Examples Approximately Solvable by Bending Theory

In Section 2.1 we already presented the DCB specimen as an example of structures in which the energy release rates may be approximately calculated by bending theory (Fig. 2.1.3). The energy release rates of

Figure 3.2.1 Structures with energy release rate approximately solvable by beam theory.

the beams or structures shown in Fig. 3.2.1 can also be solved in this way if they are slender and the cracks are assumed to grow straight ahead. The solutions are asymptotically exact as the slenderness tends to infinity.

The approximation, which has already been used in Examples 2.1.1 -2.1.4, consists of using the classical beam theory to determine the load-point displacement and the elastic energy for the arms at both sides of the crack, assuming fixed ends at the crack root sections. Two basic approaches may be used. In the first, the bending moment distribution is computed; then the energy per unit length of beam, M2/2Е1, is integrated to find the total clastic energy or the complementary energy and the energy release rate is determined by differentiation with respect to crack length according to Hqs. (2.1.15) or (2.1.21). This procedure was illustrated in Examples 2.1.1 and 2.1.2. The second approach is to compute first the compliance and use Eq. (2.1.32) to determine the energy release rate in the manner in Examples 2.1.3 and 2.1.4.

All the structures shown in Fig. 3.2.1 can be solved in either way, although some structures are statically indeterminate and then the redundant forces must be solved first. Care should be taken regarding the value of the crack length in these structures, which is a when a single crack tip exists, but Na when N crack tips are present. Hence, partial derivatives must be with respect to Na to obtain the energy release rate per crack tip.