snap back, or it may initially creep in an inverted state for a period of time. In particular, these regimes cannot be inferred by only considering the equilibrium states of the system. We note that Nachbar and Huang (1967) have analysed a similar truss using a Kelvin-Voigt model, and determined the onset of snap-through to an inverted state when a constant indentation force is suddenly applied. Here, we are interested in the dynamics of the snap-back when the indentation force is removed.
2.1 Governing equations
As shown in figure 2c, in the natural state the central springs are assumed to be inclined at an angle to the horizontal, and the springs are at their rest length; the distance between the pin joints at each base is . We assume that the central springs are linearly elastic with constant stiffness . The rest length of the vertical SLS element is , the dashpot has viscosity , and the upper springs have modulus (for the spring in parallel with the dashpot) and (for the spring in series).
Let be the downward displacement of the point mass from the natural state. To obtain an equation of motion for , we calculate the various forces exerted on the point mass. We write and for the corresponding inclination angle and change in length of the central springs, respectively. For simplicity, we assume that the truss remains shallow in shape, i.e. and . Neglecting terms of , simple geometry gives that (Gomez, 2018)
Because the central springs are linearly elastic, the vertical component of the total force exerted on the point mass by the central springs (directed downwards) is approximately .
The displacement also leads to a strain in the upper SLS element of size . The corresponding stress satisfies the constitutive law of a standard linear solid (Lakes, 1998) (with denoting time)
Note that the limiting case of a Maxwell material is recovered by setting , while the constitutive law for a Kelvin-Voigt material is recovered in the limit . The stress in the SLS element leads to a vertical force exerted on the point mass (directed downwards), where is the cross-sectional area of each element.
To make the problem dimensionless, it is natural to scale the displacement with the initial height of the truss in the small-angle approximation, i.e. . We scale time with the characteristic timescale of stress relaxation, , obtained by balancing the final two terms in (1). Balancing the remaining terms in equations (1)–(2), we introduce the dimensionless variables
Here we have chosen the stress scale so that the constitutive equation for an elastic solid is simply in dimensionless variables. Inserting these scalings into (1), and eliminating the strain for the dimensionless displacement , we obtain
where we define
The parameter plays a key role in the stability of viscoelastic structures (Urbach and Efrati, 2018), as it measures the degree of stress relaxation that occurs in response to a step increase in strain; we shall discuss this further below when considering the behaviour of the truss when the indenter is applied. The parameter may also be interpreted as the ratio of the timescale of stress relaxation () to the characteristic timescale over which creep occurs (), obtained by balancing the first two terms in equation (1). The related parameter is also referred to as the relaxation strength. The value of is governed by the physical mechanisms causing viscoelastic behaviour, such as molecular processes (e.g. molecular rearrangement in polymers) or the effects of coupled field variables (e.g. fluid flow in poroelastic materials); for a detailed discussion see Lakes (1998). Here we assume that is a known material constant, which can be measured experimentally using relaxation tests (Urbach and Efrati, 2017). We also note that in this non-dimensionalisation, the limit corresponds to a Maxwell material, since this is equivalent to setting in equation (1). This limiting case more closely resembles fluid-like behaviour in which the material has no preferred natural state and simply relaxes to the current configuration (Urbach and Efrati, 2018). The opposite limit corresponds to a purely elastic material, in which stress relaxation does not occur and the solution of (3) is simply for all times. Note that the Kelvin-Voigt model (i.e. sending in (1)) cannot be obtained in this non-dimensionalisation, since we have scaled time by the relaxation timescale . (This limiting case can only be obtained by first rescaling time by the creep timescale before sending .)
In terms of dimensionless variables, the momentum equation (2) can be written as
where we have introduced the dimensionless parameters
Here, the Deborah number measures the ratio of the timescale of stress relaxation to the characteristic timescale of the experiment (Howell et al., 2009), which here is the timescale of elastic oscillations (). (Hence, in this non-dimensionalisation, the viscous timescale is while the elastic timescale is .) We may interpret as the relative stiffness of the upper SLS element compared to the central springs. The cubic term on the right-hand side of (5) represents the dimensionless force due to the central springs. As expected, this vanishes in the undeformed state , the reflected state (when the central springs are also at their natural length), and the intermediate displacement when the springs are aligned horizontally — in this state they are compressed but do not contribute any vertical force.
