Direct FE2 based on single integration point for modeling damage evolution of materials with micro-cracks

The Direct FE 2 method is a recently developed concurrent multi-scale simulation approach with a monolithic solution scheme, where the macroscopic and microscopic models are solved in a unit iteration process. The conventional Direct FE 2 using quadrilateral element has some limitations in determining the failure state of a macro-element. In this work, we develop a new Direct FE 2 method based on the Single-integration Point triangle element (Direct SP-FE 2 ) and apply it to simulate the micro-crack-induced fracture problems for the first time. The cohesive element is inserted into a representative volume element (RVE), so that the failure process of a macro-element can be determined by the micro-crack propagation on RVE. Accordingly, the relationship between macro-deformations and microstructures can be established directly, thus avoiding complex derivation and utilization of a multi-scale damage constitutive model. In Direct SP-FE 2 , the failure state of a macro-element can be directly predicted by the RVE at the single-integration point inside, avoiding the uncertainty associated with predictions by multiple RVEs. As a result, compared to the conventional Direct FE 2 with a quadrilateral element, the Direct SP-FE 2 shows advantages in capturing complex geometric boundaries and predicting the damage evolution process and failure state more accurately. Various numerical examples are conducted to comprehensively validate the effectiveness of the present method in describing the influence of micro-cracks on macro-structure deformations. For the macro-damage behaviors, the simulation results by the Direct SP-FE 2 show good agreement with the direct numerical simulation (DNS) results of the full FE model at a significantly lower computational time. With the developed Direct SP-FE 2 , a variety of complex micro-crack fracture problems in composite materials can be successfully modeled by manipulating the interior RVE structure.