Application of the Classical Beam Theory for Studying Lengthwise Fracture of Functionally Graded Beams

Abstract

The present paper deals with analysis of lengthwise cracks in linear-elastic functionally graded beam configurations. A general approach for deriving of the strain energy release rate is developed by applying the classical beam theory. A crack located arbitrary along the beam thickness is considered, i.e. the crack arms have different thicknesses. The approach holds for beams which are functionally graded in the thickness direction (the modulus of elasticity can be distributed arbitrary along the thickness of the beam). The approach is applied to analyze the strain energy release rate for a lengthwise crack in a functionally graded cantilever beam. The beam is loaded by one concentrated force applied at the free end of the upper crack arm. An exponential law is used to describe the continuous variation of the modulus of elasticity along the beam thickness. The solution to the strain energy release rate in the cantilever beam is verified by applying the Jintegral approach. The solution is verified further by using the compliance method for deriving the strain energy release rate. The effects of crack location along the beam thickness, crack length and material gradient on the strain energy release rate in the functionally graded cantilever beam are analyzed by applying the solution derived.