Kevin T. Andrews

Thermoelastic contact with Barber's heat exchange condition

We consider a nonlinear parabolic problem that models the evolution of a one-dimensional thermoelastic system that may come into contact with a rigid obstacle. The mathematical problem is reduced to solving a nonlocal heat equation with a nonlinear and nonlocal boundary condition. This boundary condition contains a heat-exchange coefficient that depends on the pressure when there is contact with the obstacle and on the size of the gap when there is no contact. We model the heat-exchange coefficient as both a single-valued function and as a measurable selection from a maximal monotone graph. Both of these models represent modified versions of so-called imperfect contact conditions found in the work of Barber. We show that strong solutions exist when the coefficient is taken to be a continuously differentiable function and that weak solutions exist when the coefficient is taken to be a measurable selection from a maximal monotone graph. The proofs of these results reveal an interesting interplay between the regularity of the initial condition and the behavior of the coefficient at infinity.

A dynamic thermoviscoelastic contact problem with friction and wear

Abstract We prove the existence of a unique solution to a dynamic frictional contact problem involving a thermoviscoelastic body that may undergo wear on the contacting surface. The viscosity is assumed to be of the Kelvin-Voigt type, the contact is modeled by the normal compliance condition, and the wear by Archards law. The problem consists of the elasticity system in the form of a variational inequality for the displacements, the heat equation for the temperature, and a first order equation for the wear function. The existence of a weak solution is established by considering a sequence of regularized approximations. These approximations may serve as the basis for a convergent numerical scheme. Under additional conditions on the data we establish the uniqueness of the weak solution.

On the dynamic vibrations of an elastic beam in frictional contact with a rigid obstacle

Existence and uniqueness results are established for weak formulations of initial-boundary value problems which model the dynamic behavior of an Euler-Bernoulli beam that may come into frictional contact with a stationary obstacle. The beam is assumed to be situated horizontally and may move both horizontally and vertically, as a result of applied loads. One end of the beam is clamped, while the other end is free. However, the horizontal motion of the free end is restricted by the presence of a stationary obstacle and when this end contacts the obstacle, the vertical motion of the end is assumed to be affected by friction. The contact and friction at this end is modelled in two different ways. The first involves the classic Signorini unilateral or nonpenetration conditions and Coulombs law of dry friction; the second uses a normal compliance contact condition and a corresponding generalization of Coulombs law. In both cases existence and uniqueness are established when the beam is subject to Kelvin-Voigt damping. In the absence of damping, existence of a solution is established for a problem in which the normal contact stress is regularized.

Vibrations of a beam with a damping tip body

We investigate a mathematical model for the dynamics of a beam with a tip body that experiences damping. The damping is due to granular material which partially fills the tip body. We establish the existence of the unique solution to the model and analyze the model. Among other things, we establish exponential energy decay when damping is present.

One-dimensional thermoelastic contact with a stress-dependent radiation condition

A one-dimensional quasistatic thermoelastic contact problem with a stress-dependent boundary condition is considered. The problem models the evolution of the temperature and the displacement of a long thin elastic rod that may come into contact with a rigid obstacle. The mathematical problem is reduced to solving a nonlocal heat equation with a nonlinear and nonlocal boundary condition. This boundary condition contains a heat exchange coefficient that depends on the pressure when there is contact with the obstacle and on the size of the gap when there is no contact. The local existence of a strong solution to the problem and local dependence on the initial-boundary data are proved. In addition, the uniqueness of the solution is established. The proof rests on an abstract result dealing with perturbations of monotone operators, as well as some a priori estimates which permit an application of Schauder’s fixed point theorem.

Thermomechanical behaviour of a damageable beam in contact with two stops

A model for the thermomechanical behaviour of a beam which allows for the general evolution of material damage is presented and investigated. One end of the beam is fixed while the other is constrained to move between two stops. The contact of the free tip with the stops is modelled by the normal compliance condition. The thermal interaction between the stops and the free tip is described by a heat exchange condition where the heat transfer coefficient is a general function of the gaps between the tip and the stops. The effects on the mechanical properties of the material due to crack expansion are described by a damage field, which measures the decrease in the load-bearing capacity of the material. The damage evolves as a constrained diffusion process in which the microcracks that develop may grow or disappear. The mathematical model consists of a coupled system of energy--elasticity equations together with a nonlinear parabolic inclusion for the damage field. The existence of a local solution is established using truncation, penalization, and a priori estimates.

