Abstract
Let \begin{document} $\Omega $\end{document} be a smooth bounded axisymmetric set in \begin{document} $\mathbb{R}^3$\end{document} . 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 \begin{document} $W^{1,2}(\Omega ,\mathbb{R}^3)$\end{document} 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 \begin{document} $\Omega $\end{document} 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 \begin{document} $W^{1,2}$\end{document} -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.
Highlights
The most commonly used model to describe the nonlinear behavior of elastic solids undergoing large deformations is that of neo-Hookean materials, whose stored energy is assumed to be of the formE(u) = |Du|2 + H(det Du), (1)Ω where Ω ⊂ R3 is the reference configuration, u : Ω → R3 is the deformation map andH : R → R+ is a convex function satisfying lim H(t) = lim H(s) = +∞. (2) t→+∞ t s→0In spite of this model being broadly used by mathematicians, physicists, materials scientists and engineers alike, and in spite of it being justifiable from statistical mechanics, the existence of stable configurations remains a mathematical challenge
We recall that there are two important steps in the method of calculus of variations in order to minimize a functional I in a space X: i) Compactness of minimizing sequences: letn be a minimizing sequence for I in X, there exists u ∈ X such that un →τ u in X, where →τ refers to the convergence for a suitable topology in X
We look for minimizers of E in a class of deformations which satisfy some invertibility conditions, which are orientation preserving and have prescribed boundary data
Summary
The most commonly used model to describe the nonlinear behavior of elastic solids undergoing large deformations is that of neo-Hookean materials, whose stored energy is assumed to be of the form (see e.g. [32]). The last term penalizes the creation of surface and cavitation They introduced a notion of invertibility called condition (INV) which is stable under weak convergence in W 1,p for p > 2 and which allows them to recover lower semicontinuity of energies of the type Ep and compactness of minimizing sequences in the appropriate space. Minimizing the energy E in a space of deformations which satisfy condition (INV) is a problem with lack of compactness. The introduction of the surface energy E allows them to recover lower semicontinuity and compactness of the problem without using a supplementary condition like (INV) They proved that under certain conditions including a uniform bound on the surface energy E, a sequence of one-to-one almost everywhere deformations converging pointwise is such that the limit is one-to-one almost everywhere (see Theorem 2 in [20]). At last we prove that the minimizers obtained satisfy the inner-variational equations of the neo-Hookean energy E
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