In this paper, we represent second-gradient internal work functionals in Lagrangian (referential) and Eulerian (spatial) descriptions, and we deduce the corresponding expressions for the Piola transformations of stress and double-stress tensors and of external forces and double-forces. We also derive, in both the Eulerian and Lagrangian description, the expression of surface and edge contact interactions (which include forces and double-forces) for second-gradient continua in terms of the normal and the curvature of contact boundary surfaces and edge shapes.

Starting from the seminal works of Toupin, Mindlin and Germain, a wide class of generalized elastic models have been proposed via the principle of virtual work, by postulating expressions of the elastic energy enriched by additional kinematic descriptors or by higher gradients of the placement. More recently, such models have been adopted to describe phenomena which are not consistent with the Cauchy-Born continuum, namely the size dependence of apparent elastic moduli observed for micro and nano-objects, wave dispersion, optical modes and band gaps in the dynamics of heterogeneous media. For those structures the mechanical response is affected by surface effects which are predominant with respect to the bulk, and the scale of the external actions interferes with the characteristic size of the heterogeneities. Generalized continua are very often referred to as media with microstructure although a rigorous deduction is lacking between the specific microstructural features and the constitutive equations. While in the forward modelling predictions of the observations are provided, the actual observations at multiple scales can be used inversely to integrate some lack of information about the model. In this review paper, generalized continua are investigated from the standpoint of inverse problems, focusing onto three topics, tightly connected and located at the border between multiscale modelling and the experimental assessment, namely: (i) parameter identification of generalized elastic models, including asymptotic methods and homogenization strategies; (ii) design of non-conventional tests, possibly integrated with full field measurements and advanced modelling; (iii) the synthesis of meta-materials, namely the identification of the microstructures which fit a target behaviour at the macroscale. The scientific literature on generalized elastic media, with the focus on the higher gradient models, is fathomed in search of questions and methods which are typical of inverse problems theory and issues related to parameter estimation, providing hints and perspectives for future research.

After the wide premise of Part I, where the equations for Cauchy’s continuum were retrieved through the energy minimization and some differential geometric perspectives were specified, the present paper as Part II outlines the variational derivation of the equilibrium equations for second gradient materials and their transformation from the Eulerian to the Lagrangian form. Volume, face and edge contributions to the inner virtual work were provided through integration by parts and by repeated applications of the divergence theorem extended to curved surfaces with border. To sustain double forces over the faces and line forces along the edges, the role of the third rank hyperstress tensor was highlighted. Special attention was devoted to the edge work, and to the evaluation of the variables discontinuous across the edge belonging to the contiguous boundary faces. The detailed expression of the contact pressures was provided, including multiple products of normal vector components, their gradient and a combination of them: in particular, the dependence on the local mean curvature was shown. The transport of the governing equations from the Eulerian to the Lagrangian configuration was developed according to two diverse strategies, exploiting novel differential geometric formulae and revealing a coupling of terms transversely to the involved domains.