In this paper we study a class of
Volterra integrodifferential equations that arise in the description of elastic liquids,
such as polymer melts and dilute or concentrated polymer solutions. The deformation
of a small cube-shaped sample of such a liquid can be approximately described by a
symmetric 3 × 3-matrix: If the material undergoes some prescribed deformation for
times t ≤ 0 and then is allowed to recover without constraints for t > 0 (stress-free
recoil), and if inertial effects are ignored, these matrices obey an ordinary first order
Volterra integrodifferential equation. Incompressibility of the material imposes the
nonlinear constraint that the determinant of the matrices remain constant. In
addition, there is a natural small parameter η > 0, proportional to Newtonian
viscosity, which multiplies the derivative. In the case η = 0, which is also
of physical interest, the problem reduces to an implicit Volterra integral
equation.
