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
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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  1. 1.
    S.V. Gryshchuk, “Commutative complex algebras of the second rank with unity and some cases of plane orthotropy. I,” Ukr. Mat. Zh., 70, No. 8, 1058–1071 (2018); English translation: Ukr. Math. J., 70, No. 8, 1221–1236 (2019).Google Scholar
  2. 2.
    S.V. Grishchuk and S. A. Plaksa, “Monogenic functions in a biharmonic algebra,” Ukr. Mat. Zh., 61, No. 12, 1587–1596 (2009); English translation: Ukr. Math. J., 61, No. 12, 1865–1876 (2009).Google Scholar
  3. 3.
    M. M. Fridman, “Mathematical theory of elasticity of anisotropic media,” Prikl. Mat. Mekh., 14, No. 3, 321–340 (1950).MathSciNetGoogle Scholar
  4. 4.
    S. G. Lekhnitskii, Theory of Elasticity of Anisotropic Bodies [in Russian], Nauka, Moscow (1977).Google Scholar
  5. 5.
    D. I. Sherman, “Plane problem of the theory of elasticity for anisotropic media,” Tr. Seism. Inst. Akad. Nauk SSSR, No. 6, 51–78 (1938).Google Scholar
  6. 6.
    Yu. A. Bogan, “Regular integral equations for the second boundary-value problem in the anisotropic two-dimensional theory of elasticity,” Izv. Ros. Akad. Nauk, Mekh. Tverd. Tela, No. 4, 17–26 (2005).Google Scholar
  7. 7.
    V. F. Kovalev and I. P. Mel’nichencko, Algebras of Functional-Invariant Solutions of the p-Biharmonic Equation [in Russian], Preprint No. 91.10, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (1991).Google Scholar
  8. 8.
    V. F. Kovalev and I. P. Mel’nichencko, “Biharmonic potentials and plane isotropic fields of displacements,” Ukr. Mat. Zh., 40, No. 2, 229–231 (1988); English translation: Ukr. Math. J., 40, No. 2, 197–199 (1988).Google Scholar
  9. 9.
    S.V. Gryshchuk, “Hypercomplex monogenic functions of biharmonic variable in some problems of the plane theory of elasticity,” Dop. Nats. Akad. Nauk Ukr., No. 6, 7–12 (2015).Google Scholar

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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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