Fiber pull-out in a flexoelectric material

This work examines the fiber pull-out problem in the context of flexoelectricity. As a first approach we examine the deformation of an infinitely long prism due to the action of a concentrated out-of-plane line force. We assume that the material of the prism responds in the context of flexoelectricity and anti-plane conditions prevail. We incorporate the approach proposed by Giannakopoulos and Zisis [1] according to which the gradient of the polarization is incorporated into the strain energy density function. Accordingly, the field equations decouple in terms of the primary variables of the problem, that is the out-of-plane displacement and the polarization. The concentrated out-of-plane line force is materialized in the form of an out-of-plane body force. For the proposed problem we develop analytical solutions with respect to the out-of-plane displacement and polarization under static and dynamic conditions. Next, we propose a finite element (FE) model based on a structural plate analogue, for the solution of anti-plane flexoelectric boundary value problems. The proposed FE model is applied to the predefined problem and the numerical and the analytical results regarding the out-of-plane displacement are compared and found to be in excellent agreement. Next, we propose yet another structural analogue (membrane analogue) for the numerical treatment of the polarization equation and again the results are compared with the analytical solutions with respect to the polarization of the predefined problem and excellent agreement is attained. Finally, for the predefined problem the electromagnetic response is investigated and analytical solutions are proposed.