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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. [ABSTRACT FROM AUTHOR]

For homogeneous higher-gradient elasticity models we discuss frame-indifference and isotropy requirements. To this end, we introduce the notions of local versus global SO(3)-invariance and identify frame-indifference (traditionally) with global left SO(3)-invariance and isotropy with global right SO(3)-invariance. For specific restricted representations, the energy may also be local left SO(3)-invariant as well as local right SO(3)-invariant. Then we turn to linear models and consider a consequence of frame-indifference together with isotropy in nonlinear elasticity and apply this joint invariance condition to some specific linear models. The interesting point is the appearance of finite rotations in transformations of a geometrically linear model. It is shown that when starting with a linear model defined already in the infinitesimal symmetric strain ε = symGrad [ u ] , the new invariance condition is equivalent to the isotropy of the linear formulation. Therefore, it may also be used in higher-gradient elasticity models for a simple check of isotropy and for extensions to anisotropy. In this respect we consider in more detail variational formulations of the linear indeterminate couple-stress model, a new variant of it with symmetric force stresses and general linear gradient elasticity. [ABSTRACT FROM AUTHOR]

In this paper we venture a new look at the linear isotropic indeterminate couple-stress model in the general framework of second-gradient elasticity and we propose a new alternative formulation which obeys Cauchy–Boltzmann’s axiom of the symmetry of the force-stress tensor. For this model we prove the existence of solutions for the equilibrium problem. Relations with other gradient elastic theories and the possibility of switching from a fourth-order (gradient elastic) problem to a second-order micromorphic model are also discussed with the view of obtaining symmetric force-stress tensors. It is shown that the indeterminate couple-stress model can be written entirely with symmetric force-stress and symmetric couple-stress. The difference of the alternative models rests in specifying traction boundary conditions of either rotational type or strain type. If rotational-type boundary conditions are used in the integration by parts, the classical anti-symmetric nonlocal force-stress tensor formulation is obtained. Otherwise, the difference in both formulations is only a divergence-free second-order stress field such that the field equations are the same, but the traction boundary conditions are different. For these results we employ an integrability condition, connecting the infinitesimal continuum rotation and the infinitesimal continuum strain. Moreover, we provide the orthogonal boundary conditions for both models. [ABSTRACT FROM AUTHOR]