Thermoviscoelastic Plate Research Articles

Spectral Analysis and Stability of the Moore-Gibson-Thompson-Fourier Model

We consider the linear evolution system uttt+αutt+βΔ2ut+γΔ2u=-ηΔθθt-κΔθ=ηΔutt+αηΔut\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} u_{ttt}+\\alpha u_{tt} + \\beta \\Delta ^2 u_t + \\gamma \\Delta ^2 u =- \\eta \\Delta \ heta \\\\ \ heta _t - \\kappa \\Delta \ heta = \\eta \\Delta u_{tt} + \\alpha \\eta \\Delta u_t \\end{array}\\right. } \\end{aligned}$$\\end{document}describing the dynamics of a thermoviscoelastic plate of MGT type with Fourier heat conduction. The focus is the analysis of the energy transfer between the two equations, particularly when the first one stands in the supercritical regime, and exhibits an antidissipative character. The principal actor becomes then the coupling constant η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta $$\\end{document}, ruling the competition between the Fourier damping and the MGT antidamping. Indeed, we will show that a sufficiently large η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta $$\\end{document} is always able to stabilize the system exponentially fast. One of the features of this model is the presence of the bilaplacian in the first equation. With respect to the analogous model with the Laplacian, this introduces some differences in the mathematical approach. From the one side, the energy estimate method does not seem to apply in a direct way, from the other side, there is a gain of regularity allowing to rely on analytic semigroup techniques.

Critical Electric Load on a Hinged Thermoviscoelastic Rectangular Plate with Piezoelectric Sensors and Actuators*

We proposed a failure criterion for the system of controlling the forced vibration of a hinged rectangular thermoviscoelastic plate with piezoelectric sensors and actuators. This criterion is associated with the self-heating temperature reaching some critical value. The critical electric load is found on the basis of this criterion.

Vibration analysis of wave motion in micropolar thermoviscoelastic plate

The aim of the present article is to study the micropolar thermoelastic interactions in an infinite Kelvin-Voigt type viscoelastic thermally conducting plate. The coupled dynamic thermoelasticity and generalized theories of thermoelasticity, namely, Lord and Shulman\'s and Green and Lindsay\'s are employed by assuming the mechanical behaviour as dynamic to study the problem. The model has been simplified by using Helmholtz decomposition technique and the resulting equations have been solved by using variable separable method to obtain the secular equations in isolated mathematical conditions for homogeneous isotropic micropolar thermo-viscoelastic plate for symmetric and skew-symmetric wave modes. The dispersion curves, attenuation coefficients, amplitudes of stresses and temperature distribution for symmetric and skew-symmetric modes are computed numerically and presented graphically for a magnesium crystal.

Active damping of vibrations of plates subject to unknown pressure

An approach to the active damping of the forced resonant vibrations of orthotropic thermoviscoelastic plates with distributed sensors and actuators is proposed. The mechanical load is assumed unknown and is determined from the sensors’ indications. The problem of active damping of an isotropic thermoviscoelastic rectangular plate with hinged edges is solved as an example. A formula for the voltage to be applied to the actuator to damp the forced vibrations in the first mode is derived. The effect of the dimensions of the sensor and actuator and the dissipative properties of the materials on the effectiveness of active damping is studied

Forced axisymmetric vibrations and self-heating of a circular viscoelastic plate, controlled with piezoelectric sensors and actuators

The forced monoharmonic vibrations and self-heating of a circular thermoviscoelastic plate with piezoelectric sensor and actuator are studied. The viscoelastic behavior of the passive (without piezoeffect) and piezoactive materials is described using the concept of complex moduli. The problems of electroelasticity and heat conduction are solved numerically, assuming that the mechanical load is unknown. The effect of self-heating on the active damping of the vibrations of the plate is analyzed

Basic equations of the theory of thermoviscoelastic plates with distributed sensors and actuators

The basic equations of the theory of thermoviscoelastic thin-walled plates with piezoelectric sensors and actuators under monoharmonic mechanical and electric loading are derived using the Kirchhoff-Love hypotheses. The thermomechanical behavior of passive and piezoactive materials is described using the concept of complex characteristics. Methods of solving nonlinear problems of active damping of thermomechanical vibrations of plates with sensors and actuators are considered. The effect of dissipative heating on the damping of axisymmetric vibrations of a thermoviscoelastic solid circular plate is analyzed as an example

Basic relations of the theory of thermoviscoelastic plates with distributed sensors

