Global existence and decay for the semilinear thermoelastic contact problem

In this paper a class of semilinear thermoelastic contact problems is considered and the existence and exponential decay of the weak solutions are obtained.     

Global existence for a nonlocal model for adhesive contact

Abstract

In this paper, we address the analytical investigation into a model for adhesive contact introduced in a paper by Freddi and Fremond, which includes nonlocal sources of damage on the contact surface, such as the elongation. The resulting PDE system features various nonlinearities rendering the unilateral contact conditions, the physical constraints on the internal variables, as well as the contributions related to the nonlocal forces. For the associated initial-boundary value problem, we obtain a global-in-time existence result by proving the existence of a local solution via a suitable approximation procedure and then by extending the local solution to a global one by a nonstandard prolongation argument.     

Numerical analysis of a quasi-static contact problem for a thermoviscoelastic beam

In this paper we revisit a quasi-static contact problem of a thermoviscoelastic beam between two rigid obstacles which was recently studied in [1]. The variational problem leads to a coupled system, composed of an elliptic variational inequality for the vertical displacement and a linear variational equation for the temperature field. Then, its numerical resolution is considered, based on the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. Error estimates are proved from which, under adequate regularity conditions, the linear convergence is derived. Finally, some numerical simulations are presented to show the accuracy of the algorithm and the behavior of the solution.     

Global existence for a contact problem with adhesion

In this paper, we analyze a contact problem with irreversible adhesion between a viscoelastic body and a rigid support. On the basis of Frémond's theory, we detail the derivation of the model and of the resulting partial differential equation system. Hence, we prove the existence of global in time solutions (to a suitable variational formulation) of the related Cauchy problem by means of an approximation procedure, combined with monotonicity and compactness tools, and with a prolongation argument. In fact the approximate problem (for which we prove a local well-posedness result) models a contact phenomenon in which the occurrence of repulsive dynamics is allowed for. We also show local uniqueness of the solutions, and a continuous dependence result under some additional assumptions. Copyright © 2007 John Wiley & Sons, Ltd.     

Global existence of solution of Cauchy problem for nonlinear pseudo-parabolic equation

In this paper, we prove that the Cauchy problem for the nonlinear pseudo-parabolic equation

vt−αvxxt−βvxx+γvx+fx(v)=φx(vx)+g(v)−αg(v)_xx     

Polynomial and exponential stability for one-dimensional problem in thermoelastic diffusion theory

We consider one-dimensional problem for the thermoelastic diffusion theory and we obtain polynomial decay estimates. Then we show that the solution decays exponentially to zero as time goes to infinity; that is, denoting by E(t) the first-order energy of the system, we show that positive constants C ₀ and c ₀ exist which satisfy E(t) ≤ C ₀ E(0)e ^{?c ₀ t} .     

Numerical analysis of a contact problem between two thermoviscoelastic beams

We propose and analyze in this article a finite element approximation, based on a penalty formulation, to a quasi‐static unilateral contact problem between two thermoviscoelastic beams. An error bound is given and some numerical experiments discussed. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 644–661, 2011     

Global existence and exponential stability for the micropolar fluid system

We consider the micropolar fluid system in a bounded domain of

Global existence and exponential stability for the micropolar fluid system

We consider the micropolar fluid system in a bounded domain of

Dynamic frictionless contact of a nonlinear beam with two stops

In this work, we consider mathematical and numerical approaches to a dynamic contact problem with a highly nonlinear beam, the so-called Gao beam. Its left end is rigidly attached to a supporting device, whereas the other end is constrained to move between two perfectly rigid stops. Thus, the Signorini contact conditions are imposed to its right end and are interpreted as a pair of complementarity conditions. We formulate a time discretization based on a truncated variational formulation. We prove the convergence of numerical trajectories and also derive a new form of energy balance. A fully discrete numerical scheme is implemented to present numerical results.     

Existence of a solution for a Signorini contact problem for Maxwell-Norton materials

The aim of this article is to study the quasistatic evolutionof a Maxwell–Norton three-dimensional viscoelastic solidwith contact constraints. After introducing the appropiate functionalframework, we will discretize the problem in time using an implicitscheme whose resultant variational inequality is well posed.By using monotonicity arguments together with compensated compactnesstechniques, we will prove that the corresponding discrete solutionconverges to a solution of the continuous problem.     

