Determination of LEFM Parameters
The application of LEFM to practical problems requires knowledge of the stress intensity factors or the energy release. rates for the actual geometry and type of loading. In many cases one further needs the evolution of Кі with crack length. The selection of the method adequate to treat a particular problem depends very much on external inputs: economical importance of the problem; time available for analysis; bibliographical, analytical, numerical, and experimental facilities available to the analyst.
Fracture mechanics literature contains a vast number of closed form solutions of various elastic bodies with cracks. If the problem at hand can be approximated by one of the cases in the handbooks or papers, the problem is solved with ease. Section 3.1.1 briefly shows the use of closed form solutions from the handbooks. Quite often the problem does not coincide with any of the cases of the handbook, but can be obtained by superposition of other cases. The superposition may range from simple two-state cases to continuous weighted superposition in the sense of Green’s function. Section 3.1.2 illustrates by some examples the use of the superposition method.
Close to the handbook cases solved with ease are the cases where the elastic energy release rate can be analytically calculated using the energetic approaches of Section 2.1, together with adequate simplifying assumptions. The stress intensity factor, usually mode I intensity factor, then follows from Irwin’s relationship К і — s/E’Q. Section 3.2 illustrates some of the available approximations.
When no closed-form solutions ate available, other strategies are at hand for the analyst to choose. The first one is to try to find an analytical solution. This is a highly specialized mathematical task out of the reach of most engineering practitioners and researchers. It can be accomplished by one of the formal approaches described in Chapter 4, and is outside the scope of this book.
When all the analytical treatments fail — because Green’s functions arc not available for the geometry of interest, for example — one may resort to numerical methods, an expedient that is getting increasingly easy to handle, increasingly reliable, and becoming readily accessible to engineers (Section 3.3.1). Alternatively, the stress intensity factor of reduced-scale elastic specimens can be experimentally measured in various ways (Section 3.3.2).
Application of LEFM to practical cases requires the fracture parameters of the given materials to be known, too. The main aspects of the determination of Kjc and G; are presented in Section 3.4.
An aspect often і nvolved in fracture problems is the determination of displacements and similar variables such as crack volume or crack opening profile. Section 3.5 shows how these displacements can be calculated when closed form expressions for the stress intensity factor as a function of crack length are known. As a corollary, the stress intensity due to a point load on the crack faces is determined from the expressions of the crack opening profile and stress intensity factor for another arbitrary loading —an expression known as Bueckner’s (1970) weight function.