PSI - Issue 41

Victor Rizov et al. / Procedia Structural Integrity 41 (2022) 134–144 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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continuously inhomogeneous materials. Widely used and relatively new continuously inhomogeneous materials are the functionally graded composites (Ahmed Keddouri et al. (2019), Chatzigeorgiou and Charalambakis (2005), Chen and Lin (2008), Chikh (2019), Ganapathi (2007), Gururaja Udupa et al. (2014), Han et al. (2001)). The latter are manufactured by continuously mixing of their constituent materials (two or more). The properties of functionally graded materials change smoothly in one or more spatial coordinates in a structural member. One of the significant advantages of functionally graded materials over the homogeneous materials consists in the fact that the material properties graded distribution can be formed technologically so as to meet specific operational requirements (Hao et al. (2002), Kieback et al. (2003), Kou et al. (2012), Kyung-Su Na and Ji-Hwan Kim (2004), Mahamood, and Akinlabi, (2017), Marae Djouda et al. (2019), Nagaral et al. (2019)). Recently, considerable attention from both engineers and researchers has been paid to functionally graded materials and their application in various areas of modern engineering (Najafizadeh and Eslami (2002), Reichardt et al. (2020), Saidi and Sahla (2019)). One of the weaknesses of the continuously inhomogeneous materials and structures is the risk of longitudinal fracture. This is due to the fact the continuously inhomogeneous (functionally graded) materials can be built-up layer by layer (Mahamood and Akinlabi (2017)). Appearance of longitudinal cracks between layers threatens the integrity, reliability and durability of structures and components. Therefore, analyzing the influence of various factors and parameters on longitudinal fracture in continuously inhomogeneous structural members represents a task of undoubted interest (Rizov (2017), Ruzov (2018), Rizov (2019), Rizov and Altenbach (2020), Rizov and Altenbach (2020)). In many engineering applications, the continuously inhomogeneous structures exhibit non-linear viscoelastic behaviour under external loadings which change smoothly with time. In this paper, an analysis of longitudinal fracture in a non-linear viscoelastic beam structure loaded in pure bending so as the angle of rotation of the free end of the upper crack arm changes smoothly with time is developed in contrast to previous papers which consider linear viscoelastic beams (Rizov (2020)). The beam is continuously inhomogeneous along its thickness. The strain energy release rate is derived. The method of the J -integral is used for verification of the strain energy release rate. The time-dependent mechanical behaviour of the beam is described by a viscoelastic model representing a combination of linear and non-linear springs and dashpots. A parametric study of the strain energy release rate is carried-out. 2. Longitudinal fracture analysis A longitudinal crack with length, a , is located in the non-linear viscoelastic beam structure shown in Fig. 1.

Fig. 1. Non-linear viscoelastic beam structure with a longitudinal crack.

The thicknesses of the lower and upper crack arms are 1 h and 2 h , respectively. The beam is clamped in its right hand end. The width, thickness and length of the beam are b , h and l , respectively. The upper crack arm is loaded in pure bending so as the angle of rotation,  , of the free end of the upper crack arm changes with time, t , according to the following law: