Sombuddha Bhattacharyya, Tuhin Ghosh
We consider the two-dimensional Kirchhoff-Love plate equation in the context of elasticity modeling the stresses and deformations in thin plates subjected to forces and moments. We establish global recovery of the material parameters like bending stiffness, Poisson coefficient, Lamé parameters from the associated boundary Cauchy data of the equation.
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We study an initial-boundary-value problem for a quasilinear thermoelastic plate of Kirchhoff \& Love-type with parabolic heat conduction due to Fourier, mechanically simply supported and held at the reference temperature on the boundary. For this problem, we show the short-time existence and uniqueness of classical solutions under appropriate regularity and compatibility assumptions on the data. Further, we use barrier techniques to prove the global existence and exponential stability of solutions under a smallness condition on the initial data. It is the first result of this kind established…
In this article we derive a doubling inequality at the boundary for solutions to the Kirchhoff-Love isotropic plate's equation satisfying supported boundary conditions. To this end, we combine the use of a suitable conformal mapping which flattens the boundary and a reflection argument which guarantees the needed regularity of the extended solution. We finally apply inequalities of Carleman type in order to derive the result. The latter implies Strong Unique Continuation Property at the boundary (SUCPB).
In this paper we consider the inverse problem of determining a rigid inclusion inside a thin plate by applying a couple field at the boundary and by measuring the induced transversal displacement and its normal derivative at the boundary of the plate. The plate is made by non-homogeneous, linearly elastic and isotropic material. Under suitable a priori regularity assumptions on the boundary of the inclusion, we prove a constructive stability estimate of log type. Key mathematical tool is a recently proved optimal three spheres inequality at the boundary for solutions to the Kirchhoff-Love plat…
The main result of this paper is a doubling inequality at the boundary for solutions to the Kirchhoff-Love isotropic plate's equation satisfying homogeneous Dirichlet conditions. This result, like the three sphere inequality with optimal exponent at the boundary proved in Alessandrini, Rosset, Vessella, Arch. Ration. Mech. Anal. (2019), implies the Strong Unique Continuation Property at the Boundary (SUCPB). Our approach is based on a suitable Carleman estimate, and involves an ad hoc reflection of the solution. We also give a simple application of our main result, by weakening the standard hy…
We prove a three sphere inequality with optimal exponent at the boundary for solutions to the Kirchhoff-Love plate's equation satisfying homogeneous Dirichlet conditions. This result implies the Strong Unique Continuation Property at the Boundary (SUCPB). Our approach is based on the method of Carleman estimates, and involves the construction of an ad hoc conformal mapping preserving the structure of the operator and the employment of a suitable reflection of the solution with respect to the flatten boundary which ensures the needed regularity of the extended solution. To the authors' knowledg…
We consider an initial-boundary-value problem for a thermoelastic Kirchhoff & Love plate, thermally insulated and simply supported on the boundary, incorporating rotational inertia and a quasilinear hypoelastic response, while the heat effects are modeled using the hyperbolic Maxwell-Cattaneo-Vernotte law giving rise to a 'second sound' effect. We study the local well-posedness of the resulting quasilinear mixed-order hyperbolic system in a suitable solution class of smooth functions mapping into Sobolev $H^{k}$-spaces. Exploiting the sole source of energy dissipation entering the system throu…
We study the ground states of the following generalization of the Kirchhoff-Love functional, $$J_σ(u)=\int_Ω\dfrac{(Δu)^2}{2} - (1-σ)\int_Ωdet(\nabla^2u)-\int_ΩF(x,u),$$ where $Ω$ is a bounded convex domain in $\mathbb{R}^2$ with $C^{1,1}$ boundary and the nonlinearities involved are of sublinear type or superlinear with power growth. These critical points correspond to least-energy weak solutions to a fourth-order semilinear boundary value problem with Steklov boundary conditions depending on $σ$. Positivity of ground states is proved with different techniques according to the range of the pa…
We extend to the anisotropic setting the existence of solutions for the Kirchhoff-Plateau problem and its dimensional reduction.
The interplay between multiscale homogenization and dimension reduction for nonlinear elastic thin plates is analyzed in the case in which the scaling of the energy corresponds to Kirchhoff's nonlinear bending theory for plates. Different limit models are deduced depending on the relative ratio between the thickness parameter $h$ and the two homogenization scales $\ep$ and $\ep^2$.
