Abstract
Determining the stability of a viscoelastic structure is a difficult task. Seemingly stable conformations of viscoelastic structures may gradually creep until their stability is lost, while a discernible creeping in viscoelastic solids does not necessarily lead to instability. In lieu of theoretical predictive tools for viscoelastic instabilities, we are presently limited to numerical simulation to predict future stability. In this work, we describe viscoelastic solids through a temporally evolving instantaneous reference metric with respect to which elastic strains are measured. We show that for incompressible viscoelastic solids, this transparent and intuitive description allows to reduce the question of future stability to static calculations. We demonstrate the predictive power of the approach by elucidating the subtle mechanism of delayed instability in thin elastomeric shells, showing quantitative agreement with experiments.
INTRODUCTION
The snapping of the Venus fly-trap leaf, one of the fastest motions in plant kingdom, is preceded by a relatively slow creeping motion (1). A similar creep is observed before the snap through of thin elastomeric shells known as jumping poppers (2). While the snap through itself lasts only a fraction of a second (2, 3), the slow creeping motion, during which the shells seems to be elastically stable, may last orders of magnitude longer. On a much larger scale, similar behavior is, in some cases, attributed to Earth’s crust before an earthquake aftershock (4, 5). The exact role of viscoelasticity in aftershocks is not fully understood, partially because of the absence of predictive theoretical framework of future stability in these systems. In all of the abovementioned systems, the slow viscoelastic flow in the material leads the system to an instability that abruptly releases the internally stored elastic energy. While in most cases we are able to explicitly write down the equations governing the viscoelastic behavior, the mechanism of delayed instabilities in viscoelastic solids remains poorly understood.
Viscoelastic solids display reversible elastic behavior when loaded at a fast rate. However, they also show a slow creeping behavior under a constant load and exhibit stress relaxation when held at constant displacement. Commonly, these solids are modeled by a constitutive law relating the stress rate to the stress, strain, and strain rate or equivalently by expressing the stress as a function of the history of strain rate and a material-dependent memory kernel (6). While these approaches capture the material response well and yield accurate results through simulations of viscoelastic structures (7, 8), they are rarely explicitly solvable and provide little intuition regarding the state of the viscoelastic material and, in particular, have little in the way of clarifying the governing processes in viscoelastic instabilities. Recent attempts to address viscoelastic instabilities by modeling the viscoelastic response as an elastic medium with temporally evolving stiffness (7–9) show only a qualitative agreement with experiment and have limited applicability as, in particular, they cannot capture creeping at zero load.
In this work, we quantitatively address the phenomenon of viscoelastic instability through a metric description. In this metric approach, we describe the materials’ behavior as a fast elastic response with respect to temporally evolving rest lengths that change because of the slow viscoelastic flow. The microscopic response in the material that leads to stress relaxation is interpreted as the evolution of the rest lengths of the system toward the lengths assumed in its present state (see Fig. 1). The instantaneous rest lengths in the system serve as a state variable accounting for the slow and inelastic evolution in the material. This approach is somewhat reminiscent of the additive decomposition of strains (10, 11); however, using the instantaneous reference metric rather than strain as the basic ingredient of the theory allows for a much more transparent treatment that, in particular, allows to predict future stability of unconstrained viscoelastic structures.
At the resting state, all three length measures on the body, its measured length g (marked red), its instantaneous reference length
The key to our analysis of stability in viscoelastic materials is the discovery of a new stationarity property; if an incompressible viscoelastic system starting at rest is brought abruptly to another locally stable state, then despite the continuous evolution of the internal stresses in the structure, it will display no motion and, in particular, will never lose stability. This result both elucidates the subtle nature of delayed instabilities in these systems and paves the path for their quantitative understanding.
