Kravchuk A.S., Kravchuk A.I. Stress-Strain State of Thick-Walled Pipes of an Elastic Composite in Average Isotropic Material

https://doi.org/10.15688/mpcm.jvolsu.2017.5.5

Aleksander  Stepanovich  Kravchuk
Doctor of Physical and Mathematical Sciences, Associate Professor,
Professor of Department of Bio- and Nanomechanics,
Belarus State University
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Prosp. Nezavisimosti, 4, 220030 Minsk, Belarus

Anzhelika  Ivanovna  Kravchuk
Candidate of Physical and Mathematical Sciences, Associate Professor,
Department of Web Technologies and Computer Simulation,
Belarus State University
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Prosp. Nezavisimosti, 4, 220030 Minsk, Belarus

Abstract. For the first time the boundary value problem is solved for a composite solid body without the application of non-local hypotheses about the smallness of representative volume of composite material, for which the effective characteristics are set. It is assumed that the variation of Poisson’s ratios for all the components of the composite material about the  mean  value  is  small,  and  therefore  it  is  possible  to  use  the  condition  of  constancy  of Poisson’s coefficients and its equality to the average value of all components of the composite material. Due to the peculiarity of this problem, it is not possible in the case of the composite material to separately consider the plane stress and the plane strain state of the pipe, because both these states participate in the evaluation of the stress-strain state of the studied object, according to the standard hypothesis of Voigt and Reuss. An approximation for Hill average value  of  stresses  and  strains  for  representative  volume  are  obtained.  Due  to  the  fact  that
stress and displacement both on the inner and outer border are constant, their average values at any section of the boundary area are constant and equal to their original values. In this connection,  from  the methodological  point  of  view to  the  Lame  problem,  it is  shown  that solutions built using Voigt and Reuss hypotheses are self-sufficient and do not require the use of additional  assumptions  about  the smallness of  the averaging  elements inside  the elastic body. It was established that a solution for the stresses depend on the Voigt and Reuss mean values of the elastic moduli, and the deformations are  determined only by the Reuss mean values of elasticity modulus. The formulas that determine the stress-strain state in composite average isotropic plane with the hole is obtained. These solutions can be used as estimates of stresses and strains in the cross section of soil around the pile using, for example, brown-pile method of their immersion in the permafrost conditions.

Key words:  composite  heterogenic  material,  discrete  random  variable,  averaging, effective deformation characteristics, Voigt hypothesis, Reuss hypothesis, Hill approximation.

Creative Commons License

Stress-Strain State of Thick-Walled Pipes of an Elastic Composite in Average Isotropic Material by Kravchuk  A.S., Kravchuk  A.I. is licensed under a Creative Commons Attribution 4.0 International License.

Citation in English: Mathematical Physics and Computer Simulation. Vol. 20 No. 5 2017 pp. 39-48

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