Count and Symmetry of Global and Local Minimizers of the Cahn-Hilliard Energy Over Cylindrical Domains

AbstractWe address the problem of minimization of the Cahn-Hilliard energy functional under a mass constraint over two and three-dimensional cylindrical domains. Although existence is presented for a general case the focus is mainly on rectangles, parallelepipeds and circular cylinders. According to the symmetry of the domain the exact numbers of global and local minimizers are given as well as their geometric profile and interface locations; all are onedimensional increasing/decreasing and odd functions for domains with lateral symmetry in all axes and also for circular cylinders. The selection of global minimizers by the energy functional is made via the smallest interface area criterion.The approach utilizes Γ−convergence techniques to prove existence of an one-parameter family of local minimizers of the energy functional for any cylindrical domain. The exact numbers of global and local minimizers as well as their geometric profiles are accomplished via suitable applications of the unique continuation principle while exploring the domain geometry in each case and also the preservation of global minimizers through the process of Γ−convergence.

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Torus-like Solutions for the Landau-de Gennes Model. Part I: The Lyuksyutov Regime

Archive for Rational Mechanics and Analysis ◽

10.1007/s00205-020-01582-8 ◽

2020 ◽

Author(s):

Federico Dipasquale ◽

Vincent Millot ◽

Adriano Pisante

Keyword(s):

Boundary Condition ◽

Level Sets ◽

Dirichlet Boundary ◽

Three Dimensional ◽

Energy Functional ◽

Axially Symmetric ◽

Global Minimizers ◽

Norm Constraint ◽

Relevant Range ◽

Nontrivial Topology

Abstract We study global minimizers of a continuum Landau-de Gennes energy functional for nematic liquid crystals, in three-dimensional domains, under a Dirichlet boundary condition. In a relevant range of parameters (which we call the Lyuksyutov regime), the main result establishes the nontrivial topology of the biaxiality sets of minimizers for a large class of boundary conditions including the homeotropic boundary data. To achieve this result, we first study minimizers subject to a physically relevant norm constraint (the Lyuksyutov constraint), and show their regularity up to the boundary. From this regularity, we rigorously derive the norm constraint from the asymptotic Lyuksyutov regime. As a consequence, isotropic melting is avoided by unconstrained minimizers in this regime, which then allows us to analyse their biaxiality sets. In the case of a nematic droplet, this also implies that the radial hedgehog is an unstable equilibrium in the same regime of parameters. Technical results of this paper will be largely employed in Dipasquale et al. (Torus-like solutions for the Landau- de Gennes model. Part II: topology of $$\mathbb {S}^1$$ S 1 -equivariant minimizers. https://arxiv.org/pdf/2008.13676.pdf; Torus-like solutions for the Landau- de Gennes model. Part III: torus solutions vs split solutions (In preparation)), where we prove that biaxiality level sets are generically finite unions of tori for smooth configurations minimizing the energy in restricted classes of axially symmetric maps satisfying a topologically nontrivial boundary condition.

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Rigorous derivation of active plate models for thin sheets of nematic elastomers

Mathematics and Mechanics of Solids ◽

10.1177/1081286517699991 ◽

2017 ◽

Vol 25 (10) ◽

pp. 1804-1830 ◽

Cited By ~ 6

Author(s):

Virginia Agostiniani ◽

Antonio DeSimone

Keyword(s):

Finite Elasticity ◽

Three Dimensional ◽

Energy Functional ◽

Variable Degree ◽

Energy Functionals ◽

Nematic Order ◽

Reduction Techniques ◽

Nematic Elastomers ◽

Plate Models ◽

Γ Convergence

In the context of finite elasticity, we propose plate models describing the spontaneous bending of nematic elastomer thin films due to variations along the thickness of the nematic order parameters. Reduced energy functionals are deduced from a three-dimensional description of the system using rigorous dimension reduction techniques, based on the theory of Γ-convergence. The two-dimensional models are non-linear plate theories, in which deviations from a characteristic target curvature tensor cost elastic energy. Moreover, the stored energy functional cannot be minimised to zero, thus revealing the presence of residual stresses, as observed in numerical simulations. Three nematic textures are considered: splay-bend and twisted orientations of the nematic director, and a uniform director perpendicular to the mid-plane of the film, with variable degree of nematic order along the thickness. These three textures realise three very different structural models: one with only one stable spontaneously bent configuration, a bistable model with two oppositely curved configurations of minimal energy, and a shell with zero stiffness to twisting.

