Computer-aided quantifier elimination in crack problems under constraints for the stress intensity factors

Abstract

It is suggested that crack problems under constraints, concerning the values of the parameters involved, be studied by the method of quantifier elimination in such a way that quantifier-free formulae can be derived assuring the validity of a restriction about the stress intensity factors (usually that such a factor does not exceed a critical value) for all possible values of the parameters under the applicable constraints. This permits safe conclusions about the possible fracture of a cracked specimen (under constraints on the parameters involved) for every set of values of the geometry/loading/fracture parameters. The elementary crack problem of a single straight crack in an infinite plane isotropic elastic medium is used as the vehicle for the illustration of the present approach and the related quantifier-free formula is easily derived. A simple case of a periodic array of cracks is also considered in brief. The classical algebraic tools of resultants and Sturm's sequences (together with Sturm's theorem) as well as the computer algebra system Maple V have been used in the present computations. More general results, with the help of Collins' famous cylindrical algebraic decomposition method, in more complicated crack problems, can also be obtained.