Instability of the third-gradient beam

We discuss the instability of a third-gradient beam, which is an inextensible elastic beam whose deformation energy depends on the first derivative of the curvature of its axis. Such one-dimensional problems arise from the homogenisation of slender, lengthy structures made of beam lattice metamaterials. The equation of equilibrium is derived using the Lagrange variational principle. Solutions to a few linear problems are presented. In addition to the linear analysis, we present an asymptotic solution for finite deformations, similar to those of Euler’s elastica. We demonstrate the correspondence between the critical forces of third-gradient beams and those of classical Euler–Bernoulli beams.