Pseudo-bistability of viscoelastic shells

Abstract

Viscoelastic shells subjected to a pressure loading exhibit rich and complex time-dependent responses. Here we focus on the phenomenon of pseudo-bistability, i.e. a viscoelastic shell can stay inverted when pressure is removed, and snap to its natural shape after a delay time. We model and explain the mechanism of pseudo-bistability with a viscoelastic shell model. It combines the small strain, moderate rotation shell theory with the standard linear solid as the viscoelastic constitutive law, and is applicable to shells with arbitrary axisymmetric shapes. As a case study, we investigate the pseudo-bistable behaviour of viscoelastic ellipsoidal shells. Using the proposed model, we successfully predict buckling of a viscoelastic ellipsoidal shell into its inverted configuration when subjected to an instantaneous pressure, creeping when the pressure is held, staying inverted after the pressure is removed, and eventually snapping back after a delay time. The stability transition of the shell from a monostable, temporarily bistable and eventually back to the monostable state is captured by examining the evolution of the instantaneous pressure–volume change relation at different time of the holding and releasing process. A systematic parametric study is conducted to investigate the effect of geometry, viscoelastic properties and loading history on the pseudo-bistable behaviour.

This article is part of the theme issue 'Probing and dynamics of shock sensitive shells'.

1. Introduction

Shell buckling has been intensively studied for almost a century [1], and will continuously attract the great attention of researchers in the field of solid mechanics. Not only are shell structures broadly used in various industrial sectors [2–4], but also the analysis of shell buckling can improve the understanding of the motion and morphogenesis of organisms [5,6], which further inspires the design of materials and devices with novel functionalities endowed by the nonlinearity and instability of shells [7,8].

Elastic shells can undergo snap-through buckling when loaded, in which they rapidly reconfigure from one state to another when no stable equilibrium state exists nearby. Such instabilities are ubiquitous in nature and our daily life: Venus flytrap, the famous plant capable of fast movement, can rapidly flip its bistable leaves to catch prey [6], while hair clips can be quickly sprung up and down by fingers. The snap-through buckling of elastic shells has been widely employed to achieve rapid deformation in various applications, such as soft actuators [9–12], logic switches [13,14], and responsive surfaces [15]. Many of these works are based on elastomeric shells, most of which exhibit viscoelasticity. The snap-through buckling of shells could be profoundly influenced by viscoelasticity manifested by creep and stress relaxation. One example is inducing the so-called ‘pseudo-bistable’ behaviour [16–21], which can be illustrated by children's jumping poppers [17]. A jumping popper is a rubber spherical cap that can be buckled into an ‘inside-out’ configuration. After being held for a while and released from the load, the inverted popper undergoes slow creeping for certain delay time, as if it is in a stable equilibrium state, before rapidly snapping back to its natural shape. The recovery time is governed by the viscous time scale of the material and is much longer than that for elastic snap-through buckling. This pseudo-bistable behaviour has been harnessed to achieve spatio-temporal control of morphing structures [16,22–24].

The mechanism of the pseudo-bistability exhibited in buckled viscoelastic shells, however, is far from being clear. The straightforward explanation is that the originally monostable shell can temporarily acquire stability while it is held in its buckled state. When the buckled shell is released, it gradually loses its stability during creeping and eventually snaps back. However, quantitative modelling of viscoelastic shells is essential to support this explanation. One effort is describing a viscoelastic shell by a discrete spring–mass–dashpot system [17,21], where the stability transition is attributed to the change in the ratio of bending to stretching energy caused by viscoelasticity. This argument is not entirely convincing, since it is unclear how the ratio of bending to stretching energy evolves in viscoelastic shells, and how it is related to the stability. Using finite-element simulations together with experiments, researchers capture the pseudo-bistable behaviour of viscoelastic shells [16,21,22]. Although the numerical simulations can accurately predict the responses of viscoelastic shells, they provide limited information on their stability. Recently, Urbach and Efrati proposed a new approach in predicting the stability of viscoelastic solids [20]. In this framework, the behaviour of viscoelastic solids is modelled as an elastic response with respect to a temporally evolving instantaneous reference metric, which determines the stability of the solids. While this approach provides insights into the process of temporarily acquiring and eventually losing stability in the pseudo-bistable behaviour, the understanding of this delayed phenomenon is still unsatisfactory since the instantaneous reference metric is complex and abstract, and moreover, the corresponding surface may not exist in three-dimensional Euclidean spaces.

In this paper, we aim to model and explain the pseudo-bistability phenomenon by developing a viscoelastic shell model. In our previous work, we have established a shell model for viscoelastic spherical shells [25], based on the small strain, moderate rotation shell theory [26,27] combined with the viscoelastic material law of standard linear solids. Here we further extend the viscoelastic shell model to shells with arbitrary axisymmetric shapes. As a case study, we will investigate the pseudo-bistable behaviour of viscoelastic ellipsoidal shells. Since pseudo-bistability tends to occur in deep shells, which have relatively high strain and rotation when fully inverted, in order to capture the pseudo-bistable behaviour, but at the same time, to limit the deformation within the assumption of small strain and moderate rotation, we will carefully investigate and properly select the geometry, viscoelastic properties and boundary conditions of the shells. Using the proposed shell model, we will predict buckling of a viscoelastic ellipsoidal shell into its inverted configuration when subjected to an instantaneous pressure load, and snapping-through after a delay time when the pressure load is held constantly for a while prior to being removed (figure 1). Moreover, we will use the model to probe the stability of the shell at different times during the holding and releasing process by plotting the corresponding instantaneous pressure–volume change relations. The evolution of the instantaneous pressure–volume change relation confirms the stability transition of the shell from a monostable state, temporarily bistable state and eventually back to the monostable state over time. Finally, the critical creeping time, the minimum time period within which the pressure is held to achieve pseudo-bistability, as well as the recovery time, the delay time before a shell snaps back when the pressure is removed, are predicted using the proposed shell model.

Figure 1.

Schematic of the pseudo-bistable behaviour exhibited in viscoelastic shells. A viscoelastic shell buckles into the inverted configuration when an instantaneous pressure ΔP is applied. It creeps under the constant ΔP for a while before being released. The shell does not recover immediately, but instead, stays inverted as if it were bistable. After a delay time, the shell snaps back to its unbuckled configuration.

This paper is structured as follows. In §2, a model for viscoelastic shells of arbitrary axisymmetric shapes is formulated by combining the small strain, moderate rotation shell theory with the linearly viscoelastic constitutive relation. The equilibrium equations are derived using the principle of virtual work. In §3, the buckling of ellipsoidal shells of different geometry and viscoelastic properties at different loading rates are investigated to provide insights into the pseudo-bistability of viscoelastic shells. In §4, the pseudo-bistable behaviour is captured by the proposed shell model. The instantaneous pressure–volume change relations at different times during the holding and releasing process are obtained to probe the stability transition. In §5, a parametric study on the critical creeping time and the recovery time is conducted. The conclusion is made in §6.

2. Modelling viscoelastic shells with arbitrary axisymmetric shapes

(a) . Small strain, moderate rotation shell theory

The schematic of a shell structure with thickness h is shown in figure 2a. Here we limit ourselves to axisymmetric shells about the e₃ axis. Two surface coordinates (θ, ω) are used to describe the mid-surface of the shell, in which θ is the meridional angle ranging from θ_min to π/2 at the pole, and ω is the circumferential angle (not shown in figure 2a). The mid-surface radius R(θ), which quantifies the shape of the shell, could be any smooth function of θ.

Figure 2.

(a) Schematic of a viscoelastic shell. The shell with thickness h is subjected to a live pressure load ΔP. The coordinate θ is the meridional angle, ranging from θ_min to π/2. R(θ) represents the middle surface radius of the undeformed shell. The shell can slide freely along the e₁ axis but has zero displacement along e₃ at θ = θ_min, and is subjected to the axisymmetric boundary condition about the e₃ axis at θ = π/2. (b) A standard linear solid containing a Maxwell model with a spring of modulus E₁ and a dashpot of viscosity η, connected in parallel to a spring of modulus E_∞.