2.3 Steady solutions
When the system is in equilibrium with elastic constitutive law , the momentum equation (5) implies that the indentation force must balance the total force exerted by the central springs and the SLS element, which we label . In particular, the force associated with a steady displacement is
When the indentation force is removed, any equilibria must satisfy , which has roots
For , there are two real non-zero solutions, which coincide and disappear at a saddle-node bifurcation when ; the corresponding displacement at this point is . For , the only real solution is the undeformed state, . This behaviour is apparent in figure 3a, which plots the force-displacement curve for different values of ; we see that increasing (corresponding to a stiffer SLS element) acts to rotate the curve anti-clockwise about the origin, until eventually the turning point of the cubic lies above the line . The corresponding behaviour of the roots to is shown in figure 3b. It can be shown that the roots in which where (solid branches on figure 3b) are linearly stable, while the root in which (dotted branch) is linearly unstable (Panovko and Gubanova, 1987).
2.4 Indentation response
In a snap-through thought experiment, we imagine indenting the truss to an inverted state by imposing the constant displacement , for a time interval of duration (for later convenience, we define to be the time at which the indenter is released). To avoid introducing additional timescales into the problem, we suppose that the indentation is suddenly applied at , i.e. over a timescale much faster than the viscous timescale . We can then approximate the behaviour for as
where is the Heaviside step function. Substituting this into the constitutive equation (3), and solving for the stress in the upper SLS element, we obtain
This solution is classical in the literature and represents the stress relaxation of a standard linear solid under a step increase in strain (Lakes, 1998; Santer, 2010): the stress initially (i.e. at ) jumps instantaneously to a fully unrelaxed value when the indentation is applied, and then decays exponentially to the fully relaxed value associated with an elastic material.
Inspecting the momentum equation (5), we see that the effect of this relaxation is to give an effective value of that changes in time. The corresponding indentation force can be written as
where the effective value of is
Note that decreases from to during relaxation. In terms of the force-displacement curve in figure 3a, this corresponds to rotating the curve clockwise as stress relaxation occurs, so that the indentation force decreases in time. From this picture, we anticipate that there are different dynamical regimes when the indenter is released, depending on the values of and . For , the turning point on the cubic lies below the line in the fully unrelaxed state (since ), and moves further below this line as relaxation occurs. Hence the truss is bistable at the moment when the indenter is released, and we do not expect snap-through to occur if the indentation displacement is sufficiently close to the stable non-zero root of . Similarly, for , the turning point lies above the line when the structure is fully relaxed, and so the truss is always monostable; we expect snap-through to occur for any value of and . For , the turning point lies above the line when the structure is fully unrelaxed (since ), but eventually decreases below this line as stress relaxation occurs. In particular, the truss is effectively bistable when the indenter is released (i.e. ) provided that , which can be re-arranged to give
We then expect snap-through to generally not occur if the inequality (6) is satisfied, and to occur otherwise. We will show that while this naïve argument correctly accounts for different dynamical regimes, it fails to quantitatively predict when snap-through occurs because of the effects of inertia.
For later reference, we shall write for the value of the indentation force just before the indenter is released, i.e. at . From above, this is given by
2.5 Dynamics of release
At , the indenter is suddenly released so that the indentation force
Due to the presence of inertia, and must be continuous across (writing ), giving the initial conditions
The jump in acceleration here is necessary to balance the discontinuity in the applied indentation force.
Currently, we have five dimensionless parameters in the problem: the Deborah number , relative stiffness , relaxation parameter , indentation depth , and indentation time . Throughout this paper, we restrict our attention to indentation depths . As baseline values we use (i.e. both springs in the SLS element have equal modulus, ) and , i.e. the displacement at the saddle-node bifurcation; in this case the initial conditions (9) are analogous to those studied by Gomez et al. (2017a) for purely elastic snap-through. We expect to recover similar behaviour here in the elastic limit , i.e. we expect the dynamics are governed by the elastic timescale and only slow down considerably near the saddle-node bifurcation at . However, for values , it is not clear when the dynamics are instead governed by viscous relaxation. To gain insight, we focus on the limit , which corresponds to a relaxation timescale that is much slower than the elastic timescale. This is the relevant regime for many structures composed of rubbery polymers, such as silicone-based elastomers typically used in morphing devices (Brinkmeyer et al., 2012, 2013; Urbach and Efrati, 2017), where molecular rearrangement underlying viscoelastic behaviour occurs over slow timescales (Lakes, 1998).
