PSI - Issue 28

Victor Rizov et al. / Procedia Structural Integrity 28 (2020) 1237–1248 Author name / Structural Integrity Procedia 00 (2019) 000–000

1238

2

properties, loading conditions, crack location and others on the fracture behaviour is necessary for the fracture mechanics based safety design of inhomogeneous structural members and components. From view point of practical engineering, study of inhomogeneous materials is important mainly because functionally graded materials, which are a kind of inhomogeneous materials, are widely used in various applications in aerospace industry, nuclear power plants, car industry, microelectronics, biomedical implants and optics. The functionally graded materials are made usually of a mixture of two constituent materials. The growing interest towards the functionally graded materials is due mostly to the fact that their microstructure and properties vary continuously along one or more spatial coordinates (Bohidar et al. (2014), 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), Neubrand and Rödel (1997), Nagaral et al. (2019), Saidi and Sahla (2019), Saiyathibrahim et al. (2016), Shrikantha and Gangadharan (2014)). The gradual change of microstructure of functionally graded materials is designed so that the structural members and components made of these inhomogeneous materials can satisfy specific operational requirements. Many researchers have analyzed fracture behaviour of inhomogeneous structural members and components (Erdogan (1995), Rousseau and Tippur (2001), Tilbrook et al. (2005), Wang and Noda (2001), Yang et al. (2008)). These analyses, however, are concerned mainly with transversal cracks in inhomogeneous structural members which exhibit linear-elastic behaviour. Recently, several works dealing with longitudinal fracture in inhomogeneous (functionally graded) beams exhibiting non-linear mechanical behaviour of material have been published (Rizov (2017), Rizov (2018), Rizov (2019)). Analyses of longitudinal fracture behaviour are highly needed since 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. The longitudinal fracture leads to considerable reduction of strength and stiffness and complicates the buckling behaviour. The analyses reported in (Rizov (2017), Rizov (2018), Rizov (2019)) are focussed on longitudinal fracture in beams of rectangular cross-section loaded mainly in bending. Various solutions for the strain energy release rate are derived for individual beam configurations. The aim of the present paper is to develop a general solution procedure to the strain energy release rate for longitudinal cylindrical cracks in inhomogeneous non-linear elastic round bars. The Ramberg-Osgood constitutive law is used for treating the material non-linearity assuming continuous variation of the modulus of elasticity in radial direction. The bars are loaded in tension. The solution procedure is applicable for longitudinal cylindrical cracks located arbitrary in radial direction of bar cross-section. The solution is used for analyzing the longitudinal fracture behaviour of a cantilever. It should be noted that beside for non-linear elastic materials, the general solution procedure developed in the present paper can be applied also for elastic-plastic materials if the external loading increases only, i.e. if the bar undergoes active deformation. 2. General solution procedure A round bar portion containing the front of a longitudinal crack is depicted in Fig. 1. The bar cross-section is a circle of radius, 2 R . The longitudinal crack is a cylindrical surface of radius, 1 R . Thus, the crack front is a circle of radius, 1 R . The internal crack arm is a round bar of radius, 1 R . The external crack arm is a ring-shaped bar of internal and external radiuses, 1 R and 2 R , respectively. The axial force in the bar cross-section ahead of the crack front is denoted by N . The strain energy release rate, G , for the cylindrical longitudinal crack in Fig. 1 is written as (Rizov (2017), Rizov (2018))

 u RdR R R 2 1 * 01

 R 2

    R 1 0

1

   ,

u RdR *

u RdR * 01

G 





(1)

0

R

1

0

where *

* 02 u and * 0 u are the complementary strain energy densities in the cross-sections of internal and external

01 u ,

crack arms behind the crack front and in the bar cross-section ahead of the crack front, respectively. The non-linear mechanical behaviour of the material is treated by the Ramberg-Osgood constitutive law