The small strain, moderate rotation shell theory [26–28] is used to describe the deformation of viscoelastic shells. The position vector x of a material point with a coordinate (θ, ω) on the mid-surface of the undeformed shell can be expressed in the three-dimensional Euclidean space as

$$\mathit{x}(\theta ,\omega )=[R(\theta )\cos \theta \cos \omega ]{\mathit{e}}_1+[R(\theta )\cos \theta \sin \omega ]{\mathit{e}}_2+[R(\theta )\sin \theta ]{\mathit{e}}_3,$$ 2.1

where {e₁, e₂, e₃} is a group of orthonormal bases in the Euclidean space. The displacement of this material point can be written as

$$\mathit{\delta }(\theta ,\omega )=u^{\beta }{\mathit{x}}_{,\beta }+w\mathit{N}\text{, }$$ 2.2

where x_,β = ∂x/∂β and N denote the covariant bases and the normal vector of the mid-surface at the undeformed state, respectively, and (u^β, w) are the corresponding displacements. A Greek index takes on values of θ and ω, and a repeated Greek index means summation over θ and ω. In this paper, we only consider axisymmetric deformations. As a result, u^θ and w are only functions of θ, u^ω = 0, and the rotation about x_,θ, φ_ω = φ^ω = 0. The corresponding non-zero mid-surface strains and curvature strains under axisymmetric deformation are [26,27]

$$\begin{array}{ll}& E_{\omega }^{\omega }=u^{\theta }{\mathrm{\Gamma }}_{\theta \omega }^{\omega }+b_{\omega }^{\omega }w,\\ & E_{\theta }^{\theta }={u^{\theta }}^{\mathrm{\prime }}+u^{\theta }{\mathrm{\Gamma }}_{\theta \theta }^{\theta }+b_{\theta }^{\theta }w+\frac{1}{2}{\phi }^2{g}_{\theta \theta },\\ & K_{\omega }^{\omega }=\phi {\mathrm{\Gamma }}_{\theta \omega }^{\omega }\\ \text{and}& K_{\theta }^{\theta }={\phi }^{\mathrm{\prime }}+\phi {\mathrm{\Gamma }}_{\theta \theta }^{\theta },\end{array}\}$$ 2.3

where ${(\cdot )}^{\mathrm{\prime }}$ denotes $\text{d}(\cdot )/\text{d}\theta$, φ = φ^θ = g^θθφ_θ, where φ_θ denotes the rotation about x_,ω,

$$\phi =-w^{\mathrm{\prime }}{g}^{\theta \theta }+b_{\theta }^{\theta }{u}^{\theta },$$ 2.4

${\mathrm{\Gamma }}_{\theta \omega }^{\omega }$ and ${\mathrm{\Gamma }}_{\theta \theta }^{\theta }$ are Christoffel symbols (equation (A 22)), g_αβ and g^αβ are the covariant and contravariant components of the first fundamental form of the mid-surface (equation (A 19)), and $b_{\alpha }^{\beta }$ are the mixed components of the second fundamental form of the mid-surface (equation A 20). Compared to spherical shells, non-spherical shells have much more complex strains in equation (2.3) due to the non-constant R(θ), since all the coefficients of u^θ, w, φ and their derivatives depend on θ. The mid-surface and curvature strains in equation (2.3) can be expressed in terms of u^θ and w, or equivalently in terms of φ and w. Here we choose φ and w as the two independent variables by replacing u^θ with a function of φ and w obtained from equation (2.4). The strain of the shell at an arbitrary position can be expressed as ${\epsilon }_{\alpha }^{\beta }=E_{\alpha }^{\beta }+z{K}_{\alpha }^{\beta }$, where z is the coordinate in the thickness direction of the shell and measured from the mid-surface.

(b) . Viscoelastic constitutive relations

Following our previous work [25], here we develop a viscoelastic constitutive relation for viscoelastic shells. We use the Boltzmann superposition principle to quantify the effect of strain history on the current stress state. The two-dimensional stress–strain relation of viscoelasticity under plane stress can be written as

$${\sigma }_{\alpha \beta }(t)={\int }_0^t\frac{E(t-\tau )}{1-{\upsilon }^2}\left[(1-\upsilon )\frac{\text{d}{\epsilon }_{\alpha \beta }(\tau )}{\text{d}\tau }+\upsilon \frac{\text{d}{\epsilon }_{\gamma }^{\gamma }(\tau )}{\text{d}\tau }{g}_{\alpha \beta }\right]\text{d}\tau ,$$ 2.5

where E(t) is the relaxation modulus as a function of time t, υ is the Poisson's ratio, assumed to be a constant, and α and β are two free indices taking on values θ and ω. We quantify the material viscoelasticity using the standard linear solid model (figure 2b), which contains a Maxwell model with a spring of modulus E₁ and a dashpot of viscosity η, connected in parallel to a spring of modulus E_∞. Accordingly, the relaxation modulus takes the following form

$$E(t)=E_{\mathrm{\infty }}+E_1{\text{e}}^{-t/{\tau }_r},$$ 2.6

where τ_r = η/E₁ represents the relaxation time.

Integrating the stresses in equation (2.5) and stresses multiplied by distance over the thickness yield the resultant membrane stresses N_αβ(t) and the bending moments M_αβ(t) at time t, respectively,

$$N_{\alpha \beta }(t)={\int }_{-h/2}^{h/2}{\sigma }_{\alpha \beta }(t)\hspace{0.17em}\text{d}z,\text{and}{M}_{\alpha \beta }(t)={\int }_{-h/2}^{h/2}{\sigma }_{\alpha \beta }(t)z\hspace{0.17em}\text{d}z.$$ 2.7

Substituting the non-zero strains in equation (2.3) and the constitutive relation in equation (2.5) into equation (2.7), we obtain the non-zero resultant membrane stresses and the bending moments,

$$\begin{array}{l}{\displaystyle N^{\omega \omega }(t)=\frac{h{g}^{\omega \omega }}{1-{\upsilon }^2}{\int }_0^{t}E(t-\tau )\left[\frac{\text{d}{E}_{\omega }^{\omega }(\tau )}{\text{d}\tau }+\upsilon \frac{\text{d}{E}_{\theta }^{\theta }(\tau )}{\text{d}\tau }\right]\text{d}\tau ,}\\ {\displaystyle N^{\theta \theta }(t)=\frac{h{g}^{\theta \theta }}{1-{\upsilon }^2}{\int }_0^{t}E(t-\tau )\left[\frac{\text{d}{E}_{\theta }^{\theta }(\tau )}{\text{d}\tau }+\upsilon \frac{\text{d}{E}_{\omega }^{\omega }(\tau )}{\text{d}\tau }\right]\text{d}\tau ,}\\ {\displaystyle M^{\omega \omega }(t)=\frac{h^3{g}^{\omega \omega }}{12(1-{\upsilon }^2)}{\int }_0^{t}E(t-\tau )\left[\frac{\text{d}{K}_{\omega }^{\omega }(\tau )}{\text{d}\tau }+\upsilon \frac{\text{d}{K}_{\theta }^{\theta }(\tau )}{\text{d}\tau }\right]\text{d}\tau }\\ \text{and}{\displaystyle M^{\theta \theta }(t)=\frac{h^3{g}^{\theta \theta }}{12(1-{\upsilon }^2)}{\int }_0^{t}E(t-\tau )\left[\frac{\text{d}{K}_{\theta }^{\theta }(\tau )}{\text{d}\tau }+\upsilon \frac{\text{d}{K}_{\omega }^{\omega }(\tau )}{\text{d}\tau }\right]\text{d}\tau .}\end{array}\}$$ 2.8

(c) . Principle of virtual work and equilibrium equations

Following the literature [26,27], here we use the principle of virtual work to derive the equilibrium equations at a given moment t. Let δu_θ and δw be the virtual displacements of the mid-surface of the shell at time t. The associated virtual strains can be expressed as δϵ_αβ = δE_αβ + zδK_αβ. The internal virtual work (IVW) of the shell is

$$\text{IVW}={\int }_S^{}\text{d}S{\int }_{-h/2}^{h/2}\text{d}z{\sigma }^{\alpha \beta }\delta {\epsilon }_{\alpha \beta }={\int }_S[N^{\alpha \beta }\delta E_{\alpha \beta }+M^{\alpha \beta }\delta K_{\alpha \beta }]\hspace{0.17em}\text{d}S,$$ 2.9

where S represents the area of the mid-surface of the shell. The external virtual work (EVW) due to a uniform live pressure ΔP acting on the shell is [26,27] (equation (A 6))

$$\text{EVW}={\int }_S[\mathrm{\Delta }P\phi \delta u_{\theta }+\mathrm{\Delta }P(1+u_{,\gamma }^{\gamma }+b_{\gamma }^{\gamma }w)\delta w]\hspace{0.17em}\text{d}S+{\oint }_C(T^{\theta }\delta u_{\theta }+Q\delta w-M_n\delta w_{,n})\hspace{0.17em}\text{d}s$$ 2.10

where T^θ represents the edge resultant traction along x_,θ, Q represents the normal edge force, and M_n = M^αβn_αn_β is the component of the edge moment, n_β denotes the components of the unit vector normal to the boundary C tangent to the shell, and s is the length of the edge of the shell. Enforcing IVW = EVW yields the following equilibrium equations (see Appendix A for details)

$$\begin{array}{rl}& \hspace{0.17em}-M_{,\alpha \beta }^{\alpha \beta }+N^{\alpha \beta }{b}_{\alpha \beta }+{(N^{\alpha \beta }{\phi }_{\alpha })}_{,\beta }=\mathrm{\Delta }P(1+u_{,\gamma }^{\gamma }+b_{\gamma }^{\gamma }w),\\ & \hspace{0.17em}-N_{,\beta }^{\theta \beta }-M_{,\beta }^{\alpha \beta }{b}_{\alpha }^{\theta }-\frac{1}{2}{(M^{\alpha \beta }{b}_{\alpha }^{\theta }-M^{\theta \alpha }{b}_{\alpha }^{\beta })}_{,\beta }+N^{\alpha \beta }{\phi }_{\alpha }{b}_{\beta }^{\theta }=\mathrm{\Delta }P\phi ,\end{array}$$ 2.11

where ()_,α and ()_,αβ are the first- and second-order covariant derivatives of ().

