Simulation of the deformation diagram of a viscoelastic material based on a structural model

A serious problem in computer simulation of the stress state of polymer structures is to ensure the adequacy of the mathematical description of the mechanical properties of materials. The structural model of a viscoelastic material has a number of advantages in describing both the rheology of the material and trajectories of the material deformation. In this model, the material is described as a structure consisting of several elements with relatively simple rheological properties. Reproduction of a complex behavior of the material under alternating non-isothermal loading is ensured through the interaction of simple elements. A technique developed for modeling a viscoelastic material is intended for strength calculations of structures made of materials operating under conditions of prolonged repeated thermomechanical exposure using the finite element method. Application of the developed procedure to a polymeric material, polymethyl methacrylate (PMMA), is considered. The results of testing the material under uniaxial compression at a constant temperature are presented. The methodology and results of identification of the developed structural model using a specialized software are described. Formulas for approximation of the deformation characteristics of the material at a constant deformation rate and the time dependence of material deformation during the holding the material at a constant stress level are obtained. Approximation is an important step in identification of the material model which facilitates the systematization of the initial experimental data and their further mathematical processing. The best approximation of the deformation characteristics of a viscoelastic material is given by a hyperbolic tangent function, whereas the logarithmic function provides the best results for deformation upon exposure. Further construction of the structural model was carried out by selection of sequential parameters of bilinear rheological functions of the individual elements the model and iterative refinement of those parameters. The simulation results were compared with the experiments carried out at different strain rates and with exposure at different stress levels. We just present the results of the initial stage of the carried out experimental and theoretical studies.

1. Shaw M. T., MacKnight W. J. Introduction to Polymer Viscoelasticity, 3rd. Edition: Wiley-Interscience, Hoboken 2005. DOI: 10.1002/0471741833

2. Ward I. M., Sweeney J. An Introduction to the Mechanical Properties of Solid Polymers: John Wiley & Sons, 2004. — 394 p.

3. Zarubin V. S., Kuvyrkin G. N. Mathematical models of mechanics and electrodynamics of a continuous medium. — Moscow: Izd. MGTU im. N. É. Baumana, 2008. — 512 p. [in Russian].

4. Rabotnov Yu. N. Creep of structural elements. — Moscow: Nauka, 1966. — 752 p. [in Russian].

5. Gokhfeld D. A., Sadakov O. S. Plasticity and creep of structural elements under repeated loading. — Moscow: Mashinostroenie, 1984. — 256 p. [in Russian].

6. Sadakov O. S. Structural model in the rheology of structures / Vestn. Yuzh.-Ural. gos. univ. Ser. Matem. Fiz. Khimiya. 2003. Issue 4. N 8. P. 88 – 98 [in Russian].

7. Lakes R. S. Viscoelastic Materials. — Cambridge and New York: Cambridge University Press, 2009. DOI: 10.1017/CBO9780511626722

8. Mulliken A. D., Boyce M. C. Mechanics of the rate-dependent elastic-plastic deformation of glassy polymers from low to high strain rates / Int. J. Solids Struct. 2006. Vol. 43. N 5. P. 1331 – 1356. DOI: 10.1016/j.ijsolstr.2005.04.016

9. Muliana A., Rajagopal K. R., Tscharnuter D., Pinter G. A nonlinear viscoelastic constitutive model for polymeric solids based on multiple natural configuration theory / Int. J. Solid Struct. 2016. Vol. 100 – 101. P. 95 – 110. DOI: 10.1016/j.ijsolstr.2016.07.017

10. Praud F., Chatzigeorgiou G., Bikard J., Meraghni F. Phenomenological multi-mechanism constitutive modelling for thermoplastic polymers, implicit implementation and experimental validation / Mech. Mater. 2017. Vol. 114. P. 9 – 29. DOI: 10.1016/j.mechmat.2017.07.001

11. Dupaix R., Boyce M. C. Constitutive modelling of the finite strain behavior of amorphous polymers in and above the glass transition / Mech. Mater. 2007. Vol. 39. P. 39 – 52.

12. Maurel-Pantel A., Baquet E., Bikard J., Bouvard J. L., Billon N. A thermo-mechanical large deformation constitutive model for polymers based on material network description: application to a semi-crystalline polyamide 66 / Int. J. Plast. 2015. Vol. 67. P. 102 – 126. DOI: 10.1016/j.ijplas.2014.10.004

13. Furmanski J., Cady C. M., Brown E. N. Time-temperature equivalence and adiabatic heating at large strains in high density polyethylene and ultrahigh molecular weight polyethylene / Polymer. 2017. Vol. 54. P. 381 – 390. DOI: 10.1016/j.polymer.2012.11.010

14. Lei D., Liang Y., Xiao R. A fractional model with parallel fractional Maxwell elements for amorphous thermoplastics / Phys. Stat. Mech. Appl. 2018. Vol. 490. P. 465 – 475.

15. Adibeig M. R., Hassanifard S., Vakili-Tahami F. Optimum creep lifetime of polymethyl methacrylate (PMMA) tube using rheological creep constitutive models based on experimental data / Polymer Testing. 2019. Vol. 75. P. 107 – 116. DOI: 10.1016/j.polymertesting.2019.01.016

16. Lejeune J. et al. Creep and recovery analysis of polymeric materials during indentation tests / Eur. J. Mech. A. Solids. 2018. Vol. 68. P. 1 – 8.

17. Christoefl P. et al. Comprehensive investigation of the viscoelastic properties of PMMA by nanoindentation / Polymer Testing. 2021. Vol. 93. P. 1 – 9. DOI: 10.1016/j.polymertesting.2020.106978

18. Wang L. et al. Measurement of viscoelastic properties for polymers by nanoindentation / Polymer Testing. 2020. Vol. 83. P. 1 – 6. DOI: 10.1016/j.polymertesting.2020.106353