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

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For an algebra \( {\mathbbm{B}}_0:= \left\{{c}_1e+{c}_2\omega :{c}_k\in \mathbb{C},k=1,2\right\} \)such that their real-valued components

*, e*^{2}=*ω*^{2}=*e, eω*=*ωe*=*ω,*over the field of complex numbers ℂ, we consider arbitrary bases (*e, e*_{2}) such that \( e+2p{e}_2^2+{e}_2^4=0 \) for any fixed*p >*1*.*We study \( {\mathbbm{B}}_0 \)-valued “analytic” functions$$ \Phi \left( xe+y{e}_2\right)={U}_1\left(x,y\right)e+{U}_2\left(x,y\right) ie+{U}_3\left(x,y\right){e}_2+{U}_4\left(x,y\right)i{e}_2 $$

*U*_{k}*,*\( k=\overline{1,4} \)*,*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*U*_{1}≡*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*U*_{k}*,*\( k=\overline{1,4} \)*,*of the function Φ for some plane orthotropic media.## Preview

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