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Local minimizers in absence of ground states for the critical NLS energy on metric graphs

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2020.36 ◽

2020 ◽

pp. 1-29 ◽

Cited By ~ 2

Author(s):

Dario Pierotti ◽

Nicola Soave ◽

Gianmaria Verzini

Keyword(s):

State Energy ◽

Ground States ◽

Energy Functional ◽

Negative Energy ◽

Topological Properties ◽

Local Minimizers ◽

Metric Graphs ◽

Global Minimizers ◽

Underlying Graph ◽

Mass Constraint

We consider the mass-critical non-linear Schrödinger equation on non-compact metric graphs. A quite complete description of the structure of the ground states, which correspond to global minimizers of the energy functional under a mass constraint, is provided by Adami, Serra and Tilli in [R. Adami, E. Serra and P. Tilli. Negative energy ground states for the L2-critical NLSE on metric graphs. Comm. Math. Phys. 352 (2017), 387–406.] , where it is proved that existence and properties of ground states depend in a crucial way on both the value of the mass, and the topological properties of the underlying graph. In this paper we address cases when ground states do not exist and show that, under suitable assumptions, constrained local minimizers of the energy do exist. This result paves the way to the existence of stable solutions in the time-dependent equation in cases where the ground state energy level is not achieved.

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A JUSTIFICATION OF THE REISSNER–MINDLIN PLATE THEORY THROUGH VARIATIONAL CONVERGENCE

Analysis and Applications ◽

10.1142/s0219530507000936 ◽

2007 ◽

Vol 05 (02) ◽

pp. 165-182 ◽

Cited By ~ 27

Author(s):

ROBERTO PARONI ◽

PAOLO PODIO-GUIDUGLI ◽

GIUSEPPE TOMASSETTI

Keyword(s):

Plate Theory ◽

Scaling Law ◽

Three Dimensional ◽

Transversely Isotropic ◽

Energy Functional ◽

Mindlin Plate ◽

Variational Convergence ◽

Mindlin Plate Theory ◽

Γ Convergence ◽

Three Dimensional Elasticity

We provide a justification of the Reissner–Mindlin plate theory, using linear three-dimensional elasticity as framework and Γ-convergence as technical tool. Essential to our developments is the selection of a transversely isotropic material class whose stored energy depends on (first and) second gradients of the displacement field. Our choices of a candidate Γ-limit and a scaling law of the basic energy functional in terms of a thinness parameter are guided by mechanical and formal arguments that our variational convergence theorem is meant to validate mathematically.

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On the Kirchhoff-Love Hypothesis (Revised and Vindicated)

Journal of Elasticity ◽

10.1007/s10659-021-09819-7 ◽

2021 ◽

Author(s):

Olivier Ozenda ◽

Epifanio G. Virga

Keyword(s):

Free Energy ◽

Dimension Reduction ◽

Three Dimensional ◽

Energy Functional ◽

The Other ◽

Variational Model ◽

Two Dimensional ◽

Elastic Plates ◽

Kinematic Constraint ◽

Free Energy Functional

AbstractThe Kirchhoff-Love hypothesis expresses a kinematic constraint that is assumed to be valid for the deformations of a three-dimensional body when one of its dimensions is much smaller than the other two, as is the case for plates. This hypothesis has a long history checkered with the vicissitudes of life: even its paternity has been questioned, and recent rigorous dimension-reduction tools (based on standard $\varGamma $ Γ -convergence) have proven to be incompatible with it. We find that an appropriately revised version of the Kirchhoff-Love hypothesis is a valuable means to derive a two-dimensional variational model for elastic plates from a three-dimensional nonlinear free-energy functional. The bending energies thus obtained for a number of materials also show to contain measures of stretching of the plate’s mid surface (alongside the expected measures of bending). The incompatibility with standard $\varGamma $ Γ -convergence also appears to be removed in the cases where contact with that method and ours can be made.

