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Ukrainian Mathematical Journal

, Volume 70, Issue 10, pp 1594–1603 | Cite as

Commutative Complex Algebras of the Second Rank with Unity and Some Cases of Plane Orthotropy. II

  • S. V. Gryshchuk
Article
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For an algebra \( {\mathbbm{B}}_0:= \left\{{c}_1e+{c}_2\omega :{c}_k\in \mathbb{C},k=1,2\right\} \), e2 = ω2 = e, eω = ωe = ω, over the field of complex numbers ℂ, we consider arbitrary bases (e, e2) such that \( e+2p{e}_2^2+{e}_2^4=0 \) for any fixed p > 1. We study \( {\mathbbm{B}}_0 \)-valued “analytic” functions
$$ \Phi \left( xe+y{e}_2\right)={U}_1\left(x,y\right)e+{U}_2\left(x,y\right) ie+{U}_3\left(x,y\right){e}_2+{U}_4\left(x,y\right)i{e}_2 $$
such that their real-valued components Uk,\( k=\overline{1,4} \), satisfy the equation for the stress function u in the case of orthotropic plane deformations \( \left(\frac{\partial^4}{\partial {x}^4}+2p\frac{\partial^4}{\partial {x}^2\partial {y}^2}+\frac{\partial^4}{\partial {y}^4}\right)u\left(x,y\right)=0 \), where x and y are real variables. All functions Φ for which U1u are described in the case of a simply connected domain. Particular solutions of the equilibrium system of equations in displacements are found in the form of linear combinations of the components Uk,\( k=\overline{1,4} \), of the function Φ for some plane orthotropic media.

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • S. V. Gryshchuk
    • 1
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKyivUkraine

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