PSI - Issue 41

Victor Rizov et al. / Procedia Structural Integrity 41 (2022) 103–114 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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1. Introduction The material properties of continuously inhomogeneous structural materials vary smoothly along different directions in the solid. Thus, the properties of these structural materials represent smooth functions of one or more coordinates. Especially in recent decades, when practical applications of functionally graded materials (the latter are a class of continuously inhomogeneous materials) have been significantly widened, an extensive literature has been assembled to elucidate the various aspects of the mechanical behaviour of these advanced engineering materials (By Xiao Kuang et al. (2019), Chikh (2019), Hirai and Chen (1999)). In fact, a functionally graded material is a continuously inhomogeneous continuum that has two or more constituent materials. During manufacturing of the functionally graded material, its constituent materials are continuously mixed in order to ensure a smooth change of the material properties along definite directions in the solid and to eliminate sharp boundaries and interfaces between the constituent materials (this is very important for guaranteeing the integrity and continuity of the functionally graded material) (Kou et al. (2012), Mahamood and Akinlabi (2017), Marae Djouda et al. (2019), Mehrali et al. (2013)). The quickly developing technologies for manufacturing of functionally graded materials give very good opportunities to tailor the material properties and their smooth distribution in the structural member (Nagaral et al. (2019), Saidi and Sahla (2019), Saiyathibrahim et al. (2016), Shrikantha and Gangadharan (2014), Zhou and Yan (2002)). This is one of the main reasons for the great popularity of the continuously inhomogeneous (functionally graded) structural materials in such highly responsible engineering areas like aeronautics, nuclear reactors and others. One of the important types of failure of various continuously inhomogeneous structural components and details is the longitudinal fracture. This is due to the fact that very often graded materials are built-up layer by layer (Mahamood and Akinlabi (2017)). As a result of this, the transversal strength in tension of graded materials is relatively low which makes possible longitudinal fracture. There are many engineering applications in which continuously inhomogeneous beam structures of viscoelastic behaviour are subjected to both mechanical loading and periodically varying temperature. Therefore, the main aim of this paper is to analyze the longitudinal fracture of a continuously inhomogeneous viscoelastic beam under mechanical loading and periodically varying temperature. Such analysis is needed since the previous publications do not consider the effect of periodically varying temperature on longitudinal fracture of continuously inhomogeneous beam structures (Rizov (2017), Ruzov (2018), Rizov (2019), Rizov and Altenbach (2020), Rizov (2020), Rizov (2020)). The beam studied in this paper is continuously inhomogeneous in both transversal and longitudinal directions. The external loading applied on the beam is a bending moment that changes continuously with time. A solution of the strain energy release rate is found by analyzing the beam compliance. The solution is verified by considering the strain energy in the beam. A particular effort is made to elucidate the effects of the periodically varying temperature on the strain energy release rate. 2. Analysis of longitudinal fracture under periodically varying temperature The viscoelastic beam configuration in Fig. 1 exhibits continuous material inhomogeneity in both transversal and longitudinal directions. The width, thickness and length of the beam are b , h and l , respectively. The beam is simply supported. A longitudinal crack of length, a , is situated in the beam as depicted in Fig.1. The thicknesses of the upper and lower crack arm are 1 h and 2 h , respectively. The external mechanical loading consists of a time dependent bending moment, M , applied on the free-end of the upper crack arm. The variation of this bending moment with time, t , is expressed as t M M   0  , (1) where 0 M is the value of the bending moment at 0  t . The parameters,  and  , control the variation of M . The viscoelastic mechanical model in Fig. 2 is used to describe the viscoelastic behaviour of the beam. The modulus of elasticity of the spring is E . The coefficients of viscosity of the two dashpots of the model are denoted by 1  and 2  (Fig. 2). The model is under stress,  , that varies with time according to the following law: t     0  , (2)