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. https://doi.org/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.
Highlights
The strain-gradient theory of elasticity has its origin in the early works of some giants of continuum mechanics: see [1,2,3,4,5,6] for historical developments in the mechanics of generalized continua, and it was developed further in the original works by Toupin [7] and Mindlin [8,9]
This opinion was shared by Hellinger, see [10,11,12] who, in his masterpiece “Fundamentals of the mechanics of continua”, showed, already with the knowledge available in 1913, that the unifying vision given by variational principles could allow for a effective presentation of all field theories
Unlike to the general framework of the strain-gradient elasticity given by Toupin–Mindlin, the model of pantographic beam lattices relates to a strain energy density which depends on functions having different differential properties in different spatial directions, see [22,23,24,28]
Summary
The strain-gradient theory of elasticity has its origin in the early works of some giants of continuum mechanics: see [1,2,3,4,5,6] for historical developments in the mechanics of generalized continua, and it was developed further in the original works by Toupin [7] and Mindlin [8,9]. The main object of this paper is to prove a result of well-posedness of the deformation problem of linear elastic pantographic sheets deforming in space: i.e., bidimensional continua generalizing standard plate models, as their deformation energy depend on the second gradient of out-of-plane displacement and on second gradients of in-plane displacements. We believe that this is an essential intermediate step in the study of large deformation of pantographic metamaterials or of composite reinforcements, in particular when wrinkling occurs. One may need more advanced techniques as used in the case of nonlinear theories of plates and shallow shells, see, e.g., [84,85,86,87]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