In equation (2.11), φ and w are the two independent variables, and the highest order terms are φ′′′ and w′′′, yielding a system of six-order nonlinear ordinary differential equations (ODEs). In order to limit the deformation within the assumption of small strain and moderate rotation for a deep shell that possesses pseudo-bistability when fully inverted, we choose the sliding boundary for the shell, i.e. on the boundary at θ = θ_min, the shell is allowed to slide freely along e₁, but not along e₃ (figure 2a). As a result, the traction along e₁ is zero

$$(T^{\theta }{\mathit{x}}_{,\theta }+Q\mathit{N})\cdot e_1=T^{\theta }(R^{\mathrm{\prime }}\cos \theta -R\sin \theta )+Q\frac{R^{\mathrm{\prime }}\sin \theta +R\cos \theta }{\sqrt{{R^{\mathrm{\prime }}}^2+R^2}}=0,$$ 2.12

where T^θ and Q are given by equation (A 8), and the displacement along e₃ is zero,

$$\mathit{\delta }\cdot e_3=u^{\theta }(R^{\mathrm{\prime }}\sin \theta +R\cos \theta )-\frac{w(R^{\mathrm{\prime }}\cos \theta -R\sin \theta )}{\sqrt{{R^{\mathrm{\prime }}}^2+R^2}}=0.$$ 2.13

In addition, the assumption of axisymmetric deformation requires w′= φ = φ′′= 0 at the pole ($\theta =\pi /2$).

$\phi$ and w at time t can be obtained by solving the above boundary value problem, using the bvp4c solver and the finite difference method in Matlab. Here we consider three types of loading: (i) pressure-controlled loading, (ii) displacement-controlled loading and (iii) volume-controlled loading. When the pressure ΔP serves as the load parameter, the equilibrium equations in equation (2.11) are solved with prescribed evolution of pressure as a function of time. Displacement-controlled loading means that the displacement at the pole $w_{\text{pole}}=w(\theta =\pi /2)$ (figure 2a) is prescribed as the load parameter. With this loading type, the pressure is treated as an extra variable. Correspondingly, an additional ODE, ΔP^′ = 0, is added to the ODE set. Volume-controlled loading is achieved by setting the volume of the shell as an additional variable and adding an extra constraining ODE relating the volume and displacement to the ODE set, where the pressure is regarded as an extra variable as well. In this paper, the pressure-controlled loading is used to demonstrate the pseudo-bistable behaviour while the volume-controlled loading is used to produce the instantaneous pressure-volume change relations to examine the stability transition during the pseudo-bistability phenomenon. Due to the low practicality in operation, the displacement-controlled loading is only used to assist finding the equilibrium pressure-volume change paths for some elastic shells, especially for those exhibiting unstable paths when the other two loading types are adopted.

3. Rate-dependent buckling behaviours

In §2, we establish a model for viscoelastic shells with arbitrary axisymmetric shapes. In this section, we will study particular examples of ellipsoidal shells with the following R(θ)

$$R(\theta )=\frac{b}{\sqrt{1-e^2\hspace{0.17em}{\cos }^2\theta }},$$ 3.1

where $e=\sqrt{1-{(b/a)}^2}$ denotes eccentricity, and a and b denote the half lengths of the major and minor axes, respectively. Using the proposed shell model, we will first conduct the buckling analysis of elastic shells to figure out how geometry influences the stability of the shells. We then examine the effect of loading rates and viscoelastic properties on the rate-dependent buckling behaviour of viscoelastic shells. The analysis in this section provides insights into choosing proper geometric parameters and material properties to achieve pseudo-bistability in viscoelastic shells.

(a) . Buckling of elastic shells

The buckling behaviour of elastic ellipsoidal shells subjected to uniform live pressures can be obtained by solving the equilibrium equations (equation (2.11)) and boundary conditions (equations (2.12), (2.13)) with the following isotropic linearly elastic material law,

$$\begin{array}{l}{N}^{\alpha \beta }=\frac{E_{0}h}{1-{\nu }^2}[(1-\upsilon )E^{\alpha \beta }+\upsilon E_{\gamma }^{\gamma }{g}^{\alpha \beta }]\\ \text{and}{M}^{\alpha \beta }=\frac{E_0{h}^3}{12(1-{\nu }^2)}[(1-\upsilon )K^{\alpha \beta }+\upsilon K_{\gamma }^{\gamma }{g}^{\alpha \beta }],\end{array}\}$$ 3.2

where E₀ denotes the Young's modulus, and υ is the Poisson's ratio, which is assumed to equal 0.5 (incompressible material) throughout this paper. Figure 3 shows the relations between the normalized pressure ΔP/E₀ and normalized displacement − w_pole/a at the pole (figure 3a), as well as the relations between the normalized pressure ΔP/E₀ and normalized volume change ΔV/V₀ (figure 3b) for elastic ellipsoidal shells with h/a = 0.02, θ_min = 17π/128, and different minor-to-major-length ratios b/a quantifying the shallowness of the shells. Here ΔV is the volume change of the shell with respect to the undeformed state at t = 0, and V₀ represent the negative volume of the shell in the undeformed state,

$$V_0=-{\int }_{{\theta }_{\text{min}}}^{\pi /2}\pi R^2{\cos }^2\theta (R^{\mathrm{\prime }}\sin \theta +R\cos \theta )\hspace{0.17em}\text{d}\theta .$$ 3.3

Figure 3.

Buckling behaviour of elastic ellipsoidal shells with h/a = 0.02, θ_min = 17π/128, and different b/a. (a) Normalized pressure ΔP/E₀ versus normalized displacement − w_pole/a at the pole. (b) Normalized pressure ΔP/E₀ versus normalized volume change ΔV/V₀, where V₀ denotes the negative volume of the undeformed shell. The solid curves represent the theoretical results for the shells with b/a = 0.28, 0.36 and 0.47, and the circular dots represent the FEA results for the shell with b/a = 0.28 and 0.36. The ΔP/E₀–ΔV/V₀ curve for b/a = 0.47 intersects with the horizontal line of ΔP = 0 (dashed line) at three points, indicating that the shell with b/a = 0.47 is bistable. (c) The deformed shapes obtained from FEA for the shell with b/a = 0.36 at different volume changes are axisymmetric. The contour represents the maximum principal strain. (Online version in colour.)

The shells with different minor-to-major-length ratios b/a exhibit quite different ΔP/E₀–ΔV/V₀ curves (figure 3b). When b/a = 0.28, ΔP/E₀ increases monotonically with the increase of ΔV/V₀. For b/a = 0.36, ΔP/E₀ initially increases, then decreases after the shell buckles, and increases again with the increase of ΔV/V₀, showing the features of snap-through buckling. The above two shells are monostable, since their ΔP/E₀ remains positive. As b/a becomes large (b/a = 0.47), both ΔP/E₀ and ΔV/V₀ change non-monotonically, forming a very complex curve. Moreover, the ΔP/E₀–ΔV/V₀ curve intersects with the horizontal line of ΔP = 0 (dashed line) at three points (figure 3b), where points 1 and 3 represent two stable equilibrium states while point 2 represents an unstable equilibrium state when ΔP = 0. Therefore, the three intersection points indicate that the shell with b/a = 0.47 is bistable. The stability of the shells can also be measured by the local minimum pressure ΔP_min/E₀ of the ΔP/E₀–ΔV/V₀ curve. A positive ΔP_min/E₀ means monostability, whereas a negative ΔP_min/E₀ indicates that the shell has more than one stable state when ΔP = 0, and therefore bistability. From figure 3 we can see that the stability of shells (monostability or bistability) can be tuned by the minor-to-major-length ratio b/a: a deeper shell with a higher b/a is more likely to be bistable. The ΔP/E₀–ΔV/V₀ curves for b/a = 0.28 and 0.36 can be obtained by prescribing a monotonic increase in the load parameter of either w_pole/a or ΔV/V₀. The two loading methods yield the exact same ΔP/E₀–ΔV/V₀ curves. However, the ΔP/E₀–ΔV/V₀ equilibrium path for b/a = 0.47 can only be achieved by treating w_pole/a as the load parameter, since w_pole/a rather than ΔV/V₀ or $\mathrm{\Delta }P/{\mu }_0$ increases monotonically along the equilibrium path.

To verify the axisymmetric deformation of the shells with the chosen geometry, we conduct finite-element analysis (FEA) for the shells without axisymmetric constraints using the commercial software Abaqus/Standard. The static Riks method is implemented to capture the unstable equilibrium ΔP/E₀–ΔV/V₀ curve of the elastic ellipsoidal shells under pressure-controlled loading and the boundary condition as shown in equations (2.12)(2.13). The shells are modelled as an incompressible linearly elastic material with 8-node doubly curved thick shell elements with reduced integration (Abaqus type S8R). We plot the ΔP/E₀–ΔV/V₀ curves from FEA (circular dots in figure 3a,b) for the shells with b/a = 0.28 and 0.36. We find very good agreement between the results from the shell model and FEA for b/a = 0.28, and the deformation of the shell in FEA is also axisymmetric. For b/a = 0.36, although there is slight deviation between the results from the shell model and FEA after the buckling, the ΔP/E₀–ΔV/V₀ curves obtained from the shell model can still reasonably capture the deformation process. Therefore, the shell model is still a good analytical tool for us to understand the mechanism of pseudo-bistaiblity. Moreover, the deformation of the shell with b/a = 0.36 in FEA is also axisymmetric. In figure 3c we plot the deformed shapes of the shell with b/a = 0.36 when it is on the edge of buckling ($\mathrm{\Delta }V/V_0=0.493$), the pressure reaches a local minimum ($\mathrm{\Delta }V/V_0=1.107$), and the shell is fully inverted ($\mathrm{\Delta }V/V_0=1.754$). The deformation mode for b/a = 0.47 from FEA, however, is no longer axisymmetric. Therefore, in the following, we will limit ourselves to shells with b/a ≤ 0.36.

