A nonlocal elastica inspired by flexural tensegrity

A nonlocal theory is presented for the bending in large deformations under applied loads of an initially straight rod. This has similarities with the classical Euler's elastica in the sense that the bending stiffness remains homogeneously constant, but it depends on an integral average of the entire curvature field. The discretized form of the equilibrium equations is identical to those governing the response of structural systems already called flexural tensegrity beams, composed of a chain of segments in unilateral contact, whose integrity under flexion is due to prestressing tendons and to the shape of the contact surfaces. An analytical method of solution is proposed modulo the calculation of elliptic integrals, which is compared in paradigmatic examples with the numerical approach, or with an approximation of the curvature field with shape functions. The comparison between the continuum theory and the discrete case of flexural tensegrity highlights the physical role of the constitutive parameters, paving the way for a tailored design of innovative devices and the modelling of complex biological structures, based on the capability of transforming the mechanical properties with very small changes at the level of the underlying micro-constituents.