Multiscale Modeling of Fracture in Quasi-Brittle Materials Using Bifurcation Analysis and Element Elimination Method

Analyzing the fracture of heterogeneous materials is a complex problem, due to the fact that the mechanical behavior of a heterogeneous material is strongly dependent on a variety of factors, such as its microstructure, the properties of each constituent, and interactions between them. Therefore, these factors must be effectively taken into account for accurate analysis, for which the multiscale method has been widely used. In this scheme, the computational homogenization method is used to obtain the effective macroscopic properties of a heterogeneous material based on the response of a Representative Volume Element (RVE). The growth of damage in an RVE can be simulated by using common damage theories (such as formation of microcracks) and treated according to standard homogenization theories, which results in degradation of the effective mechanical properties of the material. In most cases, increasing the loading further causes microcracks to accumulate and to consequently form a localized band within the RVE, which may become sufficiently large as compared to the size of the RVE. Standard homogenization approaches have several theoretical shortcomings in dealing with localized RVE that bring into question their viability. This study aims to develop and implement methods to account for localization of RVE and then reflecting it as a discontinuity on the macroscale model within a two-way coupled multiscale framework. In the proposed method, localization of RVE is assessed by bifurcation analysis, which is performed on the anisotropic tangent stiffness tensor of the RVE. The anisotropic tangent stiffness tensor is obtained by separately applying normal and shear displacement boundary conditions on the damaged RVE at each time step. Once the bifurcation analysis meets the onset of weak discontinuity requirement, a discontinuity is inserted on the macroscale model. The element elimination method is used to simulate the discrete representation of cracks on the macroscale model. The entire algorithm was implemented in the form of a two-way linked multiscale code in FORTRAN. Additionally, certain examples were solved using the developed code to demonstrate the viability of the proposed method. The results show that this approach can successfully simulate fracture in a heterogeneous quasi-brittle material without losing its key microstructural details