Third-gradient continua: nonstandard equilibrium equations and selection of work conjugate variables

Abstract

This paper outlines the variational derivation of the Lagrangian equilibrium equations for the third-gradient materials, stemming from the minimization of the total potential energy functional, and the selection of suitable dual variables to represent the inner work in the Eulerian configuration. Volume, face, edge and wedge contributions were provided through integration by parts of the inner virtual work and by repeated applications of the divergence theorem extended to embedded submanifolds with codimension one and two. Detailed expressions were provided for the contact pressures and the edge loading, revealing the complex dependence on the face normals and on the mean curvature. Relationships were specified among the Lagrangian (hyper-)stress tensors of rank lower or equal to four, and their Eulerian counterparts.

Deformation-induced coupling of the generalized external actions in third-gradient materials

“…The above symbols possess the following meaning: F ext Ω i (X), F ext Σ i (X) , F ext L i (X) and F ext P w i denote Eulerian vectors defined in the Lagrangian domain (Ω ), over its boundary face (Σ ), along the its edges (L ) and at relevant wedges ({P w } with coordinates P w , w = 1, • • • , ntotwedge), dimensionally equal to force densities per unit volume, per unit surface, per unit length and to a point force, respectively; F ext Σ N i (X) and F ext Σ NN i (X) indicate Eulerian vector fields defined over the Lagrangian boundary surfaces, referred to as external (surface) double and triple force densities, respectively, dimensionally equal to a force per unit surface multiplied by a length (or a work per unit surface), and to a force per unit surface multiplied by a length squared; symbols F ext L N i (X) and F ext L B i (X) indicate Eulerian vector fields defined over the Lagrangian boundary edge, referred to as external (edge) double force densities, dimensionally equal to a force. It can be proven that, selecting suitable pairs of work conjugate variables, the governing equations exhibit the same mathematical form in the Lagrangian and in the Eulerian configuration, see [5,6]. Hence, the expression of the external virtual work results to be the same, on condition of substituting the Lagrangian variables and the integration domains with their Eulerian counterparts (without the subscript ).…”

“…In third-gradient continua, the deformation energy density is assumed of the form W F, ∇F, ∇ (2) F , see e.g. [6]. In the postulation of mechanics based on the virtual work principle, the contributions to the external work can be specified only after that the inner virtual work, first variation of the stored energy functional, has been represented in a unique and no more reducible way as the sum of volume, surface, edge and wedge terms.…”

“…As detailed in the next sections, the Eulerian loading will be denoted by lowercase symbols, maintaining the same convention of Lagrangian actions, for instance f ext Ω i will indicate the Eulerian counterpart of the volume loading density F ext Ω i , f ext Σ i will be used in the spatial configuration for the surface loading density F ext Σ i , etc . Equation (1), deduced by a variational approach in [6], must be regarded as a particularization of the general representation formula for the external virtual work, provided in [7] for the Hth gradient material occupying the volume Ω , namely…”

“…[3]. By a variational approach, through iterated applications of the integration by parts and of the divergence theorem extended to submanifolds with boundary [4,5], the inner virtual work, namely the first variation of the deformation energy functional, can be represented by the sum of volume, surface, edge and wedge integrals [6]. In this way, the Cauchy representation theorem for external loads in terms of stress tensor is extended, so that generalized external forces are equated to hyperstress tensors applied to the shape of Cauchy cuts, see [7,8].…”

“…Through some novel results of differential geometry serving as intermediate steps [4], a strategy was developed to transform such second-gradient governing equations from the Eulerian to the Lagrangian description [5,9]. More recently, in [6] the third-gradient equilibrium equations were deduced, endowed by suitable relationships between Lagrangian and Eulerian hyperstress tensors of order lower or equal to four. In the present study, the transport of the external actions from the Eulerian to the Lagrangian form is carried out for the first time with reference to such third-gradient continua.…”