Interval computations in the formulae for the stress intensity factors at crack tips using the method of quantifier elimination

• Κύριος συγγραφέας: Ioakimidis, Nikolaos
• Άλλοι συγγραφείς: Ιωακειμίδης, Νικόλαος
• Μορφή: Technical Report
• Γλώσσα: English
• Έκδοση: 2019
• Θέματα: Straight cracks; Slant cracks; Arrays of cracks; Periodic arrays of cracks; Collinear cracks; Stress intensity factors; Exponential loading; Plane isotropic elasticity; Fracture mechanics; Weight functions; Numerical integration; Minimax approximation; Singular integral equations; Symbolic computations; Ranges; Intervals; Interval variables; Interval arithmetic; Interval analysis; Crisp variables; Deterministic variables; Quantifiers; Existential quantifier; Quantified formulae; Quantifier elimination; Quantifier-free formulae; Semi-analytical-numerical computations; Computer algebra systems; Mathematica; Ευθύγραμμες ρωγμές; Λοξές ρωγμές; Διατάξεις ρωγμών; Περιοδικές διατάξεις ρωγμών; Συγγραμμικές ρωγμές; Συντελεστές εντάσεως τάσεων; Εκθετική φόρτιση; Επίπεδη ισότροπη ελαστικότητα; Θραυστομηχανική; Συναρτήσεις βάρους; Αριθμητική ολοκλήρωση; Προσέγγιση minimax; Ιδιόμορφες ολοκληρωτικές εξισώσεις; Συμβολικοί υπολογισμοί; Πεδία τιμών; Διαστήματα; Μεταβλητές διαστήματος; Αριθμητική διαστημάτων; Ανάλυση διαστημάτων; Σαφείς μεταβλητές; Προσδιορίσιμες μεταβλητές; Ποσοδείκτες; Υπαρξιακός ποσοδείκτης; Τύποι με ποσοδείκτες; Απαλοιφή ποσοδεικτών; Τύποι χωρίς ποσοδείκτες; Ημιαναλυτικοί-αριθμητικοί υπολογισμοί; Συστήματα υπολογιστικής άλγεβρας

The concept of the stress intensity factor at a crack tip is extremely well known and it plays a very important role in fracture mechanics. On the other hand, uncertainty is often present in engineering problems mainly because of measurement errors and it is frequently represented with the help of interval variables. Here we consider the case of formulae for the computation of stress intensity factors at crack tips with one or more than one variable in such a formula being an interval variable. In this case, we compute the related intervals for the stress intensity factors, which, naturally, are also interval variables. This computation is based on the related existentially quantified formulae and it is made with the help of the interesting computational method of quantifier elimination as this method is efficiently implemented in the computer algebra system Mathematica. More explicitly, here the following four classical crack problems are studied: (i) the problem of a straight crack in an infinite plane isotropic elastic medium under a tensile loading at infinity normal to the crack, (ii) the related problem of a slant straight crack with respect to the loading at infinity, (iii) the problem of a crack in a similar medium now under an exponential normal loading on its edges and (iv) the problem of a periodic array of collinear straight cracks again in an infinite plane isotropic elastic medium under a tensile loading at infinity normal to the cracks. In the third and the fourth problems, approximate formulae for the stress intensity factors are used. The present results permit the efficient evaluation of the intervals (the ranges) for the stress intensity factors at crack tips when interval variables instead of crisp (deterministic) variables are present in the related formulae without any overestimation of the intervals for the stress intensity factors. Naturally, the present method is also applicable to more difficult crack problems provided, of course, that the total number of variables in the existentially quantified formulae used for quantifier elimination is small (generally up to five or six variables); otherwise, quantifier elimination may fail to yield a QFF (quantifier-free formula) at least in a reasonable time interval. The present results constitute one more application of quantifier elimination and interval analysis to applied mechanics, here to fracture mechanics.