On the existence of minimizers for the neo-Hookean energy in the axisymmetric setting

Abstract

Let Ω be a smooth bounded axisymmetric set in R 3 . In this paper we investigate the existence of minimizers of the so-called neo-Hookean energy among a class of axisymmetric maps. Due to the appearance of a critical exponent in the energy we must face a problem of lack of compactness. Indeed as shown by an example of Conti-De Lellis in [12, Section 6], a phenomenon of concentration of energy can occur preventing the strong convergence in W 1,2 (Ω, R 3 ) of a minimizing sequence along with the equi-integrability of the cofactors of that sequence. We prove that this phenomenon can only take place on the axis of symmetry of the domain. Thus if we consider domains that do not contain the axis of symmetry then minimizers do exist. We also provide a partial description of the lack of compactness in terms of Cartesian currents. Then we study the case where Ω is not necessarily axisymmetric but the boundary data is affine. In that case if we do not allow cavitation (nor in the interior neither at the boundary) then the affine extension is the unique minimizer, that is, quadratic polyconvex energies are W 1,2 -quasiconvex in our admissible space. At last, in the case of an axisymmetric domain not containing its symmetry axis, we obtain for the first time the existence of weak solutions of the energy-momentum equations for 3D neo-Hookean materials.

On the lack of compactness problem in the axisymmetric neo-Hookean model

“…cof Du in the sense of measures and not necessarily weakly in L 1 (nevertheless, we do have det Du n det Du in L 1 (Ω), as proved in [31]). From the point of view of the condition…”

“…The two previous paragraphs indicate that to construct a sequence showing the lack of compactness in A r s with optimal loss of energy for E we must construct a sequence such that both cof Du n and Du n concentrate, and that inequality (3.3) becomes asymptotically an equality, thus involving conformality. Note that, because of the axisymmetry, the only place where the sequence can concentrate is the symmetry axis, as shown in [31]. To construct our recovery sequence we will use the maps B n in (3.2).…”

“…Use of cylindrical-cylindrical coordinates. We recall from the Appendix in [31] that if u : Ω → R 3 is axisymmetric and is given in cylindrical coordinates by u(r cos θ, r sin θ,…”

Infinite energy cavitating solutions: a variational approach

“…In all of these works, the Dirichlet part of the stored energy function grows like ∇u p with n − 1 < p < n, where u is a deformation and n is the space dimension. The case p = n − 1 for non cavitating deformations and for a three dimensional compressible neo-Hookean material ( [17]), has been studied in [10] for axisymmetric bodies.…”

A Neohookean Model of Plates

“…The term neohookean refers to a stored energy function E which increases to infinity when J h approaches zero. The neohookean materials have gained a lot of interest in mathematical models of nonlinear elasticity [7,8,11,13,16,32,33,35]. In particular, one minimizes functionals which are composed by the sum of the L 2 -norm of the deformation gradient and a nonlinear function of J h ; see [36].…”