3 Snap-through dynamics:
3.1 Numerical solution
Typical dimensionless trajectories in the limit are shown in figures 4a–d. These are obtained by integrating the ODE (8) with initial conditions (9) numerically in matlab (routine ode45, error tolerances here and throughout). Figure 4 shows that the initial jump in acceleration causes oscillations to occur on the fast elastic timescale ; these oscillations persist due to the absence of external damping in our model. As anticipated from the discussion in §2.4, there are different regimes depending on the size of . For ( with ), the truss appears never to snap and instead approaches the stable non-zero root of (figure 4a). For , the truss snaps back to the natural state for small enough values of , but remains in an inverted state indefinitely for larger (figure 4b). For , the truss appears to snap for any value of . However, the dynamics slow down considerably when and is sufficiently large; see figure 4c. In this regime the oscillations are rapidly damped out, and the trajectory features an initial plateau before abruptly accelerating towards the natural configuration (highlighted in the lower panel of figure 4c), reminiscent of the dynamical bottleneck caused by a saddle-node ghost (Gomez et al., 2017a). For larger values of , the dependence on decreases and this initial bottleneck phase is not observed (figure 4d).
These regimes are confirmed when we analyse the snap-through time, (defined as the time at which the displacement first crosses the natural displacement, ); the computed times are shown on the -plane for the baseline parameter values in figure 5a. The blank regions on the figure correspond to regions where snap-through does not occur (after integrating the equations up to , which was found to be sufficient due to the limited amount of slowing down in figure 5a). This shows that the critical value of at which snap-through no longer occurs with increases nonlinearly as increases, and appears to approach a finite value as . For comparison, we have also plotted the naïve prediction (6) based on whether the truss is effectively bistable at the moment when the indenter is released (green dashed curve). This provides a good approximation for smaller values of , but increasingly over-predicts the critical value of as increases, with the predicted value diverging as . (For later comparison, the boundary predicted by the multiple-scale analysis in §3.2 is shown as a red dotted curve).
Another key feature of figure 5a is that the snap-through time is very small throughout most of the parameter space. In fact, we will show that here the elastic oscillations cause the truss to immediately cross , so that . Figure 5a also confirms that the snap-through time only becomes or larger in a very narrow region of the parameter space, where and . A zoom of this region is provided in figure 5b, which shows that considerable slowing down can occur. In fact, the snap-through time appears to increase without bound as we take in this region. We will show that this is precisely the pseudo-bistable regime: here the displacement initially oscillates around an inverted state and does not immediately cross . As with the trajectories in figure 4c (lower panel), this inverted state also undergoes a slow creeping motion until the truss rapidly accelerates towards the natural state, so that . This difference in timescales (i.e. a slow creep followed by a rapid snap-through event) is considered to be a distinguishing feature of pseudo-bistable behaviour (Brinkmeyer et al., 2012, 2013).
Computed snap-through times for different values of are shown in figures 6a–b. These show that the boundary at which snap-through no longer occurs is qualitatively different depending on whether or . For a shallower indentation , the boundary appears to be shifted entirely to the left of the line , and there is no longer a region where the dynamics slow down considerably (figure 6a). The truss also snaps at values when is sufficiently small. In contrast, for deeper indentations , the boundary intercepts the line and the size of the pseudo-bistable region may be significantly larger compared to the case (figure 6b). For different values of the relaxation parameter , we observe a qualitatively similar picture provided ; see Appendix A.
