The paper studies the elastic equilibrium of a homogeneous isotropic incompressible
elliptic cylinder with a hole, when normal or tangential stresses are applied on its
internal and external surfaces. The cylinder is in a state of plane deformation. Thus,
the boundary value problems are set and solved for an incompressible confocal
elliptic ring in an elliptic coordinate system. The boundary value problems for a
confocal elliptic ring are given with the superposition of the internal and external
problems of an ellipse. For incompressible bodies, equilibrium equations and Hooke’s
law are written in the elliptic coordinate system, boundary value problems are set
and solutions are presented with two harmonic functions, which are obtained by a
method of separation of variables. Two test problems for a confocal elliptic semiring
are solved and the graphs relevant to the numerical values are drafted. One problem
concerns the change in the deformed state of the incompressible confocal elliptic
semiring in relation with the change in the axes of the elliptical hole, while in the
second problem the deformation process of the rubber shaft with the elliptical hole is
investigated.
Keywords
incompressible elliptic cylinder, elliptic coordinates,
separation of variables, harmonic function, homogeneous
isotropic body
