Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model

In the present paper a two-dimensional solid consisting of a homogeneous linear elastic orthotropic material, that is invariant under 90 degrees rotation, invariant for mirror transformation, and for which the deformation energy depends on the rst and on the second gradient of the displacement, is considered. Such anisotropy is the most general for pantographic structures that are composed by two identical orthogonal families of bers. It is well known in the literature that the corresponding strain energy depends on 8 constitutive parameters: 3 parameters related to the rst gradient part of the strain energy and 5 parameters related to the second gradient part of the strain energy. In this paper, analytical solutions for simple problems, that are here referred to the heavy sheet, to the non conventional bending and to the trapezoidal cases, are developed and shown. On the basis of such analytical solutions, gedanken experiments have been developed in such a way the whole set of the 8 constitutive parameters are completely characterized in terms of the materials of which the bers are built (i.e. of the Young modulus of the bers material), of their cross sections (i.e. of the area and of the moment of inertia of the cross sections of the bers) and of the distance between the nearest pivots. On the basis of these considerations, a remarkable form of the strain energy is derived in terms of the displacement elds, that closely resembles the strain energy of simple Euler beams. Numerical simulations con rm the validity of the presented results. Classic bone-shaped deformation are derived in classic bias numerical test and the presence of oppy mode is also made numerically evident in the present continuum model.