Semi-inverse method à la Saint-Venant for two-dimensional linear isotropic homogeneous second-gradient elasticity

Abstract

This semi-inverse method is similar to that used in the so-called Saint-Venant problem for cylindrical three-dimensional first-gradient linear homogeneous and isotropic materials. This semi-inverse method is similar to that used by Saint-Venant to solve the omonimus problem for cylindrical three-dimensional first-gradient linear homogeneous and isotropic materials. Two examples are also presented. It is found that wedge forces are necessary to maintain the body in equilibrium and that these are not an artefact of the double application of the divergence theorem in the second-gradient material derivations.

Pantographic lattices with non-orthogonal fibres: Experiments and their numerical simulations

“…(1) Although pantographic structures were conceived to give an example of second gradient metamaterial, see e.g. [17,18,19,11,20,21,22,23,24,25,26], the development of 3D printing technology allowed for the practical synthesis of such metamaterials. It deserves to be investigated how to improve the design of 3D printed fabrics in order to fully exploit the exotic behaviour of higher gradient metamaterials.…”

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“…Finally, for those points of [∂∂Ꮾ] where kinematical constraints on u are not given, i.e., where δu = 0, we have the natural boundary conditions f − f ext = 0. For further details, i.e., for proper definitions of the contact actions t, τ , and f , the reader is referred to the complete formulation in Placidi and El Dhaba 2017].…”

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

“…The procedure to find the minimum of a deformation energy functional E is standard, see [55]. The result is given by reporting the variation of the deformation energy functional,…”

“…Adopting the general constitutive relation proposed by Midlin for second gradient 3D solids and specializing it to the 2D case, the strain energy is demonstrated to depend on 6 constitutive parameters: the 2 Lamé constants (λ and μ) and other 4 (instead of 5 for the 3D case) constants (A, B, C and D). Analytical solutions of the same problem can be found in [55], see also [66]. However, in this paper a method is outlined, which is able to identify the six constitutive parameters (see [69] for another identification technique) and to design some ideal experiments that allow to write equations having as unknowns the six constants and as known terms the values of the experimental measurements of appropriately selected quantities.…”