Analytical inverse solution for coupled thermoelastic problem for the evaluation of contact stress during steel strip rolling

Knowledge of the contact stress between roll and strip is a critical factor in modern, high-speed rolling mills. Previously two inverse analytical methods have been developed to determine the elastic contact stress on the one hand and the heat flux or the temperature in the whole roll (and especially at the surface) on the other hand, by measuring the stress tensor inside the roll body with fibre optics and by measuring the temperature with a thermocouple fully embedded at only one point inside the roll. However measurements done by fibre optics take into account the elastic stress and the thermal stress. However the contact stress was determined under isothermal assumption, which is strongly incorrect for hot rolling conditions. In this paper, the coupled thermoelastic problem is solved analytically using the theorem of superposition and the expression of the temperature field exhibited previously. A significant improvement of the accuracy of the inverse method for reconstructing the contact stress is observed by taking into account thermal stress. Hot rolling simulation is given to demonstrate this result. The computation time is studied to rapidly optimise the industrial parameters during the rolling process, and considering that both inverse methods have been run, the computation of thermal stress does not cost significant additional CPU times.     

Global existence and uniform stability of solutions for a quasilinear viscoelastic problem

In this paper the nonlinear viscoelastic wave equation in canonical form with Dirichlet boundary condition is considered. By introducing a new functional and using the potential well method, we show that the damping induced by the viscoelastic term is enough to ensure global existence and uniform decay of solutions provided that the initial data are in some stable set. Copyright © 2006 John Wiley & Sons, Ltd.     

Convexity considerations and spatial behavior for the harmonic vibrations in thermoelastic plates

In this paper we study the spatial behavior of the steady-state solutions for the approach of thin thermoelastic plates developed by Lagnese and Lions [J.E. Lagnese, J.-L. Lions, Modelling, Analysis and Control of Thin Plates, Collection RMA, vol. 6, Masson, Paris, 1988]. The model leads to a coupled complex system of partial differential equations, one of fourth order in terms of the amplitude of the vertical deflection and the other of second-order in terms of the amplitude of temperature field. Coupling in an appropriate way the two equations in an integral identity we are able to identify some cross-sectional line integral measures associated with the amplitudes of the vertical deflection and temperature vibrations, provided that the exciting frequency is less than a certain critical frequency. Furthermore, we are able to establish a second-order differential inequality whose integration furnishes a Saint-Venant type decay estimate for a bounded strip and an alternative of Phragmén-Lindelöf type for a semi-infinite strip. The critical frequency is individuated by means of the use of some Wirtinger and Knowles inequalities.     

Global existence and asymptotic behavior for a multidimensional nonisentropic hydrodynamic semiconductor model with the heat source

In this paper, a multidimensional nonisentropic hydrodynamic model for semiconductors with the nonconstant lattice temperature is studied. The model is self-consistent in the sense that the electric field, which forms a forcing term in the momentum equation, is determined by the coupled Poisson equation. Global existence to the Cauchy problem for the multidimensional nonisentropic hydrodynamic semiconductor model with the small perturbed initial data is established, and the asymptotic behavior of these smooth solutions is investigated, namely, that the solutions converge to the general steady-state solution exponentially fast as t→+∞ is obtained. Moreover, the existence and uniqueness of the stationary solutions are investigated.     

Stability of contact discontinuities to 1-D piston problem for the compressible Euler equations

We consider 1-D piston problem for the compressible Euler equations when the piston is static relatively to the gas in the tube. By a modified wave front tracking method, we prove that a contact discontinuity is structurally stable under the assumptions that the total variation of the initial data and the perturbation of the piston velocity are both sufficiently small. Meanwhile, we study the asymptotic behavior of the solutions by the generalized characteristic method and approximate conservation law theory as $t\to +\infty$.     

Existence, stability and smoothness of a bounded solution for nonlinear time-varying thermoelastic plate equations

In this paper we study the existence, stability and the smoothness of a bounded solution of the following nonlinear time-varying thermoelastic plate equation with homogeneous Dirichlet boundary conditions