Study On Multiscale Finite Element Method For Mechanical Behavior And Failure Analyses Of Heterogeneous Materials

Almost all natural materials, as well as industrial and engineering materials, have multiple scale natures. At the same time, they are heterogeneous at a certain scale. For these materials, the heterogeneous nature can drastically impact on their macroscopic behaviors. In fact, most observed physical and mechanical properties of micro-heterogeneous materials depend on their heterogeneous micro-structures. Taking the composites which have been widely used in aerospace industry for example, the macroscopic mechanics properties of composites are closely related to the material properties, size, shape, and spatial distribution of the microstructural constituents and their respective interfaces. On the other hand, the macroscopic failures are usually induced by the microscopic defects and microscopic cracks. Studying the relations between the microscopic phenomena and macroscopic behavior has become an essentical problem in engineering applications. When the traditional numerical method (such as finite element method, FEM) is directly adopted for solving these multiscale problems, it will encounter difficulties due to the tremendous requirement of computer memory and computing time. Multiscale computational method servers as an effective approach for sovling these problems and has become a hot area of research in recent years.Firstly, an extended multiscale finite element method (EMsFEM) is proposed for solving the mechanical problems of heterogeneous materials in elasticity. Based on the idea of multiscale finite element method (MsFEM), the multiscale base functions (MBFs) for vector field are constructed numerically which can reflect the small-scale heterogeneities within a coarse element. Unlike the scalar field problem, the bulk expansion/contraction phenomena (Poisson effect) have to be considered in the construction of MBFs in the vector field. In other words, the displacement fields in different directions in an element are coupled. To deal with this coupling relation, a new technique is proposed for the construction of MBFs, in which the additional coupling terms are considered. The performances of MBFs with different construction techniques are compared and the results show that the new type of MBFs can exactly reflect the coupling deformations within the coarse element and can obtain an identical micro-deformation field with the traditional FEM, thus, the EMsFEM can improve the computational accuracy dramatically. Several different kinds of boundary conditions are introduced for the construction of MBFs and their influences are investigated. Moreover, the reasons for the scale effect and error are discussed. The method proposed can be implemented conveniently and has a good precision. Comparing with the traditional homogenization method, the EMsFEM here does not need the assumptions of the scale separation and microstructural periodicity, and can perform the downscaling computations easily.Secondly, a new kind of rectangular elements (i.e., Generalized Elements) is proposed based on the idea of the EMsFEM. By means of the Galerkin equation in elastic mechanics, the relations between the displacement field inside the element and the displacements at nodal points of the element are deduced theoretically. The new element can consider the coupling effect of the displacements more reasonable and obtain more accuracy results then the traditional one. Thus, our research can provide ideas for constructing new kinds of elements, which have better precision.The EMsFEM is further extended for modeling the elasto-plastic behavior of heterogeneous materials under small deformation. When the material nonlinearity is considered in the multiscale analysis, it will induce unbalanced nodal forces at the small scales with the emergence of plastic deformation. To deal with the microscopic unbalanced nodal forces, a displacement decomposition technique is proposed. In this context, the microscopic unbalanced nodal forces can be treated as the combined effects of the macroscopic equivalent forces and microscopic perturbed forces, in which the macroscopic equivalent forces are used to solve the macroscopic displamcents and the microscopic perturbed forces are used to obtain the local perturebed displacement field in microscopic scale. Then, a two-scale concurrent computational modeling with successive iteration scheme is proposed. Lastly, we make an estimate of the computer memory and CPU time for the EMsFEM, and compare them with those of the traditional FEM. Extensive numerical experiments have shown that the method developed provides excellent precision of the nonlinear response for the heterogeneous materials and can reduce the comtational cost dramatically.Also, an adaptive multiscale method (AMM) is developed for strain localization analysis of periodic lattice truss materials. In terms of practical applications, localization phenomena can be recognized as the main reasons for structure failure. Volume average based homogenization methods have some limitations in simulating localization phenomena, stemming from their basic assumptions, i.e., scale separation and microstructural periodicity. During the computional processes, the overall computional region in the AMM is adaptively decomposed into two parts:i.e., fine-scale region and coarse-scale region. The EMsFEM serves as a multiscale framework. Since the direct fine-scale simulation is used to model the localized zone, the motion of the truss elements which have severe deformations can be accurately modeled. On the other hand, the EMsFEM is adopted to capture the macroscopic nonlinear response on the coarse region whose strains are moderate. The degrees of freedom on the two different scales are then bridged at the scale-interface by virtue of the master-slave constraints. A displacement gradient based mesh indicator is proposed for the adaptive scheme. The results of the AMM show insensitivity to the coarse-scale discrtization and can capture well the localization phenomena. Moreover, the computational cost is reduced dramatically.At last, a new multiscale method for simulating crack propagation is proposed based on the AMM. The combination of the extended finite element method (XFEM) and level set method (LSM) is employed to model the discontinuities (including strong and weak discontinuities), in which the XFEM is used for computing the stress and displacement fields necessary for determining the rate of crack growth, and the LSM is used for geometrical description of the discontinuities. In this way, the method allows the crack geometry to be represented independently of the finite element mesh, thus, remeshing is not needed. For the subregions which surround the refined zone, the EMsFEM is used and resolved on a coarse-scale mesh. A displacement parameters based mesh indicator is proposed to identify adaptively the fine-scale zone. The parameters can be obtained by the level set functions. The method developed can be well used for the simulation of evolving discontinuities and can reduce the computational cost dramatically.