PSI - Issue 28

Victor Rizov et al. / Procedia Structural Integrity 28 (2020) 1212–1225 Author name / Structural Integrity Procedia 00 (2019) 000–000

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Among them, the dependence of material properties on coordinates could be particularly mentioned. The accuracy of the analysis of inhomogeneous materials and structures is influenced substantially by the adequate treatment of the effects of the dependence of material properties on coordinates. It should be mentioned that in the recent decades a great interest has been shown towards the functionally graded materials which are continuously inhomogeneous composite materials composed by two or more phases (Butcher et al. (1999), 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 material property graded distribution of functionally graded materials can be formed technologically by composition of different phases in different ratios during manufacturing. In this way the requirements for specific material properties in different parts of a structural member can be satisfied. Therefore, it is not surprising that the functionally graded materials have gained significant attention as modern structural materials in many engineering applications in aeronautics, nuclear power stations, robotics, biomedicine and others. The successful use and application of continuously inhomogeneous (functionally graded) materials especially in load-bearing structures and components requires a thorough studying of their fracture behaviour under various loading conditions (Erdogan (1995), Tilbrook et al. (2005)). The importance of analyzing the fracture of inhomogeneous (functionally graded) materials has been discussed in (Erdogan (1995)). The applicability of the methods of linear-elastic fracture mechanics when investigating the fracture behaviour of functionally graded materials has been substantiated. Some fundamental problems of fracture mechanics of functionally graded materials have been formulated and considered. Significant attention has been given to analyzing the debonding of functionally graded coatings. Surface fracture behaviour of functionally graded materials has also been studied. Works on fracture in composite materials with continuously graded material composition have been reviewed in (Tilbrook et al. (2005)). Fracture behaviour under cyclic fatigue crack loading conditions has also been discussed. Various analyses of cracks oriented parallel or perpendicular to the direction of the material gradient have been summarized. Different aspects of failure resistance behaviour of graded composites have been studied too. Besides rectilinear cracks, works on arc cracks and slightly curved cracks in graded materials have also been discussed. Solutions of various crack problems obtained by applying linear-elastic fracture mechanics have been presented. It should be noted that certain kinds of continuously inhomogeneous materials such as functionally graded materials can be manufactured layer-by-layer (Mahamood and Akinlabi (2017)) which is a premise for appearance of longitudinal cracks between layers. Therefore, analyzing longitudinal fracture of continuously inhomogeneous (functionally graded) structural members and components under various loading conditions is an important research topic (Rizov (2017), Rizov (2018), Rizov (2019)). The aim of the present paper is to analyze longitudinal fracture of a continuously inhomogeneous beam of circular cross-section loaded in torsion with considering the effects of the stress relaxation. The radius of the cross-section varies linearly along the beam length. The fracture is studied in terms of the strain energy release rate. The stress relaxation is treated by using two linear viscoelastic models under a constant applied shear strain. The balance of the energy is analyzed to verify the solution of the strain energy release rate. It should be mentioned that the basic novelty in the present paper is the analysis of the effects of stress relaxation in contrast to (Rizov (2017), Rizov (2018), Rizov (2019)) where the longitudinal fracture of inhomogeneous beams loaded in torsion is studied without considering the time-dependent behaviour due to viscoelasticity of the material. Besides, the present paper deals with a beam whose cross-section varies continuously in the length direction while the previous papers (Rizov (2017), Rizov (2018), Rizov (2019)) are focussed on beams of a constant cross-section. 2. Solutions to the strain energy release rate A cantilever beam of circular cross-section is shown schematically in Fig. 1. The beam is clamped in section, B . The radius of the cross-section varies linearly from 1 R at the free end to 2 R at the clamped end of the beam. The length of the beam is denoted by l . A longitudinal crack of length, a , is located in the beam as shown in Fig. 1. The crack is a circular cylindrical surface of radius, 3 R . Thus, the crack front is a circle of radius, 3 R . The internal crack arm is treated in the analysis as a beam of circular cross-section of radius, 3 R , and length, a . The external crack arm