Optimal design of micro/nano structures for metamaterials

Progress in manufacturing processes is going so far that now one can design and fabricate materials for engineering practice exhibiting properties that are not directly found in nature – i.e. metamaterials. Metamaterials are often composite with micro/nano sized structures. The composition is usually arranged in specific periodic patterns. Therefore, metamaterials gain their specific excellent properties. The 3D printing, electro-spinning, self-assembly and many other advanced manufacturing techniques are raising a large number of scientific questions, which have to be answered if the potential of the existing and foreseeable manufacturing novelties is to be fully realized. A typical example for metamaterials there are materials, in which the piezoelectric or piezomagnetic response of the materials originates from their special geometries or structures and flexoelectricity or flexomagneticity is utilized. Both flexoelectricity and flexomagneticity represent electromechanical coupling between the electric polarization or magnetization and strain gradients, respectively. New sensors based on flexoelectricity or flexomagneticity can be designed for structural health monitoring with better accuracy than earlier one since strain gradient can give more convenient info than a strain. For a successful application of gradient theory in flexoelectricity it is needed to determine the higher-order elastic and flexoelectric parameters staying at strain gradients in constitutive equations for higher-order stresses and electric displacements (polarization). Due to some experimental difficulties there are missing experimental measurements of these material parameters. In this project an attempt is made to correlate the results of atomistic calculations with results by the gradient theory of continuum involving unknown flexoelectric coefficient. The finite element method (FEM) and meshless formulations are developed for the derived governing equations of flexoelectricity. The mixed FEM in the project uses the C0 continuous interpolation independently for displacement and displacement gradients. Similarly the electric potential and electric intensity vector are approximated by C0 elements. The kinematic constraints between strains and displacements are satisfied by collocation at some cleverly chosen internal points in finite elements. The constraint between the electric potential and electric intensity vector is satisfied analogically by collocation method. The present collocation method reduces the number of DOFs with respect to the approach based on the concept of Lagrange multipliers. Finally, the aim of the present project is also to investigate the influence of surface stress effects in nano-sized structural elements made from metamaterials.



Fig.1. The geometry and boundary conditions of hollow cylinder (left) and the variation of temperature vs. non-dimensional radius r/L in the hollow cylinder (right)