Lack-of-contact conditions for a penny-shaped crack under a polynomial normal loading

Summary The classical problem of a penny-shaped crack inside an infinite three-dimensional isotropic elastic medium and under a polynomial normal loading (with axial symmetry) on both crack faces is reconsidered. By using elementary results from computational quantifier elimination techniques in computer algebra and applied logic, such as cylindrical algebraic decomposition and Sturm (or Sturm-Habicht) sequences, it is possible to satisfy the funcdamental inequality constraint about the positivity of the crack opening displacement inside the whole crack. This constraint assures us about the lack of contact of the crack faces, due to the loading of the crack, and the derived quantifier-free formula constitutes the related necessary and sufficient condition involving the loading parameters, that is the coefficients of the loading polynomial. Several such low-degree polynomial loadings are considered in detail (with the help of elementary and well-known solutin techniques for the present penny-shaped crack problem) as an application of the approach. Further possibilities for generalizations are also discussed in brief.