Commutative сomplex algebras of the second rank with unity and some cases of the plane orthotropy. II

Abstract

For an algebra $B_0 = \{ c_1e + c_2\omega : c_k \in C, k = 1, 2\} , e_2 = \omega 2 = e, e\omega = \omega e = \omega$, over the field of complex numbers $C$, we сonsider arbitrary bases $(e, e_2)$, such that$e + 2pe^2_2 + e^4_2 = 0$ for any fixed $p > 1$. We study $B_0$ -valued “analytic” functions $\Phi (xe+ye_2) = U_1(x, y)e + U_2(x, y)ie + U_3(x, y)e_2 + U_4(x, y)ie_2$ such that their real-valued components $U_k, k = 1, 4$, satisfy the equation for the stress function $u$ in the case of orthotropic plane deformations $$\biggl( frac{\partial^4}{\partial x^4} + 2p\frac{\partial^4}{\partial x^2 \partial y^2} + \frac{\partial^4}{\partial y^4}\biggr)u(x, y) = 0,$$ here, $x$ and $y$ are real variables. All functions $\Phi$ for which $U_1 \equiv u$ 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 $U_k , k = 1, 4$, of the function $\Phi$ for some plane orthotropic media.