Fracture initiation at an elastic crack tip: a computational implementation of the T-criterion
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Date
1999-07
Authors
Ioakimidis, Nikolaos
Journal Title
Journal ISSN
Volume Title
Publisher
Kluwer Academic Publishers
Abstract
The classical problem of fracture initiation at an elastic crack tip in two-dimensional isotropic static elasticity and under mixed-mode loading conditions is revisited with the aim to provide simultaneously necessary and sufficient logical–algebraic conditions including all of the material and stress-state parameters involved (the mode I and II stress intensity factors, the Poisson ratio of the material and fracture-related material constants) and concerning either the avoidance or, completely equivalently, the initiation of fracture. Among several popular fracture criteria, the very efficient and sufficiently physically justified Theocaris–Andrianopoulos modified T-criterion, based on the maximum value of the dilatational strain energy density TV along an appropriate elastoplastic boundary about the crack tip of variable polar radius rcr = rcr(θ) (θ denoting the polar angle), determined in accordance with the von Mises classical yield criterion and corresponding to a constant value of the distortional strain energy density TD, has been adopted in the present algebraic approach. The computer algebra methods of Lazard, based on classical considerations about the quartic polynomial, and of González-Vega, based on standard Sturm–Habicht
sequences, have been employed as the related computational quantifier elimination algebraic tools for the present computations concerning the positivity of an appropriate quartic polynomial. The present results permit the direct decision about fracture avoidance/initiation under concrete experimental (geometry–material–loading) conditions just by using the final quantifier-free formula, which is independent of the polar angle θ about the crack tip, and, therefore, without the need of numerical investigation of the maximum of the dilatational strain energy density TV along the preselected von Mises elastoplastic boundary. The present results seem also generalizable to several algebraically similar additional fracture criteria in plane elasticity under mixed-mode conditions.