A convergence result is proved for the equilibrium configurations of a three-dimensional thin elastic beam, as the diameter h of the cross-section goes to zero. More precisely, we show that stationary points of the nonlinear elastic functional E^h, whose energies (per unit cross-section) are bounded by Ch^2, converge to stationary points of the Gamma-limit of E^h/h^2. This corresponds to a nonlinear one-dimensional model for inextensible rods, describing bending and torsion effects. The proof is based on the rigidity estimate for low-energy deformations by Friesecke, James, and Müller and on a compensated compactness argument in a singular geometry. In addition, possible concentration effects of the strain are controlled by a careful truncation argument.
We prove that, in the limit of vanishing thickness, equilibrium configurations of inhomogeneous, three-dimensional non-linearly elastic rods converge to equilibrium configurations of the variational limit theory. More precisely, we show that, as $h \to 0$, stationary points of the energy $E^h$, for a rod $Ω_h \subset \mathbb R^3$ with cross-sectional diameter $h$, subconverge to stationary points of the $Γ$-limit of $E^h$, provided that the bending energy of the sequence scales appropriately. This generalizes earlier results for homogeneous materials to the case of materials with (not necessar…
We consider the problem of a one dimensional elastic filament immersed in a two dimensional steady Stokes fluid. Immersed boundary problems in which a thin elastic structure interacts with a surrounding fluid are prevalent in science and engineering, a class of problems for which Peskin has made pioneering contributions. Using boundary integrals, we first reduce the fluid equations to an evolution equation solely for the immersed filament configuration. We then establish local well-posedness for this equation with initial data in low-regularity Hölder spaces. This is accomplished by first extr…
We consider a thin elastic sheet in the shape of a disk that is clamped at its boundary such that the displacement and the deformation gradient coincide with a conical deformation with no stretching there. We define the free elastic energy as a variation of the von Kármán energy, that penalizes bending energy in $L^p$ with $p\in (2,\frac83)$ (instead of, as usual, $p=2$). We prove ansatz free upper and lower bounds for the elastic energy that scale like $h^{p/(p-1)}$, where $h$ is the thickness of the sheet.
We show that the linear brittle Griffith energy on a thin rectangle $Γ$-converges after rescaling to the linear one-dimensional brittle Euler-Bernoulli beam energy. In contrast to the existing literature, we prove a corresponding sharp compactness result, namely a suitable weak convergence after subtraction of piecewise rigid motions with the number of jumps bounded by the energy
Modelling of thin isotropic elastic plates with small piezoelectric inclusions and distributed electric circuits. Models for inclusions larger or comparable to the thickness of the plate
We consider a geometrically fully nonlinear variational model for thin elastic sheets that contain a single disclination. The free elastic energy contains the thickness $h$ as a small parameter. We give an improvement of a recently proved energy scaling law, removing the next-to leading order terms in the lower bound. Then we prove the convergence of (almost-)minimizers of the free elastic energy towards the shape of a radially symmetric cone, up to Euclidean motions, weakly in the spaces $W^{2,2}(B_1\setminus B_ρ;\mathbb{R}^3)$ for every $0<ρ<1$, as the thickness $h$ is sent to 0.
Here, we address a dimension-reduction problem in the context of nonlinear elasticity where the applied external surface forces induce bending-torsion moments. The underlying body is a multi-structure in $\mathbb{R}^3$ consisting of a thin tube-shaped domain placed upon a thin plate-shaped domain. The problem involves two small parameters, the radius of the cross-section of the tube-shaped domain and the thickness of the plate-shaped domain. We characterize the different limit models, including the limit junction condition, in the membrane-string regime according to the ratio between these two…
We discuss a 1D variational problem modeling an elastic sheet on water, lifted at one end. Its terms include the membrane and bending energy of the sheet as well as terms due to gravity and surface tension. By studying a suitable Gamma-limit, we identify a parameter regime in which the sheet is inextensible, and the bending energy and weight of the sheet are negligible. In this regime, the problem simplifies to one with a simple and explicit solution.
