On existence and uniqueness of weak solutions for linear pantographic beam lattices models

Abstract

In this paper, we discuss well-posedness of the boundary-value problems arising in some “gradient-incomplete” strain-gradient elasticity models, which appear in the study of homogenized models for a large class of metamaterials whose microstructures can be regarded as beam lattices constrained with internal pivots. We use the attribute “gradient-incomplete” strain-gradient elasticity for a model in which the considered strain energy density depends on displacements and only on some specific partial derivatives among those constituting displacements first and second gradients. So, unlike to the models of strain-gradient elasticity considered up-to-now, the strain energy density which we consider here is in a sense degenerated, since it does not contain the full set of second derivatives of the displacement field. Such mathematical problem was motivated by a recently introduced new class of metamaterials (whose microstructure is constituted by the so-called pantographic beam lattices) and by woven fabrics. Indeed, as from the physical point of view such materials are strongly anisotropic, it is not surprising that the mathematical models to be introduced must reflect such property also by considering an expression for deformation energy involving only some among the higher partial derivatives of displacement fields. As a consequence, the differential operators considered here, in the framework of introduced models, are neither elliptic nor strong elliptic as, in general, they belong to the class so-called hypoelliptic operators. Following (Eremeyev et al. in J Elast 132:175–196, 2018. 10.1007/s10659-017-9660-3) we present well-posedness results in the case of the boundary-value problems for small (linearized) spatial deformations of pantographic sheets, i.e., 2D continua, when deforming in 3D space. In order to prove the existence and uniqueness of weak solutions, we introduce a class of subsets of anisotropic Sobolev’s space defined as the energy space E relative to specifically assigned boundary conditions. As introduced by Sergey M. Nikolskii, an anisotropic Sobolev space consists of functions having different differential properties in different coordinate directions.

Weak Solutions within the Gradient-Incomplete Strain-Gradient Elasticity

“…Following [15,16,[21][22][23] let us briefly introduce the strain energy density of a three-dimensional pantographic lattice. It consists of three orthogonal families of elastic beams connected through small soft pivots, see Fig.…”

“…As a result, the standard definition of coercitivity is violated and should be modified properly. In order to prove the existence and uniqueness of weak solutions for in-plane and out-of-plane deformations of pantographic sheets in [15,16] the anisotropic Sobolev spaces were used. This class of functional spaces was introduced by Sergey Nikolskii [17], see also [18][19][20].…”

“…This class of functional spaces was introduced by Sergey Nikolskii [17], see also [18][19][20]. The aim of this paper is to extend the results of [15,16] to three-dimensional pantographic beam lattices with perfect pivots which undergo infinitesimal deformations. The paper is organized as follows.…”

On the well posedness of static boundary value problem within the linear dilatational strain gradient elasticity

“…The cited examples are not exhausting the possibilities explored up to now. Indeed, as already mentioned, it has been proven that also some materials having a beam-lattice micro-structure can be modelled using gradient incomplete models [2,7,22,24,26,37,62], for pantographic beam lattice microstructured materials see [30,32]. In all aforementioned cases of incomplete second gradient models, the analysis of the well posedness of boundary value problems requires a proper modification of the standard techniques [18,33,39].…”

Two-dimensional continua capable of large elastic extension in two independent directions: Asymptotic homogenization, numerical simulations and experimental evidence

“…After the 2000s, in addition to the study on the synthesis of rst gradient continua [23,24], the synthesis of second gradient continua has been discussed and solved in the linear and conservative case for 1D, 2D and 3D continua (see [2,4,1,5] and [7,8]). In [9] and then in [10] a synthesis of couple-stress 3D and 2D continua has been obtained.…”