Euler-Bernoulli Plate Research Articles

Gevrey regularity of the semigroup corresponding to an Euler-Bernoulli plate equation with localized structural damping

Gevrey regularity of the semigroup corresponding to an Euler-Bernoulli plate equation with localized structural damping

Longtime behavior of the weakly coupled Euler-Bernoulli plate system with structural damping

In this paper, we study the longtime behavior of the weakly coupled Euler-Bernoulli plate system with one structural damping. The system consists of two plate equations coupled by zero order terms. First, we show that when the coupling coefficient function is a nonzero constant and the damping coefficient function is a positive constant, the optimal energy decay rate of the system is t−23. Then we consider the case that the coupling and damping coefficient functions have compact supports. By Carleman estimate and the frequency domain method, we show that the energy of the system decays logarithmically when the supports of the coupling and damping coefficient functions satisfy suitable assumptions.

Stability in inverse problem of an elastic plate with a curved middle surface

We consider stability in an inverse problem of determining three spatially varying functions including the source term and the mass density for a curved plate by the Riemannian geometrical approach. The stability is derived by the Carleman estimates and observability inequalities. Two kinds of boundary conditions are considered: one is the hinged boundary conditions and the other is the clamped boundary conditions. In particular, the case of the Euler–Bernoulli plate is included.

The coupling system of Kirchhoff and Euler-Bernoulli plates with logarithmic source terms: Strong damping versus weak damping of variable-exponent type

<abstract><p>In this paper, we study the asymptotic behavior of solutions of the dissipative coupled system where we have interactions between a Kirchhoff plate and a Euler-Bernoulli plate. We investigate the interaction between the internal strong damping acting in the Kirchhoff equation and internal weak damping of variable-exponent type acting in the Euler-Bernoulli equation. By using the potential well, the energy method (multiplier method) combined with the logarithmic Sobolev inequality, we prove the global existence and derive the stability results. We show that the solutions of this system decay to zero sometimes exponentially and other times polynomially. We find explicit decay rates that depend on the weak damping of the variable-exponent type. This outcome extends earlier results in the literature.</p></abstract>

Stabilization for vibrating plate with singular structural damping

We consider the dynamic elasticity equation, modeled by the Euler-Bernoulli plate equation, with a locally distributed singular structural (or viscoelastic ) damping in a boundary domain. Using a frequency domain method combined, based on the Burq's result, combined with an estimate of Carleman type we provide precise decay estimate showing that the energy of the system decays logarithmically as the type goes to the infinity.

Stabilization by memory effects: Kirchhoff plate versus Euler–Bernoulli plate

In this paper we study the asymptotic behavior of solutions of a dissipative coupled system where we have interactions between a Kirchhoff plate and an Euler–Bernoulli plate. The dissipative mechanism is given by memory terms that act either collaboratively (in both equations) or unilaterally (in only one equation). We show that the solutions of this system decay to zero sometimes exponentially and other times polynomially. We found explicit decay rates that depend on the fractional exponents of the memory in each of the following cases: when the memory only acts in the Kirchhoff equation, or only in the Euler–Bernoulli equation, or in both. We also show that all decay rates found are the best. The results obtained are surprising for the following facts: in the collaborative case, the best decay rates of the system are given by the worst decay rates of the uncoupled equations, and in the unilateral case, we conclude that the memory effects in the Euler–Bernoulli equation dissipate the system more slowly than memory effects in the Kirchhoff equation.

Weak solutions for a poro-elastic plate system

ABSTRACT We consider a recent plate model obtained as a scaled limit of the three-dimensional Biot system of poro-elasticity. The result is a ‘2.5’-dimensional linear system that couples traditional Euler–Bernoulli plate dynamics to a pressure equation in three dimensions, where diffusion acts only transversely. We allow the permeability function to be time dependent, making the problem non-autonomous and disqualifying much of the standard abstract theory. Weak solutions are defined in the so-called quasi-static case, and the problem is framed abstractly as an implicit, degenerate evolution problem. Utilizing the theory for weak solutions for implicit evolution equations, we obtain existence of solutions. Uniqueness is obtained under additional hypotheses on the regularity of the permeability function. We address the inertial case in an appendix, by way of semigroup theory. The work here provides a baseline theory of weak solutions for the poro-elastic plate and exposits a variety of interesting related models and associated analytical investigations.

Stabilization of the Weakly Coupled Wave-Plate System with One Internal Damping

We study a stabilization problem of a system coupled by a wave and a Euler–Bernoulli plate equation. Only one of the two equations is directly damped. Under some assumptions on the damping and the coupling terms, we prove that sufficiently smooth solutions of the system decay logarithmically without any geometric conditions on the damping domain. The proofs of these decay results rely on the interpolation inequalities for a coupled elliptic-parabolic system and the estimate of the resolvent operator for that system. The main tools to derive the desired interpolation inequalities are global Carleman estimates.

Indirect stabilization of weakly coupled Kirchhoff plate and wave equations with frictional damping

We investigate the indirect stabilization of a weakly coupled system consisting of a Kirchhoff plate equation involving free boundary conditions and the wave equation with Dirichlet boundary conditions in a bounded domain. The distributed damping is frictional and appears in one of the equations only. First, we consider the case where the damping occurs in the wave equation, and using the frequency domain method combined with an interpolation inequality, we derive a polynomial decay estimate for the associated semigroup. Afterwards, we consider the case where the frictional damping occurs in the Kirchhoff plate equation, and we use the same technique to derive a polynomial decay estimate for the underlying semigroup. That latter polynomial decay estimate is similar to the one obtained earlier by different authors in the case of a coupling between a frictionally damped Euler–Bernoulli plate equation and an undamped wave equation, which is quite surprising since the operator defining the damping in the Kirchhoff plate equation is compact.