3.2 Multiple-scale analysis
To understand the above observations, we now perform a detailed analysis of the dynamics in the limit . The trajectories in figures 4a–d indicate that the displacement undergoes fast oscillations (on an timescale) around a value that varies on an timescale. This suggests that the dynamics can be understood asymptotically using the method of multiple scales (Hinch, 1991). We introduce the fast timescale defined by . Treating and as independent, the chain rule implies that
We seek an asymptotic expansion of the solution in the form
3.2.1 Leading order problem
We insert the expansion (11) into the ODE (8), re-scaling time in terms of . After Taylor expanding the and force terms about , and expanding the derivatives using (10), we obtain to leading order in the homogeneous problem
The initial conditions (9) become
Integrating the above equation for with these conditions, and simplifying using the expression (7) for , we obtain
where is unknown and satisfies .
To reveal the role that plays in the dynamics, we decompose the leading order solution into a “slow part” and a “fast part” respectively:
We specify that satisfies the “slow part” of the leading order equation (13), i.e.
Under the assumption that the “fast part” (see Appendix B), we Taylor expand the force term in (13) about and retain only linear terms in to obtain
Provided that (as justified later in Appendix C), the solutions are periodic; we denote the period by , which will vary on the slow timescale as varies. Integrating the equation from to then shows that for each
Returning to the way we decomposed the solution in (14), we see that corresponds to the mean value of and varies on the slow timescale . This evolution is captured by the variable . The variable describes the oscillations around this mean displacement that occur on the fast timescale ; the property (17) guarantees that these oscillations do not influence the mean value if their amplitude is small. We now show that it is possible to obtain an evolution equation for without requiring detailed knowledge of , using only the zero-mean property (17). (While it is possible to obtain an analytical expression for using the simplified equation (16), we do not pursue this here, as this introduces a further unknown function of the slow timescale — knowledge of will be sufficient to determine when snap-through occurs and the associated snap-through time.)
3.2.2 First order problem
At , the ODE (8) in terms of can be written
This represents a linear, inhomogeneous problem for . Setting the right-hand side to zero, we see that the homogeneous problem can be solved approximately whenever by taking , as in this case and so all derivatives vanish. The Fredholm Alternative Theorem then implies that we determine from the solvability condition associated with the approximate homogeneous solution (Keener, 1988). To formulate this condition, we simply integrate the first order problem from to . We assume that for each fixed , the solution is also a periodic function with period ; this is reasonable, since is forced by the terms that have period . It follows that all terms vanish in the integration and we are left with
This equation is exactly the original ODE (8) in the limit , i.e. neglecting the terms associated with inertia. This is perhaps not surprising: when the zero-mean property (17) holds, the fast elastic oscillations ‘cancel out’ on the slow viscous timescale and so do not affect the leading order dynamics. However, the above analysis does show that the correct initial condition is not the indentation displacement , as might be expected. Instead, from (15), satisfies
where the second equality follows from (7). This correction arises from the initial transient around in which inertia is always important; physically, the above equation states that the change in spring force in moving from to (when the SLS element is fully unrelaxed with effective stiffness ) must balance the discontinuity in the indentation force. When viewed on the slow timescale, the mean value then appears to change discontinuously from the indentation displacement .
To check our multiple-scale analysis, we integrate the simplified ODE (18) numerically subject to the initial condition (19). In figure 4c solutions are superimposed (as black dotted curves) onto the trajectories obtained by integrating the full ODE (8) for the baseline parameter values. We see that the agreement is excellent, with the multiple-scale solution indeed capturing the average behaviour of the displacement during snap-through (see lower panel in figure 4c). (The slight disagreement when the mean value changes rapidly on a timescale comparable to is because the multiple-scale analysis is no longer applicable.) Figure 4c also shows that the initial value may be much smaller than depending on the indentation time . We postpone a detailed analysis of to section §3.2.3 below and Appendix C.
3.2.3 Snap-through dynamics
We have shown that while the amplitude of the oscillations is small compared to the mean displacement, the leading order behaviour is given by
subject to the initial condition (19). Since (20) represents a first-order autonomous ODE for , the dynamics can be understood by considering the phase plane, for which the qualitative features are determined by the roots of and . For , there are two distinct non-zero stationary points, which correspond to the stable and unstable roots of ; for , the only stationary point is . The roots of correspond to vertical asymptotes where . Denoting these roots by , we find that
These roots are real and distinct if and only if .
