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

  • S. V. Gryshchuk


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.
How to Cite
GryshchukS. V. “Commutative сomplex Algebras of the Second Rank With Unity and Some cases of the Plane Orthotropy. II”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, no. 10, Oct. 2018, pp. 1382-9,
Research articles