arXiv subjects
Publications and source records attributed to Duvan Henao.
We exhibit a Lavrentiev gap phenomenon for the neo-Hookean energy in three-dimensional nonlinear elasticity. More precisely, we construct boundary data for which the infimum of the neo-Hookean energy over deformations satisfying a natural regularity and invertibility condition is strictly larger than the infimum over the weak $H^1$-closure of that class. The mechanism underlying the gap is a deformation with a dipole-type singularity.
Let $(u_j)_j$ be a sequence of maps in $W^{1,2}(\Omega;\mathbb R^3)$, where $\Omega$ is a domain in $\mathbb R^3$. When can we conclude that its weak limit $u$ has non-negative Jacobian a.e.? Hencl and Onninen shows that it is sufficient that each $u_j$ is an orientation-preserving homeomorphism, using an ingenious analysis of a topological invariant called the linking number. Following their approach, we show that if each $u_j$ is a generalised axisymmetric map that has positive Jacobian a.e. and is one-to-one a.e., then $\det Du \ge 0$ a.e. Our proof is based on using the divergence identities to control the sign of the linking numbers of the images of links in $\Omega$ under $u_j$.
Despite its high significance in nonlinear elasticity, the neo-Hookean energy is still not known to admit minimisers in some appropriate admissible class. Using ideas from relaxation theory, we propose a larger minimisation space and a modified functional that coincides with the neo-Hookean energy on the original space. This modified energy is the sum of the neo-Hookean energy and a term penalising the singularities of the inverse deformation. The new functional attains its minimum in the larger space, so the initial question of existence of minimisers of the neo-Hookean energy is thus transformed into a question of regularity of minimisers of this new energy.
We provide a fine description of the weak limit of sequences of regular axisymmetric maps with equibounded neo-Hookean energy, under the assumption that they have finite surface energy. We prove that these weak limits have a dipole structure, showing that the singular map described by Conti-- De Lellis is generic in some sense. On this map we provide the explicit relaxation of the neo-Hookean energy. We also make a link with Cartesian currents showing that the candidate for the relaxation we obtained presents strong similarities with the relaxed energy in the context of \(\mathbb{S}^2\)-valued harmonic maps.
We consider the problem of minimizing the neo-Hookean energy in \(3D\). The difficulty of this problem is that the space of maps without cavitation is not compact, as shown by Conti \& De Lellis with a pathological example involving a dipole. In order to rule out this behaviour we consider the relaxation of the neo-Hookean energy in the space of axisymmetric maps without cavitation. We propose a minimization space and a new explicit energy penalizing the creation of dipoles. This new energy, which is a lower bound of the relaxation of the original energy, bears strong similarities with the relaxed energy of Bethuel-Brezis-Hélein in the context of harmonic maps into the sphere.
The problem of the sudden growth and coalescence of voids in elastic media is considered. The Dirichlet energy is minimized among incompressible and invertible Sobolev deformations of a two-dimensional domain having $n$ microvoids of radius $\varepsilon$. The constraint is added that the cavities should reach at least certain minimum areas $v_{1},...,v_{n}$ after the deformation takes place. They can be thought of as the current areas of the cavities during a quasistatic loading, the variational problem being the way to determine the state to be attained by the elastic body in a subsequent time step. It is proved that if each $v_{i}$ is smaller than the area of a disk having a certain well defined radius, which is comparable to the distance, in the reference configuration, to either the boundary of the domain or the nearest cavity (whichever is closer), then there exists a range of external loads for which the cavities opened in the body are circular in the $\varepsilon \rightarrow 0$ limit.
In this paper we study the dependence of the Hölder estimates on the geometry of a domain with holes for the Neumann problem. For this, we study the Hölder regularity of the solutions to the Dirichlet and Neumann problems in the disk (and in the exterior of the disk), from which we get a relation between harmonic extensions and harmonic functions with prescribed Neumann condition on the boundary of the disk (for both interior and exterior problems).
We extend the existence theorems in [Barchiesi, Henao \& Mora-Corral; ARMA 224], for models of nematic elastomers and magnetoelasticity, to a larger class in the scale of Orlicz spaces. These models consider both an elastic term where a polyconvex energy density is composed with an unknown state variable defined in the deformed configuration, and a functional corresponding to the nematic energy (or the exchange and magnetostatic energies in magnetoelasticity) where the energy density is integrated over the deformed configuration. In order to obtain the desired compactness and lower semicontinuity, we show that the regularity requirement that maps create no new surface can still be imposed when the gradients are in an Orlicz class with an integrability just above the space dimension minus one. We prove that the fine properties of orientation-preserving maps satisfying that regularity requirement (namely, being weakly 1-pseudomonotone, $\mathcal H^1$-continuous, a.e.\ differentiable, and a.e.\ locally invertible) are still valid in the Orlicz-Sobolev setting.