THEORETICAL RESULTS
Linear viscoelastic continua
All linear viscoelastic material constitutive relations and, in particular, the relations for every spring dashpot model, can be expressed through a convolution of the strain rate with a stress relaxation function,
Here, C is the material stiffness and ϕ(s) is the normalized memory kernel that satisfies ϕ(0) = 1, ϕ(∞) = 0, and as we expect, the memory to only decay in time also satisfies
Here, Cijkl is the isotropic stiffness tensor, which, in three dimensions, depends only on two constants, often interpreted as the Young’s modulus, Y, and the Poisson’s ratio, ν. As the material is assumed to be incompressible, ν = 1/2 for both the elastic and viscoelastic response tensors, rendering them proportional to each other. In this case, the viscoelastic evolution is fully captured by a single scalar function. Sij(t) denotes the second Piola-Kirchhoff stress tensor, which, through the equation above, depends on the full history of the the strain tensor
The metric gij measures lengths in the material and uniquely describes the configuration of the elastic body. We have also introduced the rest reference metric,
Equation 2 predicts that instantaneous incremental deformations, Δgij, lead to linear stress increments,
The temporal evolution of the instantaneous reference metric can be deduced from Eqs. 2 to 4 and reads
It is important to stress that Eqs. 4 and 5 are completely equivalent to Eq. 2. However, the notion of an instantaneous reference metric provides a more transparent description of the viscoelastic system and allows a more intuitive understanding of its dynamics. Note that
Therefore, β also measures the material’s propensity to change its instantaneous reference metric toward the realized metric in its present configuration. β = 0 describes a purely elastic material, where the instantaneous reference metric remains
The quasi static approximation
Viscoelastic systems are dissipative; thus, the notion of an elastic free energy is ill defined. Nonetheless, the virtual work of displacements performed over a short period Δt → 0, coincides with the instantaneous elastic energy functional (16)
Viscoelastic instabilities through the metric description
A given instantaneous reference metric
Locally stable configuration stationarity. The first result, termed locally stable configuration stationarity, states that if a system at rest is brought instantaneously to a locally stable configuration, then not only it will never lose stability but also the configuration will remain stationary; i.e., the configuration will display no temporal variation despite the continuous evolution of the instantaneous reference metric and the relaxation of the corresponding stress. To prove this result, consider a system at rest, for which
This evolution of the reference metric, in turn, causes the stress to diminish by a uniform time dependent factor according to
It is thus straightforward to see that g* remains the mechanical equilibrium despite the evolution of
The minimization of the metric g (marked by a full black circle) is constrained and performed with respect to the subset of metrics that correspond to realizable configurations (thick black line). These metrics are in, particular, orientation preserving and Euclidean. Given an instantaneous reference metric,
Conversely, if a system converges to a fixed stable state (wherein the configuration and the instantaneous reference metric do not evolve), then the corresponding configuration must also be extremal with respect to the rest reference metric. This again could be shown by substituting Eq. 6 into Eq. 8. We note that these results are not limited to a particular memory kernel. Intuitively, both claims are due to collinearity of
Transient acquired elastic stability. The notion of locally stable state stationarity does not imply that viscoelastic systems cannot creep into instabilities, but rather constrains the settings in which these instabilities can occur. Starting from rest, a configuration may be elastically stable or elastically unstable. Neither case shows a slow creeping motion. Thus, if a system displays creeping, then it was not brought to its state from rest; it must have been held under external load for some time, allowing its instantaneous reference metric to evolve. This evolution of the instantaneous reference metric may lead unstable configurations to acquire stability. These acquired stability configurations are configurations that, if arrived to from rest, will be unstable (7, 8). However, after a long enough holding period under external load, they will become locally elastically stable. The instantaneous reference metric in this case would not remain stationary, and its evolution will manifest as the slow creeping of the configuration. It is straightforward to see that this creeping motion must lead to instability, rendering the acquired stability transient. Assume in contradiction that an acquired stability configuration becomes stationary, i.e., both g and
EXPERIMENTAL RESULTS
The results presented above explain many of the qualitative phenomenology of viscoelastic instabilities. We next come to test the quantitative predictions of the theory by experimentally examining the response of silicone-rubber conical poppers. We cast silicon rubber poppers in the geometry of truncated conical shells (Fig. 3A; see also the Supplementary Materials). The conical popper geometry behaves similarly to the spherical poppers, yet this geometry allows simpler control over the thickness of the sheet. The conical shells have an apex angle of 45∘, inner and outer radius radii rmin,rmax, and thickness h. Sufficiently thin poppers show bistability with the inverted shape close to a mirror image of the rest state. As the thickness is increased, this bistability is lost, and when brought from rest to the inverted state, the popper immediately snaps back. If the thickness is large enough, then this instability will persist regardless of how long we hold the conical shell in its inverted state. For intermediate values of the thickness, we expect to observe unstable states that could acquire stability if held long enough in their inverted state. This acquired stability is expected to be lost in a finite time. These three phases are plotted in Fig. 3 (B and C).
