Deciding in elasticity problems by using Sturm's theorem

Abstract

The use of the classical Sturm theorem (based on Sturm's sequences) in elementary algorithmic algebra for the determination of the number of distinct real zeros of a polynomial in a real finite or infinite interval, is suggested for making decisions in elementary elasticity problems, including polynomial equations and inequalities. In these cases, the decision really depends on whether a polynomial equation has a real zero or not in a real interval. The use of this method is generalized from elementary problems in computational/algebraic geometry to applied mechanics. The approach is illustrated in four elasticity problems: an elementary straight beam problem above an obstacle (to decide whether the beam comes into contact with the obstacle or not), a straight crack problem under a known loading distribution (to decide whether the crack edges come into contact or not), a problem concerning a stress component in a finite circular elastic medium (to decide whether this component reaches a critical value or not) and finally, a problem concerning the classical caustic about a crack tip in fracture mechanics (to decide whether the caustic is surrounded by the circumference of a concrete circle or not). Additional applications and possibilities are also suggested in brief.