Monogenic functions in commutative complex algebras of the second rank and the Lamé equilibrium system for some class of plane orthotropy

Abstract

We consider a class of plane orthotropic deformations of the form \(\varepsilon_{x} = \sigma_x + a_{12} \sigma_y\), \(\gamma_{xy} = 2 \left(p-a_{12}\right) \tau_{xy}\), \(\varepsilon_{y}= a_{12}\sigma_x+\sigma_y\), where \(\sigma_x\), \(\tau_{xy}\), \(\sigma_y\) and \(\varepsilon_{x}\), \(\frac{\gamma_{xy}}{2}\), \(\varepsilon_{y}\) are components of the stress tensor and the deformation tensor, respectively, real parameters \(p\) and \(a_{12}\) satisfy the inequalities: \(-1 \lt p \lt 1\), \(-1 \lt a_{12} \lt p\). A class of solutions of the Lamé equilibrium system for displacements is built in the form of linear combinations of components of ''analytic'' functions which take values in commutative and associative two-dimensional algebras with unity over the field of complex numbers.