The Static Response of Axisymmetric Conical Shells Exhibiting Bistable Behavior

Abstract Belleville springs are widely used in a variety of mechanical systems. Recent advances in the field of multi-stable structures suggest that these conical axisymmetric washers may be extremely useful as bistable building blocks for multi-stable architected metamaterials. In this paper, we examine the ability of existing analytical models to accurately predict the bistable behavior of Belleville springs, namely, a nonmonotonous force–displacement relation with two branches of positive stiffness separated by a branch of negative stiffness. By comparing with results of finite element (FE) simulations, we find that current analytical models may suffer from significant inaccuracies associated with the assumption of rigid rotation. According to this assumption, adopted by all analytical models of Belleville springs, the cross section of the spring rotates without bending, i.e., maintains zero curvature as the spring deforms. Motivated by this insight, we relax the rigid-rotation assumption and approximate the radial displacement field by a linear relation in terms of the distance from the spring axis. We find, based on extensive finite element simulations, that the functional dependence of the radial displacement on the geometry of the springs is indifferent to the stage of deformation and can be expressed in terms of three geometrical parameters. These findings enable us to derive closed-form expressions that are simple and straightforward to use, yet are significantly more accurate than existing analytical models.