Direct FE² based on Single Integration Point for Modeling Damage Evolution of Materials with Micro-Cracks

The Direct FE² method is a recently developed concurrent multi-scale simulation approach with a monolithicsolution scheme, where the macroscopic and microscopic models are solved in a unit iteration process. The conventionalDirect FE² using quadrilateral element has some limitations in determining the failure state of a macro element. In this work,we develop a new Direct FE² method based on the Single-integration Point triangle element (Direct SP-FE²) and apply it tosimulate the micro-crack induced fracture problems for the first time. The cohesive element is inserted into a representativevolume element (RVE), so that the failure process of a macro element can be determined by the micro-crack propagationon RVE. Accordingly, the relationship between macro deformations and micro-structures can be established directly, thusavoiding complex derivation and utilization of a multi-scale damage constitutive model. In Direct SP-FE², the failure stateof a macro element can be directly predicted by the RVE at the single integration point inside, avoiding the uncertaintyassociated with predictions by multiple RVEs. As a result, compared to the conventional Direct FE² with a quadrilateralelement, the Direct SP-FE² shows advantages in capturing complex geometric boundaries and predicting the damageevolution process and failure state more accurately. Various numerical examples are conducted to comprehensively validatethe effectiveness of the present method in describing the influence of micro-cracks on macro structure deformations. Forthe macro damage behaviors, the simulation results by the Direct SP-FE² show good agreements with the direct numericalsimulation (DNS) results of the full FE model at a significantly lower computational time. With the developed Direct SP-FE², a variety of complex micro-crack fracture problems in composite materials can be successfully modeled bymanipulating the interior RVE structure.