Numerical calculation of the effective coefficient in the problem of linear elasticity of a composite material

  • Zakharov Petr E., zapetch@gmail.com M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 42, Kulakovsky St., Yakutsk 677000, Russia
  • Sivtsev Petr V., sivkapetr@mail.ru M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 42, Kulakovsky St., Yakutsk 677000, Russia
Keywords: numerical homogenization, linear elasticity, composite material, mathematical modeling, effective coefficient

Abstract

We consider the numerical calculation of the effective coefficient of the linear elasticity problem on a representative volume element having a similar volume fraction of fibers. The obtained effective coefficient is used to solve the complete problem on a coarse grid. As an example, the problem of calculating the deformation of a concrete block with the inclusion of steel fibers under the action of a three-point bending is considered. The computational implementation of the problem is carried out by the finite element method using the FEniCS computing platform.

References

[1]Christensen R. M., Lo K. H., “Solutions for effective shear properties in three phase sphere and cylinder models”, J. Mechanics and Physics of Solids, 27:4 (1979), 315–330  crossref  scopus
[2]Li L. X., Wang T. J., “A unified approach to predict overall properties of composite materials”, Materials Characterization, 54:1 (2005), 49–62  crossref  elib  scopus
[3]Gusev A. A., “Representative volume element size for elastic composites: a numerical study”, J. Mechanics and Physics of Solids, 45:9 (1997), 1449–1459  crossref
[4]Kari S., Berger H., Gabbert U., “Numerical evaluation of effective material properties of randomly distributed short cylindrical fibre composites”, Comput. Materials Sci., 39:1 (2007), 198–204  crossref  elib  scopus
[5]Theocaris P. S., Stavroulakis G. E., Panagiotopoulos P. D., “Calculation of effective transverse elastic moduli of fiber-reinforced composites by numerical homogenization”, Composites Sci. Technol., 57:5 (1997), 573–586  crossref  scopus
[6]Kolesov A. E. et al., “Numerical analysis of reinforced concrete deep beams”, Numerical Analysis and Its Applications: 6th International Conference, NAA 2016 (Lozenetz, Bulgaria, June 15-22, 2016), Springer-Verl., 2016, 414–421  mathscinet
[7]Smarzewski P., Por ba J., Rentflejsz A., “Experimental testing of high performance fibre reinforced concrete deep beams”, Civil Engineering and Architecture, 10:1 (2012), 15–26
[8]Sivtsev P. V., “Numerical simulation of the elasticity problem of reinforced concrete slabs”, Vestn. Severo-Vost. Feder. Univ., 2015, no. 4, 98–109
[9]Automated solution of differential equations by the finite element method: The FEniCS book, Lecture Notes in Computational Science and Engineering, 84, eds. Logg A., Mardal K. A., Wells G., Springer-Verl., 2012  mathscinet
How to Cite
Zakharov, P. and Sivtsev, P. ( ) “Numerical calculation of the effective coefficient in the problem of linear elasticity of a composite material”, Mathematical notes of NEFU, 24(2), pp. 75-84. doi: https://doi.org/10.25587/SVFU.2017.2.9247.
Section
Mathematical Modeling