Virtual Convolved Action for Continuum Solid Mechanics and Electromagnetics

In recent years, convolved action principles have provided attractive variational formulations for a variety of linear problems in engineering science, including those with dissipative elements. The inclusion of a temporal convolution circumvents the main issues of Hamilton’s principle relating to an inconsistent set of end point conditions in time and its inability to address dissipative processes. The present theory paper introduces the concept of virtual convolved action to extend the applicability of convolved action principles to quite general nonlinear problems. This can also be viewed as an extension of the principle of virtual work to produce novel weak forms of the governing equations over space and time for dynamical phenomena. Initially, this idea is used to develop the principle of virtual convolved action for the dynamics of continuum solids. Afterwards, the same concept of virtual convolved action is applied to Maxwell’s theory of electromagnetism with promising theoretical results for both linear and nonlinear media. With that established, a new principle of stationary convolved action is derived for the linear case, including definition of the action, the weak form and the corresponding Euler–Lagrange equations that recover the original statement of the initial/boundary value problem of electromagnetics.