Longitudinal fracture analysis of inhomogeneous beams with continuously changing radius of cross-section along the beam length

Abstract

This paper focusses on deriving a general solution to the strain energy release rate for a longitudinal circular cylindrical crack in non-linear elastic inhomogeneous beams with circular cross-section. The radius of the cross-section changes continuously in the length direction of the beam. The solution to the strain energy release rate is derived assuming continuous (smooth) material inhomogeneity in radial direction. The solution holds for arbitrary law for continuous change of the radius of the cross-section along the length of the beam. The solution is valid for a longitudinal circular cylindrical crack located arbitrary in radial direction. The solution is applied to analyze the longitudinal fracture behaviour of a cantilever inhomogeneous beam configuration assuming that the radius of the cross-section changes continuously along the beam length in cotangent law. The longitudinal fracture behaviour of the cantilever beam is also analysed by considering the complementary strain energy in order to verify the solution to the strain energy release rate. The influence of the continuously changing radius of the cross-section along the beam length on the fracture behaviour is investigated. It is shown that the general solution to the strain energy release rate can also be used when the beams exhibit continuous material inhomogeneity in both radial and length directions.