Effects of boundary conditions on bistable behaviour in axisymmetrical shallow shells

Abstract

Multistable shells are thin-walled structures that have more than one stable state of self-stress. We consider isotropic axisymmetrical shallow shells of arbitrary polynomial shapes using a Föppl-von Kármán analytical model. By employing a Rayleigh-Ritz approach, we identify stable shapes from local minima in the strain energy formulation, and we formally characterize the level of influence of the boundary conditions on the critical geometry for achieving bistable inversion-an effect not directly answered in the literature. Systematic insight is afforded by connecting the boundary to ground through sets of extensional and rotational linear springs. For typical cap-like shells, it is shown that bistability is generally enhanced when the extensional spring stiffness increases and when the rotational spring stiffness decreases, i.e. when boundary movements in-plane are resisted but when their rotations are not; however, for certain other shapes and large in-plane stiffness values, bistability can be enhanced by resisting but not entirely preventing edge rotations. Our predictions are furnished as detailed regime maps of the critical geometry, which are accurately correlated against finite-element analysis. Furthermore, the suitabilities of single degree-of-freedom models, for which solutions are achieved in closed form, are evaluated and compared to our more accurate predictions.

Bistable polar-orthotropic shallow shells

“…In general, in-plane supports strongly favour bistable inversion, which confirms a recent observation in [34]. More interestingly, the influence of the stiffness ratio differs significantly, depending on the support conditions.…”

“…To satisfy the boundary conditions of w ( a ) = 0, m r ( a ) = 0, substitute the following values in equation (3.3): A1=−a1+βfalse(η1+aη2+a2η3+a3η4false)2emfalse0and2emA4=η1a(1+β)(β+ν)+η2a2(2+β)(1+β+ν)+η3a3(3+β)(2+β+ν)−a4(4+β)(3+β+ν).}To account for the degenerate case of β = 1, we have to consider the term η 2 ρ 3 causing a non-vanishing shear-force at the centre. For this particular case, we substitute this term with the next one in the series in equation (3.3), η 2 ρ 6 , and achieve results barely distinguishable from those in our isotropic study [34].…”

Dome‐Patterned Metamaterial Sheets

“…For a given set of geometric boundary conditions, the domes' energy landscape has been shown to depend only on the dome height, the sheet thickness, and the Poisson's ratio of the constituent material. [ 43,44 ] Domes may be patterned to further increase their potential to realize shape adaptation. [ 45 ] Previous work has also revealed that structures featuring a periodic array of domes can be fabricated from plastics or by stamping metal sheets at high temperatures.…”

Numerical methods for static shallow shells lying over an obstacle

“…In this paper, we study the displacement of a static shallow shell lying over a planar obstacle from the numerical point of view, using a suitable finite element method. Shallow shells theory, which is extensively described, for instance, in the books [ 16 ] and [ 44 ], is widely used in engineering (see, e.g., the papers [ 31 , 41 – 43 , 46 ]). According to this theory, the problem under examination is modelled in terms of a fourth-order differential operator (cf., e.g., [ 16 ]).…”