(b) . Buckling of viscoelastic shells

Next, we will examine the buckling behaviours of viscoelastic ellipsoidal shells under volume-controlled loading over a wide range of loading rates. The influence of the relative modulus of relaxation, E_rel = E₁/E₀, which is the ratio of the modulus in the Maxwell element E₁ to the instantaneous modulus E₀ = E₁ + E_∞, on the buckling behaviour is also studied. We define a dimensionless loading rate, γ_V, to quantify the rate of changes in the volume,

$${\gamma }_V=\frac{\text{d}(\mathrm{\Delta }V/V_0)}{\text{d}(t/{\tau }_r)},$$ 3.4

which indicates that in the relaxation time scale τ_r the volume change is V₀γ_V. In the following we will take the shell with b/a = 0.36 as an example and study its rate-dependent buckling behaviour. Other geometric parameters are h/a = 0.02, θ_min = 17π/128.

We first examine the influence of the loading rates on the ΔP/E₀–ΔV/V₀ curve under volume-controlled loading. The curves corresponding to different volume loading rates γ_V ranging from 0.01 to 10 are plotted in figure 4a when the relative modulus of relaxation E_rel is fixed at 0.5. When γ_V is very low (γ_V = 0.01), almost full relaxation occurs, and the response of the shell is governed by the long-term modulus, E_∞. As a result, the ΔP/E₀–ΔV/V₀ curves at very low γ_V approach that of the elastic shell with modulus E = E_∞ (dot-dashed line in figure 4a). On the other hand, the very high γ_V (γ_V = 10) results in little relaxation. Correspondingly, the effective modulus of the shell is close to the instantaneous modulus E₀ = E₁ + E_∞, and thus the ΔP/E₀–ΔV/V₀ curves at very high γ_V approach that of the elastic shell with modulus E₀ = E₁ + E_∞ (dashed line in figure 4a). The ΔP/E₀–ΔV/V₀ curves at moderate γ_V are located in between the two extreme cases, and the resultant pressure ΔP/E₀ for a given volume change ΔV/V₀ increases as γ_V increases. The buckling pressure $\mathrm{\Delta }{P}_{\text{max}}/E_0$ at very high (low) γ_V approaches that of the elastic shell with E₁ + E_∞ (E_∞) (figure 4b). In between the very low and very high γ_V, the increase of γ_V results in a notable increase in $\mathrm{\Delta }{P}_{\text{max}}/E_0$ (figure 4b). The middle-surface profiles of the shell under different volume changes ΔV/V₀ at the volume loading rate γ_V = 0.5 is shown in figure 4c. The profile of the shells stays concave before the pressure reaches the critical pressure for buckling when $\mathrm{\Delta }V/V_0=0.475$, and transitions from concave to convex as the pressure decreases and the volume change increases ($0.475<\mathrm{\Delta }V/V_0<1.12$). Finally, the profile keeps convex while the pressure increases again with the increase of the volume change ($\mathrm{\Delta }V/V_0\ge 1.12$).

Figure 4.

Buckling behaviours of viscoelastic shells with b/a = 0.36 under volume-controlled loading. (a) Normalized pressure ΔP/E₀–volume change ΔV/V₀ relations at different volume loading rates $(0.01\le {\gamma }_V\le 10)$ when E_rel = 0.5. (b) Buckling pressure ΔP_max/E₀ as a function of γ_V. The dashed and dot-dashed lines represent ΔP_max/E₀ for elastic shells with moduli E_∞ + E₁ and E_∞, respectively. (c) Middle-surface profiles of the shell under different volume changes ΔV/V₀ when E_rel = 0.5 and γ_V = 0.5. (d) Normalized pressure ΔP/E₀–volume change ΔV/V₀ relations at different relative modulus of relaxation $(0\le E_{\text{rel}}\le 1)$ when γ_V = 0.5. (e) Buckling pressure ΔP_max/E₀ as a function of E_rel. The dashed and dot-dashed lines in (a) and (d) represent the pressure–volume change relations for elastic shells with moduli E_∞ + E₁ and E_∞, respectively. (Online version in colour.)

The ΔP/E₀–ΔV/V₀ curves for viscoelastic shells also highly depend on the relative modulus of relaxation, E_rel. We consider a moderate loading rate γ_V = 0.5, and plot the ΔP/E₀–ΔV/V₀ curves for different E_rel ranging from 0 to 1, as shown in figure 4d. When E_rel = 0, no relaxation occurs, and thus the corresponding ΔP/E₀–ΔV/V₀ curve coincides with that of the elastic shell with modulus E₀ = E₁ + E_∞ (dashed line in figure 4d). As E_rel increases from 0, the resultant pressure ΔP/E₀ reduces notably for a given volume change ΔV/V₀, leading to a reduction in $\mathrm{\Delta }{P}_{\text{max}}/E_0$ (figure 4e).

4. Mechanism of pseudo-bistability

In this section, we will first use the viscoelastic shell model formulated in Section 2 to demonstrate the pseudo-bistability phenomenon, in which an inverted viscoelastic ellipsoidal shell snaps back to its natural state with a delay time after a pressure load is held constantly for a while prior to being released. We will then probe the stability of the shell at different time during this holding and releasing process by plotting the corresponding instantaneous pressure–volume change relations. The obtained stability transition provides insights into the mechanism of the pseudo-bistability.

(a) . Predicting pseudo-bistable behaviour

Demonstrating pseudo-bistability in a viscoelastic shell requires a careful choice of geometry and viscoelastic properties. The geometry should result in a monostable pressure ΔP/E₀-volume change ΔV/V₀ relation if the shell were elastic, but the minimum normalized pressure $\mathrm{\Delta }{P}_{\text{min}}/E_0$ is not too far away from zero. On the other hand, the viscoelastic effects should be large enough to trigger pseudo-bistability. We choose a viscoelastic shell with the geometric parameters as b/a = 0.36, h/a = 0.02, θ_min = 17π/128, which corresponds to a monostable shell if it were elastic, and material parameter E_rel = 0.5. We apply an instantaneous pressure load $\mathrm{\Delta }P/E_0=2.67\times {10}^{-5}$, which is above its buckling pressure $\mathrm{\Delta }{P}_{\text{max}}/E_0$, and release this pressure after holding it for t_creep = τ_r (figure 5a). The corresponding volume change ΔV/V₀ as a function of time t is computed based on the proposed shell model (figure 5b and Supplementary video [29]). We observe that the shell immediately buckles into an inverted shape once the pressure is applied (figure 5b,c, moment 2), and creeps with a small increase in volume change for t_creep = τ_r (figure 5b,c, from moment 2 to moment 4). After the pressure is removed, the viscoelastic shell can temporarily stay inverted for t_rec = 1.136τ_r (figure 5b,c, from moment 5 to moment 7). At moment 7, a solution of the inverted state can no longer be found using the solution of the last iteration as the initial guess with the ODE solver, but only a solution of the unbuckled state can be found using the undeformed configuration as the initial guess. Accordingly, the shell snaps from the inverted configuration (moment 7) back to the unbuckled configuration (moment 8). After this snapping deformation, the shell gradually recovers its undeformed shape, with ΔV/V₀ slowly decreasing to zero. The characteristics of this observed deformation history agree with those of FEA simulations and experiments reported in literature [16,21,22], indicating that the proposed viscoelastic shell model can capture the pseudo-bistability exhibited in viscoelastic shells.

Figure 5.

Pseudo-bistable behaviour of a viscoelastic shell. (a) Applied pressure-time relation and (b) the corresponding volume change-time relation for a viscoelastic ellipsoidal shell with b/a = 0.36, h/a = 0.02, θ_min = 17π/128 and E_rel = 0.5. The time period within which a constant pressure is held is defined as the creeping time t_creep, and the time period within which the shell stays inverted after the pressure is removed is defined as the recovery time t_rec. τ_r = η/E₁ denotes the relaxation time constant of the viscoelastic material. (c) Middle-surface profiles of the shell at different time moments as labelled in (b).

(b) . Stability transition during delayed snap-through

Having successfully predicted the pseudo-bistable behaviour of a viscoelastic shell using the proposed shell model, we next investigate the stability transition of the shell during this holding and releasing process. For an elastic shell, the number of the intersection points of its pressure–volume change curve with the horizontal line of zero pressure determines its stability (figure 3b). One intersection point indicates that the shell is monostable, since there is only one stable equilibrium state when no pressure is applied. Three intersection points, on the other hand, indicate that the shell is bistable, since there are two stable and one unstable equilibrium states at zero pressure. To probe the stability evolution of the viscoelastic shell during the hold and releasing process, we need to plot the instantaneous pressure-volume change relation at different time moments and check the number of intersection points with the horizontal line of zero pressure. In §3, we have learned that an extremely fast loading γ_V ≫ 1 can eliminate the viscoelastic relaxation effects and yield an instantaneous pressure-volume change response of a shell. Therefore, in the following we will conduct volume-controlled loading to different time moments of interest in the holding and releasing process, and unload (or load for some cases) at an extremely high rate to obtain the corresponding instantaneous pressure–volume change responses, which provide information on the stability evolution of the viscoelastic shell.