A Gevrey class semigroup, exponential decay and Lack of analyticity for a system formed by a Kirchhoff-Love plate equation and the equation of a membrane-like electric network with indirect fractional damping
本文研究 Kirchhoff-Love 板与膜状电网耦合系统的半群性质，证明其在 $θ\in[0,1[$ 非解析、$θ=1$ 解析，指数衰减，且 $θ\in]0,1[$ 时为 Gevrey 类 $s>1/θ$。
In this paper, we derive a linearized Kirchhoff model from three dimensional nonlinear elastic energy of plates with incompatible prestrain as its thickness $h$ tends to zero and its elastic energy scales like $h^β$ with $2<β<4.$ The incompatible prestrain is given as a Riemannian metric $G(x')$ in the three dimensional thin plate which only depends on mid-plate of the thin plates. The problem is studied rigorously by using a variational approach and establishing the $Γ-$ limit of the non-Euclidean version of the nonlinear elasticity functional when the gauss curvature of the mid-plate $(Ω, g=…
The inf-sup condition, also called the Ladyzhenskaya--Babu\v ska--Brezzi (LBB) condition, ensures the existence, uniqueness and well-posedness of a saddle point problem, relative to a partial differential equation. Discretization by the finite element method gives the discrete problem which must satisfy the discrete inf-sup condition. But, depending on the choice of finite elements, the discrete condition may fail. This paper attempts to explain why it fails from an engineer's perspective, and reviews current methods to work around this failure. The last part recalls the mathematical bases.
In this paper, we consider the Kirchhoff plate equation with delay terms on the boundary control are added (see system \eqref{p5-2.1} below). we give some instability examples of system \eqref{p5-2.1} for some choices of delays. Finally, we prove its well-posedness, strong stability without any geometric condition and exponential stability under a multiplier geometric control condition.
Effects of surface tension and elasticity on critical points of the Kirchhoff-Plateau problem
本文提出修改后的Kirchhoff-Plateau问题，通过添加能量项惩罚横截面形变，推导了平面版本的Euler-Lagrange方程，并证明了在物理参数范围内存在唯一临界点，最后分析了表面张力对平衡时横截面形状的影响。
The mechanical model of a thin plate with boundary control and observation is presented as a port-Hamiltonian system (pHs), both in vectorial and tensorial forms: the Kirchhoff-Love model of a plate is described by using a Stokes-Dirac structure and this represents a novelty with respect to the existing literature. This formulation is carried out both in vectorial and tensorial forms. Thanks to tensorial calculus, this model is found to mimic the interconnection structure of its one-dimensional counterpart, i.e. the Euler-Bernoulli beam. The Partitioned Finite Element Method (PFEM) is then ext…
In this paper, we consider a weakly coupled system consisting of a viscoelastic Kirchhoff plate equation involving free boundary conditions and the viscoelastic wave equation with Dirichlet boundary conditions in a bounded domain. By assuming a more general type of relaxation functions, we establish explicit and general decay rate results, using the multiplier method and some properties of the convex functions.
Derivation of Kirchhoff-type plate theories for elastic materials with voids
本文从含Willmore曲率惩罚的三维模型严格推导出含孔隙薄板的Blake-Zisserman-Kirchhoff理论，推广了Santili和Schmidt的结果，处理一般孔隙几何并构造所有容许极限构型的恢复序列。
In this paper we investigate the limit behavior of the solution to quasi-static Biot's equations in thin poroelastic plates as the thickness tends to zero. We choose Terzaghi's time corresponding to the plate thickness and obtain the strong convergence of the three-dimensional solid displacement, fluid pressure and total poroelastic stress to the solution of the new class of plate equations. In the new equations the in-plane stretching is described by the 2D Navier's linear elasticity equations, with elastic moduli depending on Gassmann's and Biot's coefficients. The bending equation is couple…
A coupled rate-dependent/rate-independent system for adhesive contact in Kirchhoff-Love plates
针对黏弹性动力学板中的耦合速率相关/速率无关黏附接触模型，提出半稳定能量解概念，证明其黏性极限收敛性，通过降维分析揭示两种阻尼情形下面内外耦合特性及极限演化规律。
We consider a two phase elastic thin film with soft inclusions subject to bending dominated deformations. The soft (void) phase may comprise asymptotically small droplets within the elastic matrix. We perform a dimension reduction analysis and obtain a novel `Blake-Zisserman-Kirchhoff' model that combines Kirchhoff's classical plate theory with Blake-Zisserman type surface energy contributions at cracks, folds and the boundary of voids.
