Continuously inhomogeneous beam with longitudinal vertical cracks: an analytical investigation

. The present paper is concerned with fracture analysis of an inhomogeneous beam with three longitudinal vertical parallel cracks. The three cracks are located symmetrically with respect to the mid-span. A notch is cut-out in the lateral surface of the beam in the mid-span. Only half of the beam is considered due to the symmetry. The material is continuously inhomogeneous in the width direction of the beam. Besides, the material exhibits non-linear elastic mechanical behavior. The three cracks are located arbitrary in the width direction so as the cross-sections of the four crack arms have different width. The longitudinal fracture behavior is studied in terms of the strain energy release rate. Three solutions to the strain energy release rate are derived by differentiating the complementary strain energy with respect to the areas of the three cracks. The strain energy release rate is determined also by analyzing the balance of the energy for verification. Further verifications are carried-out by applying the J -integral approach. The influences of the locations of the three cracks, the material inhomogeneity and the beam geometry on the longitudinal fracture behavior are appraised. Results of analyses of a beam that is continuously inhomogeneous in both width and length directions are also presented.