PSI - Issue 82

Victor Rizov et al. / Procedia Structural Integrity 82 (2026) 246–252 V. Rizov/ Structural Integrity Procedia 00 (2026) 000–000

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2

is of great importance in the context of safety design of a wide class of constructions and mechanisms in different spheres of engineering. Application of multilayered and functionally graded materials for manufacturing of components of various engineering structures including beams under torsion has become a power means for perfecting of structures in the last decades (Araki et al. (1992), Dolgov (2005), Dolgov (2016), Kaul (2014), Toudehdehghan et al. (2017), Gandra et al. (2011)). The reason for continuously increasing use of multilayered and functionally graded materials in load bearing structures is the fact that these modern materials have superior properties in comparison with the classical homogeneous structural materials (Lloyd and Molina-Aldareguia (2003), Mahamood and Akinlabi (2017), Nikbakht et al. (2019), Nagaral et al. (2019), Radhika et al. (2020), Riov (2018), Gururaja Udupa et al. (2014), Fanani et al. (2021)). One of the problems which has significant impact on the integrity and reliability of structural members with viscoelastic behavior is the energy dissipation (Narisawa (1987)). In view of this fact, the aim of the present paper is to develop analysis of the energy dissipation in multilayered functionally graded beam member of circular cross section under torsion (the loading of the beam is applied so that the angle of twist of the beam free end increases linearly with time). The beam under consideration exhibits linear viscoelastic behavior. One of the novel aspect of the present paper is that the coefficient of viscosity of the viscous component of the rheological model used for describing the mechanical behavior of the beam under torsion increases with time in contrast to prior publications which deal with energy dissipation analysis assuming that the coefficient of viscosity does not change with time (Narisawa (1987), Rizov (2021)). Besides, the prior publications consider energy dissipations in beam structures under bending (Narisawa (1987), Rizov (2021)). It should also be specified that there are previous studies analyzing continuously inhomogeneous viscoelastic beam structures whit coefficient of viscosity changing with time (Rizov (2020)). However, the analysis presented in (Rizov (2020)) is concerned with longitudinal fracture problem (the energy dissipation is not analyzed in (Rizov (2020))). 2. Deriving of dissipated energy The coefficient of viscosity of the viscous component and the shear modulus of the linear-elastic component (the spring) of the Maxwell rheological model are marked by and , respectively (Fig. 1).

! "

! !

Fig. 1. Rheological model.

! !

!

The dependency of

on time, , is expressed as

"! # # BA = ! ! " ! " #

,

(1)

! "

! !

where and are material properties ( regulates the variation of the material characteristic, , with time). The external influence on the model represents shear strain, , that varies with time according to the linear law , (2) where is the strain velocity. In order to derive the stress-strain-time relationship of the model, the following dependencies are written: , (3) ! ! " ! # # ! ! = ! " !

! ! ! ! ! " + = " !