The complementary energy principle and generalized variational principles of quasi-static electro-magneto-thermo-elasticity

The basic equations of coupled electro-magneto-thermo-elasticity are very complicated. Therefore, it is hard to obtain analytical solutions, even in the simplest conditions, so approximate computational methods are used. However, variational principles are the foundation of the finite element method and other approximate computational methods. According to the corresponding relations between generalized forces and generalized displacements, basic equations were multiplied by corresponding virtual quantities, then integrated with volume and area and added algebraically. Complementary energy principles and the first H-R generalized variational principles of quasi-static electro-magneto-thermo-elasticity were established, offering theoretical support for approximate calculation of multi-physics field problems of electro-magneto-thermo-elasticity. Results of stationary value conditions show that the suitable differential equations are composed of stationary value conditions and various pre-conditions. All differential equations are composed of these suitable differential equations, including temperature field equations and supplementary conditions. Thus the validity of these two variational principles is verified.