Proceedings of the 7th World Congress on Mechanical, Chemical, and Material Engineering (MCM'21)
Prague, Czech Republic Virtual Conference – August, 2021
Paper No. ICMIE 129
DOI: 10.11159/icmie21.129

Analytical and Numerical Solutions of Contact Problem for theMicroperiodic Slant Layered Composite Body in A Plane Strain State
Piotr Sebestianiuk1, Paweł Mościcki1, Dariusz Perkowski1,1Faculty of Mechanical Engineering/Bialystok University of Technology
Wiejska 45C street, 15-351 Białystok, Poland
p.sebestianiuk@doktoranci.pb.edu.pl; pawel.moscicki.96@gmail.com
d.perkowski@pb.edu.pl;

Extended Abstract
Contact problems of elastic solids are widely studied due to their engineering applications when considering contact offriction pairs or other processes related to the transfer of load from one body to another for both homogeneous andheterogeneous materials - including materials with structural gradation of properties or composites. For this type of materials,
solving problems and describing mechanical properties often comes down to the use of various averaging methods, e.g.
homogenization methods. One of the examples of such methods is the homogenization method with microlocal parameters
used, among others, for heterogeneous composite materials.
Author’s main scope of research are composite materials with layered structure – especially microperiodic layered
composites [1–4]. For this type of materials, solutions for layers perpendicular and parallel to the edge are widely known,
and the inclined layering is still under investigation. This work presents a plane contact problem for the microperiodic
composite half space with slant layering. Presented work discuss a contact of a rigid punch pressed into the composite bodywith normal forces in the state of limit equilibrium. Two punch shapes were considered – cylindrical and rectangular.
Microperiodic composite under consideration is made from thin layers arranged periodically. Each layer is made of two
homogenous components differing in their Young’s modules.
Presented problem has been solved analytically using the homogenized model with microlocal parameters andnumerically using a FEM. Analytical results in the form of the contact pressures and stress distributions are presented forboth components of the composite body in the integral form. The integrals have been computed numerically. Characteristicsof maximum pressures and width of the contact zone are presented for different angles of stratification and for different
stiffness ratios of composite components. The results are presented for both punch shapes. Authors are testing theapplicability of the homogenized model based on FEM solutions, because analytical solutions under the classical theory ofelasticity are almost impossible to obtain, and the time spent on it is too long.
In the case of the per-component contact pressures it could be seen that their maximum values differ the more the higher
stiffness ratios are, so considering only the averaged values is not justified and an approach that allows to determine thestresses in each layer separately should be used – as is possible with the homogenized model with microlocal parameters.

Acknowledgements
The research has been conducted as part of projects at the Faculty of Mechanical Engineering of the Bialystok Universityof Technology. Project number WZ/WM-IIM/3/2020.

References[1] P. Sebestianiuk, D. M. Perkowski, R. Kulchytsky-Zhyhailo, “On contact problem for the microperiodic composite
half-plane with slant layering,” Int. J. Mechanical Sci., vol. 182 , pp.1-16, 2020.
[2] C. Woźniak, “A nonstandard method of modelling of thermoelastic periodic composites,” Int. J. Eng. Sci., vol. 25, no. 4,pp. 483-498, 1987.
[3] R. Kulchytsky-Zhyhailo, S. J. Matysiak, “On three-dimensional problems in a semi-infinite periodically layered body,”
J. Theoretical an Applied Mech., vol. 33, pp. 771-781, 1995.
[4] D. M. Perkowski, S. J. Matysiak, „On heat conduction in a semi-infinite laminated layer. Comparative results for twoapproaches”, Int. Comm. Heat and Mass Transfer”, vol. 37, no.4, pp. 343-349, 2010.

ICMIE 129-1