Browsing by Author "Henao, Duvan"

Now showing 1 - 9 of 9
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    A lower bound for the void coalescence load in nonlinearly elastic solids
    (2019) Canulef-Aguilar, Victor; Henao, Duvan
    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 epsilon. 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 epsilon -> 0 limit. In light of the results by Sivalonagathan & Spector and Henao & Serfaty that cavities always prefer to have a circular shape (unless prevented to do so by the constraint of incompressibility), our theorem suggests that the elongation and coalescence of the cavities experimentally and numerically observed for large loads can only take place after all the cavities have attained a volume comparable to the space they have available in the reference configuration. Based on the previous work of Henao & Serfaty, who apply the Ginzburg-Landau theory for superconductivity to the cavitation problem, this paper shows how the study of the interaction of the cavities is connected to the following more basic question: for what cavitation sites a(1), ..., a(n) and areas v(1), ..., v(n) does there exist an incompressible and invertible deformation producing cavities of those areas originating from those points. In order to use the incompressible flow of Dacorogna & Moser to answer that question, it is necessary to study first how do the elliptic regularity estimates for the Neumann problem in domains with circular holes depend on the domain geometry.
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    A NUMERICAL SCHEME AND VALIDATION OF THE ASYMPTOTIC ENERGY RELEASE RATE FORMULA FOR A 2D GEL THIN-FILM DEBONDING PROBLEM
    (2024) Calderer, Maria carme; Henao, Duvan; Sanchez, Manuel a.; Siegel, Ronald a.; Song, Sichen
    This article presents a numerical scheme for the variational model formulated by Calderer et al. [J. Elast., 141 (2020), pp. 51--73] for the debonding of a hydrogel film from a rigid substrate upon exposure to solvent, in the two-dimensional case of a film placed between two parallel walls. It builds upon the scheme introduced by Song et al. [J. Elast., 153 (2023), pp. 651--679] for completely bonded gels, which fails to be robust in the case of gels that are already debonded. The new scheme is used to compute the energy release rate function, based on which predictions are offered for the threshold thickness below which the gel/substrate system is stable against debonding. This study, in turn, makes it possible to validate a theoretical estimate for the energy release rate obtained in the cited works, which is based on a thin-film asymptotic analysis and which, due to its explicit nature, is potentially valuable in medical device development. An existence theorem and rigorous justifications of some approximations made in our numerical scheme are also provided.
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    Experiments, Modelling, and Simulations for a Gel Bonded to a Rigid Substrate
    (2023) Song, Sichen; Siegel, Ronald A.; Sanchez, Manuel A.; Carme Calderer, M.; Henao, Duvan
    In preparation for a more thorough study based on our own experimental work of the debonding of a thin film gel by stress concentration on the interface with a rigid substrate, in this article we revisit, from the viewpoint of the synergy between mathematics, experiments, and finite element simulations, the problem of the swelling of a thin rectangular polyacrylamide gel covalently bonded on the bottom surface to a glass slide. With methods of the calculus of variations and perturbation theory we show that the solution to the corresponding zero-displacement boundary value problem converges, in the thin film limit, to a uniquely defined uniform uniaxial extension on the direction normal to the substrate. Both the experiments and the finite element simulations that we perform confirm that the amount of lateral swelling is very small, with a very good quantitative agreement between the two approaches. The proposed model of minimizing an energy functional comprising both a term for the elastic distortion and the Flory-Huggins expression for the entropy of mixing is thus experimentally and numerically validated, with parameters coming from experimental measurements, including the initial polymer volume fraction of the hydrogel synthesized in the laboratory (which is taken as the reference configuration instead of the dry polymer).
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    Gels: Energetics, Singularities, and Cavitation
    (2024) Calderer, M. Carme; Henao, Duvan; Sanchez, Manuel A.; Siegel, Ronald A.; Song, Sichen
    This article studies equilibrium singular configurations of gels and addresses open questions concerning gel energetics. We model a gel as an incompressible, immiscible and saturated mixture of a solid polymer and a solvent that sustain chemical interactions at the molecular level. We assume that the energy of the gel consists of the elastic energy of its polymer network plus the Flory-Huggins energy of mixing. The latter involves the entropic energies of the individual components plus that of interaction between polymer and solvent, with the temperature dependent Flory parameter, ?, encoding properties of the solvent. In particular, a good solvent promoting the mixing regime, is found below the threshold value ? = 0.5, whereas the phase separating regime develops above that critical value. We show that cavities and singularities develop in the latter regime. We find two main classes of singularities: (i) drying out of the solvent, with water possibly exiting the gel domain through the boundary, leaving behind a core of exposed polymer at the centre of the gel; (ii) cavitation, in response to traction on the boundary or some form of negative pressure, with a cavity that can be either void or flooded by the solvent. The straightforward and unified mathematical approach to treat all such singularities is based on the construction of appropriate test functions, inspired by the particular states of uniform swelling or compression. The last topic of the article addresses a statistical mechanics rooted controversy in the research community, providing an experimental and analytic study in support of the phantom elastic energy versus the affine one.
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    Global invertibility of Sobolev maps
    (2021) Henao, Duvan; Mora-Corral, Carlos; Oliva, Marcos
    We define a class of Sobolev W1-p(Omega, R-n) functions, with p > n -1, such that its trace on de is also Sobolev, and do not present cavitation in the interior or on the boundary. We show that if a function in this class has positive Jacobian and coincides on the boundary with an injective map, then the function is itself injective. We then prove the existence of minimizers within this class for the type of functionals that appear in nonlinear elasticity.
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    Harmonic Dipoles and the Relaxation of the Neo-Hookean Energy in 3D Elasticity
    (2023) Barchiesi, Marco; Henao, Duvan; Mora-Corral, Carlos; Rodiac, Remy
    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-Helein in the context of harmonic maps into the sphere.
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    HOLDER ESTIMATES FOR THE NEUMANN PROBLEM IN A DOMAIN WITH HOLES AND A RELATION FORMULA BETWEEN THE DIRICHLET AND NEUMANN PROBLEMS
    (2020) Canulef-Aguilar, Victor; Henao, Duvan
    In this paper we study the dependence of the Holder estimates on the geometry of a domain with holes for the Neumann problem. For this, we study the Holder 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). A novelty of this work is that we deal directly with the Holder regularity of the single layer potentials of the Dirichlet and Neumann problems for the Poisson equation, something that most of the times seems to be avoided by studying the Newtonian potential.
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    On the lack of compactness in the axisymmetric neo-Hookean model
    (2024) Barchiesi, Marco; Henao, Duvan; Mora-Corral, Carlos; Rodiac, Remy
    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 and 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.
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    SYMMETRY OF UNIAXIAL GLOBAL LANDAU-DE GENNES MINIMIZERS IN THE THEORY OF NEMATIC LIQUID CRYSTALS
    (SIAM PUBLICATIONS, 2012) Henao, Duvan; Majumdar, Apala
    We extend the recent radial symmetry results by Pisante [J. Funct. Anal., 260 (2011), pp. 892-905] and Millot and Pisante [J. Eur. Math. Soc. (JEMS), 12 (2010), pp. 1069-1096] (who show that the equivariant solutions are the only entire solutions of the three-dimensional Ginzburg-Landau equations in superconductivity theory) to the Landau-de Gennes framework in the theory of nematic liquid crystals. In the low temperature limit, we obtain a characterization of global Landau-de Gennes minimizers, in the restricted class of uniaxial tensors, in terms of the well-known radial-hedgehog solution. We use this characterization to prove that global Landau-de Gennes minimizers cannot be purely uniaxial for sufficiently low temperatures.