Quaternionic Operator Calculus for Boundary Value Problems of Micropolar Elasticity

Abstract

Micropolar elasticity is a refined version of the classical elasticity. Equations of micropolar elasticity are not given only by a single differential equation w.r.t. a vector field of displacement, but by a coupled system of differential equations connecting fields of displacements and rotations. However, construction of solution methods for boundary value problems of micropolar elasticity is still an open mathematical task, mostly due to the coupled nature of the resulting system of partial differential equations. Especially, only few results are available for spatial problems of micropolar elasticity. Therefore, in this paper, we present a quaternionic operator calculus-based approach to construct general solutions to three-dimensional problems of micropolar elasticity. Moreover, we prove solvability of the boundary value problem of micropolar elasticity, as well as we provide an explicit estimate for the difference between the classical elasticity and the micropolar model.