Dynamic Gao Beam in Contact with a Reactive or Rigid Foundation

This chapter constructs and analyzes a model for the dynamic behavior of nonlinear viscoelastic beam, which is acted upon by a horizontal traction, that may come in contact with a rigid or reactive foundation underneath it. We use a model, first developed and studied by D.Y. Gao, that allows for the buckling of the beam when the horizontal traction is sufficiently large. In contrast with the behavior of the standard Euler–Bernoulli linear beam, it can have three steady states, two of which are buckled. Moreover, the Gao beam can vibrate about such buckled states, which makes it important in engineering applications. We describe the contact process with either the normal compliance condition when the foundation is reactive, or with the Signorini condition when the foundation is perfectly rigid. We use various tools from the theory of pseudomonotone operators and variational inequalities to establish the existence and uniqueness of the weak or variational solution to the dynamic problem with the normal compliance contact condition. The main step is in the truncation of the nonlinear term and then establishing the necessary a priori estimates. Then, we show that when the viscosity of the material approaches zero and the stiffness of the foundation approaches infinity, making it perfectly rigid, the associated solutions of the problem with normal compliance converge to a solution of the elastic problem with the Signorini condition.

A one-dimensional spot welding model

A one-dimensional model is proposed for the simulations of resistance spot welding, which is a common industrial method used to join metallic plates by electrical heating. The model consists of the Stefan problem, in enthalpy form, coupled with the equation of charge conservation for the electrical potential. The temperature dependence of the density, thermal conductivity, specific heat, and electrical conductivity are taken into account, since the process generally involves a large temperature range, on the order of 1000K. The model is general enough to allow for the welding of plates of different thicknesses or dissimilar materials and to account for variations in the Joule heating through the material thickness due to the dependence of electrical resistivity on the temperature. A novel feature in the model is the inclusion of the effects of interface resistance between the plates which is also assumed to be temperature dependent. In addition to constructing the model, a finite difference scheme for its numerical approximations is described, and representative computer simulations are depicted. These describe welding processes involving different interface resistances, different thicknesses, different materials, and differentvoltageforms.The differencesin the processdue toAC orDC currents are depicted as well.

Asymptotic approximations to one-dimensional problems of quasistatic thermoelastic contact

We compare solutions to coupled and uncoupled versions of three one-dimensional problems of quasistatic thermoelastic contact which model, respectively, the mechanical behaviour of one rod, of two rods and of a cylinder. We show that if a is a nondimensional parameter which is proportional to the coefficient of thermal expansion, then the difference between the temperatures is of order O(a) when the heat exchange is modeled by a coefficient k that depends on the gap size and the contact stress. The difference is O(a^2) when k is constant. The differences in the corresponding displacements are O(a^2) and O(a^3), respectively. These results lend support to the usual practice in thermoelastic problems of neglecting the term corresponding to the work of internal forces in the heat equation.

Dynamic Evolution of an Elastic Beam in Frictional Contact with an Obstacle

Problems involving contact and friction phenomena have received a great deal of attention in recent years and by now there is a considerable body of engineering literature devoted to this subject. In contrast, there are relatively few general mathematical results available in this area, due to the substantial difficulties encountered in establishing existence results for initial-boundary value problems that model these phenomena. Moreover, in both cases, most of the existing literature deals with static situations, or, occasionally, with a sequence of static problems, which arise from the time discretization of an evolution problem. Modeling and mathematical analysis of such problems can be found in Duvaut and Lions [DL], Moreau et al. [MPS], Kikuchi and Oden [KO], and Telega [Tel], and the references therein (see also Curnier [Cu]). There are, however, some recent results on quasistatic and dynamic behavior in Andersson [An], Telega [Te2], Klarbring et al. [KMS2] and Oden and Martins [OM].