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Optical properties of three-dimensional woodpile photonic crystals composed of circular cylinders with planar defect structures

Applied Optics ◽

10.1364/ao.50.006657 ◽

2011 ◽

Vol 50 (36) ◽

pp. 6657

Author(s):

Shih-Hsuan Chung ◽

Jaw-Yen Yang

Keyword(s):

Optical Properties ◽

Photonic Crystals ◽

Three Dimensional ◽

Planar Defect ◽

Circular Cylinders ◽

Defect Structures

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Three-Dimensional Vibration Analysis of Twisted Cylinder With Sectorial Cross Section

Journal of Applied Mechanics ◽

10.1115/1.4007475 ◽

2013 ◽

Vol 80 (2) ◽

Cited By ~ 1

Author(s):

D. Zhou ◽

S. H. Lo

Keyword(s):

Cross Section ◽

Chebyshev Polynomial ◽

Numerical Solutions ◽

Twist Angle ◽

Ritz Method ◽

Three Dimensional ◽

Cylindrical Domain ◽

Linear Elasticity Theory ◽

Boundary Functions ◽

Polynomial Series

The three-dimensional (3D) free vibration of twisted cylinders with sectorial cross section or a radial crack through the height of the cylinder is studied by means of the Chebyshev–Ritz method. The analysis is based on the three-dimensional small strain linear elasticity theory. A simple coordinate transformation is applied to map the twisted cylindrical domain into a normal cylindrical domain. The product of a triplicate Chebyshev polynomial series along with properly defined boundary functions is selected as the admissible functions. An eigenvalue matrix equation can be conveniently derived through a minimization process by the Rayleigh–Ritz method. The boundary functions are devised in such a way that the geometric boundary conditions of the cylinder are automatically satisfied. The excellent property of Chebyshev polynomial series ensures robustness and rapid convergence of the numerical computations. The present study provides a full vibration spectrum for thick twisted cylinders with sectorial cross section, which could not be determined by 1D or 2D models. Highly accurate results presented for the first time are systematically produced, which can serve as a benchmark to calibrate other numerical solutions for twisted cylinders with sectorial cross section. The effects of height-to-radius ratio and twist angle on frequency parameters of cylinders with different subtended angles in the sectorial cross section are discussed in detail.

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Numerical exploration on snap buckling of a pre-stressed hemispherical gridshell

Journal of Applied Mechanics ◽

10.1115/1.4052289 ◽

2021 ◽

pp. 1-11

Author(s):

Weicheng Huang ◽

Longhui Qin ◽

Qiang Chen

Keyword(s):

Three Dimensional ◽

Energy Functional ◽

Elastic Rods ◽

Geometrically Nonlinear ◽

Local Response ◽

Micro Electro Mechanical Systems ◽

One Dimensional ◽

Numerical Exploration ◽

Planar Grid ◽

Fundamental Understanding

Abstract Motivated by the observations of snap-through phenomena in pre-stressed strips and curved shells, we numerically investigate the snapping of a pre-buckled hemispherical gridshell under apex load indentation. Our experimentally validated numerical framework on elastic gridshell simulation combines two components: (i) Discrete Elastic Rods method, for the geometrically nonlinear description of one dimensional rods; and (ii) a naive penalty-based energy functional, to perform the non-deviation condition between two rods at joint. An initially planar grid of slender rods can be actuated into a three dimensional hemispherical shape by loading its extremities through a prescribed path, known as buckling induced assembly; next, this pre-buckled structure can suddenly change its bending direction at some threshold points when compressing its apex to the other side. We find that the hemispherical gridshell can undergo snap-through buckling through two different paths based on two different apex loading conditions. The first critical snap-through point slightly increases as the number of rods in gridshell structure becomes denser, which emphasizes the mechanically nonlocal property in hollow grids, in contrast to the local response of continuum shells. The findings may bridge the gap among rods, grids, knits, and shells, for a fundamental understanding of a group of thin elastic structures, and inspire the design of novel micro-electro-mechanical systems and functional metamaterials.

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