Fracture of inhomogeneous beam with three parallel lengthwise cracks

Abstract

A lengthwise fracture analysis of an inhomogeneous beam with three parallel cracks is developed. The cracks are located symmetrically with respect to the mid-span. The loading consists of two axial forces applied at the ends of the beam. A vertical notch is cut-out in the mid-span so as the upper three crack arms are free of stresses. Only half of the beam is analyzed due to the symmetry. The cracks are located arbitrary along the beam height. Therefore, the heights of the cross-sections of crack arms are different. The beam exhibits continuous material inhomogeneity in the height direction. The material has non-linear elastic behaviour. Three solutions to the strain energy release rate are derived. For this purpose, the complementary strain energy in the beam is differentiated with respect to the areas of the three cracks. The J-integral approach is applied in order to verify the solutions to the strain energy release rate. The influence of the locations of the three cracks along the beam height on the lengthwise fracture behaviour is studied in detail. The influence of the material inhomogeneity on the lengthwise fracture behaviour of the beam is also analyzed.