It is important to have fracture toughness parameter as an inherent mechanical property of materials just like yield stress. The mode-I fracture toughness (KIc) meets that definition with certain limitations. As the analysis involved still depends on the linear elastic fracture mechanics theories, the testing procedures followed are applicable to materials with limited ductility, such as high-strength steels, some titanium and aluminum alloys, and, of course, other brittle materials like ceramics.

The elastic stress field around the crack tip can be described by a single parame­ter known as “stress intensity factor (K).’’ This factor depends on many factors such as the geometry of the crack-containing solids, the size and location of the crack, and the magnitude and distribution of the loads applied. It can be reasonably assumed that an unstable rapid failure would occur if a critical value of K is reached. There are three modes of testing, as shown in Figure 5.22. In the opening mode (mode-I), the displacement is perpendicular to the crack faces. In mode-II (sliding mode), the displacement is made parallel to the crack faces, but perpendic­ular to the leading edge. In mode-III (tearing mode), the displacement is parallel to the crack faces and the leading edge.

In reality, opening mode (i. e., mode-I) is most important. That is why the conven­tional tests for fracture toughness are done under mode-I (i. e., opening mode of loading), and the critical value of K is called KIc, the plain strain fracture toughness. For a given type of loading and geometry, the relation is

KIc — Y opPOc, (5.32)

where Y is a parameter that depends on the specimen and crack geometry, ac is the critical crack length, and o is the applied stress. If KIc and applied stress are known, one can compute the maximum crack length tolerable. In other words, maximum

allowable stress can be computed for a given crack size provided one knows the KIc value of the material. KIc generally decreases with decreasing temperature and increasing strain rate, and vice versa. It also strongly depends on metallurgical vari­ables such as heat treatment, crystallographic texture, impurities, inclusions, grain size, and so on.

A notch in a thick plate is far more damaging than that in a thin plate because it leads to a plane strain state of stress with a high degree of triaxiality. The fracture toughness measured under plane strain conditions is obtained under maximum constraint or material brittleness. The plane strain fracture toughness is thus desig­nated as KIc and is a true material property. A mixed mode (ductile-brittle) fracture occurs with thin specimens. Once the specimen has the critical thickness, the frac­ture surface becomes flat and the fracture toughness reaches a constant minimum value with increasing specimen thickness (Figure 5.23). The minimum thickness (B for breadth) to achieve plane strain condition is given by

B > 2.5(KIC/s,)2, (5.33)

where s0 is 0.2% yield stress.