On the manifold effects of Maxwell stress in dielectrics and consequences of the Einstein–Laub tensor for crack problems

Source abstract

For over 100 years, accurate expressions of electromagnetic forces and couples acting on polarizable matter have been subject to heated debate. The so-called Maxwell stress tensor plays a key role in this regard, as its flux through the body’s surface provides the resultant force and couple in electro- and magnetostatic settings, respectively. In this work, a careful localization of these flux integrals is performed, disclosing novel concentrated loads originating in singularities of the Maxwell stress tensor. As such, singularities are inherent to, inter alia, problems involving edges, corners or cracks, the latter taken as the basis to explore the manifold action of the Maxwell stress model of Einstein and Laub. On the example of a Griffith crack in a linear elastic dielectric under remote electromechanical loading, induced electrostatic body and surface forces as well as previously unnoticed concentrated line forces along the crack fronts are investigated analytically. The mechanical solution exhibits uncommon features such as a higher-order stress singularity at the crack tips, whereupon an obligatory extension of conventional fracture criteria is discussed. Overall, electrostatic body and surface forces are found to effectuate significant negative mode I loading for a wide range of parameters and electric crack face conditions.