In this paper a one-dimensional continuum whose configurations are curves in a plane
and whose deformation energy depends on the gradient of curvature is considered: a
third-gradient 1D continuum. We find Euler–Lagrange stationary conditions valid for
a class of deformation energies depending quadratically on elongation, gradient of
elongation, curvature and gradient of curvature, including the “compatible” essential
and natural boundary conditions. To expedite the algebraic calculation we introduce
a constraint linking the used measures of deformation to the placement and its
gradient and the corresponding Lagrange multiplier. This formulation of energy
minimality condition allows us to start exploring the mechanical properties of the
introduced continuum through numerical simulations carried out by using the finite
element method and the commercial software COMSOL Multiphysics. Here,
we prove the existence of coupled constant-elongation/constant-curvature
floppy modes for the considered continuum. Some equilibrium configurations
are found imposing the displacement of three or five material points of the
third-gradient 1D continuum. The generalized cantilever third-gradient beam
problem is also considered and some exotic mechanical behaviors are shown to be
possible.
