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

Abstract

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, nonstandard 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 subsystems , 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.

Effective strain gradient continuum model of metamaterials and size effects analysis

“…The size effects can also be examined by making use of the homogenization method for different metamaterials under different boundary conditions, for example, metamaterials of D4-invariant, D6invariant, Z2-invariant, etc., as shown in [9] under stretching, shearing, transverse bending [86]. In a specific case, the pantographic metamaterial which has been largely studied by dell'Isola and his colleagues [7,29,31,32,68,70,72,73,78,79] is a so-called higher gradient material [2][3][4]16,27,28]. It is also possible to determine the effective material parameters of a pantographic metamaterial by using this method.…”

Large in-plane elastic deformations of bi-pantographic fabrics: asymptotic homogenization and experimental validation

“…Pantographic beam discrete model. The assembly and kinematics of a discrete pantographic beam slightly generalizing that presented in [33,36] are sketched in Fig. 1.…”

“…which, together with Piola's micro-macro identication (36), are used to establish a correspondence between boundary conditions (53) for the continuum model and those for the discrete one. Such a correspondence is reported in Tab.…”

Equilibrium of Two-Dimensional Cycloidal Pantographic Metamaterials in Three-Dimensional Deformations

“…This arrangement of the fibers has been specially conceived with the purpose of employing the superior capability of curved fibers to bear loads (fibers tend to behave like arches) and to have reinforcement in long boundaries (see Figure 1). Therefore, the sample has a notably anisotropic behavior with an extremely soft core (due to the pantographic effect [16,17,[48][49][50][51]) and a stiffer part at the borders.…”