PSI - Issue 28

Victor Rizov et al. / Procedia Structural Integrity 28 (2020) 1226–1236 Author name / Structural Integrity Procedia 00 (2019) 000–000

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That is why such structural materials are widely used in practical engineering. As a typical example, we can mention the functionally graded materials which are composed of two or more constituent materials (Chikh (2019), Gasik (2010), Hirai and Chen (1999), Kawasaki and Watanabe (1997), Kou et al. (2012), Levashov et al. (2002), Marae Djouda at al. (2019), Nemat-Allal et al. (2011), Nagaral et al. (2019), Saidi and Sahla (2019), Saiyathibrahim et al. (2016), Shrikantha and Gangadharan (2014)). The properties of functionally graded materials can be formed technologically during the manufacturing by gradually changing their composition and microstructure. In this way, smooth variation of the properties of the functionally graded materials along one or more spatial directions can be obtained. By allowing continuous variation of microstructure in the solid, the material properties can be modified to meet different performance requirements in different parts of a structural member. The advancement of methodology for fracture analysis of inhomogeneous structural members and components is very important for adequate assessment of operational performance and integrity of load-bearing structures. Thus, fracture analyses of inhomogeneous materials present a great deal of interest for both academicians and practising engineers. However, the material inhomogeneity brings significant difficulties into mathematical treatment of fracture problems. For example, the fact that the material properties are functions of coordinates complicates the fracture analyses of inhomogeneous materials and structures (Erdogan (1995), Tilbrook et al. (2005)). Investigations of fracture behaviour of inhomogeneous (functionally graded) materials by using linear-elastic fracture mechanics have been reviewed in (Tilbrook et al. (2005)). Various publications which deal with different aspects of fracture of inhomogeneous materials and structures have been presented. Analyses of cracks which have different orientation with respect to the gradient direction have been reviewed. The effects of compositional and microstructural gradation on the fracture behaviour have been discussed. Besides under static crack loading conditions, studies of the fracture behaviour of inhomogeneous materials under cyclic fatigue crack loading conditions have also been summarized (Tilbrook et al. (2005)). One of the actual problems in the theory of fracture of inhomogeneous materials is the analysis of longitudinal fracture behaviour of inhomogeneous structural members. This is so because a certain kind of inhomogeneous materials such as functionally graded materials can be built-up layer by layer (Mahamood and Akinlabi (2017)) which is a premise for appearance of longitudinal cracks between layers. Therefore, analyzing longitudinal fracture of inhomogeneous structural members under various external loadings is an up-to-date problem. The goal of the present paper is to develop a longitudinal fracture analysis of an inhomogeneous non-linear elastic rod with an internal crack loaded in torsion. For this purpose, a solution to the strain energy release rate is derived. It should be noted that previous papers in this area deal with inhomogeneous rods with a longitudinal crack located in the end of the rod (Rizov (2017), Rizov (2018), Rizov (2019)). When the crack is in the end of the rod, the torsion moments in the two crack arms which are needed to obtain the strain energy release rate are obvious. However, in practice, very often the crack is located inside the rod which brings difficulties in the analysis. These difficulties are resolved by the approach reported in the present paper. 2. Analysis of the strain energy release rate An inhomogeneous non-linear elastic rod with one internal longitudinal crack is shown schematically in Fig. 1. The external loading consists of two torsion moments, T , applied at the ends of the rod. The rod has a circular cross section of radius, 2 R . The length of the rod is l 2 . An internal longitudinal crack of length, a 2 , is located symmetrically with respect to the middle of the rod. The longitudinal crack represents a circular cylindrical surface of radius, 1 R . Thus, the internal crack arm has a circular cross-section of radius, 1 R . The external crack arm has a ring shaped cross-section of internal and external radiuses, 1 R and 2 R , respectively. The rod exhibits continuous material inhomogeneity in radial direction. The longitudinal fracture behaviour is studied in terms of the strain energy release rate, G . Due to the symmetry, only half of the rod, l l x 2   , is analyzed. In order to derive of the strain energy release rate, the complementary strain energy,  U , is differentiated with respect to the crack area