Theoretical fracture mechanics and elliptic boundary value problems

Shepherd, Duncan Paul (1998) Theoretical fracture mechanics and elliptic boundary value problems. University of Southampton, Doctoral Thesis.

Abstract

In the engineering theory of fracture mechanics, the prediction of crack growth and failure relies principally on the value of the coefficients in the asymptotic expansion of the stress field at the crack tip. Consequently, the form of this expansion, and the means for calculating the coefficients which result from this, constitute the foundations of the entire subject. Although demonstrated previously for certain special problems, the first general derivation of the form of this expansion is usually attributed to Williams (On the stress distribution at the base of a stationary crack, J. App. Mech. 24. pp109-114), using the so-called eigenfunction expansion technique. Curiously, however, although Williams paper clearly implies that the expansion is completely general in nature, no mathematical justification is provided. This is all the more surprising in view of the fact that the basic technique has subsequently been exploited to study several different types of crack problem, and is generally recognised as a standard tool in fracture mechanics analysis.

There is, however, a completely different approach to the study of linear elastic crack tip stress fields, through which we can seek to address these issues. This involves utilising the abstract theory of partial differential operators, and in particular, the study of elliptic boundary value problems on non-smooth domains. Since the fundamental problems of classical elasticity are elliptic in nature, the general theory can be exploited to yield specific results relating to fracture analysis. Moreover, since the mathematics involved is very different from that normally encountered in engineering analysis, this approach potentially offers new insight into some much studied problems.

The purpose of this thesis is both to describe the mathematics involved in this type of analysis, and to consider the implications from a fracture mechanics perspective. In fact, since the analysis yields precisely the problem considered by Williams, it turns out that it provides the mechanism through which a full mathematical justification if his methodology can be given. Moreover, this can be carried out in a more general framework than considered previously, and at the same time, a unified approach to the study of weight functions in two dimensions follows automatically. Finally, in three-dimensional situations, it is demonstrated that the asymptotic stress field assumed in practical engineering calculations is quite general in nature, a problem which had previously been considered open by fracture mechanics authors.

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