ABSTRACT
In addition to the nonlinear problems of the previous chapters, we now consider fully dynamical and quasi-static thermoelastic contact problems, respectively, which model the evolution of temperature and displacement in an elastic body that may come into contact with a rigid foundation. The system will consist of the linearized equations together with nonlinear boundary conditions which reflect the contact situation. If 0 C W1 (n > 2) denotes the reference configuration, we assume that the smooth boundary 90 consists of three mutually disjoint parts Td ^ n ^ c such that 90 = r D UTN U r c, and VD ^ 0. The body is held fixed on Tp, tractions are zero on and Tc is the part which may have contact with a rigid foundation. The temperature is held fixed on 90. Then the dynamical initial boundary value problem for the displacement U = U(t,x) and the temperature difference 6 = 8(t, x), where t > 0 and x E 0, to be considered is the following:
The comma notation j denotes the differentiation d/dxj with respect to Xj in this chapter, p and 5 will be assumed in the sequel to be equal to one without loss of generality. The tensors Cijkh Cij and kij are assumed to
210 8. Contact Problems
satisfy
The initial values U°, U1 and 6° are prescribed with regularity to be made more precise in the final formulation of the system. The stress tensor is given by a = (<7#)
Since on the boundary 6 = 0 we have there
The unit normal vector in x G dtt is again denoted by v — v(x) and the normal component of U by Uv:
The normal component gv of the stress tensor is given by
and the tangential part &t is
The function g describes the initial gap between the part Tc of the reference configuration and the rigid foundation and is assumed to satisfy
Hence the boundary conditions (8.5) describe Signorini’s1 contact condi tions on Tc for a frictionless (<rT = 0) contact.
