On electromagnetic forming processes in finitely strained solids: Theory and examples

The process of electromagnetic forming (EMF) is a high velocity manufacturing technique that uses electromagnetic (Lorentz) body forces to shape sheet metal parts. EMF holds several advantages over conventional forming techniques: speed, repeatability, one-sided tooling, and most importantly considerable ductility increase in several metals. Current modeling techniques for EMF processes are not based on coupled variational principles to simultaneously account for electromagnetic and mechanical effects. Typically, separate solutions to the electromagnetic (Maxwell) and motion (Newton) equations are combined in staggered or lock-step methods, sequentially solving the mechanical and electromagnetic problems.The present work addresses these issues by introducing a fully coupled Lagrangian (reference configuration) least-action variational principle, involving magnetic flux and electric potentials and the displacement field as independent variables. The corresponding Euler–Lagrange equations are Maxwell's and Newton's equations in the reference configuration, which are shown to coincide with their current configuration counterparts obtained independently by a direct approach. The general theory is subsequently simplified for EMF processes by considering the eddy current approximation.Next, an application is presented for axisymmetric EMF problems. It is shown that the proposed variational principle forms the basis of a variational integration numerical scheme that provides an efficient staggered solution algorithm. As an illustration a number of such processes are simulated, inspired by recent experiments of freely expanding uncoated and polyurea-coated aluminum tubes.