Mindlin Timoshenko Research Articles

Nonlinear damping effects for the 2D Mindlin–Timoshenko system

AbstractIn this article, we investigate the asymptotic behavior of the Mindlin–Timoshenko system under the influence of nonlinear dissipation affecting the rotation angle equations. Initially, we provide a concise review of the system’s solution existence. Subsequently, we demonstrate that the energy associated with the solution of the Mindlin–Timoshenko setup follows a dissipation. Furthermore, under the condition of equal wave speeds, we establish a comprehensive decay theorem for the energy, offering explicit insights into its general behavior.

Polynomial stabilization for thermoelastic Reissner–Mindlin–Timoshenko plates with structural damping

In this paper, we are interested in studying the well-posedness, optimal polynomial stability, and the lack of exponential stability for a class of thermoelastic system of Reissner–Mindlin–Timoshenko plates with structural damping, that is, with the dissipation of Kelvin–Voigt type on the equations for the rotation angles. We also consider the thermal effect with thermal variables described by Fourier’s law of heat conduction.

Dynamics of nonlinear Reissner–Mindlin–Timoshenko plate systems

AbstractThe problem of Reissner–Mindlin–Timoshenko plate systems with nonlinear damping terms is considered. The main result is the existence of global attractors. By showing the system is gradient and asymptotic smoothness via a stabilizability inequality, we establish the existence of global attractors with finite fractal dimension. The continuity of global attractors regarding the parameter in a residual dense set is also proved. The above results allow the damping terms with polynomial growth.

Controllability of a thermoelastic system

In this paper we present a null controllability result for a thermoelastic Rayleigh system. Instead of working directly with the control system, we obtain the controlled system as the modulus of elasticity in shear tends to infinity in the corresponding thermoelastic Mindlin–Timoshenko system. Our results follow the seminal book of Lagnese and Lions (Rech. Math. Appl. 6(1988)) where the controllability of a Kirkhhoff model is proposed as the limit of a controlled Mindlin–Timoshenko one. We use estimates for some eigenvalues of the beam model that were obtained in (SIAM J. Control Optim. 47 (2008) 1909–1938) and the recent paper of Komornik and Tenenbaum (Evolution Equations and Control Theory 4(3) (2015) 297–314) where explicit estimates for systems with real and complex eigenvalues are proposed.

Decay of solution for a 2D irrotational Mindlin–Timoshenko plate system

Decay of solution for a 2D irrotational Mindlin–Timoshenko plate system

Recovering density for the Mindlin–Timoshenko system by means of a single boundary measurement

In this paper, we consider an inverse problem for the Mindlin–Timoshenko plate system, which is a strongly coupled two-dimensional system consisting of a wave equation and a system of isotropic elasticity, that arises in modeling plate vibrations especially at high frequencies and thicker plates. More precisely, we prove the global uniqueness of recovering the plate density from a single boundary measurement under appropriate geometrical assumptions. Our approach relies on diagonalizing the principal part of the system and making it a system of wave-like equations.

Lifting plate incompressibility condition for stability of a magnetoelastic Mindlin–Timoshenko model

In this article we are concerned with the stabilization of a magnetoelastic Mindlin–Timoshenko plate model. We consider the model within the framework of semigroup theory and employ a recently developed resolvent criterion to accomplish polynomial stability. An ingenious choice of multiplier enables us to lift the condition of incompressibility of the plate material that was imperative in our earlier work.

Exact controllability for the semilinear Mindlin-Timoshenko system

We consider the dynamical one-dimensional Mindlin–Timoshenko system for beams. We obtain a global exact controllability result for this semilinear system with superlinear nonlinearities. For this purpose, we establish an observability estimate for the linearized system with bounded potentials. Moreover, we obtain an explicit estimate of the observability constant in terms of the norms of the potentials. Such an estimate is obtained by a combination of a pointwise estimate, a global Carleman estimate for hyperbolic differential operators and an analysis of the regularity of solutions of a auxiliary optimal control problem. The controllability of the semilinear system is proved using the Hilbert Uniqueness Method (HUM) and a fixed point technique.

