PSI - Issue 47

Victor Rizov / Procedia Structural Integrity 47 (2023) 3–12 Author name / Structural Integrity Procedia 00 (2019) 000–000

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manufacturing to meet exactly the performance requirements by providing a unique combination of properties (Nagaral et al. (2019), Saidi and Sahla (2019), Saiyathibrahim et al. (2016), Yan et al. (2020), Zhang et al. (2019)). Since certain kinds of inhomogeneous materials such as the functionally graded materials can be built-up layer by layer (Mahamood and Akinlabi (2017)), a high risk of appearance of parallel longitudinal cracks between layers exists. Various aspects of longitudinal fracture in continuously inhomogeneous structural components have been studied recently in (Rizov (2017), Ruzov (2018), Rizov (2019), Rizov and Altenbach (2020), Rizov (2020), Rizov (2020)). One of the important factors that should be taken into account when analyzing longitudinal fracture of beam structures made of continuously inhomogeneous materials is the non-linear viscoelastic behaviour. However, the publications on longitudinal fracture of continuously inhomogeneous beams with parallel longitudinal cracks usually deal with beams which exhibit linear viscoelastic behaviour (Rizov (2020)). Therefore, the objective of the present paper is to analyze an inhomogeneous non-linear viscoelastic cantilever beam of circular cross-section with three longitudinal circular cylindrical cracks. 2. Theoretical investigation An inhomogeneous non-linear viscoelastic beam with three longitudinal circular cylindrical cracks is shown in Fig. 1.

Fig. 1. Geometry and loading of an inhomogenous beam with three longitudinal circular cylindrical cracks.

The beam has a circular cross-section of radius, 4 R . The length of the beam is l . The beam is clamped in its right hand end. The loading consists of an axial force, F , and a bending moment, M , applied at the free end of the beam. The cracks 1, 2 and 3 are circular cylindrical surfaces of radiuses, 1 R , 2 R and 3 R , respectively. The lengths of cracks 1, 2 and 3 are 1 a , 2 a and 3 a , respectively. In portion, 1 2 DD , the beam is divided by the three cracks in four concentric crack arms. Crack arm 1 has a circular cross-section of radius, 1 R . Crack arm 2 has a ring-shaped cross-section of internal and external radiuses, 1 R and 2 R , respectively. Crack arm 3 has a ring-shaped cross section of internal and external radiuses, 2 R and 3 R , respectively. Crack arm 4 has a ring-shaped cross-section of internal and external radiuses, 3 R and 4 R , respectively. The beam exhibits continuous material inhomogeneity in radial direction. The material of the beam has non-linear viscoelastic behaviour that is treated by using the following stress-strain relation (Kishkilov and Apostolov (1994)):