Thermoelastic System Research Articles

Exponential decay of solution to the viscoelastic porous‐thermoelastic system of type III with boundary time‐varying delay

In this paper, we are concerned with a one‐dimensional porous‐thermoelastic system of type III with a viscoelastic damping and boundary time‐varying delay. Under suitable assumptions on relaxation function and time delay, we establish the exponential decay result of the system in which the damping is strong enough to stabilize the thermoelastic system in the presence of time delay. Copyright © 2015 John Wiley & Sons, Ltd.

Stability of abstract thermo-elastic semigroups

In this paper we characterize the stabilization for some thermo-elastic type system with Cattaneo law and we prove that the exponential or polynomial stability of this system implies a polynomial stability of the corresponding thermoelastic system with the Fourier law. The proof of the main results uses, respectively, the methodology introduced by Haraux in [11] and generalized by Ammari and Tucsnak in [8], where the exponential stability for the closed loop problem is reduced to an observability estimate for the corresponding uncontrolled system, and a characterization of the polynomial stability for a C0-semigroup, in a Hilbert space, by a polynomial estimation of the resolvent of its generator obtained by Borichev and Tomilov [9]. Illustrating examples are given.

On the decay rates of Timoshenko system with second sound

In this work, we study the well‐posedness and the asymptotic stability of a one‐dimensional linear thermoelastic Timoshenko system, where the heat conduction is given by Cattaneo's law and the coupling is via the displacement equation. We prove that the system is exponentially stable provided that the stability number χτ=0. Otherwise, we show that the system lacks exponential stability. Furthermore, in the latter case, we show that the solution decays polynomially. Copyright © 2015 John Wiley & Sons, Ltd.

Plane strain and three-dimensional analyses for thermo-mechanical behavior of multilayered transversely isotropic materials

An extended precise integration model for analysis of thermo-mechanical behavior of multilayered transversely isotropic materials in Cartesian coordinate system is established in this paper. From the governing equations of plane strain and three-dimensional transversely isotropic thermo-elastic materials, ordinary differential equations are derived by employing Laplace–Fourier transform. An extended precise integration model is built and solved in the transformed domain by extending the precise integration method (PIM) to multilayered thermo-elastic material system, and the actual solutions are obtained by adopting numerical quadrature schemes for the Laplace–Fourier transform inversion. The accuracy and feasibility of the proposed model is demonstrated by selected numerical examples, and then the influences of heat source’s shapes, layering, mechanical and thermal parameters of the transversely isotropic material on thermo-mechanical response are investigated.

Exponential-stability and super-stability of a thermoelastic system of type II with boundary damping

In this paper, the stability of a one-dimensional thermoelastic system with boundary damping is considered. The theory of thermoelasticity under consideration is developed by Green and Naghdi, which is named as ``thermoelasticity of type II''. This system consists of two strongly coupled wave equations. By the frequency domain method, we prove that the energy of this system generally decays to zero exponentially. Furthermore, by showing the spectrum of the system is empty under certain condition and estimating the norm of the resolvent operator, we give a sufficient condition on the super-stability of this thermoelastic system. Under this condition, the solution to the system is identical to zero after finite time. Moreover, we also estimate the maximum existence time of the nonzero part of the solution. Finally, we give some numerical simulations.

A nonlinear thermoelastic system with nonlinear boundary conditions

In this paper, we give a mathematical treatment of a model for small vertical vibrations of an elastic membrane coupled with a heat equation and subject to nonlinear mixed boundary conditions. We establish the existence, uniqueness, and a uniform decay rate for global solutions to this nonlinear nonlocal thermoelastic coupled system under nonlinear boundary conditions.

On a nonlinear thermoelastic system with nonlocal coefficients

This paper deals with the global existence and uniqueness of solutions, and uniform stabilization of the energy of an initial–boundary value problem for a thermoelastic system with nonlinear terms of nonlocal kind. The asymptotic stabilization of the energy of system (1.1) is obtained without any dissipation mechanism acting in the displacement variable u of the membrane equation.

Exact Controllability of Semilinear Thermoelastic Systems with Monotone Nonlinearity

This article is concerned with the exact controllability of the semilinear thermoelastic system, which can be written as with clamped/Dirichlet boundary conditions. Here, a relation between the reachable set of a semilinear thermoelastic system and that of the corresponding linear system is proved using the results given.

Asymptotic stability of thermoelastic systems of Bresse type

We provide a comprehensive stability analysis of the thermoelastic Bresse system (also known as the circular arch problem). In particular, assuming a temperature evolution of Gurtin–Pipkin type, we establish a necessary and sufficient condition for exponential stability in terms of the structural parameters of the problem. As a byproduct, a complete characterization of the longtime behavior of Bresse-type systems with Fourier, Maxwell–Cattaneo and Coleman–Gurtin thermal laws is obtained. Our main theorem also subsumes some recent achievements in the stability properties of thermoelastic Timoshenko systems with classical and nonclassical heat conduction.

A conservative finite difference scheme for the Falk model system of shape memory alloys

In this article we propose a new finite difference scheme for the Falk model system. The Falk model system is a thermoelastic system describing the phase transition occurring on shape memory alloys. Our scheme inherits three important properties: the energy conservative law, the momentum conservation law and the law of increasing entropy. In addition, we show the existence of solution for the scheme and positivity of temperature under some conditions.

