PSI - Issue 26

Victor Rizov et al. / Procedia Structural Integrity 26 (2020) 63–74 Rizov / Structural Integrity Procedia 00 (2019) 000 – 000

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increasing application of inhomogeneous structures, especially in aggressive environment, is facilitated by a superior performance in comparison with the conventional homogeneous metal structures. It should be mentioned that the growing interest towards the inhomogeneous materials and structures is due mainly to the increased use of certain kinds of inhomogeneous materials, such as functionally graded materials in various applications in practical engineering. Functionally graded materials are inhomogeneous composites introduced in Japan in 1980-s. They are made of two or more constituent materials. The composition and microstructure of functionally graded materials are varied continuously during the manufacturing process. Thus, a graded continuous distribution of material properties in one or more spatial directions is formed. In this way, requirements of different material properties in different parts of a structural member can be satisfied (Bohidar et al. (2014), Gasik (2010), Hirai and Chen (1999), Kawasaki and Watanabe (1997), Kou et al. (2012), Levashov et al. (2002), Nemat-Allal et al. (2011), Neubrand and Rödel (1997), Saiyathibrahim et al. (2016), Shrikantha and Gangadharan (2014) ). Recently, due to significant advantages in material properties and functional characteristics, functionally graded materials have become inreplaceable in aeronautics, nuclear reactors, biomedicine, chemical and mechanical engineering especially when non-uniformly distributed external influences (thermal fields, mechanical loading and chemical agents) are present. The strength, stability, load-bearing capacity and proper functioning of inhomogeneous structural members and components depend to a large degree on fracture behaviour. Therefore, development of methods for analyzing the fracture behaviour of inhomogeneous materials and structures is very important for the practical engineering. It should be noted that the fact that material properties are functions of coordinates imposes a great difficulty in analyzing fracture of inhomogeneous materials. Nevertheless, problems of fracture of inhomogeneous materials and structures have received worldwide attention (Erdogan (1995), Wang and Noda (2001)). Problems of longitudinal fracture behaviour of inhomogeneous beam structures are particularly important since certain kinds 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. Thus, several works on longitudinal fracture of inhomogeneous (functionally graded) beams have been published recently (Rizov (2017), Rizov (2018), Rizov (2019)). In these works, the longitudinal fracture has been studied in terms of the strain energy release rate. However, these works have been concerned with analyses of individual beam configurations of constant sizes of the cross section along the beam length (Rizov (2017), Rizov (2018), Rizov (2019)). Therefore, the aim of the present work is to develop a general approach for analyzing the strain energy release rate for a longitudinal crack in inhomogeneous beams of circular cross-section assuming continuous variation of the radius of the cross-section along the beam length. The longitudinal crack presents a circular cylindrical surface. The beams are loaded in tension by axial forces. The beams exhibit continuous (smooth) material inhomogeneity in radial direction. The material has non-linear elastic mechanical behaviour. The general approach is applied to analyze the strain energy release rate for a longitudinal crack in a cantilever beam under axial forces. The strain energy release rate is derived also by considering the complementary strain energy for verification. 2. Analysis of the strain energy release rate An inhomogeneous rectilinear beam of a circular cross-section is shown in Fig. 1. The radius, R , of the cross section varies continuously along the beam length from 1 R at the left-hand end of the beam to 2 R at the right-hand end of the beam. Thus, R is a continuous function of x = ( ) , (1) where x is the longitudinal centroidal axis. A longitudinal crack presenting a circular cylindrical surface of radius, R 3 , is located in the beam (Fig. 1). The crack length is α . The beam length is l . The internal crack arm is treated as a beam of circular cross-section of radius, R 3 , and length, α . The external crack arm is treated as a beam of ring-shaped cross-section of internal and external radiuses, R 3 and R , and length, α . The beam is loaded in tension by an arbitrary number of axial forces, F i . The internal crack arm is loaded in tension by an axial force, F b , as shown in Fig. 1. Under these axial forces, the beam is in state of equilibrium. The beam exhibits continuous (smooth) material inhomogeneity in radial direction. Besides, the material has non-linear elastic mechanical behaviour.