Сommutative сomplex algebras of the second rank with unity and some cases of plane orthotropy. I

Abstract

Among all two-dimensional algebras of the second rank with unity $e$ over the field of complex numbers $C$, we find a semisimple algebra $B_0 = \{ c_1e + c_2\omega: c_k \in C, k = 1, 2\} , \omega^2 = e$, containing bases $(e_1, e_2)$, such that $e^4_1 + 2pe^2_1e^2_2 + e^4_2 = 0$ for every fixed $p > 1$. A domain $\{ (e1, e2)\}$ is described in the explicit form. We construct $B_0$ -valued “analytic” functions $\Phi$ such that their real-valued components satisfy the equation for the stress function $u$ in the case of orthotropic plane deformations $$\biggl(\frac{\partial^4}{\partial x^4} + 2p \frac{\partial^4}{\partial x^2 \partial y^2} + \frac{\partial^4}{\partial y^4}\biggr) u(x, y) = 0,$$ where $x, y$ are real variables.