Basic criteria to design and produce multistable shells

Abstract

A shell can have multiple stable equilibria either if its initial curvature is sufficiently high or if a suitably strong pre-stress is applied. Under the hypotheses of a thin and shallow shell, we derive closed form results for the critical values of curvatures and pre-stresses leading to bistability and tristability. These analytical expressions allow to easily provide guidelines to build shells with different stability properties.

Appendix: Identification of stability boundaries for multistable shells by FE simulations

In Fig. 3, we resume the prediction of the stability boundaries for multistable shells: results obtained by applying the UC and QC models (solid and dashed lines, respectively) can be compared to the ones computed by refined Finite Element approximations (marked as black dots). In particular, we performed FE simulations with Abaqus CAE using the S4R element, which is a shell element with four nodes and five degrees of freedom per node with reduced integration. Geometrical non linearities are taken into account in a Full-Newton scheme. In order to validate the analytical results as well as to locate the experimental response of the prototypes described in Sect. 5, results of simulations correspond to the same material system as in experiments, that is 2 mm-thick bilayered shells made of Green$-$(Green$+$Pink) silicone material. The non-dimensional diagram of Fig. 3 is valid for both silicone and rubber materials since they have the same Poisson’s ratio ($\mu =0.5$) which is the only parameter affecting its boundaries. The planform of the shell is circular, but simulations based on square planforms were also performed and gave similar results. The diameter was initially fixed at $D=10\, \text { cm}$ in order to reproduce the experimental results. Simulations were also performed with a diameter of $D=20\, \text { cm}$ in order to apply sufficiently high initial curvatures $c_0$ (points with $c_0 > 5$ in Fig. 3).

In the analytical models, the shell is free and no boundary condition is applied; thus, in the FE simulations we only clamped the central node in order to avoid rigid body motions.

In order to reproduce the stability diagram of Fig. 3, the two parameters to be varied in the FE study are the natural curvature $c_0$ and the inelastic curvature $c_i$ (we remind that $c_0$ and $c_i$ are adimensional curvatures, according to definitions given in Sect. 3). Different levels of natural curvature $c_0$ simply correspond to different paraboloid shell surfaces $\frac{c_0}{R}(x^2 + y^2)$, where R is the characteristic radius defined by Eq. (10): changing $c_0$ required to build a new mesh for the initial stress-free configuration, the number of elements of the mesh varying from 1456 (9114 d.o.f.) for a disk ($c_0=0$) to 1828 (11346 d.o.f.) for the deepest shell considered ($c_0=9$). Different levels of inelastic curvature $c_i$ were induced by applying an equivalent uniform temperature change $\varDelta T$ to the shell (different expansion coefficients are defined for the two layers, thus inducing an inelastic curvature of the shell: values of the ficticious expansion coefficients are identified from the experimental behaviour of the silicone material system, shown in Fig. 8).

From an operational point of view, we fixed a discrete set of $c_0$ values (see dots on Fig. 3). For each value of $c_0$, we built the corresponding mesh (stress-free configuration) of the shell and we found the corresponding everted configuration at $c_i=0$, if it existed, by applying imposed displacements onto the shell: clearly there is a range of values of $c_0$ ($c_0 \in [-c_0^{*},c_0^{*}]$) where the everted configuration cannot be found. Then, starting from each natural configuration and from each everted configuration, we have progressively increased the inelastic curvature value $c_i$ (in practice, the temperature change $\varDelta T$) until the stability was lost, thus marking the critical values of inelastic curvature in order to obtain bifurcation or loss of equilibrium.

In order to find the different points on the stability diagram (see Fig. 14), we followed the procedures described hereafter.

• 1. Intersection between solid gray and dashed gray lines (black point, $c_i^{*}$): start from a flat disk and increase the inelastic curvature $c_i$ until finding the critical value $c_i^{*}$ inducing bistability through a pitchfork bifurcation (evolution of curvatures is shown in Fig. 2a).

• 2. Intersection between dashed black line and the horizontal axis $c_i=0$ (gray point, $c_0^{*}$): start from a spherical shallow shell and impose displacements onto the shell in order to evert the curvature; repeat the simulations by increasing the initial curvature $c_0$ until the existence of the everted configuration is established at $c_0=c_0^{*}$ (see results on Fig. 1).

• 3. Dashed gray boundary ($c_0>0$): start from a spherical shell with initial curvature $c_0>0$ and apply a positive inelastic curvature $c_i$ until finding the value corresponding to the pitchfork bifurcation of the natural configuration into two quasi-cylindrical shapes.

• 4. Dashed black boundary ($c_0>0$): first find the everted configuration, then apply a positive inelastic curvature $c_i$ until it looses stability (above the dashed black line the everted configuration is no longer stable: tristability is attested above the dashed gray line and below the dashed black line).

• 5. Solid black boundary ($c_0<0$): start from a curved shell with initial negative $c_0$ curvature, and apply a positive inelastic curvature $c_i$ until it looses stability.

• 6. Solid gray boundary ($c_0<0$): apply a positive inelastic curvature $c_i$ onto an everted configuration corresponding to intial negative natural curvature $c_0$ and increase $c_i$ until the everted configuration bifurcates into two quasi-cylindrical shapes.