We follow the same loading process up to the different time moments as in figure 5a, and unload at a very high rate of changes in the volume γ_V = 10 to obtain the instantaneous pressure ΔP/E₀-volume change ΔV/V₀ relations (figure 6a–f). When the shell is unloaded at moment 2, right after the instantaneous pressure is applied (figure 5a), the instantaneous ΔP/E₀-ΔV/V₀ relation (figure 6a) is exactly the same as the ΔP/E₀-ΔV/V₀ curve for the elastic shell with the same geometry (the red curve in figure 3b). This agreement is due to the fact that creeping has not yet started and thus viscoelasticity plays no role. At moment 2, the local minimum pressure of the instantaneous ΔP/E₀–ΔV/V₀ curve, $\mathrm{\Delta }{P}_{\text{min}}/E_0$, is larger than zero, so the shell is monostable. When the pressure is held until moment 3, $\mathrm{\Delta }{P}_{\text{min}}/E_0$ of the instantaneous ΔP/E₀–ΔV/V₀ curve decreases to zero (figure 6b). When the pressure is held for an even longer time, for example until moment 4, $\mathrm{\Delta }{P}_{\text{min}}/E_0$ of the instantaneous ΔP/E₀–ΔV/V₀ curve becomes negative (figure 6c). Accordingly, the number of the intersection points between the instantaneous ΔV/V₀–ΔV/V₀ curve and the horizontal line of $\mathrm{\Delta }P/E_0=0$ (dashed line) changes from one (figure 6a) to two (figure 6b), and eventually to three (figure 6c). Thus, the stability of the shell transitions from a monostable state to bistable state due to viscoelastic creeping, with moment 3 as the critical transition time, at which $\mathrm{\Delta }{P}_{\text{min}}/E_0=0$.

Figure 6.

(a–f) The instantaneous pressure–volume change relations at the time moments labelled in figure 5b. The red dots represent the states of the shells for the corresponding time moments. (Online version in colour.)

Right after the pressure is removed, the shell jumps from the configuration at moment 4 to the closest stable configuration (moment 5 in figure 5b), which corresponds to the third intersection point between the instantaneous ΔP/E₀–ΔV/V₀ curve and the horizontal line of $\mathrm{\Delta }P/E_0=0$ (point 5 in figure 6c). This jump results in a sudden drop in ΔV/V₀ (from moment 4 to 5 in figure 5b). After the load is released ($\mathrm{\Delta }P/E_0=0$), the instantaneous ΔP/E₀–ΔV/V₀ curve further evolves: $\mathrm{\Delta }{P}_{\text{min}}/E_0$ starts to increase (figure 6d,e), and the third intersection point gradually shifts to the left (from point 5 in figure 6c to point 6 in figure 6d), resulting in a slow decrease in ΔV/V₀ (from moment 5 to 6 in figure 5b). The shell is bistable and stays inverted as long as $\mathrm{\Delta }{P}_{\text{min}}/E_0<0$. When $\mathrm{\Delta }{P}_{\text{min}}/E_0$ increases back to zero at moment 7 (figure 6e), the third and the second intersection points merge into a single point (point 7 in figure 6e) tangent to the horizontal line of $\mathrm{\Delta }P/E_0=0$. The shell at moment 7 is unstable and thus snaps to the only stable configuration (point 8 in figure 6e). Correspondingly, the shell snaps from the inverted state (moment 7) to unbuckled state (moment 8). Therefore, moment 7 is the critical moment at which the stability transitions from the bistable state back to the monostable state. As the creeping process continues, $\mathrm{\Delta }{P}_{\text{min}}/E_0$ keeps increasing. As a result, only one intersection point exists and shifts to the left (from point 8 in figure 6e to point 9 in figure 6f), leading to a decrease in ΔV/V₀ (from moment 8 to 9 in figure 5b). At moment 9, the instantaneous ΔP/E₀–ΔV/V₀ curve almost recovers the one at moment 2, the intersection point almost overlaps the origin, and the shell almost recovers the stress-free shape and volume (figure 6f).

We summarize the $\mathrm{\Delta }{P}_{\text{min}}/E_0$–time relation in figure 7, from which we can clearly observe the stability transition of the viscoelastic shell from a monostable state to a bistable state, and back to the monostable state during the holding and releasing process. $\mathrm{\Delta }{P}_{\text{min}}/E_0$, starting with a positive value, monotonically decreases while the pressure is held constantly, and reaches its minimum when the pressure is removed at t/τ_r = t_creep/τ_r = 1. Accordingly, the shell is initially monostable, and switches to bistable when $\mathrm{\Delta }{P}_{\text{min}}/E_0$ flips its sign from positive to negative, and stays bistable. Here we define the time period within which $\mathrm{\Delta }{P}_{\text{min}}/E_0$ decreases to zero as the critical creeping time $t_{\text{creep}}^{\text{cr}}$, representing the minimum creeping time required for the stability transition. Only if the pressure is held for a time period longer than $t_{\text{creep}}^{\text{cr}}$ can the shell exhibit pseudo-bistability. After the shell is released at t/τ_r = 1, $\mathrm{\Delta }{P}_{\text{min}}/E_0$ starts to increase. When $\mathrm{\Delta }{P}_{\text{min}}/E_0$ flips its sign back to positive, the acquired stability is lost and the shell recovers the monostable state, triggering snapping from the inverted configuration to the unbuckled configuration. The time period within which $\mathrm{\Delta }{P}_{\text{min}}/E_0$ increases from its minimum to zero is the recovery time t_rec defined in figure 5b. As time goes on, $\mathrm{\Delta }{P}_{\text{min}}/E_0$ continues increasing and approaches its initial value.

Figure 7.

Local minimum pressure ΔP_min/E₀ of the instantaneous pressure–volume change curve as a function of time. The time period within which ΔP_min/E₀ decreases to zero is defined as the critical creeping time $t_{\text{creep}}^{\text{cr}}$, indicating the minimum creeping time required for pseudo-bistability.

5. Influence of geometry, viscoelastic property and loading history on pseudo-bistability

In this section, we will investigate how the geometry and viscoelastic property of ellipsoidal shells, and the loading history influence their pseudo-bistable behaviour. Specifically, the minor-to-major-length ratio b/a, relative modulus of relaxation E_rel, and the holding time t_creep/τ_r are considered. A higher b/a (a deeper shell) results in a smaller local minimum pressure $\mathrm{\Delta }{P}_{\text{min}}/E_0$ in the instantaneous ΔP/E₀-ΔV/V₀ curve (figure 3b), and thus leads to a shell closer to a bistable one if it were elastic. A larger E_rel causes a stronger viscoelastic effect, while a longer t_creep/τ_r results in a longer creeping process. Other parameters such as h/a = 0.02, θ_min = 17π/128, and the applied pressure $\mathrm{\Delta }P/E_0=2.67\times {10}^{-5}$ are fixed. The critical creeping time $t_{\text{creep}}^{\text{cr}}$ and the recovery time t_rec will be investigated with respect to different values of the parameters mentioned above.

We first examine the effect of the minor-to-major-length ratio b/a on the pseudo-bistable behaviour. We apply an instantaneous pressure and release this pressure after holding it for t_creep/τ_r = 1 (figure 5a). The volume change ΔV/V₀ as a function of time t, as well as the local minimum pressure $\mathrm{\Delta }{P}_{\text{min}}/E_0$ of the instantaneous ΔP/E₀-ΔV/V₀ curve as a function of time t for shells with relative modulus of relaxation E_rel = 0.5 and different b/a are plotted in figure 8. From the ΔV/V₀-t curves (figure 8a), we find that the shells with b/a = 0.35 and 0.36 exhibit pseudo-bistability while the shell with b/a = 0.34 does not, and that the shell with b/a = 0.36 has a longer delay time than the one with 0.35. This is because $\mathrm{\Delta }{P}_{\text{min}}/E_0$ decreases slower for a shallower shell (lower b/a) (figure 8b) during the holding process. At t/τ_r = t_creep/τ_r = 1, the shells with b/a = 0.35 and 0.36 reach negative $\mathrm{\Delta }{P}_{\text{min}}/E_0$, indicating that they are temporally bistable. The shell with b/a = 0.36 has a smaller $\mathrm{\Delta }{P}_{\text{min}}/E_0$ than the one with b/a = 0.35. Thus, it takes longer for the $\mathrm{\Delta }{P}_{\text{min}}/E_0$ of the shell with b/a = 0.36 to recover a positive value after the pressure is removed, leading to a longer recovery time t_rec. In addition, the critical creeping time $t_{\text{creep}}^{\text{cr}}$, the intersection point between the $\mathrm{\Delta }{P}_{\text{min}}/E_0$–t curve and the horizontal line of $\mathrm{\Delta }{P}_{\text{min}}/E_0=0$ (dashed line in figure 8b), for b/a = 0.36 is smaller than the one for b/a = 0.35. The $\mathrm{\Delta }{P}_{\text{min}}/E_0$ for the shell with b/a = 0.34, however, remains positive at t/τ_r = t_creep/τ_r = 1, indicating that it stays monostable during the holding process and thus immediately snaps back after it is released. The shell with b/a = 0.34 needs longer t_creep/τ_r to reduce $\mathrm{\Delta }{P}_{\text{min}}/E_0$ to a negative value in order to trigger pseudo-bistability. For all the three shells, their $\mathrm{\Delta }{P}_{\text{min}}/E_0$ recovers the initial values as t approaches 5τ_r.

Figure 8.

(a) Volume change ΔV/V₀-time t/τ_r relations and (b) local minimum pressure ΔP_min/E₀ of the instantaneous pressure-volume change curves as a function of time t/τ_r for viscoelastic ellipsoidal shells with relative modulus of relaxation E_rel = 0.5 and different minor-to-major-length ratios b/a during the holding for t_creep/τ_r = 1 and releasing process. (Online version in colour.)