Different cases for the phase plane are illustrated in figure 7. In Appendix C, we show that when (so that may diverge), the initial value must satisfy either or , i.e. always starts outside the interval between the two vertical asymptotes. We have therefore lightly shaded this region in figure 7. Figure 7 shows how when , there are no vertical asymptotes and the trajectories smoothly approach the natural state; this explains the observation that pseudo-bistable behaviour does not occur for larger values of , even though the snap-through time (discussed more in Appendix A). We see that pseudo-bistability is only possible if : the asymptotes on the phase plane correspond to rapid snap-through events, which occur after a slow creeping motion provided is sufficiently large (top right panel in figure 7). Since we are primarily interested in this pseudo-bistable regime, we restrict our attention to the case in the following analysis; in fact, we will restrict to so that we are safely in this regime in considering values of up to and slightly beyond 222As indicated by figure 1b in Appendix A, the truss may show different behaviour when is slightly below the critical value . Further analysis shows that as the left asymptote on the phase plane approaches (i.e. as ), the precise definition of becomes important; for example, the mean value may start to the left of both asymptotes so that the truss immediately snaps to near the natural configuration, but if is close to the truss does not cross (our definition of ) on the elastic timescale. By restricting to , we are able to bypass these technicalities for values of in the interval of interest for pseudo-bistable behaviour..
Focussing on , we deduce from figure 7 that there are three possibilities depending on where the solution starts in the phase plane:
If and , then the mean value starts to the right of both vertical asymptotes and approaches the stable non-zero root of . Because , the truss remains in an inverted position and there is no snap-through333An additional case is possible in which the unstable root of is larger than the vertical asymptote at , so that two stationary points lie to the right of both asymptotes on the phase plane (unlike the top left panel of figure 7, in which the unstable root lies in the lightly shaded region between the asymptotes). However, in focussing on values , this regime is found to occur only for values of extremely close to and so can generally be ignored..
If and , then the mean value starts to the right of both vertical asymptotes. It then decreases to the vertical asymptote at where rapid snap-through occurs and inertial effects again become significant — because this approach to the asymptote occurs on the slow timescale , the total time taken to snap-through is at least . This regime corresponds to pseudo-bistable behaviour, in which the snap-through time is governed by the timescale of viscous relaxation.
If , then the mean value starts to the left of the vertical asymptotes and smoothly decays to zero. Because , the truss immediately snaps back to near its natural configuration during the initial transient in which inertia is important. The amplitude of the elastic oscillations will therefore be large compared to the mean value in this case, and our assumption is no longer valid; nevertheless, we expect the truss to pass on an timescale so that .
The final task is to determine when the initial value satisfies . This is not obvious because (19) implies that is the root of a cubic polynomial, for which there may be multiple real solutions. The relevant solution can be found by analysing the phase portrait of equation (13): setting , this equation governs the elastic behaviour of the truss at very early times, and hence determines which root of (19) the solution approaches. When viewed on the slow timescale, this root corresponds to the relevant value of . The full analysis is provided in Appendix C. The key result is that if , then if and only if ; otherwise we have . Physically, this states that the indentation force needs to be adhesive for the truss to remain in an inverted state. This is intuitive: if the truss has to be ‘pulled’ upwards to the imposed indentation depth, it should move further downwards (increasing ) when the indenter is released. Using the expression (7) for , the condition can be expressed as
In the alternative case , the condition turns out to no longer be relevant. Instead, in Appendix C we show that
Physically, this condition arises from bounding the amplitude of the elastic oscillations at very early times, so that these do not ‘push’ the truss sufficiently far from the inverted state and cause an immediate snap-back.
