Frame-Invariant Polyconvex Strain-Energy Functions for Some Anisotropic Solids

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At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: An underestimated and still topical contribution of Gabrio Piola (2013, background): “Note that δρ(2) and Λ(X, X̄, ρ) are invariant (see [127]) under any change of observer and as Piola had repeadedly remarked, see e.”
Analytical continuum mechanics à la Hamilton–Piola least action principle for second gradient continua and capillary fluids (2013, unknown): “Therefore (interesting connections can be seen in this context with the considerations developed in Steigmann [151, 150]) we must have Ŵ cap (ρ,∇ρ) = W̌ cap (ρ, β) (25) where we introduced the scalar β := ∇ρ · ∇ρ.”
Anisotropic polyconvex energies on the basis of crystallographic motivated structural tensors (2008, methodology): “, the Ogden-, Mooney-Rivlin- and Neo-Hooke-type free energy functions, which satisfy the polyconvexity conditions; in this context see also Steigmann [43], Hartmann & Neff [15] and Mielke [26].”