PSI - Issue 26

Victor Rizov et al. / Procedia Structural Integrity 26 (2020) 75–85 Rizov / Structural Integrity Procedia 00 (2019) 000 – 000

84 10

The longitudinal fracture behaviour of the inhomogeneous non-linear elastic beam is analyzed also assuming that the crack reaches to the left-hand end of the beam as shown in Fig. 6. The strain energy release rate for the case when the crack reaches the left-hand end of the beam is presented in non-dimensional form as a function of 0 / H B ratio in Fig. 7. The strain energy release rate for the case when the crack is internal is presented also in Fig. 7. It can be observed in Fig. 7 that the strain energy release rate increases with increasing 0 / H B ratio. The curves in Fig. 7 indicate also that the strain energy release rate for the case when the crack reaches the left-hand end of the beam is higher than when the crack is internal.

0 / H B ratio (curve 1 – for internal crack, curve 2 –

Fig. 7. The strain energy release rate in non-dimensional form presented as a function of

for crack reaching the left-hand end of the beam).

4. Conclusions

A fracture analysis of an inhomogeneous beam configuration with an internal longitudinal crack is developed. The beam is clamped in its right-hand end. The beam exhibits continuous material inhomogeneity in the height direction. Besides, the material of the beam has non-linear elastic behavior. The crack is located arbitrary along the height of the cross-section of the beam. Thus, the two crack arms have different heights. The external loading consists of one axial force and one bending moment applied at the left-hand end of the beam. The main novelty in the present paper is the fact that the longitudinal crack is internal. Since the crack is internal, the axial forces and the bending moments in the two crack arms can not be determined directly. In the present paper, the beam is treated as a structure of two degrees of internal static indeterminacy (the axial force and the bending moment in the upper crack arm are taken as redundants). The static indeterminacy is resolved by applying the theorem of Castigliano for structures which exhibit material non-linearity. The longitudinal fracture behavior of the beam is studied in terms of the strain energy release rate. For this purpose, the complementary strain energy is considered. The solution to the strain energy release rate derived in the present paper is verified by applying the J -integral approach. Effects of the material inhomogeneity and the crack location along the height of the beam are investigated. The analysis reveals that the strain energy release rate decreases with increasing of 0 / B B K ratio (this ratio characterizes the material inhomogeneity). Concerning the effect of the crack location, it is found that the strain energy release rate has maximum at / 0.58 1 = h h for the considered loading conditions and material behavior. The analysis indicates that strain energy release rate increases with increasing of 0 / D B and 0 / H B ratios. The longitudinal fracture behavior of the inhomogeneous beam is analyzed also assuming that the crack reaches the left-hand end of the beam. It is found that the strain energy release rate in the beam with crack reaching the left-hand end is higher that that in the beam with internal longitudinal crack. References

Broek, D., 1986. Elementary engineering fracture mechanics. Springer. Gasik, M.M., 2010. Functionally graded materials: bulk processing techniques. International Journal of Materials and Product Technology 39, 20-29.