Factorization of Equilibrium Equation in Toupin–Mindlin Strain-Gradient Elasticity in Quaternion Analysis

In this paper, we introduce a factorization of the equilibrium equation in the theory of Toupin–Mindlin strain-gradient elasticity in the framework of quaternion analysis by using the inframonogenic operator \(\mathcal {F}u=DuD\) $\mathcal{F}u=DuD$ and the Dirac operator of mixed sides \(D_au=\cos aDu+\sin a uD\) $D_{a}u=cosaDu+sinauD$ , where D is the Dirac operator in \(\mathbb {R}^3\) ${\mathbb{R}}^3$ . We construct some integral representation formulae for the displacement fields. In application, we obtain a decomposition formula of the displacement fields.