# Determining the change of stress concentration with time in a 3-D viscoelastic transverse isotropic plate

## Keywords:

stress concentration, viscoelastic solid, transverse isotropic solid, incremental viscoelastic formulation, finite element method.

## Abstract

The 3-D problem of linear viscoelasticity for a transversely isotropic structural member (a 3-D plate with a circular hole) is solved. The constitutive equations in the Boltzmann-Volterra integral form were used. The integrals in the constitutive equations are converted to the incremental form on the time grid. At each time interval, the problem is solved with respect to displacement increments. The relaxation functions of viscoelastic transversally isotropic material modules are described in exponential form. Analytical expressions for the constitutive matrix of the finite element method were constructed for these modules using the principle of elastic–viscoelastic analogy. The changes of stresses in the plane of the plate and transverse stresses with time on the concentration line are illustrated. Numerical examples are constructed for the middle of the stress concentration line and its ends.

## References

Zobeiry, N. and all (2016). A differential approach to finite element modelling of isotropic and transversely isotropic viscoelastic materials. J. Mechanics of Materials, 97, pp. 76-91. https://doi.org/10.1016/j.mechmat.2016.02.013

Zocher, M. A., Groves, S. E. & Allen, D.H. (1997). A three dimensional finite element formulation for ther moviscoelastic orthotropic media. Int. J. Numer. Meth. Engng., 40, No. 12, pp. 2267-2288. https://doi.org/10.1002/(SICI)1097-0207(19970630)40:12<2267::AID-NME156>3.0.CO;2-P

Chazal, C. & Pitti, R. M. (2011). Incremental constitutive formulation for time dependent materials: creep integral approach. J. Mech. Time-Dep. Mater., 15, pp. 239-253. https://doi.org/10.1007/s11043-011-9135-z

Selivanov, M., Kulbachnyy, Y. & Onishchenko, D. (2020). Determining the change of stress concentration with time in a viscoelastic orthotropic solid. Dopov. Nac. akad. nauk Ukr., 10, pp. 28-34 (in Ukrainian). https://doi.org/10.15407/dopovidi2020.10.028

09.03.2023

## How to Cite

Selivanov М. ., & Fernati П. . (2023). Determining the change of stress concentration with time in a 3-D viscoelastic transverse isotropic plate. Reports of the National Academy of Sciences of Ukraine, (1), 33–39. https://doi.org/10.15407/dopovidi2023.01.033

Mechanics