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
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 U1 ≡ u 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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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, No. 10, pp. 1382–1389, October, 2018.
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Gryshchuk, S.V. Commutative Complex Algebras of the Second Rank with Unity and Some Cases of Plane Orthotropy. II. Ukr Math J 70, 1594–1603 (2019). https://doi.org/10.1007/s11253-019-01592-0
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DOI: https://doi.org/10.1007/s11253-019-01592-0