Commutative Complex 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 𝔹0: ={c1e + c2ω : ck ∈ ℂ, k = 1, 2}, ω2 = e, containing bases (e1, e2) such that $$ {e}_1^4+2p{e}_1^2{e}_2^2+{e}_2^4=0 $$ for any fixed p > 1. A domain {(e1, e2)} is described in the explicit form. We construct 𝔹0-valued “analytic” functions Φ such that their real-valued components 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.