Pantographic beam: a complete second gradient 1D-continuum in plane

There is a class of planar 1D-continua which can be described exclusively by their placement functions which in turn are curves in a two-dimensional space. In contrast to the Elastica for which the deformation energy depends on the projection of the second gradient to the normal vector of the placement function, i.e. the material curvature, the proposed continuum does also depend on the projection onto the tangent vector, introduced as the stretch gradient. Thus, the deformation energy takes into account the complete second gradient of the placement function. In such a model, non-standard boundary conditions and more generalized forces such as double forces do appear. The deformation energy of the continuum is obtained by applying a heuristic homogenization procedure to a family of slender discrete pantographic structures constituted by extensional and rotational springs. Within the homogenization process, the overall length of the system is kept fixed, the number of the periodically appearing sub-systems, called cells, is increased, and the stiffnesses are appropriately scaled. For two examples, we numerically compare the family of discrete systems with the continuum. The analysis shows that the continuum represents the behaviour of the discrete system already for a relatively moderate number of cells. In particular, the behaviour of the deformation energy error between the discrete and the continuum models when the number of cells tends to infinity is determined by the homogenization process.