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, a phenomenon of concentration of energy can occur preventing the strong convergence in $W^{1,2}(Ω,\mathbb{R}^3)$ 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 $3$D neo-Hookean materials.
We study global minimizers of the Landau-de Gennes (LdG) energy functional for nematic liquid crystals, on arbitrary three-dimensional simply connected geometries with topologically non-trivial and physically relevant Dirichlet boundary conditions. Our results are specific to the low-temperature limit. We prove (i) that (re-scaled) global LdG minimizers converge uniformly to a (minimizing) limiting harmonic map, away from the singular set of the limiting map; (ii) there exist both a point of maximal biaxiality and a nonempty Lebesgue-null set of uniaxial points near each singular point of the limiting harmonic map (this improves the recent results of \cite{contreraslamy}); (iii) estimates for the size of "strongly biaxial" regions in terms of the reduced temperature $t$. We further show that global LdG minimizers in the restricted class of uniaxial $\Qvec$-tensors cannot be stable critical points of the LdG energy for low temperatures.
This work is devoted so show the appearance of different cracking modes in linearly elastic thin film systems by means of an asymptotic analysis as the thickness tends to zero. By superposing two thin plates, and upon suitable scaling law assumptions on the elasticity and fracture parameters, it is proven that either debonding or transverse cracks can emerge in the limit. A model coupling debonding, transverse cracks and delamination is also discussed.
We study uniaxial energy-minimizers within the Landau-de Gennes theory for nematic liquid crystals on a three-dimensional spherical droplet subject to homeotropic boundary conditions. We work in the low-temperature regime and show that uniaxial energy-minimizers necessarily have the structure of the well-studied radial-hedgehog solution in the low-temperature limit. An immediate consequence of this result is that Landau-de Gennes energy minimizers cannot be purely uniaxial for sufficiently low temperatures.
We consider the minimization of $\int_{Ω_{\ep}} |D\vec u|^p \dd\vec x$ in a perforated domain $Ω_{\ep}:= Ω\setminus \bigcup_{i=1}^M B_{\ep}(\vec a_i)$ of $\R^n$, among maps $\vec u \in W^{1,p}(Ω_{\ep}, \R^n)$ that are incompressible ($\det D\vec u\equiv 1$), invertible, and satisfy a Dirichlet boundary condition $\vec u= \vec g$ on $\partial Ω$. If the volume enclosed by $\vec g (\partial Ω)$ is greater than $|Ω|$, any such deformation $\vec u$ is forced to map the small holes $B_{\ep}(\vec a_i)$ onto macroscopically visible cavities (which do not disappear as $\ep\to 0$). We restrict our attention to the critical exponent $p=n$, where the energy required for cavitation is of the order of $\sum_{i=1}^M v_i |\log \ep|$ and the model is suited, therefore, for an asymptotic analysis ($v_1,..., v_M$ denote the volumes of the cavities). In the spirit of the analysis of vortices in Ginzburg-Landau theory, we obtain estimates for the "renormalized" energy $\frac{1}{n}\int_{Ω_{\ep}} |\frac{D\vec u}{\sqrt{n-1}}|^p \dd\vec x - \sum_i v_i |\log \ep|$, showing its dependence on the size and the shape of the cavities, on the initial distance between the cavitation points $\vec a_1,..., \vec a_M$, and on the distance from these points to the outer boundary $\partial Ω$. Based on those estimates we conclude, for the case of two cavities, that either the cavities prefer to be spherical in shape and well separated, or to be very close to each other and appear as a single equivalent round cavity. This is in agreement with existing numerical simulations, and is reminiscent of the interaction between cavities in the mechanism of ductile fracture by void growth and coalescence.
We consider the problem of uniqueness of positive solutions to boundary value problems containing the equation: -Δ_p u =K(|x|)f(u), p>1. f is positive, is locally Lipschitz and satisfies some superlinear growth condition after u_0, a zero of f before which it is non positive and not identically 0. We show that the Sturmnian theory arguments used by Coffman and Kwong are valid for the equation containing the p-Laplacian operator, even though they were thought to be unextendable beyond the semilinear equation. We obtain a monotone separation result which finally yields the desired uniqueness results.