A finite element formulation for a simplified, relaxed micromorphic continuum model

Abstract

We discuss a simplified problem derived from the relaxed micromorphic continuum model in two dimensions. The model captures important aspects of the micromorphic approach even as a degeneration of the bulk model. Typically, the employed mechanical strain combines the gradient of displacements with the microdistortion field. The interaction between both fields is ruled by the minimization of the overall free energy, where we employ the Curl of the microdistortion. The Curl significantly influences the resulting equations for the balance of linear and angular momentum. Further, we explain the necessity of an extended finite element method. Finite elements based on solely the H1‐Hilbert space are not sufficient for the efficient approximation of the Curl based microdistortion. Therefore, we suggest using a hybrid scheme employing both, H1 and H(Curl) based functions. The resulting hybrid element formulation is successfully tested for a problem with a predefined Dirichlet boundary condition.