Identification of two-dimensional pantographic structures with a linear d4 orthotropic second gradient elastic model accounting for external bulk double forces

The present paper deals with the identification of the nine constitutive parameters appearing in the strain energy density of a linear elastic second gradient D4 orthotropic two-dimensional continuum model accounting for an external bulk double force mext. The aim is to specialize the model for the description of pantographic fabrics, which show such a kind of anisotropy. Analytical solutions for model problems, which are here referred to as the heavy sheet, the non-conventional bending and the trapezoidal cases are recalled from a previous paper and further elaborated in order to perform gedanken experiments. We completely characterize the set of nine constitutive parameters in terms of the materials the fibers are made of (i.e. of the Young’s modulus of the fiber materials), of their cross section (i.e. of the area and of themoment of inertia of the fiber cross sections), of the internal rotational spring positioned at each intersection point between the two families of fibers and of the pitch, i.e. the distance between adjacent pivots. Finally, the remarkable form of the strain energy, derived in terms of the displacement field, is shortly discussed.