We then investigate how the viscoelastic effect influences the pseudo-bistable behaviour. We apply the same loading process as shown in figure 5a, and plot the ΔV/V₀–t/τ_r curves (figure 9a) and corresponding $\mathrm{\Delta }{P}_{\text{min}}/E_0$–t/τ_r relations (figure 9b) for viscoelastic shells with b/a = 0.36 and different E_rel. Figure 9a shows that the shells with high (E_rel = 0.5) and intermediate (E_rel = 0.4) viscoelastic effects exhibit pseudo-bistable behaviour while the shell with a low viscoelastic effect (E_rel = 0.05) snaps back immediately after the pressure is removed. In addition, a higher viscoelastic effect results in a longer recovery time t_rec. In figure 9b, we can clearly see that the stability transitions from a monostable state ($\mathrm{\Delta }{P}_{\text{min}}/E_0>0$) to a bistable state ($\mathrm{\Delta }{P}_{\text{min}}/E_0<0$) for the shells with E_rel = 0.4 and 0.5, whereas the shell with E_rel = 0.05 remains monostable during the holding process. The $\mathrm{\Delta }{P}_{\text{min}}/E_0$ for the shell with E_rel = 0.05 reduces more and more slowly as t increases and reaches a plateau at t/τ_r = t_creep/τ_r = 1, meaning that increasing the holding time can never reduce $\mathrm{\Delta }{P}_{\text{min}}/E_0$ to a negative value, and thus leads to no pseudo-bistable behaviour. Therefore, there exists a critical value of E_rel for shells to achieve pseudo-bistability.

Figure 9.

(a) Volume change ΔV/V₀-time t/τ_r relations and (b) local minimum pressure ΔP_min/E₀ of the instantaneous pressure-volume change curves as a function of time t/τ_r for viscoelastic ellipsoidal shells with minor-to-major-length ratio b/a = 0.36 and different relative moduli of relaxation E_rel during the holding for t_creep/τ_r = 1 and releasing process. (Online version in colour.)

Moreover, we study the effect of time period of the holding process, t_creep/τ_r, on the pseudo-bistable behaviour of viscoelastic shells. We fix b/a = 0.36 and E_rel = 0.5, and vary t_creep/τ_r (figure 5a) from 0.6134, 1 to 5, where t_creep/τ_r = 0.6134 is the critical creeping time $t_{\text{creep}}^{\text{cr}}$ (moment 3 in figures 5b and 6b). Therefore, the ΔV/V₀-t/τ_r curve (figure 10a) shows no delay time after the shell is released. Correspondingly, the $\mathrm{\Delta }{P}_{\text{min}}/E_0$ decreases to zero at t_creep/τ_r = 0.6134 and starts to increase, indicating that the shell stays monostable during the holding process. The shells for both t_creep/τ_r = 1 and 5 exhibit pseudo-bistable behaviour (figure 10a), and the shell for t_creep/τ_r = 5 shows a longer recovery time t_rec than the one for t_creep/τ_r = 1. This is because a longer time of holding process results in a smaller $\mathrm{\Delta }{P}_{\text{min}}/E_0$, and therefore a longer time is needed for $\mathrm{\Delta }{P}_{\text{min}}/E_0$ to recover positive (figure 10b). In addition, figure 10b shows that $\mathrm{\Delta }{P}_{\text{min}}/E_0$ decreases more and more slowly as the time of holding process increases and almost reaches a plateau when t_creep/τ_r = 5. This indicates that t_rec also approaches a plateau as t_creep/τ_r becomes very long.

Figure 10.

(a) Volume change ΔV/V₀-time t/τ_r relations and (b) local minimum pressure ΔP_min/E₀ of the instantaneous pressure-volume change curves as a function of time t/τ_r for viscoelastic ellipsoidal shells with minor-to-major-length ratio b/a = 0.36 and relative modulus of relaxation E_rel = 0.5 during the holding for different t_creep/τ_r and releasing process. (Online version in colour.)

We summarize the effect of geometry, viscoelastic property and loading history on the pseudo-bistable behaviour in figures 11 and 12. In figure 11, we show the effect of minor-to-major-length ratio b/a and relative modulus of relaxation E_rel on the critical creeping time $t_{\text{creep}}^{\text{cr}}$. When b/a = 0.36, $t_{\text{creep}}^{\text{cr}}$ increases with the decrease of E_rel, and goes to infinity as E_rel approaches 0.214 (figure 11), indicating an infinite time of holding required for pseudo-bistability. The viscoelastic shells with E_rel below 0.214 can never exhibit pseudo-bistability no matter how long the pressure is held, since the viscoelastic effect is not strong enough. As b/a decreases, the corresponding asymptotic line (dashed line in figure 11) is shifted to the right, meaning that a shallow shell (lower b/a) needs a stronger viscoelastic effect (larger E_rel) for the stability transition to occur. For a given E_rel, a deeper shell has a shorter $t_{\text{creep}}^{\text{cr}}$, and thus requires a shorter holding time to acquire pseudo-bistability. Figure 12 illustrates the influence of t_creep/τ_r, b/a and E_rel on the recovery time t_rec/τ_r. We find t_rec/τ_r increases with t_creep/τ_r and saturates when t_creep/τ_r becomes much longer than 1, regardless of b/a and E_rel. For fixed b/a = 0.36, a higher E_rel results in a longer t_rec/τ_r, and requires a shorter t_creep/τ_r to trigger the pseudo-bistable behaviour (t_rec/τ_r > 0) (figure 12a). Moreover, b/a also has a strong influence on t_rec/τ_r (figure 12b). We find that a deeper monostable shell with higher b/a, which is closer to that of bistable shells, leads to a more significant delay time with longer t_rec/τ_r. The effects of t_creep/τ_r, b/a and E_rel on t_rec/τ_r mentioned above are consistent with the FEA simulations and experimental observations reported in literature [16,18,21,22].

Figure 11.

Dependence of the critical creeping time $t_{\text{creep}}^{\text{cr}}$ on relative modulus of relaxation E_rel and minor-to-major-length ratios b/a. The dashed lines represent the critical values of E_rel for $t_{\text{creep}}^{\text{cr}}$ to asymptotically reach infinity.

Figure 12.

Dependence of the recovery time t_rec on the creep time t_creep for different (a) relative modulus of relaxation E_rel and (b) minor-to-major-length ratios b/a.

6. Conclusion

In this paper, we model and explain the pseudo-bistable behaviour of viscoelastic shells with a viscoelastic shell model. The model combines the small strain, moderate rotation shell theory with the standard linear solid as the viscoelastic constitutive law. The equilibrium equations are derived by using the principle of virtual work based on the assumption of axisymmetric deformation. By numerically solving the equilibrium equations, the time-dependent buckling behaviors of viscoelastic shells far beyond the buckling point are obtained.

As an example, we apply the proposed model to investigate viscoelastic ellipsoidal shells. Time-dependent buckling analyses are conducted for them under volume-controlled loading conditions. The viscoelastic shells loaded extremely fast (slow) exhibit pressure-volume change relations approaching those of the elastic shells with the short-time elastic modulus E₁ + E_∞ (long-time elastic modulus E_∞). For a moderate loading rate, the pressure-volume change curve shifts downward as either the loading rate decreases or the relative relaxation modulus E_rel increases. Correspondingly, the critical pressure for buckling decreases.

Using the proposed viscoelastic shell model, we successfully predict the pseudo-bistable behaviour and reveal its mechanism by quantitatively probing the stability transition of viscoelastic shells during a process of holding and releasing a pressure. We first apply an instantaneous pressure sufficient to buckle a monostable shell, hold the pressure for a certain amount of time, and then remove it. With an appropriate choice of shallowness and viscoelasticity, the buckled shell creeps while the pressure is held, stays inverted after the pressure is removed, and finally recovers from its inverted state after a delay time. The characteristics of this time-dependent deformation agree with those obtained from FEA and experiments in literature. Moreover, the viscoelastic shell model allows us to produce the evolution of the instantaneous pressure-volume change relation, which indicates the stability of the shell, at different times during the holding and releasing process. We observe that the shell's stability transitions from a monostable state, temporarily bistable state and eventually back to the monostable state. This observation confirms the mechanism of the pseudo-bistability phenomenon. Finally, we conduct a parametric study to investigate the influence of geometry, viscoelastic property and loading history on the pseudo-bistable behaviour. We find that a shallower shell requires a longer time of holding to achieve pseudo-bistability, and that the recovery time can be increased by either enlarging the viscoelastic relaxation or reducing the shallowness closer to that of bistable shells.

