Linear pantographic sheets: existence and uniqueness of weak solutions

Victor A. Eremeyev (1, 2, 3) , Francesco Dell'Isola (4, 1) , Claude Boutin (5) , David Steigmann (6)

1 M&MoCS - International Research Centre on Mathematics & Mechanics of Complex Systems
2 Rzeszów University of Technology
3 SFEDU - Southern Federal University [Rostov-on-Don]
4 DISG - Dipartimento di Ingegneria Strutturale e Geotecnica
5 Université de Lyon
6 LBNL - Lawrence Berkeley National Laboratory [Berkeley]

The well-posedness of the boundary value problem for second gradient elasticity has been studied under the assumption of strong ellipticity of the dependence on the second placement gradients (see e.g. [9,32]). In [18] the equilibrium of pantographic lattices is studied via a homogenised second gradient deformation energy and the predictions obtained with such a model are successfully compared with experiments. This energy is not strongly elliptic in its dependence on second gradients. This circumstance motivates the present paper, where we address the well-posedness for the equilibrium problem for homogenised pantographic lattices. To do so: i) we introduce a class of subsets of anisotropic Sobolevs space as 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 is the strain energy is vanishing (so-called floppy modes) 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.

Domaines: Mécanique [physics]

Victor A. Eremeyev, Francesco Dell'Isola, Claude Boutin, David Steigmann. Linear pantographic sheets: existence and uniqueness of weak solutions. 2017. ⟨hal-01466979⟩