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-convex state space. We derive the existence of weak-solutions up to the point of a potential collision.
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.
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 analyze a linear 3D/3D fluid-structure interaction problem between a thin layer of a viscous fluid and a thin elastic plate-like structure with the aim of deriving a simplified reduced model. Based on suitable energy dissipation inequalities quantified in terms of two small parameters, thickness of the fluid layer and thickness of the elastic structure, we identify the right relation between the system coefficients and small parameters which eventually provide a reduced model on the vanishing limit. The reduced model is a linear sixth-order thin-film equation describing the out-of-plane dis…
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 …
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…
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 study a planar thin brittle beam subject to elastic deformations and cracks described in terms of a nonlinear Griffith energy functional acting on $SBV$ deformations of the beam. In particular we consider the case in which elastic bulk contributions due to finite bending of the beam are comparable to the surface energy which is necessary to completely break the beam into several large pieces. In the limit of vanishing aspect ratio we rigorously derive an effective Griffith-Euler-Bernoulli functional which acts on piecewise $W^{2,2}$ regular curves representing the midline of the beam. The e…
We consider the axial compression of a thin elastic cylinder placed about a hard cylindrical core. Treating the core as an obstacle, we prove upper and lower bounds on the minimum energy of the cylinder that depend on its relative thickness and the magnitude of axial compression. We focus exclusively on the setting where the radius of the core is greater than or equal to the natural radius of the cylinder. We consider two cases: the "large mandrel" case, where the radius of the core exceeds that of the cylinder, and the "neutral mandrel" case, where the radii of the core and cylinder are the s…
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…
