PSI - Issue 77

Victor Rizov et al. / Procedia Structural Integrity 77 (2026) 382–388 Author name / Structural Integrity Procedia 00 (2026) 000–000

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mainly by excellent properties of these relatively new composite materials with graded microstructure (Mahamood and Akinlabi (2017), Nagaral et al. (2019), Reichardt et al. (2020), Shrikantha and Gangadharan (2014)). For instance, the continuous distribution of properties of these materials in a structural member can be designed in order to satisfy specific requirements with respect to strength, stiffness, stability, fracture behavior, etc. (Najafizadeh and Eslami (2020), Rizov (2020), Rizov and Altenbach (2020), Saidi and Sahla (2019), Saiyathibrahim et al. (2016)). This fact makes the functionally graded materials a powerful means for enhancement of load-bearing capacity and improvement of performance of structures and machines. The present paper explores theoretically how the longitudinal fracture in beam-type functionally graded engineering constructions is biased by the time factor by using a non-linear viscoelastic model. Two parameters of the model (the modulus of elasticity and the coefficient of viscosity of the non-linear spring and dashpot) are continuously varying with time. Actually, the basic motive for the present study consists in the fact that previous longitudinal fracture studies dealing with time-dependent modulus of elasticity (Rizov (2022)) or coefficient of viscosity (Rizov (2020)) use linear viscoelastic mechanical models. The time-dependent strain energy release rate is derived and analyzed in the present paper. It is found the strain energy release rate – time curve has rising mode. Another finding is that the strain energy release rate reduces when the values of the parameters used for treating the dependency of the modulus of elasticity and the coefficient of viscosity on time rise. 2. Theoretical model We are focused on a functionally graded beam construction of length, l , and thickness, h , as displayed in Fig. 1.

F ig. 8. Geometry and loading of functionally graded beam. The beam has non-linear viscoelastic mechanical behavior. A bending moment, M , acts in section, 3 Q , of the beam.

Fig. 2. Viscoelastic mechanical model.