Bistable polar-orthotropic shallow shells

Abstract

We investigate stabilizing and eschewing factors on bistability in polar-orthotropic shells in order to enhance morphing structures. The material law causes stress singularities when the circumferential stiffness is smaller than the radial stiffness ( β < 1), requiring a careful choice of the trial functions in our Ritz approach, which employs a higher-order geometrically nonlinear analytical model. Bistability is found to strongly depend on the orthotropic ratio, β , and the in-plane support conditions. An investigation of their interaction offers a new perspective on the effect of the hoop stiffness on bistability: while usually perceived as promoting, it is shown to be only stabilizing insofar as it prevents radial expansions; however, if in-plane supports are present, it becomes a redundant feature. Closed-form approximations of the bistable threshold are then provided by single-curvature-term approaches. For significantly stiffer values of the radial stiffness, a strong coupling of the orthotropic ratio and the support conditions is revealed: while roller-supported shells are monostable, fixed-pinned ones are most disposed to stable inversions; insight is given by comparing to a simplified beam model. Eventually, we show that cutting a central hole is a suitable method to deal with stress singularities: while fixed-pinned shells are barely affected by a hole, the presence of a hole strongly favours bistable inversions in roller-supported shells.

Architectured Lattice Materials with Tunable Anisotropy: Design and Analysis of the Material Property Space with the Aid of Machine Learning

“…[ 24 ] Of special interest are 2D adaptive systems whose functioning is associated with the existence of a set of stable states that are attained depending on the applied load. [ 25–27 ] This set is determined by the elasticity constants, notably by the compliance coefficients . That is why an expansion of the region Ω enables an extension of the set of possible stable states and, hence, the additivity of the structure.…”

Programmable Multistable Perforated Shellular

“…Due to their highly nonlinear deformation behavior caused by their intriguing structural instability, elastic shells can be invoked to design multistable unit cells for mechanical metamaterials that can adapt themselves to loading conditions. [ 31 ] Conventional bistable or tristable shells (e.g., half tennis ball [ 32 ] and Venus flytrap [ 33 ] ) display buckling [ 18 ] and snapping [ 34 ] configurations, determined by their curvature, prestress, and residual stress. [ 31 ] In addition, surface patterning, like corrugation, [ 35 ] or varied thickness surfaces, [ 34 ] can impose structural multistability in elastic shells.…”

“…[ 31 ] Conventional bistable or tristable shells (e.g., half tennis ball [ 32 ] and Venus flytrap [ 33 ] ) display buckling [ 18 ] and snapping [ 34 ] configurations, determined by their curvature, prestress, and residual stress. [ 31 ] In addition, surface patterning, like corrugation, [ 35 ] or varied thickness surfaces, [ 34 ] can impose structural multistability in elastic shells. Based on polyhedron templates and high degrees of freedom of soft hinges, assembled prismatic metamaterials can exhibit multistable behavior along multiple directions; [ 36 ] nevertheless, to the best of the authors knowledge, none of the existing polyhedron‐based multidirectional multistable cells can achieve the stable states independently along different directions, as their multidirectional stable states are not independent of each other, and the realization of a new stable state in one direction can break the multistability in other directions.…”

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][42][43]46]). According to this theory, the problem under examination is modelled in terms of a fourth-order differential operator (cf., e.g., [16]).…”