(A) Straight and inverted conical poppers. Photo credit: Erez Y. Urbach, Weizmann Institute. (B) The two axes span the dimensionless geometrical properties of the truncated conical poppers. The background colors represent the theoretically predicted regions of each of the phases. Each marker corresponds to a different popper; different shaped (and colored) markers indicate the different phases observed in experiment. (C) Numerically calculated flipping time as a function of the normalized thickness of the conical popper for immediate release and long holding time. The different poppers were simulated by varying their thicknesses and constant radii rmin = 10 mm, rmax = 25 mm. The material properties taken were β = 0.1, and the memory kernel was assumed to be exponential with τ = 0.1 s, Young’s modulus E = 2.5 MPa, and Poisson’s ratio v = 0.47. Varying the kernel may lead to varying rate of divergence of the flipping time between the stable and acquired stability region, yet the location of this divergence will remain unchanged. The divergence of flipping times was addressed in (8), and more recently, the rate of divergence was studied in (18).
The conical shell popper geometry is completely determined by two dimensionless variables, h/rmin and rmax/rmin. The boundaries between the different stability phases of Fig. 3C depend on these geometric parameters and on the material constant β. We produced ∼50 different conical popper of different geometries and tested their phases. All popper were produced from the same material and share the same β. Starting from rest
Theoretically determining the phase boundaries of delayed stability can be achieved by performing only static calculations, which we solved numerically (see the Supplementary Materials). Determining the line separating the stable configurations from unstable configurations (top line separating the white and blue regions in Fig. 3) is carried out by stability analysis with respect to
DISCUSSION
The metric description of viscoelasticity presented here is closely related to the additive decomposition of strains in elastoplasticity (11). One can further generalize the theory presented here to account for plasticity or growth within the material by allowing
When implemented to isotropic incompressible viscoelastic solids, the metric theory provides the basic rules for viscoelastic instabilities. For a given structure to creep into instability, the creeping must have been preceded by an aging stage where the structures were held under external load for a finite time. When viscoelastic structures are held fixed in a nonextremal configuration, stress relaxation could lead to new acquired stable configurations, yet this acquired stability must be lost in time. The general metric theory of viscoelasticity applies to all linear-viscoelastic materials regardless of the form of their memory kernel. However, the result of locally stable configuration stationarity was obtained under the restricting assumptions that the Poisson’s ratio associated with the elastic and viscoelastic response are similar, and that the system is brought instantaneously from its rest state to the locally stable state. The former assumption is plausible for incompressible elastomers and gels. The latter assumption, however, is expected to be violated by every physical experiment. While one could not prove full stationarity if these assumptions are lifted, the material’s response is continuous in the deviations from the idealized conditions and small deviations will result in very little motion. In particular, in the experiments that we performed, no discernible motion was observed for locally stable states.
The theory proves particularly powerful when applied to describe the delayed instability in elastomeric thin shells. We focused on stability classification in the present work, yet the theory is capable of quantitatively describing the nontrivial dependence of jumping time on the holding time in the inverted state and, in particular, its divergence near the boundary between the unstable regimes and stable regime in Fig. 3C. In the vicinity of this boundary, very short holding times could lead to arbitrarily long delays before the inevitable instability.
While in some specific cases we have compelling evidence for the role of viscoelasticity in the delayed triggering of earthquake aftershocks (4), in the general case, its role as the instigator of delayed instabilities is still unknown. In lieu of a theoretical framework for viscoelastic instabilities, the evidence for the relevance of viscoelasticity to aftershocks mostly relies on numerical simulations relevant to a specific setting. The metric description proposed here aims to provide a theoretical framework for these delayed viscoelastic instabilities. Slip events are described as fast variations in
SUPPLEMENTARY MATERIALS
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/36/eabb2948/DC1
This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.
REFERENCES AND NOTES
- Copyright © 2020 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).