Appendix A. Derivation of equilibrium equations using the principle of virtual work

In this section, the derivation for the equilibrium equations in equation (2.11) is presented in detail. The internal virtual work (IVW) can be expressed as,

$$\text{IVW}={\int }_S[N^{\alpha \beta }\delta E_{\alpha \beta }+M^{\alpha \beta }\delta K_{\alpha \beta }]\hspace{0.17em}\text{d}S,$$ A 1

where δE_αβ and δK_αβ are the virtual strain components, S denotes the area of the mid-surface of the shell. Based on the small strain, moderate rotation shell theory [26,27], δE_αβ and δK_αβ can be written as

$$\begin{array}{l}\delta E_{\alpha \beta }=\frac{1}{2}(\delta u_{\alpha ,\beta }+\delta u_{\beta ,\alpha })+b_{\alpha \beta }\delta w+{\phi }_{\alpha }\delta {\phi }_{\beta }+g_{\alpha \beta }\varphi \delta \varphi \\ \text{and}\delta K_{\alpha \beta }=-\delta w_{,\alpha \beta }+b_{\alpha \gamma ,\beta }\delta u^{\gamma }-\frac{1}{4}(b_{\beta }^{\gamma }\delta u_{\alpha ,\gamma }+b_{\alpha }^{\gamma }\delta u_{\beta ,\gamma })+\frac{3}{4}(b_{\beta \gamma }\delta u_{,\alpha }^{\gamma }+b_{\alpha \gamma }\delta u_{,\beta }^{\gamma }),\end{array}\}$$ A 2

where

$$\begin{array}{ll}& {\phi }_{\alpha }=-w_{,\alpha }+b_{\alpha }^{\gamma }{u}_{\gamma },\delta {\phi }_{\alpha }=-\delta w_{,\alpha }+b_{\alpha }^{\gamma }\delta u_{\gamma },\varphi =\frac{1}{2}{ϵ}^{\alpha \beta }{u}_{\beta ,\alpha },\delta \varphi =\frac{1}{2}{ϵ}^{\alpha \beta }\delta u_{\beta ,\alpha },\\ & {ϵ}^{\alpha \beta }=\{\begin{array}{lll}1/\sqrt{g},& \text{when}& \alpha =1,\beta =2\\ 0,& \text{when}& \alpha =\beta \\ -1/\sqrt{g},& \text{when}& \alpha =2,\beta =1\end{array},g=|g_{\alpha \beta }|.\end{array}$$ A 3

Substituting equation (A 2) into equation (A 1), the internal virtual work can be rewritten as

$$\begin{array}{rl}\text{IVW}& \hspace{0.17em}={\int }_S\{[-M_{,\alpha \beta }^{\alpha \beta }+N^{\alpha \beta }{b}_{\alpha \beta }+{(N^{\alpha \beta }{\phi }_{\alpha })}_{,\beta }]\delta w+[-N_{,\beta }^{\gamma \beta }-M_{,\beta }^{\alpha \beta }{b}_{\alpha }^{\gamma }-\frac{1}{2}{(M^{\alpha \beta }{b}_{\alpha }^{\gamma }-M^{\gamma \alpha }{b}_{\alpha }^{\beta })}_{,\beta }\\ & +N^{\alpha \beta }{\phi }_{\alpha }{b}_{\beta }^{\gamma }-\frac{1}{2}{(N_{\alpha }^{\alpha }\varphi {ϵ}^{\mu \gamma })}_{,\mu }]\delta u_{\gamma }\}\text{d}S\\ & \hspace{0.17em}+{\oint }_C\{-M^{\alpha \beta }{n}_{\beta }{n}_{\alpha }\delta w_{,n}+[M_{,\beta }^{\alpha \beta }{n}_{\alpha }+{(M^{\alpha \beta }{n}_{\beta }{t}_{\alpha })}_{,t}-N^{\alpha \beta }{\phi }_{\alpha }{n}_{\beta }]\delta w\\ & +\left(N^{\gamma \beta }{n}_{\beta }+\frac{3}{2}{M}^{\alpha \beta }{n}_{\beta }{b}_{\alpha }^{\gamma }-\frac{1}{2}{M}^{\gamma \beta }{b}_{\beta }^{\alpha }{n}_{\alpha }+\frac{1}{2}{N}_{\alpha }^{\alpha }\varphi {ϵ}^{\mu \gamma }{n}_{\mu }\right)\delta u_{\gamma }\}\text{d}s-M^{\alpha \beta }{n}_{\beta }{t}_{\alpha }\delta w{|}_{\text{corners}},\end{array}$$ A 4

where ()_,α and ()_,αβ are the first- and second-order covariant derivatives of (), C is the boundary of the mid-surface, n_α and t_α represent the components of the unit vectors normal and tangent to the edge C, respectively. The external virtual work (EVW) due to a uniform pressure acting on the shell in the deformed state (called live pressure) is [27]

$$\text{EVW}={\int }_{\overline{S}}\mathrm{\Delta }P\overline{\mathit{N}}\cdot (\delta u^{\beta }{\mathit{x}}_{,\beta }+\delta w\mathit{N})\hspace{0.17em}\text{d}\overline{S}+{\oint }_C(T^{\gamma }\delta u_{\gamma }+Q\delta w-M_n\delta w_{,n})\hspace{0.17em}\text{d}s,$$ A 5

where $\overline{\mathit{N}}$ denotes the unit vector normal to the mid-surface of the deformed shell, $\overline{S}$ denotes the area of the mid-surface of the shell in the deformed state. Under the condition of small strain and moderate rotation, the EVW becomes

$$\begin{array}{rl}\text{EVW}& \hspace{0.17em}={\int }_S[\mathrm{\Delta }P(1+u_{,\gamma }^{\gamma }+b_{\gamma }^{\gamma }w)\delta w+\mathrm{\Delta }P({\phi }^{\gamma }+\varphi {\phi }^{\eta }{ϵ}_{\eta }^{\gamma })\delta u_{\gamma }]\hspace{0.17em}\text{d}S+{\oint }_C(T^{\gamma }\delta u_{\gamma }+Q\delta w-M_n\delta w_{,n})\hspace{0.17em}\text{d}s.\end{array}$$ A 6

By enforcing IVW = EVW, we can obtain the following equilibrium equations

$$\begin{array}{rl}& \hspace{0.17em}-M_{,\alpha \beta }^{\alpha \beta }+N^{\alpha \beta }{b}_{\alpha \beta }+{(N^{\alpha \beta }{\phi }_{\alpha })}_{,\beta }=\mathrm{\Delta }P(1+u_{,\gamma }^{\gamma }+b_{\gamma }^{\gamma }w),\\ & \hspace{0.17em}-N_{,\beta }^{\gamma \beta }-M_{,\beta }^{\alpha \beta }{b}_{\alpha }^{\gamma }-\frac{1}{2}{(M^{\alpha \beta }{b}_{\alpha }^{\gamma }-M^{\gamma \alpha }{b}_{\alpha }^{\beta })}_{,\beta }+N^{\alpha \beta }{\phi }_{\alpha }{b}_{\beta }^{\gamma }-\frac{1}{2}{(N_{\alpha }^{\alpha }\varphi {ϵ}^{\mu \gamma })}_{,\mu }\\ & \hspace{0.17em}=\mathrm{\Delta }P({\phi }^{\gamma }+\varphi {\phi }^{\eta }{ϵ}_{\eta }^{\gamma }),\end{array}$$ A 7

and boundary conditions:

$$\begin{array}{rl}& \text{Specify}{N}^{\gamma \beta }{n}_{\beta }+\frac{3}{2}{M}^{\alpha \beta }{n}_{\beta }{b}_{\alpha }^{\gamma }-\frac{1}{2}{M}^{\gamma \beta }{b}_{\beta }^{\alpha }{n}_{\alpha }+\frac{1}{2}{N}_{\alpha }^{\alpha }\varphi {ϵ}^{\mu \gamma }{n}_{\mu }=T^{\gamma }\hspace{0.17em}\text{or}\hspace{0.17em}{u}_{\gamma }\\ & \text{Specify}{M}^{\alpha \beta }{n}_{\beta }{n}_{\alpha }=M_n\hspace{0.17em}\text{or}\hspace{0.17em}{w}_{,n}\\ & \text{Specify}{M}_{,\beta }^{\alpha \beta }{n}_{\alpha }+{(M^{\alpha \beta }{n}_{\beta }{t}_{\alpha })}_{,t}-N^{\alpha \beta }{\phi }_{\alpha }{n}_{\beta }=Q\hspace{0.17em}\text{or}\hspace{0.17em}w.\end{array}$$ A 8

The term − M^αβn_βt_αδw|_corners in equation (A 4) is related to the virtual work of concentrated loads at any corners.

Since we only consider axisymmetric deformation, u^θ and w are only functions of θ, u^ω = 0, φ^ω = φ_ω = 0, ϕ = 0. With the assumption of axisymmetric deformation, each term in equation (A 7) can be written as

$$\begin{array}{rl}-M_{,\alpha \beta }^{\alpha \beta }& \hspace{0.17em}=M^{\omega \omega }(-{{\mathrm{\Gamma }}_{\omega \omega }^{\theta }}^{\mathrm{\prime }}-{\mathrm{\Gamma }}_{\theta \theta }^{\theta }{\mathrm{\Gamma }}_{\omega \omega }^{\theta }-{\mathrm{\Gamma }}_{\omega \omega }^{\theta }{\mathrm{\Gamma }}_{\theta \omega }^{\omega })\\ & \hspace{0.17em}+M^{\theta \theta }[-{({\mathrm{\Gamma }}_{\theta \omega }^{\omega })}^2-2{{\mathrm{\Gamma }}_{\theta \theta }^{\theta }}^{\mathrm{\prime }}-2{({\mathrm{\Gamma }}_{\theta \theta }^{\theta })}^2-{{\mathrm{\Gamma }}_{\theta \omega }^{\omega }}^{\mathrm{\prime }}-3{\mathrm{\Gamma }}_{\theta \theta }^{\theta }{\mathrm{\Gamma }}_{\theta \omega }^{\omega }]\\ & \hspace{0.17em}-{M^{\omega \omega }}^{\mathrm{\prime }}{\mathrm{\Gamma }}_{\omega \omega }^{\theta }+{M^{\theta \theta }}^{\mathrm{\prime }}(-3{\mathrm{\Gamma }}_{\theta \theta }^{\theta }-2{\mathrm{\Gamma }}_{\theta \omega }^{\omega })-{M^{\theta \theta }}^{\mathrm{\prime }\mathrm{\prime }},\end{array}$$ A 9

