Multiscale modeling of ductile fracture in solids
Ductile fracture occurs in many situations of engineering relevance, for example metal cutting and crashworthiness applications, where the fracture process is important to understand and predict. Increased understanding can be gained by using multiscale modeling, where the effective response of the material is computed from microscale simulations on Statistical Volume Elements (SVEs) 1 containing explicit models for the nucleation and propagation of microscopic cracks. However, development of accurate and numerically stable models for failure is challenging already on a single scale. In a multiscale setting, the modeling of propagating cracks leads to additional difficulties. Choosing suitable boundary conditions on the SVE is particularly challenging, because conventional boundary conditions (Dirichlet, Neumann and strong periodic) are inaccurate when cracks are present in the SVE. Furthermore, the scale transition relations, i.e. the coupling between the macroscale and the microscale, need to account for the effect of strain localization due to the formation of macroscopic cracks. Even though several approaches to overcome these difficulties have been proposed in the literature, previously proposed models frequently involve explicit assumptions on the constitutive models adopted on the microscale, and require explicit tracking of an effective discontinuity inside the SVE. For the general situation, such explicit discontinuity tracking is cumbersome. Therefore, a multiscale scheme that employs less restrictive assumptions on the microscale constitutive model would be very valuable. To this end, a two-scale model for fracturing solids is developed, whereby macroscale discontinuities are modeled by the eXtended Finite Element Method (XFEM). The model has two key ingredients: i) boundary conditions on the SVE that are accurate also when crack propagation occurs in the microstructure, and ii) suitable scale transition relations when cracks are present on both scales. Starting from a previously proposed mixed formulation for weakly periodic boundary conditions, effective boundary conditions are developed to obtain accurate results also in the presence of cracks. The modified boundary conditions are combined with smeared macro-to-micro discontinuity transitions, leading to a multiscale modeling scheme capable of handling cracks on both scales. Several numerical examples are given, demonstrating that the proposed scheme is accurate in terms of convergence with increasing SVE size. Furthermore, the good performance of the proposed scheme is demonstrated by comparisons with Direct Numerical Simulations (DNS).