A generalized formulation of nonlinear effects of biasing fields on vibrations and propagation of elastic waves in electromagnetic solids

The nonlinear analysis of functional solids and structures subjected to various loadings, including mechanical, thermal, electric, and magnetic fields-both static and dynamic in nature-is of significant practical importance in contemporary engineering applications involving structures, machinery, and electrical systems. The rapid miniaturization of electromagnetic components amplifies the effects of these biasing fields, leading to substantial mechanical deformations and subsequent alterations in other associated fields. To accurately assess the impact of biasing fields, it is essential to employ the nonlinear theory of elasticity alongside wave propagation principles for solids undergoing relatively large deformations. This approach necessitates the consideration from both kinematic and constitutive perspectives, which invariably results in couplings with multiphysical phenomena that include the quasi-static Maxwell equations. Building upon previous efforts concerning nonlinear equations as well as coupled analyses for both linear and nonlinear solids through established methods, a systematic formulation addressing large deformation scenarios will yield fundamental equations governing elastic solids under strong biasing influences due to generalized loadings. These equations are integral to the nonlinear theory of wave propagation and form part of the foundational framework for functional elastic solids. They also facilitate practical procedures for obtaining approximate solutions via the finite element method. Besides analytical procedures for various biasing fields, an illustrative example featuring an elastic beam subjected to initial deformation is provided herein to demonstrate both the formulation process and solution methodology applicable to analogous vibration issues encountered in newer solids and structures such as soft materials and flexible structures.