PSI - Issue 26

Victor Rizov et al. / Procedia Structural Integrity 26 (2020) 75–85 Rizov / Structural Integrity Procedia 00 (2019) 000 – 000

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safety. The inhomogeneous materials are ideal also for structural applications in aggressive environment. Functionally graded materials are typical example of novel inhomogeneous composite materials which are manufactured by continuously mixing of two or more constituent materials (Gasik (2010), Hirai and Chen (1999), Knoppers et al. (2003), Kou et al. (2012), Mao et al. (2014), Le (2017), Saiyathibrahim et al. (2016), Shrikantha and Gangadharan (2014), Uslu Uysal and Kremzer (2015 ), Usly Uysal (2016), Uslu Uysal and Güven (2016), Wu et al. (2014), Zhang et al. (2010)). As a matter of fact, the increased interest towards the inhomogeneous materials in the recent years around the globe is due mainly to the widespread applications of functionally graded materials in aerospace, aeronautics, microelectronics, robotics, optics and biomedicine. The most important advantage of functionally graded materials in comparison with the traditional homogeneous structural materials is the fact that the properties of functionally graded materials can be formed technologically during the manufacturing process so as to satisfy specific performance requirements. Smooth spatial variation of microstructure and properties of functionally graded materials can be obtained by gradually varying the composition of constituent materials along one or more spatial coordinates. By tailoring of the spatial variation of the material properties during manufacturing, the performance of functionally graded structural members and components can be modified to meet different exploitation requirements in different parts of the solid and to improve the operational capacity, durability and reliability of the structure. The problems associated with fracture behaviour of inhomogeneous materials especially in the context of their load-bearing structural applications have been of great concern. Since certain kinds of inhomogeneous materials such as functionally graded materials can be built-up layer by layer (Mahamood and Akinlabi (2017)) there is a high risk of appearance of longitudinal cracks between layers. The longitudinal cracks reduce significantly the load-bearing capacity and even can lead to catastrophic failure of the entire structure. Therefore, developing of longitudinal fracture analyses of different inhomogeneous structural members and components under various externally applied loadings is of great interest for both academicians and practitioners. The present paper is concerned with a longitudinal fracture analysis of an inhomogeneous non-linear elastic cantilever beam configuration loaded by axial force and bending moment. In contrast to previous papers which consider longitudinal cracks located in the end of the beam (Rizov (2017), Rizov (2018), Rizov (2019)), the present paper analyzes an internal longitudinal crack. Since the crack is internal, the bending moments and axial forces in the two crack arms which are needed in order to derive the strain energy release rate can not be obtained directly. In the present paper, an approach that treats the beam as statically undetermined structure with two internal redundants is developed. By using the axial forces and bending moments obtained by resolving the static indeterminacy, a solution to the strain energy release rate is derived. The J -integral approach is applied for verification of the solution.

2. Theoretical model

An inhomogeneous beam configuration with an internal longitudinal crack is shown in Fig. 1.

Fig. 1. Geometry and loading of inhomogeneous beam with internal longitudinal crack.

The beam has a rectangular cross-section of width, b , and height, h . The length of the beam is l . The beam is clamped in its right-hand end. The external loading consists of one bending moment, M , and one axial force, N , applied at the left-hand end of the beam. An internal longitudinal crack of length, a , is located in the beam as shown