Inverse problems for nonlinear Kirchhoff plate equations with multiple unknown parameters
本文研究非线性基尔霍夫板方程的逆边值问题，建立正问题整体适定性，针对被动、主动测量情形分别实现初始数据、初始数据与系数的恢复，提出Runge逼近等方法，可推广至多类板模型。
A fully averaged poroelastic Kirchhoff plate interacting with an incompressible, viscous fluid: analysis and numerical simulation
本文研究了一种新的完全平均的孔隙弹性Kirchhoff板模型，该模型与由时间依赖Stokes方程描述的不可压缩粘性流体流动耦合。完全平均的公式相较于经典Biot孔隙弹性板模型具有优势：弹性动力学和压力方程都位于一个协维数为一的界面上，所得到的数值方案更易于实现且计算更高效，流体-结构耦合更自然。我们分析了一个线性耦合的流体-结构相互作用问题，强制法向速度连续性，Beavers-Joseph-Saffman滑动条件下的切向速度，以及流体和孔隙弹性结构之间的力平衡。我们利用能量方法建立了弱解的存在性，然后利用相关空间算子的扇形性和由此产生的Cauchy问题的最大L^p正则性，证明了正则化问题的全局时间唯一强解的存在性。对于指数衰减的数据，我们证明了解的指数衰减。最后，我们开发了一种有限元方法用于耦合系统的数值逼近，并证明它在薄结构极限下能很好地逼近完整的Biot-Stokes系统。该模型的主要优势在于其显著的简单实现，因为孔隙弹性板方程构成一个表面模型，边界一个体积流体域。
Upper-semicontinuity of uniform attractors for the non-autonomous viscoelastic Kirchhoff plate equation with memory
本文研究非自治粘弹性Kirchhoff板方程的记忆效应下均匀吸引子的上半连续性，通过能量估计和反证法证明其存在性及连续性。
Regularity of Euler-Bernoulli and Kirchhoff-Love Thermoelastic Plates with Fractional Coupling
本文研究具有分数阶耦合的 Euler-Bernoulli 与 Kirchhoff-Love 热弹性板抽象系统，直接确定了 Gevrey 尖锐类并证明了相应半群的解析性。
For the Generalized Plane Stress (GPS) problem in linear elasticity, we obtain an optimal stability estimate of logarithmic type for the inverse problem of determining smooth cavities inside a thin isotropic cylinder {}from a single boundary measurement of traction and displacement. The result is obtained by reformulating the GPS problem as a Kirchhoff-Love plate-like problem in terms of the Airy's function, and by using the strong unique continuation at the boundary for a Kirchhoff-Love plate operator under homogeneous Dirichlet conditions, which has been recently obtained in \cite{l:arv}.
Soap films: from the Plateau problem to deformable boundaries
该研究综述了经典Plateau问题，梳理了Kirchhoff-Plateau问题的最新研究进展，并提出了该领域未来可能的研究方向。
The Kirchhoff-Plateau problem concerns the equilibrium shapes of a system in which a flexible filament in the form of a closed loop is spanned by a liquid film, with the filament being modeled as a Kirchhoff rod and the action of the spanning surface being solely due to surface tension. We establish the existence of an equilibrium shape that minimizes the total energy of the system under the physical constraint of non-interpenetration of matter, but allowing for points on the surface of the bounding loop to come into contact. In our treatment, the bounding loop retains a finite cross-sectional…
We consider a conservative system consisting of an elastic plate interacting with a gas filling a semi-infinite tube. The plate is placed on the bottom of the tube. The dynamics of the gas velocity potential is governed by the linear wave equation. The plate displacement satisfies the linear Kirchhoff equation. We show that this system possesses an infinite number of periodic solutions with the frequencies tending to infinity. This means that the well-known property of decaying of local wave energy in tube domains does not hold for the system considered.
