Conservation and Balance Laws in Linear Elasticity of Grade Three

Abstract

In the present paper we investigate conservation and balance laws in the framework of linear elastodynamics considering the strain energy density depending on the gradients of the displacement up to the third order, as originally proposed by Mindlin (Int. J. Solids Struct. 1, 417-438, 1965). The conservation and balance laws that correspond to the symmetries of translation, rotation, scaling and addition of solutions are derived using Noether's theorem. Also, the formulas of the dynamical J , L and M-integrals are presented for the problem under study. Moreover, the balance law of addition of solutions gives rise to explore the dynamical reciprocal theorem as well as the restrictions under which it is valid.

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 the Nonlinear Continuum Theory of Dislocations: A Gauge Field Theoretical Approach

“…where the integrand L(χ (1) , φ (1) ) is a smooth function of χ , φ and the derivatives of χ , φ up to first order, χ (1) , φ (1) , and is an open, connected subset of B R with smooth boundary ∂ and dX = dX 1 dX 2 dX 3 . The Euler-Lagrange equations are…”

“…One can read about the theorem of Noether in the books of Olver [19] and Bluman and Kumei [2]. The reader can also find a more extensive application of Noether's theorem to generalized elasticity as a third order variational problem in Agiasofitou and Lazar [1].…”

“…Since the functional (19) contains derivatives up to the first order, we need the first prolongation of v R , pr (1) …”

“…The well-known infinitesimal variational symmetry condition [19] for a first order variational problem states that the group of transformations G is a variational symmetry of the functional (19) if and only if pr (1) …”