PSI - Issue 33

Victor Rizov et al. / Procedia Structural Integrity 33 (2021) 402–415 Author name / Structural Integrity Procedia 00 (2019) 000–000

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1. Introduction The use of functionally graded materials in various load-bearing structural applications has been increased in the recent decades. The properties of these materials vary spatially in the solid. The functionally graded materials are very attractive alternative to the conventional homogeneous structural materials such as metals. By using functionally graded materials, one can achieve high performance and efficiency of structures. The functionally graded materials open excellent possibilities for reducing the weight of structures without compromising their strength, stiffness and stability (Andrew et al. (2007), Bao-Lin Wang et al. (2004), Chikh (2019), Ganapathi (2007), Hao et al. (2002), Hirai and Chen (1999), Kou et al. (2012), Mahamood, and Akinlabi, (2017), Marae Djouda et al. (2019), Mehrali et al. (2013), Nagaral et al. (2019)). The functionally graded materials are novel inhomogeneous composites have two or more constituent materials. Graded distribution of the material properties of functionally graded materials is formed technologically during the manufacturing process so as to satisfy different performance requirements in different parts of a structural member. In recent years, the international academic circles have paid a significant attention to the development of the functionally graded materials and structures (Reddy and Chin (1998), Saidi and Sahla (2019), Saiyathibrahim et al. (2016), Shrikantha and Gangadharan (2014)). Multilayered functionally graded materials are made of adhesively bonded layers. The material of each layer is functionally graded in the thickness direction. Due to their high strength-to-weight and stiffness-to-weight ratios, the multilayered functionally graded materials are excellent candidates for light-weight load-bearing structural applications in various areas of the practical engineering. The basic drawback of multilayered materials and structures is the delamnation fracture. The delamination, or separation of layers, sharply deteriorates the operational performance of multilayered structural members and components. The delamination behaviour is of primary importance for the structural integrity and reliability of multilayered structures. It should be mentioned that the delamination analyses of multilayered functionally graded beams usually are concerned mainly with instantaneous reaction of the structure (Rizov (2017), Rizov (2018), Rizov (2019), Rizov (2020), Rizov and Altenbach (2020)). The time-dependent effects of the viscoelastic behaviour of material on the delamination are relatively less investigated (Rizov (2020)). However, the multilayered functionally graded structural members and components sometimes exhibit viscoelastic behaviour that has to be considered in the delamination studies. Therefore, the purpose of the present paper is to develop a strain energy release rate analysis for a delamination crack in a multilayered functionally graded viscoelastic beam that is made of vertical layers. The viscoelastic behaviour is treated by using a linear viscoelastic model consisting of two springs and a dashpot. The modulii of elasticity and the coefficient of viscosity are distributed continuously in the width direction of each layer. A time dependent solution to the strain energy release rate is obtained by analyzing the strain energy cumulated in the beam. The solution derived is verified by applying the compliance method. The variation of the strain energy release rate with the time due to viscoelastic behaviour of the multilayered material is studied. 2. Solution of the strain energy release rate A multilayered functionally graded cantilever beam configuration is depicted in Fig. 1. The beam is made of adhesively bonded vertical longitudinal layers. The number of layers is arbitrary. Each layer has individual width and material properties. The beam is clamped in section, B . The length of the beam is denoted by l . The beam has rectangular cross-section of width, b , and thickness, h . A delamination crack of length, a , is located arbitrary between layers. The widths of the left-hand and right-hand delamination crack arms are denoted by 1 b and 2 b , respectively. The right-hand delamination crack arm is loaded in pure bending so as the angle of rotation,  , of free end of this crack arm increases with the time, t , at constant speed,  v , i.e. v t    . (1) The left-hand delamination crack arm is free of stresses. The present analysis is carried-out assuming validity of the Bernoulli’s hypothesis for plane sections since the beam under consideration has a high length to thickness ratio. The longitudinal displacements, u , at the free end of