PSI - Issue 33

Victor Rizov et al. / Procedia Structural Integrity 33 (2021) 428–442 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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1. Introduction The application of functionally graded structural materials has increased in the recent decades. The properties of these materials are continuous functions of coordinates. A graded distribution of the material properties of the functionally graded materials is formed by gradually changing the composition of the constituent materials during the manufacturing process. In this way, the properties of the functionally graded materials can be tailored technologically in order to meet the performance requirements (Bao-Lin Wang et al. (2004), Chatzigeorgiou and Charalambakis (2005), Chikh (2019), Ganapathi (2007), Han et al. (2001), 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), Mehrali et al. (2013), Nagaral et al. (2019), Najafizadeh and Eslami (2002)). Therefore, it is not surprising that recently the functionally graded materials have gained considerable attention as very promising structural materials in many applications in smart structures, aerospace and aeronautics, nuclear reactors, electronics and optoelectronics, biomedical implants and others (Reddy and Chin (1998), Reichardt et al. (2020), Saidi and Sahla (2019), Saiyathibrahim et al. (2016)). The use of the functionally graded materials in various load-bearing structures sets high requirements with respect to the fracture behaviour. Appearance of cracks considerably deteriorates the integrity and reliability and reduces the lifetime of the structural members and components made of functionally graded materials. Therefore, fracture analysis of functionally graded materials and structures is an important research topic. A specific problem is the lengthwise fracture. The reason for appearance of lengthwise cracks is the fact that the functionally graded materials can be built-up layer by layer. Therefore, recently, several papers focused on lengthwise fracture analysis of functionally graded beam structures in terms of the strain energy release rate have been published (Rizov (2017), Rizov (2018), Rizov (2019), Rizov and Altenbach (2020)). These analyses (Rizov (2017), Rizov (2018), Rizov (2019), Rizov and Altenbach (2020), Rizov (2020)), however, consider the instantaneous fracture while the effects of the creep and relaxation on the time-dependent lengthwise fracture behaviour have got less attention (Rizov (2020)). That’s why the main goal of the present paper is to derive a time -dependent solution to the strain energy release rate for a lengthwise crack in a functionally graded cantilever beam configuration that exhibits stress relaxation. A linear viscoelastic model is applied for treating the stress relaxation. The modulus of elasticity and the coefficient of viscosity are distributed continuously along the thickness of the beam. The balance of the energy is analyzed in order to obtain a time-dependent solution to the strain energy release rate. The solution is verified by applying the compliance method. The evolution of the strain energy release rate with the time is investigated by using the solution derived. The effects of the crack location along the beam thickness and the material gradient on the time dependent strain energy release rate are studied. 2. Time-dependent solution of the strain energy release rate The functionally graded beam shown in Fig. 1 is under consideration. The beam is clamped in its right-hand end. The length of the beam is l . The beam has a rectangular cross-section of width, b , and thickness, h . The material is functionally graded along the beam thickness. A lengthwise crack of length, a , is located in the beam as shown in Fig. 1. The lower and the upper crack arms have different thicknesses denoted by 1 h and 2 h , respectively. The beam exhibits stress relaxation, i.e. the stresses decrease with the time while the strains do not change. For this purpose, it is assumed that the free end of the lower crack arm is quickly loaded in centric tension and then held. In this way, the strains in the beam remain constant with the time. The horizontal displacement of the lower crack arm free end is F u (Fig. 1). It is obvious that the upper crack arm is free of stresses. The stress relaxation is treated by applying a linear viscoelsatic model that consists of a spring and a dashpot. The model is shown schematically in Fig. 2. The modulus of elasticity of the spring and the coefficient of viscosity of the dashpot are denoted by E and  , respectively. Since the beam is functionally graded in the thickness direction, the distributions of E and  along the thickness of the beam are expressed as