PSI - Issue 41

Victor Rizov et al. / Procedia Structural Integrity 41 (2022) 125–133 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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complicated. In recent decades, the interest to the continuously inhomogeneous structural materials has increased considerably. The reason for this is the fast development of the functionally graded materials since their introduction in 1980-s (Chatzigeorgiou and Charalambakis (2005), Chen and Lin (2008), Chikh (2019), Ganapathi (2007), Han et al. (2001)). The functionally graded materials are advanced continuously inhomogeneous composites manufactured by mixing of two or more constituent materials (Hao et al. (2002), 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)). The microstructure of functionally graded materials is formed technologically so as to satisfy specific operational requirements (Najafizadeh and Eslami (2002), Reichardt et al. (2020), Saidi and Sahla (2019)). One of the factors which have strong influence on structural integrity and the operational reliability and durability of continuously inhomogeneous structural members and components is the fracture behaviour (Rizov (2017), Ruzov (2018), Rizov (2019), Rizov and Altenbach (2020)). Therefore, the profound knowledge in the field of fracture mechanics of continuously inhomogeneous materials is of paramount importance to the practising engineers. The objective of this paper is to derive a solution of the problem of the strain energy release rate for a lengthwise crack in a statically undetermined continuously inhomogeneous beam structure of circular cross-section exhibiting non-linear viscoelastic behaviour under torsion. The emphasis of the paper is on the non-linear viscoelastic behaviour and its influence on the fracture because the continuously inhomogeneous structural members and components usually have non-linear viscoelastic behaviour. It should also be noted that the previous publications deal with lengthwise fracture in beams which have linear viscoelastic behaviour (Rizov (2020), Rizov (2021)). This paper represents a further development of the investigations described in (Rizov (2020), Rizov (2021)) by taking into account the non-linear viscoelastic behaviour. A mechanical model consisting of non-linear springs and non-linear dashpots is used for treating the viscoelastic behaviour of beam. 2. Analysis of lengthwise fracture Consider the viscoelastic model depicted in Fig. 1.

Fig. 1. Non-linear viscoelastic model. The model consists of n non-linear springs and n non-linear dashpots. The non-linear stress-strain relationship of the i -th spring is expressed as i i b i G G    , (1)