Uniqueness of Equilibrium Solutions in Second-Order Gradient Nonlinear Elasticity

Geometrically nonlinear higher-gradient elasticity with energetic boundaries

“…• The papers which are investigating the mathematical structure of "generalized" (with respect to NavierCauchy's) continuum theories: Germain (1972Germain ( , 1973Seppecher (1987Seppecher ( , 1989Seppecher (1995, 1997); Suiker and Chang (2000); Charlotte and Truskinovsky (2002);Polizzotto (2003); Mareno (2004); Kirchner and Steinmann (2005); ; Rambert et al (2007); Kirchner and Steinmann (2007); Van and Papenfuss (2008); Charlotte and Truskinovsky (2008); Agiasofitou and Lazar (2009);Ganghoffer (2010); Yi et al (2010); Ciarletta et al (2012).…”

On uniqueness in the affine boundary value problem of the nonlinear elastic dielectric

“…The complete set of equations governing the deformation of the dielectric may be obtained by considering the stationary points of the electric enthalpy (2-7) with respect to independent variations of , F, and P. (See, for example, [Trimarco 2002;2003;Maugin and Trimarco 1991;2001;Maugin 1993;Pack and Herrmann 1986;Yu 1995] and the important discussion in [Ericksen 2006, to appear].) This yields the Euler-Lagrange equations:…”

“…Lemma 2.1 [Ericksen 2006, to appear;Maugin 1993;Maugin and Epstein 1991;Trimarco 2002;2003]. Let ⊂ IR n have smooth boundary ∂ with unit outward normal N .…”

“…Another application of the basic proof is by Mareno [2004] who investigated uniqueness in the second order theory of nonlinear elasticity. As a consequence of these previous studies, the intrinsic mathematical interest of the present paper is seen as lying not so much in the details required to extend the proof, but rather in the electromagnetic problems under consideration, for which the uniqueness results are new.…”

“…The consequent enlarged set of Euler-Lagrange equations contain the transformed MaxwellLorentz equations, and enable certain properties of the energy-momentum tensor to be derived. These in turn lead to an identity, analogous to the elastostatic conservation law, whose construction, while nontrivial, appears to be more direct than is usual; compare for example, [Ericksen 2006, to appear;Maugin 1993;Maugin and Trimarco 1991;2001, Trimarco 20022003;and Pack and Herrmann 1986]. The Maxwell-Lorentz equations also imply that the electric enthalpy is independent of the polarization, so that the conditions for the electric enthalpy to be quasiconvex and rank-one convex, both essential for the proof, need only be defined in terms of the deformation gradient and the electric field.…”