Galerkin’s Method and Multivariate Newton’s Method for the Nonlinear Deformation of the Magneto-Electro-Elastic Bi-Layered Laminates

Abstract

In this paper, mathematical modelling for the large deformation of a magneto-electro-elastic rectangular bi-layered laminate with general boundary conditions is presented. Constitutive equations involving the magneto-electro-elastic (MEE) material properties are introduced, Maxwell equations accounts for the electric and magnetic effects are also utilized. First-order shear deformation theory (FSDT) considering the von Karman nonlinear strain is adopted, and the plain strain/stress assumption applicable for thin plate analysis is used. A rather compact set of governing equations related to kinematical variables, electric/magnetic potentials and the Airy stress function is obtained as a consequence of the preliminary condensation for the electro-magnetic state to the plate kinematics. Semi-analytic solution for a bi-layered BaTiO3-CoFe2O4 laminate with specified boundary conditions subjected to various external applied loads is performed. By employing the Bubnov-Galerkin method, the set of nonlinear partial differential equations is transformed to a set of third-order nonlinear algebraic equations for the static deformation due to applied load. Numerical results are carried out by using the multivariate Newton's method with respect to various volume fractions indicating the volume ratio between piezoelectric BaTiO3 layer and piezomagnetic CoFe2O4. From the result, the nonlinearity of the von Karman strain appears to enhance system rigidity as smaller deformations will be detected when external load is applied. Also, some other interesting results are obtained which could be useful to future analysis and design of multiphase composite plates.