Asymptotic Stability of Memory-Type Euler-Bernoulli Plate with Variable Coefficients and Time Delay

This paper considers a viscoelastic Euler-Bernoulli plate system with variable coefficients and time-varying delay. By the multiplier method, Riemannian geometry method, and under suitable assumptions on viscoelastic term and time delay, the authors obtain the general stability of the solution for the system which depends on the behavior of the relaxation function.

General stability of a viscoelastic variable coefficients plate equation with time-varying delay in localized nonlinear internal feedback

This paper is concerned with the decay properties of a memory-type Euler–Bernoulli plate equation with variable coefficients and time-varying delay in localized nonlinear internal feedback. By using piecewise multiplier method and Riemannian geometric analysis, we establish the explicit and general decay rates of system whose nonlinear internal feedbacks act on a suitable subregion.

Numerical Solution to the Dirichlet Control Problem on a Part of the Boundary for the Petrovsky system

Abstract For the Euler-Bernoulli plate equation the Dirichlet control problem is considered. To solve this problem numerically, a combination of classical regularization methods and a scheme of compact embedding of the spaces of optimal regularity into an auxiliary pair of spaces is used. Estimates of the uniform proximity of the exact and approximate operators in the new pair of spaces are obtained.

Existence and energy decay rates of solutions to the variable-coefficient Euler–Bernoulli plate with a delay in localized nonlinear internal feedback

This paper is concerned with the existence and energy decay estimates of solutions to the variable-coefficient Euler–Bernoulli plate with a delay in localized nonlinear internal feedback. The existence of solutions to the nonlinear system is obtained by using Galerkin method combined with some energy estimates. Furthermore, applying Riemannian geometry method, we establish uniform decay rates of the plate equation with variable coefficients and a nonlinear delay term.

Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping

Let a fourth and a second order evolution equations be coupled via the interface by transmission conditions, and suppose that the first one is stabilized by a localized distributed feedback. What will then be the effect of such a partial stabilization on the decay of solutions at infinity? Is the behavior of the first component sufficient to stabilize the second one? The answer given in this paper is that sufficiently smooth solutions decay logarithmically at infinity even the feedback dissipation affects an arbitrarily small open subset of the interior. The method used, in this case, is based on a frequency method, and this by combining a contradiction argument with the Carleman estimates technique to carry out a special analysis for the resolvent.

Qualitative results on the dynamics of a Berger plate with nonlinear boundary damping

The dynamics of a (nonlinear) Berger plate in the absence of rotational inertia are considered with inhomogeneous boundary conditions. In our analysis, we consider boundary damping in two scenarios: (i) free plate boundary conditions, or (ii) hinged-type boundary conditions. In either situation, the nonlinearity gives rise to complicating boundary terms. In the case of free boundary conditions we show that well-posedness of finite-energy solutions can be obtained via highly nonlinear boundary dissipation. Additionally, we show the existence of a compact global attractor for the dynamics in the presence of hinged-type boundary dissipation (assuming a geometric condition on the entire boundary (Lagnese, 1989)). To obtain the existence of the attractor we explicitly construct the absorbing set for the dynamics by employing energy methods that: (i) exploit the structure of the Berger nonlinearity, and (ii) utilize sharp trace results for the Euler–Bernoulli plate in Lasiecka and Triggiani (1993).We provide a parallel commentary (from a mathematical point of view) to the discussion of modeling with Berger versus von Karman nonlinearities: to wit, we describe the derivation of each nonlinear dynamics and a discussion of the validity of the Berger approximation. We believe this discussion to be of broad value across engineering and applied mathematics communities.

Existence of smooth global attractors for nonlinear viscoelastic equations with memory

We consider the memory relaxation of an Euler–Bernoulli plate equation with nonlinear source term and internal frictional damping of arbitrary polynomial growth. The main focus is the existence of a smooth global attractor for the associated dynamical system.

Global existence of a coupled Euler-Bernoulli plate system with variable coefficients

In this paper, we study the initial boundary value problem of a coupled Euler-Bernoulli plate system with spatially varying coefficients of viscosity, damping and source term in a bounded domain. We prove that under some conditions on the initial value, the growth orders of the damping terms and the source terms the solution to the problem exists globally.

Stabilization of the Euler–Bernoulli plate with variable coefficients by nonlinear internal feedback

We consider the energy decay for solutions of the Euler–Bernoulli plate equation with variable coefficients where a nonlinear internal feedback acts in a suitable subregion of the domain. The Riemannian geometric method is used to deal with variable coefficient problems. When the feedback region has a structure similar to that for the wave equation with constant coefficients, we establish the stabilization of the system in the case of fixed boundary conditions. Several energy decay rates are established according to various growth restrictions on the nonlinear feedback near the origin and at infinity. We further show that, unlike for the case of constant coefficients, choices of such feedback regions depend not only on the type of boundary conditions but also on the curvature of a Riemannian metric, based on the coefficients of the system.

Global Existence and Blow-Up for the Euler-Bernoulli Plate Equation with Variable Coefficients

We prove the local existence, blow-up, global existence, and stability of solutions to the initial boundary value problem for Euler-Bernoulli plate equation with variable coefficients.

Solving Inverse Source Problems Using Observability. Applications to the Euler–Bernoulli Plate Equation

The aim of this paper is to provide a general framework for solving a class of inverse source problems by using exact observability of infinite dimensional systems. More precisely, we show that if a system is exactly observable, then a source term in this system can be identified by knowing its intensity and appropriate observations which often correspond to measurements of some boundary traces. This abstract theory is then applied to obtain new identifiability results for a system governed by the Euler-Bernoulli plate equation. Using a different methodology, we show that exact observability can be used to identify both the locations and the intensities of combinations of point sources in the plate equation.