Pseudo-Membrane Method For Analysis Of Two-Dimensional Heterogeneous Materials/Structures With Periodic Unit Cells

Pseudomembrane method is presented to analysis the material design and to predict the elastic properties of two-dimensional heterogeneous materials (or structures) with periodic microstructures (or unit cells).Generally, the scales of the unit cells of the heterogeneous materials may be diverse. The unit cells may be macrostructures, be microstructures or even be the nanostructures. In numerical simulation, it is a big trouble to analyze those materials directly on the theories of traditional continuity mechanics. For this reason, the homogenization analysis method is suggested: firstly, the unit cell of a heterogeneous material is analyzed to obtain the equivalent properties of the initial heterogeneous material, and then the response of the initial material is solved by using those equivalent properties with traditional theories. The pioneer of the homogenization analysis method is the approximation of the lattice structure by a continuum model in civil engineering. It was an effective approach to solve such lattice structure when the computational conditions were poor. Even under the current conditions of computational ability, it is still hard to analyze the lattice structures which contain large number of bars by finite element method directly. The continuum model is necessary to be adopted. The so-called homogenization method, which was presented by applied mathematicians in the end of 1970s, reveals the essential of the idea and promotes the developments both of composites and micromechanical theory. From a mathematical point of view, the theory of homogenization is a limit theory which uses the asymptotic expansion and the assumption of periodicity to substitute the differential equations with rapidly oscillating coefficients are constant or slowly varying in such a way that the solution are close to the initial problem. At present, homogenization method is evaluated to be multiscale method and widely applied in many fields of physics and engineering. Besides homogenization method in micromechanics, representative volume element (RVE) method is another excellent one, which was proposed associating with material experiments almost a decade ago. Its basic idea is summarized as: for any point of a specimen, it has a neighborhood, in which the relationship between the average strain and the average stress is independent of the loading conditions, then the relationship is the elastic constitutive property of this point, and the neighborhood is its representative volume element.