Couette flows of rigid/plastic solids: analytical examples of the interaction of constitutive and frictional laws

The exact analytical solution under plane strain conditions is found for a hollow circular cylinder subjected to rotational friction on the inner surface and an uniform pressure on the outer surface. Two constitutive laws, a rigid/plastic hardening model with a saturation stress (Voce-Palm material) and a rigid/viscoplastic model (Bingham material), are considered as well as three interfacial laws, sticking, Coulomb sliding friction and Tresca shearing friction. The study focuses on general behavior of the solutions for various pairs of constitutive and interfacial models, such as existence and uniqueness and qualitative differences in solution. For the Voce-Palm material, it is shown that solutions do not exist for certain boundary conditions if only sticking or Coulomb friction are permitted. On the other hand, multiple solutions exist for certain boundary conditions if only sticking and Tresca friction are permitted. Existence is achieved under all boundary conditions if sticking and both frictional laws permitted simultaneously. Uniqueness and unambiguity are achieved if two interface selection principles are invoked: (1) sticking must occur if it is possible and (2) Coulomb sliding must occur if it is possible and sticking is not. Geometrical interpretation of results in the form of friction maps is provided to illustrate possible regimes of interfacial behavior for different boundary conditions.