Variational theory for linear magneto-electro-elasticity

To describe the physical behavior of a magneto-electro-elastic medium, the fundamental equations, including equilibrium equations, strain-displacement relations, and constitutive relations, and all boundary conditions are expressed as stationary condition (Euler equations and natural conditions) of a generalized variational principle, which is obtained by the semi-inverse method proposed by He. The principle is deduced from an energy-like trial functional with a certain unknown function, which can be identified step by step. A family of various variational principles for the discussed problem is also obtained for differential applications. Present theory provides a quite straightforward tool to the search for various variational principles for physical problems. This paper aims at providing a more complete theoretical basis for the finite element applications, meshfree particle methods, and other direct variational methods such as Ritz's, Trefftz's and Kantorovitch's methods.