The basic relations of the thermomechanics of thin-walled viscoelastic plates with distributed piezoelectric sensors under monogarmonic mechanical loading are presented. To describe the thermomechanical behavior of materials, the concept of complex characteristics is used. Numerical and variational methods are used to study the forced resonant vibrations of viscoelastic plates with distributed piezoelectric sensors. The effect of dissipative heating on the readings of the sensors of a circular viscoelastic plate undergoing axisymmetric resonant bending vibrations is analyzed as an example

Basic equations for thermoviscoelastic plates with distributed actuators under monoharmonic loading

The basic equations for thin-walled thermoviscoelastic plates with distributed piezoelectric actuators under monoharmonic mechanical and electric loads are derived. The thermomechanical behavior of materials is described using the concept of complex characteristics. Variational methods of solving nonlinear problems of active damping of the bending vibrations of plates are considered. The effect of dissipative heating on the damping of the axisymmetric bending vibrations of a circular plate is examined as an example

Asymptotic behavior of a thermoviscoelastic plate with memory effects

We consider a coupled linear system describing a thermoviscoelastic plate with hereditary effects. The system con- sists of a hyperbolic integrodifferential equation, governing the temperature, which is linearly coupled with the partial differ- ential equation ruling the evolution of the vertical deflection, presenting a convolution term accounting for memory effects. It is also assumed that the thermal power contains a memory term characterized by a relaxation kernel. We prove that the system is exponentially stable and we obtain a closeness estimate between the system with memory effects and the corresponding memory-free limiting system, as the kernels fade in a suitable sense.

Vibration analysis of linear coupled thermoviscoelastic thin plates by a variational approach

According to the integral type constitutive relation of linear coupled thermoviscoelasticity, a mathematical model of thin plates is set up by the introduction of “structural functions” and “thermal functions” in the sense of the Kirchhoff’s hypothesis. The corresponding integral type variational formulations are presented by means of modern convolution bilinear forms as well as classical Cartesian bilinear forms. The Ritz method in the spatial domain and the differentiating method in the temporal domain are used to approximate the mathematical model in a system of rectangular Cartesian coordinates. By properties of inequality and parabola, the structure of dynamic solution to vibration of a thermoviscoelastic thin plate under a harmonic thermal load is studied in the space splayed by material parameter and loading parameter. The influences of thermal excitation frequency, mechanical relaxation time and thermal relaxation time on amplitude and phase difference of steady-state vibration of a square plate are investigated by amplitude-frequency analysis and phase-frequency analysis. Double-peak resonance vibration of thermoviscoelastic plates exists for given parameters.

Mathematical Models of Reissner–Mindlin Thermoviscoelastic Plates

In this paper we investigate mathematical models describing deformations and thermal variations of a thin homogeneous thermoviscoelastic plate. A hereditary non-Fourier constitutive law for the heat flux and some heat power constitutive equation with linear memory are considered. The resulting models are derived in the framework of the well-established theory, due to Gurtin and Pipkin, and according to the standard approximation procedure for the Reissner–Mindlin plate model.

Mathematical models of thin thermoviscoelastic plates

In this paper we investigate mathematical models of a thin homogeneous thermoviscoelastic plate subject to thermal deformations. A non-Fourier constitutive law for the heat flux and some heat power constitutive equation with linear memory are considered here. The resulting models are derived in the framework of the well-established theory of heat flow with memory due to Gurtin and Pipkin and according to the standard approximation procedure for the Kirchhoff plate.

Existence and Decay to Contact Problems for Thermoviscoelastic Plates

We consider the dynamic contact problem for thermoviscoelastic plates. We show the existence of weak solutions as well as exponential decay of the solution.

Decay rates of solutions to thermoviscoelastic plates with memory

We consider the thermoelastic plate under the presence of a long range memory. We find uniform rates of decay (in time) of the energy, provided that suitable assumptions on the relaxation functions are given. Namely, if the relaxation decays exponentially then the first order energy also decays exponentially. When the relaxation g satisfies -c1g(t)1+1/p ⩽ g′(t) ⩽ -cog(t)1+1/p; and g,g1-1/p ∈ L1 (R) with p > 2 then the energy decays as 1/(1+t)p. A new Liapunov functional is built for this problem.

SOLUTION OF DYNAMIC PROBLEMS FOR THERMOVISCOELASTIC PLATES BY THE FINITE-ELEMENT METHOD

A generalized variational principle for dynamic problems of thermoviscoelastic plates is formulated. The solution of the problem by the Laplace transform-finite-element method is then proposed, and properties of eigenvalues and the convergence of the method are analyzed.