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

• S. V. Gryshchuk
Article
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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