Finite Element Analysis of Micromorphic Electrodynamics

The key points of micromorphic theory, including the balance laws and entropy principle, are briefly introduced. Maxwell’s equations and the Lorentz Transformation of $\mathbf{\text{E}}\text{ and }\mathbf{\text{B}}$ fields in both relativistic and non-relativistic electromagnetic theory are discussed. The link between the thermomechanical part and the electromagnetic part of the micromorphic electromagnetic theory is established through the body force, body moment, and energy source. The constitutive theory for thermo-visco-elastic-plastic-electromagnetic (TVEP-EM) materials is formulated. Then the constitutive relations are reduced to the materially linear constitutive equations. Onsager’s postulate is utilized for the derivation of viscosity. Return-Mapping-Algorithm is invoked for plasticity. It is a well-known physical fact that the electric field $\mathbf{\text{E}}$ and the magnetic flux $\mathbf{\text{B}}$ are not independent of each other. To resolve this problem, the scalar potential $\varphi$ and the vector potential $\mathbf{\text{A}}$ are introduced and derived, which are related to the electric field and magnetic flux as $\mathbf{\text{B}}\equiv \mathrm{\nabla }\times \mathbf{\text{A}}$ and $\mathbf{\text{E}}\equiv -\mathrm{\nabla }\varphi -\frac{1}{c}\frac{\mathrm{\partial }\mathbf{\text{A}}}{\mathrm{\partial }t}$. Finite element formulations are rigorously derived. On each node, there are displacements, micromotions, temperature, scalar, and vector potentials. It is numerically impossible and physically meaningless to solve the five sets of finite element equations simultaneously. We propose to solve the problem of a hollow cylinder subjected to twist in two stages. In the first stage, the static or nearly static solutions for displacements, micromotions, plastic strains, and temperatures are obtained. In the second stage, the propagation of scalar and vector potentials under the influence of deformations and temperature gradients is investigated. The material of micromorphic theory can contain more complex substances, so it can be utilized to treat blood, bubbly fluids, liquid crystals, etc. Incorporating the coupling between thermomechanics and electromagnetics in micromorphic theory can further enhance the understanding and prediction of large classes of physical phenomena and provide many technological applications. Phenomenologically important cross-effects, such as Peltier, Seebeck, Hall, Ettingshausen, Righi-Leduc, and Nernst effects, can now be studied theoretically and numerically.