Piola’s approach to the equilibrium problem for bodies with second gradient energies. Part I: First gradient theory and differential geometry

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

“…Assuming for the Jacobian determinant the condition J = det (F) . 0, the same regularity must be guaranteed for the inverse placement map: hence, the deformation process establishes a diffeomorphism between the reference and the current domain, regarded as differential submanifolds with boundary [12,82]. The objectivity of the energy can be ensured by prescribing a dependence on the right Cauchy-Green tensor C = F T F and on its gradient rC and r (2) C (or on C À1 , rC À1 , and r (2) C À1 ), as illustrated in Fortune and Vallee [83] and Auffray et al [84].…”

“…A involving complementary surface projectors [12] (see Appendix 1), the gradient of the virtual placement map in Lagrangian form can be additively decomposed into a normal and and a tangential contribution, namely…”

“…Considering the Schwarz theorem on the permutation of the mixed derivatives, we can further manipulate the first addend in equation (12) obtaining…”

“…For instance, the Lagrangian inner virtual work in equation ( 2) can now be expressed as follows (see e.g. [11,12])…”

“…In recent studies [11][12][13][14], the equilibrium equations for the second-gradient materials were derived in a fully variational approach. A viable strategy was drawn to transform such equations from the Lagrangian configuration to the Eulerian configuration, based on remarkable transformation formulae for the tangent and normal to the boundary edges and on a novel formulation for the divergence theorem applied to curved surfaces with border, relating material and spatial expressions, which generalizes Piola's bulk transformation.…”