PSI - Issue 51

Victor Rizov et al. / Procedia Structural Integrity 51 (2023) 44–50 V. Rizov / Structural Integrity Procedia 00 (2022) 000–000

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et al. (2012)). The functionally graded materials (FGM) are relatively new (first they have been used in 1980-s) inhomogeneous composites with two or more constituents. The spatial variation of the properties of FGM in the solid can be controlled to meet the performance requirements (Mahamood and Akinlabi (2017), Saidi and Sahla (2019), Saiyathibrahim et al. (2016)). The design of inhomogeneous (functionally graded) load-bearing structural members and components requires an understanding of their fracture behaviour under various loading conditions. However, fracture analyses are complicated since the material properties vary continuously (Erdogan (1995)). Here the fracture in an inhomogeneous non-linear elastic beam with three parallel lengthwise cracks axially loaded in tension is analyzed. The paper is motivated by the fact that beams subjected to axial loading are widely used in engineering structures. Besides, FGM can be built-up layer by layer which favours lengthwise fracture. It should be noted here that fracture of inhomogeneous beams with one lengthwise crack has already been analyzed in the previous papers (Rizov (2017), Rizov and Altenbach (2020)). However, the fact that the layered inhomogeneous beams are very weak in the transverse direction creates conditions for rise of more than one parallel lengthwise cracks. Thus, the present paper attempts to address this problem by analyzing a beam with three cracks under axial loading. 2. Theoretical model The inhomogeneous beam configuration with three parallel lengthwise cracks depicted in Fig. 1 is studied in the present paper.

Fig. 1. Beam with three lengthwise cracks under axial loading.

The width, height and span of the beam are h , b and l 2 . Two axial forces, F , are applied at the beam (Fig. 1). The material of the beam is continuously inhomogeneous in the height direction. The three lengthwise cracks are symmetric. The lengths of crack 1, 2 and 3 are 1 2 a , 2 2 a and 3 2 a , respectively. There is a vertical notch of depth 3 1 2 h h h   in the beam (Fig. 1). The notch divides each of the three cracks in two parts which are symmetric. Only half of beam, l l x 2 1   , is analyzed (the 1 x -axis is defined in Fig. 1). In portion, 1 2 S S , there are four crack arms. The heights of crack arms 1, 2, 3 and 4 are 1 h , 2 h , 3 h and 4 h , respectively. It should be noted that due to