Nonlinear deformation for thermo–magneto–electro–elastic (TMEE) laminates
摘要
In this study, static deformation for the thermo–magneto–electro–elastic (TMEE) laminates subjected to uniform mechanical load is implemented by considering all the coupled effects including elastic stresses, electric displacements, magnetic inductions, and thermal entropy. The conventional constitutive equations are recapped and converted into their corresponding alternative forms, von Karman nonlinear strains for large deflection are adopted in accordance with the kinematic of classical plate theory. Maxwell equations accounting for the relations between electric/magnetic fields and electric/magnetic potentials are introduced, in addition, Fourier’s law for heat conduction is illustrated to present the relation between the thermal flux and the temperature change. In-plane electric fields and magnetic fields are assumed to be neglectable in comparison with the transverse ones, furthermore, the temperature field is deemed to be various along the thickness direction only. In such doing, the Gauss’ laws and the heat equation will thus be reduced into three simplified partial differential equations with respect solely to the thickness variable. By solving these equations simultaneously along with appropriate boundary conditions, exact solutions for the electric/magnetic potentials as well as thermal entropy can be achieved analytically. After substituting these quantities into stress components, the stress resultants of the TMEE thin plate can be easily evaluated, and thus the equilibrium equation for the plate bending can be attained in a nonlinear form. By employing the Bubnov–Galerkin method, the nonlinear governing equation will be transformed into a cubic equation in terms of the maximum deflection for the rectangular TMEE plate with simply supported boundary conditions. Two kinds of composite laminates are demonstrated as examples of TMEE heterogeneous materials, one is a bi-layered