In this paper we investigate the interaction of fluid flow with a thin porous elastic layer. We consider two fluid-filled bulk domains which are separated by a thin periodically perforated layer consisting of a fluid and an elastic solid part. Thickness and periodicity of the layer are of order $ε$, where $ε$ is small compared to the size of the bulk domains. The fluid flow is described by an instationary Stokes equation and the solid via linear elasticity. The main contribution of this paper is the rigorous homogenization of the porous structure in the layer and the reduction of the layer to …
A developable cone ("d-cone") is the shape made by an elastic sheet when it is pressed at its center into a hollow cylinder by a distance $ε$. Starting from a nonlinear model depending on the thickness $h > 0$ of the sheet, we prove a $Γ$-convergence result as $h \rightarrow 0$ to a fourth-order obstacle problem for curves in $\mathbb{S}^2$. We then describe the exact shape of minimizers of the limit problem when $ε$ is small. In particular, we rigorously justify previous results in the physics literature.
Nematic equilibria on a two-dimensional annulus: defects and energies
我们研究了强和弱切线锚定的二维环上的平面双折射平衡，在Oseen-Frank和Landau-de Gennes双折射液晶理论。我们分析了Oseen-Frank框架中的无缺陷状态，并得到了弹性各向异性、环长宽比和锚固强度的解析稳定性准则。我们分别考虑无缺陷态的径向和方位扰动，得到了无缺陷态的完整稳定性图。我们构建了一个任意数量的缺陷上的二维环与强切线锚定和计算它们的能量，这些平衡是广义的对角和旋转状态观察到的一个正方形。这提供了新的见解之间的相关性的首选数量的缺陷，它们的位置和几何形状。在Landau-de Gennes框架中，我们将Mironescu在Ginzburg-Landau框架中的强大稳定性结果（P. Mironescu，On the stability of radial solutions of the Ginzburg-Landau equation，1995）用于计算无缺陷状态在温度和几何形状方面的局部稳定性的定量标准。
In this paper we discuss the motion of a beam in interaction with fluids. We allow the beam to move freely in all coordinate directions. We consider the case of a beam situated in between two different fluids as well as the case where the beam is attached only to one fluid. In both cases the fluid-domain is time changing. The fluid is governed by the incompressible Navier-Stokes equations. The beam is elastic and governed by a hyperbolic partial differential equation. In order to allow for large deformations the elastic potential of the beam is non-quadratic and naturally possesses a non-conve…
In this paper we extend the asymptotic analysis in [LLOO], performed on a structure consisting of two linearly elastic bodies connected by a thin soft nonlinear Kelvin-Voigt viscoelastic adhesive layer, to the case in which the total mass of the layer remains strictly positive as its thickness tends to zero. We obtain convergence results by means of a nonlinear version of Trotter's theory of approximation of semigroups acting on variable Hilbert spaces. Differently from the limit models derived in [LLOO], in the present analysis the dynamic effects on the surface to which the layer shrinks do …
We study the stable configurations of a thin three-dimensional weakly prestrained rod subject to a terminal load as the thickness of the section vanishes. By $Γ$-convergence we derive a one-dimensional limit theory and show that isolated local minimizers of the limit model can be approached by local minimizers of the three-dimensional model. In the case of isotropic materials and for two-layers prestrained three-dimensional models the limit energy further simplifies to that of a Kirchhoff rod-model of an intrinsically curved beam. In this case we study the limit theory and investigate global a…
An asymptotic analysis is performed for thin anisotropic elastic plate clamped along its lateral side and also supported at a small area $θ_{h}$ of one base with diameter of the same order as the plate thickness $h\ll1.$ A three-dimensional boundary layer in the vicinity of the support $θ_{h}$ is involved into the asymptotic form which is justified by means of the previously derived weighted inequality of Korn's type provides an error estimate with the bound $ ch^{1/2} \left | \ln h\right| .$ Ignoring this boundary layer effect reduces the precision order down to $\left| \ln h\right| ^{-1/2}.$…
