Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions

Abstract

The well-posedness of the boundary value problems for second gradient elasticity has been studied under the assumption of strong ellipticity of the dependence on the second placement gradients (see, e.g., Chambon and Moullet in Comput. Methods Appl. Mech. Eng. 193:2771-2796 and Mareno and Healey in SIAM J. Math. Anal. 38:103-115, 2006.The study of the equilibrium of planar pantographic lattices has been approached in two different ways: in dell 'Isola et al. (Proc. R. Soc. Lond. Ser. A 472:20150, 2016) a discrete model was introduced involving extensional and rotational springs which is also valid in large deformations regimes while in Boutin et al. (Math. Mech. Complex Syst. 5:127-162, 2017) the lattice has been modelled as a set of beam elements interconnected by internal pivots, but the analysis was restricted to the linear case. In both papers a homogenized second gradient deformation energy, quadratic in the neighbourhood of non deformed configuration, is obtained via perturbative methods and the predictions obtained with the obtained continuum model are successfully compared with experiments. This energy is not strongly elliptic in its dependence on second gradients. We consider in this paper also the important particular case of pantographic lattices whose first gradient energy does not depend on shear deformation: this could be considered either a pathological case or an important exceptional case (see Stillwell et al. in Am. Math. Mon. 105:850-858, 1998 and Turro in Angew. Chem., Int. Ed. Engl. 39:2255Engl. 39: -2259Engl. 39: , 2000. In both cases we believe that such a particular case deserves some attention because of what we can understand by studying it (see Dyson in Science 200:677-678, 1978). This circumstance motivates the present paper, where we address the well-posedness of the planar linearized equilibrium problem for homogenized pantographic lattices. To do so: (i) we introduce a class of subsets of anisotropic Sobolev's space as the most suitable energy space E relative to assigned boundary conditions; (ii) we prove that the considered strain energy density is coercive and positive definite in E; (iii) we prove that the set of placements for which the strain energy is vanishing (the so-called floppy modes) must strictly include rigid motions; (iv) we determine the restrictions on displacement boundary conditions which assure existence and uniqueness of linear static problems. The presented results represent one of the first mechanical applications of the concept of Anisotropic Sobolev space, initially introduced only on the basis of purely abstract mathematical considerations.

Pantographic metamaterials: an example of mathematically driven design and of its technological challenges

“…The well-posedness of linearized equilibrium equations deriving from the stationarity of the energy functional (18), which is valid in the neighborhood of a stress free configuration for pantographic sheets, cannot be immediately studied by using the results available in the literature. It has been proven that the standard strategy involving the use of Poincaré inequality, Lax-Milgram Theorem, and coercivity of bilinear strain energy form also apply in the context of linear elastic pantographic sheets [44]. The key idea is the exploitation of an unusual energy space, where the solutions relative to well-posed boundary conditions are looked for.…”

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

“…Substituting (39), (40) and (41) into (32) together with ρ α (x i , y j ) = ∂χ ∂α , the desired expansion of the micro-model energy E ε is derived as a function of the kinematic descriptors χ andl µν α as…”

Determination of metamaterial parameters by means of a homogenization approach based on asymptotic analysis

“…Using gamma-convergence, homogenization results have been obtained in [4,5,70]. A remarkable class of structures being described at the macroscale by using a second gradient elasticity theory in pantographic structures [18,83,86], which have received a notable follow-up in the literature [30,34,76,91,93], also from a mathematically rigorous standpoint, regarding fundamental issues such as well-posedness [36].…”