Multiscale Finite Element Method For Gradient Elastic And Magneto-electro-mechanically Coupling Analyses Of Heterogeneous Materials

Most of the natural materials, as well as the composites materials that have been extensively applied in engineering practices, are usually heterogeneous, such as rock and soils, bones, alloys and functionally graded materials. The minimum characteristic size of these materials is of lower order than the size of the resulting macroscopic structure, meaning that they have multiscale features. The microscopic heterogeneities have a great influence on their overall behaviors. Due to geometric complexity of microstructures and the spatial variations of its material properties, the element sizes are required to be much smaller than the minimum characteristic size when using the traditional computational methods such as the finite element method (FEM) to get meaningful solution in microscale. This is bound to bring the demand of huge amouts of computer resources and computation time, and always make the computation infeasible. The aim of this dissertation is to develop efficient multiscale computational methods that can predict the macroscopic behaviors effectively and give the accurate microscopic responses of the heterogeneous materials.Firstly, the extended multiscale finite element method (EMsFEM) for solving the linear static mechanical problems of heterogeneous materials is introduced briefly. In this method, numerical displacement multiscale base functions reflect the heterogeneities of the microstructure and act the role of information transfer between microscale and macroscale. We summarised serveral kinds of boundary conditions to construct the multiscale base functions, and the procedures of macroscopic and downscaling computations. Moreover, the deformation features of unit cell with different boundary conditions are analysed, and the influence of the updates of the multiscale base functions during the computation process on the computational accuracy is investigated.Secondly, a multiscale computational formulation for gradient elasticity problems of heterogeneous structures is developed. This formulation solves the problem that the classical continuum model cannot describe the size-dependent response and the removal of singularities at the crack tips. On the basis of the derived staggered form of the finite element discretization, perform the macroscopic and downscaling computation after the multiscale base functions are constructed, the final microscopic displacement field is determined as the superposition of the microscopic one obtained by downscaling computation and the local perturbation. And then the gradient-enriched stress and strain fields are computed within each coarse element or on microlevel.Thirdly, an efficient multiscale computational method is developed for the geometric nonlinear analysis of heterogeneous piezoelectric materials and dielectric polymers. In order to capture the electro-mechanical coupling effect, displacement base functions and electric potential base functions are constructed, the additional coupling terms between displacement and electric potential are considered. A heterogeneous piezoelectric unit cell is equivalent into a macroscopic piezoelectric element by means of the numerical base functions. The equivalent stiffness matrix of the macroscopic element involves the microscopic heterogeneities. The co-rotational coordinate method is applied to describe the motion of macroscopic elements, so the transition matrix between the local and global coordinates, as well as the equivalent tangent stiffness matrix could be derived. Then the original nonlinear problems are solved in the global coordinate system by iterative calculation. After the convergence condition is satisfied on the macrolevel, the obtained macroscopic nodal variables are converted into the local coordinate system, and the microscopic responses are gained through the downscaling computation, such as stress, strain and electric field. The influences of the geometric and physical properties of microstructures on the macroscopic behaviors are studied, and it is proved that the developed multiscale method can capture the nonlinear mechanical and electric responses efficiently and accuratly. Besides, the simulations of heterogeneous piezoelectric structures under varying driven patterns show that this method has wide suitability.Finally, a multiscale computational method is also proposed for the simulation of magneto-electro-mechanically coupled problems. We give the way to construct the three types of numerical base functions with linear boundary condition, oversampling boundary condition and periodic boundary condition, respectively. The three types of base functions include displacement base function, electric potential base function and magnetic potential base function. These functions contain the coupling effects between different physical fields and reflect the heterogeneities within macroscopic elements. The derived macroscopic stiffness matrix brings the heterogeneities from microscale to macroscale implicitly. The original coupling problem can be solved at the macroscopic level directly. According to the macroscopic nodal solutions, the microscopic stress, strain, electric and magnetic fields will be obtained by downscaling computation using the base functions once again. The proposed method not only can provide excellent precision of the coupling responses for the magneto-electro-elastic materials but also has high computational efficiency.