We study thin films with residual strain by analyzing the $Γ-$limit of non-Euclidean elastic energy functionals as the material's thickness tends to $0.$ We begin by extending prior results \cite{bhattacharya2016plates} \cite{agostiniani2018heterogeneous} \cite{lewicka2018dimension} \cite{schmidt2007plate}, to a wider class of films, whose prestrain depends on both the midplate and the transversal variables. The ansatz for our $Γ-$convergence result uses a specific type of wrinkling, which is built on exotic solutions to the Monge-Ampere equation, constructed via convex integration \cite{lewic…
We rigorously derive a Kirchhoff plate theory, via $Γ$-convergence, from a three-di\-men\-sio\-nal model that describes the finite elasticity of an elastically heterogeneous, thin sheet. The heterogeneity in the elastic properties of the material results in a spontaneous strain that depends on both the thickness and the plane variables $x'$. At the same time, the spontaneous strain is $h$-close to the identity, where $h$ is the small parameter quantifying the thickness. The 2D Kirchhoff limiting model is constrained to the set of isometric immersions of the mid-plane of the plate into $\mathbb…
For two different scenarios regarding thin elastic structures, described by 2d-Föppl-von Kármán plate models, we obtain energy scaling laws. Firstly, assuming the reference geometry being that of a singular excess-cone, we obtain fairly optimal upper- and lower energy bounds and we highlight how those bounds scale wrt. the thickness-parameter $h.$ Secondly, we consider the half sphere, while being indented by a thin object at the top and perpendicular to its surface. In this situation we provide an energy-scaling law, for radial symmetric admissible maps, this time, depending on $h$ and the in…
We prove a relation between the scaling $h^β$ of the elastic energies of shrinking non-Euclidean bodies $S_h$ of thickness $h\to 0$, and the curvature along their mid-surface $S$. This extends and generalizes similar results for plates [BLS16, LRR] to any dimension and co-dimension. In particular, it proves that the natural scaling for non-Euclidean rods with smooth metric is $h^4$, as claimed in [AAE+12] using a formal asymptotic expansion. The proof involves calculating the $Γ$-limit for the elastic energies of small balls $B_h(p)$, scaled by $h^4$, and showing that the limit infimum energy …
How much energy does it take to stamp a thin elastic shell flat? Motivated by recent experiments on the wrinkling patterns of floating shells, we develop a rigorous method via $Γ$-convergence for answering this question to leading order in the shell's thickness and other small parameters. The observed patterns involve "ordered" regions of well-defined wrinkles alongside "disordered" regions whose local features are less robust; as little to no tension is applied, the preference for order is not a priori clear. Rescaling by the energy of a typical pattern, we derive a limiting variational probl…
We analyse a system of partial differential equations describing the behaviour of an elastic plate with periodic moduli in the two planar directions, in the asymptotic regime when the period and the plate thickness are of the same order of smallness. Assuming that the displacement gradients of the points of the plate are small enough for the equations of linearised elasticity to be a suitable approximation of the material response, such as the case in e.g. acoustic wave propagation, we derive a class of "hybrid", homogenisation dimension-reduction, norm-resolvent estimates for the plate, under…
We present equivalent conditions and asymptotic models for the diffraction problem of elastic and acoustic waves in a solid medium surrounded by a thin layer of fluid medium. Due to the thinness of the layer with respect to the wavelength, this problem is well suited for the notion of equivalent conditions and the effect of the fluid medium on the solid is as a first approximation local. We derive and validate equivalent conditions up to the fourth order for the elastic displacement. These conditions approximate the acoustic waves which propagate in the fluid region. This approach leads to sol…