On a model for the magnetoelastic vibrations of a perfectly conducting Mindlin–Timoshenko plate

This article is a continuation of our earlier work on models for the magnetoelastic interactions of an elastic field and a magnetic field in an electrically conducting Mindlin–Timoshenko plate. In the present work our focus is on the role of the electrical conductivity. It turns out that when the plate exhibits ‘infinite’ electrical conductivity, i.e. the electrical resistance is negligible compared to other effects, then the dissipative property of the energy associated with the original model is forfeited and the model reduces to a Mindlin–Timoshenko plate model augmented by a Lorentz force term. We establish well-posedness of the model without imposing the condition of negligibly small normal magnetic stresses at the plate faces in our earlier work. This entails the analysis of a perturbed evolution equation. Finally polynomial stabilization of the unperturbed model is achieved by incorporating damping of Kelvin–Voigt type only in the system for the shear angles of plate filaments.

Transmission problems for Mindlin–Timoshenko plates: frictional versus viscous damping mechanisms

In this article, we make a comparative analysis of the stabilizing effect of the frictional dissipation with the dissipation produced by viscous materials of Kelvin–Voigt type both located in a part of a Mindlin–Timoshenko plate. We model these dissipative mechanisms through transmission problems and show that localized frictional damping, when effective over a strategic component of the plate, produces exponential stability of the corresponding semigroup. On the other hand, although the dissipation of Kelvin–Voigt is considered a strong dissipation, we prove that it loses its uniform stabilizing properties when localized over a component of the material and provides only a slower polynomial decay.

Asymptotic limits and stabilization for the 2D nonlinear Mindlin–Timoshenko system

Asymptotic limits and stabilization for the 2D nonlinear Mindlin–Timoshenko system

Polynomial stability of a magneto-thermoelastic Mindlin–Timoshenko plate model

In this paper, we consider the magneto-thermoelastic interactions in a two-dimensional Mindlin–Timoshenko plate. Our main result is concerned with the strong asymptotic stabilization of the model. In particular, we determine the rate of polynomial decay of the associated energy. In contrast with what was observed in other related articles, geometrical hypotheses on the plate configuration (such as radial symmetry) are not imposed in this study nor any kind of frictional damping mechanism. A suitable multiplier is instrumental in establishing the polynomial stability with the aid of a recent result due to Borichev and Tomilov (Math Ann 347(2):455–478, 2010).

Dynamic contact of a thermoelastic Mindlin–Timoshenko beam with a rigid obstacle

We concentrate on the dynamics of a thermoelastic Mindlin–Timoshenko beam striking a rigid obstacle. We state classical formulations involving complementarity conditions. Weak formulations are in the form of systems consisting of a hyperbolic variational inequality for a deflection, a hyperbolic and a parabolic equation for an angle of rotation and a thermal strain, respectively. The penalization method is applied to solve the unilateral problem. The time derivative of the function representing the deflection of the beam’s middle line is not continuous due to the hitting the obstacle. The acceleration term has the form of a vector measure.

Stability of weakly dissipative Reissner–Mindlin–Timoshenko plates: A sharp result

In the present article, we show that there exists a critical number that stabilizes the Reissner–Mindlin–Timoshenko system with frictional dissipation acting on rotation angles. We identify two speed characteristics v12:=K/ρ1 and v22:=D/ρ2, and we show that the system is exponentially stable if and only if \begin{equation*} v_{1}^{2}=v_{2}^{2}. \end{equation*} For v12 ≠ v22, we prove that the system is polynomially stable and determine an optimal estimate for the decay. To confirm our analytical results, we compute the numerical solutions by means of several numerical experiments by using a finite difference method.

Indirect stabilization of a Mindlin–Timoshenko plate

In this note, we investigate stability issues for a Mindlin–Timoshenko plate with internal dissipation distributed everywhere in the domain under consideration. The damping occurs only in the elasticity equations describing the motion of the angles of rotation of a filament; the vertical deflection equation has no damping, but the effect of the damping is transmitted there by the coupling. A natural question is then: is the coupling robust enough to induce the exponential stability of the Mindlin–Timoshenko plate? A positive response to that question is given in this note, provided that the plate is clamped (Dirichlet boundary conditions), and the speed of propagation of the wave generated by the longitudinal component of the rotation angles and that of the wave generated by the vertical deflection are identical. This result strongly improves earlier results where only polynomial stability was established. When those speeds are distinct, an improved polynomial stability of the system is established. The proofs of those stability results are constructive, and they rely on the frequency domain method combined with the multipliers technique. In particular, the exponential stability result seems to be the first one for the Mindlin–Timoshenko system where the mechanical damping is present in the rotation angle equations only.

