Advanced Aspects of LEFM
In (his chapter, wc summarize some advanced topics in LEFM that were not covered in depth in the preceding chapters. First, we present th^thcoretical framework to analytically handle plane elasticity problems with cracks. Emphasis is put on the understanding of various methods of solution, such as the complex potentials (expounded in Section 4.1), Wcstcrgaard stress functions (presented in Section 4.2), and Airy stress functions (developed as exercises). The presentation does not aim at complete, formal presentations (for this purpose, see, e. g., England 1971). Neither it aims at teaching the skills to obtain the solution from scratch. It only aims at facilitating insight into the use of complex potentials and Westergaard stress functions to obtain stress and displacement lields. As a basic example, these methods are applied to the analysis of the infinite center-cracked panel.
The complex potentials are next used to analyze the near-tip fields (Section 4.3). The in-plane case, involving fracture modes 1 (pure opening) and II (in-plane shear), is discussed first. Then (he formalism to handle the antiplane case of mode III (antiplane shear) is introduced and the general antiplane stress and displacement near-tip fields are obtained.
The next topic covered is that of the path-independent integrals, of which the./-integral is the most important. Section 4.4 shows formally that Rice’s./-integral is path-independent under certain assumptions’; it introduces a further path-independent integral for the LEFM case which is based on the reciprocity theorem and is used to provide another derivation of Irwin’s relation. Finally, other path independent integrals are briefly discussed (/-,L-, and AZ-integrals).
The last section deals with the topic of mixed mode fracture in LEFM. The existing fracture criteria are briefly described, with emphasis put on the single-parameter models, especially the maximum principal stress criterion (Erdogan and Sill 1963).