Three-dimensional vibrations of cracked rectangular parallelepipeds of functionally graded material

A novel examination of the three-dimensional (3-D) vibrations of rectangular parallelepipeds of functionally graded material (FGM) having side cracks is summarized. Employing 3-D theory of elasticity and a variational Ritz methodology, new hybrid series of mathematically complete orthogonal polynomials and crack functions as the assumed displacement fields are proposed to enhance the convergence modeling of the stress singular behavior of the crack terminus edge front in a rectangular FGM parallelepiped. The proposed admissible hybrid series properly describe the ϑ(1/r)3-D stress singularities at the terminus edge front of the crack, allowing for displacement discontinuities across the crack sufficient to explain the most general 3-D “mixed modes” of local crack-edge deformation and stress fields typically seen in fracture mechanics. The correctness and validity of the vibration analysis are confirmed through comprehensive convergence studies and comparisons with published results for cracked rectangular FGM parallelepipeds modeled as homogeneous rectangular plates with side cracks and FGM rectangular plates with no cracks based on various plate theories. Two types of FGM parallelepipeds, Al/Al2O3 and Al/ZrO2, are included in the study. The locally effective material properties are estimated by a simple power law and the effects of the volume fraction on the frequencies are investigated. For the first time in the published literature, this work reports frequency data and nodal patterns for FGM rectangular parallelepipeds modeled as moderately thick plates with several combinations of hinged, clamped, and completely free kinematic and stress conditions along the four side faces, and having side cracks with varying crack size effects implying flaw-size influence in FGM parallelepiped vibration and fracture, including crack length ratios (d/a and d/b), crack positions (cx/a and cy/b), and crack inclination angles (α).

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