Stability of Mindlin–Timoshenko plate with nonlinear boundary damping and boundary sources

Presented here is a study of long-term behavior of Mindlin–Timoshenko (RMT) plate systems, focusing on the interplay between nonlinear viscous boundary damping and boundary source terms. This work complements [28] which established local well-posedness of this problem, and global well-posedness when the boundary damping dominates the boundary sources (in an appropriate sense). The current paper develops the potential well theory for the RMT system: global existence for potential well solutions without restricting the boundary source exponents, and explicit energy decay rates dependent on the boundary damping exponents. This work along with [26–28] provides the fundamental well-posedness and stability theory for MT plates under the interplay of damping and source terms acting either in the interior or on the boundary of the plate.

Well-posedness of Mindlin–Timoshenko Plate with Nonlinear Boundary Damping and Sources

This study addresses well-posedness of a Mindlin---Timoshenko (MT) plate model that incorporates nonlinear viscous damping and nonlinear source term in Neumann boundary conditions. The main results verify local and global existence of solutions as well as their continuous dependence on the initial data in appropriate function spaces. Along with (Pei et al. in J Math Anal Appl 418(2):535---568, 2014, in Nonlinear Anal 105:62---85, 2014) this work completes the fundamental well-posedness theory for MT plates under the interplay of damping and source terms acting either in the interior or on the boundary of the plate.

An analytic semigroup for the magnetoelastic Mindlin–Timoshenko plate model

Along with the development of magnetoelastic sensor technology, the subject of magnetoelastic vibrations in plates has attracted considerable interest during the past few decades. More recently, models for the magnetoelastic interactions of a Mindlin–Timoshenko plate and a magnetic field in an electrically conducting plate have been investigated by Grobbelaar-Van Dalsen in [Grobbelaar-Van Dalsen M. On the dissipative effect of a magnetic field in a Mindlin–Timoshenko plate model. Z. Angew. Math. Phys. 2012;63:1047–1065] and [Grobbelaar-Van Dalsen M. Exponential stabilization of magnetoelastic waves in a Mindlin–Timoshenko plate by localized internal damping. Z. Angew. Math. Phys. 2015;66:1751–1776]. In the cited contributions, we accomplished, respectively, polynomial stability of the model, with no mechanical damping, and exponential stability of the model, equipped with nonlinear locally distributed damping that exhibits linear behaviour near the origin. In this article, we complete the cycle by showing that inclusion of Kelvin–Voigt damping in the interior of the plate furnishes the property of analyticity of the semigroup of contractions associated with the model. This provides a direct route to the exponential uniform stability of the model.

Stability to the dissipative Reissner–Mindlin–Timoshenko acting on displacement equation

In this paper, we show that there exists a critical number that stabilises the Reissner–Mindlin–Timoshenko system with frictional dissipation acting only on the equation for the transverse displacement. We identify that the Reissner–Mindlin–Timoshenko system has two speed characteristics v12 := K/ρ1 and v22 := D/ρ2 and we show that the system is exponentially stable if only if \begin{equation*} v_{1}^{2}=v_{2}^{2}. \end{equation*}In the general case, we prove that the system is polynomially stable with optimal decay rate. Numerical experiments using finite differences are given to confirm our analytical results. Our numerical results are qualitatively in agreement with the corresponding results from dynamical in infinite dimensional.

Reinforcement of a Mindlin–Timoshenko plate by a thin layer

We consider a problem of reinforcement of a Mindlin–Timoshenko plate with a thin stiffener of thickness \({\delta }\), on a portion of its boundary. We investigate the case where the mass density, the rigidity and the shear modulus of the material constituting the stiffener vary as \({\delta ^{-a}}\), where \({a\in \mathbb{R_{+}^{\ast }}}\). We perform an asymptotic analysis of the solution as \({\delta }\) goes to zero. We shall show that different limit behaviors occur when a vary in \({\mathbb{R_{+}^{\ast }}}\). The situation where the stiffener is of variable thickness is also investigated. It is also shown how the Kirchhoff–Love model, with Ventcel boundary conditions, is obtained, when the shear modulus approaches +\({\infty}\) (when it behaves as specific power of \({\delta}\)) in both the plate and the stiffener.