Using the quarter-point crack tip element, a simple formula is derived which provides a quick and expedient means to evaluate the elastic T-stress in the finite element method (FEM) fracture mechanics analysis of functionally graded materials (FGMs). The formula is derived by comparing the variation of displacements in the quarter-point element with the corresponding asymptotic expression for the displacement field in the vicinity of a crack-tip. The asymptotic fields in the crack-tip vicinity in a FGM with continuously varying material properties are the same as in a homogeneous one. The simplicity of the use of this formula for T-stress evaluation in FEM fracture mechanics analysis is demonstrated by its application to some transient elastodynamic and thermoelastic problems.

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Journal Theoretical and Applied Fracture Mechanics
Sladek, J., Sladek, V., Repka, M., & Tan, C. (2016). Evaluation of the T-stress for cracks in functionally graded materials by the FEM. Theoretical and Applied Fracture Mechanics. doi:10.1016/j.tafmec.2016.09.004