Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients

In the present paper a two-dimensional solid consisting of a linear elastic isotropic material, for which the deformation energy depends on the second gradient of the displacement, is considered. The strain energy is demonstrated to depend on 6 constitutive parameters: the 2 Lamé constants ( and ) and 4 more parameters (instead of 5 as it is in the 3D-case). Analytical solutions for classical problems such as heavy sheet, bending and exure are provided. The idea is very simple: the solutions of the corresponding problem of rst gradient classical case are imposed and the corresponding forces, biforces and wedge forces are found out. On the basis of such solutions, a method is outlined which is able to identify the six constitutive parameters. Ideal (or gedanken) experiments are designed to allow to write equations that have as unknowns the six constants and as known terms the values of the experimental measurements of appropriately selected quantities.