Energy decay in thermoelasticity with viscoelastic damping of general type

In this paper we consider an n-dimensional thermoelastic system with viscoelastic damping. We establish an explicit and general decay rate result without imposing restrictive assumptions on the behavior of the relaxation function at infinity. Our result allows a larger class of relaxation functions and generalizes previous results existing in the literature.

On the boundary control of thermoviscoelastic systems of type III

In this paper, we consider an n-dimensional thermoelastic system of type III with viscoelastic damping localized on a part of the boundary. We establish an explicit and general decay rate result that allows a larger class of relaxation functions and generalizes previous results existing in the literature.

Multiscale Computation and Convergence for Coupled Thermoelastic System in Composite Materials

This paper discusses the multiscale computation for linear time-dependent thermoelastic equations in composite materials with a periodic microstructure. The new contributions presented in this study are the formulation of the correction terms and the strong convergence result with an explicit rate for the second-order multiscale solutions. By application of the Laplace transform, the multiscale method is developed for the static coupled thermoelastic system instead of the original time-dependent equations. Numerical results are reported to demonstrate that the proposed method has a competitive advantage for solving the dynamic thermoelastic system in three-dimensional composite materials. Moreover, the method is highly suitable for parallel computation.

Exponential decay for linear damped porous thermoelastic systems with second sound

In this paper, we investigate two problems in porous thermoelasticity where the heat conduction is given by Cattaneo's law and prove exponential decay results in the presence of both macro- and micro-dissipations.

Dynamic modeling for thermal error in motorized spindles

This paper proposes a new modeling methodology to predict thermal error in motorized spindles. The dynamic model predicts thermal errors that are caused by deformation in the motorized spindle structure due to heat flow from internal sources. These thermally induced errors become more serious and dominate the total error when it comes to high speed and high precision. If these thermal errors can be predicted, they can be compensated in real time. In this paper, a new thermal errors model (ARX model) is presented which capitalizes on the notion that the motorized spindle thermoelastic system has very complicated dynamics. Furthermore, the selection principle of temperature key points, which are indispensable for building a robust thermal error model, is provided using the thermal error sensitivity technology. At last, an experiment on the thermal error in a motorized spindle is conducted to verify the effectiveness of the ARX model, the experimental results show that above 80 % of axial thermal errors are predicted for a variety of motorized spindle cycles and the dynamic model has good accuracy and robustness.

Stability for a nonlinear coupled system of elasticity and thermoelasticity with second sound

In this paper, we study the qualitative properties of solutions to a nonlinear system describing the motion of a bar in which the middle part is sensitive to the thermal change, while the outer parts are insensible. By the energy method, we show that the initial boundary value problem for this coupled system of wave equations and thermoelastic equations with second sound in one space variable is well-posed globally in time, and it is also stable exponentially as the time goes to infinity when the wave speed of the outer parts is properly large, under certain restrictions on the initial data and the growth rate of the nonlinear terms.

A stability result of a Timoshenko system in thermoelasticity of second sound with a delay term in the internal feedback

Abstract In this paper, we consider a one-dimensional linear thermoelastic system of Timoshenko type with a delay term in the feedback. The heat conduction is given by Cattaneo's law. Under an appropriate assumption between the weight of the delay and the weight of the damping, we prove the well-posedness of the problem. Furthermore, an exponential stability result is shown without the usual assumption on the wave speeds. To achieve our goals, we make use of the semigroup method and the energy method.

Well-posedness and stability results in a Timoshenko-type system of thermoelasticity of type III with delay

In this paper, we consider a one-dimensional linear thermoelastic system of Timoshenko type with delay, where the heat conduction is given by Green and Naghdi’s theory. We establish the well-posedness and the stability of the system for the cases of equal and nonequal speeds of wave propagation.

Recovery of a space-dependent vector source in thermoelastic systems

In this contribution, an inverse problem of determining a space-dependent vector source in a thermoelastic system of type-I, type-II and type-III is studied using information from a supplementary measurement at a fixed time. These thermoelastic systems consist of two equations that are coupled: a parabolic equation for the temperature and a vectorial hyperbolic equation for the displacement . In this latter one, the source is unknown, but solely space dependent. A spacewise-dependent additional measurement at the final time ensures that the inverse problem corresponding with each type of thermoelasticity has a unique solution when a damping term (with componentwise strictly monotone increasing) is present in the hyperbolic equation. Despite the ill-posed nature of these inverse problems, a stable iterative algorithm is proposed to recover the unknown source in the case that is also linear. This method is based on a sequence of well-posed direct problems, which are numerically solved at each iteration, step by step, using the finite element method. The instability of the inverse source problem is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results support the theoretically obtained results.

An Exponential Stability Result of a Timoshenko System with Thermoelasticity with Second Sound and in the Presence of Delay

In this paper, we consider a one-dimensional linear thermoelastic system of Timoshenko type with a delay, where the heat flux is given by Cattaneo's law. We prove an exponential decay result under a smallness condition on the delay and a stability number introduced first in Santos et al. (J Diff Eqs 253:2715---2733, 2012), using a method different from that of Santos et al. (J Diff Eqs 253:2715---2733, 2012). We also reproduce the polynomial decay of Santos et al. (J Diff Eqs 253:2715---2733, 2012) using the multiplier method in the case of absence of delay. The polynomial decay issue in the presence of a small delay is an open question.