On weak solutions of the boundary value problem within linear dilatational strain gradient elasticity for polyhedral Lipschitz domains

Victor A Eremeyev (1, 2, 3) , Francesco Dell'Isola (3, 4, 5, 6)

1 UniCa - Università degli Studi di Cagliari = University of Cagliari
2 Faculty of Civil and Environmental Engineering Gdańsk University of Technology Gdańsk Poland
3 Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences,
4 DICEAA - Dipartimento di Ingegneria Civile, Edile-Architettura, Ambientale
5 MEMOCS - International Research Center for the Mathematics & Mechanics of Complex Systems
6 UNIVAQ - Università degli Studi dell'Aquila = University of L'Aquila

We provide the proof of an existence and uniqueness theorem for weak solutions of the equilibrium problem in linear dilatational strain gradient elasticity for bodies occupying, in the reference configuration, Lipschitz domains with edges. The considered elastic model belongs to the class of so-called incomplete strain gradient continua whose potential energy density depends quadratically on linear strains and on the gradient of dilatation only. Such a model has many applications, e.g., to describe phenomena of interest in poroelasticity or in some situations where media with scalar microstructure are necessary. We present an extension of the previous results by Eremeyev et al. (2020 Z angew Math Phys 71(6): 1-16) to the case of domains with edges and when external line forces are applied. Let us note that the interest paid to Lipschitz polyhedra-type domains is at least twofold. First, it is known that geometrical singularity of the boundary may essentially influence singularity of solutions. On the other hand, the analysis of weak solutions in polyhedral domains is of great significance for design of optimal computations using a finite-element method and for the analysis of convergence of numerical solutions.

Mots clés: weak solution; dilatational strain gradient elasticity; Sobolev spaces; Lipschitz polyhedra; edge forces; H(div; V)

Domaines: Mécanique des matériaux [physics.class-ph]

Victor A Eremeyev, Francesco Dell'Isola. On weak solutions of the boundary value problem within linear dilatational strain gradient elasticity for polyhedral Lipschitz domains. Mathematics and Mechanics of Solids, 2021. ⟨hal-03427363⟩