Combining this with the phase-plane discussion above, the predicted dynamical regimes are shown schematically in figure 8. This explains how the qualitative features of the dynamics are very different in the two cases and . When , it may be shown that the boundary predicted by (21) reaches a vertical asymptote on the -plane when . For values below the boundary we have , and the above discussion implies that the truss immediately snaps with (shaded blue in figure 8). Above the boundary, and, because here, the truss does not snap-through. Pseudo-bistable behaviour therefore cannot be obtained when . Conversely, when , the boundary predicted by (22) reaches a vertical asymptote when . Hence, there is a region where and , in which pseudo-bistable behaviour occurs (shaded red in figure 8). We deduce that precisely when
To check the validity of the picture presented in figure 8, we have superimposed the boundaries predicted by (21)–(22) (purple dashed curves, red dotted curves respectively) onto the numerical snap-through times in figures 5–6 (and figure 1 of Appendix A). We observe that the agreement with the numerics is excellent when , despite the fact that the assumption made in the multiple-scale analysis is not formally valid throughout the range of values shown.
Figure 8 explains many basic features of pseudo-bistability that have been observed previously in experiments and numerical simulations (Santer, 2010; Brinkmeyer et al., 2012, 2013; Madhukar et al., 2014; Urbach and Efrati, 2017, 2018). We see that pseudo-bistability occurs only in a narrow parameter range, near the threshold at which snap-through no longer occurs (i.e. here) and the width of the pseudo-bistable region grows as the amount of stress relaxation increases (increasing ). Pseudo-bistable behaviour is not obtained if is too small, nor if the indentation depth is below a critical value. In addition, the phase plane in figure 7 explains how the truss initially creeps in an inverted state before abruptly accelerating, leading to the difference in timescales that is characteristic of pseudo-bistable behaviour (Brinkmeyer et al., 2012, 2013). We emphasise that the analytical understanding of these features presented here is, to the best of our knowledge, new.
The importance of inertial effects immediately after the indenter is released has not been appreciated previously. Because this causes the displacement of the truss to change rapidly from the indentation displacement, the effective stiffness will also change rapidly from its value just before the indenter is released. This is in direct contrast to the viscoelastic models used by Santer (2010) and Brinkmeyer et al. (2012, 2013), which assume that (i) the stiffness reverses back to its fully unrelaxed value when the indenter is released, and (ii) there is no rapid change in the stiffness caused by the discontinuity in the applied indentation force. In Appendix D we show that when we make assumptions (i)–(ii) in the framework of the truss model, we obtain radically different predictions of when snap-through and pseudo-bistability occur. In the type of snap-through experiment considered here (a structure is allowed to relax in a specified displacement before being abruptly released), assumptions (i)–(ii) cannot therefore be derived from first principles starting from the constitutive law for a standard linear solid. Instead, we believe it is necessary to couple the stress within the structure to its displacement and account for inertial effects when the indenter is removed.
3.2.4 Snap-through time in the pseudo-bistable regime
Another key feature of the dynamics is that the snap-through time increases considerably as in the pseudo-bistable regime, becoming much larger than (figure 5b). This slowing down does not require (since it can be observed when , figure 6b). The phase plane in figure 7 (top right panel) suggests that this slowing down is due to a saddle-node ghost: when the non-zero stationary point no longer exists, but as the trajectory passes increasingly close to the line at . Because the velocity becomes very small but non-zero, this will lead to a slow passage through a bottleneck.
To analyse this slowing down in detail, we set
where and ; here we anticipate an scaling for the change in displacement during the bottleneck phase, which is the generic scaling for overdamped dynamics near a saddle-node bifurcation (Strogatz, 2014). (We have also introduced a minus sign since we expect the displacement to decrease during snap-through.) Expanding the force terms appearing in the simplified ODE (20), we obtain
which (up to numerical pre-factors) is the normal form for overdamped dynamics near a saddle-node bifurcation (Strogatz, 2014); here the neglected terms are small compared to at least one retained term provided . The solution is
The snap-through time is dominated by the time spent passing through the bottleneck, , which can be determined by finding the time at which first reaches ; after this point, we no longer have and so the truss is moving rapidly. Using the expansion as , we have that when
It follows that there are different distinguished limits, depending on the size and sign of ; these are discussed in Appendix E. For example, for the baseline case , we insert the expansions (23) into the initial condition (19). Upon neglecting terms quadratic in and (e.g. with , figure 5b implies we have in the pseudo-bistable regime, so that ), we obtain
The above expression for the bottleneck duration then gives the leading order estimate
The distinguished limits correspond to and , and we obtain