Summary: In the present work the mesoscopic stress, strain rate and strain states of axisymmetric cells under two types of boundary loadings are formulated. Then, the stress triaxiality of axisymmetric cells is expressed in terms of the axial and radial mesoscopic stress components. Based on the formulations of the mesoscopic stress, three strategies for numerical realization of constant stress triaxiality are presented. The advantages and disadvantages of these strategies are discussed. These numerical strategies are implemented on the platform of the general-purpose finite element programme ABAQUS. They can be applied for representative volume element (RVE) calculations under constant triaxiality, monotonous and cyclic loading controlled by displacement, force, traction and the mesoscopic equivalent strain of the RVE. Several numerical examples are shown to prove the effectivity of these strategies and programme.