Micromorphic Model Of Graphene-like Crystals And Its Constitutive Constants

In recent years, with graphene-like crystals being applied in nano composite materials, nano electromechanical systems and other fields gradually, developing an effective forecast method used to describe the mechanical properties of graphene-like crystals has become the frontiers of researchers currently. However, due to the classical continuum theory ignoring the discrete nature of the particles, hence the existing theoretical models of graphene-like crystals were mainly established by various equivalent methods. Micromorphic theory assumes that the particles have the additional nine micro-deformation degrees of freedom to describe the internal deformations and micro-rotations of the microstructure of finite size, and the material intrinsic scale parameter can be introduced, which is more applicable for describing the mechanical behaviors of graphene-like crystals. Therefore, a novel model of graphene-like crystals is presented by micromorphic theory.First of all, the Bravais cells of graphene-like crystals are selected as the basal element according to the interrelations between micromorphic theory and graphene-like crystals. Then the governing equations of the model are derived from the basic equations of micromorphic theory in global coordinates.Secondly, for the Bravais cell of graphene-like crystals containing two atoms, the secular equations of phonon dispersions are then obtained in micromorphic form by analyzing the relations between the modes of phonons and the independent degrees of freedom of the basal elements, and they are further simplified according to the properties of phonon dispersions of two-dimensional crystals, thus the constitutive equations of the model are conformed.Finally, with graphene and monolayer hexagonal boron nitride as two examples, the constitutive constants are determined respectively by fitting their experimental and theoretical data of the in-plane phonon dispersion relations with the simplified secular equations. The obtained equivalent elastic constants and the phonon speeds of our models both show good agreements with available experimental and theoretical values.Moreover, the constitutive constants are further verified by strain energy density function, uniaxial tensile solution of micromorphic model of graphene-like crystals and stress wave velocity of finite element model.