A new graded singular finite element for crack problems in functionally graded materials

Abstract

A new graded singular finite element is proposed for analyzing crack problems in linear elastic isotropic functionally graded materials (FGMs) with spatially varying elastic parameters. The general formulation of the suggested singular element is obtained by analyzing the crack tip stress field using the Westergaard stress function method. The general shape function is generated by integrating the strains obtained from the stress function. The stiffness matrix for the singular element is then determined using the principle of minimum potential energy. Using the displacement continuity between the singular and the adjacent regular elements, stiffness matrices are assembled. This new element is characterized by containing both singular and higher order terms, which provides more precise description of the crack tip fields. The validity of the new element is demonstrated by comparing with existing solutions. This element is implemented for simulating crack problems in FGMs whose elastic properties vary normal to the crack line. Stress intensity factors, energy release rates, levels of non-singular stresses, and stress distributions near the crack tip are investigated. Numerical results reveal that the introduced graded singular element is more efficient than conventional finite elements as it provides more accurate description of the crack tip field with less efforts.