Weak Solutions within the Gradient-Incomplete Strain-Gradient Elasticity

Abstract

In this paper we consider existence and uniqueness of the three-dimensional static boundary-value problems in the framework of so-called gradient-incomplete strain-gradient elasticity. We call the straingradient elasticity model gradient-incomplete such model where the considered strain energy density depends on displacements and only on some specific partial derivatives of displacements of first-and second-order. Such models appear as a result of homogenization of pantographic beam lattices and in some physical models. Using anisotropic Sobolev spaces we analyze the mathematical properties of weak solutions. Null-energy solutions are discussed.

The consistent coupling boundary condition for the classical micromorphic model: existence, uniqueness and interpretation of parameters

“…In this paper we will fully investigate the new boundary condition (14) for the classical Mindlin-Eringen micromorphic model. We will show existence and uniqueness under this weaker requirement, we will derive the missing normal Neumann-type condition on the Dirichlet boundary Γ for the third order moment stress tensor m DP and we will compare analytical solutions for (12) versus (14). This allows us to better understand the different Dirichlet conditions on P (which both lead to existence and uniqueness).…”

“…The variations δP in the bulk and at the boundary are independent, therefore δF The second gradient linearized elasticity model (without mixed terms) reads (see for example [12,13,14,15,16,17,26,39,49,64])…”

Shear rupture mechanism and dissipation phenomena in bias-extension test of pantographic sheets: Numerical modeling and experiments

“…Pantographic structures have been intensively studied in recent years [1][2][3][4][5][6] because of their peculiar behavior, consisting of two main features within the family of lightweight structures: (1) the capability to exploit large deformations in the elastic regime; (2) hierarchical behavior in the damage mechanisms. The research lines can be roughly classified into those devoted to the analysis of theoretical aspects [7][8][9][10][11][12], to the formulation of computational methods in discrete [13][14][15][16] and continuum framework [17][18][19][20], as well as to the investigations of experimental evidence [21][22][23][24][25][26][27]. The genesis in conceiving the pantographic sheet can be traced back in the grounding idea of non-local materials, and in particular of second or higher gradient theories [28][29][30][31][32][33][34][35][36][37], in which the constitutive law is a function also of the second or higher gradient of the displacement field.…”

Inverse analysis of metamaterials and parameter determination by means of an automatized optimization problem

“…Through a homogenization procedure [60], the behavior of the structure is estimated by the reduced‐order model [61, 62]. The existence and uniqueness of weak solutions for linear pantographic structures is presented in [63, 64]. The wave dispersion in nonlinear pantographic beams has been studied [65].…”