On nonlinear dilatational strain gradient elasticity

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

1 Department of Civil and Environmental Engineering and Architecture
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 call nonlinear dilatational strain gradient elasticity the theory in which the specific class of dilatational second gradient continua is considered: those whose deformation energy depends, in an objective way, on the gradient of placement and on the gradient of the determinant of the gradient of placement. It is an interesting particular case of complete Toupin-Mindlin nonlinear strain gradient elasticity: indeed, in it, the only second gradient effects are due to the inhomogeneous dilatation state of the considered deformable body. The dilatational second gradient continua are strictly related to other generalized models with scalar (onedimensional) microstructure as those considered in poroelasticity. They could be also regarded to be the result of a kind of "solidification" of the strain gradient fluids known as Korteweg or Cahn-Hilliard fluids. Using the variational approach we derive, for dilatational second gradient continua the Euler-Lagrange equilibrium conditions in both Lagrangian and Eulerian descriptions. In particular, we show that the considered continua can support contact forces concentrated on edges but also on surface curves in the faces of piecewise orientable contact surfaces. The conditions characterizing the possible externally applicable double forces and curve forces are found and examined in detail. As a result of linearization the case of small deformations is also presented. The peculiarities of the model is illustrated through axial deformations of a thick-walled elastic tube and the propagation of dilatational waves. Keywords Strain gradient elasticity • Dilatational elasticity • Variational principle • Poroelasticity Communicated by Marcus Aßmus and Andreas Öchsner. V.A.E. and F.d'I. acknowledge the support by the Russian Science Foundation under Grant 20-41-04404 issued to the Institute of Applied Mechanics of Russian Academy of Sciences.

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

Victor A Eremeyev, Antonio Cazzani, Francesco Dell'Isola. On nonlinear dilatational strain gradient elasticity. Continuum Mechanics and Thermodynamics, 2021. ⟨hal-03427360⟩