$$\begin{array}{r}{N}^{\alpha \beta }{b}_{\alpha \beta }=N^{\omega \omega }{b}_{\omega }^{\omega }{g}_{\omega \omega }+N^{\theta \theta }{b}_{\theta }^{\theta }{g}_{\theta \theta },\end{array}$$ A 10

$$\begin{array}{rl}{(N^{\alpha \beta }{\phi }_{\alpha })}_{,\beta }& \hspace{0.17em}=-N^{\omega \omega }\phi g_{\theta \theta }{\mathrm{\Gamma }}_{\omega \omega }^{\theta }+({N^{\theta \theta }}^{\mathrm{\prime }}+N^{\theta \theta }{\mathrm{\Gamma }}_{\theta \theta }^{\theta })\phi g_{\theta \theta }+N^{\theta \theta }({\phi }^{\mathrm{\prime }}{g}_{\theta \theta }+\phi {g_{\theta \theta }}^{\mathrm{\prime }})\\ & \hspace{0.17em}+\phi g_{\theta \theta }(N^{\omega \omega }{\mathrm{\Gamma }}_{\omega \omega }^{\theta }+N^{\theta \theta }{\mathrm{\Gamma }}_{\theta \omega }^{\omega }),\end{array}$$ A 11

$$\begin{array}{rl}& \mathrm{\Delta }P(1+u_{,\gamma }^{\gamma }+b_{\gamma }^{\gamma }w)=\mathrm{\Delta }P[1+u^{\theta }{\mathrm{\Gamma }}_{\theta \omega }^{\omega }+u^{{\theta }^{\mathrm{\prime }}}+u^{\theta }{\mathrm{\Gamma }}_{\theta \theta }^{\theta }+(b_{\omega }^{\omega }+b_{\theta }^{\theta })w],\end{array}$$ A 12

$$\begin{array}{rl}& \hspace{0.17em}-N_{,\beta }^{\gamma \beta }=-N_{,\beta }^{\theta \beta }=-N^{\omega \omega }{\mathrm{\Gamma }}_{\omega \omega }^{\theta }-N^{\theta \theta }{\mathrm{\Gamma }}_{\theta \omega }^{\omega }-{N^{\theta \theta }}^{\mathrm{\prime }}-2N^{\theta \theta }{\mathrm{\Gamma }}_{\theta \theta }^{\theta },\end{array}$$ A 13

$$\begin{array}{rl}& \hspace{0.17em}-M_{,\beta }^{\alpha \beta }{b}_{\alpha }^{\gamma }=-M_{,\beta }^{\alpha \beta }{b}_{\alpha }^{\theta }=-b_{\theta }^{\theta }(M^{\omega \omega }{\mathrm{\Gamma }}_{\omega \omega }^{\theta }+M^{\theta \theta }{\mathrm{\Gamma }}_{\theta \omega }^{\omega }+{M^{\theta \theta }}^{\mathrm{\prime }}+2M^{\theta \theta }{\mathrm{\Gamma }}_{\theta \theta }^{\theta }),\end{array}$$ A 14

$$\begin{array}{rl}& \hspace{0.17em}-\frac{1}{2}{(M^{\alpha \beta }{b}_{\alpha }^{\gamma }-M^{\gamma \alpha }{b}_{\alpha }^{\beta })}_{,\beta }=-\frac{1}{2}{(M^{\alpha \beta }{b}_{\alpha }^{\theta }-M^{\theta \alpha }{b}_{\alpha }^{\beta })}_{,\beta }=0,\end{array}$$ A 15

$$\begin{array}{rl}& N^{\alpha \beta }{\phi }_{\alpha }{b}_{\beta }^{\gamma }=N^{\alpha \beta }{\phi }_{\alpha }{b}_{\beta }^{\theta }=N^{\theta \theta }{b}_{\theta }^{\theta }\phi g_{\theta \theta },\end{array}$$ A 16

$$\begin{array}{rl}& \hspace{0.17em}-\frac{1}{2}{(N_{\alpha }^{\alpha }\varphi {ϵ}^{\mu \gamma })}_{,\mu }=-\frac{1}{2}{(N_{\alpha }^{\alpha }\varphi {ϵ}^{\mu \theta })}_{,\mu }=0\end{array}$$ A 17

$$\begin{array}{r}\text{and}\mathrm{\Delta }P({\phi }^{\gamma }+\varphi {\phi }^{\eta }{ϵ}_{\eta }^{\gamma })=\mathrm{\Delta }P({\phi }^{\theta }+\varphi {\phi }^{\eta }{\epsilon }_{\eta }^{\theta })=\mathrm{\Delta }P\phi ,\end{array}$$ A 18

where (·)′ denotes d(·)/dθ, φ = φ^θ, g_αβ and g^αβ are the covariant and contravariant components of the first fundamental form of the mid-surface,

$$\begin{array}{rl}{g}_{\omega \omega }& \hspace{0.17em}=\frac{1}{g^{\omega \omega }}={\mathit{x}}_{,\omega }\cdot {\mathit{x}}_{,\omega }=R^2{\cos }^2\theta ,g_{\theta \theta }=\frac{1}{g^{\theta \theta }}={\mathit{x}}_{,\theta }\cdot {\mathit{x}}_{,\theta }={R^{\mathrm{\prime }}}^2+R^2.\\ g_{\omega \theta }& \hspace{0.17em}=g_{\theta \omega }=g^{\omega \theta }=g^{\theta \omega }=0,\end{array}$$ A 19

b_αβ and $b_{\beta }^{\alpha }$ are the covariant and the mixed components of the second fundamental form of the mid-surface,

$$\begin{array}{rl}{b}_{\omega \omega }& \hspace{0.17em}=-\mathit{N}\cdot {\mathit{x}}_{,\omega \omega }=\frac{R\cos \theta (R^{\mathrm{\prime }}\sin \theta +R\cos \theta )}{\sqrt{{R^{\mathrm{\prime }}}^2+R^2}},b_{\omega }^{\omega }=b_{\omega \omega }{g}^{\omega \omega }=\frac{R^{\mathrm{\prime }}\sin \theta +R\cos \theta }{R\cos \theta \sqrt{{R^{\mathrm{\prime }}}^2+R^2}}\\ b_{\theta \theta }& \hspace{0.17em}=-\mathit{N}\cdot {\mathit{x}}_{,\theta \theta }=\frac{2{R^{\mathrm{\prime }}}^2-R{R}^{\mathrm{\prime }\mathrm{\prime }}+R^2}{\sqrt{{R^{\mathrm{\prime }}}^2+R^2}},b_{\theta }^{\theta }=b_{\theta \theta }{g}^{\theta \theta }=\frac{(2{R^{\mathrm{\prime }}}^2-R{R}^{\mathrm{\prime }\mathrm{\prime }}+R^2)}{{({R^{\mathrm{\prime }}}^2+R^2)}^{(3/2)}},\\ b_{\omega \theta }& \hspace{0.17em}=b_{\theta \omega }=b_{\theta }^{\omega }=b_{\omega }^{\theta }=0,\end{array}$$ A 20

and ${\mathrm{\Gamma }}_{\alpha \beta }^{\gamma }$ denotes the Christoffel symbols,

$${\mathrm{\Gamma }}_{\alpha \beta }^{\gamma }=\frac{1}{2}{g}^{\gamma \lambda }\left(\frac{\mathrm{\partial }{g}_{\alpha \lambda }}{\mathrm{\partial }\beta }+\frac{\mathrm{\partial }{g}_{\beta \lambda }}{\mathrm{\partial }\alpha }-\frac{\mathrm{\partial }{g}_{\alpha \beta }}{\mathrm{\partial }\lambda }\right)$$ A 21

yielding the following non-zero components

$${\mathrm{\Gamma }}_{\theta \omega }^{\omega }={\mathrm{\Gamma }}_{\omega \theta }^{\omega }=\frac{R^{\mathrm{\prime }}\cos \theta -R\sin \theta }{R\cos \theta },{\mathrm{\Gamma }}_{\theta \theta }^{\theta }=\frac{R^{\mathrm{\prime }}{R}^{\mathrm{\prime }\mathrm{\prime }}+R{R}^{\mathrm{\prime }}}{{R^{\mathrm{\prime }}}^2+R^2},{\mathrm{\Gamma }}_{\omega \omega }^{\theta }=\frac{-R{R}^{\mathrm{\prime }}{\cos }^2\theta +R^2\cos \theta \sin \theta }{{R^{\mathrm{\prime }}}^2+R^2}.$$ A 22

Data accessibility

The codes and the supplementary video are provided in electronic supplementary material [29].

Authors' contributions

Y.C.: conceptualization, data curation, methodology, software, visualization, writing—original draft, writing—review and editing; T.L.: methodology, software, writing—review and editing; L.J.: conceptualization, funding acquisition, methodology, supervision, validation, writing—review and editing.

All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Conflict of interest declaration

We declare we have no competing interests.

Funding

This work is supported by the startup fund from Henry Samueli School of Engineering and Applied Science at the University of California, Los Angeles and National Science Foundation through a CAREER Award no. CMMI-2048219. T. L. acknowledges the support from the Fundamental Research Funds for the Central Universities (no. 2242022R20022) and Jiangsu Funding Program for Excellent Postdoctoral Talent (no. 2022ZB133).