Nonlinear Kirchhoff--Love shell models derived from the Ciarlet--Geymonat energy: modelling and existence of minimizers
基于Ciarlet-Geymonat能量的三维模型，推导出非线性壳模型，分析其几何和弹性参数对壳体行为的影响，并证明能量泛函的 coercivity 和 lower semicontinuity 性质。
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…
We present and summarize the results of recent studies on non-Euclidean plates with imposed constant negative Gaussian curvature in both the Föppl - von Kármán and Kirchhoff approximations. Motivated by experimental results we focus on annuli with a periodic profile. We show that in the Föppl - von Kármán approximation there are only two types of global minimizers -- flat and saddle shaped deformations with localized regions of stretching near the boundary of the annulus. We also show that there exists local minimizers with $n$-waves that have regions of stretching near their lines of inflecti…
Stable determination of the Winkler subgrade coefficient in a nanoplate
本文研究了边界固定的弹性基础纳米板中Winkler系数的反问题，基于简化应变梯度弹性理论，通过单次内部挠度测量证明了基床系数的全局Hölder稳定性估计。
Derivation of a Biot-Plate-System for a thin poroelastic layer
针对薄孔隙弹性层的不可压缩流体流动问题，采用多尺度技术同步进行均匀化与降维，严格推导出由广义Darcy定律和含Darcy压力的Kirchhoff-Love型板方程耦合的Biot板宏观模型。
Stability for an interface transmission problem of wave-plate equations with dynamical boundary controls
该研究针对带动态边界控制的二维波-Kirchhoff板耦合传输系统，首次开展稳定性分析，证明了无几何条件下的强稳定性，指出系统不具备指数稳定性，并建立了光滑初值下1/t型的多项式能量衰减估计。
Infinitely many solutions to Kirchhoff-Boussinesq-type equations on Riemannian manifolds
该研究针对闭黎曼流形上的Kirchhoff-Boussinesq型方程，在对称性假设下证明了临界非线性项的多等变解与次临界非线性项的基态解存在性，还得到高阶Sobolev空间的Gagliardo-Nirenberg插值不等式与等价范数。
Well-posedness for the biharmonic scattering problem for a penetrable obstacle
本文研究了可渗透障碍物在弹性板中的双调和波散射问题，通过算子分解方法分析了适定性及互易关系，并通过数值示例验证了理论结果。
We study a nonlinear fluid-structure interaction problem in which the fluid is described by the three-dimensional incompressible Navier-Stokes equations, and the elastic structure is modeled by the nonlinear plate equation which includes a generalization of Kirchhoff, von Kármán and Berger plate models. The fluid and the structure are fully coupled via kinematic and dynamic boundary conditions. The existence of a weak solution is obtained by designing a hybrid approximation scheme that successfully deals with the nonlinearities of the system. We combine time-discretization and operator splitti…
The present work intends to complement the study of the regularity of the solutions of the thermoelastic plate with rotacional forces. The rotational forces involve the spectral fractional Laplacian, with power parameter $τ\in [0,1]$ ( $γ(-Δ)^τu_{tt}$). Previous research regarding regularity showed that, as for the analyticity of the semigroup $S(t)=e^{\mathbb{B}t}$ for the Euler-Bernoulli Plate($τ=0$) model, the first result was established by Liu and Renardy, \cite{LiuR95} in the case of hinged and clamped boundary conditions, for the case $τ=1$ (Plate Kirchoff-Love) Lasiecka and Triggiani s…
Symmetry and uniqueness for a hinged plate problem in a ball
针对源于 Kirchhoff-Love 薄板模型的二维球内半线性四阶边值问题，本文结合 ODE 技术与 Talenti 重排论证，证明径向单调性、全参数范围唯一性及基态对称性。
Steady periodic hydroelastic waves in polar regions
本文构造了极地地区有限深度水域中具有涡量的二维稳态周期水弹性波，利用特殊Cosserat超弹性壳体理论建模漂浮弹性板，并给出了局部分岔的充要条件。
Homogenization of elasto-plastic plate equations with vanishing hardening
本文研究薄异质弹性塑性板在线性化弹性塑性框架下的渐进行为，聚焦于板厚远快于材料周期微结构特征尺度消失的 regime。通过进化 Γ-收敛的维度约简方法，推导出具有等向和运动硬化特性的异质板模型，并通过双尺度均质化得到有效弹性完美塑性板模型，刻画了不同相界面的耗散势。
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 …
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