Peter Shi

Weak solution to an evolution problem with a nonlocal constraint

A parabolic problem is considered with a nonlocal constraint in place of one of the standard boundary conditions. Well-posedness of the problem is proved in a weighted, fractional Sobolev space with the problem data proposed in related weighted spaces. This comes as an intrinsic requirement of the problem. The proof uses an interpolation inequality for norms of fractional Sobolev spaces and is based on an interesting choice of the test function.

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.

Uniqueness and stability of the solution to a thermoelastic contact problem

Uniqueness and continuous dependence on the initial temperature are proved for a onedimensional, quasistatic and frictionless contact problem in linear thermoelasticity. First the problem is reformulated in such a way that it decouples. The resulting problem for the temperature is a nonlinear integro-differential equation. Once the temperature is known the displacement is recovered from an appropriate variational inequality. Uniqueness is proved by considering an integral transform of the temperature. The steady solution is obtained and the asymptotic stability is shown. It turns out that the asymptotic behaviour and the steady state are determined by a relation between the coupling constant a and the initial gap.

NUMERICAL SOLUTIONS TO ONE-DIMENSIONAL PROBLEMS OF THERMOELASTIC CONTACT

Abstract A numerical method, based on the Crank-Nicolson discretisation, is applied to one-dimensional thermoelastic contact problems. Such problems can be reduced, using an appropriate transformation, to the heat equation with a nonlinear and nonlocal source term. It is proved that the method converges. A number of numerical experiments are presented to illustrate the method and the behavior of the solutions to the problem.

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.

Decoupling of the quasistatic system of thermoelasticity on the unit disk

The quasi-static, linearized thermoclastic system on the unit disk is decoupied, such that the temperature satisfies an integro-differential equation. The result, based on the function theoretic method, is of both theoretical and numerical interest.

Dynamic plasticity models

The evolution of an elastic-plastic material is modeled as an initial boundary value problem consisting of the dynamic momentum equation coupled with a constitutive law for which the hysteretic dependence between stress and strain is described by a system of variational inequalities. This system is posed as an evolution equation in Hilbert space for which is proved the existence and uniqueness of three classes of solutions which are distinguished by their regularity.

Regularity of solutions to a contact problem

We consider a variational inequality for the Lamé system which models an elastic body in contact with a rigid foundation. We give conditions on the domain and the contact set which allow us to prove regularity of solutions to the variational inequality. In particular, we show that the gradient of the solution is a square integrable function on the boundary.